id	sid	tid	token	lemma	pos
ap-7583	1	1	acta	acta	PROPN
ap-7583	1	2	polytechnica	polytechnica	PROPN
ap-7583	1	3	https://doi.org/10.14311/ap.2022.62.0165	https://doi.org/10.14311/ap.2022.62.0165	PROPN
ap-7583	1	4	acta	acta	PROPN
ap-7583	1	5	polytechnica	polytechnica	PROPN
ap-7583	1	6	62(1):165–189	62(1):165–189	PROPN
ap-7583	1	7	,	,	PUNCT
ap-7583	1	8	2022	2022	NUM
ap-7583	1	9	©	©	ADP
ap-7583	1	10	2022	2022	NUM
ap-7583	1	11	the	the	DET
ap-7583	1	12	author(s	author(s	NOUN
ap-7583	1	13	)	)	PUNCT
ap-7583	1	14	.	.	PUNCT
ap-7583	2	1	licensed	license	VERB
ap-7583	2	2	under	under	ADP
ap-7583	2	3	a	a	DET
ap-7583	2	4	cc	cc	NOUN
ap-7583	2	5	-	-	PUNCT
ap-7583	2	6	by	by	ADP
ap-7583	2	7	4.0	4.0	NUM
ap-7583	2	8	licence	licence	NOUN
ap-7583	2	9	published	publish	VERB
ap-7583	2	10	by	by	ADP
ap-7583	2	11	the	the	DET
ap-7583	2	12	czech	czech	PROPN
ap-7583	2	13	technical	technical	PROPN
ap-7583	2	14	university	university	PROPN
ap-7583	2	15	in	in	ADP
ap-7583	2	16	prague	prague	PROPN
ap-7583	2	17	on	on	ADP
ap-7583	2	18	generalized	generalized	ADJ
ap-7583	2	19	heun	heun	NOUN
ap-7583	2	20	equation	equation	NOUN
ap-7583	2	21	with	with	ADP
ap-7583	2	22	some	some	DET
ap-7583	2	23	mathematical	mathematical	ADJ
ap-7583	2	24	properties	property	NOUN
ap-7583	2	25	nasser	nasser	PROPN
ap-7583	2	26	saad	saad	PROPN
ap-7583	2	27	university	university	PROPN
ap-7583	2	28	of	of	ADP
ap-7583	2	29	prince	prince	PROPN
ap-7583	2	30	edward	edward	PROPN
ap-7583	2	31	island	island	PROPN
ap-7583	2	32	,	,	PUNCT
ap-7583	2	33	department	department	NOUN
ap-7583	2	34	of	of	ADP
ap-7583	2	35	mathematics	mathematics	PROPN
ap-7583	2	36	and	and	CCONJ
ap-7583	2	37	statistics	statistic	NOUN
ap-7583	2	38	,	,	PUNCT
ap-7583	2	39	550	550	NUM
ap-7583	2	40	university	university	NOUN
ap-7583	2	41	avenue	avenue	NOUN
ap-7583	2	42	,	,	PUNCT
ap-7583	2	43	charlottetown	charlottetown	PROPN
ap-7583	2	44	,	,	PUNCT
ap-7583	2	45	pei	pei	PROPN
ap-7583	2	46	,	,	PUNCT
ap-7583	2	47	canada	canada	PROPN
ap-7583	2	48	c1a	c1a	AUX
ap-7583	2	49	4p3	4p3	X
ap-7583	2	50	.	.	PUNCT
ap-7583	3	1	correspondence	correspondence	NOUN
ap-7583	3	2	:	:	PUNCT
ap-7583	3	3	nsaad@upei.ca	nsaad@upei.ca	NOUN
ap-7583	3	4	abstract	abstract	ADJ
ap-7583	3	5	.	.	PUNCT
ap-7583	4	1	we	we	PRON
ap-7583	4	2	study	study	VERB
ap-7583	4	3	the	the	DET
ap-7583	4	4	analytic	analytic	ADJ
ap-7583	4	5	solutions	solution	NOUN
ap-7583	4	6	of	of	ADP
ap-7583	4	7	the	the	DET
ap-7583	4	8	generalized	generalize	VERB
ap-7583	4	9	heun	heun	NOUN
ap-7583	4	10	equation	equation	NOUN
ap-7583	4	11	,	,	PUNCT
ap-7583	4	12	(	(	PUNCT
ap-7583	4	13	α0	α0	ADJ
ap-7583	4	14	+	+	NUM
ap-7583	4	15	α1	α1	PROPN
ap-7583	4	16	r	r	NOUN
ap-7583	4	17	+	+	CCONJ
ap-7583	4	18	α2	α2	ADJ
ap-7583	4	19	r2	r2	PROPN
ap-7583	4	20	+	+	CCONJ
ap-7583	4	21	α3	α3	PROPN
ap-7583	4	22	r3	r3	PROPN
ap-7583	4	23	)	)	PUNCT
ap-7583	4	24	y′′	y′′	PROPN
ap-7583	5	1	+	+	CCONJ
ap-7583	5	2	(	(	PUNCT
ap-7583	5	3	β0	β0	NOUN
ap-7583	5	4	+	+	CCONJ
ap-7583	5	5	β1	β1	PROPN
ap-7583	5	6	r	r	NOUN
ap-7583	5	7	+	+	CCONJ
ap-7583	5	8	β2	β2	NOUN
ap-7583	5	9	r2	r2	NOUN
ap-7583	5	10	)	)	PUNCT
ap-7583	5	11	y′	y′	PUNCT
ap-7583	6	1	+	+	CCONJ
ap-7583	6	2	(	(	PUNCT
ap-7583	6	3	ε0	ε0	PROPN
ap-7583	6	4	+	+	CCONJ
ap-7583	6	5	ε1	ε1	PROPN
ap-7583	6	6	r	r	NOUN
ap-7583	6	7	)	)	PUNCT
ap-7583	6	8	y	y	PROPN
ap-7583	6	9	=	=	SYM
ap-7583	6	10	0	0	PROPN
ap-7583	6	11	,	,	PUNCT
ap-7583	6	12	where	where	SCONJ
ap-7583	6	13	|α3|	|α3|	NOUN
ap-7583	6	14	+	+	CCONJ
ap-7583	6	15	|β2|	|β2|	VERB
ap-7583	6	16	=	=	NOUN
ap-7583	6	17	̸	̸	NUM
ap-7583	6	18	0	0	NUM
ap-7583	6	19	,	,	PUNCT
ap-7583	6	20	and	and	CCONJ
ap-7583	6	21	{	{	PUNCT
ap-7583	6	22	αi}3	αi}3	NOUN
ap-7583	6	23	i=0	i=0	ADJ
ap-7583	6	24	,	,	PUNCT
ap-7583	6	25	{	{	PUNCT
ap-7583	6	26	βi}2	βi}2	PUNCT
ap-7583	6	27	i=0	i=0	ADJ
ap-7583	6	28	,	,	PUNCT
ap-7583	6	29	{	{	PUNCT
ap-7583	6	30	εi}1	εi}1	NOUN
ap-7583	6	31	i=0	i=0	PROPN
ap-7583	6	32	are	be	AUX
ap-7583	6	33	real	real	ADJ
ap-7583	6	34	parameters	parameter	NOUN
ap-7583	6	35	.	.	PUNCT
ap-7583	7	1	the	the	DET
ap-7583	7	2	existence	existence	NOUN
ap-7583	7	3	conditions	condition	NOUN
ap-7583	7	4	for	for	ADP
ap-7583	7	5	the	the	DET
ap-7583	7	6	polynomial	polynomial	ADJ
ap-7583	7	7	solutions	solution	NOUN
ap-7583	7	8	are	be	AUX
ap-7583	7	9	given	give	VERB
ap-7583	7	10	.	.	PUNCT
ap-7583	8	1	a	a	DET
ap-7583	8	2	simple	simple	ADJ
ap-7583	8	3	procedure	procedure	NOUN
ap-7583	8	4	based	base	VERB
ap-7583	8	5	on	on	ADP
ap-7583	8	6	a	a	DET
ap-7583	8	7	recurrence	recurrence	NOUN
ap-7583	8	8	relation	relation	NOUN
ap-7583	8	9	is	be	AUX
ap-7583	8	10	introduced	introduce	VERB
ap-7583	8	11	to	to	PART
ap-7583	8	12	evaluate	evaluate	VERB
ap-7583	8	13	these	these	DET
ap-7583	8	14	polynomial	polynomial	ADJ
ap-7583	8	15	solutions	solution	NOUN
ap-7583	8	16	explicitly	explicitly	ADV
ap-7583	8	17	.	.	PUNCT
ap-7583	9	1	for	for	ADP
ap-7583	9	2	α0	α0	ADJ
ap-7583	9	3	=	=	SYM
ap-7583	9	4	0	0	NUM
ap-7583	9	5	,	,	PUNCT
ap-7583	9	6	α1	α1	PROPN
ap-7583	9	7	̸=	̸=	PROPN
ap-7583	9	8	0	0	NUM
ap-7583	9	9	,	,	PUNCT
ap-7583	9	10	we	we	PRON
ap-7583	9	11	prove	prove	VERB
ap-7583	9	12	that	that	SCONJ
ap-7583	9	13	the	the	DET
ap-7583	9	14	polynomial	polynomial	ADJ
ap-7583	9	15	solutions	solution	NOUN
ap-7583	9	16	of	of	ADP
ap-7583	9	17	the	the	DET
ap-7583	9	18	corresponding	corresponding	ADJ
ap-7583	9	19	differential	differential	ADJ
ap-7583	9	20	equation	equation	NOUN
ap-7583	9	21	are	be	AUX
ap-7583	9	22	sources	source	NOUN
ap-7583	9	23	of	of	ADP
ap-7583	9	24	finite	finite	ADJ
ap-7583	9	25	sequences	sequence	NOUN
ap-7583	9	26	of	of	ADP
ap-7583	9	27	orthogonal	orthogonal	ADJ
ap-7583	9	28	polynomials	polynomial	NOUN
ap-7583	9	29	.	.	PUNCT
ap-7583	10	1	several	several	ADJ
ap-7583	10	2	mathematical	mathematical	ADJ
ap-7583	10	3	properties	property	NOUN
ap-7583	10	4	,	,	PUNCT
ap-7583	10	5	such	such	ADJ
ap-7583	10	6	as	as	ADP
ap-7583	10	7	the	the	DET
ap-7583	10	8	recurrence	recurrence	NOUN
ap-7583	10	9	relation	relation	NOUN
ap-7583	10	10	,	,	PUNCT
ap-7583	10	11	christoffel	christoffel	NOUN
ap-7583	10	12	-	-	PUNCT
ap-7583	10	13	darboux	darboux	VERB
ap-7583	10	14	formulas	formula	NOUN
ap-7583	10	15	and	and	CCONJ
ap-7583	10	16	the	the	DET
ap-7583	10	17	norms	norm	NOUN
ap-7583	10	18	of	of	ADP
ap-7583	10	19	these	these	DET
ap-7583	10	20	polynomials	polynomial	NOUN
ap-7583	10	21	,	,	PUNCT
ap-7583	10	22	are	be	AUX
ap-7583	10	23	discussed	discuss	VERB
ap-7583	10	24	.	.	PUNCT
ap-7583	11	1	we	we	PRON
ap-7583	11	2	shall	shall	AUX
ap-7583	11	3	also	also	ADV
ap-7583	11	4	show	show	VERB
ap-7583	11	5	that	that	SCONJ
ap-7583	11	6	they	they	PRON
ap-7583	11	7	exhibit	exhibit	VERB
ap-7583	11	8	a	a	DET
ap-7583	11	9	factorization	factorization	NOUN
ap-7583	11	10	property	property	NOUN
ap-7583	11	11	that	that	PRON
ap-7583	11	12	permits	permit	VERB
ap-7583	11	13	the	the	DET
ap-7583	11	14	construction	construction	NOUN
ap-7583	11	15	of	of	ADP
ap-7583	11	16	other	other	ADJ
ap-7583	11	17	infinite	infinite	ADJ
ap-7583	11	18	sequences	sequence	NOUN
ap-7583	11	19	of	of	ADP
ap-7583	11	20	orthogonal	orthogonal	ADJ
ap-7583	11	21	polynomials	polynomial	NOUN
ap-7583	11	22	.	.	PUNCT
ap-7583	12	1	keywords	keyword	NOUN
ap-7583	12	2	:	:	PUNCT
ap-7583	12	3	heun	heun	PROPN
ap-7583	12	4	equation	equation	NOUN
ap-7583	12	5	,	,	PUNCT
ap-7583	12	6	confluent	confluent	ADJ
ap-7583	12	7	forms	form	NOUN
ap-7583	12	8	of	of	ADP
ap-7583	12	9	heun	heun	PROPN
ap-7583	12	10	’s	’s	PART
ap-7583	12	11	equation	equation	NOUN
ap-7583	12	12	,	,	PUNCT
ap-7583	12	13	polynomial	polynomial	ADJ
ap-7583	12	14	solutions	solution	NOUN
ap-7583	12	15	,	,	PUNCT
ap-7583	12	16	sequences	sequence	NOUN
ap-7583	12	17	of	of	ADP
ap-7583	12	18	orthogonal	orthogonal	ADJ
ap-7583	12	19	polynomials	polynomial	NOUN
ap-7583	12	20	.	.	PUNCT
ap-7583	13	1	1	1	X
ap-7583	13	2	.	.	X
ap-7583	13	3	introduction	introduction	NOUN
ap-7583	13	4	it	it	PRON
ap-7583	13	5	seems	seem	VERB
ap-7583	13	6	as	as	ADP
ap-7583	13	7	a	a	DET
ap-7583	13	8	simple	simple	ADJ
ap-7583	13	9	question	question	NOUN
ap-7583	13	10	to	to	PART
ap-7583	13	11	ask	ask	VERB
ap-7583	13	12	:	:	PUNCT
ap-7583	13	13	under	under	ADP
ap-7583	13	14	what	what	DET
ap-7583	13	15	conditions	condition	NOUN
ap-7583	13	16	does	do	AUX
ap-7583	13	17	the	the	DET
ap-7583	13	18	differential	differential	ADJ
ap-7583	13	19	equation	equation	NOUN
ap-7583	13	20	π3(r	π3(r	NOUN
ap-7583	13	21	)	)	PUNCT
ap-7583	13	22	y′′	y′′	NOUN
ap-7583	13	23	+	+	CCONJ
ap-7583	13	24	π2(r	π2(r	NOUN
ap-7583	13	25	)	)	PUNCT
ap-7583	13	26	y′	y′	PUNCT
ap-7583	14	1	+	+	PUNCT
ap-7583	14	2	π1(r	π1(r	X
ap-7583	14	3	)	)	PUNCT
ap-7583	14	4	y	y	NOUN
ap-7583	14	5	=	=	PUNCT
ap-7583	14	6	(	(	PUNCT
ap-7583	14	7	λn	λn	X
ap-7583	14	8	+	+	CCONJ
ap-7583	14	9	µnπ0(r	µnπ0(r	NUM
ap-7583	14	10	)	)	PUNCT
ap-7583	14	11	)	)	PUNCT
ap-7583	15	1	y	y	NOUN
ap-7583	15	2	,	,	PUNCT
ap-7583	15	3	where	where	SCONJ
ap-7583	15	4	λn	λn	NOUN
ap-7583	15	5	and	and	CCONJ
ap-7583	15	6	µn	µn	PROPN
ap-7583	15	7	are	be	AUX
ap-7583	15	8	constants	constant	NOUN
ap-7583	15	9	and	and	CCONJ
ap-7583	15	10	πj(r	πj(r	NOUN
ap-7583	15	11	)	)	PUNCT
ap-7583	15	12	,	,	PUNCT
ap-7583	15	13	j	j	PROPN
ap-7583	15	14	=	=	SYM
ap-7583	15	15	0	0	NUM
ap-7583	15	16	,	,	PUNCT
ap-7583	15	17	1	1	NUM
ap-7583	15	18	,	,	PUNCT
ap-7583	15	19	2	2	NUM
ap-7583	15	20	,	,	PUNCT
ap-7583	15	21	3	3	NUM
ap-7583	15	22	are	be	AUX
ap-7583	15	23	polynomials	polynomial	NOUN
ap-7583	15	24	of	of	ADP
ap-7583	15	25	unknown	unknown	ADJ
ap-7583	15	26	degree	degree	NOUN
ap-7583	15	27	to	to	PART
ap-7583	15	28	be	be	AUX
ap-7583	15	29	found	find	VERB
ap-7583	15	30	,	,	PUNCT
ap-7583	15	31	has	have	VERB
ap-7583	15	32	n	n	CCONJ
ap-7583	15	33	-	-	PUNCT
ap-7583	15	34	degree	degree	NOUN
ap-7583	15	35	monic	monic	ADJ
ap-7583	15	36	polynomial	polynomial	ADJ
ap-7583	15	37	solutions	solution	NOUN
ap-7583	15	38	yn	yn	X
ap-7583	16	1	=	=	PUNCT
ap-7583	16	2	∑n	∑n	PROPN
ap-7583	16	3	k=0	k=0	PROPN
ap-7583	16	4	ck	ck	ADJ
ap-7583	16	5	rk	rk	NOUN
ap-7583	16	6	,	,	PUNCT
ap-7583	16	7	c0	c0	PROPN
ap-7583	16	8	̸=	̸=	PROPN
ap-7583	16	9	0	0	NUM
ap-7583	16	10	,	,	PUNCT
ap-7583	16	11	ck	ck	NOUN
ap-7583	16	12	=	=	NOUN
ap-7583	16	13	1	1	X
ap-7583	16	14	?	?	PUNCT
ap-7583	16	15	a	a	DET
ap-7583	16	16	simple	simple	ADJ
ap-7583	16	17	approach	approach	NOUN
ap-7583	16	18	to	to	PART
ap-7583	16	19	deduce	deduce	VERB
ap-7583	16	20	the	the	DET
ap-7583	16	21	possible	possible	ADJ
ap-7583	16	22	degrees	degree	NOUN
ap-7583	16	23	of	of	ADP
ap-7583	16	24	πj	πj	NOUN
ap-7583	16	25	,	,	PUNCT
ap-7583	16	26	j	j	PROPN
ap-7583	16	27	=	=	SYM
ap-7583	16	28	0	0	NUM
ap-7583	16	29	,	,	PUNCT
ap-7583	16	30	1	1	NUM
ap-7583	16	31	,	,	PUNCT
ap-7583	16	32	2	2	NUM
ap-7583	16	33	,	,	PUNCT
ap-7583	16	34	3	3	NUM
ap-7583	16	35	,	,	PUNCT
ap-7583	16	36	is	be	AUX
ap-7583	16	37	to	to	PART
ap-7583	16	38	examine	examine	VERB
ap-7583	16	39	the	the	DET
ap-7583	16	40	degrees	degree	NOUN
ap-7583	16	41	for	for	ADP
ap-7583	16	42	the	the	DET
ap-7583	16	43	(	(	PUNCT
ap-7583	16	44	possible	possible	ADJ
ap-7583	16	45	)	)	PUNCT
ap-7583	16	46	polynomial	polynomial	ADJ
ap-7583	16	47	solutions	solution	NOUN
ap-7583	16	48	yn	yn	NOUN
ap-7583	16	49	:	:	PUNCT
ap-7583	16	50	for	for	ADP
ap-7583	16	51	n	n	NOUN
ap-7583	16	52	=	=	SYM
ap-7583	16	53	0	0	NUM
ap-7583	16	54	,	,	PUNCT
ap-7583	16	55	y0(r	y0(r	NOUN
ap-7583	16	56	)	)	PUNCT
ap-7583	16	57	=	=	SYM
ap-7583	16	58	1	1	NUM
ap-7583	16	59	,	,	PUNCT
ap-7583	16	60	we	we	PRON
ap-7583	16	61	must	must	AUX
ap-7583	16	62	have	have	VERB
ap-7583	16	63	π1(r	π1(r	NOUN
ap-7583	16	64	)	)	PUNCT
ap-7583	16	65	=	=	SYM
ap-7583	16	66	λ0	λ0	NOUN
ap-7583	16	67	+	+	CCONJ
ap-7583	16	68	µ0π0(r	µ0π0(r	NOUN
ap-7583	16	69	)	)	PUNCT
ap-7583	16	70	and	and	CCONJ
ap-7583	16	71	the	the	DET
ap-7583	16	72	degree	degree	NOUN
ap-7583	16	73	of	of	ADP
ap-7583	16	74	the	the	DET
ap-7583	16	75	polynomial	polynomial	ADJ
ap-7583	16	76	π1(r	π1(r	NOUN
ap-7583	16	77	)	)	PUNCT
ap-7583	16	78	must	must	AUX
ap-7583	16	79	have	have	VERB
ap-7583	16	80	the	the	DET
ap-7583	16	81	same	same	ADJ
ap-7583	16	82	degree	degree	NOUN
ap-7583	16	83	as	as	ADP
ap-7583	16	84	that	that	PRON
ap-7583	16	85	of	of	ADP
ap-7583	16	86	π0(r	π0(r	PROPN
ap-7583	16	87	)	)	PUNCT
ap-7583	16	88	,	,	PUNCT
ap-7583	16	89	so	so	SCONJ
ap-7583	16	90	we	we	PRON
ap-7583	16	91	may	may	AUX
ap-7583	16	92	combine	combine	VERB
ap-7583	16	93	the	the	DET
ap-7583	16	94	same	same	ADJ
ap-7583	16	95	degree	degree	NOUN
ap-7583	16	96	polynomial	polynomial	ADJ
ap-7583	16	97	coefficients	coefficient	NOUN
ap-7583	16	98	of	of	ADP
ap-7583	16	99	y	y	PROPN
ap-7583	16	100	and	and	CCONJ
ap-7583	16	101	write	write	VERB
ap-7583	16	102	the	the	DET
ap-7583	16	103	equation	equation	NOUN
ap-7583	16	104	as	as	ADP
ap-7583	16	105	π3(r)y′′	π3(r)y′′	NOUN
ap-7583	16	106	+	+	CCONJ
ap-7583	16	107	π2(r)y′	π2(r)y′	NOUN
ap-7583	17	1	+	+	CCONJ
ap-7583	17	2	π1(r)y	π1(r)y	SYM
ap-7583	17	3	=	=	SYM
ap-7583	17	4	0	0	X
ap-7583	17	5	.	.	PUNCT
ap-7583	18	1	next	next	ADV
ap-7583	18	2	,	,	PUNCT
ap-7583	18	3	for	for	ADP
ap-7583	18	4	a	a	DET
ap-7583	18	5	polynomial	polynomial	ADJ
ap-7583	18	6	solution	solution	NOUN
ap-7583	18	7	of	of	ADP
ap-7583	18	8	degree	degree	NOUN
ap-7583	18	9	one	one	NUM
ap-7583	18	10	,	,	PUNCT
ap-7583	18	11	say	say	VERB
ap-7583	18	12	y1(r	y1(r	NOUN
ap-7583	18	13	)	)	PUNCT
ap-7583	19	1	=	=	SYM
ap-7583	19	2	r	r	NOUN
ap-7583	19	3	+	+	NUM
ap-7583	19	4	α	α	NOUN
ap-7583	19	5	,	,	PUNCT
ap-7583	19	6	the	the	DET
ap-7583	19	7	differential	differential	ADJ
ap-7583	19	8	equation	equation	NOUN
ap-7583	19	9	reduces	reduce	VERB
ap-7583	19	10	to	to	ADP
ap-7583	19	11	π2(r	π2(r	PROPN
ap-7583	19	12	)	)	PUNCT
ap-7583	20	1	+	+	CCONJ
ap-7583	20	2	π1(r)(r	π1(r)(r	NOUN
ap-7583	20	3	+	+	CCONJ
ap-7583	20	4	α	α	NOUN
ap-7583	20	5	)	)	PUNCT
ap-7583	20	6	=	=	SYM
ap-7583	20	7	0	0	NUM
ap-7583	20	8	and	and	CCONJ
ap-7583	20	9	the	the	DET
ap-7583	20	10	degree	degree	NOUN
ap-7583	20	11	of	of	ADP
ap-7583	20	12	π2	π2	NOUN
ap-7583	20	13	should	should	AUX
ap-7583	20	14	be	be	AUX
ap-7583	20	15	the	the	DET
ap-7583	20	16	degree	degree	NOUN
ap-7583	20	17	of	of	ADP
ap-7583	20	18	π1(r	π1(r	NOUN
ap-7583	20	19	)	)	PUNCT
ap-7583	20	20	plus	plus	CCONJ
ap-7583	20	21	one	one	NUM
ap-7583	20	22	.	.	PUNCT
ap-7583	21	1	similarly	similarly	ADV
ap-7583	21	2	,	,	PUNCT
ap-7583	21	3	for	for	ADP
ap-7583	21	4	a	a	DET
ap-7583	21	5	second	second	ADJ
ap-7583	21	6	-	-	PUNCT
ap-7583	21	7	order	order	NOUN
ap-7583	21	8	polynomial	polynomial	ADJ
ap-7583	21	9	solution	solution	NOUN
ap-7583	21	10	,	,	PUNCT
ap-7583	21	11	say	say	VERB
ap-7583	21	12	y(r	y(r	NOUN
ap-7583	21	13	)	)	PUNCT
ap-7583	21	14	=	=	SYM
ap-7583	21	15	r2	r2	PROPN
ap-7583	21	16	+	+	CCONJ
ap-7583	21	17	α	α	NOUN
ap-7583	21	18	r	r	NOUN
ap-7583	22	1	+	+	NUM
ap-7583	22	2	β	β	X
ap-7583	22	3	,	,	PUNCT
ap-7583	22	4	it	it	PRON
ap-7583	22	5	follows	follow	VERB
ap-7583	22	6	by	by	ADP
ap-7583	22	7	substitution	substitution	NOUN
ap-7583	22	8	that	that	SCONJ
ap-7583	22	9	π3(r	π3(r	VERB
ap-7583	22	10	)	)	PUNCT
ap-7583	22	11	+	+	NUM
ap-7583	22	12	π2(r)(2r	π2(r)(2r	X
ap-7583	22	13	+	+	CCONJ
ap-7583	22	14	α	α	X
ap-7583	22	15	)	)	PUNCT
ap-7583	22	16	+	+	CCONJ
ap-7583	22	17	π1(r)(r2	π1(r)(r2	VERB
ap-7583	22	18	+	+	CCONJ
ap-7583	22	19	αr	αr	NUM
ap-7583	22	20	+	+	NOUN
ap-7583	22	21	β	β	X
ap-7583	22	22	)	)	PUNCT
ap-7583	22	23	=	=	SYM
ap-7583	22	24	0	0	NUM
ap-7583	22	25	which	which	PRON
ap-7583	22	26	indicates	indicate	VERB
ap-7583	22	27	that	that	SCONJ
ap-7583	22	28	the	the	DET
ap-7583	22	29	degree	degree	NOUN
ap-7583	22	30	of	of	ADP
ap-7583	22	31	π3(r	π3(r	X
ap-7583	22	32	)	)	PUNCT
ap-7583	22	33	should	should	AUX
ap-7583	22	34	be	be	AUX
ap-7583	22	35	the	the	DET
ap-7583	22	36	degree	degree	NOUN
ap-7583	22	37	of	of	ADP
ap-7583	22	38	π2	π2	NOUN
ap-7583	22	39	plus	plus	CCONJ
ap-7583	22	40	one	one	NUM
ap-7583	22	41	,	,	PUNCT
ap-7583	22	42	which	which	PRON
ap-7583	22	43	,	,	PUNCT
ap-7583	22	44	in	in	ADP
ap-7583	22	45	turn	turn	NOUN
ap-7583	22	46	,	,	PUNCT
ap-7583	22	47	is	be	AUX
ap-7583	22	48	a	a	DET
ap-7583	22	49	polynomial	polynomial	NOUN
ap-7583	22	50	of	of	ADP
ap-7583	22	51	π1	π1	ADJ
ap-7583	22	52	degree	degree	NOUN
ap-7583	22	53	plus	plus	CCONJ
ap-7583	22	54	one	one	NUM
ap-7583	22	55	.	.	PUNCT
ap-7583	23	1	this	this	DET
ap-7583	23	2	simple	simple	ADJ
ap-7583	23	3	argument	argument	NOUN
ap-7583	23	4	shows	show	VERB
ap-7583	23	5	that	that	SCONJ
ap-7583	23	6	for	for	ADP
ap-7583	23	7	the	the	DET
ap-7583	23	8	polynomial	polynomial	ADJ
ap-7583	23	9	solutions	solution	NOUN
ap-7583	23	10	of	of	ADP
ap-7583	23	11	the	the	DET
ap-7583	23	12	linear	linear	ADJ
ap-7583	23	13	second	second	ADJ
ap-7583	23	14	-	-	PUNCT
ap-7583	23	15	order	order	NOUN
ap-7583	23	16	differential	differential	ADJ
ap-7583	23	17	equation	equation	NOUN
ap-7583	23	18	with	with	ADP
ap-7583	23	19	polynomial	polynomial	ADJ
ap-7583	23	20	coefficients	coefficient	NOUN
ap-7583	23	21	,	,	PUNCT
ap-7583	23	22	the	the	DET
ap-7583	23	23	degree	degree	NOUN
ap-7583	23	24	of	of	ADP
ap-7583	23	25	the	the	DET
ap-7583	23	26	polynomial	polynomial	ADJ
ap-7583	23	27	coefficients	coefficient	NOUN
ap-7583	23	28	πj(r	πj(r	PRON
ap-7583	23	29	)	)	PUNCT
ap-7583	23	30	,	,	PUNCT
ap-7583	23	31	j	j	PROPN
ap-7583	23	32	=	=	SYM
ap-7583	23	33	3	3	NUM
ap-7583	23	34	,	,	PUNCT
ap-7583	23	35	2	2	NUM
ap-7583	23	36	,	,	PUNCT
ap-7583	23	37	1	1	NUM
ap-7583	23	38	must	must	AUX
ap-7583	23	39	be	be	AUX
ap-7583	23	40	of	of	ADP
ap-7583	23	41	degree	degree	NOUN
ap-7583	23	42	n	n	CCONJ
ap-7583	23	43	,	,	PUNCT
ap-7583	23	44	n	n	CCONJ
ap-7583	23	45	−	−	PROPN
ap-7583	23	46	1	1	NUM
ap-7583	23	47	165	165	NUM
ap-7583	23	48	https://doi.org/10.14311/ap.2022.62.0165	https://doi.org/10.14311/ap.2022.62.0165	PROPN
ap-7583	23	49	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
ap-7583	23	50	https://www.cvut.cz/en	https://www.cvut.cz/en	PROPN
ap-7583	23	51	nasser	nasser	PROPN
ap-7583	23	52	saad	saad	PROPN
ap-7583	23	53	acta	acta	PROPN
ap-7583	23	54	polytechnica	polytechnica	PROPN
ap-7583	23	55	and	and	CCONJ
ap-7583	23	56	n	n	CCONJ
ap-7583	23	57	−	−	PROPN
ap-7583	23	58	2	2	NUM
ap-7583	23	59	,	,	PUNCT
ap-7583	23	60	respectively	respectively	ADV
ap-7583	23	61	.	.	PUNCT
ap-7583	24	1	so	so	ADV
ap-7583	24	2	,	,	PUNCT
ap-7583	24	3	without	without	ADP
ap-7583	24	4	the	the	DET
ap-7583	24	5	loss	loss	NOUN
ap-7583	24	6	of	of	ADP
ap-7583	24	7	generality	generality	NOUN
ap-7583	24	8	,	,	PUNCT
ap-7583	24	9	we	we	PRON
ap-7583	24	10	may	may	AUX
ap-7583	24	11	direct	direct	VERB
ap-7583	24	12	out	out	ADP
ap-7583	24	13	attention	attention	NOUN
ap-7583	24	14	to	to	ADP
ap-7583	24	15	the	the	DET
ap-7583	24	16	following	following	NOUN
ap-7583	24	17	:	:	PUNCT
ap-7583	24	18	under	under	ADP
ap-7583	24	19	what	what	DET
ap-7583	24	20	conditions	condition	NOUN
ap-7583	24	21	on	on	ADP
ap-7583	24	22	the	the	DET
ap-7583	24	23	equation	equation	NOUN
ap-7583	24	24	parameters	parameter	NOUN
ap-7583	24	25	αk	αk	VERB
ap-7583	24	26	,	,	PUNCT
ap-7583	24	27	βk	βk	NOUN
ap-7583	24	28	,	,	PUNCT
ap-7583	24	29	and	and	CCONJ
ap-7583	24	30	εk	εk	NOUN
ap-7583	24	31	,	,	PUNCT
ap-7583	24	32	for	for	ADP
ap-7583	24	33	k	k	PROPN
ap-7583	24	34	=	=	SYM
ap-7583	24	35	0	0	NUM
ap-7583	24	36	,	,	PUNCT
ap-7583	24	37	1	1	NUM
ap-7583	24	38	,	,	PUNCT
ap-7583	24	39	.	.	PUNCT
ap-7583	24	40	.	.	PUNCT
ap-7583	25	1	.	.	PUNCT
ap-7583	26	1	,	,	PUNCT
ap-7583	26	2	n	n	CCONJ
ap-7583	26	3	,	,	PUNCT
ap-7583	26	4	does	do	VERB
ap-7583	26	5	the	the	DET
ap-7583	26	6	differential	differential	ADJ
ap-7583	26	7	equation	equation	NOUN
ap-7583	26	8	(	(	PUNCT
ap-7583	26	9	n∑	n∑	NOUN
ap-7583	26	10	k=0	k=0	PROPN
ap-7583	26	11	αk	αk	AUX
ap-7583	26	12	rk	rk	NOUN
ap-7583	26	13	)	)	PUNCT
ap-7583	26	14	y′′(r	y′′(r	NOUN
ap-7583	26	15	)	)	PUNCT
ap-7583	26	16	+	+	CCONJ
ap-7583	26	17	(	(	PUNCT
ap-7583	26	18	n−1∑	n−1∑	PROPN
ap-7583	26	19	k=0	k=0	PROPN
ap-7583	26	20	βk	βk	ADP
ap-7583	26	21	rk	rk	PROPN
ap-7583	26	22	)	)	PUNCT
ap-7583	26	23	y′(r	y′(r	SYM
ap-7583	26	24	)	)	PUNCT
ap-7583	27	1	+	+	CCONJ
ap-7583	27	2	(	(	PUNCT
ap-7583	27	3	n−2∑	n−2∑	NUM
ap-7583	27	4	k=0	k=0	PROPN
ap-7583	27	5	εk	εk	PROPN
ap-7583	27	6	rk	rk	PROPN
ap-7583	27	7	)	)	PUNCT
ap-7583	27	8	y(r	y(r	PROPN
ap-7583	27	9	)	)	PUNCT
ap-7583	27	10	=	=	SYM
ap-7583	27	11	0	0	NUM
ap-7583	27	12	,	,	PUNCT
ap-7583	27	13	n	n	PRON
ap-7583	27	14	≥	≥	NOUN
ap-7583	27	15	2	2	NUM
ap-7583	27	16	(	(	PUNCT
ap-7583	27	17	1	1	NUM
ap-7583	27	18	)	)	PUNCT
ap-7583	27	19	has	have	VERB
ap-7583	27	20	polynomial	polynomial	ADJ
ap-7583	27	21	solutions	solution	NOUN
ap-7583	27	22	y	y	PROPN
ap-7583	27	23	=	=	SYM
ap-7583	27	24	∑m	∑m	PROPN
ap-7583	27	25	k=0	k=0	PROPN
ap-7583	27	26	cjrj	cjrj	VERB
ap-7583	27	27	?	?	PUNCT
ap-7583	28	1	a	a	DET
ap-7583	28	2	logical	logical	ADJ
ap-7583	28	3	approach	approach	NOUN
ap-7583	28	4	is	be	AUX
ap-7583	28	5	to	to	PART
ap-7583	28	6	examine	examine	VERB
ap-7583	28	7	the	the	DET
ap-7583	28	8	differential	differential	ADJ
ap-7583	28	9	equation	equation	NOUN
ap-7583	28	10	using	use	VERB
ap-7583	28	11	the	the	DET
ap-7583	28	12	series	series	NOUN
ap-7583	28	13	solution	solution	PROPN
ap-7583	28	14	y	y	NOUN
ap-7583	29	1	=	=	PROPN
ap-7583	30	1	∞∑	∞∑	NUM
ap-7583	30	2	j=0	j=0	PROPN
ap-7583	30	3	cj	cj	PROPN
ap-7583	30	4	rj	rj	PROPN
ap-7583	30	5	,	,	PUNCT
ap-7583	30	6	y′	y′	X
ap-7583	30	7	=	=	SYM
ap-7583	30	8	∞∑	∞∑	NUM
ap-7583	30	9	j=0	j=0	PROPN
ap-7583	30	10	j	j	PROPN
ap-7583	30	11	cj	cj	PROPN
ap-7583	30	12	rj−1	rj−1	PROPN
ap-7583	30	13	,	,	PUNCT
ap-7583	30	14	y′′	y′′	NOUN
ap-7583	30	15	=	=	PUNCT
ap-7583	31	1	∞∑	∞∑	NUM
ap-7583	31	2	j=0	j=0	PROPN
ap-7583	31	3	j	j	PROPN
ap-7583	31	4	(	(	PUNCT
ap-7583	31	5	j	j	PROPN
ap-7583	31	6	−	−	PROPN
ap-7583	31	7	1)cj	1)cj	PROPN
ap-7583	31	8	rj−2	rj−2	PROPN
ap-7583	31	9	in	in	ADP
ap-7583	31	10	(	(	PUNCT
ap-7583	31	11	1	1	NUM
ap-7583	31	12	)	)	PUNCT
ap-7583	31	13	and	and	CCONJ
ap-7583	31	14	enforce	enforce	VERB
ap-7583	31	15	the	the	DET
ap-7583	31	16	coefficients	coefficient	NOUN
ap-7583	31	17	cj	cj	PROPN
ap-7583	31	18	=	=	SYM
ap-7583	31	19	0	0	NUM
ap-7583	31	20	for	for	ADP
ap-7583	31	21	all	all	DET
ap-7583	31	22	j	j	PROPN
ap-7583	31	23	≥	≥	NUM
ap-7583	31	24	m	m	VERB
ap-7583	31	25	+	+	ADJ
ap-7583	31	26	1	1	NUM
ap-7583	31	27	,	,	PUNCT
ap-7583	31	28	m	m	VERB
ap-7583	31	29	=	=	NOUN
ap-7583	31	30	0	0	NUM
ap-7583	31	31	,	,	PUNCT
ap-7583	31	32	1	1	NUM
ap-7583	31	33	,	,	PUNCT
ap-7583	31	34	2	2	NUM
ap-7583	31	35	,	,	PUNCT
ap-7583	31	36	·	·	PUNCT
ap-7583	31	37	·	·	PUNCT
ap-7583	31	38	·	·	PUNCT
ap-7583	31	39	to	to	PART
ap-7583	31	40	find	find	VERB
ap-7583	31	41	the	the	DET
ap-7583	31	42	condition	condition	NOUN
ap-7583	31	43	so	so	SCONJ
ap-7583	31	44	that	that	SCONJ
ap-7583	31	45	cm	cm	NOUN
ap-7583	31	46	=	=	SYM
ap-7583	31	47	̸	̸	NUM
ap-7583	31	48	0	0	NUM
ap-7583	31	49	.	.	PUNCT
ap-7583	32	1	this	this	DET
ap-7583	32	2	approach	approach	NOUN
ap-7583	32	3	leads	lead	VERB
ap-7583	32	4	to	to	ADP
ap-7583	32	5	a	a	DET
ap-7583	32	6	conclusion	conclusion	NOUN
ap-7583	32	7	that	that	SCONJ
ap-7583	32	8	for	for	ADP
ap-7583	32	9	equation	equation	NOUN
ap-7583	32	10	(	(	PUNCT
ap-7583	32	11	1	1	NUM
ap-7583	32	12	)	)	PUNCT
ap-7583	32	13	to	to	PART
ap-7583	32	14	have	have	AUX
ap-7583	32	15	m	m	NOUN
ap-7583	32	16	degree	degree	NOUN
ap-7583	32	17	polynomial	polynomial	ADJ
ap-7583	32	18	solution	solution	NOUN
ap-7583	32	19	,	,	PUNCT
ap-7583	32	20	it	it	PRON
ap-7583	32	21	is	be	AUX
ap-7583	32	22	necessary	necessary	ADJ
ap-7583	32	23	that	that	SCONJ
ap-7583	32	24	εn−2	εn−2	PROPN
ap-7583	32	25	=	=	SYM
ap-7583	32	26	−m	−m	PROPN
ap-7583	32	27	(	(	PUNCT
ap-7583	32	28	m	m	NOUN
ap-7583	32	29	−	−	NOUN
ap-7583	32	30	1	1	NUM
ap-7583	32	31	)	)	PUNCT
ap-7583	32	32	αn	αn	NOUN
ap-7583	32	33	−	−	PROPN
ap-7583	32	34	m	m	VERB
ap-7583	32	35	βn−1	βn−1	ADJ
ap-7583	32	36	,	,	PUNCT
ap-7583	32	37	n	n	NOUN
ap-7583	32	38	=	=	SYM
ap-7583	32	39	2	2	NUM
ap-7583	32	40	,	,	PUNCT
ap-7583	32	41	3	3	NUM
ap-7583	32	42	,	,	PUNCT
ap-7583	32	43	·	·	PUNCT
ap-7583	32	44	·	·	PUNCT
ap-7583	32	45	·	·	PUNCT
ap-7583	32	46	.	.	PUNCT
ap-7583	33	1	(	(	PUNCT
ap-7583	33	2	2	2	X
ap-7583	33	3	)	)	PUNCT
ap-7583	33	4	did	do	AUX
ap-7583	33	5	this	this	PRON
ap-7583	33	6	answer	answer	VERB
ap-7583	33	7	the	the	DET
ap-7583	33	8	question	question	NOUN
ap-7583	33	9	?	?	PUNCT
ap-7583	34	1	indeed	indeed	ADV
ap-7583	34	2	,	,	PUNCT
ap-7583	34	3	no	no	INTJ
ap-7583	34	4	.	.	PUNCT
ap-7583	35	1	consider	consider	VERB
ap-7583	35	2	,	,	PUNCT
ap-7583	35	3	for	for	ADP
ap-7583	35	4	example	example	NOUN
ap-7583	35	5	,	,	PUNCT
ap-7583	35	6	this	this	DET
ap-7583	35	7	simple	simple	ADJ
ap-7583	35	8	equation	equation	NOUN
ap-7583	35	9	r3y′′	r3y′′	NOUN
ap-7583	36	1	+	+	CCONJ
ap-7583	36	2	2	2	NUM
ap-7583	36	3	r2y′	r2y′	NOUN
ap-7583	36	4	+	+	CCONJ
ap-7583	36	5	(	(	PUNCT
ap-7583	36	6	−2	−2	NOUN
ap-7583	36	7	r	r	NOUN
ap-7583	36	8	+	+	NUM
ap-7583	36	9	5)y	5)y	NUM
ap-7583	36	10	=	=	SYM
ap-7583	36	11	0	0	X
ap-7583	36	12	.	.	PUNCT
ap-7583	37	1	clearly	clearly	ADV
ap-7583	37	2	,	,	PUNCT
ap-7583	37	3	the	the	DET
ap-7583	37	4	necessary	necessary	ADJ
ap-7583	37	5	condition	condition	NOUN
ap-7583	37	6	(	(	PUNCT
ap-7583	37	7	2	2	NUM
ap-7583	37	8	)	)	PUNCT
ap-7583	37	9	is	be	AUX
ap-7583	37	10	satisfied	satisfied	ADJ
ap-7583	37	11	for	for	ADP
ap-7583	37	12	m	m	PROPN
ap-7583	37	13	=	=	SYM
ap-7583	37	14	1	1	NUM
ap-7583	37	15	and	and	CCONJ
ap-7583	37	16	one	one	NUM
ap-7583	37	17	expects	expect	VERB
ap-7583	37	18	the	the	DET
ap-7583	37	19	existence	existence	NOUN
ap-7583	37	20	of	of	ADP
ap-7583	37	21	a	a	DET
ap-7583	37	22	first	first	ADJ
ap-7583	37	23	degree	degree	NOUN
ap-7583	37	24	polynomial	polynomial	ADJ
ap-7583	37	25	solution	solution	NOUN
ap-7583	37	26	,	,	PUNCT
ap-7583	37	27	say	say	VERB
ap-7583	37	28	y	y	NOUN
ap-7583	37	29	=	=	PUNCT
ap-7583	37	30	r	r	NOUN
ap-7583	37	31	+	+	NUM
ap-7583	37	32	b	b	NOUN
ap-7583	37	33	,	,	PUNCT
ap-7583	37	34	for	for	ADP
ap-7583	37	35	an	an	DET
ap-7583	37	36	arbitrary	arbitrary	ADJ
ap-7583	37	37	value	value	NOUN
ap-7583	37	38	of	of	ADP
ap-7583	37	39	b	b	PROPN
ap-7583	37	40	∈	∈	PROPN
ap-7583	37	41	r	r	NOUN
ap-7583	37	42	,	,	PUNCT
ap-7583	37	43	however	however	ADV
ap-7583	37	44	,	,	PUNCT
ap-7583	37	45	2	2	NUM
ap-7583	37	46	r2	r2	NOUN
ap-7583	37	47	+	+	CCONJ
ap-7583	37	48	(	(	PUNCT
ap-7583	37	49	−2r	−2r	PROPN
ap-7583	37	50	+	+	PROPN
ap-7583	37	51	5)(r	5)(r	NUM
ap-7583	37	52	+	+	CCONJ
ap-7583	37	53	b	b	X
ap-7583	37	54	)	)	PUNCT
ap-7583	37	55	̸=	̸=	PROPN
ap-7583	37	56	0	0	NUM
ap-7583	37	57	for	for	ADP
ap-7583	37	58	any	any	DET
ap-7583	37	59	real	real	ADJ
ap-7583	37	60	value	value	NOUN
ap-7583	37	61	of	of	ADP
ap-7583	37	62	b.	b.	PROPN
ap-7583	37	63	therefore	therefore	ADV
ap-7583	37	64	,	,	PUNCT
ap-7583	37	65	for	for	ADP
ap-7583	37	66	n	n	PRON
ap-7583	37	67	≥	≥	NUM
ap-7583	37	68	3	3	NUM
ap-7583	37	69	,	,	PUNCT
ap-7583	37	70	the	the	DET
ap-7583	37	71	condition	condition	NOUN
ap-7583	37	72	(	(	PUNCT
ap-7583	37	73	2	2	X
ap-7583	37	74	)	)	PUNCT
ap-7583	37	75	is	be	AUX
ap-7583	37	76	necessary	necessary	ADJ
ap-7583	37	77	but	but	CCONJ
ap-7583	37	78	not	not	PART
ap-7583	37	79	sufficient	sufficient	ADJ
ap-7583	37	80	for	for	ADP
ap-7583	37	81	the	the	DET
ap-7583	37	82	existence	existence	NOUN
ap-7583	37	83	of	of	ADP
ap-7583	37	84	polynomial	polynomial	ADJ
ap-7583	37	85	solutions	solution	NOUN
ap-7583	37	86	of	of	ADP
ap-7583	37	87	the	the	DET
ap-7583	37	88	differential	differential	ADJ
ap-7583	37	89	equation	equation	NOUN
ap-7583	37	90	(	(	PUNCT
ap-7583	37	91	1	1	NUM
ap-7583	37	92	)	)	PUNCT
ap-7583	37	93	.	.	PUNCT
ap-7583	38	1	note	note	VERB
ap-7583	38	2	,	,	PUNCT
ap-7583	38	3	for	for	ADP
ap-7583	38	4	n	n	NOUN
ap-7583	38	5	=	=	SYM
ap-7583	38	6	2	2	NUM
ap-7583	38	7	,	,	PUNCT
ap-7583	38	8	equation	equation	NOUN
ap-7583	38	9	(	(	PUNCT
ap-7583	38	10	1	1	X
ap-7583	38	11	)	)	PUNCT
ap-7583	38	12	is	be	AUX
ap-7583	38	13	the	the	DET
ap-7583	38	14	classical	classical	ADJ
ap-7583	38	15	hypergeometric	hypergeometric	ADJ
ap-7583	38	16	-	-	PUNCT
ap-7583	38	17	type	type	NOUN
ap-7583	38	18	differential	differential	ADJ
ap-7583	38	19	equation	equation	NOUN
ap-7583	38	20	[	[	X
ap-7583	38	21	1–4	1–4	X
ap-7583	38	22	]	]	X
ap-7583	38	23	(	(	PUNCT
ap-7583	38	24	α2	α2	PROPN
ap-7583	38	25	r2	r2	PROPN
ap-7583	38	26	+	+	CCONJ
ap-7583	38	27	α1	α1	PROPN
ap-7583	38	28	r	r	NOUN
ap-7583	38	29	+	+	CCONJ
ap-7583	38	30	α0	α0	ADJ
ap-7583	38	31	)	)	PUNCT
ap-7583	38	32	y′′	y′′	NOUN
ap-7583	38	33	+	+	CCONJ
ap-7583	38	34	(	(	PUNCT
ap-7583	38	35	β1	β1	PROPN
ap-7583	38	36	r	r	NOUN
ap-7583	38	37	+	+	NOUN
ap-7583	38	38	β0	β0	NOUN
ap-7583	38	39	)	)	PUNCT
ap-7583	38	40	y′	y′	PUNCT
ap-7583	39	1	+	+	CCONJ
ap-7583	39	2	ε0	ε0	PROPN
ap-7583	39	3	y	y	NOUN
ap-7583	39	4	=	=	SYM
ap-7583	39	5	0	0	PUNCT
ap-7583	40	1	(	(	PUNCT
ap-7583	40	2	3	3	NUM
ap-7583	40	3	)	)	PUNCT
ap-7583	40	4	with	with	ADP
ap-7583	40	5	the	the	DET
ap-7583	40	6	necessary	necessary	ADJ
ap-7583	40	7	and	and	CCONJ
ap-7583	40	8	sufficient	sufficient	ADJ
ap-7583	40	9	condition	condition	NOUN
ap-7583	41	1	[	[	X
ap-7583	41	2	2	2	NUM
ap-7583	41	3	]	]	PUNCT
ap-7583	41	4	for	for	SCONJ
ap-7583	41	5	the	the	DET
ap-7583	41	6	polynomial	polynomial	ADJ
ap-7583	41	7	solutions	solution	NOUN
ap-7583	41	8	ε0	ε0	NOUN
ap-7583	41	9	=	=	SYM
ap-7583	41	10	−m	−m	NOUN
ap-7583	41	11	(	(	PUNCT
ap-7583	41	12	m	m	NOUN
ap-7583	41	13	−	−	NOUN
ap-7583	41	14	1	1	NUM
ap-7583	41	15	)	)	PUNCT
ap-7583	41	16	α2	α2	ADJ
ap-7583	41	17	−	−	NOUN
ap-7583	41	18	m	m	VERB
ap-7583	41	19	β1	β1	PROPN
ap-7583	41	20	,	,	PUNCT
ap-7583	41	21	m	m	VERB
ap-7583	41	22	=	=	NOUN
ap-7583	41	23	0	0	NUM
ap-7583	41	24	,	,	PUNCT
ap-7583	41	25	1	1	NUM
ap-7583	41	26	,	,	PUNCT
ap-7583	41	27	2	2	NUM
ap-7583	41	28	,	,	PUNCT
ap-7583	41	29	·	·	PUNCT
ap-7583	41	30	·	·	PUNCT
ap-7583	41	31	·	·	PUNCT
ap-7583	41	32	.	.	PUNCT
ap-7583	42	1	for	for	ADP
ap-7583	42	2	n	n	NOUN
ap-7583	42	3	=	=	SYM
ap-7583	42	4	3	3	NUM
ap-7583	42	5	,	,	PUNCT
ap-7583	42	6	the	the	DET
ap-7583	42	7	differential	differential	ADJ
ap-7583	42	8	equation	equation	NOUN
ap-7583	42	9	(	(	PUNCT
ap-7583	42	10	1	1	X
ap-7583	42	11	)	)	PUNCT
ap-7583	42	12	assumes	assume	VERB
ap-7583	42	13	the	the	DET
ap-7583	42	14	form	form	NOUN
ap-7583	42	15	p3(r	p3(r	NOUN
ap-7583	42	16	)	)	PUNCT
ap-7583	42	17	y′′	y′′	PROPN
ap-7583	42	18	+	+	CCONJ
ap-7583	42	19	p2(r	p2(r	PROPN
ap-7583	42	20	)	)	PUNCT
ap-7583	42	21	y′	y′	PUNCT
ap-7583	43	1	+	+	PUNCT
ap-7583	43	2	p1(r	p1(r	X
ap-7583	43	3	)	)	PUNCT
ap-7583	43	4	y	y	NOUN
ap-7583	43	5	=	=	PUNCT
ap-7583	43	6	0,	0,	NUM
ap-7583	43	7	p3(r	p3(r	NOUN
ap-7583	43	8	)	)	PUNCT
ap-7583	43	9	=	=	SYM
ap-7583	43	10	3∑	3∑	PROPN
ap-7583	43	11	j=0	j=0	VERB
ap-7583	43	12	αjrj	αjrj	NOUN
ap-7583	43	13	,	,	PUNCT
ap-7583	43	14	p2(r	p2(r	NOUN
ap-7583	43	15	)	)	PUNCT
ap-7583	43	16	=	=	PUNCT
ap-7583	44	1	2∑	2∑	NUM
ap-7583	44	2	j=0	j=0	PROPN
ap-7583	44	3	βj	βj	PROPN
ap-7583	44	4	rj	rj	PROPN
ap-7583	44	5	,	,	PUNCT
ap-7583	44	6	p1(r	p1(r	PROPN
ap-7583	44	7	)	)	PUNCT
ap-7583	44	8	=	=	SYM
ap-7583	44	9	1∑	1∑	PROPN
ap-7583	44	10	j=0	j=0	PROPN
ap-7583	44	11	εjrj	εjrj	NOUN
ap-7583	44	12	,	,	PUNCT
ap-7583	44	13	αj	αj	X
ap-7583	44	14	,	,	PUNCT
ap-7583	44	15	βj	βj	PRON
ap-7583	44	16	,	,	PUNCT
ap-7583	44	17	εj	εj	NOUN
ap-7583	44	18	∈	∈	PROPN
ap-7583	44	19	r	r	NOUN
ap-7583	44	20	,	,	PUNCT
ap-7583	44	21	(	(	PUNCT
ap-7583	44	22	4	4	X
ap-7583	44	23	)	)	PUNCT
ap-7583	44	24	which	which	PRON
ap-7583	44	25	includes	include	VERB
ap-7583	44	26	as	as	ADP
ap-7583	44	27	a	a	DET
ap-7583	44	28	special	special	ADJ
ap-7583	44	29	case	case	NOUN
ap-7583	44	30	or	or	CCONJ
ap-7583	44	31	with	with	ADP
ap-7583	44	32	elementary	elementary	ADJ
ap-7583	44	33	substitutions	substitution	NOUN
ap-7583	44	34	,	,	PUNCT
ap-7583	44	35	the	the	DET
ap-7583	44	36	classical	classical	ADJ
ap-7583	44	37	heun	heun	NOUN
ap-7583	44	38	differential	differential	NOUN
ap-7583	44	39	equation	equation	NOUN
ap-7583	44	40	[	[	X
ap-7583	44	41	5	5	NUM
ap-7583	44	42	,	,	PUNCT
ap-7583	44	43	6	6	NUM
ap-7583	44	44	]	]	PUNCT
ap-7583	44	45	y′′	y′′	PROPN
ap-7583	44	46	+	+	CCONJ
ap-7583	44	47	(	(	PUNCT
ap-7583	44	48	γ	γ	X
ap-7583	44	49	r	r	NOUN
ap-7583	44	50	+	+	NUM
ap-7583	44	51	δ	δ	PROPN
ap-7583	44	52	r	r	NOUN
ap-7583	44	53	−	−	PROPN
ap-7583	44	54	1	1	NUM
ap-7583	44	55	+	+	NUM
ap-7583	44	56	ε	ε	PROPN
ap-7583	44	57	r	r	NOUN
ap-7583	44	58	−	−	PROPN
ap-7583	44	59	a	a	PRON
ap-7583	44	60	)	)	PUNCT
ap-7583	44	61	y′	y′	NOUN
ap-7583	45	1	+	+	CCONJ
ap-7583	45	2	α	α	PRON
ap-7583	45	3	β	β	NOUN
ap-7583	45	4	r	r	NOUN
ap-7583	45	5	−	−	NOUN
ap-7583	45	6	q	q	NOUN
ap-7583	45	7	r(r	r(r	NOUN
ap-7583	45	8	−	−	PROPN
ap-7583	46	1	1)(r	1)(r	NUM
ap-7583	46	2	−	−	NOUN
ap-7583	46	3	a	a	X
ap-7583	46	4	)	)	PUNCT
ap-7583	46	5	y	y	PROPN
ap-7583	46	6	=	=	SYM
ap-7583	46	7	0	0	PROPN
ap-7583	46	8	,	,	PUNCT
ap-7583	46	9	(	(	PUNCT
ap-7583	46	10	5	5	X
ap-7583	46	11	)	)	PUNCT
ap-7583	46	12	subject	subject	NOUN
ap-7583	46	13	to	to	ADP
ap-7583	46	14	the	the	DET
ap-7583	46	15	regularity	regularity	NOUN
ap-7583	46	16	(	(	PUNCT
ap-7583	46	17	at	at	ADP
ap-7583	46	18	infinity	infinity	NOUN
ap-7583	46	19	)	)	PUNCT
ap-7583	46	20	condition	condition	NOUN
ap-7583	46	21	α	α	NOUN
ap-7583	47	1	+	+	X
ap-7583	47	2	β	β	X
ap-7583	47	3	+	+	ADJ
ap-7583	47	4	1	1	NUM
ap-7583	47	5	=	=	SYM
ap-7583	47	6	γ	γ	X
ap-7583	47	7	+	+	PROPN
ap-7583	47	8	δ	δ	PROPN
ap-7583	47	9	+	+	CCONJ
ap-7583	47	10	ε	ε	PROPN
ap-7583	47	11	,	,	PUNCT
ap-7583	47	12	and	and	CCONJ
ap-7583	47	13	its	its	PRON
ap-7583	47	14	four	four	NUM
ap-7583	47	15	confluent	confluent	ADJ
ap-7583	47	16	forms	form	NOUN
ap-7583	47	17	(	(	PUNCT
ap-7583	47	18	confluent	confluent	ADJ
ap-7583	47	19	,	,	PUNCT
ap-7583	47	20	doubly	doubly	ADV
ap-7583	47	21	-	-	PUNCT
ap-7583	47	22	confluent	confluent	ADJ
ap-7583	47	23	,	,	PUNCT
ap-7583	47	24	biconfluent	biconfluent	ADJ
ap-7583	47	25	and	and	CCONJ
ap-7583	47	26	triconfluent	triconfluent	ADJ
ap-7583	47	27	heun	heun	PROPN
ap-7583	47	28	equations	equations	PROPN
ap-7583	47	29	)	)	PUNCT
ap-7583	47	30	.	.	PUNCT
ap-7583	48	1	these	these	DET
ap-7583	48	2	equations	equation	NOUN
ap-7583	48	3	are	be	AUX
ap-7583	48	4	indispensable	indispensable	ADJ
ap-7583	48	5	from	from	ADP
ap-7583	48	6	the	the	DET
ap-7583	48	7	point	point	NOUN
ap-7583	48	8	of	of	ADP
ap-7583	48	9	view	view	NOUN
ap-7583	48	10	of	of	ADP
ap-7583	48	11	a	a	DET
ap-7583	48	12	mathematical	mathematical	ADJ
ap-7583	48	13	analysis	analysis	NOUN
ap-7583	49	1	[	[	X
ap-7583	49	2	5–11	5–11	X
ap-7583	49	3	]	]	PUNCT
ap-7583	49	4	and	and	CCONJ
ap-7583	49	5	for	for	ADP
ap-7583	49	6	its	its	PRON
ap-7583	49	7	valuable	valuable	ADJ
ap-7583	49	8	applications	application	NOUN
ap-7583	49	9	in	in	ADP
ap-7583	49	10	many	many	ADJ
ap-7583	49	11	areas	area	NOUN
ap-7583	49	12	of	of	ADP
ap-7583	49	13	theoretical	theoretical	ADJ
ap-7583	49	14	physics	physics	NOUN
ap-7583	49	15	[	[	X
ap-7583	49	16	5	5	NUM
ap-7583	49	17	,	,	PUNCT
ap-7583	49	18	6	6	NUM
ap-7583	49	19	,	,	PUNCT
ap-7583	49	20	12–20	12–20	NUM
ap-7583	49	21	]	]	PUNCT
ap-7583	49	22	.	.	PUNCT
ap-7583	50	1	in	in	ADP
ap-7583	50	2	the	the	DET
ap-7583	50	3	present	present	ADJ
ap-7583	50	4	work	work	NOUN
ap-7583	50	5	166	166	NUM
ap-7583	50	6	vol	vol	NOUN
ap-7583	50	7	.	.	PUNCT
ap-7583	51	1	62	62	NUM
ap-7583	51	2	no	no	INTJ
ap-7583	51	3	.	.	PUNCT
ap-7583	52	1	1/2022	1/2022	NUM
ap-7583	52	2	on	on	ADP
ap-7583	52	3	generalized	generalized	ADJ
ap-7583	52	4	heun	heun	NOUN
ap-7583	52	5	equation	equation	NOUN
ap-7583	52	6	with	with	ADP
ap-7583	52	7	some	some	DET
ap-7583	52	8	mathematical	mathematical	NOUN
ap-7583	52	9	.	.	PUNCT
ap-7583	52	10	.	.	PUNCT
ap-7583	53	1	.	.	PUNCT
ap-7583	54	1	•	•	NUM
ap-7583	54	2	from	from	ADP
ap-7583	54	3	equation	equation	NOUN
ap-7583	54	4	(	(	PUNCT
ap-7583	54	5	4	4	NUM
ap-7583	54	6	)	)	PUNCT
ap-7583	54	7	,	,	PUNCT
ap-7583	54	8	we	we	PRON
ap-7583	54	9	will	will	AUX
ap-7583	54	10	extract	extract	VERB
ap-7583	54	11	the	the	DET
ap-7583	54	12	possible	possible	ADJ
ap-7583	54	13	differential	differential	ADJ
ap-7583	54	14	equations	equation	NOUN
ap-7583	54	15	that	that	PRON
ap-7583	54	16	can	can	AUX
ap-7583	54	17	be	be	AUX
ap-7583	54	18	solved	solve	VERB
ap-7583	54	19	using	use	VERB
ap-7583	54	20	two	two	NUM
ap-7583	54	21	-	-	PUNCT
ap-7583	54	22	term	term	NOUN
ap-7583	54	23	recurrence	recurrence	NOUN
ap-7583	54	24	formulas	formula	NOUN
ap-7583	54	25	.	.	PUNCT
ap-7583	55	1	•	•	NUM
ap-7583	55	2	from	from	ADP
ap-7583	55	3	equation	equation	NOUN
ap-7583	55	4	(	(	PUNCT
ap-7583	55	5	4	4	NUM
ap-7583	55	6	)	)	PUNCT
ap-7583	55	7	,	,	PUNCT
ap-7583	55	8	we	we	PRON
ap-7583	55	9	will	will	AUX
ap-7583	55	10	extract	extract	VERB
ap-7583	55	11	all	all	DET
ap-7583	55	12	the	the	DET
ap-7583	55	13	differential	differential	ADJ
ap-7583	55	14	equations	equation	NOUN
ap-7583	55	15	whose	whose	DET
ap-7583	55	16	series	series	NOUN
ap-7583	55	17	solutions	solution	NOUN
ap-7583	55	18	can	can	AUX
ap-7583	55	19	be	be	AUX
ap-7583	55	20	evaluated	evaluate	VERB
ap-7583	55	21	with	with	ADP
ap-7583	55	22	a	a	DET
ap-7583	55	23	three	three	NUM
ap-7583	55	24	-	-	PUNCT
ap-7583	55	25	term	term	NOUN
ap-7583	55	26	recurrence	recurrence	NOUN
ap-7583	55	27	formula	formula	NOUN
ap-7583	55	28	.	.	PUNCT
ap-7583	56	1	•	•	NUM
ap-7583	56	2	for	for	ADP
ap-7583	56	3	a0	a0	PROPN
ap-7583	56	4	̸=	̸=	PROPN
ap-7583	56	5	0	0	NUM
ap-7583	56	6	,	,	PUNCT
ap-7583	56	7	we	we	PRON
ap-7583	56	8	shall	shall	AUX
ap-7583	56	9	devise	devise	VERB
ap-7583	56	10	a	a	DET
ap-7583	56	11	procedure	procedure	NOUN
ap-7583	56	12	based	base	VERB
ap-7583	56	13	on	on	ADP
ap-7583	56	14	the	the	DET
ap-7583	56	15	asymptotic	asymptotic	ADJ
ap-7583	56	16	iteration	iteration	NOUN
ap-7583	56	17	method	method	NOUN
ap-7583	57	1	[	[	X
ap-7583	57	2	21	21	NUM
ap-7583	57	3	]	]	PUNCT
ap-7583	57	4	to	to	PART
ap-7583	57	5	find	find	VERB
ap-7583	57	6	the	the	DET
ap-7583	57	7	series	series	NOUN
ap-7583	57	8	and	and	CCONJ
ap-7583	57	9	polynomial	polynomial	ADJ
ap-7583	57	10	solutions	solution	NOUN
ap-7583	57	11	of	of	ADP
ap-7583	57	12	the	the	DET
ap-7583	57	13	differential	differential	ADJ
ap-7583	57	14	equation	equation	NOUN
ap-7583	57	15	(	(	PUNCT
ap-7583	57	16	4	4	NUM
ap-7583	57	17	)	)	PUNCT
ap-7583	57	18	.	.	PUNCT
ap-7583	58	1	•	•	NOUN
ap-7583	58	2	in	in	ADP
ap-7583	58	3	the	the	DET
ap-7583	58	4	neighbordhoud	neighbordhoud	NOUN
ap-7583	58	5	of	of	ADP
ap-7583	58	6	a	a	DET
ap-7583	58	7	singular	singular	ADJ
ap-7583	58	8	point	point	NOUN
ap-7583	58	9	r	r	NOUN
ap-7583	58	10	=	=	SYM
ap-7583	58	11	0	0	NUM
ap-7583	58	12	,	,	PUNCT
ap-7583	58	13	i.e.	i.e.	X
ap-7583	58	14	,	,	PUNCT
ap-7583	58	15	with	with	ADP
ap-7583	58	16	a0	a0	PROPN
ap-7583	58	17	=	=	SYM
ap-7583	58	18	0	0	PROPN
ap-7583	58	19	,	,	PUNCT
ap-7583	58	20	we	we	PRON
ap-7583	58	21	will	will	AUX
ap-7583	58	22	prove	prove	VERB
ap-7583	58	23	that	that	SCONJ
ap-7583	58	24	the	the	DET
ap-7583	58	25	series	series	NOUN
ap-7583	58	26	solution	solution	NOUN
ap-7583	58	27	can	can	AUX
ap-7583	58	28	be	be	AUX
ap-7583	58	29	written	write	VERB
ap-7583	58	30	as	as	ADP
ap-7583	58	31	y(r	y(r	NOUN
ap-7583	58	32	)	)	PUNCT
ap-7583	58	33	=	=	PUNCT
ap-7583	59	1	∞∑	∞∑	NUM
ap-7583	59	2	k=0	k=0	PROPN
ap-7583	59	3	(	(	PUNCT
ap-7583	59	4	−1)k	−1)k	PROPN
ap-7583	59	5	pk;s(ε0	pk;s(ε0	NOUN
ap-7583	59	6	)	)	PUNCT
ap-7583	59	7	αk	αk	CCONJ
ap-7583	59	8	1	1	NUM
ap-7583	59	9	(	(	PUNCT
ap-7583	59	10	β0	β0	PROPN
ap-7583	59	11	α1	α1	PROPN
ap-7583	59	12	+	+	SYM
ap-7583	59	13	s	s	NOUN
ap-7583	59	14	)	)	PUNCT
ap-7583	59	15	k	k	NOUN
ap-7583	59	16	(	(	PUNCT
ap-7583	59	17	1	1	NUM
ap-7583	59	18	+	+	NUM
ap-7583	59	19	s)k	s)k	NOUN
ap-7583	59	20	rk+s	rk+s	NOUN
ap-7583	59	21	,	,	PUNCT
ap-7583	59	22	where	where	SCONJ
ap-7583	59	23	s	s	NOUN
ap-7583	59	24	is	be	AUX
ap-7583	59	25	a	a	DET
ap-7583	59	26	root	root	NOUN
ap-7583	59	27	of	of	ADP
ap-7583	59	28	the	the	DET
ap-7583	59	29	indicial	indicial	ADJ
ap-7583	59	30	equation	equation	NOUN
ap-7583	59	31	.	.	PUNCT
ap-7583	60	1	also	also	ADV
ap-7583	60	2	,	,	PUNCT
ap-7583	60	3	we	we	PRON
ap-7583	60	4	show	show	VERB
ap-7583	60	5	that	that	SCONJ
ap-7583	60	6	{	{	PUNCT
ap-7583	60	7	pk;s(ε0)}∞	pk;s(ε0)}∞	NOUN
ap-7583	60	8	k=0	k=0	PROPN
ap-7583	60	9	is	be	AUX
ap-7583	60	10	an	an	DET
ap-7583	60	11	infinite	infinite	ADJ
ap-7583	60	12	sequence	sequence	NOUN
ap-7583	60	13	of	of	ADP
ap-7583	60	14	orthogonal	orthogonal	ADJ
ap-7583	60	15	polynomials	polynomial	NOUN
ap-7583	60	16	with	with	ADP
ap-7583	60	17	several	several	ADJ
ap-7583	60	18	interesting	interesting	ADJ
ap-7583	60	19	properties	property	NOUN
ap-7583	60	20	.	.	PUNCT
ap-7583	61	1	•	•	NUM
ap-7583	61	2	by	by	ADP
ap-7583	61	3	imposing	impose	VERB
ap-7583	61	4	the	the	DET
ap-7583	61	5	termination	termination	NOUN
ap-7583	61	6	conditions	condition	NOUN
ap-7583	61	7	,	,	PUNCT
ap-7583	61	8	we	we	PRON
ap-7583	61	9	study	study	VERB
ap-7583	61	10	the	the	DET
ap-7583	61	11	mathematical	mathematical	ADJ
ap-7583	61	12	properties	property	NOUN
ap-7583	61	13	of	of	ADP
ap-7583	61	14	the	the	DET
ap-7583	61	15	finite	finite	ADJ
ap-7583	61	16	sequences	sequence	NOUN
ap-7583	61	17	of	of	ADP
ap-7583	61	18	the	the	DET
ap-7583	61	19	orthogonal	orthogonal	ADJ
ap-7583	61	20	polynomials	polynomial	NOUN
ap-7583	61	21	{	{	PUNCT
ap-7583	61	22	pk;s(ε0)}n	pk;s(ε0)}n	NOUN
ap-7583	61	23	k=0	k=0	PROPN
ap-7583	61	24	and	and	CCONJ
ap-7583	61	25	explore	explore	VERB
ap-7583	61	26	the	the	DET
ap-7583	61	27	factorization	factorization	NOUN
ap-7583	61	28	property	property	NOUN
ap-7583	61	29	associated	associate	VERB
ap-7583	61	30	with	with	ADP
ap-7583	61	31	these	these	DET
ap-7583	61	32	polynomials	polynomial	NOUN
ap-7583	61	33	.	.	PUNCT
ap-7583	62	1	2	2	X
ap-7583	62	2	.	.	X
ap-7583	62	3	elementary	elementary	ADJ
ap-7583	62	4	observations	observation	NOUN
ap-7583	62	5	the	the	DET
ap-7583	62	6	classical	classical	ADJ
ap-7583	62	7	approach	approach	NOUN
ap-7583	62	8	to	to	PART
ap-7583	62	9	study	study	VERB
ap-7583	62	10	the	the	DET
ap-7583	62	11	analytical	analytical	ADJ
ap-7583	62	12	solutions	solution	NOUN
ap-7583	62	13	of	of	ADP
ap-7583	62	14	equation	equation	NOUN
ap-7583	62	15	(	(	PUNCT
ap-7583	62	16	4	4	X
ap-7583	62	17	)	)	PUNCT
ap-7583	62	18	relies	rely	VERB
ap-7583	62	19	on	on	ADP
ap-7583	62	20	the	the	DET
ap-7583	62	21	nature	nature	NOUN
ap-7583	62	22	of	of	ADP
ap-7583	62	23	the	the	DET
ap-7583	62	24	singular	singular	ADJ
ap-7583	62	25	points	point	NOUN
ap-7583	62	26	of	of	ADP
ap-7583	62	27	the	the	DET
ap-7583	62	28	leading	lead	VERB
ap-7583	62	29	polynomial	polynomial	ADJ
ap-7583	62	30	coefficients	coefficient	NOUN
ap-7583	62	31	l	l	PROPN
ap-7583	62	32	≡	≡	PROPN
ap-7583	63	1	α0	α0	ADJ
ap-7583	63	2	+	+	CCONJ
ap-7583	63	3	α1	α1	PROPN
ap-7583	63	4	r	r	NOUN
ap-7583	63	5	+	+	CCONJ
ap-7583	63	6	α2	α2	ADJ
ap-7583	63	7	r2	r2	PROPN
ap-7583	63	8	+	+	CCONJ
ap-7583	63	9	α3	α3	PROPN
ap-7583	63	10	r3	r3	PROPN
ap-7583	63	11	in	in	ADP
ap-7583	63	12	addition	addition	NOUN
ap-7583	63	13	to	to	ADP
ap-7583	63	14	the	the	DET
ap-7583	63	15	point	point	NOUN
ap-7583	63	16	r	r	NOUN
ap-7583	63	17	=	=	SYM
ap-7583	63	18	∞	∞	PROPN
ap-7583	63	19	in	in	ADP
ap-7583	63	20	the	the	DET
ap-7583	63	21	extended	extended	ADJ
ap-7583	63	22	plane	plane	NOUN
ap-7583	63	23	.	.	PUNCT
ap-7583	64	1	for	for	ADP
ap-7583	64	2	real	real	ADJ
ap-7583	64	3	coefficients	coefficient	NOUN
ap-7583	64	4	and	and	CCONJ
ap-7583	64	5	α0	α0	ADJ
ap-7583	64	6	̸=	̸=	PROPN
ap-7583	64	7	0	0	NUM
ap-7583	64	8	,	,	PUNCT
ap-7583	64	9	the	the	DET
ap-7583	64	10	odd	odd	ADJ
ap-7583	64	11	-	-	PUNCT
ap-7583	64	12	degree	degree	NOUN
ap-7583	64	13	polynomial	polynomial	ADJ
ap-7583	64	14	l	l	NOUN
ap-7583	64	15	is	be	AUX
ap-7583	64	16	factored	factor	VERB
ap-7583	64	17	into	into	ADP
ap-7583	64	18	either	either	CCONJ
ap-7583	64	19	a	a	DET
ap-7583	64	20	product	product	NOUN
ap-7583	64	21	of	of	ADP
ap-7583	64	22	a	a	DET
ap-7583	64	23	linear	linear	ADJ
ap-7583	64	24	polynomial	polynomial	NOUN
ap-7583	64	25	and	and	CCONJ
ap-7583	64	26	an	an	DET
ap-7583	64	27	irreducible	irreducible	ADJ
ap-7583	64	28	quadratic	quadratic	ADJ
ap-7583	64	29	polynomial	polynomial	NOUN
ap-7583	64	30	or	or	CCONJ
ap-7583	64	31	a	a	DET
ap-7583	64	32	product	product	NOUN
ap-7583	64	33	of	of	ADP
ap-7583	64	34	three	three	NUM
ap-7583	64	35	linear	linear	ADJ
ap-7583	64	36	factors	factor	NOUN
ap-7583	64	37	.	.	PUNCT
ap-7583	65	1	in	in	ADP
ap-7583	65	2	the	the	DET
ap-7583	65	3	first	first	ADJ
ap-7583	65	4	case	case	NOUN
ap-7583	65	5	,	,	PUNCT
ap-7583	65	6	the	the	DET
ap-7583	65	7	polynomial	polynomial	ADJ
ap-7583	65	8	l	l	NOUN
ap-7583	65	9	can	can	AUX
ap-7583	65	10	be	be	AUX
ap-7583	65	11	written	write	VERB
ap-7583	65	12	as	as	ADP
ap-7583	65	13	l	l	NOUN
ap-7583	65	14	=	=	PUNCT
ap-7583	66	1	α3(r	α3(r	NUM
ap-7583	66	2	−	−	NOUN
ap-7583	66	3	ξ)(r2	ξ)(r2	NOUN
ap-7583	67	1	+	+	CCONJ
ap-7583	67	2	br	br	NOUN
ap-7583	67	3	+	+	CCONJ
ap-7583	67	4	c	c	X
ap-7583	67	5	)	)	PUNCT
ap-7583	67	6	where	where	SCONJ
ap-7583	67	7	r2	r2	PROPN
ap-7583	67	8	+	+	CCONJ
ap-7583	67	9	br	br	PROPN
ap-7583	68	1	+	+	CCONJ
ap-7583	68	2	c	c	NOUN
ap-7583	68	3	is	be	AUX
ap-7583	68	4	an	an	DET
ap-7583	68	5	irreducible	irreducible	ADJ
ap-7583	68	6	polynomial	polynomial	NOUN
ap-7583	68	7	.	.	PUNCT
ap-7583	69	1	in	in	ADP
ap-7583	69	2	this	this	DET
ap-7583	69	3	case	case	NOUN
ap-7583	69	4	,	,	PUNCT
ap-7583	69	5	ξ	ξ	PROPN
ap-7583	69	6	is	be	AUX
ap-7583	69	7	regular	regular	ADJ
ap-7583	69	8	,	,	PUNCT
ap-7583	69	9	real	real	ADJ
ap-7583	69	10	,	,	PUNCT
ap-7583	69	11	singular	singular	ADJ
ap-7583	69	12	point	point	NOUN
ap-7583	69	13	and	and	CCONJ
ap-7583	69	14	∞	∞	NUM
ap-7583	69	15	is	be	AUX
ap-7583	69	16	irregular	irregular	ADJ
ap-7583	69	17	for	for	ADP
ap-7583	69	18	otherwise	otherwise	ADV
ap-7583	69	19	,	,	PUNCT
ap-7583	69	20	the	the	DET
ap-7583	69	21	differential	differential	ADJ
ap-7583	69	22	equation	equation	NOUN
ap-7583	69	23	can	can	AUX
ap-7583	69	24	be	be	AUX
ap-7583	69	25	solved	solve	VERB
ap-7583	69	26	in	in	ADP
ap-7583	69	27	terms	term	NOUN
ap-7583	69	28	of	of	ADP
ap-7583	69	29	elementary	elementary	ADJ
ap-7583	69	30	functions	function	NOUN
ap-7583	69	31	according	accord	VERB
ap-7583	69	32	to	to	ADP
ap-7583	69	33	the	the	DET
ap-7583	69	34	classical	classical	ADJ
ap-7583	69	35	theory	theory	NOUN
ap-7583	69	36	of	of	ADP
ap-7583	69	37	ordinary	ordinary	ADJ
ap-7583	69	38	differential	differential	ADJ
ap-7583	69	39	equations	equation	NOUN
ap-7583	69	40	.	.	PUNCT
ap-7583	70	1	in	in	ADP
ap-7583	70	2	this	this	DET
ap-7583	70	3	case	case	NOUN
ap-7583	70	4	,	,	PUNCT
ap-7583	70	5	the	the	DET
ap-7583	70	6	differential	differential	ADJ
ap-7583	70	7	equation	equation	NOUN
ap-7583	70	8	can	can	AUX
ap-7583	70	9	be	be	AUX
ap-7583	70	10	written	write	VERB
ap-7583	70	11	as	as	ADP
ap-7583	70	12	d2y	d2y	PROPN
ap-7583	70	13	dr2	dr2	PROPN
ap-7583	70	14	+	+	CCONJ
ap-7583	70	15	(	(	PUNCT
ap-7583	70	16	µ1	µ1	PROPN
ap-7583	70	17	r	r	NOUN
ap-7583	70	18	−	−	PROPN
ap-7583	70	19	ξ	ξ	SYM
ap-7583	71	1	+	+	PUNCT
ap-7583	71	2	µ2	µ2	PROPN
ap-7583	71	3	r2	r2	NOUN
ap-7583	71	4	+	+	CCONJ
ap-7583	71	5	br	br	PROPN
ap-7583	71	6	+	+	CCONJ
ap-7583	71	7	c	c	NOUN
ap-7583	71	8	)	)	PUNCT
ap-7583	71	9	dy	dy	PROPN
ap-7583	71	10	dr	dr	PROPN
ap-7583	71	11	+	+	PROPN
ap-7583	71	12	ε1	ε1	PROPN
ap-7583	71	13	r	r	NOUN
ap-7583	71	14	+	+	CCONJ
ap-7583	71	15	ε0	ε0	PROPN
ap-7583	71	16	α3(r	α3(r	NUM
ap-7583	71	17	−	−	PROPN
ap-7583	71	18	ξ)(r2	ξ)(r2	NOUN
ap-7583	72	1	+	+	NUM
ap-7583	72	2	b	b	NOUN
ap-7583	72	3	r	r	NOUN
ap-7583	72	4	+	+	NOUN
ap-7583	72	5	c)y	c)y	X
ap-7583	72	6	=	=	X
ap-7583	72	7	0	0	X
ap-7583	72	8	.	.	PUNCT
ap-7583	73	1	(	(	PUNCT
ap-7583	73	2	6	6	NUM
ap-7583	73	3	)	)	PUNCT
ap-7583	73	4	the	the	DET
ap-7583	73	5	second	second	ADJ
ap-7583	73	6	case	case	NOUN
ap-7583	73	7	,	,	PUNCT
ap-7583	73	8	the	the	DET
ap-7583	73	9	polynomial	polynomial	ADJ
ap-7583	73	10	l	l	NOUN
ap-7583	73	11	can	can	AUX
ap-7583	73	12	be	be	AUX
ap-7583	73	13	written	write	VERB
ap-7583	73	14	as	as	ADP
ap-7583	73	15	l	l	NOUN
ap-7583	73	16	=	=	PUNCT
ap-7583	74	1	α3(r	α3(r	NUM
ap-7583	74	2	−	−	NOUN
ap-7583	75	1	ξ1)(r	ξ1)(r	NUM
ap-7583	75	2	−	−	PROPN
ap-7583	75	3	ξ2)(r	ξ2)(r	PROPN
ap-7583	75	4	−	−	NOUN
ap-7583	76	1	ξ3	ξ3	NOUN
ap-7583	76	2	)	)	PUNCT
ap-7583	76	3	where	where	SCONJ
ap-7583	76	4	ξj	ξj	NOUN
ap-7583	76	5	,	,	PUNCT
ap-7583	76	6	j	j	PROPN
ap-7583	76	7	=	=	SYM
ap-7583	76	8	1	1	NUM
ap-7583	76	9	,	,	PUNCT
ap-7583	76	10	2	2	NUM
ap-7583	76	11	,	,	PUNCT
ap-7583	76	12	3	3	NUM
ap-7583	76	13	and	and	CCONJ
ap-7583	76	14	∞	∞	NUM
ap-7583	76	15	are	be	AUX
ap-7583	76	16	all	all	ADV
ap-7583	76	17	regular	regular	ADJ
ap-7583	76	18	singular	singular	ADJ
ap-7583	76	19	points	point	NOUN
ap-7583	76	20	,	,	PUNCT
ap-7583	76	21	i.e.	i.e.	X
ap-7583	76	22	,	,	PUNCT
ap-7583	76	23	the	the	DET
ap-7583	76	24	differential	differential	ADJ
ap-7583	76	25	equation	equation	NOUN
ap-7583	76	26	of	of	ADP
ap-7583	76	27	fushsian	fushsian	ADJ
ap-7583	76	28	type	type	NOUN
ap-7583	76	29	,	,	PUNCT
ap-7583	76	30	d2y	d2y	PROPN
ap-7583	76	31	dr2	dr2	NOUN
ap-7583	76	32	+	+	CCONJ
ap-7583	76	33			PROPN
ap-7583	76	34	3∑	3∑	NUM
ap-7583	76	35	j=1	j=1	NOUN
ap-7583	77	1	µj	µj	ADP
ap-7583	77	2	r	r	NOUN
ap-7583	77	3	−	−	PROPN
ap-7583	78	1	ξj	ξj	NOUN
ap-7583	78	2			PROPN
ap-7583	78	3	dy	dy	PROPN
ap-7583	78	4	dr	dr	PROPN
ap-7583	78	5	+	+	PROPN
ap-7583	78	6	ε1	ε1	PROPN
ap-7583	78	7	r	r	NOUN
ap-7583	78	8	+	+	CCONJ
ap-7583	78	9	ε0	ε0	PROPN
ap-7583	78	10	α3(r	α3(r	NUM
ap-7583	78	11	−	−	PROPN
ap-7583	79	1	ξ1)(r	ξ1)(r	NUM
ap-7583	79	2	−	−	PROPN
ap-7583	79	3	ξ2)(r	ξ2)(r	PROPN
ap-7583	79	4	−	−	NOUN
ap-7583	79	5	ξ3)y	ξ3)y	ADJ
ap-7583	79	6	=	=	NOUN
ap-7583	79	7	0	0	NUM
ap-7583	79	8	.	.	PUNCT
ap-7583	80	1	(	(	PUNCT
ap-7583	80	2	7	7	NUM
ap-7583	80	3	)	)	PUNCT
ap-7583	80	4	where	where	SCONJ
ap-7583	80	5	µj	µj	PROPN
ap-7583	80	6	are	be	AUX
ap-7583	80	7	constants	constant	NOUN
ap-7583	80	8	depending	depend	VERB
ap-7583	80	9	on	on	ADP
ap-7583	80	10	the	the	DET
ap-7583	80	11	differential	differential	ADJ
ap-7583	80	12	equation	equation	NOUN
ap-7583	80	13	parameters	parameter	NOUN
ap-7583	80	14	.	.	PUNCT
ap-7583	81	1	one	one	PRON
ap-7583	81	2	can	can	AUX
ap-7583	81	3	then	then	ADV
ap-7583	81	4	study	study	VERB
ap-7583	81	5	the	the	DET
ap-7583	81	6	series	series	NOUN
ap-7583	81	7	solutions	solution	NOUN
ap-7583	81	8	of	of	ADP
ap-7583	81	9	equations	equation	NOUN
ap-7583	81	10	(	(	PUNCT
ap-7583	81	11	6	6	NUM
ap-7583	81	12	)	)	PUNCT
ap-7583	81	13	and	and	CCONJ
ap-7583	81	14	(	(	PUNCT
ap-7583	81	15	7	7	X
ap-7583	81	16	)	)	PUNCT
ap-7583	81	17	using	use	VERB
ap-7583	81	18	the	the	DET
ap-7583	81	19	classical	classical	ADJ
ap-7583	81	20	frobenius	frobenius	NOUN
ap-7583	81	21	method	method	NOUN
ap-7583	81	22	.	.	PUNCT
ap-7583	82	1	another	another	DET
ap-7583	82	2	approach	approach	NOUN
ap-7583	82	3	,	,	PUNCT
ap-7583	82	4	recently	recently	ADV
ap-7583	82	5	adopted	adopt	VERB
ap-7583	82	6	,	,	PUNCT
ap-7583	82	7	to	to	PART
ap-7583	82	8	study	study	VERB
ap-7583	82	9	(	(	PUNCT
ap-7583	82	10	4	4	NUM
ap-7583	82	11	)	)	PUNCT
ap-7583	82	12	,	,	PUNCT
ap-7583	82	13	depends	depend	VERB
ap-7583	82	14	on	on	ADP
ap-7583	82	15	the	the	DET
ap-7583	82	16	possible	possible	ADJ
ap-7583	82	17	combination	combination	NOUN
ap-7583	82	18	of	of	ADP
ap-7583	82	19	the	the	DET
ap-7583	82	20	parameters	parameter	NOUN
ap-7583	82	21	αj	αj	PROPN
ap-7583	82	22	,	,	PUNCT
ap-7583	82	23	j	j	PROPN
ap-7583	82	24	=	=	SYM
ap-7583	82	25	0	0	NUM
ap-7583	82	26	,	,	PUNCT
ap-7583	82	27	1	1	NUM
ap-7583	82	28	,	,	PUNCT
ap-7583	82	29	2	2	NUM
ap-7583	82	30	,	,	PUNCT
ap-7583	82	31	3	3	NUM
ap-7583	82	32	such	such	ADJ
ap-7583	82	33	that	that	SCONJ
ap-7583	82	34	the	the	DET
ap-7583	82	35	polynomial	polynomial	ADJ
ap-7583	82	36	l	l	NOUN
ap-7583	82	37	does	do	AUX
ap-7583	82	38	not	not	PART
ap-7583	82	39	vanish	vanish	VERB
ap-7583	82	40	identically	identically	ADV
ap-7583	82	41	.	.	PUNCT
ap-7583	83	1	there	there	PRON
ap-7583	83	2	are	be	VERB
ap-7583	83	3	fifteen	fifteen	NUM
ap-7583	83	4	possible	possible	ADJ
ap-7583	83	5	combinations	combination	NOUN
ap-7583	83	6	in	in	ADP
ap-7583	83	7	total	total	NOUN
ap-7583	83	8	.	.	PUNCT
ap-7583	84	1	these	these	DET
ap-7583	84	2	fifteen	fifteen	ADJ
ap-7583	84	3	combinations	combination	NOUN
ap-7583	84	4	can	can	AUX
ap-7583	84	5	be	be	AUX
ap-7583	84	6	classified	classify	VERB
ap-7583	84	7	into	into	ADP
ap-7583	84	8	two	two	NUM
ap-7583	84	9	main	main	ADJ
ap-7583	84	10	classes	class	NOUN
ap-7583	84	11	:	:	PUNCT
ap-7583	84	12	the	the	DET
ap-7583	84	13	first	first	ADJ
ap-7583	84	14	class	class	NOUN
ap-7583	84	15	is	be	AUX
ap-7583	84	16	characterized	characterize	VERB
ap-7583	84	17	by	by	ADP
ap-7583	84	18	α0	α0	ADJ
ap-7583	84	19	=	=	SYM
ap-7583	84	20	̸	̸	NUM
ap-7583	84	21	0	0	NUM
ap-7583	84	22	,	,	PUNCT
ap-7583	84	23	which	which	PRON
ap-7583	84	24	has	have	VERB
ap-7583	84	25	eight	eight	NUM
ap-7583	84	26	equations	equation	NOUN
ap-7583	84	27	in	in	ADP
ap-7583	84	28	total	total	NOUN
ap-7583	84	29	,	,	PUNCT
ap-7583	84	30	the	the	DET
ap-7583	84	31	second	second	ADJ
ap-7583	84	32	class	class	NOUN
ap-7583	84	33	characterized	characterize	VERB
ap-7583	84	34	by	by	ADP
ap-7583	84	35	α0	α0	PROPN
ap-7583	84	36	=	=	SYM
ap-7583	84	37	0	0	NUM
ap-7583	84	38	includes	include	VERB
ap-7583	84	39	the	the	DET
ap-7583	84	40	remaining	remain	VERB
ap-7583	84	41	seven	seven	NUM
ap-7583	84	42	equations	equation	NOUN
ap-7583	84	43	.	.	PUNCT
ap-7583	85	1	each	each	PRON
ap-7583	85	2	of	of	ADP
ap-7583	85	3	these	these	DET
ap-7583	85	4	two	two	NUM
ap-7583	85	5	classes	class	NOUN
ap-7583	85	6	will	will	AUX
ap-7583	85	7	be	be	AUX
ap-7583	85	8	studied	study	VERB
ap-7583	85	9	in	in	ADP
ap-7583	85	10	the	the	DET
ap-7583	85	11	next	next	ADJ
ap-7583	85	12	sections	section	NOUN
ap-7583	85	13	.	.	PUNCT
ap-7583	86	1	first	first	ADV
ap-7583	86	2	,	,	PUNCT
ap-7583	86	3	we	we	PRON
ap-7583	86	4	consider	consider	VERB
ap-7583	86	5	some	some	DET
ap-7583	86	6	elementary	elementary	ADJ
ap-7583	86	7	observations	observation	NOUN
ap-7583	86	8	regarding	regard	VERB
ap-7583	86	9	the	the	DET
ap-7583	86	10	differential	differential	ADJ
ap-7583	86	11	equation	equation	NOUN
ap-7583	86	12	(	(	PUNCT
ap-7583	86	13	4	4	NUM
ap-7583	86	14	)	)	PUNCT
ap-7583	86	15	.	.	PUNCT
ap-7583	87	1	167	167	NUM
ap-7583	87	2	nasser	nasser	PROPN
ap-7583	87	3	saad	saad	PROPN
ap-7583	87	4	acta	acta	PROPN
ap-7583	87	5	polytechnica	polytechnica	PROPN
ap-7583	87	6	we	we	PRON
ap-7583	87	7	assume	assume	VERB
ap-7583	87	8	no	no	DET
ap-7583	87	9	common	common	ADJ
ap-7583	87	10	factor	factor	NOUN
ap-7583	87	11	among	among	ADP
ap-7583	87	12	the	the	DET
ap-7583	87	13	polynomial	polynomial	ADJ
ap-7583	87	14	coefficients	coefficient	NOUN
ap-7583	87	15	pj(r	pj(r	PROPN
ap-7583	87	16	)	)	PUNCT
ap-7583	87	17	,	,	PUNCT
ap-7583	87	18	j	j	PROPN
ap-7583	87	19	=	=	SYM
ap-7583	87	20	1	1	NUM
ap-7583	87	21	,	,	PUNCT
ap-7583	87	22	2	2	NUM
ap-7583	87	23	,	,	PUNCT
ap-7583	87	24	3	3	NUM
ap-7583	87	25	,	,	PUNCT
ap-7583	87	26	we	we	PRON
ap-7583	87	27	start	start	VERB
ap-7583	87	28	our	our	PRON
ap-7583	87	29	study	study	NOUN
ap-7583	87	30	of	of	ADP
ap-7583	87	31	equation	equation	NOUN
ap-7583	87	32	(	(	PUNCT
ap-7583	87	33	4	4	NUM
ap-7583	87	34	)	)	PUNCT
ap-7583	87	35	by	by	ADP
ap-7583	87	36	asking	ask	VERB
ap-7583	87	37	the	the	DET
ap-7583	87	38	following	follow	VERB
ap-7583	87	39	simple	simple	ADJ
ap-7583	87	40	question	question	NOUN
ap-7583	87	41	:	:	PUNCT
ap-7583	87	42	under	under	ADP
ap-7583	87	43	what	what	DET
ap-7583	87	44	conditions	condition	NOUN
ap-7583	87	45	the	the	DET
ap-7583	87	46	series	series	NOUN
ap-7583	87	47	solutions	solution	NOUN
ap-7583	87	48	of	of	ADP
ap-7583	87	49	the	the	DET
ap-7583	87	50	differential	differential	ADJ
ap-7583	87	51	equation	equation	NOUN
ap-7583	87	52	(	(	PUNCT
ap-7583	87	53	4	4	X
ap-7583	87	54	)	)	PUNCT
ap-7583	87	55	can	can	AUX
ap-7583	87	56	be	be	AUX
ap-7583	87	57	evaluated	evaluate	VERB
ap-7583	87	58	using	use	VERB
ap-7583	87	59	a	a	DET
ap-7583	87	60	two	two	NUM
ap-7583	87	61	-	-	PUNCT
ap-7583	87	62	term	term	NOUN
ap-7583	87	63	recurrence	recurrence	NOUN
ap-7583	87	64	relation	relation	NOUN
ap-7583	88	1	[	[	X
ap-7583	88	2	22	22	NUM
ap-7583	88	3	]	]	PUNCT
ap-7583	88	4	?	?	PUNCT
ap-7583	89	1	for	for	ADP
ap-7583	89	2	,	,	PUNCT
ap-7583	89	3	in	in	ADP
ap-7583	89	4	this	this	DET
ap-7583	89	5	case	case	NOUN
ap-7583	89	6	,	,	PUNCT
ap-7583	89	7	the	the	DET
ap-7583	89	8	two	two	NUM
ap-7583	89	9	linearly	linearly	ADV
ap-7583	89	10	independent	independent	ADJ
ap-7583	89	11	series	series	NOUN
ap-7583	89	12	solutions	solution	NOUN
ap-7583	89	13	can	can	AUX
ap-7583	89	14	be	be	AUX
ap-7583	89	15	found	find	VERB
ap-7583	89	16	explicitly	explicitly	ADV
ap-7583	89	17	.	.	PUNCT
ap-7583	90	1	theorem	theorem	VERB
ap-7583	90	2	2.1	2.1	NUM
ap-7583	90	3	.	.	PUNCT
ap-7583	91	1	the	the	DET
ap-7583	91	2	necessary	necessary	ADJ
ap-7583	91	3	and	and	CCONJ
ap-7583	91	4	sufficient	sufficient	ADJ
ap-7583	91	5	conditions	condition	NOUN
ap-7583	91	6	for	for	ADP
ap-7583	91	7	the	the	DET
ap-7583	91	8	linear	linear	PROPN
ap-7583	91	9	differential	differential	ADJ
ap-7583	91	10	equation	equation	NOUN
ap-7583	91	11	p2(r	p2(r	NOUN
ap-7583	91	12	)	)	PUNCT
ap-7583	91	13	u′′(r	u′′(r	NOUN
ap-7583	91	14	)	)	PUNCT
ap-7583	91	15	+	+	PUNCT
ap-7583	91	16	p1(r	p1(r	NOUN
ap-7583	91	17	)	)	PUNCT
ap-7583	91	18	u′(r	u′(r	PROPN
ap-7583	91	19	)	)	PUNCT
ap-7583	91	20	+	+	CCONJ
ap-7583	91	21	p0(r	p0(r	NOUN
ap-7583	91	22	)	)	PUNCT
ap-7583	91	23	u(r	u(r	NOUN
ap-7583	91	24	)	)	PUNCT
ap-7583	91	25	=	=	SYM
ap-7583	91	26	0	0	NUM
ap-7583	91	27	,	,	PUNCT
ap-7583	91	28	(	(	PUNCT
ap-7583	91	29	8)	8)	NUM
ap-7583	91	30	to	to	PART
ap-7583	91	31	have	have	AUX
ap-7583	91	32	a	a	DET
ap-7583	91	33	two	two	NUM
ap-7583	91	34	-	-	PUNCT
ap-7583	91	35	term	term	NOUN
ap-7583	91	36	recurrence	recurrence	NOUN
ap-7583	91	37	relationship	relationship	NOUN
ap-7583	91	38	that	that	PRON
ap-7583	91	39	relates	relate	VERB
ap-7583	91	40	the	the	DET
ap-7583	91	41	successive	successive	ADJ
ap-7583	91	42	coefficients	coefficient	NOUN
ap-7583	91	43	in	in	ADP
ap-7583	91	44	its	its	PRON
ap-7583	91	45	series	series	NOUN
ap-7583	91	46	solution	solution	NOUN
ap-7583	91	47	is	be	AUX
ap-7583	91	48	that	that	SCONJ
ap-7583	91	49	in	in	ADP
ap-7583	91	50	the	the	DET
ap-7583	91	51	neighbourhood	neighbourhood	NOUN
ap-7583	91	52	of	of	ADP
ap-7583	91	53	the	the	DET
ap-7583	91	54	singular	singular	ADJ
ap-7583	91	55	regular	regular	ADJ
ap-7583	91	56	point	point	NOUN
ap-7583	91	57	r0	r0	NOUN
ap-7583	91	58	(	(	PUNCT
ap-7583	91	59	where	where	SCONJ
ap-7583	91	60	p2(r0	p2(r0	ADV
ap-7583	91	61	)	)	PUNCT
ap-7583	91	62	=	=	SYM
ap-7583	91	63	0	0	NUM
ap-7583	91	64	)	)	PUNCT
ap-7583	91	65	,	,	PUNCT
ap-7583	91	66	the	the	DET
ap-7583	91	67	equation	equation	NOUN
ap-7583	91	68	(	(	PUNCT
ap-7583	91	69	8)	8)	NUM
ap-7583	91	70	can	can	AUX
ap-7583	91	71	be	be	AUX
ap-7583	91	72	written	write	VERB
ap-7583	91	73	as	as	ADP
ap-7583	91	74	:	:	PUNCT
ap-7583	92	1	[	[	X
ap-7583	92	2	q2,0	q2,0	X
ap-7583	92	3	+	+	X
ap-7583	92	4	q2,h	q2,h	PROPN
ap-7583	92	5	(	(	PUNCT
ap-7583	92	6	r	r	NOUN
ap-7583	92	7	−	−	PROPN
ap-7583	92	8	r0)h]︸	r0)h]︸	PROPN
ap-7583	92	9	︷︷	︷︷	PROPN
ap-7583	92	10	︸	︸	ADP
ap-7583	92	11	q2(r	q2(r	PROPN
ap-7583	92	12	)	)	PUNCT
ap-7583	92	13	(	(	PUNCT
ap-7583	92	14	r	r	NOUN
ap-7583	92	15	−	−	NOUN
ap-7583	92	16	r0)2−m	r0)2−m	NOUN
ap-7583	92	17	u′′(r	u′′(r	NOUN
ap-7583	92	18	)	)	PUNCT
ap-7583	92	19	+	+	CCONJ
ap-7583	93	1	[	[	X
ap-7583	93	2	q1,0	q1,0	NOUN
ap-7583	93	3	+	+	NUM
ap-7583	93	4	q1,h	q1,h	NOUN
ap-7583	93	5	(	(	PUNCT
ap-7583	93	6	r	r	NOUN
ap-7583	93	7	−	−	PROPN
ap-7583	93	8	r0)h]︸	r0)h]︸	PROPN
ap-7583	93	9	︷︷	︷︷	PROPN
ap-7583	93	10	︸	︸	ADP
ap-7583	93	11	q1(r	q1(r	PROPN
ap-7583	93	12	)	)	PUNCT
ap-7583	93	13	r1−m	r1−m	NOUN
ap-7583	93	14	u′(r	u′(r	PROPN
ap-7583	93	15	)	)	PUNCT
ap-7583	94	1	+	+	CCONJ
ap-7583	95	1	[	[	X
ap-7583	95	2	q0,0	q0,0	NOUN
ap-7583	95	3	+	+	X
ap-7583	95	4	q0,h	q0,h	PROPN
ap-7583	95	5	(	(	PUNCT
ap-7583	95	6	r	r	NOUN
ap-7583	95	7	−	−	PROPN
ap-7583	95	8	r0)h]︸	r0)h]︸	PROPN
ap-7583	95	9	︷︷	︷︷	PROPN
ap-7583	95	10	︸	︸	X
ap-7583	95	11	q0(r	q0(r	NOUN
ap-7583	95	12	)	)	PUNCT
ap-7583	95	13	(	(	PUNCT
ap-7583	95	14	r	r	NOUN
ap-7583	95	15	−	−	PROPN
ap-7583	95	16	r0)−m	r0)−m	NOUN
ap-7583	95	17	u(r	u(r	NOUN
ap-7583	95	18	)	)	PUNCT
ap-7583	95	19	=	=	SYM
ap-7583	95	20	0	0	NUM
ap-7583	95	21	,	,	PUNCT
ap-7583	95	22	(	(	PUNCT
ap-7583	95	23	9	9	NUM
ap-7583	95	24	)	)	PUNCT
ap-7583	95	25	where	where	SCONJ
ap-7583	95	26	,	,	PUNCT
ap-7583	95	27	for	for	ADP
ap-7583	95	28	m	m	PROPN
ap-7583	95	29	∈	∈	PROPN
ap-7583	95	30	z	z	NOUN
ap-7583	95	31	,	,	PUNCT
ap-7583	95	32	h	h	PROPN
ap-7583	95	33	∈	∈	PROPN
ap-7583	95	34	z+	z+	PRON
ap-7583	95	35	,	,	PUNCT
ap-7583	95	36	j	j	PROPN
ap-7583	95	37	=	=	SYM
ap-7583	95	38	0	0	NUM
ap-7583	95	39	,	,	PUNCT
ap-7583	95	40	1	1	NUM
ap-7583	95	41	,	,	PUNCT
ap-7583	95	42	2	2	NUM
ap-7583	95	43	,	,	PUNCT
ap-7583	95	44	qj(r	qj(r	NOUN
ap-7583	95	45	)	)	PUNCT
ap-7583	95	46	≡	≡	PROPN
ap-7583	95	47	∞∑	∞∑	PRON
ap-7583	95	48	k=0	k=0	PROPN
ap-7583	95	49	qj	qj	PROPN
ap-7583	95	50	,	,	PUNCT
ap-7583	95	51	k(r	k(r	PROPN
ap-7583	95	52	−	−	PROPN
ap-7583	95	53	r0)k	r0)k	NOUN
ap-7583	95	54	=	=	PUNCT
ap-7583	95	55	pj(r	pj(r	PROPN
ap-7583	95	56	)	)	PUNCT
ap-7583	95	57	(	(	PUNCT
ap-7583	95	58	r	r	NOUN
ap-7583	95	59	−	−	PROPN
ap-7583	95	60	r0)m−j	r0)m−j	PROPN
ap-7583	95	61	,	,	PUNCT
ap-7583	95	62	(	(	PUNCT
ap-7583	95	63	10	10	NUM
ap-7583	95	64	)	)	PUNCT
ap-7583	95	65	when	when	SCONJ
ap-7583	95	66	at	at	ADV
ap-7583	95	67	least	least	ADV
ap-7583	95	68	one	one	NUM
ap-7583	95	69	of	of	ADP
ap-7583	95	70	qj,0	qj,0	PROPN
ap-7583	95	71	,	,	PUNCT
ap-7583	95	72	j	j	PROPN
ap-7583	95	73	=	=	SYM
ap-7583	95	74	0	0	NUM
ap-7583	95	75	,	,	PUNCT
ap-7583	95	76	1	1	NUM
ap-7583	95	77	,	,	PUNCT
ap-7583	95	78	2	2	NUM
ap-7583	95	79	and	and	CCONJ
ap-7583	95	80	qj	qj	PROPN
ap-7583	95	81	,	,	PUNCT
ap-7583	95	82	h	h	NOUN
ap-7583	95	83	,	,	PUNCT
ap-7583	95	84	j	j	PROPN
ap-7583	95	85	=	=	SYM
ap-7583	95	86	0	0	NUM
ap-7583	95	87	,	,	PUNCT
ap-7583	95	88	1	1	NUM
ap-7583	95	89	,	,	PUNCT
ap-7583	95	90	2	2	NUM
ap-7583	95	91	,	,	PUNCT
ap-7583	95	92	is	be	AUX
ap-7583	95	93	different	different	ADJ
ap-7583	95	94	from	from	ADP
ap-7583	95	95	zero	zero	NUM
ap-7583	95	96	.	.	PUNCT
ap-7583	96	1	in	in	ADP
ap-7583	96	2	this	this	DET
ap-7583	96	3	case	case	NOUN
ap-7583	96	4	,	,	PUNCT
ap-7583	96	5	the	the	DET
ap-7583	96	6	two	two	NUM
ap-7583	96	7	-	-	PUNCT
ap-7583	96	8	term	term	NOUN
ap-7583	96	9	recurrence	recurrence	NOUN
ap-7583	96	10	formula	formula	NOUN
ap-7583	96	11	is	be	AUX
ap-7583	96	12	given	give	VERB
ap-7583	96	13	by	by	ADP
ap-7583	96	14	ck	ck	PROPN
ap-7583	96	15	ck−h	ck−h	PROPN
ap-7583	96	16	=	=	PUNCT
ap-7583	97	1	−	−	PROPN
ap-7583	97	2	(	(	PUNCT
ap-7583	97	3	k	k	PROPN
ap-7583	97	4	+	+	NOUN
ap-7583	97	5	λ	λ	X
ap-7583	97	6	−	−	NOUN
ap-7583	98	1	h)[q2,h	h)[q2,h	INTJ
ap-7583	98	2	(	(	PUNCT
ap-7583	98	3	k	k	X
ap-7583	98	4	+	+	CCONJ
ap-7583	98	5	λ	λ	PROPN
ap-7583	98	6	−	−	NOUN
ap-7583	98	7	h	h	NOUN
ap-7583	98	8	−	−	NOUN
ap-7583	98	9	1	1	NUM
ap-7583	98	10	)	)	PUNCT
ap-7583	99	1	+	+	CCONJ
ap-7583	100	1	q1,h	q1,h	NOUN
ap-7583	100	2	]	]	PUNCT
ap-7583	101	1	+	+	CCONJ
ap-7583	101	2	q0,h	q0,h	PROPN
ap-7583	101	3	(	(	PUNCT
ap-7583	101	4	k	k	NOUN
ap-7583	101	5	+	+	PROPN
ap-7583	101	6	λ)[q2,0	λ)[q2,0	PROPN
ap-7583	101	7	(	(	PUNCT
ap-7583	101	8	k	k	PROPN
ap-7583	101	9	+	+	PROPN
ap-7583	101	10	λ	λ	X
ap-7583	101	11	−	−	NOUN
ap-7583	101	12	1	1	NUM
ap-7583	101	13	)	)	PUNCT
ap-7583	101	14	+	+	CCONJ
ap-7583	101	15	q1,0	q1,0	NOUN
ap-7583	101	16	]	]	X
ap-7583	101	17	+	+	NUM
ap-7583	101	18	q0,0	q0,0	NOUN
ap-7583	101	19	,	,	PUNCT
ap-7583	101	20	(	(	PUNCT
ap-7583	101	21	11	11	NUM
ap-7583	101	22	)	)	PUNCT
ap-7583	101	23	where	where	SCONJ
ap-7583	101	24	c0	c0	PROPN
ap-7583	101	25	̸=	̸=	PROPN
ap-7583	101	26	0	0	NUM
ap-7583	101	27	,	,	PUNCT
ap-7583	101	28	and	and	CCONJ
ap-7583	101	29	λ	λ	X
ap-7583	101	30	=	=	SYM
ap-7583	101	31	λ1	λ1	PROPN
ap-7583	101	32	,	,	PUNCT
ap-7583	101	33	λ2	λ2	NOUN
ap-7583	101	34	are	be	AUX
ap-7583	101	35	the	the	DET
ap-7583	101	36	roots	root	NOUN
ap-7583	101	37	of	of	ADP
ap-7583	101	38	the	the	DET
ap-7583	101	39	indicial	indicial	ADJ
ap-7583	101	40	equation	equation	NOUN
ap-7583	101	41	q2,0	q2,0	PROPN
ap-7583	101	42	λ	λ	INTJ
ap-7583	101	43	(	(	PUNCT
ap-7583	101	44	λ	λ	X
ap-7583	101	45	−	−	PROPN
ap-7583	101	46	1	1	NUM
ap-7583	101	47	)	)	PUNCT
ap-7583	101	48	+	+	CCONJ
ap-7583	101	49	q1,0	q1,0	PROPN
ap-7583	101	50	λ	λ	X
ap-7583	101	51	+	+	CCONJ
ap-7583	101	52	q0,0	q0,0	NOUN
ap-7583	101	53	=	=	SYM
ap-7583	101	54	0	0	NUM
ap-7583	101	55	.	.	PUNCT
ap-7583	101	56	(	(	PUNCT
ap-7583	101	57	12	12	NUM
ap-7583	101	58	)	)	PUNCT
ap-7583	101	59	the	the	DET
ap-7583	101	60	closed	closed	ADJ
ap-7583	101	61	form	form	NOUN
ap-7583	101	62	of	of	ADP
ap-7583	101	63	the	the	DET
ap-7583	101	64	series	series	NOUN
ap-7583	101	65	solution	solution	NOUN
ap-7583	101	66	generated	generate	VERB
ap-7583	101	67	by	by	ADP
ap-7583	101	68	(	(	PUNCT
ap-7583	101	69	11	11	NUM
ap-7583	101	70	)	)	PUNCT
ap-7583	101	71	can	can	AUX
ap-7583	101	72	be	be	AUX
ap-7583	101	73	written	write	VERB
ap-7583	101	74	in	in	ADP
ap-7583	101	75	terms	term	NOUN
ap-7583	101	76	of	of	ADP
ap-7583	101	77	the	the	DET
ap-7583	101	78	generalized	generalized	ADJ
ap-7583	101	79	hypergeometric	hypergeometric	ADJ
ap-7583	101	80	function	function	NOUN
ap-7583	101	81	as	as	ADP
ap-7583	101	82	u(r	u(r	NOUN
ap-7583	101	83	;	;	PUNCT
ap-7583	101	84	λ	λ	X
ap-7583	101	85	)	)	PUNCT
ap-7583	102	1	=	=	NOUN
ap-7583	102	2	zλ	zλ	NOUN
ap-7583	102	3	∞∑	∞∑	PRON
ap-7583	102	4	k=0	k=0	PROPN
ap-7583	102	5	chkrhk	chkrhk	PROPN
ap-7583	102	6	=	=	SYM
ap-7583	102	7	rλ	rλ	ADP
ap-7583	102	8	3f2	3f2	NUM
ap-7583	102	9	(	(	PUNCT
ap-7583	102	10	1	1	NUM
ap-7583	102	11	,	,	PUNCT
ap-7583	102	12	2λ−1	2λ−1	NUM
ap-7583	102	13	2h	2h	NUM
ap-7583	102	14	+	+	CCONJ
ap-7583	102	15	q1,h	q1,h	NOUN
ap-7583	102	16	2	2	NUM
ap-7583	102	17	h	h	NOUN
ap-7583	102	18	q2,h	q2,h	ADV
ap-7583	102	19	−	−	NOUN
ap-7583	103	1	√	√	NOUN
ap-7583	103	2	(	(	PUNCT
ap-7583	103	3	q1,h−q2,h)2−4q0,hq2,h	q1,h−q2,h)2−4q0,hq2,h	PROPN
ap-7583	103	4	2	2	NUM
ap-7583	103	5	h	h	NOUN
ap-7583	103	6	q2,h	q2,h	PROPN
ap-7583	103	7	,	,	PUNCT
ap-7583	103	8	2λ−1	2λ−1	NUM
ap-7583	103	9	2h	2h	NUM
ap-7583	103	10	+	+	CCONJ
ap-7583	103	11	q1,h	q1,h	NOUN
ap-7583	103	12	2	2	NUM
ap-7583	103	13	h	h	NOUN
ap-7583	103	14	q2,h	q2,h	PROPN
ap-7583	103	15	+	+	NUM
ap-7583	103	16	√	√	NUM
ap-7583	103	17	(	(	PUNCT
ap-7583	103	18	q1,h−q2,h)2−4q0,hq2,h	q1,h−q2,h)2−4q0,hq2,h	PROPN
ap-7583	103	19	2	2	NUM
ap-7583	103	20	h	h	NOUN
ap-7583	103	21	q2,h	q2,h	PROPN
ap-7583	103	22	;	;	PUNCT
ap-7583	103	23	1	1	NUM
ap-7583	104	1	+	+	SYM
ap-7583	104	2	2λ−1	2λ−1	NUM
ap-7583	104	3	2	2	NUM
ap-7583	104	4	h	h	NOUN
ap-7583	104	5	+	+	CCONJ
ap-7583	104	6	q1,0	q1,0	NUM
ap-7583	104	7	2	2	NUM
ap-7583	104	8	h	h	NOUN
ap-7583	104	9	q2,0	q2,0	NOUN
ap-7583	104	10	−	−	PROPN
ap-7583	104	11	√	√	PROPN
ap-7583	105	1	(	(	PUNCT
ap-7583	105	2	q1,0−q2,h)2−4q0,0q2,0	q1,0−q2,h)2−4q0,0q2,0	PROPN
ap-7583	105	3	2	2	NUM
ap-7583	105	4	h	h	NOUN
ap-7583	105	5	q2,0	q2,0	PROPN
ap-7583	105	6	,	,	PUNCT
ap-7583	105	7	1	1	NUM
ap-7583	106	1	+	+	SYM
ap-7583	106	2	2λ−1	2λ−1	NUM
ap-7583	106	3	2h	2h	NUM
ap-7583	106	4	+	+	CCONJ
ap-7583	106	5	q1,0	q1,0	NUM
ap-7583	106	6	2	2	NUM
ap-7583	106	7	h	h	NOUN
ap-7583	106	8	q2,0	q2,0	PROPN
ap-7583	106	9	+	+	CCONJ
ap-7583	106	10	√	√	PROPN
ap-7583	106	11	(	(	PUNCT
ap-7583	106	12	q1,0−q2,h)2−4q0,0q2,0	q1,0−q2,h)2−4q0,0q2,0	PROPN
ap-7583	106	13	2	2	NUM
ap-7583	106	14	h	h	NOUN
ap-7583	106	15	q2,0	q2,0	PROPN
ap-7583	106	16	;	;	PUNCT
ap-7583	106	17	−	−	PROPN
ap-7583	106	18	q2,h	q2,h	PROPN
ap-7583	106	19	q2,0	q2,0	PROPN
ap-7583	106	20	rh	rh	PROPN
ap-7583	106	21	)	)	PUNCT
ap-7583	106	22	.	.	PUNCT
ap-7583	107	1	(	(	PUNCT
ap-7583	107	2	13	13	X
ap-7583	107	3	)	)	PUNCT
ap-7583	107	4	applying	apply	VERB
ap-7583	107	5	this	this	DET
ap-7583	107	6	theorem	theorem	NOUN
ap-7583	107	7	,	,	PUNCT
ap-7583	107	8	equation	equation	NOUN
ap-7583	107	9	(	(	PUNCT
ap-7583	107	10	4	4	X
ap-7583	107	11	)	)	PUNCT
ap-7583	107	12	generates	generate	VERB
ap-7583	107	13	the	the	DET
ap-7583	107	14	following	follow	VERB
ap-7583	107	15	solvable	solvable	ADJ
ap-7583	107	16	equations	equation	NOUN
ap-7583	107	17	:	:	PUNCT
ap-7583	107	18	•	•	NUM
ap-7583	107	19	differential	differential	NOUN
ap-7583	107	20	equation	equation	NOUN
ap-7583	107	21	:	:	PUNCT
ap-7583	107	22	r2	r2	PROPN
ap-7583	107	23	(	(	PUNCT
ap-7583	107	24	α2	α2	PROPN
ap-7583	107	25	+	+	CCONJ
ap-7583	107	26	α3	α3	ADJ
ap-7583	107	27	r	r	NOUN
ap-7583	107	28	)	)	PUNCT
ap-7583	107	29	u′′(r	u′′(r	NOUN
ap-7583	107	30	)	)	PUNCT
ap-7583	108	1	+	+	CCONJ
ap-7583	108	2	r	r	NOUN
ap-7583	108	3	(	(	PUNCT
ap-7583	108	4	β1	β1	NOUN
ap-7583	108	5	+	+	CCONJ
ap-7583	108	6	β2	β2	NOUN
ap-7583	108	7	r	r	NOUN
ap-7583	108	8	)	)	PUNCT
ap-7583	108	9	u′(r	u′(r	PROPN
ap-7583	108	10	)	)	PUNCT
ap-7583	109	1	+	+	CCONJ
ap-7583	109	2	(	(	PUNCT
ap-7583	109	3	ε0	ε0	PROPN
ap-7583	109	4	+	+	CCONJ
ap-7583	109	5	ε1	ε1	PROPN
ap-7583	109	6	r	r	NOUN
ap-7583	109	7	)	)	PUNCT
ap-7583	109	8	u(r	u(r	NOUN
ap-7583	109	9	)	)	PUNCT
ap-7583	110	1	=	=	SYM
ap-7583	110	2	0	0	NUM
ap-7583	110	3	,	,	PUNCT
ap-7583	110	4	ε0	ε0	PROPN
ap-7583	110	5	̸=	̸=	PROPN
ap-7583	110	6	0	0	NUM
ap-7583	110	7	,	,	PUNCT
ap-7583	110	8	(	(	PUNCT
ap-7583	110	9	14	14	NUM
ap-7583	110	10	)	)	PUNCT
ap-7583	110	11	recurrence	recurrence	NOUN
ap-7583	110	12	relation	relation	NOUN
ap-7583	110	13	:	:	PUNCT
ap-7583	110	14	for	for	ADP
ap-7583	110	15	k	k	PROPN
ap-7583	110	16	=	=	SYM
ap-7583	110	17	1	1	NUM
ap-7583	110	18	,	,	PUNCT
ap-7583	110	19	2	2	NUM
ap-7583	110	20	,	,	PUNCT
ap-7583	110	21	·	·	PUNCT
ap-7583	110	22	·	·	PUNCT
ap-7583	110	23	·	·	PUNCT
ap-7583	110	24	,	,	PUNCT
ap-7583	110	25	and	and	CCONJ
ap-7583	110	26	c0	c0	PROPN
ap-7583	110	27	=	=	SYM
ap-7583	110	28	1	1	NUM
ap-7583	110	29	,	,	PUNCT
ap-7583	110	30	ck	ck	PROPN
ap-7583	110	31	ck−1	ck−1	NOUN
ap-7583	110	32	=	=	SYM
ap-7583	110	33	−	−	PROPN
ap-7583	111	1	(	(	PUNCT
ap-7583	111	2	k	k	PROPN
ap-7583	111	3	+	+	NOUN
ap-7583	111	4	λ	λ	PROPN
ap-7583	111	5	−	−	PROPN
ap-7583	111	6	1)[α3	1)[α3	NUM
ap-7583	111	7	(	(	PUNCT
ap-7583	111	8	k	k	PROPN
ap-7583	111	9	+	+	PROPN
ap-7583	111	10	λ	λ	X
ap-7583	111	11	−	−	NOUN
ap-7583	111	12	2	2	NUM
ap-7583	111	13	)	)	PUNCT
ap-7583	111	14	+	+	CCONJ
ap-7583	111	15	β2	β2	VERB
ap-7583	111	16	]	]	X
ap-7583	111	17	+	+	CCONJ
ap-7583	112	1	ε1	ε1	PROPN
ap-7583	112	2	(	(	PUNCT
ap-7583	112	3	k	k	PROPN
ap-7583	112	4	+	+	PROPN
ap-7583	112	5	λ)[α2	λ)[α2	VERB
ap-7583	112	6	(	(	PUNCT
ap-7583	112	7	k	k	X
ap-7583	112	8	+	+	NOUN
ap-7583	112	9	λ	λ	X
ap-7583	112	10	−	−	NOUN
ap-7583	112	11	1	1	NUM
ap-7583	112	12	)	)	PUNCT
ap-7583	112	13	+	+	CCONJ
ap-7583	112	14	β1	β1	X
ap-7583	112	15	]	]	PUNCT
ap-7583	113	1	+	+	CCONJ
ap-7583	113	2	ε0	ε0	PROPN
ap-7583	113	3	,	,	PUNCT
ap-7583	113	4	(	(	PUNCT
ap-7583	113	5	15	15	NUM
ap-7583	113	6	)	)	PUNCT
ap-7583	113	7	where	where	SCONJ
ap-7583	113	8	λ	λ	NOUN
ap-7583	113	9	=	=	PRON
ap-7583	113	10	λ+	λ+	NOUN
ap-7583	113	11	,	,	PUNCT
ap-7583	113	12	λ−	λ−	PROPN
ap-7583	113	13	are	be	AUX
ap-7583	113	14	the	the	DET
ap-7583	113	15	roots	root	NOUN
ap-7583	113	16	of	of	ADP
ap-7583	113	17	the	the	DET
ap-7583	113	18	indical	indical	ADJ
ap-7583	113	19	equation	equation	NOUN
ap-7583	113	20	α2	α2	PROPN
ap-7583	113	21	λ	λ	PROPN
ap-7583	113	22	(	(	PUNCT
ap-7583	113	23	λ	λ	X
ap-7583	113	24	−	−	NOUN
ap-7583	113	25	1	1	NUM
ap-7583	113	26	)	)	PUNCT
ap-7583	113	27	+	+	CCONJ
ap-7583	113	28	β1	β1	PROPN
ap-7583	113	29	λ	λ	PROPN
ap-7583	113	30	+	+	CCONJ
ap-7583	113	31	ε0	ε0	PROPN
ap-7583	113	32	=	=	SYM
ap-7583	113	33	0	0	NUM
ap-7583	113	34	,	,	PUNCT
ap-7583	113	35	namely	namely	ADV
ap-7583	113	36	λ±	λ±	X
ap-7583	113	37	=	=	SYM
ap-7583	113	38	α2	α2	PROPN
ap-7583	113	39	−	−	PROPN
ap-7583	113	40	β1	β1	PROPN
ap-7583	113	41	±	±	PROPN
ap-7583	113	42	√	√	PROPN
ap-7583	113	43	(	(	PUNCT
ap-7583	113	44	α2	α2	PROPN
ap-7583	113	45	−	−	PROPN
ap-7583	113	46	β1)2	β1)2	NUM
ap-7583	113	47	−	−	PROPN
ap-7583	113	48	4α2ε0	4α2ε0	PROPN
ap-7583	113	49	2α2	2α2	NUM
ap-7583	113	50	.	.	PUNCT
ap-7583	114	1	168	168	NUM
ap-7583	114	2	vol	vol	NOUN
ap-7583	114	3	.	.	PUNCT
ap-7583	115	1	62	62	NUM
ap-7583	115	2	no	no	INTJ
ap-7583	115	3	.	.	PUNCT
ap-7583	116	1	1/2022	1/2022	NUM
ap-7583	116	2	on	on	ADP
ap-7583	116	3	generalized	generalized	ADJ
ap-7583	116	4	heun	heun	NOUN
ap-7583	116	5	equation	equation	NOUN
ap-7583	116	6	with	with	ADP
ap-7583	116	7	some	some	DET
ap-7583	116	8	mathematical	mathematical	NOUN
ap-7583	116	9	.	.	PUNCT
ap-7583	116	10	.	.	PUNCT
ap-7583	116	11	.	.	PUNCT
ap-7583	117	1	the	the	DET
ap-7583	117	2	two	two	NUM
ap-7583	117	3	linearly	linearly	ADV
ap-7583	117	4	independent	independent	ADJ
ap-7583	117	5	solutions	solution	NOUN
ap-7583	117	6	generated	generate	VERB
ap-7583	117	7	by	by	ADP
ap-7583	117	8	(	(	PUNCT
ap-7583	117	9	15	15	NUM
ap-7583	117	10	)	)	PUNCT
ap-7583	117	11	,	,	PUNCT
ap-7583	117	12	in	in	ADP
ap-7583	117	13	terms	term	NOUN
ap-7583	117	14	of	of	ADP
ap-7583	117	15	the	the	DET
ap-7583	117	16	gauss	gauss	ADJ
ap-7583	117	17	hypergeometric	hypergeometric	ADJ
ap-7583	117	18	functions	function	NOUN
ap-7583	117	19	,	,	PUNCT
ap-7583	117	20	are	be	AUX
ap-7583	117	21	:	:	PUNCT
ap-7583	117	22	u±	u±	PROPN
ap-7583	117	23	=	=	PUNCT
ap-7583	117	24	r	r	NOUN
ap-7583	117	25	α2−β1±	α2−β1±	NUM
ap-7583	117	26	√	√	NUM
ap-7583	117	27	(	(	PUNCT
ap-7583	117	28	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	117	29	2α2	2α2	NUM
ap-7583	117	30	2f1	2f1	NUM
ap-7583	117	31	(	(	PUNCT
ap-7583	117	32	β2	β2	VERB
ap-7583	117	33	2α3	2α3	NUM
ap-7583	117	34	−	−	PROPN
ap-7583	118	1	β1	β1	VERB
ap-7583	118	2	2α2	2α2	NUM
ap-7583	118	3	±	±	NOUN
ap-7583	118	4	√	√	PROPN
ap-7583	118	5	(	(	PUNCT
ap-7583	118	6	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	118	7	2α2	2α2	NUM
ap-7583	119	1	−	−	NOUN
ap-7583	119	2	√	√	PROPN
ap-7583	119	3	(	(	PUNCT
ap-7583	119	4	α3−β2)2−4α3ε1	α3−β2)2−4α3ε1	NUM
ap-7583	119	5	2α3	2α3	NUM
ap-7583	119	6	,	,	PUNCT
ap-7583	119	7	β2	β2	VERB
ap-7583	119	8	2α3	2α3	NUM
ap-7583	119	9	−	−	PROPN
ap-7583	120	1	β1	β1	VERB
ap-7583	120	2	2α2	2α2	NUM
ap-7583	120	3	±	±	NOUN
ap-7583	120	4	√	√	PROPN
ap-7583	120	5	(	(	PUNCT
ap-7583	120	6	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	120	7	2α2	2α2	NUM
ap-7583	121	1	+	+	CCONJ
ap-7583	121	2	√	√	NUM
ap-7583	121	3	(	(	PUNCT
ap-7583	121	4	α3−β2)2−4α3c1	α3−β2)2−4α3c1	NUM
ap-7583	121	5	2α3	2α3	NUM
ap-7583	121	6	;	;	PUNCT
ap-7583	121	7	α2±	α2±	NUM
ap-7583	121	8	√	√	PROPN
ap-7583	121	9	(	(	PUNCT
ap-7583	121	10	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	121	11	α2	α2	PROPN
ap-7583	121	12	;	;	PUNCT
ap-7583	121	13	−	−	PROPN
ap-7583	121	14	α3	α3	ADJ
ap-7583	121	15	α2	α2	ADJ
ap-7583	121	16	r	r	NOUN
ap-7583	121	17	)	)	PUNCT
ap-7583	121	18	.	.	PUNCT
ap-7583	122	1	(	(	PUNCT
ap-7583	122	2	16	16	NUM
ap-7583	122	3	)	)	PUNCT
ap-7583	122	4	•	•	NUM
ap-7583	122	5	differential	differential	NOUN
ap-7583	122	6	equation	equation	NOUN
ap-7583	122	7	:	:	PUNCT
ap-7583	122	8	(	(	PUNCT
ap-7583	122	9	α1	α1	NOUN
ap-7583	122	10	r	r	NOUN
ap-7583	122	11	+	+	CCONJ
ap-7583	122	12	α3	α3	PROPN
ap-7583	122	13	r3	r3	NOUN
ap-7583	122	14	)	)	PUNCT
ap-7583	122	15	u′′(r	u′′(r	NOUN
ap-7583	122	16	)	)	PUNCT
ap-7583	122	17	+	+	CCONJ
ap-7583	122	18	(	(	PUNCT
ap-7583	122	19	β0	β0	PROPN
ap-7583	122	20	+	+	CCONJ
ap-7583	122	21	β2	β2	NOUN
ap-7583	122	22	r2	r2	NOUN
ap-7583	122	23	)	)	PUNCT
ap-7583	122	24	u′(r	u′(r	VERB
ap-7583	122	25	)	)	PUNCT
ap-7583	123	1	+	+	CCONJ
ap-7583	123	2	ε1	ε1	VERB
ap-7583	123	3	r	r	NOUN
ap-7583	123	4	u(r	u(r	NOUN
ap-7583	123	5	)	)	PUNCT
ap-7583	124	1	=	=	SYM
ap-7583	124	2	0	0	NUM
ap-7583	124	3	,	,	PUNCT
ap-7583	124	4	(	(	PUNCT
ap-7583	124	5	β0	β0	NOUN
ap-7583	124	6	̸=	̸=	PROPN
ap-7583	124	7	0	0	NUM
ap-7583	124	8	)	)	PUNCT
ap-7583	124	9	,	,	PUNCT
ap-7583	124	10	(	(	PUNCT
ap-7583	124	11	17	17	NUM
ap-7583	124	12	)	)	PUNCT
ap-7583	124	13	recurrence	recurrence	NOUN
ap-7583	124	14	relation	relation	NOUN
ap-7583	124	15	:	:	PUNCT
ap-7583	124	16	ck	ck	PROPN
ap-7583	125	1	ck−2	ck−2	NOUN
ap-7583	125	2	=	=	SYM
ap-7583	126	1	−	−	PROPN
ap-7583	126	2	(	(	PUNCT
ap-7583	126	3	k	k	PROPN
ap-7583	126	4	+	+	PROPN
ap-7583	127	1	λ	λ	PROPN
ap-7583	127	2	−	−	PROPN
ap-7583	127	3	2)[α3	2)[α3	NUM
ap-7583	127	4	(	(	PUNCT
ap-7583	127	5	k	k	PROPN
ap-7583	127	6	+	+	PROPN
ap-7583	127	7	λ	λ	PROPN
ap-7583	127	8	−	−	NOUN
ap-7583	127	9	3	3	NUM
ap-7583	127	10	)	)	PUNCT
ap-7583	127	11	+	+	CCONJ
ap-7583	127	12	β2	β2	VERB
ap-7583	127	13	]	]	X
ap-7583	127	14	+	+	CCONJ
ap-7583	127	15	ε1	ε1	PROPN
ap-7583	127	16	(	(	PUNCT
ap-7583	127	17	k	k	NOUN
ap-7583	127	18	+	+	CCONJ
ap-7583	127	19	λ)[α1(k	λ)[α1(k	NOUN
ap-7583	127	20	+	+	CCONJ
ap-7583	127	21	λ	λ	NOUN
ap-7583	127	22	−	−	NOUN
ap-7583	127	23	1	1	NUM
ap-7583	127	24	)	)	PUNCT
ap-7583	127	25	+	+	CCONJ
ap-7583	127	26	β0	β0	ADJ
ap-7583	127	27	]	]	PUNCT
ap-7583	127	28	,	,	PUNCT
ap-7583	127	29	(	(	PUNCT
ap-7583	127	30	c0	c0	NOUN
ap-7583	127	31	,	,	PUNCT
ap-7583	127	32	c1	c1	PROPN
ap-7583	127	33	̸=	̸=	PROPN
ap-7583	127	34	0	0	NUM
ap-7583	127	35	,	,	PUNCT
ap-7583	127	36	k	k	NOUN
ap-7583	127	37	=	=	SYM
ap-7583	127	38	2	2	NUM
ap-7583	127	39	,	,	PUNCT
ap-7583	127	40	3	3	NUM
ap-7583	127	41	,	,	PUNCT
ap-7583	127	42	·	·	PUNCT
ap-7583	127	43	·	·	PUNCT
ap-7583	127	44	·	·	PUNCT
ap-7583	127	45	)	)	PUNCT
ap-7583	127	46	,	,	PUNCT
ap-7583	127	47	(	(	PUNCT
ap-7583	127	48	18	18	NUM
ap-7583	127	49	)	)	PUNCT
ap-7583	127	50	where	where	SCONJ
ap-7583	127	51	λ	λ	NOUN
ap-7583	127	52	=	=	PRON
ap-7583	127	53	λ+	λ+	NOUN
ap-7583	127	54	,	,	PUNCT
ap-7583	127	55	λ−	λ−	PROPN
ap-7583	127	56	are	be	AUX
ap-7583	127	57	the	the	DET
ap-7583	127	58	roots	root	NOUN
ap-7583	127	59	of	of	ADP
ap-7583	127	60	the	the	DET
ap-7583	127	61	indicial	indicial	ADJ
ap-7583	127	62	equation	equation	NOUN
ap-7583	127	63	α1	α1	PROPN
ap-7583	127	64	λ(λ	λ(λ	X
ap-7583	127	65	−	−	PROPN
ap-7583	127	66	1	1	NUM
ap-7583	127	67	)	)	PUNCT
ap-7583	127	68	+	+	CCONJ
ap-7583	127	69	β0	β0	PROPN
ap-7583	127	70	λ	λ	X
ap-7583	127	71	=	=	SYM
ap-7583	127	72	0	0	NUM
ap-7583	127	73	,	,	PUNCT
ap-7583	127	74	i.e	i.e	PRON
ap-7583	127	75	λ+	λ+	PUNCT
ap-7583	127	76	=	=	SYM
ap-7583	127	77	0	0	NUM
ap-7583	127	78	,	,	PUNCT
ap-7583	127	79	λ−	λ−	PROPN
ap-7583	127	80	=	=	NOUN
ap-7583	127	81	1	1	NUM
ap-7583	127	82	−	−	PROPN
ap-7583	127	83	β0	β0	PROPN
ap-7583	127	84	/	/	SYM
ap-7583	127	85	α1	α1	PROPN
ap-7583	127	86	.	.	PUNCT
ap-7583	128	1	the	the	DET
ap-7583	128	2	two	two	NUM
ap-7583	128	3	linearly	linearly	ADV
ap-7583	128	4	independent	independent	ADJ
ap-7583	128	5	series	series	NOUN
ap-7583	128	6	solutions	solution	NOUN
ap-7583	128	7	generated	generate	VERB
ap-7583	128	8	by	by	ADP
ap-7583	128	9	(	(	PUNCT
ap-7583	128	10	18	18	NUM
ap-7583	128	11	)	)	PUNCT
ap-7583	128	12	are	be	AUX
ap-7583	128	13	:	:	PUNCT
ap-7583	128	14	u+(r	u+(r	X
ap-7583	128	15	)	)	PUNCT
ap-7583	129	1	=	=	SYM
ap-7583	129	2	2f1	2f1	NUM
ap-7583	130	1	(	(	PUNCT
ap-7583	130	2	β2	β2	VERB
ap-7583	130	3	4α3	4α3	NOUN
ap-7583	130	4	−	−	NUM
ap-7583	130	5	1	1	NUM
ap-7583	130	6	4	4	NUM
ap-7583	130	7	−	−	NOUN
ap-7583	130	8	√	√	NUM
ap-7583	130	9	(	(	PUNCT
ap-7583	130	10	α3	α3	NOUN
ap-7583	130	11	−	−	PROPN
ap-7583	130	12	β2)2	β2)2	PUNCT
ap-7583	130	13	−	−	PROPN
ap-7583	130	14	4α3ε1	4α3ε1	PROPN
ap-7583	131	1	4α3	4α3	NUM
ap-7583	131	2	,	,	PUNCT
ap-7583	131	3	β2	β2	NOUN
ap-7583	131	4	4α3	4α3	NOUN
ap-7583	131	5	−	−	NUM
ap-7583	131	6	1	1	NUM
ap-7583	131	7	4	4	NUM
ap-7583	131	8	+	+	NUM
ap-7583	131	9	√	√	PROPN
ap-7583	131	10	(	(	PUNCT
ap-7583	131	11	α3	α3	NOUN
ap-7583	131	12	−	−	PROPN
ap-7583	131	13	β2)2	β2)2	PUNCT
ap-7583	131	14	−	−	PROPN
ap-7583	131	15	4α3ε1	4α3ε1	PROPN
ap-7583	132	1	4α3	4α3	NUM
ap-7583	132	2	;	;	PUNCT
ap-7583	132	3	1	1	NUM
ap-7583	132	4	2	2	NUM
ap-7583	132	5	+	+	CCONJ
ap-7583	132	6	β0	β0	ADJ
ap-7583	132	7	2α1	2α1	NUM
ap-7583	132	8	;	;	PUNCT
ap-7583	132	9	−α3	−α3	PROPN
ap-7583	132	10	α1	α1	PROPN
ap-7583	132	11	r2	r2	PROPN
ap-7583	132	12	)	)	PUNCT
ap-7583	132	13	,	,	PUNCT
ap-7583	132	14	(	(	PUNCT
ap-7583	132	15	19	19	NUM
ap-7583	132	16	)	)	PUNCT
ap-7583	132	17	and	and	CCONJ
ap-7583	132	18	u−(r	u−(r	ADJ
ap-7583	132	19	)	)	PUNCT
ap-7583	133	1	=	=	PROPN
ap-7583	133	2	r1−	r1−	NUM
ap-7583	133	3	β0	β0	ADJ
ap-7583	133	4	α1	α1	PROPN
ap-7583	133	5	×	×	PROPN
ap-7583	133	6	2f1	2f1	NUM
ap-7583	133	7	(	(	PUNCT
ap-7583	133	8	1	1	NUM
ap-7583	133	9	4	4	NUM
ap-7583	133	10	+	+	CCONJ
ap-7583	133	11	β2	β2	NOUN
ap-7583	134	1	4α3	4α3	NUM
ap-7583	134	2	−	−	PROPN
ap-7583	135	1	β0	β0	ADJ
ap-7583	135	2	2α1	2α1	NUM
ap-7583	135	3	−	−	NOUN
ap-7583	135	4	√	√	NUM
ap-7583	135	5	(	(	PUNCT
ap-7583	135	6	α3	α3	NOUN
ap-7583	135	7	−	−	PROPN
ap-7583	135	8	β2)2	β2)2	PUNCT
ap-7583	135	9	−	−	PROPN
ap-7583	135	10	4α3ε1	4α3ε1	PROPN
ap-7583	136	1	4α3	4α3	NUM
ap-7583	136	2	,	,	PUNCT
ap-7583	136	3	1	1	NUM
ap-7583	136	4	4	4	NUM
ap-7583	136	5	+	+	CCONJ
ap-7583	136	6	β2	β2	NOUN
ap-7583	137	1	4α3	4α3	NUM
ap-7583	137	2	−	−	PROPN
ap-7583	138	1	β0	β0	ADJ
ap-7583	138	2	2α1	2α1	NUM
ap-7583	138	3	+	+	CCONJ
ap-7583	138	4	√	√	INTJ
ap-7583	138	5	(	(	PUNCT
ap-7583	138	6	α3	α3	NOUN
ap-7583	138	7	−	−	PROPN
ap-7583	138	8	β2)2	β2)2	PUNCT
ap-7583	138	9	−	−	PROPN
ap-7583	138	10	4α3ε1	4α3ε1	PROPN
ap-7583	139	1	4α3	4α3	NUM
ap-7583	139	2	;	;	PUNCT
ap-7583	139	3	3	3	NUM
ap-7583	139	4	2	2	NUM
ap-7583	139	5	−	−	NOUN
ap-7583	139	6	β0	β0	ADJ
ap-7583	139	7	2α1	2α1	NUM
ap-7583	139	8	;	;	PUNCT
ap-7583	139	9	−α3	−α3	PROPN
ap-7583	139	10	α1	α1	PROPN
ap-7583	139	11	r2	r2	PROPN
ap-7583	139	12	)	)	PUNCT
ap-7583	139	13	.	.	PUNCT
ap-7583	140	1	(	(	PUNCT
ap-7583	140	2	20	20	NUM
ap-7583	140	3	)	)	PUNCT
ap-7583	140	4	•	•	NOUN
ap-7583	140	5	differential	differential	NOUN
ap-7583	140	6	equation	equation	NOUN
ap-7583	140	7	(	(	PUNCT
ap-7583	140	8	α0	α0	ADJ
ap-7583	140	9	+	+	CCONJ
ap-7583	140	10	α3	α3	ADJ
ap-7583	140	11	r3	r3	PROPN
ap-7583	140	12	)	)	PUNCT
ap-7583	140	13	u′′(r	u′′(r	NOUN
ap-7583	140	14	)	)	PUNCT
ap-7583	140	15	+	+	CCONJ
ap-7583	140	16	β2	β2	NOUN
ap-7583	140	17	r2	r2	PROPN
ap-7583	140	18	u′(r	u′(r	ADP
ap-7583	140	19	)	)	PUNCT
ap-7583	141	1	+	+	CCONJ
ap-7583	141	2	ε1r	ε1r	PRON
ap-7583	141	3	u(r	u(r	NOUN
ap-7583	141	4	)	)	PUNCT
ap-7583	141	5	=	=	SYM
ap-7583	142	1	0	0	NUM
ap-7583	142	2	,	,	PUNCT
ap-7583	142	3	α0	α0	ADJ
ap-7583	142	4	̸=	̸=	PROPN
ap-7583	142	5	0	0	NUM
ap-7583	142	6	,	,	PUNCT
ap-7583	142	7	(	(	PUNCT
ap-7583	142	8	21	21	NUM
ap-7583	142	9	)	)	PUNCT
ap-7583	142	10	recurrence	recurrence	NOUN
ap-7583	142	11	relation	relation	NOUN
ap-7583	142	12	:	:	PUNCT
ap-7583	142	13	ck	ck	PROPN
ap-7583	142	14	ck−3	ck−3	NOUN
ap-7583	142	15	=	=	PUNCT
ap-7583	142	16	−	−	PROPN
ap-7583	143	1	(	(	PUNCT
ap-7583	143	2	k	k	PROPN
ap-7583	143	3	+	+	PROPN
ap-7583	143	4	λ	λ	PROPN
ap-7583	144	1	−	−	X
ap-7583	144	2	3)[α3(k	3)[α3(k	NOUN
ap-7583	144	3	+	+	CCONJ
ap-7583	144	4	λ	λ	NOUN
ap-7583	144	5	−	−	NOUN
ap-7583	144	6	4	4	NUM
ap-7583	144	7	)	)	PUNCT
ap-7583	144	8	+	+	CCONJ
ap-7583	144	9	β2	β2	VERB
ap-7583	144	10	]	]	X
ap-7583	144	11	+	+	CCONJ
ap-7583	144	12	ε1	ε1	VERB
ap-7583	144	13	α0	α0	PROPN
ap-7583	144	14	(	(	PUNCT
ap-7583	144	15	k	k	PROPN
ap-7583	144	16	+	+	PROPN
ap-7583	144	17	λ)(k	λ)(k	PROPN
ap-7583	145	1	+	+	CCONJ
ap-7583	145	2	λ	λ	X
ap-7583	145	3	−	−	NOUN
ap-7583	145	4	1	1	NUM
ap-7583	145	5	)	)	PUNCT
ap-7583	145	6	,	,	PUNCT
ap-7583	145	7	(	(	PUNCT
ap-7583	145	8	c0	c0	PROPN
ap-7583	145	9	̸=	̸=	PROPN
ap-7583	145	10	0	0	NUM
ap-7583	145	11	)	)	PUNCT
ap-7583	145	12	,	,	PUNCT
ap-7583	145	13	(	(	PUNCT
ap-7583	145	14	22	22	NUM
ap-7583	145	15	)	)	PUNCT
ap-7583	145	16	where	where	SCONJ
ap-7583	145	17	λ	λ	PROPN
ap-7583	145	18	=	=	SYM
ap-7583	145	19	λ1	λ1	PROPN
ap-7583	145	20	,	,	PUNCT
ap-7583	145	21	λ2	λ2	NOUN
ap-7583	145	22	are	be	AUX
ap-7583	145	23	the	the	DET
ap-7583	145	24	roots	root	NOUN
ap-7583	145	25	of	of	ADP
ap-7583	145	26	the	the	DET
ap-7583	145	27	indicial	indicial	ADJ
ap-7583	145	28	equation	equation	NOUN
ap-7583	145	29	α0	α0	PROPN
ap-7583	145	30	λ	λ	PROPN
ap-7583	145	31	(	(	PUNCT
ap-7583	145	32	λ	λ	X
ap-7583	145	33	−	−	NOUN
ap-7583	145	34	1	1	NUM
ap-7583	145	35	)	)	PUNCT
ap-7583	145	36	=	=	SYM
ap-7583	145	37	0	0	NUM
ap-7583	145	38	,	,	PUNCT
ap-7583	145	39	namely	namely	ADV
ap-7583	145	40	,	,	PUNCT
ap-7583	145	41	λ1	λ1	PROPN
ap-7583	145	42	=	=	SYM
ap-7583	145	43	0	0	NUM
ap-7583	145	44	,	,	PUNCT
ap-7583	145	45	λ2	λ2	NOUN
ap-7583	145	46	=	=	SYM
ap-7583	145	47	1	1	NUM
ap-7583	145	48	.	.	PUNCT
ap-7583	146	1	the	the	DET
ap-7583	146	2	two	two	NUM
ap-7583	146	3	linearly	linearly	ADV
ap-7583	146	4	independent	independent	ADJ
ap-7583	146	5	series	series	NOUN
ap-7583	146	6	solutions	solution	NOUN
ap-7583	146	7	are	be	AUX
ap-7583	146	8	:	:	PUNCT
ap-7583	146	9	u1(r	u1(r	X
ap-7583	146	10	)	)	PUNCT
ap-7583	146	11	=	=	SYM
ap-7583	146	12	2f1	2f1	NUM
ap-7583	146	13	(	(	PUNCT
ap-7583	146	14	−	−	PUNCT
ap-7583	146	15	α3	α3	NOUN
ap-7583	146	16	−	−	PROPN
ap-7583	146	17	β2	β2	NOUN
ap-7583	146	18	+	+	CCONJ
ap-7583	146	19	√	√	PROPN
ap-7583	146	20	(	(	PUNCT
ap-7583	146	21	α3	α3	NOUN
ap-7583	146	22	−	−	PROPN
ap-7583	146	23	β2)2	β2)2	PUNCT
ap-7583	146	24	−	−	PROPN
ap-7583	146	25	4α3ε1	4α3ε1	PROPN
ap-7583	146	26	6α3	6α3	NUM
ap-7583	146	27	,	,	PUNCT
ap-7583	146	28	−α3	−α3	PROPN
ap-7583	146	29	+	+	CCONJ
ap-7583	146	30	β2	β2	NOUN
ap-7583	146	31	+	+	CCONJ
ap-7583	146	32	√	√	PROPN
ap-7583	146	33	(	(	PUNCT
ap-7583	146	34	α3	α3	NOUN
ap-7583	146	35	−	−	PROPN
ap-7583	146	36	β2)2	β2)2	PUNCT
ap-7583	146	37	−	−	PROPN
ap-7583	146	38	4α3ε1	4α3ε1	PROPN
ap-7583	146	39	6α3	6α3	NUM
ap-7583	146	40	;	;	PUNCT
ap-7583	146	41	2	2	NUM
ap-7583	146	42	3	3	NUM
ap-7583	146	43	;	;	PUNCT
ap-7583	146	44	−α3	−α3	PROPN
ap-7583	146	45	α0	α0	PROPN
ap-7583	146	46	r3	r3	PROPN
ap-7583	146	47	)	)	PUNCT
ap-7583	146	48	,	,	PUNCT
ap-7583	146	49	(	(	PUNCT
ap-7583	146	50	23	23	NUM
ap-7583	146	51	)	)	PUNCT
ap-7583	146	52	and	and	CCONJ
ap-7583	146	53	u2(r	u2(r	X
ap-7583	146	54	)	)	PUNCT
ap-7583	147	1	=	=	SYM
ap-7583	147	2	r	r	NOUN
ap-7583	147	3	2f1	2f1	NUM
ap-7583	147	4	(	(	PUNCT
ap-7583	147	5	α3	α3	PROPN
ap-7583	147	6	+	+	CCONJ
ap-7583	147	7	β2	β2	NOUN
ap-7583	147	8	−	−	PROPN
ap-7583	147	9	√	√	PROPN
ap-7583	147	10	(	(	PUNCT
ap-7583	147	11	α3	α3	NOUN
ap-7583	147	12	−	−	PROPN
ap-7583	147	13	β2)2	β2)2	PUNCT
ap-7583	147	14	−	−	PROPN
ap-7583	147	15	4α3ε1	4α3ε1	PROPN
ap-7583	147	16	6α3	6α3	NUM
ap-7583	147	17	,	,	PUNCT
ap-7583	147	18	α3	α3	PROPN
ap-7583	147	19	+	+	CCONJ
ap-7583	147	20	β2	β2	NOUN
ap-7583	147	21	+	+	CCONJ
ap-7583	147	22	√	√	PROPN
ap-7583	147	23	(	(	PUNCT
ap-7583	147	24	α3	α3	NOUN
ap-7583	147	25	−	−	PROPN
ap-7583	147	26	β2)2	β2)2	PUNCT
ap-7583	147	27	−	−	PROPN
ap-7583	147	28	4α3ε1	4α3ε1	PROPN
ap-7583	147	29	6α3	6α3	NUM
ap-7583	147	30	;	;	PUNCT
ap-7583	147	31	4	4	NUM
ap-7583	147	32	3	3	NUM
ap-7583	147	33	;	;	PUNCT
ap-7583	147	34	−α3	−α3	PROPN
ap-7583	147	35	α0	α0	PROPN
ap-7583	147	36	r3	r3	PROPN
ap-7583	147	37	)	)	PUNCT
ap-7583	147	38	.	.	PUNCT
ap-7583	148	1	(	(	PUNCT
ap-7583	148	2	24	24	NUM
ap-7583	148	3	)	)	PUNCT
ap-7583	148	4	out	out	ADP
ap-7583	148	5	of	of	ADP
ap-7583	148	6	the	the	DET
ap-7583	148	7	three	three	NUM
ap-7583	148	8	generic	generic	ADJ
ap-7583	148	9	equations	equation	NOUN
ap-7583	148	10	(	(	PUNCT
ap-7583	148	11	14	14	NUM
ap-7583	148	12	)	)	PUNCT
ap-7583	148	13	,	,	PUNCT
ap-7583	148	14	(	(	PUNCT
ap-7583	148	15	17	17	NUM
ap-7583	148	16	)	)	PUNCT
ap-7583	148	17	and	and	CCONJ
ap-7583	148	18	(	(	PUNCT
ap-7583	148	19	21	21	NUM
ap-7583	148	20	)	)	PUNCT
ap-7583	148	21	,	,	PUNCT
ap-7583	148	22	five	five	NUM
ap-7583	148	23	exactly	exactly	ADV
ap-7583	148	24	solvable	solvable	ADJ
ap-7583	148	25	differential	differential	ADJ
ap-7583	148	26	equations	equation	NOUN
ap-7583	148	27	(	(	PUNCT
ap-7583	148	28	cases	case	NOUN
ap-7583	148	29	1	1	NUM
ap-7583	148	30	,	,	PUNCT
ap-7583	148	31	4	4	NUM
ap-7583	148	32	,	,	PUNCT
ap-7583	148	33	5	5	NUM
ap-7583	148	34	,	,	PUNCT
ap-7583	148	35	8	8	NUM
ap-7583	148	36	,	,	PUNCT
ap-7583	148	37	and	and	CCONJ
ap-7583	148	38	10	10	NUM
ap-7583	148	39	)	)	PUNCT
ap-7583	148	40	of	of	ADP
ap-7583	148	41	the	the	DET
ap-7583	148	42	type	type	NOUN
ap-7583	148	43	(	(	PUNCT
ap-7583	148	44	4	4	NUM
ap-7583	148	45	)	)	PUNCT
ap-7583	148	46	follows	follow	VERB
ap-7583	148	47	and	and	CCONJ
ap-7583	148	48	other	other	ADJ
ap-7583	148	49	five	five	NUM
ap-7583	148	50	(	(	PUNCT
ap-7583	148	51	cases	case	NOUN
ap-7583	148	52	2	2	NUM
ap-7583	148	53	,	,	PUNCT
ap-7583	148	54	3	3	NUM
ap-7583	148	55	,	,	PUNCT
ap-7583	148	56	6	6	NUM
ap-7583	148	57	,	,	PUNCT
ap-7583	148	58	7	7	NUM
ap-7583	148	59	,	,	PUNCT
ap-7583	148	60	9	9	NUM
ap-7583	148	61	)	)	PUNCT
ap-7583	148	62	that	that	PRON
ap-7583	148	63	can	can	AUX
ap-7583	148	64	be	be	AUX
ap-7583	148	65	derived	derive	VERB
ap-7583	148	66	directly	directly	ADV
ap-7583	148	67	from	from	ADP
ap-7583	148	68	them	they	PRON
ap-7583	148	69	by	by	ADP
ap-7583	148	70	taking	take	VERB
ap-7583	148	71	the	the	DET
ap-7583	148	72	limits	limit	NOUN
ap-7583	148	73	of	of	ADP
ap-7583	148	74	the	the	DET
ap-7583	148	75	equation	equation	NOUN
ap-7583	148	76	parameters	parameter	NOUN
ap-7583	148	77	.	.	PUNCT
ap-7583	149	1	for	for	ADP
ap-7583	149	2	direct	direct	ADJ
ap-7583	149	3	use	use	NOUN
ap-7583	149	4	,	,	PUNCT
ap-7583	149	5	the	the	DET
ap-7583	149	6	ten	ten	NUM
ap-7583	149	7	equations	equation	NOUN
ap-7583	149	8	are	be	AUX
ap-7583	149	9	listed	list	VERB
ap-7583	149	10	in	in	ADP
ap-7583	149	11	table	table	NOUN
ap-7583	149	12	1	1	NUM
ap-7583	149	13	.	.	SYM
ap-7583	149	14	169	169	NUM
ap-7583	149	15	nasser	nasser	PROPN
ap-7583	149	16	saad	saad	PROPN
ap-7583	149	17	acta	acta	PROPN
ap-7583	149	18	polytechnica	polytechnica	PROPN
ap-7583	149	19	des	des	PROPN
ap-7583	149	20	and	and	CCONJ
ap-7583	149	21	their	their	PRON
ap-7583	149	22	linearly	linearly	ADV
ap-7583	149	23	independent	independent	ADJ
ap-7583	149	24	solutions	solution	NOUN
ap-7583	149	25	1	1	NUM
ap-7583	149	26	α2	α2	ADJ
ap-7583	149	27	r2u′′	r2u′′	PROPN
ap-7583	149	28	+	+	CCONJ
ap-7583	149	29	(	(	PUNCT
ap-7583	149	30	β1r	β1r	VERB
ap-7583	149	31	+	+	NUM
ap-7583	149	32	β2r2	β2r2	NOUN
ap-7583	149	33	)	)	PUNCT
ap-7583	149	34	u′	u′	PROPN
ap-7583	150	1	+	+	CCONJ
ap-7583	150	2	(	(	PUNCT
ap-7583	150	3	ε0	ε0	PROPN
ap-7583	150	4	+	+	CCONJ
ap-7583	150	5	ε1r	ε1r	PROPN
ap-7583	150	6	)	)	PUNCT
ap-7583	150	7	u	u	NOUN
ap-7583	150	8	=	=	SYM
ap-7583	150	9	0	0	NUM
ap-7583	150	10	u	u	NOUN
ap-7583	150	11	=	=	NOUN
ap-7583	150	12	r	r	NOUN
ap-7583	150	13	1	1	NUM
ap-7583	150	14	2	2	NUM
ap-7583	150	15	−	−	NOUN
ap-7583	150	16	β1	β1	NOUN
ap-7583	150	17	2α2	2α2	NUM
ap-7583	151	1	+	+	CCONJ
ap-7583	151	2	1	1	NUM
ap-7583	151	3	2α2	2α2	NUM
ap-7583	151	4	√	√	NUM
ap-7583	151	5	(	(	PUNCT
ap-7583	151	6	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	151	7	1f1	1f1	NUM
ap-7583	151	8	(	(	PUNCT
ap-7583	151	9	1	1	NUM
ap-7583	151	10	2	2	NUM
ap-7583	151	11	−	−	PRON
ap-7583	152	1	β1	β1	PROPN
ap-7583	152	2	2α2	2α2	NUM
ap-7583	152	3	+	+	CCONJ
ap-7583	152	4	ε1	ε1	PROPN
ap-7583	152	5	β2	β2	NOUN
ap-7583	152	6	+	+	CCONJ
ap-7583	152	7	√	√	PROPN
ap-7583	152	8	(	(	PUNCT
ap-7583	152	9	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	152	10	2α2	2α2	NUM
ap-7583	152	11	;	;	PUNCT
ap-7583	152	12	1	1	NUM
ap-7583	152	13	+	+	NUM
ap-7583	152	14	√	√	PROPN
ap-7583	152	15	(	(	PUNCT
ap-7583	152	16	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	152	17	α2	α2	PROPN
ap-7583	152	18	;	;	PUNCT
ap-7583	152	19	−	−	PROPN
ap-7583	152	20	β2	β2	NOUN
ap-7583	152	21	α2	α2	NOUN
ap-7583	152	22	r	r	PROPN
ap-7583	152	23	)	)	PUNCT
ap-7583	152	24	,	,	PUNCT
ap-7583	152	25	u	u	NOUN
ap-7583	152	26	=	=	PROPN
ap-7583	152	27	r−	r−	PROPN
ap-7583	152	28	1	1	NUM
ap-7583	152	29	2	2	NUM
ap-7583	152	30	+	+	CCONJ
ap-7583	152	31	β1	β1	VERB
ap-7583	152	32	2α2	2α2	NUM
ap-7583	153	1	−	−	NUM
ap-7583	153	2	1	1	NUM
ap-7583	153	3	2α2	2α2	NUM
ap-7583	153	4	√	√	NUM
ap-7583	153	5	(	(	PUNCT
ap-7583	153	6	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	153	7	1f1	1f1	NUM
ap-7583	153	8	(	(	PUNCT
ap-7583	153	9	1	1	NUM
ap-7583	153	10	2	2	NUM
ap-7583	153	11	−	−	PRON
ap-7583	153	12	β1	β1	PROPN
ap-7583	153	13	2α2	2α2	NUM
ap-7583	154	1	+	+	CCONJ
ap-7583	154	2	ε1	ε1	PROPN
ap-7583	154	3	β2	β2	NOUN
ap-7583	154	4	−	−	PROPN
ap-7583	154	5	√	√	PROPN
ap-7583	154	6	(	(	PUNCT
ap-7583	154	7	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	154	8	2α2	2α2	NUM
ap-7583	154	9	;	;	PUNCT
ap-7583	154	10	1	1	NUM
ap-7583	154	11	−	−	NOUN
ap-7583	154	12	√	√	NUM
ap-7583	154	13	(	(	PUNCT
ap-7583	154	14	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	154	15	α2	α2	PROPN
ap-7583	154	16	;	;	PUNCT
ap-7583	154	17	−	−	PROPN
ap-7583	154	18	β2	β2	NOUN
ap-7583	154	19	α2	α2	NOUN
ap-7583	154	20	r	r	NOUN
ap-7583	154	21	)	)	PUNCT
ap-7583	154	22	.	.	PUNCT
ap-7583	155	1	2	2	NUM
ap-7583	155	2	α2	α2	ADJ
ap-7583	155	3	r2u′′	r2u′′	PROPN
ap-7583	155	4	+	+	CCONJ
ap-7583	155	5	β1r	β1r	VERB
ap-7583	155	6	u′	u′	PROPN
ap-7583	155	7	+	+	CCONJ
ap-7583	155	8	(	(	PUNCT
ap-7583	155	9	ε0	ε0	PROPN
ap-7583	155	10	+	+	CCONJ
ap-7583	155	11	ε1r	ε1r	PROPN
ap-7583	155	12	)	)	PUNCT
ap-7583	155	13	u	u	NOUN
ap-7583	155	14	=	=	SYM
ap-7583	155	15	0	0	NUM
ap-7583	155	16	u	u	NOUN
ap-7583	155	17	=	=	NOUN
ap-7583	155	18	r	r	NOUN
ap-7583	155	19	1	1	NUM
ap-7583	155	20	2	2	NUM
ap-7583	155	21	−	−	NOUN
ap-7583	155	22	β1	β1	NOUN
ap-7583	155	23	2α2	2α2	NUM
ap-7583	156	1	+	+	CCONJ
ap-7583	156	2	1	1	NUM
ap-7583	156	3	2α2	2α2	NUM
ap-7583	156	4	√	√	NUM
ap-7583	156	5	(	(	PUNCT
ap-7583	156	6	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	156	7	0f1	0f1	NUM
ap-7583	157	1	(	(	PUNCT
ap-7583	157	2	−	−	NOUN
ap-7583	157	3	;	;	PUNCT
ap-7583	157	4	1	1	NUM
ap-7583	157	5	+	+	NUM
ap-7583	157	6	√	√	PROPN
ap-7583	157	7	(	(	PUNCT
ap-7583	157	8	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	157	9	α2	α2	PROPN
ap-7583	157	10	;	;	PUNCT
ap-7583	157	11	−	−	PROPN
ap-7583	157	12	ε1	ε1	VERB
ap-7583	157	13	α2	α2	ADJ
ap-7583	157	14	r	r	NOUN
ap-7583	157	15	)	)	PUNCT
ap-7583	157	16	,	,	PUNCT
ap-7583	157	17	u	u	NOUN
ap-7583	157	18	=	=	PROPN
ap-7583	157	19	r−	r−	PROPN
ap-7583	157	20	1	1	NUM
ap-7583	157	21	2	2	NUM
ap-7583	157	22	+	+	CCONJ
ap-7583	157	23	β1	β1	VERB
ap-7583	157	24	2α2	2α2	NUM
ap-7583	157	25	−	−	NUM
ap-7583	157	26	1	1	NUM
ap-7583	157	27	2α2	2α2	NUM
ap-7583	157	28	√	√	PROPN
ap-7583	157	29	(	(	PUNCT
ap-7583	157	30	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	157	31	0f1	0f1	NUM
ap-7583	158	1	(	(	PUNCT
ap-7583	158	2	−	−	NOUN
ap-7583	158	3	;	;	PUNCT
ap-7583	158	4	1	1	NUM
ap-7583	158	5	−	−	NOUN
ap-7583	158	6	√	√	NUM
ap-7583	158	7	(	(	PUNCT
ap-7583	158	8	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	158	9	α2	α2	PROPN
ap-7583	158	10	;	;	PUNCT
ap-7583	158	11	−	−	PROPN
ap-7583	158	12	β2	β2	NOUN
ap-7583	158	13	α2	α2	NOUN
ap-7583	158	14	r	r	NOUN
ap-7583	158	15	)	)	PUNCT
ap-7583	158	16	.	.	PUNCT
ap-7583	159	1	3	3	NUM
ap-7583	159	2	α2	α2	ADJ
ap-7583	159	3	r2u′′	r2u′′	PROPN
ap-7583	159	4	+	+	CCONJ
ap-7583	159	5	β2r2u′	β2r2u′	X
ap-7583	159	6	+	+	CCONJ
ap-7583	159	7	(	(	PUNCT
ap-7583	159	8	ε0	ε0	PROPN
ap-7583	159	9	+	+	CCONJ
ap-7583	159	10	ε1r	ε1r	PROPN
ap-7583	159	11	)	)	PUNCT
ap-7583	159	12	u	u	NOUN
ap-7583	159	13	=	=	SYM
ap-7583	159	14	0	0	NUM
ap-7583	159	15	u	u	NOUN
ap-7583	159	16	=	=	NOUN
ap-7583	159	17	r	r	NOUN
ap-7583	159	18	1	1	NUM
ap-7583	159	19	2	2	NUM
ap-7583	159	20	−	−	NOUN
ap-7583	159	21	√	√	NUM
ap-7583	159	22	α2−4ε0	α2−4ε0	PROPN
ap-7583	159	23	2√	2√	PROPN
ap-7583	159	24	α2	α2	PROPN
ap-7583	159	25	1f1	1f1	NUM
ap-7583	159	26	(	(	PUNCT
ap-7583	159	27	1	1	NUM
ap-7583	159	28	2	2	NUM
ap-7583	159	29	+	+	NUM
ap-7583	159	30	ε1	ε1	PROPN
ap-7583	159	31	β2	β2	NOUN
ap-7583	159	32	−	−	PROPN
ap-7583	159	33	√	√	NUM
ap-7583	159	34	α2−4ε0	α2−4ε0	PROPN
ap-7583	159	35	2√	2√	PROPN
ap-7583	159	36	α2	α2	NOUN
ap-7583	159	37	;	;	PUNCT
ap-7583	159	38	1	1	NUM
ap-7583	159	39	−	−	NUM
ap-7583	159	40	√	√	NUM
ap-7583	159	41	α2−4ε0√	α2−4ε0√	NOUN
ap-7583	159	42	α2	α2	PROPN
ap-7583	159	43	;	;	PUNCT
ap-7583	159	44	−	−	PROPN
ap-7583	159	45	β2	β2	NOUN
ap-7583	159	46	α2	α2	NOUN
ap-7583	159	47	r	r	PROPN
ap-7583	159	48	)	)	PUNCT
ap-7583	159	49	,	,	PUNCT
ap-7583	159	50	u	u	NOUN
ap-7583	160	1	=	=	SYM
ap-7583	160	2	r	r	NOUN
ap-7583	160	3	1	1	NUM
ap-7583	160	4	2	2	NUM
ap-7583	160	5	+	+	CCONJ
ap-7583	160	6	√	√	PUNCT
ap-7583	160	7	α2−4ε0	α2−4ε0	PROPN
ap-7583	160	8	2√	2√	PROPN
ap-7583	160	9	α2	α2	PROPN
ap-7583	160	10	1f1	1f1	NUM
ap-7583	160	11	(	(	PUNCT
ap-7583	160	12	1	1	NUM
ap-7583	160	13	2	2	NUM
ap-7583	160	14	+	+	NUM
ap-7583	160	15	ε1	ε1	PROPN
ap-7583	160	16	β2	β2	NOUN
ap-7583	160	17	+	+	CCONJ
ap-7583	160	18	√	√	PUNCT
ap-7583	160	19	α2−4ε0	α2−4ε0	PROPN
ap-7583	160	20	2√	2√	PROPN
ap-7583	160	21	α2	α2	NOUN
ap-7583	160	22	;	;	PUNCT
ap-7583	160	23	1	1	NUM
ap-7583	160	24	+	+	CCONJ
ap-7583	160	25	√	√	NOUN
ap-7583	160	26	α2−4ε0√	α2−4ε0√	NOUN
ap-7583	160	27	α2	α2	PROPN
ap-7583	160	28	;	;	PUNCT
ap-7583	160	29	−	−	PROPN
ap-7583	160	30	β2	β2	NOUN
ap-7583	160	31	α2	α2	NOUN
ap-7583	160	32	r	r	NOUN
ap-7583	160	33	)	)	PUNCT
ap-7583	160	34	.	.	PUNCT
ap-7583	161	1	4	4	NUM
ap-7583	161	2	(	(	PUNCT
ap-7583	161	3	α2	α2	ADJ
ap-7583	161	4	r2	r2	PROPN
ap-7583	161	5	+	+	CCONJ
ap-7583	161	6	α3	α3	ADJ
ap-7583	161	7	r3)u′′	r3)u′′	NOUN
ap-7583	161	8	+	+	CCONJ
ap-7583	161	9	β1	β1	PROPN
ap-7583	161	10	r	r	NOUN
ap-7583	161	11	u′	u′	PROPN
ap-7583	161	12	+	+	CCONJ
ap-7583	161	13	(	(	PUNCT
ap-7583	161	14	ε0	ε0	PROPN
ap-7583	161	15	+	+	CCONJ
ap-7583	161	16	ε1r	ε1r	PROPN
ap-7583	161	17	)	)	PUNCT
ap-7583	161	18	u	u	NOUN
ap-7583	161	19	=	=	SYM
ap-7583	161	20	0	0	NUM
ap-7583	161	21	u	u	NOUN
ap-7583	161	22	=	=	NOUN
ap-7583	161	23	r	r	NOUN
ap-7583	161	24	1	1	NUM
ap-7583	161	25	2	2	NUM
ap-7583	161	26	−	−	NOUN
ap-7583	161	27	β1	β1	NOUN
ap-7583	161	28	2α2	2α2	NUM
ap-7583	162	1	+	+	CCONJ
ap-7583	162	2	1	1	NUM
ap-7583	162	3	2α2	2α2	NUM
ap-7583	162	4	√	√	NUM
ap-7583	162	5	(	(	PUNCT
ap-7583	162	6	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	162	7	2f1	2f1	NUM
ap-7583	162	8	(	(	PUNCT
ap-7583	162	9	1	1	NUM
ap-7583	162	10	2	2	NUM
ap-7583	162	11	√	√	NOUN
ap-7583	162	12	α3−4ε1	α3−4ε1	PROPN
ap-7583	162	13	α3	α3	PROPN
ap-7583	162	14	−	−	PROPN
ap-7583	163	1	β1	β1	PROPN
ap-7583	163	2	2α2	2α2	NUM
ap-7583	164	1	+	+	CCONJ
ap-7583	164	2	√	√	PROPN
ap-7583	164	3	(	(	PUNCT
ap-7583	164	4	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	164	5	2α2	2α2	NUM
ap-7583	164	6	,	,	PUNCT
ap-7583	164	7	−	−	PROPN
ap-7583	164	8	1	1	NUM
ap-7583	164	9	2	2	NUM
ap-7583	164	10	√	√	NOUN
ap-7583	164	11	α3−4ε1	α3−4ε1	PROPN
ap-7583	164	12	α3	α3	PROPN
ap-7583	164	13	−	−	PROPN
ap-7583	164	14	β1	β1	PROPN
ap-7583	164	15	2α2	2α2	NUM
ap-7583	165	1	+	+	CCONJ
ap-7583	165	2	√	√	PROPN
ap-7583	165	3	(	(	PUNCT
ap-7583	165	4	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	165	5	2α2	2α2	NUM
ap-7583	165	6	;	;	PUNCT
ap-7583	166	1	1	1	NUM
ap-7583	166	2	+	+	NUM
ap-7583	166	3	√	√	PROPN
ap-7583	166	4	(	(	PUNCT
ap-7583	166	5	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	166	6	α2	α2	PROPN
ap-7583	166	7	;	;	PUNCT
ap-7583	166	8	−	−	PROPN
ap-7583	166	9	α3	α3	ADJ
ap-7583	166	10	α2	α2	ADJ
ap-7583	166	11	r	r	NOUN
ap-7583	166	12	)	)	PUNCT
ap-7583	166	13	,	,	PUNCT
ap-7583	167	1	u	u	NOUN
ap-7583	167	2	=	=	SYM
ap-7583	167	3	r	r	NOUN
ap-7583	167	4	1	1	NUM
ap-7583	167	5	2	2	NUM
ap-7583	167	6	−	−	PRON
ap-7583	167	7	β1	β1	NOUN
ap-7583	167	8	2α2	2α2	NUM
ap-7583	167	9	−	−	NUM
ap-7583	167	10	1	1	NUM
ap-7583	167	11	2α2	2α2	NUM
ap-7583	167	12	√	√	NUM
ap-7583	167	13	(	(	PUNCT
ap-7583	167	14	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	167	15	2f1	2f1	NUM
ap-7583	167	16	(	(	PUNCT
ap-7583	167	17	1	1	NUM
ap-7583	167	18	2	2	NUM
ap-7583	167	19	√	√	NOUN
ap-7583	167	20	α3−4ε1	α3−4ε1	PROPN
ap-7583	167	21	α3	α3	PROPN
ap-7583	167	22	−	−	PROPN
ap-7583	167	23	β1	β1	PROPN
ap-7583	167	24	2α2	2α2	NUM
ap-7583	168	1	−	−	NOUN
ap-7583	168	2	√	√	PROPN
ap-7583	168	3	(	(	PUNCT
ap-7583	168	4	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	168	5	2α2	2α2	NUM
ap-7583	168	6	,	,	PUNCT
ap-7583	168	7	−	−	PROPN
ap-7583	168	8	1	1	NUM
ap-7583	168	9	2	2	NUM
ap-7583	168	10	√	√	NOUN
ap-7583	168	11	α3−4ε1	α3−4ε1	PROPN
ap-7583	168	12	α3	α3	PROPN
ap-7583	168	13	−	−	PROPN
ap-7583	169	1	β1	β1	PROPN
ap-7583	169	2	2α2	2α2	NUM
ap-7583	170	1	−	−	NOUN
ap-7583	171	1	√	√	PROPN
ap-7583	172	1	(	(	PUNCT
ap-7583	172	2	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	172	3	2α2	2α2	NUM
ap-7583	172	4	;	;	PUNCT
ap-7583	172	5	1	1	NUM
ap-7583	172	6	−	−	NOUN
ap-7583	172	7	√	√	NUM
ap-7583	172	8	(	(	PUNCT
ap-7583	172	9	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	X
ap-7583	172	10	α2	α2	PROPN
ap-7583	172	11	;	;	PUNCT
ap-7583	172	12	−	−	PROPN
ap-7583	172	13	α3	α3	ADJ
ap-7583	172	14	α2	α2	ADJ
ap-7583	172	15	r	r	NOUN
ap-7583	172	16	)	)	PUNCT
ap-7583	172	17	.	.	PUNCT
ap-7583	173	1	5	5	NUM
ap-7583	173	2	(	(	PUNCT
ap-7583	173	3	α2	α2	ADJ
ap-7583	173	4	r2	r2	PROPN
ap-7583	173	5	+	+	CCONJ
ap-7583	173	6	α3	α3	ADJ
ap-7583	173	7	r3)u′′	r3)u′′	NOUN
ap-7583	173	8	+	+	X
ap-7583	173	9	β2r2	β2r2	PUNCT
ap-7583	173	10	u′	u′	X
ap-7583	173	11	+	+	CCONJ
ap-7583	173	12	(	(	PUNCT
ap-7583	173	13	ε0	ε0	PROPN
ap-7583	173	14	+	+	CCONJ
ap-7583	173	15	ε1r	ε1r	PROPN
ap-7583	173	16	)	)	PUNCT
ap-7583	173	17	u	u	NOUN
ap-7583	173	18	=	=	SYM
ap-7583	173	19	0	0	NUM
ap-7583	173	20	u	u	NOUN
ap-7583	173	21	=	=	NOUN
ap-7583	173	22	r	r	NOUN
ap-7583	173	23	1	1	NUM
ap-7583	173	24	2	2	NUM
ap-7583	173	25	−	−	NUM
ap-7583	173	26	1	1	NUM
ap-7583	173	27	2√	2√	NUM
ap-7583	173	28	α2	α2	ADJ
ap-7583	173	29	√	√	PROPN
ap-7583	173	30	α2−4ε0	α2−4ε0	PROPN
ap-7583	173	31	2f1	2f1	NUM
ap-7583	173	32	(	(	PUNCT
ap-7583	173	33	β2	β2	VERB
ap-7583	173	34	2α3	2α3	NUM
ap-7583	173	35	−	−	NUM
ap-7583	173	36	1	1	NUM
ap-7583	173	37	2	2	NUM
ap-7583	173	38	√	√	NOUN
ap-7583	173	39	α2−4ε0	α2−4ε0	NOUN
ap-7583	173	40	α2	α2	PROPN
ap-7583	173	41	+	+	CCONJ
ap-7583	173	42	√	√	PROPN
ap-7583	173	43	(	(	PUNCT
ap-7583	173	44	α3−β2)2−4α3ε1	α3−β2)2−4α3ε1	NUM
ap-7583	173	45	2α3	2α3	NUM
ap-7583	173	46	,	,	PUNCT
ap-7583	173	47	β2	β2	VERB
ap-7583	173	48	2α3	2α3	NUM
ap-7583	173	49	−	−	NUM
ap-7583	173	50	1	1	NUM
ap-7583	173	51	2	2	NUM
ap-7583	173	52	√	√	NOUN
ap-7583	173	53	α2−4ε0	α2−4ε0	NOUN
ap-7583	173	54	α2	α2	PROPN
ap-7583	173	55	−	−	PROPN
ap-7583	174	1	√	√	PROPN
ap-7583	174	2	(	(	PUNCT
ap-7583	174	3	α3−β2)2−4α3ε1	α3−β2)2−4α3ε1	NUM
ap-7583	174	4	2α3	2α3	NUM
ap-7583	174	5	;	;	PUNCT
ap-7583	174	6	1	1	NUM
ap-7583	174	7	−	−	NUM
ap-7583	174	8	√	√	NUM
ap-7583	174	9	α2−4ε0√	α2−4ε0√	NOUN
ap-7583	174	10	α2	α2	PROPN
ap-7583	174	11	;	;	PUNCT
ap-7583	174	12	−	−	PROPN
ap-7583	174	13	α3	α3	ADJ
ap-7583	174	14	α2	α2	ADJ
ap-7583	174	15	r	r	NOUN
ap-7583	174	16	)	)	PUNCT
ap-7583	174	17	,	,	PUNCT
ap-7583	175	1	u	u	NOUN
ap-7583	175	2	=	=	SYM
ap-7583	175	3	r	r	NOUN
ap-7583	175	4	1	1	NUM
ap-7583	175	5	2	2	NUM
ap-7583	175	6	+	+	NUM
ap-7583	175	7	1	1	NUM
ap-7583	175	8	2√	2√	NUM
ap-7583	175	9	α2	α2	ADJ
ap-7583	175	10	√	√	PROPN
ap-7583	175	11	α2−4ε0	α2−4ε0	PROPN
ap-7583	175	12	2f1	2f1	NUM
ap-7583	176	1	(	(	PUNCT
ap-7583	176	2	β2	β2	VERB
ap-7583	176	3	2α3	2α3	NUM
ap-7583	177	1	+	+	CCONJ
ap-7583	177	2	1	1	NUM
ap-7583	177	3	2	2	NUM
ap-7583	177	4	√	√	NOUN
ap-7583	177	5	α2−4ε0	α2−4ε0	NOUN
ap-7583	177	6	α2	α2	PROPN
ap-7583	177	7	−	−	PROPN
ap-7583	178	1	√	√	PROPN
ap-7583	178	2	(	(	PUNCT
ap-7583	178	3	α3−β2)2−4α3ε1	α3−β2)2−4α3ε1	NUM
ap-7583	178	4	2α3	2α3	NUM
ap-7583	178	5	,	,	PUNCT
ap-7583	178	6	β2	β2	VERB
ap-7583	178	7	2α3	2α3	NUM
ap-7583	179	1	+	+	CCONJ
ap-7583	179	2	1	1	NUM
ap-7583	179	3	2	2	NUM
ap-7583	179	4	√	√	NOUN
ap-7583	179	5	α2−4ε0	α2−4ε0	NOUN
ap-7583	179	6	α2	α2	PROPN
ap-7583	179	7	+	+	CCONJ
ap-7583	179	8	√	√	PROPN
ap-7583	179	9	(	(	PUNCT
ap-7583	179	10	α3−β2)2−4α3ε1	α3−β2)2−4α3ε1	NUM
ap-7583	179	11	2α3	2α3	NUM
ap-7583	179	12	;	;	PUNCT
ap-7583	179	13	1	1	NUM
ap-7583	179	14	+	+	CCONJ
ap-7583	179	15	√	√	NOUN
ap-7583	179	16	α2−4ε0√	α2−4ε0√	NOUN
ap-7583	179	17	α2	α2	PROPN
ap-7583	179	18	;	;	PUNCT
ap-7583	179	19	−	−	PROPN
ap-7583	179	20	α3	α3	ADJ
ap-7583	179	21	α2	α2	ADJ
ap-7583	179	22	r	r	NOUN
ap-7583	179	23	)	)	PUNCT
ap-7583	179	24	.	.	PUNCT
ap-7583	180	1	6	6	NUM
ap-7583	180	2	(	(	PUNCT
ap-7583	180	3	α2	α2	ADJ
ap-7583	180	4	r2	r2	PROPN
ap-7583	180	5	+	+	CCONJ
ap-7583	180	6	α3	α3	ADJ
ap-7583	180	7	r3)u′′	r3)u′′	NOUN
ap-7583	180	8	+	+	CCONJ
ap-7583	180	9	(	(	PUNCT
ap-7583	180	10	ε0	ε0	PROPN
ap-7583	180	11	+	+	CCONJ
ap-7583	180	12	ε1	ε1	PROPN
ap-7583	180	13	r	r	NOUN
ap-7583	180	14	)	)	PUNCT
ap-7583	180	15	u	u	NOUN
ap-7583	180	16	=	=	SYM
ap-7583	180	17	0	0	NUM
ap-7583	180	18	u	u	NOUN
ap-7583	180	19	=	=	NOUN
ap-7583	180	20	r	r	NOUN
ap-7583	180	21	1	1	NUM
ap-7583	180	22	2	2	NUM
ap-7583	180	23	−	−	NUM
ap-7583	180	24	1	1	NUM
ap-7583	180	25	2√	2√	NUM
ap-7583	180	26	α2	α2	ADJ
ap-7583	180	27	√	√	PROPN
ap-7583	180	28	α2−4ε0	α2−4ε0	PROPN
ap-7583	180	29	2f1	2f1	NUM
ap-7583	180	30	(	(	PUNCT
ap-7583	180	31	−	−	PROPN
ap-7583	180	32	1	1	NUM
ap-7583	180	33	2	2	NUM
ap-7583	180	34	√	√	NOUN
ap-7583	180	35	α2−4ε0	α2−4ε0	NOUN
ap-7583	180	36	α2	α2	PROPN
ap-7583	180	37	−	−	PROPN
ap-7583	180	38	√	√	PROPN
ap-7583	180	39	α3−4ε1	α3−4ε1	PROPN
ap-7583	180	40	2√	2√	NUM
ap-7583	180	41	α3	α3	NOUN
ap-7583	180	42	,	,	PUNCT
ap-7583	180	43	−	−	PROPN
ap-7583	180	44	1	1	NUM
ap-7583	180	45	2	2	NUM
ap-7583	180	46	√	√	NOUN
ap-7583	180	47	α2−4ε0	α2−4ε0	NOUN
ap-7583	180	48	α2	α2	PROPN
ap-7583	180	49	−	−	PROPN
ap-7583	180	50	√	√	PROPN
ap-7583	180	51	α3−4ε1	α3−4ε1	PROPN
ap-7583	180	52	2√	2√	NUM
ap-7583	180	53	α3	α3	NOUN
ap-7583	180	54	;	;	PUNCT
ap-7583	180	55	1	1	NUM
ap-7583	180	56	−	−	NUM
ap-7583	180	57	√	√	NUM
ap-7583	180	58	α2−4ε0√	α2−4ε0√	NOUN
ap-7583	180	59	α2	α2	PROPN
ap-7583	180	60	;	;	PUNCT
ap-7583	180	61	−	−	PROPN
ap-7583	180	62	α3	α3	ADJ
ap-7583	180	63	α2	α2	ADJ
ap-7583	180	64	r	r	NOUN
ap-7583	180	65	)	)	PUNCT
ap-7583	180	66	,	,	PUNCT
ap-7583	180	67	u	u	NOUN
ap-7583	181	1	=	=	SYM
ap-7583	181	2	r	r	NOUN
ap-7583	181	3	1	1	NUM
ap-7583	181	4	2	2	NUM
ap-7583	181	5	+	+	NUM
ap-7583	181	6	1	1	NUM
ap-7583	181	7	2√	2√	NUM
ap-7583	181	8	α2	α2	ADJ
ap-7583	181	9	√	√	PROPN
ap-7583	182	1	α2−4ε0	α2−4ε0	PROPN
ap-7583	182	2	2f1	2f1	NUM
ap-7583	182	3	(	(	PUNCT
ap-7583	182	4	1	1	NUM
ap-7583	182	5	2	2	NUM
ap-7583	182	6	√	√	NOUN
ap-7583	182	7	α2−4ε0	α2−4ε0	NOUN
ap-7583	182	8	α2	α2	PROPN
ap-7583	182	9	−	−	PROPN
ap-7583	182	10	√	√	PROPN
ap-7583	182	11	α3−4ε1	α3−4ε1	PROPN
ap-7583	182	12	2√	2√	NUM
ap-7583	182	13	α3	α3	NOUN
ap-7583	182	14	,	,	PUNCT
ap-7583	182	15	1	1	NUM
ap-7583	182	16	2	2	NUM
ap-7583	182	17	√	√	NOUN
ap-7583	182	18	α2−4ε0	α2−4ε0	NOUN
ap-7583	182	19	α2	α2	PROPN
ap-7583	182	20	+	+	CCONJ
ap-7583	182	21	√	√	PUNCT
ap-7583	182	22	α3−4ε1	α3−4ε1	PROPN
ap-7583	182	23	2√	2√	NUM
ap-7583	182	24	α3	α3	NOUN
ap-7583	182	25	;	;	PUNCT
ap-7583	182	26	1	1	NUM
ap-7583	182	27	+	+	CCONJ
ap-7583	182	28	√	√	NOUN
ap-7583	182	29	α2−4ε0√	α2−4ε0√	NOUN
ap-7583	182	30	α2	α2	PROPN
ap-7583	182	31	;	;	PUNCT
ap-7583	183	1	−	−	PROPN
ap-7583	183	2	α3	α3	ADJ
ap-7583	183	3	α2	α2	ADJ
ap-7583	183	4	r	r	NOUN
ap-7583	183	5	)	)	PUNCT
ap-7583	183	6	.	.	PUNCT
ap-7583	184	1	7	7	NUM
ap-7583	184	2	α2	α2	ADJ
ap-7583	184	3	r2	r2	PROPN
ap-7583	184	4	u′′	u′′	PROPN
ap-7583	184	5	+	+	CCONJ
ap-7583	184	6	(	(	PUNCT
ap-7583	184	7	ε0	ε0	PROPN
ap-7583	184	8	+	+	CCONJ
ap-7583	184	9	ε1	ε1	PROPN
ap-7583	184	10	r	r	NOUN
ap-7583	184	11	)	)	PUNCT
ap-7583	184	12	u	u	NOUN
ap-7583	184	13	=	=	SYM
ap-7583	184	14	0	0	NUM
ap-7583	184	15	u	u	NOUN
ap-7583	184	16	=	=	NOUN
ap-7583	184	17	r	r	NOUN
ap-7583	184	18	1	1	NUM
ap-7583	184	19	2	2	NUM
ap-7583	184	20	+	+	CCONJ
ap-7583	184	21	1	1	NUM
ap-7583	184	22	2	2	NUM
ap-7583	184	23	√	√	NUM
ap-7583	184	24	1−	1−	NUM
ap-7583	184	25	4ε0	4ε0	NUM
ap-7583	184	26	α2	α2	ADJ
ap-7583	184	27	0f1	0f1	NUM
ap-7583	185	1	(	(	PUNCT
ap-7583	185	2	;	;	PUNCT
ap-7583	185	3	1	1	NUM
ap-7583	185	4	+	+	CCONJ
ap-7583	185	5	√	√	NUM
ap-7583	185	6	1	1	NUM
ap-7583	185	7	+	+	CCONJ
ap-7583	185	8	4ε0	4ε0	NUM
ap-7583	185	9	α2	α2	ADJ
ap-7583	185	10	;	;	PUNCT
ap-7583	185	11	−	−	PROPN
ap-7583	185	12	ε1	ε1	VERB
ap-7583	185	13	α2	α2	ADJ
ap-7583	185	14	r	r	NOUN
ap-7583	185	15	)	)	PUNCT
ap-7583	185	16	,	,	PUNCT
ap-7583	185	17	u	u	NOUN
ap-7583	185	18	=	=	SYM
ap-7583	185	19	r	r	NOUN
ap-7583	185	20	1	1	NUM
ap-7583	185	21	2	2	NUM
ap-7583	185	22	−	−	NOUN
ap-7583	185	23	1	1	NUM
ap-7583	185	24	2	2	NUM
ap-7583	185	25	√	√	NUM
ap-7583	185	26	1−	1−	NUM
ap-7583	185	27	4ε0	4ε0	NUM
ap-7583	186	1	α2	α2	ADJ
ap-7583	186	2	0f1	0f1	NUM
ap-7583	187	1	(	(	PUNCT
ap-7583	187	2	;	;	PUNCT
ap-7583	187	3	1	1	NUM
ap-7583	187	4	−	−	NOUN
ap-7583	187	5	√	√	NOUN
ap-7583	187	6	1	1	NUM
ap-7583	187	7	−	−	NOUN
ap-7583	187	8	4ε0	4ε0	NUM
ap-7583	187	9	α2	α2	ADJ
ap-7583	187	10	;	;	PUNCT
ap-7583	187	11	−	−	PROPN
ap-7583	187	12	ε1	ε1	VERB
ap-7583	187	13	α2	α2	ADJ
ap-7583	187	14	r	r	NOUN
ap-7583	187	15	)	)	PUNCT
ap-7583	187	16	8	8	NUM
ap-7583	187	17	α1	α1	PROPN
ap-7583	187	18	r	r	NOUN
ap-7583	187	19	u′′	u′′	PROPN
ap-7583	187	20	+	+	CCONJ
ap-7583	187	21	(	(	PUNCT
ap-7583	187	22	β0	β0	NOUN
ap-7583	187	23	+	+	CCONJ
ap-7583	187	24	β2r2	β2r2	NOUN
ap-7583	187	25	)	)	PUNCT
ap-7583	187	26	u′	u′	PROPN
ap-7583	188	1	+	+	CCONJ
ap-7583	188	2	ε1	ε1	PROPN
ap-7583	188	3	r	r	NOUN
ap-7583	188	4	u	u	NOUN
ap-7583	188	5	=	=	PROPN
ap-7583	188	6	0	0	NUM
ap-7583	188	7	,	,	PUNCT
ap-7583	188	8	u	u	NOUN
ap-7583	188	9	=	=	PROPN
ap-7583	188	10	1f1	1f1	NUM
ap-7583	188	11	(	(	PUNCT
ap-7583	188	12	ε1	ε1	PROPN
ap-7583	188	13	2β2	2β2	NUM
ap-7583	188	14	;	;	PUNCT
ap-7583	188	15	1	1	NUM
ap-7583	188	16	2	2	NUM
ap-7583	188	17	+	+	CCONJ
ap-7583	188	18	β0	β0	ADJ
ap-7583	188	19	2α1	2α1	NUM
ap-7583	188	20	;	;	PUNCT
ap-7583	188	21	−	−	PROPN
ap-7583	188	22	β2	β2	VERB
ap-7583	188	23	2α1	2α1	NUM
ap-7583	188	24	r2	r2	PROPN
ap-7583	188	25	)	)	PUNCT
ap-7583	188	26	,	,	PUNCT
ap-7583	188	27	u	u	NOUN
ap-7583	188	28	=	=	PUNCT
ap-7583	188	29	r1−	r1−	PROPN
ap-7583	188	30	β0	β0	PROPN
ap-7583	188	31	α1	α1	PROPN
ap-7583	188	32	1f1	1f1	NUM
ap-7583	188	33	(	(	PUNCT
ap-7583	188	34	1	1	NUM
ap-7583	188	35	2	2	NUM
ap-7583	188	36	−	−	NOUN
ap-7583	188	37	β0	β0	ADJ
ap-7583	188	38	2α1	2α1	NUM
ap-7583	188	39	+	+	CCONJ
ap-7583	188	40	ε1	ε1	VERB
ap-7583	188	41	2β2	2β2	NUM
ap-7583	188	42	;	;	PUNCT
ap-7583	188	43	3	3	NUM
ap-7583	188	44	2	2	NUM
ap-7583	188	45	−	−	NOUN
ap-7583	188	46	β0	β0	ADJ
ap-7583	188	47	2α1	2α1	NUM
ap-7583	188	48	;	;	PUNCT
ap-7583	188	49	−	−	PROPN
ap-7583	188	50	β2	β2	VERB
ap-7583	188	51	2α1	2α1	NUM
ap-7583	188	52	r2	r2	PROPN
ap-7583	188	53	)	)	PUNCT
ap-7583	188	54	.	.	PUNCT
ap-7583	189	1	9	9	NUM
ap-7583	189	2	α1	α1	PROPN
ap-7583	189	3	r	r	NOUN
ap-7583	189	4	u′′	u′′	PROPN
ap-7583	189	5	+	+	CCONJ
ap-7583	189	6	β0	β0	PROPN
ap-7583	189	7	u′	u′	PROPN
ap-7583	189	8	+	+	CCONJ
ap-7583	189	9	ε1	ε1	PROPN
ap-7583	189	10	r	r	NOUN
ap-7583	189	11	u	u	NOUN
ap-7583	189	12	=	=	PROPN
ap-7583	189	13	0	0	NUM
ap-7583	189	14	,	,	PUNCT
ap-7583	189	15	u	u	NOUN
ap-7583	189	16	=	=	PROPN
ap-7583	189	17	0f1	0f1	NUM
ap-7583	189	18	(	(	PUNCT
ap-7583	189	19	−	−	NOUN
ap-7583	189	20	;	;	PUNCT
ap-7583	189	21	1	1	NUM
ap-7583	189	22	2	2	NUM
ap-7583	189	23	+	+	CCONJ
ap-7583	189	24	β0	β0	ADJ
ap-7583	189	25	2α1	2α1	NUM
ap-7583	189	26	;	;	PUNCT
ap-7583	189	27	−	−	PROPN
ap-7583	189	28	ε1	ε1	PROPN
ap-7583	189	29	4α1	4α1	PROPN
ap-7583	189	30	z2	z2	PROPN
ap-7583	189	31	)	)	PUNCT
ap-7583	189	32	,	,	PUNCT
ap-7583	189	33	u	u	NOUN
ap-7583	189	34	=	=	PUNCT
ap-7583	189	35	r1−	r1−	PROPN
ap-7583	189	36	β0	β0	PROPN
ap-7583	189	37	α1	α1	PROPN
ap-7583	189	38	0f1	0f1	NUM
ap-7583	190	1	(	(	PUNCT
ap-7583	190	2	−	−	NOUN
ap-7583	190	3	;	;	PUNCT
ap-7583	190	4	3	3	NUM
ap-7583	190	5	2	2	NUM
ap-7583	190	6	−	−	NOUN
ap-7583	190	7	β0	β0	ADJ
ap-7583	190	8	2α1	2α1	NUM
ap-7583	190	9	;	;	PUNCT
ap-7583	190	10	−	−	PROPN
ap-7583	190	11	ε1	ε1	VERB
ap-7583	190	12	4α1	4α1	NUM
ap-7583	190	13	r2	r2	PROPN
ap-7583	190	14	)	)	PUNCT
ap-7583	190	15	.	.	PUNCT
ap-7583	191	1	10	10	NUM
ap-7583	191	2	α0	α0	ADJ
ap-7583	191	3	u′′	u′′	PROPN
ap-7583	191	4	+	+	CCONJ
ap-7583	191	5	β2	β2	NOUN
ap-7583	191	6	r2u′	r2u′	VERB
ap-7583	191	7	+	+	CCONJ
ap-7583	191	8	ε1	ε1	PROPN
ap-7583	191	9	r	r	NOUN
ap-7583	191	10	u	u	NOUN
ap-7583	191	11	=	=	PROPN
ap-7583	191	12	0	0	NUM
ap-7583	191	13	,	,	PUNCT
ap-7583	191	14	u	u	NOUN
ap-7583	191	15	=	=	PROPN
ap-7583	191	16	1f1	1f1	NUM
ap-7583	191	17	(	(	PUNCT
ap-7583	191	18	ε1	ε1	PROPN
ap-7583	191	19	3β2	3β2	NUM
ap-7583	191	20	;	;	PUNCT
ap-7583	191	21	2	2	NUM
ap-7583	191	22	3	3	NUM
ap-7583	191	23	;	;	PUNCT
ap-7583	191	24	−	−	PROPN
ap-7583	191	25	β2	β2	VERB
ap-7583	191	26	3α0	3α0	NUM
ap-7583	191	27	r3	r3	PROPN
ap-7583	191	28	)	)	PUNCT
ap-7583	191	29	,	,	PUNCT
ap-7583	191	30	u	u	NOUN
ap-7583	192	1	=	=	SYM
ap-7583	192	2	r	r	NOUN
ap-7583	192	3	1f1	1f1	NUM
ap-7583	192	4	(	(	PUNCT
ap-7583	192	5	1	1	NUM
ap-7583	192	6	3	3	NUM
ap-7583	192	7	+	+	CCONJ
ap-7583	192	8	ε1	ε1	VERB
ap-7583	192	9	3β2	3β2	NUM
ap-7583	192	10	;	;	PUNCT
ap-7583	192	11	4	4	NUM
ap-7583	192	12	3	3	NUM
ap-7583	192	13	;	;	PUNCT
ap-7583	192	14	−	−	PROPN
ap-7583	192	15	β2	β2	VERB
ap-7583	192	16	3α0	3α0	NUM
ap-7583	192	17	r3	r3	PROPN
ap-7583	192	18	)	)	PUNCT
ap-7583	192	19	.	.	PUNCT
ap-7583	193	1	table	table	NOUN
ap-7583	193	2	1	1	NUM
ap-7583	193	3	.	.	X
ap-7583	193	4	ten	ten	NUM
ap-7583	193	5	solvable	solvable	ADJ
ap-7583	193	6	equations	equation	NOUN
ap-7583	193	7	of	of	ADP
ap-7583	193	8	the	the	DET
ap-7583	193	9	type	type	NOUN
ap-7583	193	10	(	(	PUNCT
ap-7583	193	11	4	4	NUM
ap-7583	193	12	)	)	PUNCT
ap-7583	193	13	that	that	PRON
ap-7583	193	14	follows	follow	VERB
ap-7583	193	15	from	from	ADP
ap-7583	193	16	the	the	DET
ap-7583	193	17	generic	generic	ADJ
ap-7583	193	18	equations	equation	NOUN
ap-7583	193	19	(	(	PUNCT
ap-7583	193	20	14	14	NUM
ap-7583	193	21	)	)	PUNCT
ap-7583	193	22	,	,	PUNCT
ap-7583	193	23	(	(	PUNCT
ap-7583	193	24	17	17	NUM
ap-7583	193	25	)	)	PUNCT
ap-7583	193	26	,	,	PUNCT
ap-7583	193	27	and	and	CCONJ
ap-7583	193	28	(	(	PUNCT
ap-7583	193	29	21	21	NUM
ap-7583	193	30	)	)	PUNCT
ap-7583	193	31	.	.	PUNCT
ap-7583	194	1	170	170	NUM
ap-7583	194	2	vol	vol	NOUN
ap-7583	194	3	.	.	PUNCT
ap-7583	195	1	62	62	NUM
ap-7583	195	2	no	no	INTJ
ap-7583	195	3	.	.	PUNCT
ap-7583	196	1	1/2022	1/2022	NUM
ap-7583	196	2	on	on	ADP
ap-7583	196	3	generalized	generalized	ADJ
ap-7583	196	4	heun	heun	NOUN
ap-7583	196	5	equation	equation	NOUN
ap-7583	196	6	with	with	ADP
ap-7583	196	7	some	some	DET
ap-7583	196	8	mathematical	mathematical	NOUN
ap-7583	196	9	.	.	PUNCT
ap-7583	196	10	.	.	PUNCT
ap-7583	196	11	.	.	PUNCT
ap-7583	197	1	3	3	X
ap-7583	197	2	.	.	X
ap-7583	198	1	the	the	DET
ap-7583	198	2	solutions	solution	NOUN
ap-7583	198	3	in	in	ADP
ap-7583	198	4	the	the	DET
ap-7583	198	5	neighbourhood	neighbourhood	NOUN
ap-7583	198	6	of	of	ADP
ap-7583	198	7	an	an	DET
ap-7583	198	8	ordinary	ordinary	ADJ
ap-7583	198	9	point	point	NOUN
ap-7583	198	10	3.1	3.1	NUM
ap-7583	198	11	.	.	PUNCT
ap-7583	199	1	series	series	PROPN
ap-7583	199	2	solutions	solution	NOUN
ap-7583	199	3	in	in	ADP
ap-7583	199	4	the	the	DET
ap-7583	199	5	case	case	NOUN
ap-7583	199	6	of	of	ADP
ap-7583	199	7	α0	α0	ADJ
ap-7583	199	8	̸=	̸=	PROPN
ap-7583	199	9	0	0	NUM
ap-7583	199	10	,	,	PUNCT
ap-7583	199	11	r	r	NOUN
ap-7583	199	12	=	=	SYM
ap-7583	199	13	0	0	NUM
ap-7583	199	14	,	,	PUNCT
ap-7583	199	15	there	there	PRON
ap-7583	199	16	is	be	VERB
ap-7583	199	17	an	an	DET
ap-7583	199	18	ordinary	ordinary	ADJ
ap-7583	199	19	point	point	NOUN
ap-7583	199	20	for	for	ADP
ap-7583	199	21	the	the	DET
ap-7583	199	22	differential	differential	ADJ
ap-7583	199	23	equations	equation	NOUN
ap-7583	199	24	(	(	PUNCT
ap-7583	199	25	4	4	NUM
ap-7583	199	26	)	)	PUNCT
ap-7583	199	27	.	.	PUNCT
ap-7583	200	1	the	the	DET
ap-7583	200	2	classical	classical	ADJ
ap-7583	200	3	theory	theory	NOUN
ap-7583	200	4	of	of	ADP
ap-7583	200	5	differential	differential	ADJ
ap-7583	200	6	equation	equation	NOUN
ap-7583	200	7	ensure	ensure	VERB
ap-7583	200	8	that	that	SCONJ
ap-7583	200	9	the	the	DET
ap-7583	200	10	(	(	PUNCT
ap-7583	200	11	4	4	NUM
ap-7583	200	12	)	)	PUNCT
ap-7583	200	13	has	have	VERB
ap-7583	200	14	two	two	NUM
ap-7583	200	15	linearly	linearly	ADV
ap-7583	200	16	independent	independent	ADJ
ap-7583	200	17	power	power	NOUN
ap-7583	200	18	series	series	NOUN
ap-7583	200	19	solutions	solution	NOUN
ap-7583	200	20	in	in	ADP
ap-7583	200	21	the	the	DET
ap-7583	200	22	neighbourhood	neighbourhood	NOUN
ap-7583	200	23	of	of	ADP
ap-7583	200	24	r	r	NOUN
ap-7583	200	25	=	=	SYM
ap-7583	200	26	0	0	NUM
ap-7583	200	27	and	and	CCONJ
ap-7583	200	28	valid	valid	ADJ
ap-7583	200	29	to	to	ADP
ap-7583	200	30	the	the	DET
ap-7583	200	31	nearest	near	ADJ
ap-7583	200	32	real	real	ADJ
ap-7583	200	33	singular	singular	ADJ
ap-7583	200	34	point	point	NOUN
ap-7583	200	35	of	of	ADP
ap-7583	200	36	the	the	DET
ap-7583	200	37	leading	lead	VERB
ap-7583	200	38	polynomial	polynomial	ADJ
ap-7583	200	39	coefficient	coefficient	NOUN
ap-7583	200	40	l	l	PROPN
ap-7583	200	41	≡	≡	PROPN
ap-7583	201	1	α0	α0	PROPN
ap-7583	201	2	+	+	CCONJ
ap-7583	201	3	α1	α1	PROPN
ap-7583	201	4	r	r	NOUN
ap-7583	201	5	+	+	CCONJ
ap-7583	201	6	α2	α2	ADJ
ap-7583	201	7	r2	r2	PROPN
ap-7583	201	8	+	+	CCONJ
ap-7583	201	9	α3	α3	PROPN
ap-7583	201	10	r3	r3	PROPN
ap-7583	201	11	=	=	SYM
ap-7583	201	12	0	0	X
ap-7583	201	13	.	.	PUNCT
ap-7583	202	1	indeed	indeed	ADV
ap-7583	202	2	,	,	PUNCT
ap-7583	202	3	the	the	DET
ap-7583	202	4	polynomial	polynomial	ADJ
ap-7583	202	5	l	l	NOUN
ap-7583	202	6	=	=	SYM
ap-7583	202	7	0	0	PUNCT
ap-7583	202	8	has	have	VERB
ap-7583	202	9	the	the	DET
ap-7583	202	10	discriminate	discriminate	NOUN
ap-7583	202	11	[	[	X
ap-7583	202	12	23	23	NUM
ap-7583	202	13	]	]	X
ap-7583	202	14	:	:	PUNCT
ap-7583	202	15	∆	∆	PROPN
ap-7583	202	16	=	=	SYM
ap-7583	202	17	18	18	NUM
ap-7583	202	18	α3	α3	ADJ
ap-7583	202	19	α2	α2	ADJ
ap-7583	202	20	α1	α1	PROPN
ap-7583	202	21	α0	α0	ADJ
ap-7583	202	22	−	−	NUM
ap-7583	202	23	4	4	NUM
ap-7583	202	24	α3	α3	NOUN
ap-7583	202	25	2	2	NUM
ap-7583	202	26	α0	α0	ADJ
ap-7583	202	27	+	+	CCONJ
ap-7583	202	28	α2	α2	ADJ
ap-7583	202	29	2	2	NUM
ap-7583	202	30	α2	α2	ADJ
ap-7583	202	31	1	1	NUM
ap-7583	202	32	−	−	NOUN
ap-7583	202	33	4	4	NUM
ap-7583	202	34	α3	α3	NOUN
ap-7583	202	35	α3	α3	ADJ
ap-7583	202	36	1	1	NUM
ap-7583	202	37	−	−	PROPN
ap-7583	202	38	27	27	NUM
ap-7583	202	39	α2	α2	NOUN
ap-7583	202	40	3	3	NUM
ap-7583	202	41	α2	α2	NOUN
ap-7583	202	42	0	0	NUM
ap-7583	202	43	.	.	PUNCT
ap-7583	203	1	(	(	PUNCT
ap-7583	203	2	25	25	NUM
ap-7583	203	3	)	)	PUNCT
ap-7583	203	4	the	the	DET
ap-7583	203	5	nature	nature	NOUN
ap-7583	203	6	of	of	ADP
ap-7583	203	7	the	the	DET
ap-7583	203	8	l	l	NOUN
ap-7583	203	9	roots	root	NOUN
ap-7583	203	10	as	as	SCONJ
ap-7583	203	11	given	give	VERB
ap-7583	203	12	by	by	ADP
ap-7583	203	13	(	(	PUNCT
ap-7583	203	14	25	25	NUM
ap-7583	203	15	)	)	PUNCT
ap-7583	203	16	along	along	ADP
ap-7583	203	17	with	with	ADP
ap-7583	203	18	the	the	DET
ap-7583	203	19	corresponding	correspond	VERB
ap-7583	203	20	eight	eight	NUM
ap-7583	203	21	differential	differential	ADJ
ap-7583	203	22	equations	equation	NOUN
ap-7583	203	23	are	be	AUX
ap-7583	203	24	summarized	summarize	VERB
ap-7583	203	25	in	in	ADP
ap-7583	203	26	table	table	NOUN
ap-7583	203	27	2	2	NUM
ap-7583	203	28	.	.	PUNCT
ap-7583	204	1	for	for	ADP
ap-7583	204	2	these	these	DET
ap-7583	204	3	differential	differential	ADJ
ap-7583	204	4	equations	equation	NOUN
ap-7583	204	5	,	,	PUNCT
ap-7583	204	6	the	the	DET
ap-7583	204	7	following	follow	VERB
ap-7583	204	8	theorem	theorem	VERB
ap-7583	204	9	,	,	PUNCT
ap-7583	204	10	that	that	PRON
ap-7583	204	11	can	can	AUX
ap-7583	204	12	be	be	AUX
ap-7583	204	13	easily	easily	ADV
ap-7583	204	14	proved	prove	VERB
ap-7583	204	15	using	use	VERB
ap-7583	204	16	frobenius	frobenius	ADJ
ap-7583	204	17	method	method	NOUN
ap-7583	204	18	,	,	PUNCT
ap-7583	204	19	holds	hold	VERB
ap-7583	204	20	.	.	PUNCT
ap-7583	205	1	theorem	theorem	VERB
ap-7583	205	2	3.1	3.1	NUM
ap-7583	205	3	.	.	PUNCT
ap-7583	206	1	(	(	PUNCT
ap-7583	206	2	formal	formal	ADJ
ap-7583	206	3	series	series	NOUN
ap-7583	206	4	solutions	solution	NOUN
ap-7583	206	5	)	)	PUNCT
ap-7583	206	6	in	in	ADP
ap-7583	206	7	the	the	DET
ap-7583	206	8	neighbourhood	neighbourhood	NOUN
ap-7583	206	9	of	of	ADP
ap-7583	206	10	the	the	DET
ap-7583	206	11	ordinary	ordinary	ADJ
ap-7583	206	12	point	point	NOUN
ap-7583	206	13	r	r	NOUN
ap-7583	206	14	=	=	SYM
ap-7583	206	15	0	0	NUM
ap-7583	206	16	,	,	PUNCT
ap-7583	206	17	the	the	DET
ap-7583	206	18	coefficients	coefficient	NOUN
ap-7583	206	19	of	of	ADP
ap-7583	206	20	the	the	DET
ap-7583	206	21	series	series	NOUN
ap-7583	206	22	solution	solution	NOUN
ap-7583	206	23	y(r	y(r	NOUN
ap-7583	206	24	)	)	PUNCT
ap-7583	206	25	=	=	NOUN
ap-7583	207	1	∑∞	∑∞	NOUN
ap-7583	207	2	k=0	k=0	PROPN
ap-7583	207	3	ck	ck	PROPN
ap-7583	207	4	rk	rk	NOUN
ap-7583	207	5	to	to	ADP
ap-7583	207	6	the	the	DET
ap-7583	207	7	differential	differential	ADJ
ap-7583	207	8	equation	equation	NOUN
ap-7583	207	9	(	(	PUNCT
ap-7583	207	10	4	4	X
ap-7583	207	11	)	)	PUNCT
ap-7583	207	12	satisfy	satisfy	VERB
ap-7583	207	13	the	the	DET
ap-7583	207	14	four	four	NUM
ap-7583	207	15	-	-	PUNCT
ap-7583	207	16	term	term	NOUN
ap-7583	207	17	recurrence	recurrence	NOUN
ap-7583	207	18	relation	relation	NOUN
ap-7583	207	19	(	(	PUNCT
ap-7583	207	20	(	(	PUNCT
ap-7583	207	21	k	k	NOUN
ap-7583	207	22	−	−	PROPN
ap-7583	207	23	1	1	NUM
ap-7583	207	24	)	)	PUNCT
ap-7583	207	25	(	(	PUNCT
ap-7583	207	26	(	(	PUNCT
ap-7583	207	27	k	k	X
ap-7583	207	28	−	−	PROPN
ap-7583	207	29	2	2	NUM
ap-7583	207	30	)	)	PUNCT
ap-7583	207	31	α3	α3	NOUN
ap-7583	207	32	+	+	CCONJ
ap-7583	207	33	β2	β2	PROPN
ap-7583	207	34	)	)	PUNCT
ap-7583	207	35	+	+	SYM
ap-7583	208	1	ε1	ε1	PROPN
ap-7583	208	2	)	)	PUNCT
ap-7583	208	3	ck−1	ck−1	NOUN
ap-7583	208	4	+	+	CCONJ
ap-7583	208	5	(	(	PUNCT
ap-7583	208	6	k((k	k((k	NOUN
ap-7583	208	7	−	−	PROPN
ap-7583	208	8	1	1	X
ap-7583	208	9	)	)	PUNCT
ap-7583	208	10	α2	α2	PROPN
ap-7583	208	11	+	+	CCONJ
ap-7583	208	12	β1	β1	PROPN
ap-7583	208	13	)	)	PUNCT
ap-7583	209	1	+	+	CCONJ
ap-7583	209	2	ε0	ε0	PROPN
ap-7583	209	3	)	)	PUNCT
ap-7583	209	4	ck	ck	PROPN
ap-7583	210	1	+	+	CCONJ
ap-7583	210	2	(	(	PUNCT
ap-7583	210	3	k	k	X
ap-7583	210	4	+	+	NUM
ap-7583	210	5	1)(kα1	1)(kα1	NUM
ap-7583	210	6	+	+	CCONJ
ap-7583	210	7	β0	β0	NOUN
ap-7583	210	8	)	)	PUNCT
ap-7583	210	9	ck+1	ck+1	PUNCT
ap-7583	211	1	+	+	CCONJ
ap-7583	211	2	(	(	PUNCT
ap-7583	211	3	k	k	X
ap-7583	211	4	+	+	PROPN
ap-7583	211	5	2	2	NUM
ap-7583	211	6	)	)	PUNCT
ap-7583	211	7	(	(	PUNCT
ap-7583	211	8	k	k	X
ap-7583	212	1	+	+	PROPN
ap-7583	212	2	1	1	X
ap-7583	212	3	)	)	PUNCT
ap-7583	212	4	α0	α0	ADJ
ap-7583	212	5	ck+2	ck+2	NOUN
ap-7583	212	6	=	=	SYM
ap-7583	212	7	0	0	NUM
ap-7583	212	8	,	,	PUNCT
ap-7583	212	9	(	(	PUNCT
ap-7583	212	10	26	26	NUM
ap-7583	212	11	)	)	PUNCT
ap-7583	212	12	where	where	SCONJ
ap-7583	212	13	k	k	NOUN
ap-7583	212	14	=	=	SYM
ap-7583	212	15	0	0	NUM
ap-7583	212	16	,	,	PUNCT
ap-7583	212	17	1	1	NUM
ap-7583	212	18	,	,	PUNCT
ap-7583	212	19	2	2	NUM
ap-7583	212	20	,	,	PUNCT
ap-7583	212	21	·	·	PUNCT
ap-7583	212	22	·	·	PUNCT
ap-7583	212	23	·	·	PUNCT
ap-7583	212	24	,	,	PUNCT
ap-7583	212	25	with	with	ADP
ap-7583	212	26	c−1	c−1	PROPN
ap-7583	212	27	=	=	SYM
ap-7583	212	28	0	0	NUM
ap-7583	212	29	and	and	CCONJ
ap-7583	212	30	arbitrary	arbitrary	ADJ
ap-7583	212	31	nonzero	nonzero	PROPN
ap-7583	212	32	constants	constant	NOUN
ap-7583	212	33	c0	c0	PROPN
ap-7583	212	34	and	and	CCONJ
ap-7583	212	35	c1	c1	PROPN
ap-7583	212	36	.	.	PUNCT
ap-7583	213	1	the	the	DET
ap-7583	213	2	radius	radius	NOUN
ap-7583	213	3	of	of	ADP
ap-7583	213	4	convergence	convergence	NOUN
ap-7583	213	5	of	of	ADP
ap-7583	213	6	these	these	DET
ap-7583	213	7	series	series	NOUN
ap-7583	213	8	solutions	solution	NOUN
ap-7583	213	9	is	be	AUX
ap-7583	213	10	extended	extend	VERB
ap-7583	213	11	from	from	ADP
ap-7583	213	12	r	r	NOUN
ap-7583	213	13	=	=	SYM
ap-7583	213	14	0	0	NUM
ap-7583	213	15	to	to	ADP
ap-7583	213	16	the	the	DET
ap-7583	213	17	nearest	near	ADJ
ap-7583	213	18	singular	singular	ADJ
ap-7583	213	19	point	point	NOUN
ap-7583	213	20	of	of	ADP
ap-7583	213	21	the	the	DET
ap-7583	213	22	leading	lead	VERB
ap-7583	213	23	polynomial	polynomial	ADJ
ap-7583	213	24	coefficient	coefficient	NOUN
ap-7583	213	25	l	l	NOUN
ap-7583	213	26	=	=	SYM
ap-7583	213	27	0	0	X
ap-7583	213	28	.	.	PUNCT
ap-7583	214	1	the	the	DET
ap-7583	214	2	first	first	ADJ
ap-7583	214	3	few	few	ADJ
ap-7583	214	4	terms	term	NOUN
ap-7583	214	5	of	of	ADP
ap-7583	214	6	the	the	DET
ap-7583	214	7	series	series	NOUN
ap-7583	214	8	solution	solution	NOUN
ap-7583	214	9	are	be	AUX
ap-7583	214	10	given	give	VERB
ap-7583	214	11	explicitly	explicitly	ADV
ap-7583	214	12	by	by	ADP
ap-7583	214	13	c2	c2	PROPN
ap-7583	214	14	=	=	PUNCT
ap-7583	214	15	−	−	PROPN
ap-7583	214	16	ε0	ε0	PROPN
ap-7583	214	17	2α0	2α0	NUM
ap-7583	214	18	c0	c0	NOUN
ap-7583	214	19	−	−	PROPN
ap-7583	214	20	β0	β0	PROPN
ap-7583	214	21	2α0	2α0	NUM
ap-7583	214	22	c1	c1	NOUN
ap-7583	214	23	,	,	PUNCT
ap-7583	214	24	c3	c3	PROPN
ap-7583	214	25	=	=	SYM
ap-7583	214	26	(	(	PUNCT
ap-7583	214	27	α1+β0	α1+β0	NOUN
ap-7583	214	28	)	)	PUNCT
ap-7583	214	29	ε0−α0	ε0−α0	NOUN
ap-7583	214	30	ε1	ε1	VERB
ap-7583	214	31	6α2	6α2	NUM
ap-7583	214	32	0	0	NUM
ap-7583	214	33	c0	c0	NOUN
ap-7583	214	34	+	+	CCONJ
ap-7583	214	35	β0(α1+β0)−α0(β1+ε0	β0(α1+β0)−α0(β1+ε0	NOUN
ap-7583	214	36	)	)	PUNCT
ap-7583	214	37	6	6	NUM
ap-7583	214	38	α2	α2	ADJ
ap-7583	214	39	0	0	NUM
ap-7583	214	40	c1	c1	PROPN
ap-7583	214	41	,	,	PUNCT
ap-7583	214	42	·	·	PUNCT
ap-7583	214	43	·	·	PUNCT
ap-7583	214	44	·	·	PUNCT
ap-7583	214	45	.	.	PUNCT
ap-7583	215	1	for	for	ADP
ap-7583	215	2	α0	α0	ADJ
ap-7583	215	3	̸=	̸=	PROPN
ap-7583	215	4	0	0	NUM
ap-7583	215	5	,	,	PUNCT
ap-7583	215	6	using	use	VERB
ap-7583	215	7	(	(	PUNCT
ap-7583	215	8	26	26	NUM
ap-7583	215	9	)	)	PUNCT
ap-7583	215	10	,	,	PUNCT
ap-7583	215	11	we	we	PRON
ap-7583	215	12	can	can	AUX
ap-7583	215	13	extract	extract	VERB
ap-7583	215	14	the	the	DET
ap-7583	215	15	following	follow	VERB
ap-7583	215	16	differential	differential	ADJ
ap-7583	215	17	equations	equation	NOUN
ap-7583	215	18	with	with	ADP
ap-7583	215	19	series	series	NOUN
ap-7583	215	20	solution	solution	NOUN
ap-7583	215	21	from	from	ADP
ap-7583	215	22	(	(	PUNCT
ap-7583	215	23	4	4	X
ap-7583	215	24	)	)	PUNCT
ap-7583	215	25	using	use	VERB
ap-7583	215	26	a	a	DET
ap-7583	215	27	three	three	NUM
ap-7583	215	28	-	-	PUNCT
ap-7583	215	29	term	term	NOUN
ap-7583	215	30	recurrence	recurrence	NOUN
ap-7583	215	31	relation	relation	NOUN
ap-7583	215	32	:	:	PUNCT
ap-7583	215	33	•	•	NUM
ap-7583	215	34	differential	differential	NOUN
ap-7583	215	35	equation	equation	NOUN
ap-7583	215	36	:	:	PUNCT
ap-7583	215	37	(	(	PUNCT
ap-7583	215	38	α0	α0	ADJ
ap-7583	215	39	+	+	NUM
ap-7583	215	40	α1	α1	PROPN
ap-7583	215	41	r	r	NOUN
ap-7583	215	42	+	+	CCONJ
ap-7583	215	43	α3	α3	NOUN
ap-7583	215	44	r3)u′′(r	r3)u′′(r	NOUN
ap-7583	215	45	)	)	PUNCT
ap-7583	216	1	+	+	CCONJ
ap-7583	216	2	(	(	PUNCT
ap-7583	216	3	β0	β0	NOUN
ap-7583	216	4	+	+	CCONJ
ap-7583	216	5	β2	β2	NOUN
ap-7583	216	6	r2)u′(r	r2)u′(r	NOUN
ap-7583	216	7	)	)	PUNCT
ap-7583	217	1	+	+	NUM
ap-7583	218	1	ε1	ε1	VERB
ap-7583	218	2	r	r	NOUN
ap-7583	218	3	u(r	u(r	NOUN
ap-7583	218	4	)	)	PUNCT
ap-7583	219	1	=	=	SYM
ap-7583	219	2	0	0	X
ap-7583	219	3	.	.	PUNCT
ap-7583	220	1	(	(	PUNCT
ap-7583	220	2	27	27	NUM
ap-7583	220	3	)	)	PUNCT
ap-7583	220	4	recurrence	recurrence	NOUN
ap-7583	220	5	formula	formula	NOUN
ap-7583	220	6	:	:	PUNCT
ap-7583	220	7	ck+2	ck+2	NUM
ap-7583	220	8	=	=	SYM
ap-7583	220	9	−	−	PROPN
ap-7583	220	10	(	(	PUNCT
ap-7583	220	11	k	k	PROPN
ap-7583	220	12	+	+	PROPN
ap-7583	220	13	1)(k	1)(k	NUM
ap-7583	220	14	α1	α1	PROPN
ap-7583	220	15	+	+	CCONJ
ap-7583	220	16	β0	β0	NOUN
ap-7583	220	17	)	)	PUNCT
ap-7583	220	18	(	(	PUNCT
ap-7583	220	19	k	k	NOUN
ap-7583	221	1	+	+	PROPN
ap-7583	221	2	1	1	X
ap-7583	221	3	)	)	PUNCT
ap-7583	221	4	(	(	PUNCT
ap-7583	221	5	k	k	X
ap-7583	221	6	+	+	PROPN
ap-7583	221	7	2	2	X
ap-7583	221	8	)	)	PUNCT
ap-7583	221	9	α0	α0	ADJ
ap-7583	221	10	ck+1	ck+1	NUM
ap-7583	221	11	−	−	PROPN
ap-7583	221	12	(	(	PUNCT
ap-7583	221	13	k	k	NOUN
ap-7583	221	14	−	−	PROPN
ap-7583	221	15	1)((k	1)((k	NOUN
ap-7583	221	16	−	−	NOUN
ap-7583	221	17	2	2	NUM
ap-7583	221	18	)	)	PUNCT
ap-7583	221	19	α3	α3	NOUN
ap-7583	221	20	+	+	CCONJ
ap-7583	221	21	β2	β2	VERB
ap-7583	221	22	)	)	PUNCT
ap-7583	221	23	+	+	CCONJ
ap-7583	222	1	ε1	ε1	PROPN
ap-7583	222	2	(	(	PUNCT
ap-7583	222	3	k	k	PROPN
ap-7583	222	4	+	+	PROPN
ap-7583	222	5	1	1	NUM
ap-7583	222	6	)	)	PUNCT
ap-7583	222	7	(	(	PUNCT
ap-7583	222	8	k	k	X
ap-7583	222	9	+	+	PROPN
ap-7583	222	10	2	2	X
ap-7583	222	11	)	)	PUNCT
ap-7583	222	12	α0	α0	ADJ
ap-7583	222	13	ck−1	ck−1	NOUN
ap-7583	222	14	.	.	PUNCT
ap-7583	223	1	(	(	PUNCT
ap-7583	223	2	28	28	NUM
ap-7583	223	3	)	)	PUNCT
ap-7583	223	4	•	•	NUM
ap-7583	223	5	differential	differential	NOUN
ap-7583	223	6	equation	equation	NOUN
ap-7583	223	7	:	:	PUNCT
ap-7583	223	8	(	(	PUNCT
ap-7583	223	9	α0	α0	ADJ
ap-7583	223	10	+	+	CCONJ
ap-7583	223	11	α2r2	α2r2	NUM
ap-7583	223	12	+	+	SYM
ap-7583	223	13	α3r3	α3r3	NOUN
ap-7583	223	14	)	)	PUNCT
ap-7583	223	15	u′′(r	u′′(r	NOUN
ap-7583	223	16	)	)	PUNCT
ap-7583	223	17	+	+	CCONJ
ap-7583	223	18	(	(	PUNCT
ap-7583	223	19	β1	β1	PROPN
ap-7583	223	20	r	r	NOUN
ap-7583	223	21	+	+	CCONJ
ap-7583	223	22	β2	β2	NOUN
ap-7583	223	23	r2)u′(r	r2)u′(r	NOUN
ap-7583	223	24	)	)	PUNCT
ap-7583	223	25	+	+	NUM
ap-7583	224	1	ε1	ε1	VERB
ap-7583	224	2	r	r	NOUN
ap-7583	224	3	u(r	u(r	NOUN
ap-7583	224	4	)	)	PUNCT
ap-7583	225	1	=	=	SYM
ap-7583	225	2	0	0	X
ap-7583	225	3	.	.	PUNCT
ap-7583	226	1	(	(	PUNCT
ap-7583	226	2	29	29	NUM
ap-7583	226	3	)	)	PUNCT
ap-7583	226	4	recurrence	recurrence	NOUN
ap-7583	226	5	formula	formula	NOUN
ap-7583	226	6	:	:	PUNCT
ap-7583	226	7	ck+2	ck+2	NUM
ap-7583	226	8	=	=	SYM
ap-7583	226	9	−k(k	−k(k	NOUN
ap-7583	226	10	−	−	PROPN
ap-7583	227	1	1)α2	1)α2	PRON
ap-7583	227	2	+	+	CCONJ
ap-7583	227	3	k	k	PROPN
ap-7583	227	4	β1	β1	PROPN
ap-7583	227	5	(	(	PUNCT
ap-7583	227	6	k	k	PROPN
ap-7583	227	7	+	+	PROPN
ap-7583	227	8	1	1	X
ap-7583	227	9	)	)	PUNCT
ap-7583	227	10	(	(	PUNCT
ap-7583	227	11	k	k	X
ap-7583	228	1	+	+	PROPN
ap-7583	228	2	2	2	X
ap-7583	228	3	)	)	PUNCT
ap-7583	228	4	α0	α0	ADJ
ap-7583	228	5	ck	ck	INTJ
ap-7583	228	6	−	−	PROPN
ap-7583	229	1	(	(	PUNCT
ap-7583	229	2	k	k	PROPN
ap-7583	229	3	−	−	PROPN
ap-7583	229	4	1)(k	1)(k	NUM
ap-7583	229	5	−	−	PROPN
ap-7583	229	6	2)α3	2)α3	NUM
ap-7583	229	7	+	+	CCONJ
ap-7583	229	8	(	(	PUNCT
ap-7583	229	9	k	k	PROPN
ap-7583	229	10	−	−	PROPN
ap-7583	229	11	1)β2	1)β2	NUM
ap-7583	229	12	+	+	NUM
ap-7583	229	13	ε1	ε1	PROPN
ap-7583	229	14	(	(	PUNCT
ap-7583	229	15	k	k	PROPN
ap-7583	229	16	+	+	PROPN
ap-7583	229	17	1)(k	1)(k	NUM
ap-7583	229	18	+	+	CCONJ
ap-7583	229	19	2	2	NUM
ap-7583	229	20	)	)	PUNCT
ap-7583	229	21	α0	α0	ADJ
ap-7583	229	22	ck−1	ck−1	NOUN
ap-7583	229	23	.	.	PUNCT
ap-7583	230	1	(	(	PUNCT
ap-7583	230	2	30	30	NUM
ap-7583	230	3	)	)	PUNCT
ap-7583	230	4	•	•	NOUN
ap-7583	230	5	differential	differential	NOUN
ap-7583	230	6	equation	equation	NOUN
ap-7583	230	7	:	:	PUNCT
ap-7583	230	8	(	(	PUNCT
ap-7583	230	9	α0	α0	ADJ
ap-7583	230	10	+	+	CCONJ
ap-7583	230	11	α2	α2	ADJ
ap-7583	230	12	r2	r2	NOUN
ap-7583	230	13	)	)	PUNCT
ap-7583	230	14	u′′(r	u′′(r	NOUN
ap-7583	230	15	)	)	PUNCT
ap-7583	230	16	+	+	CCONJ
ap-7583	230	17	(	(	PUNCT
ap-7583	230	18	β1	β1	PROPN
ap-7583	230	19	r	r	NOUN
ap-7583	230	20	+	+	CCONJ
ap-7583	230	21	β2	β2	NOUN
ap-7583	230	22	r2	r2	NOUN
ap-7583	230	23	)	)	PUNCT
ap-7583	230	24	u′(r	u′(r	VERB
ap-7583	230	25	)	)	PUNCT
ap-7583	231	1	+	+	CCONJ
ap-7583	231	2	ε1	ε1	VERB
ap-7583	231	3	r	r	NOUN
ap-7583	231	4	u(r	u(r	NOUN
ap-7583	231	5	)	)	PUNCT
ap-7583	232	1	=	=	SYM
ap-7583	232	2	0	0	X
ap-7583	232	3	.	.	PUNCT
ap-7583	233	1	(	(	PUNCT
ap-7583	233	2	31	31	NUM
ap-7583	233	3	)	)	PUNCT
ap-7583	233	4	recurrence	recurrence	NOUN
ap-7583	233	5	formula	formula	NOUN
ap-7583	233	6	:	:	PUNCT
ap-7583	233	7	ck+2	ck+2	NUM
ap-7583	233	8	=	=	X
ap-7583	233	9	−k	−k	PROPN
ap-7583	233	10	(	(	PUNCT
ap-7583	233	11	k	k	PROPN
ap-7583	233	12	−	−	PROPN
ap-7583	234	1	1)α2	1)α2	PRON
ap-7583	234	2	+	+	CCONJ
ap-7583	234	3	k	k	PROPN
ap-7583	234	4	β1	β1	PROPN
ap-7583	234	5	(	(	PUNCT
ap-7583	234	6	k	k	PROPN
ap-7583	234	7	+	+	PROPN
ap-7583	234	8	1	1	X
ap-7583	234	9	)	)	PUNCT
ap-7583	234	10	(	(	PUNCT
ap-7583	234	11	k	k	X
ap-7583	235	1	+	+	PROPN
ap-7583	235	2	2	2	X
ap-7583	235	3	)	)	PUNCT
ap-7583	235	4	α0	α0	ADJ
ap-7583	235	5	ck	ck	INTJ
ap-7583	235	6	−	−	PROPN
ap-7583	236	1	(	(	PUNCT
ap-7583	236	2	k	k	PROPN
ap-7583	236	3	−	−	PROPN
ap-7583	236	4	1)β2	1)β2	NUM
ap-7583	236	5	+	+	NUM
ap-7583	236	6	ε1	ε1	PROPN
ap-7583	236	7	(	(	PUNCT
ap-7583	236	8	k	k	PROPN
ap-7583	236	9	+	+	PROPN
ap-7583	236	10	1	1	NUM
ap-7583	236	11	)	)	PUNCT
ap-7583	236	12	(	(	PUNCT
ap-7583	236	13	k	k	X
ap-7583	236	14	+	+	PROPN
ap-7583	236	15	2	2	X
ap-7583	236	16	)	)	PUNCT
ap-7583	236	17	α0	α0	ADJ
ap-7583	236	18	ck−1	ck−1	NOUN
ap-7583	236	19	.	.	PUNCT
ap-7583	237	1	(	(	PUNCT
ap-7583	237	2	32	32	NUM
ap-7583	237	3	)	)	PUNCT
ap-7583	237	4	171	171	NUM
ap-7583	237	5	nasser	nasser	PROPN
ap-7583	237	6	saad	saad	PROPN
ap-7583	237	7	acta	acta	PROPN
ap-7583	237	8	polytechnica	polytechnica	PROPN
ap-7583	237	9	de	de	PROPN
ap-7583	237	10	α3	α3	PROPN
ap-7583	237	11	α2	α2	ADJ
ap-7583	237	12	α1	α1	PROPN
ap-7583	237	13	α0	α0	ADJ
ap-7583	237	14	discriminant	discriminant	ADJ
ap-7583	237	15	roots	root	NOUN
ap-7583	237	16	of	of	ADP
ap-7583	237	17	l	l	NOUN
ap-7583	237	18	domain	domain	NOUN
ap-7583	237	19	definition	definition	NOUN
ap-7583	237	20	i	i	PRON
ap-7583	237	21	∆3	∆3	PROPN
ap-7583	237	22	>	>	SYM
ap-7583	237	23	0	0	NUM
ap-7583	238	1	ξ1	ξ1	PROPN
ap-7583	238	2	̸=	̸=	PROPN
ap-7583	238	3	ξ2	ξ2	NOUN
ap-7583	238	4	̸=	̸=	PROPN
ap-7583	238	5	ξ3	ξ3	PROPN
ap-7583	238	6	|r|	|r|	NOUN
ap-7583	238	7	<	<	X
ap-7583	238	8	mini=1,2,3	mini=1,2,3	PROPN
ap-7583	238	9	ξi	ξi	NUM
ap-7583	238	10	α3	α3	NOUN
ap-7583	238	11	α2	α2	ADJ
ap-7583	238	12	α1	α1	PROPN
ap-7583	238	13	α0	α0	ADJ
ap-7583	238	14	∆3	∆3	NOUN
ap-7583	238	15	=	=	SYM
ap-7583	238	16	0	0	NUM
ap-7583	238	17	ξ1	ξ1	NOUN
ap-7583	238	18	=	=	SYM
ap-7583	238	19	ξ2	ξ2	NOUN
ap-7583	238	20	=	=	SYM
ap-7583	238	21	ξ3	ξ3	PROPN
ap-7583	238	22	=	=	SYM
ap-7583	238	23	ξ	ξ	SYM
ap-7583	238	24	|r|	|r|	NOUN
ap-7583	238	25	<	<	X
ap-7583	238	26	ξ	ξ	X
ap-7583	238	27	∆3	∆3	X
ap-7583	238	28	<	<	X
ap-7583	238	29	0	0	PUNCT
ap-7583	238	30	ξ	ξ	X
ap-7583	238	31	∈	∈	PROPN
ap-7583	238	32	r	r	NOUN
ap-7583	238	33	|r|	|r|	NOUN
ap-7583	238	34	<	<	X
ap-7583	238	35	ξ	ξ	PROPN
ap-7583	238	36	differential	differential	NOUN
ap-7583	238	37	equation	equation	NOUN
ap-7583	238	38	:	:	PUNCT
ap-7583	238	39	(	(	PUNCT
ap-7583	238	40	α0	α0	ADJ
ap-7583	238	41	+	+	NUM
ap-7583	238	42	α1	α1	PROPN
ap-7583	238	43	r	r	NOUN
ap-7583	238	44	+	+	CCONJ
ap-7583	238	45	α2	α2	ADJ
ap-7583	238	46	r2	r2	PROPN
ap-7583	238	47	+	+	CCONJ
ap-7583	238	48	α3	α3	PROPN
ap-7583	238	49	r3	r3	PROPN
ap-7583	238	50	)	)	PUNCT
ap-7583	238	51	y′′	y′′	PROPN
ap-7583	238	52	+	+	CCONJ
ap-7583	238	53	(	(	PUNCT
ap-7583	238	54	β0	β0	NOUN
ap-7583	238	55	+	+	CCONJ
ap-7583	238	56	β1	β1	PROPN
ap-7583	238	57	r	r	NOUN
ap-7583	238	58	+	+	CCONJ
ap-7583	238	59	β2	β2	NOUN
ap-7583	238	60	r2	r2	NOUN
ap-7583	238	61	)	)	PUNCT
ap-7583	238	62	y′	y′	PUNCT
ap-7583	239	1	+	+	CCONJ
ap-7583	239	2	(	(	PUNCT
ap-7583	239	3	ε0	ε0	PROPN
ap-7583	239	4	+	+	CCONJ
ap-7583	239	5	ε1	ε1	PROPN
ap-7583	239	6	r	r	NOUN
ap-7583	239	7	)	)	PUNCT
ap-7583	239	8	y	y	NOUN
ap-7583	239	9	=	=	SYM
ap-7583	239	10	0	0	NUM
ap-7583	239	11	discriminant	discriminant	NOUN
ap-7583	239	12	:	:	PUNCT
ap-7583	239	13	∆3	∆3	NOUN
ap-7583	239	14	=	=	SYM
ap-7583	239	15	18	18	NUM
ap-7583	239	16	α3	α3	ADJ
ap-7583	239	17	α2	α2	ADJ
ap-7583	239	18	α1	α1	PROPN
ap-7583	239	19	α0	α0	ADJ
ap-7583	239	20	−	−	NUM
ap-7583	239	21	4	4	NUM
ap-7583	239	22	α2	α2	ADJ
ap-7583	239	23	3	3	NUM
ap-7583	239	24	α0	α0	ADJ
ap-7583	239	25	+	+	CCONJ
ap-7583	239	26	α2	α2	ADJ
ap-7583	239	27	2α1	2α1	NOUN
ap-7583	239	28	2	2	NUM
ap-7583	239	29	−	−	NUM
ap-7583	239	30	4	4	NUM
ap-7583	239	31	α3	α3	NOUN
ap-7583	239	32	α1	α1	PROPN
ap-7583	239	33	3	3	NUM
ap-7583	239	34	−	−	NUM
ap-7583	239	35	27	27	NUM
ap-7583	239	36	α3	α3	NOUN
ap-7583	239	37	2	2	NUM
ap-7583	239	38	α0	α0	ADJ
ap-7583	239	39	2	2	NUM
ap-7583	239	40	ii	ii	NOUN
ap-7583	239	41	∆3	∆3	ADP
ap-7583	239	42	>	>	SYM
ap-7583	239	43	0	0	NUM
ap-7583	239	44	ξ1	ξ1	PROPN
ap-7583	239	45	̸=	̸=	PROPN
ap-7583	239	46	ξ2	ξ2	VERB
ap-7583	239	47	|r|	|r|	NOUN
ap-7583	239	48	<	<	X
ap-7583	239	49	mini=1,2	mini=1,2	NOUN
ap-7583	239	50	ξi	ξi	NOUN
ap-7583	239	51	0	0	NUM
ap-7583	239	52	α2	α2	ADJ
ap-7583	239	53	α1	α1	PROPN
ap-7583	240	1	α0	α0	ADJ
ap-7583	240	2	∆3	∆3	NOUN
ap-7583	240	3	=	=	SYM
ap-7583	240	4	0	0	NUM
ap-7583	240	5	ξ1	ξ1	NOUN
ap-7583	240	6	=	=	SYM
ap-7583	240	7	ξ2	ξ2	NOUN
ap-7583	240	8	=	=	SYM
ap-7583	240	9	ξ	ξ	SYM
ap-7583	240	10	|r|	|r|	NOUN
ap-7583	240	11	<	<	X
ap-7583	240	12	ξ	ξ	X
ap-7583	240	13	∆3	∆3	X
ap-7583	240	14	<	<	X
ap-7583	240	15	0	0	PUNCT
ap-7583	240	16	none	none	NOUN
ap-7583	240	17	|r|	|r|	NOUN
ap-7583	240	18	<	<	X
ap-7583	240	19	∞	∞	PROPN
ap-7583	240	20	differential	differential	NOUN
ap-7583	240	21	equation	equation	NOUN
ap-7583	240	22	:	:	PUNCT
ap-7583	240	23	(	(	PUNCT
ap-7583	240	24	α0	α0	ADJ
ap-7583	240	25	+	+	NUM
ap-7583	240	26	α1	α1	PROPN
ap-7583	240	27	r	r	NOUN
ap-7583	240	28	+	+	CCONJ
ap-7583	240	29	α2	α2	ADJ
ap-7583	240	30	r2	r2	PROPN
ap-7583	240	31	)	)	PUNCT
ap-7583	240	32	y′′	y′′	PROPN
ap-7583	241	1	+	+	CCONJ
ap-7583	241	2	(	(	PUNCT
ap-7583	241	3	β0	β0	NOUN
ap-7583	241	4	+	+	CCONJ
ap-7583	241	5	β1	β1	PROPN
ap-7583	241	6	r	r	NOUN
ap-7583	241	7	+	+	CCONJ
ap-7583	241	8	β2	β2	NOUN
ap-7583	241	9	r2	r2	NOUN
ap-7583	241	10	)	)	PUNCT
ap-7583	241	11	y′	y′	PUNCT
ap-7583	242	1	+	+	CCONJ
ap-7583	242	2	(	(	PUNCT
ap-7583	242	3	ε0	ε0	PROPN
ap-7583	242	4	+	+	CCONJ
ap-7583	242	5	ε1	ε1	PROPN
ap-7583	242	6	r	r	NOUN
ap-7583	242	7	)	)	PUNCT
ap-7583	242	8	y	y	NOUN
ap-7583	242	9	=	=	SYM
ap-7583	242	10	0	0	NUM
ap-7583	242	11	discriminant	discriminant	NOUN
ap-7583	242	12	:	:	PUNCT
ap-7583	242	13	∆3	∆3	NOUN
ap-7583	242	14	=	=	SYM
ap-7583	242	15	α2	α2	PROPN
ap-7583	242	16	2(−4	2(−4	NUM
ap-7583	242	17	α2	α2	ADJ
ap-7583	243	1	α0	α0	PROPN
ap-7583	243	2	+	+	CCONJ
ap-7583	243	3	α1	α1	PROPN
ap-7583	243	4	2	2	NUM
ap-7583	243	5	)	)	PUNCT
ap-7583	243	6	iii	iii	NOUN
ap-7583	243	7	0	0	NUM
ap-7583	243	8	0	0	NUM
ap-7583	243	9	α1	α1	PROPN
ap-7583	243	10	α0	α0	ADJ
ap-7583	243	11	α1α0	α1α0	NUM
ap-7583	243	12	>	>	SYM
ap-7583	243	13	0	0	PUNCT
ap-7583	243	14	r	r	NOUN
ap-7583	243	15	=	=	SYM
ap-7583	243	16	−α0	−α0	NOUN
ap-7583	243	17	/	/	SYM
ap-7583	243	18	α1	α1	PROPN
ap-7583	243	19	−∞	−∞	ADP
ap-7583	243	20	<	<	X
ap-7583	243	21	r	r	NOUN
ap-7583	243	22	<	<	X
ap-7583	243	23	−α0	−α0	NOUN
ap-7583	243	24	/	/	SYM
ap-7583	243	25	α1	α1	PROPN
ap-7583	243	26	α1α0	α1α0	X
ap-7583	243	27	<	<	X
ap-7583	243	28	0	0	NUM
ap-7583	243	29	r	r	NOUN
ap-7583	243	30	=	=	SYM
ap-7583	243	31	−α0	−α0	NOUN
ap-7583	243	32	/	/	SYM
ap-7583	243	33	α1	α1	PROPN
ap-7583	243	34	−α0	−α0	NOUN
ap-7583	243	35	/	/	SYM
ap-7583	243	36	α1	α1	PROPN
ap-7583	243	37	<	<	X
ap-7583	243	38	r	r	NOUN
ap-7583	243	39	<	<	X
ap-7583	243	40	∞	∞	PROPN
ap-7583	243	41	differential	differential	NOUN
ap-7583	243	42	equation	equation	NOUN
ap-7583	243	43	:	:	PUNCT
ap-7583	243	44	(	(	PUNCT
ap-7583	243	45	α0	α0	ADJ
ap-7583	243	46	+	+	NUM
ap-7583	243	47	α1	α1	PROPN
ap-7583	243	48	r	r	NOUN
ap-7583	243	49	)	)	PUNCT
ap-7583	243	50	y′′	y′′	NOUN
ap-7583	243	51	+	+	CCONJ
ap-7583	243	52	(	(	PUNCT
ap-7583	243	53	β0	β0	NOUN
ap-7583	243	54	+	+	CCONJ
ap-7583	243	55	β1	β1	PROPN
ap-7583	243	56	r	r	NOUN
ap-7583	243	57	+	+	CCONJ
ap-7583	243	58	β2	β2	NOUN
ap-7583	243	59	r2	r2	NOUN
ap-7583	243	60	)	)	PUNCT
ap-7583	243	61	y′	y′	PUNCT
ap-7583	244	1	+	+	CCONJ
ap-7583	244	2	(	(	PUNCT
ap-7583	244	3	ε0	ε0	PROPN
ap-7583	244	4	+	+	CCONJ
ap-7583	244	5	ε1	ε1	PROPN
ap-7583	244	6	r	r	NOUN
ap-7583	244	7	)	)	PUNCT
ap-7583	244	8	y	y	NOUN
ap-7583	244	9	=	=	SYM
ap-7583	244	10	0	0	NUM
ap-7583	244	11	discriminant	discriminant	NOUN
ap-7583	244	12	:	:	PUNCT
ap-7583	244	13	∆3	∆3	NOUN
ap-7583	244	14	=	=	SYM
ap-7583	244	15	0	0	NUM
ap-7583	244	16	iv	iv	NUM
ap-7583	244	17	0	0	NUM
ap-7583	244	18	0	0	NUM
ap-7583	244	19	0	0	NUM
ap-7583	245	1	α0	α0	ADJ
ap-7583	245	2	none	none	NOUN
ap-7583	245	3	none	none	NOUN
ap-7583	245	4	−∞	−∞	ADP
ap-7583	245	5	<	<	X
ap-7583	245	6	r	r	X
ap-7583	245	7	<	<	X
ap-7583	245	8	∞	∞	PROPN
ap-7583	245	9	differential	differential	NOUN
ap-7583	245	10	equation	equation	NOUN
ap-7583	245	11	:	:	PUNCT
ap-7583	245	12	α0	α0	ADJ
ap-7583	245	13	y′′	y′′	PROPN
ap-7583	245	14	+	+	CCONJ
ap-7583	245	15	(	(	PUNCT
ap-7583	245	16	β0	β0	NOUN
ap-7583	245	17	+	+	CCONJ
ap-7583	245	18	β1	β1	PROPN
ap-7583	245	19	r	r	NOUN
ap-7583	245	20	+	+	CCONJ
ap-7583	245	21	β2	β2	NOUN
ap-7583	245	22	r2	r2	NOUN
ap-7583	245	23	)	)	PUNCT
ap-7583	245	24	y′	y′	PUNCT
ap-7583	246	1	+	+	CCONJ
ap-7583	246	2	(	(	PUNCT
ap-7583	246	3	ε0	ε0	PROPN
ap-7583	246	4	+	+	CCONJ
ap-7583	246	5	ε1	ε1	PROPN
ap-7583	246	6	r	r	NOUN
ap-7583	246	7	)	)	PUNCT
ap-7583	246	8	y	y	NOUN
ap-7583	246	9	=	=	SYM
ap-7583	246	10	0	0	NUM
ap-7583	246	11	discriminant	discriminant	NOUN
ap-7583	246	12	:	:	PUNCT
ap-7583	246	13	∆3	∆3	X
ap-7583	246	14	=	=	SYM
ap-7583	246	15	0	0	NUM
ap-7583	246	16	v	v	NOUN
ap-7583	246	17	∆3	∆3	PROPN
ap-7583	246	18	>	>	SYM
ap-7583	246	19	0	0	NUM
ap-7583	247	1	ξ1	ξ1	PROPN
ap-7583	247	2	̸=	̸=	PROPN
ap-7583	247	3	ξ2	ξ2	NOUN
ap-7583	247	4	̸=	̸=	PROPN
ap-7583	247	5	ξ3	ξ3	PROPN
ap-7583	247	6	|r|	|r|	NOUN
ap-7583	247	7	<	<	X
ap-7583	247	8	mini=1,2,3	mini=1,2,3	PROPN
ap-7583	247	9	ξi	ξi	NOUN
ap-7583	247	10	α3	α3	NOUN
ap-7583	247	11	0	0	NUM
ap-7583	247	12	α1	α1	PROPN
ap-7583	247	13	α0	α0	ADJ
ap-7583	247	14	∆3	∆3	NOUN
ap-7583	247	15	=	=	SYM
ap-7583	247	16	0	0	NUM
ap-7583	247	17	ξ1	ξ1	NOUN
ap-7583	247	18	=	=	SYM
ap-7583	247	19	ξ2	ξ2	NOUN
ap-7583	247	20	=	=	SYM
ap-7583	247	21	ξ3	ξ3	PROPN
ap-7583	247	22	=	=	SYM
ap-7583	247	23	ξ	ξ	SYM
ap-7583	247	24	|r|	|r|	NOUN
ap-7583	247	25	<	<	X
ap-7583	247	26	ξ	ξ	X
ap-7583	247	27	∆3	∆3	X
ap-7583	247	28	<	<	X
ap-7583	247	29	0	0	PUNCT
ap-7583	247	30	ξ	ξ	X
ap-7583	247	31	∈	∈	PROPN
ap-7583	247	32	r	r	NOUN
ap-7583	247	33	|r|	|r|	NOUN
ap-7583	247	34	<	<	X
ap-7583	247	35	ξ	ξ	PROPN
ap-7583	247	36	differential	differential	NOUN
ap-7583	247	37	equation	equation	NOUN
ap-7583	247	38	:	:	PUNCT
ap-7583	247	39	(	(	PUNCT
ap-7583	247	40	α0	α0	ADJ
ap-7583	247	41	+	+	NUM
ap-7583	247	42	α1	α1	PROPN
ap-7583	247	43	r	r	NOUN
ap-7583	247	44	+	+	CCONJ
ap-7583	247	45	α3	α3	PROPN
ap-7583	247	46	r3	r3	PROPN
ap-7583	247	47	)	)	PUNCT
ap-7583	247	48	y′′	y′′	PROPN
ap-7583	247	49	+	+	CCONJ
ap-7583	247	50	(	(	PUNCT
ap-7583	247	51	β0	β0	NOUN
ap-7583	247	52	+	+	CCONJ
ap-7583	247	53	β1	β1	PROPN
ap-7583	247	54	r	r	NOUN
ap-7583	247	55	+	+	CCONJ
ap-7583	247	56	β2	β2	NOUN
ap-7583	247	57	r2	r2	NOUN
ap-7583	247	58	)	)	PUNCT
ap-7583	247	59	y′	y′	PUNCT
ap-7583	248	1	+	+	CCONJ
ap-7583	248	2	(	(	PUNCT
ap-7583	248	3	ε0	ε0	PROPN
ap-7583	248	4	+	+	CCONJ
ap-7583	248	5	ε1	ε1	PROPN
ap-7583	248	6	r	r	NOUN
ap-7583	248	7	)	)	PUNCT
ap-7583	248	8	y	y	NOUN
ap-7583	248	9	=	=	SYM
ap-7583	248	10	0	0	NUM
ap-7583	248	11	discriminant	discriminant	NOUN
ap-7583	248	12	:	:	PUNCT
ap-7583	248	13	∆3	∆3	PROPN
ap-7583	248	14	=	=	SYM
ap-7583	248	15	−4	−4	PROPN
ap-7583	248	16	α3	α3	PROPN
ap-7583	248	17	α1	α1	PROPN
ap-7583	248	18	3	3	NUM
ap-7583	248	19	−	−	NUM
ap-7583	248	20	27	27	NUM
ap-7583	248	21	α3	α3	NOUN
ap-7583	248	22	2	2	NUM
ap-7583	248	23	α0	α0	ADJ
ap-7583	248	24	2	2	NUM
ap-7583	248	25	vi	vi	NOUN
ap-7583	248	26	∆3	∆3	PRON
ap-7583	248	27	>	>	SYM
ap-7583	248	28	0	0	NUM
ap-7583	249	1	ξ1	ξ1	PROPN
ap-7583	249	2	̸=	̸=	PROPN
ap-7583	249	3	ξ2	ξ2	NOUN
ap-7583	249	4	̸=	̸=	PROPN
ap-7583	249	5	ξ3	ξ3	PROPN
ap-7583	249	6	|r|	|r|	NOUN
ap-7583	249	7	<	<	X
ap-7583	249	8	mini=1,2,3	mini=1,2,3	PROPN
ap-7583	249	9	ξi	ξi	NOUN
ap-7583	249	10	α3	α3	NOUN
ap-7583	249	11	α2	α2	NOUN
ap-7583	249	12	0	0	PUNCT
ap-7583	250	1	α0	α0	ADJ
ap-7583	250	2	∆3	∆3	NOUN
ap-7583	250	3	=	=	SYM
ap-7583	250	4	0	0	NUM
ap-7583	250	5	ξ1	ξ1	NOUN
ap-7583	250	6	=	=	SYM
ap-7583	250	7	ξ2	ξ2	NOUN
ap-7583	250	8	=	=	SYM
ap-7583	250	9	ξ3	ξ3	PROPN
ap-7583	250	10	=	=	SYM
ap-7583	250	11	ξ	ξ	SYM
ap-7583	250	12	|r|	|r|	NOUN
ap-7583	250	13	<	<	X
ap-7583	250	14	ξ	ξ	X
ap-7583	250	15	∆3	∆3	X
ap-7583	250	16	<	<	X
ap-7583	250	17	0	0	PUNCT
ap-7583	250	18	ξ	ξ	X
ap-7583	250	19	∈	∈	PROPN
ap-7583	250	20	r	r	NOUN
ap-7583	250	21	|r|	|r|	NOUN
ap-7583	250	22	<	<	X
ap-7583	250	23	ξ	ξ	PROPN
ap-7583	250	24	differential	differential	NOUN
ap-7583	250	25	equation	equation	NOUN
ap-7583	250	26	:	:	PUNCT
ap-7583	250	27	(	(	PUNCT
ap-7583	250	28	α0	α0	ADJ
ap-7583	250	29	+	+	CCONJ
ap-7583	250	30	α2	α2	ADJ
ap-7583	250	31	r2	r2	PROPN
ap-7583	250	32	+	+	CCONJ
ap-7583	250	33	α3	α3	PROPN
ap-7583	250	34	r3	r3	PROPN
ap-7583	250	35	)	)	PUNCT
ap-7583	250	36	y′′	y′′	PROPN
ap-7583	250	37	+	+	CCONJ
ap-7583	250	38	(	(	PUNCT
ap-7583	250	39	β0	β0	NOUN
ap-7583	250	40	+	+	CCONJ
ap-7583	250	41	β1	β1	PROPN
ap-7583	250	42	r	r	NOUN
ap-7583	250	43	+	+	CCONJ
ap-7583	250	44	β2	β2	NOUN
ap-7583	250	45	r2	r2	NOUN
ap-7583	250	46	)	)	PUNCT
ap-7583	250	47	y′	y′	PUNCT
ap-7583	251	1	+	+	CCONJ
ap-7583	251	2	(	(	PUNCT
ap-7583	251	3	ε0	ε0	PROPN
ap-7583	251	4	+	+	CCONJ
ap-7583	251	5	ε1	ε1	PROPN
ap-7583	251	6	r	r	NOUN
ap-7583	251	7	)	)	PUNCT
ap-7583	251	8	y	y	NOUN
ap-7583	251	9	=	=	SYM
ap-7583	251	10	0	0	NUM
ap-7583	251	11	discriminant	discriminant	NOUN
ap-7583	251	12	:	:	PUNCT
ap-7583	251	13	∆3	∆3	NOUN
ap-7583	251	14	=	=	SYM
ap-7583	251	15	−4	−4	PROPN
ap-7583	251	16	α2	α2	ADJ
ap-7583	251	17	3	3	NUM
ap-7583	252	1	α0	α0	ADJ
ap-7583	252	2	−	−	NUM
ap-7583	252	3	27	27	NUM
ap-7583	252	4	α3	α3	NOUN
ap-7583	252	5	2	2	NUM
ap-7583	252	6	α0	α0	PROPN
ap-7583	252	7	2	2	NUM
ap-7583	252	8	vii	vii	NOUN
ap-7583	252	9	α3	α3	PROPN
ap-7583	252	10	0	0	NUM
ap-7583	252	11	0	0	NUM
ap-7583	252	12	α0	α0	ADJ
ap-7583	252	13	α0	α0	ADJ
ap-7583	252	14	α3	α3	NOUN
ap-7583	252	15	<	<	X
ap-7583	252	16	0	0	NUM
ap-7583	252	17	or	or	CCONJ
ap-7583	252	18	α0	α0	ADJ
ap-7583	252	19	α3	α3	NOUN
ap-7583	252	20	>	>	X
ap-7583	252	21	0	0	PUNCT
ap-7583	253	1	ξ	ξ	X
ap-7583	253	2	=	=	SYM
ap-7583	253	3	3	3	NUM
ap-7583	253	4	√	√	PROPN
ap-7583	253	5	−α0	−α0	NOUN
ap-7583	253	6	/	/	SYM
ap-7583	253	7	α3	α3	NOUN
ap-7583	253	8	|r|	|r|	NOUN
ap-7583	253	9	<	<	X
ap-7583	253	10	ξ	ξ	PROPN
ap-7583	253	11	differential	differential	NOUN
ap-7583	253	12	equation	equation	NOUN
ap-7583	253	13	:	:	PUNCT
ap-7583	253	14	(	(	PUNCT
ap-7583	253	15	α0	α0	ADJ
ap-7583	253	16	+	+	CCONJ
ap-7583	253	17	α3	α3	ADJ
ap-7583	253	18	r3	r3	PROPN
ap-7583	253	19	)	)	PUNCT
ap-7583	253	20	y′′	y′′	PROPN
ap-7583	253	21	+	+	CCONJ
ap-7583	253	22	(	(	PUNCT
ap-7583	253	23	β0	β0	NOUN
ap-7583	253	24	+	+	CCONJ
ap-7583	253	25	β1	β1	PROPN
ap-7583	253	26	r	r	NOUN
ap-7583	253	27	+	+	CCONJ
ap-7583	253	28	β2	β2	NOUN
ap-7583	253	29	r2	r2	NOUN
ap-7583	253	30	)	)	PUNCT
ap-7583	253	31	y′	y′	PUNCT
ap-7583	254	1	+	+	CCONJ
ap-7583	254	2	(	(	PUNCT
ap-7583	254	3	ε0	ε0	PROPN
ap-7583	254	4	+	+	CCONJ
ap-7583	254	5	ε1	ε1	PROPN
ap-7583	254	6	r	r	NOUN
ap-7583	254	7	)	)	PUNCT
ap-7583	254	8	y	y	NOUN
ap-7583	254	9	=	=	SYM
ap-7583	254	10	0	0	NUM
ap-7583	254	11	discriminant	discriminant	NOUN
ap-7583	254	12	:	:	PUNCT
ap-7583	254	13	∆3	∆3	X
ap-7583	254	14	=	=	SYM
ap-7583	254	15	−27	−27	NUM
ap-7583	254	16	α3	α3	NOUN
ap-7583	254	17	2	2	NUM
ap-7583	254	18	α0	α0	ADJ
ap-7583	254	19	2	2	NUM
ap-7583	254	20	viii	viii	NOUN
ap-7583	254	21	0	0	NUM
ap-7583	254	22	α2	α2	ADJ
ap-7583	254	23	0	0	NUM
ap-7583	255	1	α0	α0	ADJ
ap-7583	255	2	α2α0	α2α0	NOUN
ap-7583	255	3	<	<	X
ap-7583	255	4	0	0	NUM
ap-7583	255	5	r	r	NOUN
ap-7583	255	6	=	=	SYM
ap-7583	255	7	±	±	NOUN
ap-7583	255	8	√	√	NOUN
ap-7583	255	9	−	−	PROPN
ap-7583	255	10	α0	α0	ADJ
ap-7583	255	11	α2	α2	NOUN
ap-7583	255	12	−	−	PROPN
ap-7583	255	13	√	√	PROPN
ap-7583	255	14	−	−	PROPN
ap-7583	256	1	α0	α0	ADJ
ap-7583	256	2	α2	α2	NOUN
ap-7583	256	3	<	<	X
ap-7583	256	4	r	r	NOUN
ap-7583	256	5	<	<	NOUN
ap-7583	256	6	√	√	NOUN
ap-7583	256	7	−	−	PROPN
ap-7583	256	8	α0	α0	ADJ
ap-7583	256	9	α2	α2	PROPN
ap-7583	256	10	α2α0	α2α0	NOUN
ap-7583	256	11	>	>	X
ap-7583	256	12	0	0	NUM
ap-7583	257	1	none	none	NOUN
ap-7583	257	2	−∞	−∞	ADP
ap-7583	257	3	<	<	X
ap-7583	257	4	r	r	X
ap-7583	257	5	<	<	X
ap-7583	257	6	∞	∞	PROPN
ap-7583	257	7	differential	differential	NOUN
ap-7583	257	8	equation	equation	NOUN
ap-7583	257	9	:	:	PUNCT
ap-7583	257	10	(	(	PUNCT
ap-7583	257	11	α0	α0	ADJ
ap-7583	257	12	+	+	CCONJ
ap-7583	257	13	α2	α2	ADJ
ap-7583	257	14	r2	r2	PROPN
ap-7583	257	15	)	)	PUNCT
ap-7583	257	16	y′′	y′′	PROPN
ap-7583	257	17	+	+	CCONJ
ap-7583	257	18	(	(	PUNCT
ap-7583	257	19	β0	β0	NOUN
ap-7583	257	20	+	+	CCONJ
ap-7583	257	21	β1	β1	PROPN
ap-7583	257	22	r	r	NOUN
ap-7583	257	23	+	+	CCONJ
ap-7583	257	24	β2	β2	NOUN
ap-7583	257	25	r2	r2	NOUN
ap-7583	257	26	)	)	PUNCT
ap-7583	257	27	y′	y′	PUNCT
ap-7583	258	1	+	+	CCONJ
ap-7583	258	2	(	(	PUNCT
ap-7583	258	3	ε0	ε0	PROPN
ap-7583	258	4	+	+	CCONJ
ap-7583	258	5	ε1	ε1	PROPN
ap-7583	258	6	r	r	NOUN
ap-7583	258	7	)	)	PUNCT
ap-7583	258	8	y	y	NOUN
ap-7583	258	9	=	=	SYM
ap-7583	258	10	0	0	NUM
ap-7583	258	11	discriminant	discriminant	NOUN
ap-7583	258	12	:	:	PUNCT
ap-7583	258	13	∆3	∆3	NOUN
ap-7583	258	14	=	=	SYM
ap-7583	258	15	−4	−4	PROPN
ap-7583	258	16	α2	α2	ADJ
ap-7583	258	17	3	3	NUM
ap-7583	258	18	α0	α0	ADJ
ap-7583	258	19	table	table	NOUN
ap-7583	258	20	2	2	NUM
ap-7583	258	21	.	.	PUNCT
ap-7583	259	1	tabulating	tabulate	VERB
ap-7583	259	2	the	the	DET
ap-7583	259	3	eight	eight	NUM
ap-7583	259	4	different	different	ADJ
ap-7583	259	5	types	type	NOUN
ap-7583	259	6	of	of	ADP
ap-7583	259	7	differential	differential	ADJ
ap-7583	259	8	equations	equation	NOUN
ap-7583	259	9	,	,	PUNCT
ap-7583	259	10	which	which	PRON
ap-7583	259	11	apply	apply	VERB
ap-7583	259	12	to	to	ADP
ap-7583	259	13	theorem	theorem	ADJ
ap-7583	259	14	3.1	3.1	NUM
ap-7583	259	15	.	.	PUNCT
ap-7583	259	16	172	172	NUM
ap-7583	259	17	vol	vol	NOUN
ap-7583	259	18	.	.	PUNCT
ap-7583	260	1	62	62	NUM
ap-7583	260	2	no	no	INTJ
ap-7583	260	3	.	.	PUNCT
ap-7583	261	1	1/2022	1/2022	NUM
ap-7583	261	2	on	on	ADP
ap-7583	261	3	generalized	generalized	ADJ
ap-7583	261	4	heun	heun	NOUN
ap-7583	261	5	equation	equation	NOUN
ap-7583	261	6	with	with	ADP
ap-7583	261	7	some	some	DET
ap-7583	261	8	mathematical	mathematical	NOUN
ap-7583	261	9	.	.	PUNCT
ap-7583	261	10	.	.	PUNCT
ap-7583	261	11	.	.	PUNCT
ap-7583	262	1	•	•	NUM
ap-7583	262	2	differential	differential	NOUN
ap-7583	262	3	equation	equation	NOUN
ap-7583	262	4	:	:	PUNCT
ap-7583	262	5	(	(	PUNCT
ap-7583	262	6	α0	α0	ADJ
ap-7583	262	7	+	+	CCONJ
ap-7583	262	8	α3	α3	NOUN
ap-7583	262	9	r3)u′′(r	r3)u′′(r	NOUN
ap-7583	262	10	)	)	PUNCT
ap-7583	262	11	+	+	CCONJ
ap-7583	262	12	(	(	PUNCT
ap-7583	262	13	β1	β1	PROPN
ap-7583	262	14	r	r	NOUN
ap-7583	262	15	+	+	CCONJ
ap-7583	262	16	β2	β2	NOUN
ap-7583	262	17	r2	r2	NOUN
ap-7583	262	18	)	)	PUNCT
ap-7583	262	19	u′(r	u′(r	SYM
ap-7583	262	20	)	)	PUNCT
ap-7583	263	1	+	+	CCONJ
ap-7583	263	2	r	r	NOUN
ap-7583	263	3	ε1	ε1	VERB
ap-7583	263	4	u(r	u(r	NOUN
ap-7583	263	5	)	)	PUNCT
ap-7583	263	6	=	=	SYM
ap-7583	264	1	0	0	X
ap-7583	264	2	.	.	PUNCT
ap-7583	264	3	(	(	PUNCT
ap-7583	264	4	33	33	NUM
ap-7583	264	5	)	)	PUNCT
ap-7583	264	6	recurrence	recurrence	NOUN
ap-7583	264	7	formula	formula	NOUN
ap-7583	264	8	:	:	PUNCT
ap-7583	264	9	ck+2	ck+2	NUM
ap-7583	264	10	=	=	SYM
ap-7583	264	11	−	−	PROPN
ap-7583	264	12	k	k	PROPN
ap-7583	264	13	β1	β1	PROPN
ap-7583	264	14	(	(	PUNCT
ap-7583	264	15	k	k	PROPN
ap-7583	265	1	+	+	PROPN
ap-7583	265	2	1	1	X
ap-7583	265	3	)	)	PUNCT
ap-7583	265	4	(	(	PUNCT
ap-7583	265	5	k	k	X
ap-7583	265	6	+	+	PROPN
ap-7583	265	7	2	2	X
ap-7583	265	8	)	)	PUNCT
ap-7583	265	9	α0	α0	ADJ
ap-7583	265	10	ck	ck	INTJ
ap-7583	265	11	−	−	PROPN
ap-7583	265	12	(	(	PUNCT
ap-7583	265	13	k	k	PROPN
ap-7583	265	14	−	−	PROPN
ap-7583	266	1	1)(k	1)(k	NUM
ap-7583	266	2	−	−	PROPN
ap-7583	266	3	2)α3	2)α3	NUM
ap-7583	266	4	+	+	CCONJ
ap-7583	267	1	(	(	PUNCT
ap-7583	267	2	k	k	PROPN
ap-7583	267	3	−	−	PROPN
ap-7583	267	4	1)β2	1)β2	NUM
ap-7583	267	5	+	+	NUM
ap-7583	267	6	ε1	ε1	PROPN
ap-7583	267	7	(	(	PUNCT
ap-7583	267	8	k	k	PROPN
ap-7583	267	9	+	+	PROPN
ap-7583	267	10	1	1	NUM
ap-7583	267	11	)	)	PUNCT
ap-7583	267	12	(	(	PUNCT
ap-7583	267	13	k	k	X
ap-7583	267	14	+	+	PROPN
ap-7583	267	15	2	2	X
ap-7583	267	16	)	)	PUNCT
ap-7583	267	17	α0	α0	ADJ
ap-7583	267	18	ck−1	ck−1	NOUN
ap-7583	267	19	.	.	PUNCT
ap-7583	268	1	(	(	PUNCT
ap-7583	268	2	34	34	NUM
ap-7583	268	3	)	)	PUNCT
ap-7583	268	4	•	•	NOUN
ap-7583	268	5	differential	differential	NOUN
ap-7583	268	6	equation	equation	NOUN
ap-7583	268	7	:	:	PUNCT
ap-7583	268	8	α0	α0	ADJ
ap-7583	268	9	u′′(r	u′′(r	NOUN
ap-7583	268	10	)	)	PUNCT
ap-7583	268	11	+	+	CCONJ
ap-7583	268	12	(	(	PUNCT
ap-7583	268	13	β1	β1	PROPN
ap-7583	268	14	r	r	NOUN
ap-7583	268	15	+	+	CCONJ
ap-7583	268	16	β2	β2	NOUN
ap-7583	268	17	r2	r2	NOUN
ap-7583	268	18	)	)	PUNCT
ap-7583	268	19	u′(r	u′(r	VERB
ap-7583	268	20	)	)	PUNCT
ap-7583	269	1	+	+	CCONJ
ap-7583	269	2	ε1	ε1	VERB
ap-7583	269	3	r	r	NOUN
ap-7583	269	4	u(r	u(r	NOUN
ap-7583	269	5	)	)	PUNCT
ap-7583	270	1	=	=	SYM
ap-7583	270	2	0	0	X
ap-7583	270	3	.	.	PUNCT
ap-7583	271	1	(	(	PUNCT
ap-7583	271	2	35	35	NUM
ap-7583	271	3	)	)	PUNCT
ap-7583	271	4	recurrence	recurrence	NOUN
ap-7583	271	5	formula	formula	NOUN
ap-7583	271	6	:	:	PUNCT
ap-7583	271	7	ck+2	ck+2	NUM
ap-7583	271	8	=	=	SYM
ap-7583	271	9	−	−	PROPN
ap-7583	271	10	k	k	PROPN
ap-7583	271	11	β1	β1	PROPN
ap-7583	271	12	(	(	PUNCT
ap-7583	271	13	k	k	PROPN
ap-7583	272	1	+	+	PROPN
ap-7583	272	2	1	1	X
ap-7583	272	3	)	)	PUNCT
ap-7583	272	4	(	(	PUNCT
ap-7583	272	5	k	k	X
ap-7583	272	6	+	+	PROPN
ap-7583	272	7	2	2	X
ap-7583	272	8	)	)	PUNCT
ap-7583	272	9	α0	α0	ADJ
ap-7583	272	10	ck	ck	INTJ
ap-7583	272	11	−	−	PROPN
ap-7583	272	12	(	(	PUNCT
ap-7583	272	13	k	k	NOUN
ap-7583	272	14	−	−	PROPN
ap-7583	272	15	1	1	X
ap-7583	272	16	)	)	PUNCT
ap-7583	272	17	β2	β2	NOUN
ap-7583	272	18	+	+	CCONJ
ap-7583	272	19	ε1	ε1	PROPN
ap-7583	272	20	(	(	PUNCT
ap-7583	272	21	k	k	PROPN
ap-7583	272	22	+	+	PROPN
ap-7583	272	23	1	1	NUM
ap-7583	272	24	)	)	PUNCT
ap-7583	272	25	(	(	PUNCT
ap-7583	272	26	k	k	X
ap-7583	272	27	+	+	PROPN
ap-7583	272	28	2	2	X
ap-7583	272	29	)	)	PUNCT
ap-7583	272	30	α0	α0	ADJ
ap-7583	272	31	ck−1	ck−1	NOUN
ap-7583	272	32	.	.	PUNCT
ap-7583	273	1	(	(	PUNCT
ap-7583	273	2	36	36	NUM
ap-7583	273	3	)	)	PUNCT
ap-7583	273	4	•	•	NUM
ap-7583	273	5	differential	differential	NOUN
ap-7583	273	6	equation	equation	NOUN
ap-7583	273	7	:	:	PUNCT
ap-7583	273	8	(	(	PUNCT
ap-7583	273	9	α0	α0	ADJ
ap-7583	273	10	+	+	CCONJ
ap-7583	273	11	α2	α2	ADJ
ap-7583	273	12	r2	r2	PROPN
ap-7583	273	13	+	+	CCONJ
ap-7583	273	14	α3	α3	PROPN
ap-7583	273	15	r3	r3	PROPN
ap-7583	273	16	)	)	PUNCT
ap-7583	273	17	u′′(r	u′′(r	NOUN
ap-7583	273	18	)	)	PUNCT
ap-7583	273	19	+	+	CCONJ
ap-7583	273	20	β1	β1	PROPN
ap-7583	273	21	r	r	NOUN
ap-7583	273	22	u′(r	u′(r	PROPN
ap-7583	273	23	)	)	PUNCT
ap-7583	274	1	+	+	CCONJ
ap-7583	274	2	ε1	ε1	VERB
ap-7583	274	3	r	r	NOUN
ap-7583	274	4	u(r	u(r	NOUN
ap-7583	274	5	)	)	PUNCT
ap-7583	275	1	=	=	SYM
ap-7583	275	2	0	0	X
ap-7583	275	3	.	.	PUNCT
ap-7583	276	1	(	(	PUNCT
ap-7583	276	2	37	37	NUM
ap-7583	276	3	)	)	PUNCT
ap-7583	276	4	recurrence	recurrence	NOUN
ap-7583	276	5	formula	formula	NOUN
ap-7583	276	6	:	:	PUNCT
ap-7583	276	7	ck+2	ck+2	NUM
ap-7583	276	8	=	=	X
ap-7583	276	9	−k	−k	PROPN
ap-7583	276	10	(	(	PUNCT
ap-7583	276	11	k	k	PROPN
ap-7583	276	12	−	−	PROPN
ap-7583	277	1	1)α2	1)α2	PRON
ap-7583	277	2	+	+	CCONJ
ap-7583	277	3	k	k	PROPN
ap-7583	277	4	β1	β1	PROPN
ap-7583	277	5	(	(	PUNCT
ap-7583	277	6	k	k	PROPN
ap-7583	277	7	+	+	PROPN
ap-7583	277	8	1	1	X
ap-7583	277	9	)	)	PUNCT
ap-7583	277	10	(	(	PUNCT
ap-7583	277	11	k	k	X
ap-7583	278	1	+	+	PROPN
ap-7583	278	2	2	2	X
ap-7583	278	3	)	)	PUNCT
ap-7583	278	4	α0	α0	ADJ
ap-7583	278	5	ck	ck	INTJ
ap-7583	278	6	−	−	PROPN
ap-7583	279	1	(	(	PUNCT
ap-7583	279	2	k	k	PROPN
ap-7583	279	3	−	−	PROPN
ap-7583	279	4	2)(k	2)(k	NUM
ap-7583	280	1	−	−	PROPN
ap-7583	281	1	1)α3	1)α3	PRON
ap-7583	281	2	+	+	NUM
ap-7583	281	3	ε1	ε1	PROPN
ap-7583	281	4	(	(	PUNCT
ap-7583	281	5	k	k	PROPN
ap-7583	281	6	+	+	PROPN
ap-7583	281	7	1	1	NUM
ap-7583	281	8	)	)	PUNCT
ap-7583	281	9	(	(	PUNCT
ap-7583	281	10	k	k	X
ap-7583	281	11	+	+	PROPN
ap-7583	281	12	2	2	X
ap-7583	281	13	)	)	PUNCT
ap-7583	281	14	α0	α0	ADJ
ap-7583	281	15	ck−1	ck−1	NOUN
ap-7583	281	16	.	.	PUNCT
ap-7583	282	1	(	(	PUNCT
ap-7583	282	2	38	38	NUM
ap-7583	282	3	)	)	PUNCT
ap-7583	282	4	•	•	NUM
ap-7583	282	5	differential	differential	NOUN
ap-7583	282	6	equation	equation	NOUN
ap-7583	282	7	:	:	PUNCT
ap-7583	282	8	(	(	PUNCT
ap-7583	282	9	α0	α0	ADJ
ap-7583	282	10	+	+	CCONJ
ap-7583	282	11	α3	α3	NOUN
ap-7583	282	12	r3)u′′(r	r3)u′′(r	NOUN
ap-7583	282	13	)	)	PUNCT
ap-7583	283	1	+	+	CCONJ
ap-7583	283	2	β1	β1	PROPN
ap-7583	283	3	r	r	NOUN
ap-7583	283	4	u′(r	u′(r	PROPN
ap-7583	283	5	)	)	PUNCT
ap-7583	284	1	+	+	CCONJ
ap-7583	284	2	ε1	ε1	VERB
ap-7583	284	3	r	r	NOUN
ap-7583	284	4	u(r	u(r	NOUN
ap-7583	284	5	)	)	PUNCT
ap-7583	285	1	=	=	SYM
ap-7583	285	2	0	0	X
ap-7583	285	3	.	.	PUNCT
ap-7583	285	4	(	(	PUNCT
ap-7583	285	5	39	39	NUM
ap-7583	285	6	)	)	PUNCT
ap-7583	285	7	recurrence	recurrence	NOUN
ap-7583	285	8	formula	formula	NOUN
ap-7583	285	9	:	:	PUNCT
ap-7583	285	10	ck+2	ck+2	NUM
ap-7583	285	11	=	=	SYM
ap-7583	285	12	−	−	PROPN
ap-7583	285	13	k	k	PROPN
ap-7583	285	14	β1	β1	PROPN
ap-7583	285	15	(	(	PUNCT
ap-7583	285	16	k	k	PROPN
ap-7583	285	17	+	+	PROPN
ap-7583	285	18	1)(k	1)(k	NUM
ap-7583	285	19	+	+	CCONJ
ap-7583	285	20	2	2	NUM
ap-7583	285	21	)	)	PUNCT
ap-7583	285	22	α0	α0	ADJ
ap-7583	285	23	ck	ck	INTJ
ap-7583	285	24	−	−	PROPN
ap-7583	286	1	(	(	PUNCT
ap-7583	286	2	k	k	NOUN
ap-7583	286	3	−	−	PROPN
ap-7583	286	4	2	2	NUM
ap-7583	286	5	)	)	PUNCT
ap-7583	286	6	(	(	PUNCT
ap-7583	286	7	k	k	NOUN
ap-7583	286	8	−	−	PROPN
ap-7583	286	9	1	1	X
ap-7583	286	10	)	)	PUNCT
ap-7583	286	11	α3	α3	NOUN
ap-7583	286	12	+	+	CCONJ
ap-7583	286	13	ε1	ε1	PROPN
ap-7583	286	14	(	(	PUNCT
ap-7583	286	15	k	k	PROPN
ap-7583	286	16	+	+	PROPN
ap-7583	286	17	1)(k	1)(k	NUM
ap-7583	286	18	+	+	CCONJ
ap-7583	286	19	2	2	NUM
ap-7583	286	20	)	)	PUNCT
ap-7583	286	21	α0	α0	ADJ
ap-7583	286	22	ck−1	ck−1	NOUN
ap-7583	286	23	.	.	PUNCT
ap-7583	287	1	(	(	PUNCT
ap-7583	287	2	40	40	NUM
ap-7583	287	3	)	)	PUNCT
ap-7583	287	4	•	•	NUM
ap-7583	287	5	differential	differential	NOUN
ap-7583	287	6	equation	equation	NOUN
ap-7583	287	7	:	:	PUNCT
ap-7583	287	8	(	(	PUNCT
ap-7583	287	9	α0	α0	ADJ
ap-7583	287	10	+	+	CCONJ
ap-7583	287	11	α2	α2	ADJ
ap-7583	287	12	r2)u′′(r	r2)u′′(r	NOUN
ap-7583	287	13	)	)	PUNCT
ap-7583	288	1	+	+	CCONJ
ap-7583	288	2	β1	β1	PROPN
ap-7583	288	3	r	r	NOUN
ap-7583	288	4	u′(r	u′(r	PROPN
ap-7583	288	5	)	)	PUNCT
ap-7583	289	1	+	+	CCONJ
ap-7583	289	2	ε1	ε1	VERB
ap-7583	289	3	r	r	NOUN
ap-7583	289	4	u(r	u(r	NOUN
ap-7583	289	5	)	)	PUNCT
ap-7583	290	1	=	=	SYM
ap-7583	290	2	0	0	X
ap-7583	290	3	.	.	PUNCT
ap-7583	291	1	(	(	PUNCT
ap-7583	291	2	41	41	NUM
ap-7583	291	3	)	)	PUNCT
ap-7583	291	4	recurrence	recurrence	NOUN
ap-7583	291	5	formula	formula	NOUN
ap-7583	291	6	:	:	PUNCT
ap-7583	291	7	ck+2	ck+2	NUM
ap-7583	291	8	=	=	SYM
ap-7583	291	9	−k(k	−k(k	NOUN
ap-7583	291	10	−	−	PROPN
ap-7583	292	1	1)α2	1)α2	PRON
ap-7583	292	2	+	+	CCONJ
ap-7583	292	3	k	k	PROPN
ap-7583	292	4	β1	β1	PROPN
ap-7583	292	5	(	(	PUNCT
ap-7583	292	6	k	k	PROPN
ap-7583	292	7	+	+	PROPN
ap-7583	292	8	1)(k	1)(k	NUM
ap-7583	293	1	+	+	SYM
ap-7583	293	2	2)α0	2)α0	NUM
ap-7583	293	3	ck	ck	NOUN
ap-7583	293	4	−	−	PROPN
ap-7583	293	5	ε1	ε1	PROPN
ap-7583	293	6	(	(	PUNCT
ap-7583	293	7	k	k	PROPN
ap-7583	293	8	+	+	PROPN
ap-7583	293	9	1)(k	1)(k	NUM
ap-7583	293	10	+	+	CCONJ
ap-7583	293	11	2	2	NUM
ap-7583	293	12	)	)	PUNCT
ap-7583	293	13	α0	α0	ADJ
ap-7583	293	14	ck−1	ck−1	NOUN
ap-7583	293	15	.	.	PUNCT
ap-7583	294	1	(	(	PUNCT
ap-7583	294	2	42	42	NUM
ap-7583	294	3	)	)	PUNCT
ap-7583	294	4	•	•	NUM
ap-7583	294	5	differential	differential	NOUN
ap-7583	294	6	equation	equation	NOUN
ap-7583	294	7	:	:	PUNCT
ap-7583	294	8	(	(	PUNCT
ap-7583	294	9	α0	α0	ADJ
ap-7583	294	10	+	+	CCONJ
ap-7583	294	11	α2	α2	ADJ
ap-7583	294	12	r2	r2	NOUN
ap-7583	294	13	)	)	PUNCT
ap-7583	294	14	u′′(r	u′′(r	NOUN
ap-7583	294	15	)	)	PUNCT
ap-7583	294	16	+	+	CCONJ
ap-7583	294	17	(	(	PUNCT
ap-7583	294	18	β1	β1	PROPN
ap-7583	294	19	+	+	CCONJ
ap-7583	294	20	β2	β2	NOUN
ap-7583	294	21	r2)u′(r	r2)u′(r	NOUN
ap-7583	294	22	)	)	PUNCT
ap-7583	294	23	+	+	NUM
ap-7583	295	1	ε1	ε1	VERB
ap-7583	295	2	r	r	NOUN
ap-7583	295	3	u(r	u(r	NOUN
ap-7583	295	4	)	)	PUNCT
ap-7583	296	1	=	=	SYM
ap-7583	296	2	0	0	X
ap-7583	296	3	.	.	PUNCT
ap-7583	297	1	(	(	PUNCT
ap-7583	297	2	43	43	NUM
ap-7583	297	3	)	)	PUNCT
ap-7583	297	4	recurrence	recurrence	NOUN
ap-7583	297	5	formula	formula	NOUN
ap-7583	297	6	:	:	PUNCT
ap-7583	297	7	ck+2	ck+2	NUM
ap-7583	297	8	=	=	SYM
ap-7583	297	9	−k(k	−k(k	NOUN
ap-7583	297	10	−	−	PROPN
ap-7583	298	1	1)α2	1)α2	PRON
ap-7583	298	2	+	+	CCONJ
ap-7583	298	3	β1	β1	PROPN
ap-7583	298	4	k	k	PROPN
ap-7583	298	5	)	)	PUNCT
ap-7583	298	6	(	(	PUNCT
ap-7583	298	7	k	k	X
ap-7583	298	8	+	+	PROPN
ap-7583	298	9	1	1	X
ap-7583	298	10	)	)	PUNCT
ap-7583	298	11	(	(	PUNCT
ap-7583	298	12	k	k	X
ap-7583	298	13	+	+	PROPN
ap-7583	298	14	2	2	X
ap-7583	298	15	)	)	PUNCT
ap-7583	298	16	α0	α0	ADJ
ap-7583	298	17	ck	ck	INTJ
ap-7583	298	18	−	−	PROPN
ap-7583	298	19	(	(	PUNCT
ap-7583	298	20	k	k	PROPN
ap-7583	298	21	−	−	PROPN
ap-7583	298	22	1)β2	1)β2	NUM
ap-7583	298	23	+	+	NUM
ap-7583	298	24	ε1	ε1	PROPN
ap-7583	298	25	(	(	PUNCT
ap-7583	298	26	k	k	PROPN
ap-7583	298	27	+	+	PROPN
ap-7583	298	28	1	1	NUM
ap-7583	298	29	)	)	PUNCT
ap-7583	298	30	(	(	PUNCT
ap-7583	298	31	k	k	X
ap-7583	298	32	+	+	PROPN
ap-7583	298	33	2	2	X
ap-7583	298	34	)	)	PUNCT
ap-7583	298	35	α0	α0	ADJ
ap-7583	298	36	ck−1	ck−1	NOUN
ap-7583	298	37	.	.	PUNCT
ap-7583	299	1	(	(	PUNCT
ap-7583	299	2	44	44	NUM
ap-7583	299	3	)	)	PUNCT
ap-7583	299	4	•	•	NUM
ap-7583	299	5	differential	differential	NOUN
ap-7583	299	6	equation	equation	NOUN
ap-7583	299	7	:	:	PUNCT
ap-7583	299	8	u′′(r	u′′(r	NOUN
ap-7583	299	9	)	)	PUNCT
ap-7583	299	10	+	+	CCONJ
ap-7583	299	11	β1	β1	PROPN
ap-7583	299	12	r	r	NOUN
ap-7583	299	13	u′(r	u′(r	PROPN
ap-7583	299	14	)	)	PUNCT
ap-7583	300	1	+	+	CCONJ
ap-7583	300	2	ε1	ε1	VERB
ap-7583	300	3	r	r	NOUN
ap-7583	300	4	u(r	u(r	NOUN
ap-7583	300	5	)	)	PUNCT
ap-7583	301	1	=	=	SYM
ap-7583	301	2	0	0	NUM
ap-7583	301	3	,	,	PUNCT
ap-7583	301	4	(	(	PUNCT
ap-7583	301	5	45	45	NUM
ap-7583	301	6	)	)	PUNCT
ap-7583	301	7	recurrence	recurrence	NOUN
ap-7583	301	8	formula	formula	NOUN
ap-7583	301	9	:	:	PUNCT
ap-7583	301	10	ck+2	ck+2	NUM
ap-7583	301	11	=	=	SYM
ap-7583	301	12	−	−	PROPN
ap-7583	301	13	k	k	PROPN
ap-7583	301	14	β1	β1	PROPN
ap-7583	301	15	(	(	PUNCT
ap-7583	301	16	k	k	PROPN
ap-7583	302	1	+	+	PROPN
ap-7583	302	2	1	1	X
ap-7583	302	3	)	)	PUNCT
ap-7583	302	4	(	(	PUNCT
ap-7583	302	5	k	k	X
ap-7583	302	6	+	+	PROPN
ap-7583	302	7	2	2	X
ap-7583	302	8	)	)	PUNCT
ap-7583	302	9	ck	ck	NOUN
ap-7583	303	1	−	−	PROPN
ap-7583	303	2	ε1	ε1	PROPN
ap-7583	303	3	(	(	PUNCT
ap-7583	303	4	k	k	PROPN
ap-7583	303	5	+	+	PROPN
ap-7583	303	6	1	1	NUM
ap-7583	303	7	)	)	PUNCT
ap-7583	303	8	(	(	PUNCT
ap-7583	303	9	k	k	X
ap-7583	303	10	+	+	PROPN
ap-7583	303	11	2	2	X
ap-7583	303	12	)	)	PUNCT
ap-7583	303	13	ck−1	ck−1	NOUN
ap-7583	303	14	.	.	PUNCT
ap-7583	303	15	(	(	PUNCT
ap-7583	303	16	46	46	NUM
ap-7583	303	17	)	)	PUNCT
ap-7583	303	18	3.2	3.2	NUM
ap-7583	303	19	.	.	PUNCT
ap-7583	304	1	polynomial	polynomial	ADJ
ap-7583	304	2	solutions	solution	NOUN
ap-7583	304	3	the	the	DET
ap-7583	304	4	series	series	NOUN
ap-7583	304	5	solution	solution	NOUN
ap-7583	304	6	y(r	y(r	NOUN
ap-7583	304	7	)	)	PUNCT
ap-7583	305	1	=	=	NOUN
ap-7583	305	2	∑∞	∑∞	NOUN
ap-7583	305	3	k=0	k=0	PROPN
ap-7583	305	4	ck	ck	INTJ
ap-7583	305	5	rk	rk	PRON
ap-7583	305	6	terminates	terminate	VERB
ap-7583	305	7	to	to	ADP
ap-7583	305	8	an	an	DET
ap-7583	305	9	nth	nth	NOUN
ap-7583	305	10	-	-	PUNCT
ap-7583	305	11	degree	degree	NOUN
ap-7583	305	12	polynomial	polynomial	NOUN
ap-7583	305	13	if	if	SCONJ
ap-7583	305	14	cn	cn	PROPN
ap-7583	305	15	̸=	̸=	PROPN
ap-7583	305	16	0	0	NUM
ap-7583	305	17	and	and	CCONJ
ap-7583	305	18	cj	cj	X
ap-7583	305	19	=	=	NOUN
ap-7583	305	20	0	0	NUM
ap-7583	305	21	for	for	ADP
ap-7583	305	22	all	all	DET
ap-7583	305	23	j	j	PROPN
ap-7583	305	24	≥	≥	NUM
ap-7583	305	25	n	n	NOUN
ap-7583	305	26	+	+	NUM
ap-7583	305	27	1	1	X
ap-7583	305	28	.	.	PUNCT
ap-7583	306	1	it	it	PRON
ap-7583	306	2	is	be	AUX
ap-7583	306	3	not	not	PART
ap-7583	306	4	difficult	difficult	ADJ
ap-7583	306	5	to	to	PART
ap-7583	306	6	show	show	VERB
ap-7583	306	7	by	by	ADP
ap-7583	306	8	direct	direct	ADJ
ap-7583	306	9	substitution	substitution	NOUN
ap-7583	306	10	that	that	PRON
ap-7583	306	11	for	for	ADP
ap-7583	306	12	polynomial	polynomial	ADJ
ap-7583	306	13	solutions	solution	NOUN
ap-7583	306	14	of	of	ADP
ap-7583	306	15	pn(r	pn(r	NOUN
ap-7583	306	16	)	)	PUNCT
ap-7583	306	17	=	=	SYM
ap-7583	307	1	∑n	∑n	PROPN
ap-7583	307	2	k=0	k=0	PROPN
ap-7583	307	3	ck	ck	ADJ
ap-7583	307	4	rk	rk	NOUN
ap-7583	307	5	,	,	PUNCT
ap-7583	307	6	it	it	PRON
ap-7583	307	7	is	be	AUX
ap-7583	307	8	necessary	necessary	ADJ
ap-7583	307	9	that	that	SCONJ
ap-7583	307	10	ε1	ε1	PROPN
ap-7583	307	11	=	=	SYM
ap-7583	307	12	−n	−n	X
ap-7583	307	13	(	(	PUNCT
ap-7583	307	14	n	n	CCONJ
ap-7583	307	15	−	−	PROPN
ap-7583	307	16	1	1	NUM
ap-7583	307	17	)	)	PUNCT
ap-7583	307	18	α3	α3	NOUN
ap-7583	307	19	−	−	PROPN
ap-7583	307	20	n	n	DET
ap-7583	307	21	β2	β2	NOUN
ap-7583	307	22	,	,	PUNCT
ap-7583	307	23	n	n	NOUN
ap-7583	307	24	=	=	SYM
ap-7583	307	25	0	0	NUM
ap-7583	307	26	,	,	PUNCT
ap-7583	307	27	1	1	NUM
ap-7583	307	28	,	,	PUNCT
ap-7583	307	29	2	2	NUM
ap-7583	307	30	,	,	PUNCT
ap-7583	307	31	·	·	PUNCT
ap-7583	307	32	·	·	PUNCT
ap-7583	307	33	·	·	PUNCT
ap-7583	307	34	.	.	PUNCT
ap-7583	308	1	(	(	PUNCT
ap-7583	308	2	47	47	NUM
ap-7583	308	3	)	)	PUNCT
ap-7583	308	4	173	173	NUM
ap-7583	308	5	nasser	nasser	PROPN
ap-7583	308	6	saad	saad	PROPN
ap-7583	308	7	acta	acta	PROPN
ap-7583	308	8	polytechnica	polytechnica	PROPN
ap-7583	308	9	furthermore	furthermore	ADV
ap-7583	308	10	,	,	PUNCT
ap-7583	308	11	the	the	DET
ap-7583	308	12	polynomial	polynomial	ADJ
ap-7583	308	13	solution	solution	NOUN
ap-7583	308	14	coefficients	coefficient	NOUN
ap-7583	308	15	{	{	PUNCT
ap-7583	308	16	ck}n	ck}n	PROPN
ap-7583	308	17	k=0	k=0	PROPN
ap-7583	308	18	satisfy	satisfy	VERB
ap-7583	308	19	a	a	DET
ap-7583	308	20	four	four	NUM
ap-7583	308	21	-	-	PUNCT
ap-7583	308	22	term	term	NOUN
ap-7583	308	23	recurrence	recurrence	NOUN
ap-7583	308	24	relation	relation	NOUN
ap-7583	308	25	,	,	PUNCT
ap-7583	308	26	see	see	VERB
ap-7583	308	27	(	(	PUNCT
ap-7583	308	28	26	26	NUM
ap-7583	308	29	)	)	PUNCT
ap-7583	308	30	,	,	PUNCT
ap-7583	308	31	(	(	PUNCT
ap-7583	308	32	(	(	PUNCT
ap-7583	308	33	k	k	X
ap-7583	308	34	−	−	PROPN
ap-7583	309	1	1	1	NUM
ap-7583	309	2	)	)	PUNCT
ap-7583	309	3	(	(	PUNCT
ap-7583	309	4	(	(	PUNCT
ap-7583	309	5	k	k	X
ap-7583	309	6	−	−	PROPN
ap-7583	309	7	2)α3	2)α3	PROPN
ap-7583	309	8	+	+	CCONJ
ap-7583	309	9	β2	β2	NOUN
ap-7583	309	10	)	)	PUNCT
ap-7583	310	1	+	+	CCONJ
ap-7583	310	2	ε1;n	ε1;n	X
ap-7583	310	3	)	)	PUNCT
ap-7583	311	1	ck−1	ck−1	NOUN
ap-7583	312	1	+	+	CCONJ
ap-7583	312	2	(	(	PUNCT
ap-7583	312	3	k	k	X
ap-7583	312	4	(	(	PUNCT
ap-7583	312	5	(	(	PUNCT
ap-7583	312	6	k	k	X
ap-7583	312	7	−	−	PROPN
ap-7583	312	8	1)α2	1)α2	PROPN
ap-7583	312	9	+	+	CCONJ
ap-7583	312	10	β1	β1	PROPN
ap-7583	312	11	)	)	PUNCT
ap-7583	313	1	+	+	CCONJ
ap-7583	313	2	ε0;n	ε0;n	NUM
ap-7583	313	3	)	)	PUNCT
ap-7583	313	4	ck	ck	PROPN
ap-7583	314	1	+	+	CCONJ
ap-7583	314	2	(	(	PUNCT
ap-7583	314	3	k	k	X
ap-7583	314	4	+	+	NUM
ap-7583	314	5	1)(kα1	1)(kα1	NUM
ap-7583	314	6	+	+	CCONJ
ap-7583	314	7	β0	β0	NOUN
ap-7583	314	8	)	)	PUNCT
ap-7583	314	9	ck+1	ck+1	PUNCT
ap-7583	315	1	+	+	CCONJ
ap-7583	315	2	(	(	PUNCT
ap-7583	315	3	k	k	PROPN
ap-7583	315	4	+	+	NOUN
ap-7583	315	5	1)(k	1)(k	NUM
ap-7583	315	6	+	+	SYM
ap-7583	315	7	2)α0	2)α0	NUM
ap-7583	315	8	ck+2	ck+2	NOUN
ap-7583	315	9	=	=	SYM
ap-7583	315	10	0	0	NUM
ap-7583	315	11	,	,	PUNCT
ap-7583	315	12	k	k	PROPN
ap-7583	315	13	=	=	SYM
ap-7583	315	14	0	0	NUM
ap-7583	315	15	,	,	PUNCT
ap-7583	315	16	1	1	NUM
ap-7583	315	17	,	,	PUNCT
ap-7583	315	18	.	.	PUNCT
ap-7583	315	19	.	.	PUNCT
ap-7583	316	1	.	.	PUNCT
ap-7583	317	1	,	,	PUNCT
ap-7583	317	2	n	n	PROPN
ap-7583	317	3	+	+	NUM
ap-7583	317	4	1	1	NUM
ap-7583	317	5	,	,	PUNCT
ap-7583	317	6	(	(	PUNCT
ap-7583	317	7	48	48	NUM
ap-7583	317	8	)	)	PUNCT
ap-7583	317	9	that	that	PRON
ap-7583	317	10	generates	generate	VERB
ap-7583	317	11	a	a	DET
ap-7583	317	12	system	system	NOUN
ap-7583	317	13	of	of	ADP
ap-7583	317	14	(	(	PUNCT
ap-7583	317	15	n	n	X
ap-7583	317	16	+	+	CCONJ
ap-7583	317	17	2	2	NUM
ap-7583	317	18	)	)	PUNCT
ap-7583	317	19	linear	linear	ADJ
ap-7583	317	20	equations	equation	NOUN
ap-7583	317	21	in	in	ADP
ap-7583	317	22	{	{	PUNCT
ap-7583	317	23	ck}n	ck}n	PROPN
ap-7583	317	24	k=0	k=0	PROPN
ap-7583	317	25	:	:	PUNCT
ap-7583	317	26	n−equations︷	n−equations︷	PROPN
ap-7583	317	27	︸︸	︸︸	PUNCT
ap-7583	317	28	︷︸	︷︸	PROPN
ap-7583	317	29	︷︷	︷︷	PROPN
ap-7583	317	30	︸	︸	X
ap-7583	317	31	(	(	PUNCT
ap-7583	317	32	n+2)−equations	n+2)−equation	NOUN
ap-7583	317	33	the	the	DET
ap-7583	317	34	first	first	ADJ
ap-7583	317	35	n	n	PRON
ap-7583	317	36	equations	equation	NOUN
ap-7583	317	37	are	are	X
ap-7583	317	38	k	k	X
ap-7583	318	1	=	=	PUNCT
ap-7583	318	2	0	0	NUM
ap-7583	318	3	,	,	PUNCT
ap-7583	318	4	→	→	SYM
ap-7583	318	5	ε0c0	ε0c0	X
ap-7583	318	6	+	+	X
ap-7583	318	7	β0c1	β0c1	PUNCT
ap-7583	318	8	+	+	SYM
ap-7583	318	9	2	2	NUM
ap-7583	318	10	α0	α0	ADJ
ap-7583	318	11	c2	c2	PROPN
ap-7583	318	12	=	=	SYM
ap-7583	318	13	0	0	PUNCT
ap-7583	319	1	k	k	NOUN
ap-7583	319	2	=	=	SYM
ap-7583	319	3	1	1	NUM
ap-7583	319	4	,	,	PUNCT
ap-7583	319	5	→	→	SYM
ap-7583	319	6	ε1c0	ε1c0	X
ap-7583	319	7	+	+	CCONJ
ap-7583	319	8	(	(	PUNCT
ap-7583	319	9	β1	β1	NOUN
ap-7583	319	10	+	+	CCONJ
ap-7583	319	11	ε0)c1	ε0)c1	NOUN
ap-7583	319	12	+	+	CCONJ
ap-7583	319	13	2(α1	2(α1	NUM
ap-7583	319	14	+	+	CCONJ
ap-7583	319	15	β0)c2	β0)c2	PUNCT
ap-7583	320	1	+	+	CCONJ
ap-7583	320	2	6	6	NUM
ap-7583	320	3	α0	α0	ADJ
ap-7583	320	4	c3	c3	NOUN
ap-7583	320	5	=	=	SYM
ap-7583	320	6	0	0	PUNCT
ap-7583	320	7	k	k	NOUN
ap-7583	320	8	=	=	SYM
ap-7583	320	9	2	2	NUM
ap-7583	320	10	,	,	PUNCT
ap-7583	320	11	→	→	PUNCT
ap-7583	320	12	(	(	PUNCT
ap-7583	320	13	β2	β2	VERB
ap-7583	320	14	+	+	CCONJ
ap-7583	320	15	ε1)c1	ε1)c1	PROPN
ap-7583	320	16	+	+	CCONJ
ap-7583	320	17	(	(	PUNCT
ap-7583	320	18	2	2	NUM
ap-7583	320	19	α2	α2	ADJ
ap-7583	320	20	+	+	CCONJ
ap-7583	320	21	2	2	NUM
ap-7583	320	22	β1	β1	NOUN
ap-7583	320	23	+	+	CCONJ
ap-7583	320	24	ε0)c2	ε0)c2	X
ap-7583	320	25	+	+	CCONJ
ap-7583	320	26	3(2	3(2	NUM
ap-7583	320	27	α1	α1	NOUN
ap-7583	320	28	+	+	CCONJ
ap-7583	320	29	β0)c3	β0)c3	NOUN
ap-7583	320	30	+	+	CCONJ
ap-7583	320	31	12	12	NUM
ap-7583	320	32	α0	α0	ADJ
ap-7583	320	33	c4	c4	NOUN
ap-7583	320	34	=	=	NOUN
ap-7583	320	35	0	0	PUNCT
ap-7583	321	1	k	k	NOUN
ap-7583	321	2	=	=	SYM
ap-7583	321	3	3	3	NUM
ap-7583	321	4	,	,	PUNCT
ap-7583	321	5	→	→	PUNCT
ap-7583	321	6	(	(	PUNCT
ap-7583	321	7	2α3	2α3	NUM
ap-7583	321	8	+	+	SYM
ap-7583	321	9	2	2	NUM
ap-7583	321	10	β2	β2	NOUN
ap-7583	321	11	+	+	CCONJ
ap-7583	321	12	ε1)c2	ε1)c2	VERB
ap-7583	322	1	+	+	CCONJ
ap-7583	322	2	(	(	PUNCT
ap-7583	322	3	6	6	NUM
ap-7583	322	4	α2	α2	ADJ
ap-7583	322	5	+	+	CCONJ
ap-7583	322	6	3	3	NUM
ap-7583	322	7	β1	β1	NOUN
ap-7583	322	8	+	+	CCONJ
ap-7583	322	9	ε0)c3	ε0)c3	NOUN
ap-7583	322	10	+	+	CCONJ
ap-7583	322	11	4(3	4(3	NUM
ap-7583	322	12	α1	α1	NOUN
ap-7583	322	13	+	+	CCONJ
ap-7583	322	14	β0)c4	β0)c4	NOUN
ap-7583	322	15	+	+	ADJ
ap-7583	322	16	20	20	NUM
ap-7583	322	17	α0	α0	ADJ
ap-7583	322	18	c5	c5	PROPN
ap-7583	322	19	=	=	PUNCT
ap-7583	322	20	0	0	NUM
ap-7583	322	21	...	...	PUNCT
ap-7583	323	1	k	k	X
ap-7583	323	2	=	=	PUNCT
ap-7583	323	3	n	n	CCONJ
ap-7583	323	4	−	−	PROPN
ap-7583	323	5	1	1	NUM
ap-7583	323	6	,	,	PUNCT
ap-7583	323	7	→	→	PUNCT
ap-7583	323	8	(	(	PUNCT
ap-7583	323	9	(	(	PUNCT
ap-7583	323	10	n	n	CCONJ
ap-7583	323	11	−	−	PROPN
ap-7583	323	12	2	2	NUM
ap-7583	323	13	)	)	PUNCT
ap-7583	323	14	(	(	PUNCT
ap-7583	323	15	(	(	PUNCT
ap-7583	323	16	n	n	CCONJ
ap-7583	323	17	−	−	NOUN
ap-7583	323	18	3	3	NUM
ap-7583	323	19	)	)	PUNCT
ap-7583	323	20	α3	α3	NOUN
ap-7583	323	21	+	+	CCONJ
ap-7583	323	22	β2	β2	NOUN
ap-7583	323	23	)	)	PUNCT
ap-7583	324	1	+	+	CCONJ
ap-7583	324	2	ε1;n	ε1;n	NOUN
ap-7583	324	3	)	)	PUNCT
ap-7583	325	1	cn−2	cn−2	PROPN
ap-7583	325	2	+	+	CCONJ
ap-7583	325	3	(	(	PUNCT
ap-7583	325	4	(	(	PUNCT
ap-7583	325	5	n	n	CCONJ
ap-7583	325	6	−	−	PROPN
ap-7583	325	7	1	1	NUM
ap-7583	325	8	)	)	PUNCT
ap-7583	325	9	(	(	PUNCT
ap-7583	325	10	(	(	PUNCT
ap-7583	325	11	n	n	CCONJ
ap-7583	325	12	−	−	PROPN
ap-7583	325	13	2	2	NUM
ap-7583	325	14	)	)	PUNCT
ap-7583	325	15	α2	α2	PROPN
ap-7583	325	16	+	+	CCONJ
ap-7583	325	17	β1	β1	PROPN
ap-7583	325	18	)	)	PUNCT
ap-7583	326	1	+	+	CCONJ
ap-7583	326	2	ε0;n	ε0;n	NOUN
ap-7583	326	3	)	)	PUNCT
ap-7583	327	1	cn−1	cn−1	PROPN
ap-7583	327	2	+	+	PROPN
ap-7583	327	3	n	n	PROPN
ap-7583	327	4	(	(	PUNCT
ap-7583	327	5	(	(	PUNCT
ap-7583	327	6	n	n	CCONJ
ap-7583	327	7	−	−	PROPN
ap-7583	327	8	1	1	NUM
ap-7583	327	9	)	)	PUNCT
ap-7583	327	10	α1	α1	PROPN
ap-7583	327	11	+	+	CCONJ
ap-7583	327	12	β0	β0	ADJ
ap-7583	327	13	)	)	PUNCT
ap-7583	327	14	cn	cn	PROPN
ap-7583	328	1	=	=	NOUN
ap-7583	328	2	0	0	PROPN
ap-7583	328	3	.	.	PUNCT
ap-7583	329	1	(	(	PUNCT
ap-7583	329	2	49	49	NUM
ap-7583	329	3	)	)	PUNCT
ap-7583	329	4	these	these	DET
ap-7583	329	5	equations	equation	NOUN
ap-7583	329	6	permit	permit	VERB
ap-7583	329	7	the	the	DET
ap-7583	329	8	evaluation	evaluation	NOUN
ap-7583	329	9	,	,	PUNCT
ap-7583	329	10	using	use	VERB
ap-7583	329	11	say	say	VERB
ap-7583	329	12	cramer	cramer	PROPN
ap-7583	329	13	’s	’s	PART
ap-7583	329	14	rule	rule	NOUN
ap-7583	329	15	,	,	PUNCT
ap-7583	329	16	of	of	ADP
ap-7583	329	17	the	the	DET
ap-7583	329	18	coefficients	coefficient	NOUN
ap-7583	329	19	{	{	PUNCT
ap-7583	329	20	ck}n	ck}n	PROPN
ap-7583	329	21	k=1	k=1	NOUN
ap-7583	329	22	of	of	ADP
ap-7583	329	23	the	the	DET
ap-7583	329	24	polynomial	polynomial	ADJ
ap-7583	329	25	solution	solution	NOUN
ap-7583	329	26	in	in	ADP
ap-7583	329	27	terms	term	NOUN
ap-7583	329	28	of	of	ADP
ap-7583	329	29	the	the	DET
ap-7583	329	30	non	non	ADJ
ap-7583	329	31	-	-	ADJ
ap-7583	329	32	zero	zero	ADJ
ap-7583	329	33	constant	constant	ADJ
ap-7583	329	34	c0	c0	NOUN
ap-7583	329	35	.	.	PUNCT
ap-7583	330	1	the	the	DET
ap-7583	330	2	(	(	PUNCT
ap-7583	330	3	n	n	NOUN
ap-7583	330	4	+	+	CCONJ
ap-7583	330	5	1)th	1)th	NUM
ap-7583	330	6	equation	equation	NOUN
ap-7583	330	7	(	(	PUNCT
ap-7583	330	8	(	(	PUNCT
ap-7583	330	9	n	n	CCONJ
ap-7583	330	10	−	−	PROPN
ap-7583	330	11	1	1	NUM
ap-7583	330	12	)	)	PUNCT
ap-7583	330	13	(	(	PUNCT
ap-7583	330	14	n	n	CCONJ
ap-7583	330	15	−	−	PROPN
ap-7583	330	16	2	2	NUM
ap-7583	330	17	)	)	PUNCT
ap-7583	330	18	α3	α3	NOUN
ap-7583	330	19	+	+	CCONJ
ap-7583	330	20	(	(	PUNCT
ap-7583	330	21	n	n	CCONJ
ap-7583	330	22	−	−	PROPN
ap-7583	330	23	1	1	NUM
ap-7583	330	24	)	)	PUNCT
ap-7583	330	25	β2	β2	NOUN
ap-7583	330	26	+	+	CCONJ
ap-7583	330	27	ε1	ε1	PROPN
ap-7583	330	28	)	)	PUNCT
ap-7583	330	29	cn−1	cn−1	PROPN
ap-7583	330	30	+	+	CCONJ
ap-7583	330	31	(	(	PUNCT
ap-7583	330	32	n(n	n(n	NOUN
ap-7583	330	33	−	−	NOUN
ap-7583	330	34	1	1	X
ap-7583	330	35	)	)	PUNCT
ap-7583	330	36	α2	α2	PROPN
ap-7583	330	37	+	+	CCONJ
ap-7583	330	38	n	n	CCONJ
ap-7583	330	39	β1	β1	NOUN
ap-7583	330	40	+	+	CCONJ
ap-7583	330	41	ε0	ε0	PROPN
ap-7583	330	42	)	)	PUNCT
ap-7583	330	43	cn	cn	PROPN
ap-7583	331	1	=	=	NOUN
ap-7583	331	2	0	0	PROPN
ap-7583	331	3	,	,	PUNCT
ap-7583	331	4	(	(	PUNCT
ap-7583	331	5	50	50	NUM
ap-7583	331	6	)	)	PUNCT
ap-7583	331	7	gives	give	VERB
ap-7583	331	8	our	our	PRON
ap-7583	331	9	sufficient	sufficient	ADJ
ap-7583	331	10	condition	condition	NOUN
ap-7583	331	11	that	that	PRON
ap-7583	331	12	relates	relate	VERB
ap-7583	331	13	ε0	ε0	PROPN
ap-7583	331	14	≡	≡	PROPN
ap-7583	331	15	ε0;n	ε0;n	PROPN
ap-7583	331	16	to	to	ADP
ap-7583	331	17	the	the	DET
ap-7583	331	18	remaining	remain	VERB
ap-7583	331	19	parameters	parameter	NOUN
ap-7583	331	20	of	of	ADP
ap-7583	331	21	the	the	DET
ap-7583	331	22	differential	differential	ADJ
ap-7583	331	23	equation	equation	NOUN
ap-7583	331	24	.	.	PUNCT
ap-7583	332	1	finally	finally	ADV
ap-7583	332	2	,	,	PUNCT
ap-7583	332	3	the	the	DET
ap-7583	332	4	(	(	PUNCT
ap-7583	332	5	n	n	NOUN
ap-7583	332	6	+	+	CCONJ
ap-7583	332	7	2)th	2)th	NOUN
ap-7583	332	8	equation	equation	NOUN
ap-7583	332	9	ε1;n	ε1;n	NOUN
ap-7583	332	10	=	=	SYM
ap-7583	332	11	−n	−n	ADJ
ap-7583	332	12	(	(	PUNCT
ap-7583	332	13	n	n	CCONJ
ap-7583	332	14	−	−	PROPN
ap-7583	332	15	1	1	NUM
ap-7583	332	16	)	)	PUNCT
ap-7583	332	17	α3	α3	NOUN
ap-7583	332	18	−	−	PROPN
ap-7583	332	19	n	n	PRON
ap-7583	332	20	β2	β2	NOUN
ap-7583	332	21	,	,	PUNCT
ap-7583	332	22	n	n	NOUN
ap-7583	332	23	=	=	SYM
ap-7583	332	24	0	0	NUM
ap-7583	332	25	,	,	PUNCT
ap-7583	332	26	1	1	NUM
ap-7583	332	27	,	,	PUNCT
ap-7583	332	28	·	·	PUNCT
ap-7583	332	29	·	·	PUNCT
ap-7583	332	30	·	·	PUNCT
ap-7583	332	31	,	,	PUNCT
ap-7583	332	32	(	(	PUNCT
ap-7583	332	33	51	51	NUM
ap-7583	332	34	)	)	PUNCT
ap-7583	332	35	re	re	VERB
ap-7583	332	36	-	-	VERB
ap-7583	332	37	establishes	establish	VERB
ap-7583	332	38	the	the	DET
ap-7583	332	39	necessary	necessary	ADJ
ap-7583	332	40	condition	condition	NOUN
ap-7583	332	41	(	(	PUNCT
ap-7583	332	42	ε1	ε1	PROPN
ap-7583	332	43	≡	≡	PROPN
ap-7583	332	44	ε1;n	ε1;n	PROPN
ap-7583	332	45	)	)	PUNCT
ap-7583	332	46	for	for	ADP
ap-7583	332	47	the	the	DET
ap-7583	332	48	existence	existence	NOUN
ap-7583	332	49	of	of	ADP
ap-7583	332	50	the	the	DET
ap-7583	332	51	n	n	CCONJ
ap-7583	332	52	-	-	PUNCT
ap-7583	332	53	degree	degree	NOUN
ap-7583	332	54	polynomial	polynomial	ADJ
ap-7583	332	55	solution	solution	NOUN
ap-7583	332	56	,	,	PUNCT
ap-7583	332	57	see	see	VERB
ap-7583	332	58	(	(	PUNCT
ap-7583	332	59	47	47	NUM
ap-7583	332	60	)	)	PUNCT
ap-7583	332	61	.	.	PUNCT
ap-7583	333	1	for	for	ADP
ap-7583	333	2	a	a	DET
ap-7583	333	3	non	non	ADJ
ap-7583	333	4	-	-	ADJ
ap-7583	333	5	zero	zero	NUM
ap-7583	333	6	solution	solution	NOUN
ap-7583	333	7	,	,	PUNCT
ap-7583	333	8	the	the	DET
ap-7583	333	9	n	n	NOUN
ap-7583	333	10	+	+	CCONJ
ap-7583	333	11	1	1	NUM
ap-7583	333	12	linear	linear	NOUN
ap-7583	333	13	equations	equation	NOUN
ap-7583	333	14	generated	generate	VERB
ap-7583	333	15	by	by	ADP
ap-7583	333	16	the	the	DET
ap-7583	333	17	recurrence	recurrence	NOUN
ap-7583	333	18	relation	relation	NOUN
ap-7583	333	19	(	(	PUNCT
ap-7583	333	20	48	48	NUM
ap-7583	333	21	)	)	PUNCT
ap-7583	333	22	require	require	VERB
ap-7583	333	23	the	the	DET
ap-7583	333	24	vanishing	vanishing	NOUN
ap-7583	333	25	of	of	ADP
ap-7583	333	26	the	the	DET
ap-7583	333	27	(	(	PUNCT
ap-7583	333	28	n	n	NOUN
ap-7583	333	29	+	+	CCONJ
ap-7583	333	30	1	1	NUM
ap-7583	333	31	)	)	PUNCT
ap-7583	333	32	×	×	NOUN
ap-7583	333	33	(	(	PUNCT
ap-7583	333	34	n	n	X
ap-7583	333	35	+	+	X
ap-7583	333	36	1)-determinant	1)-determinant	NUM
ap-7583	333	37	(	(	PUNCT
ap-7583	333	38	with	with	ADP
ap-7583	333	39	four	four	NUM
ap-7583	333	40	main	main	ADJ
ap-7583	333	41	diagonals	diagonal	NOUN
ap-7583	333	42	and	and	CCONJ
ap-7583	333	43	all	all	DET
ap-7583	333	44	other	other	ADJ
ap-7583	333	45	entries	entry	NOUN
ap-7583	333	46	being	be	AUX
ap-7583	333	47	zeros	zero	NOUN
ap-7583	333	48	)	)	PUNCT
ap-7583	333	49	∆n+1	∆n+1	PROPN
ap-7583	333	50	=	=	SYM
ap-7583	333	51	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	333	52	s0	s0	PROPN
ap-7583	333	53	t1	t1	PROPN
ap-7583	333	54	η1	η1	PROPN
ap-7583	333	55	γ1	γ1	PROPN
ap-7583	333	56	s1	s1	PROPN
ap-7583	333	57	t2	t2	PROPN
ap-7583	333	58	η2	η2	VERB
ap-7583	333	59	γ2	γ2	PROPN
ap-7583	333	60	s2	s2	PROPN
ap-7583	333	61	t3	t3	PROPN
ap-7583	333	62	η3	η3	PROPN
ap-7583	333	63	.	.	PUNCT
ap-7583	333	64	.	.	PUNCT
ap-7583	333	65	.	.	PUNCT
ap-7583	333	66	.	.	PUNCT
ap-7583	333	67	.	.	PUNCT
ap-7583	333	68	.	.	PUNCT
ap-7583	333	69	.	.	PUNCT
ap-7583	333	70	.	.	PUNCT
ap-7583	333	71	.	.	PUNCT
ap-7583	333	72	.	.	PUNCT
ap-7583	333	73	.	.	PUNCT
ap-7583	333	74	.	.	PUNCT
ap-7583	334	1	γn−2	γn−2	PROPN
ap-7583	334	2	sn−2	sn−2	ADV
ap-7583	334	3	tn−1	tn−1	PROPN
ap-7583	334	4	ηn−1	ηn−1	PROPN
ap-7583	334	5	γn−1	γn−1	PROPN
ap-7583	334	6	sn−1	sn−1	PROPN
ap-7583	334	7	tn	tn	PROPN
ap-7583	334	8	γn	γn	ADP
ap-7583	334	9	sn	sn	PROPN
ap-7583	334	10	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	334	11	,	,	PUNCT
ap-7583	334	12	where	where	SCONJ
ap-7583	334	13	sk	sk	ADP
ap-7583	334	14	=	=	PROPN
ap-7583	334	15	ε0;n	ε0;n	PROPN
ap-7583	334	16	+	+	CCONJ
ap-7583	334	17	k	k	X
ap-7583	334	18	(	(	PUNCT
ap-7583	334	19	(	(	PUNCT
ap-7583	334	20	k	k	X
ap-7583	334	21	−	−	PROPN
ap-7583	334	22	1)α2	1)α2	PROPN
ap-7583	334	23	+	+	CCONJ
ap-7583	334	24	β1	β1	PROPN
ap-7583	334	25	)	)	PUNCT
ap-7583	334	26	,	,	PUNCT
ap-7583	334	27	tk	tk	PROPN
ap-7583	334	28	=	=	SYM
ap-7583	334	29	k	k	PROPN
ap-7583	335	1	(	(	PUNCT
ap-7583	335	2	(	(	PUNCT
ap-7583	335	3	k	k	X
ap-7583	335	4	−	−	PROPN
ap-7583	335	5	1)α1	1)α1	NUM
ap-7583	335	6	+	+	CCONJ
ap-7583	335	7	β0	β0	PROPN
ap-7583	335	8	)	)	PUNCT
ap-7583	335	9	,	,	PUNCT
ap-7583	335	10	γk	γk	X
ap-7583	335	11	=	=	PUNCT
ap-7583	335	12	ε1;n	ε1;n	PROPN
ap-7583	335	13	+	+	X
ap-7583	336	1	(	(	PUNCT
ap-7583	336	2	k	k	NOUN
ap-7583	336	3	−	−	PROPN
ap-7583	336	4	1	1	NUM
ap-7583	336	5	)	)	PUNCT
ap-7583	336	6	(	(	PUNCT
ap-7583	336	7	(	(	PUNCT
ap-7583	336	8	k	k	X
ap-7583	336	9	−	−	PROPN
ap-7583	336	10	2)α3	2)α3	PROPN
ap-7583	336	11	+	+	CCONJ
ap-7583	336	12	β2	β2	NOUN
ap-7583	336	13	)	)	PUNCT
ap-7583	336	14	,	,	PUNCT
ap-7583	336	15	ηk	ηk	X
ap-7583	336	16	=	=	SYM
ap-7583	336	17	k(k	k(k	PROPN
ap-7583	337	1	+	+	CCONJ
ap-7583	337	2	1)α0	1)α0	NUM
ap-7583	337	3	,	,	PUNCT
ap-7583	337	4	and	and	CCONJ
ap-7583	337	5	for	for	ADP
ap-7583	337	6	fixed	fixed	ADJ
ap-7583	337	7	n	n	NOUN
ap-7583	337	8	,	,	PUNCT
ap-7583	337	9	ε1;n	ε1;n	NOUN
ap-7583	337	10	=	=	SYM
ap-7583	337	11	−n	−n	NOUN
ap-7583	337	12	(	(	PUNCT
ap-7583	337	13	n	n	CCONJ
ap-7583	337	14	−	−	PROPN
ap-7583	337	15	1	1	NUM
ap-7583	337	16	)	)	PUNCT
ap-7583	337	17	α3	α3	NOUN
ap-7583	337	18	−	−	PROPN
ap-7583	337	19	n	n	PRON
ap-7583	337	20	β2	β2	NOUN
ap-7583	337	21	.	.	PUNCT
ap-7583	338	1	(	(	PUNCT
ap-7583	338	2	52	52	NUM
ap-7583	338	3	)	)	PUNCT
ap-7583	338	4	174	174	NUM
ap-7583	338	5	vol	vol	NOUN
ap-7583	338	6	.	.	PUNCT
ap-7583	339	1	62	62	NUM
ap-7583	339	2	no	no	INTJ
ap-7583	339	3	.	.	PUNCT
ap-7583	340	1	1/2022	1/2022	NUM
ap-7583	340	2	on	on	ADP
ap-7583	340	3	generalized	generalized	ADJ
ap-7583	340	4	heun	heun	NOUN
ap-7583	340	5	equation	equation	NOUN
ap-7583	340	6	with	with	ADP
ap-7583	340	7	some	some	DET
ap-7583	340	8	mathematical	mathematical	NOUN
ap-7583	340	9	.	.	PUNCT
ap-7583	340	10	.	.	PUNCT
ap-7583	340	11	.	.	PUNCT
ap-7583	341	1	a	a	DET
ap-7583	341	2	simple	simple	ADJ
ap-7583	341	3	relation	relation	NOUN
ap-7583	341	4	to	to	PART
ap-7583	341	5	evaluate	evaluate	VERB
ap-7583	341	6	this	this	DET
ap-7583	341	7	determinant	determinant	ADJ
ap-7583	341	8	in	in	ADP
ap-7583	341	9	terms	term	NOUN
ap-7583	341	10	of	of	ADP
ap-7583	341	11	lower	low	ADJ
ap-7583	341	12	-	-	PUNCT
ap-7583	341	13	degree	degree	NOUN
ap-7583	341	14	determinants	determinant	NOUN
ap-7583	341	15	is	be	AUX
ap-7583	341	16	given	give	VERB
ap-7583	341	17	by	by	ADP
ap-7583	341	18	∆k+1	∆k+1	NOUN
ap-7583	341	19	=	=	PUNCT
ap-7583	341	20	sk	sk	PROPN
ap-7583	341	21	∆k	∆k	PROPN
ap-7583	341	22	−	−	PROPN
ap-7583	341	23	γk	γk	PROPN
ap-7583	341	24	tk	tk	PROPN
ap-7583	342	1	∆k−1	∆k−1	PROPN
ap-7583	342	2	+	+	PROPN
ap-7583	342	3	γk	γk	PROPN
ap-7583	342	4	γk−1	γk−1	PROPN
ap-7583	342	5	ηk−1	ηk−1	PROPN
ap-7583	342	6	∆k−2	∆k−2	PRON
ap-7583	342	7	,	,	PUNCT
ap-7583	342	8	(	(	PUNCT
ap-7583	342	9	∆−2	∆−2	X
ap-7583	342	10	=	=	SYM
ap-7583	342	11	∆−1	∆−1	PROPN
ap-7583	342	12	=	=	SYM
ap-7583	342	13	0	0	PROPN
ap-7583	342	14	,	,	PUNCT
ap-7583	342	15	∆0	∆0	NOUN
ap-7583	342	16	=	=	VERB
ap-7583	342	17	1	1	NUM
ap-7583	342	18	,	,	PUNCT
ap-7583	342	19	k	k	NOUN
ap-7583	342	20	=	=	SYM
ap-7583	342	21	0	0	NUM
ap-7583	342	22	,	,	PUNCT
ap-7583	342	23	1	1	NUM
ap-7583	342	24	,	,	PUNCT
ap-7583	342	25	.	.	PUNCT
ap-7583	342	26	.	.	PUNCT
ap-7583	343	1	.	.	PUNCT
ap-7583	343	2	,	,	PUNCT
ap-7583	343	3	n	n	CCONJ
ap-7583	343	4	)	)	PUNCT
ap-7583	343	5	.	.	PUNCT
ap-7583	344	1	(	(	PUNCT
ap-7583	344	2	53	53	NUM
ap-7583	344	3	)	)	PUNCT
ap-7583	344	4	although	although	SCONJ
ap-7583	344	5	there	there	PRON
ap-7583	344	6	is	be	VERB
ap-7583	344	7	a	a	DET
ap-7583	344	8	classical	classical	ADJ
ap-7583	344	9	theorem	theorem	NOUN
ap-7583	344	10	[	[	X
ap-7583	344	11	24	24	NUM
ap-7583	344	12	]	]	PUNCT
ap-7583	344	13	that	that	PRON
ap-7583	344	14	guarantees	guarantee	VERB
ap-7583	344	15	the	the	DET
ap-7583	344	16	simple	simple	ADJ
ap-7583	344	17	distinct	distinct	ADJ
ap-7583	344	18	real	real	ADJ
ap-7583	344	19	roots	root	NOUN
ap-7583	344	20	of	of	ADP
ap-7583	344	21	the	the	DET
ap-7583	344	22	three	three	NUM
ap-7583	344	23	diagonal	diagonal	ADJ
ap-7583	344	24	matrix	matrix	NOUN
ap-7583	344	25	,	,	PUNCT
ap-7583	344	26	to	to	ADP
ap-7583	344	27	the	the	DET
ap-7583	344	28	best	good	ADJ
ap-7583	344	29	of	of	ADP
ap-7583	344	30	our	our	PRON
ap-7583	344	31	knowledge	knowledge	NOUN
ap-7583	344	32	,	,	PUNCT
ap-7583	344	33	there	there	PRON
ap-7583	344	34	is	be	VERB
ap-7583	344	35	no	no	DET
ap-7583	344	36	such	such	ADJ
ap-7583	344	37	theorem	theorem	NOUN
ap-7583	344	38	available	available	ADJ
ap-7583	344	39	for	for	ADP
ap-7583	344	40	the	the	DET
ap-7583	344	41	matrix	matrix	NOUN
ap-7583	344	42	-	-	PUNCT
ap-7583	344	43	type	type	NOUN
ap-7583	344	44	(	(	PUNCT
ap-7583	344	45	52	52	NUM
ap-7583	344	46	)	)	PUNCT
ap-7583	344	47	.	.	PUNCT
ap-7583	345	1	however	however	ADV
ap-7583	345	2	,	,	PUNCT
ap-7583	345	3	we	we	PRON
ap-7583	345	4	shall	shall	AUX
ap-7583	345	5	assume	assume	VERB
ap-7583	345	6	,	,	PUNCT
ap-7583	345	7	in	in	ADP
ap-7583	345	8	the	the	DET
ap-7583	345	9	following	follow	VERB
ap-7583	345	10	example	example	NOUN
ap-7583	345	11	,	,	PUNCT
ap-7583	345	12	that	that	SCONJ
ap-7583	345	13	the	the	DET
ap-7583	345	14	matrix	matrix	NOUN
ap-7583	345	15	entries	entry	NOUN
ap-7583	345	16	allow	allow	VERB
ap-7583	345	17	for	for	ADP
ap-7583	345	18	the	the	DET
ap-7583	345	19	distinct	distinct	ADJ
ap-7583	345	20	real	real	ADJ
ap-7583	345	21	roots	root	NOUN
ap-7583	345	22	of	of	ADP
ap-7583	345	23	the	the	DET
ap-7583	345	24	resulting	result	VERB
ap-7583	345	25	polynomial	polynomial	NOUN
ap-7583	345	26	of	of	ADP
ap-7583	345	27	ε0;n	ε0;n	PROPN
ap-7583	345	28	.	.	PUNCT
ap-7583	346	1	illustrative	illustrative	ADJ
ap-7583	346	2	example	example	NOUN
ap-7583	346	3	:	:	PUNCT
ap-7583	346	4	•	•	NOUN
ap-7583	346	5	for	for	ADP
ap-7583	346	6	the	the	DET
ap-7583	346	7	zero	zero	NUM
ap-7583	346	8	-	-	PUNCT
ap-7583	346	9	degree	degree	NOUN
ap-7583	346	10	polynomial	polynomial	ADJ
ap-7583	346	11	solution	solution	NOUN
ap-7583	346	12	p0	p0	NOUN
ap-7583	346	13	(	(	PUNCT
ap-7583	346	14	r	r	NOUN
ap-7583	346	15	)	)	PUNCT
ap-7583	346	16	=	=	SYM
ap-7583	346	17	1	1	NUM
ap-7583	346	18	,	,	PUNCT
ap-7583	346	19	i.e.	i.e.	X
ap-7583	346	20	,	,	PUNCT
ap-7583	346	21	n	n	PROPN
ap-7583	346	22	=	=	SYM
ap-7583	346	23	0	0	NUM
ap-7583	346	24	,	,	PUNCT
ap-7583	346	25	the	the	DET
ap-7583	346	26	coefficients	coefficient	NOUN
ap-7583	346	27	cj	cj	X
ap-7583	346	28	=	=	SYM
ap-7583	346	29	0	0	NUM
ap-7583	346	30	for	for	ADP
ap-7583	346	31	all	all	DET
ap-7583	346	32	j	j	PROPN
ap-7583	346	33	≥	≥	NUM
ap-7583	346	34	1	1	NUM
ap-7583	346	35	and	and	CCONJ
ap-7583	346	36	the	the	DET
ap-7583	346	37	recurrence	recurrence	NOUN
ap-7583	346	38	relation	relation	NOUN
ap-7583	346	39	(	(	PUNCT
ap-7583	346	40	28	28	NUM
ap-7583	346	41	)	)	PUNCT
ap-7583	346	42	for	for	ADP
ap-7583	346	43	k	k	PROPN
ap-7583	346	44	=	=	SYM
ap-7583	346	45	0	0	NUM
ap-7583	346	46	,	,	PUNCT
ap-7583	346	47	1	1	NUM
ap-7583	346	48	gives	give	NOUN
ap-7583	346	49	,	,	PUNCT
ap-7583	346	50	respectively	respectively	ADV
ap-7583	346	51	,	,	PUNCT
ap-7583	346	52	the	the	DET
ap-7583	346	53	necessary	necessary	ADJ
ap-7583	346	54	and	and	CCONJ
ap-7583	346	55	sufficient	sufficient	ADJ
ap-7583	346	56	conditions	condition	NOUN
ap-7583	346	57	ε1;0	ε1;0	PROPN
ap-7583	346	58	=	=	SYM
ap-7583	346	59	0	0	NUM
ap-7583	346	60	,	,	PUNCT
ap-7583	346	61	ε0;0	ε0;0	PROPN
ap-7583	346	62	=	=	SYM
ap-7583	346	63	0	0	X
ap-7583	346	64	.	.	PUNCT
ap-7583	347	1	(	(	PUNCT
ap-7583	347	2	54	54	NUM
ap-7583	347	3	)	)	PUNCT
ap-7583	347	4	•	•	NOUN
ap-7583	347	5	for	for	ADP
ap-7583	347	6	a	a	DET
ap-7583	347	7	first	first	ADJ
ap-7583	347	8	-	-	PUNCT
ap-7583	347	9	degree	degree	NOUN
ap-7583	347	10	polynomial	polynomial	ADJ
ap-7583	347	11	solution	solution	NOUN
ap-7583	347	12	,	,	PUNCT
ap-7583	347	13	n	n	NOUN
ap-7583	347	14	=	=	SYM
ap-7583	347	15	1	1	NUM
ap-7583	347	16	,	,	PUNCT
ap-7583	347	17	the	the	DET
ap-7583	347	18	coefficients	coefficient	NOUN
ap-7583	347	19	cj	cj	X
ap-7583	348	1	=	=	SYM
ap-7583	348	2	0	0	NUM
ap-7583	348	3	for	for	ADP
ap-7583	348	4	all	all	DET
ap-7583	348	5	j	j	PROPN
ap-7583	348	6	≥	≥	NUM
ap-7583	348	7	2	2	NUM
ap-7583	349	1	where	where	SCONJ
ap-7583	349	2	k	k	PROPN
ap-7583	349	3	=	=	SYM
ap-7583	349	4	0	0	NUM
ap-7583	349	5	,	,	PUNCT
ap-7583	349	6	1	1	NUM
ap-7583	349	7	,	,	PUNCT
ap-7583	349	8	2	2	NUM
ap-7583	349	9	give	give	VERB
ap-7583	349	10	the	the	DET
ap-7583	349	11	following	follow	VERB
ap-7583	349	12	three	three	NUM
ap-7583	349	13	equations	equation	NOUN
ap-7583	349	14			PROPN
ap-7583	349	15	ε0;1	ε0;1	PROPN
ap-7583	349	16	c0	c0	NOUN
ap-7583	349	17	+	+	CCONJ
ap-7583	349	18	β0	β0	PROPN
ap-7583	349	19	c1	c1	PROPN
ap-7583	349	20	=	=	SYM
ap-7583	349	21	0	0	PROPN
ap-7583	349	22	,	,	PUNCT
ap-7583	349	23	ε1;1	ε1;1	PROPN
ap-7583	349	24	c0	c0	NOUN
ap-7583	349	25	+	+	CCONJ
ap-7583	349	26	(	(	PUNCT
ap-7583	349	27	β1	β1	PROPN
ap-7583	349	28	+	+	CCONJ
ap-7583	349	29	ε0;1	ε0;1	PROPN
ap-7583	349	30	)	)	PUNCT
ap-7583	349	31	c1	c1	NOUN
ap-7583	349	32	=	=	PUNCT
ap-7583	349	33	0	0	PROPN
ap-7583	349	34	,	,	PUNCT
ap-7583	349	35	(	(	PUNCT
ap-7583	349	36	β2	β2	PROPN
ap-7583	349	37	+	+	CCONJ
ap-7583	349	38	ε1;1	ε1;1	PROPN
ap-7583	349	39	)	)	PUNCT
ap-7583	349	40	c1	c1	NOUN
ap-7583	349	41	=	=	PUNCT
ap-7583	349	42	0	0	PROPN
ap-7583	349	43	.	.	PUNCT
ap-7583	350	1	(	(	PUNCT
ap-7583	350	2	55	55	NUM
ap-7583	350	3	)	)	PUNCT
ap-7583	350	4	so	so	ADV
ap-7583	350	5	,	,	PUNCT
ap-7583	350	6	for	for	ADP
ap-7583	350	7	c0	c0	NOUN
ap-7583	350	8	=	=	SYM
ap-7583	350	9	1	1	NUM
ap-7583	350	10	,	,	PUNCT
ap-7583	350	11	it	it	PRON
ap-7583	350	12	is	be	AUX
ap-7583	350	13	necessary	necessary	ADJ
ap-7583	351	1	that	that	SCONJ
ap-7583	351	2	ε1;1	ε1;1	PROPN
ap-7583	351	3	=	=	SYM
ap-7583	351	4	−β2	−β2	PROPN
ap-7583	351	5	and	and	CCONJ
ap-7583	351	6	therefore	therefore	ADV
ap-7583	351	7	,	,	PUNCT
ap-7583	351	8	c1	c1	PROPN
ap-7583	351	9	=	=	PUNCT
ap-7583	351	10	−ε0;1	−ε0;1	PROPN
ap-7583	351	11	/	/	SYM
ap-7583	351	12	β0	β0	NOUN
ap-7583	351	13	where	where	SCONJ
ap-7583	351	14	ε0;1	ε0;1	PROPN
ap-7583	351	15	are	be	AUX
ap-7583	351	16	now	now	ADV
ap-7583	351	17	the	the	DET
ap-7583	351	18	roots	root	NOUN
ap-7583	351	19	of	of	ADP
ap-7583	351	20	the	the	DET
ap-7583	351	21	quadratic	quadratic	ADJ
ap-7583	351	22	equation	equation	NOUN
ap-7583	351	23	β0	β0	PROPN
ap-7583	351	24	β2	β2	PROPN
ap-7583	351	25	+	+	CCONJ
ap-7583	351	26	β1ε0;1	β1ε0;1	PROPN
ap-7583	351	27	+	+	CCONJ
ap-7583	351	28	ε2	ε2	ADJ
ap-7583	351	29	0;1	0;1	NOUN
ap-7583	351	30	=	=	SYM
ap-7583	351	31	0	0	X
ap-7583	351	32	.	.	PUNCT
ap-7583	352	1	let	let	VERB
ap-7583	352	2	εℓ	εℓ	PROPN
ap-7583	352	3	0;1	0;1	NOUN
ap-7583	352	4	,	,	PUNCT
ap-7583	352	5	ℓ	ℓ	PROPN
ap-7583	352	6	=	=	SYM
ap-7583	352	7	1	1	NUM
ap-7583	352	8	,	,	PUNCT
ap-7583	352	9	2	2	NUM
ap-7583	352	10	,	,	PUNCT
ap-7583	352	11	denote	denote	NOUN
ap-7583	352	12	,	,	PUNCT
ap-7583	352	13	if	if	SCONJ
ap-7583	352	14	any	any	PRON
ap-7583	352	15	,	,	PUNCT
ap-7583	352	16	the	the	DET
ap-7583	352	17	two	two	NUM
ap-7583	352	18	distinct	distinct	ADJ
ap-7583	352	19	real	real	ADJ
ap-7583	352	20	roots	root	NOUN
ap-7583	352	21	ε0	ε0	ADJ
ap-7583	352	22	0;1	0;1	NOUN
ap-7583	352	23	̸=	̸=	PROPN
ap-7583	352	24	ε1	ε1	VERB
ap-7583	352	25	0;1	0;1	NOUN
ap-7583	352	26	of	of	ADP
ap-7583	352	27	this	this	DET
ap-7583	352	28	quadratic	quadratic	ADJ
ap-7583	352	29	equation	equation	NOUN
ap-7583	352	30	.	.	PUNCT
ap-7583	353	1	then	then	ADV
ap-7583	353	2	,	,	PUNCT
ap-7583	353	3	for	for	ADP
ap-7583	353	4	the	the	DET
ap-7583	353	5	two	two	NUM
ap-7583	353	6	(	(	PUNCT
ap-7583	353	7	distinct	distinct	ADJ
ap-7583	353	8	)	)	PUNCT
ap-7583	353	9	differential	differential	NOUN
ap-7583	353	10	equations	equation	NOUN
ap-7583	353	11	(	(	PUNCT
ap-7583	353	12	α0	α0	ADJ
ap-7583	353	13	+	+	NUM
ap-7583	353	14	α1	α1	PROPN
ap-7583	353	15	r	r	NOUN
ap-7583	353	16	+	+	CCONJ
ap-7583	353	17	α2	α2	ADJ
ap-7583	353	18	r2	r2	NOUN
ap-7583	353	19	+	+	CCONJ
ap-7583	353	20	α3	α3	ADJ
ap-7583	353	21	r3)p′′	r3)p′′	NOUN
ap-7583	353	22	1;ℓ	1;ℓ	NUM
ap-7583	353	23	(	(	PUNCT
ap-7583	353	24	r	r	NOUN
ap-7583	353	25	)	)	PUNCT
ap-7583	353	26	+	+	CCONJ
ap-7583	353	27	(	(	PUNCT
ap-7583	353	28	β0	β0	NOUN
ap-7583	353	29	+	+	CCONJ
ap-7583	353	30	β1	β1	PROPN
ap-7583	353	31	r	r	NOUN
ap-7583	353	32	+	+	CCONJ
ap-7583	353	33	β2	β2	NOUN
ap-7583	353	34	r2)p′	r2)p′	PROPN
ap-7583	353	35	1;ℓ	1;ℓ	NOUN
ap-7583	353	36	(	(	PUNCT
ap-7583	353	37	r	r	NOUN
ap-7583	353	38	)	)	PUNCT
ap-7583	353	39	+	+	CCONJ
ap-7583	353	40	(	(	PUNCT
ap-7583	353	41	εℓ	εℓ	X
ap-7583	353	42	0;1	0;1	NOUN
ap-7583	353	43	−	−	PROPN
ap-7583	354	1	β2	β2	NOUN
ap-7583	354	2	r	r	NOUN
ap-7583	354	3	)	)	PUNCT
ap-7583	354	4	p1;ℓ	p1;ℓ	NOUN
ap-7583	354	5	(	(	PUNCT
ap-7583	354	6	r	r	NOUN
ap-7583	354	7	)	)	PUNCT
ap-7583	354	8	=	=	SYM
ap-7583	354	9	0	0	NUM
ap-7583	354	10	,	,	PUNCT
ap-7583	354	11	ℓ	ℓ	NOUN
ap-7583	354	12	=	=	SYM
ap-7583	354	13	1	1	NUM
ap-7583	354	14	,	,	PUNCT
ap-7583	354	15	2	2	NUM
ap-7583	354	16	,	,	PUNCT
ap-7583	354	17	(	(	PUNCT
ap-7583	354	18	56	56	NUM
ap-7583	354	19	)	)	PUNCT
ap-7583	354	20	the	the	DET
ap-7583	354	21	first	first	ADJ
ap-7583	354	22	-	-	PUNCT
ap-7583	354	23	order	order	NOUN
ap-7583	354	24	polynomial	polynomial	ADJ
ap-7583	354	25	solutions	solution	NOUN
ap-7583	354	26	arep1;ℓ	arep1;ℓ	NOUN
ap-7583	354	27	(	(	PUNCT
ap-7583	354	28	r	r	NOUN
ap-7583	354	29	)	)	PUNCT
ap-7583	354	30	=	=	SYM
ap-7583	355	1	1	1	NUM
ap-7583	355	2	−	−	NOUN
ap-7583	355	3	εℓ	εℓ	NOUN
ap-7583	355	4	0;1	0;1	NOUN
ap-7583	355	5	β0	β0	NOUN
ap-7583	355	6	r	r	PROPN
ap-7583	355	7	,	,	PUNCT
ap-7583	355	8	β0	β0	NOUN
ap-7583	355	9	β2	β2	PROPN
ap-7583	355	10	+	+	CCONJ
ap-7583	355	11	β1	β1	PROPN
ap-7583	355	12	εℓ	εℓ	NOUN
ap-7583	355	13	0;1	0;1	NOUN
ap-7583	355	14	+	+	CCONJ
ap-7583	355	15	(	(	PUNCT
ap-7583	355	16	εℓ	εℓ	X
ap-7583	355	17	0;1)2	0;1)2	NOUN
ap-7583	355	18	=	=	SYM
ap-7583	355	19	0	0	NUM
ap-7583	355	20	,	,	PUNCT
ap-7583	355	21	ℓ	ℓ	NOUN
ap-7583	355	22	=	=	SYM
ap-7583	355	23	1	1	NUM
ap-7583	355	24	,	,	PUNCT
ap-7583	355	25	2	2	NUM
ap-7583	355	26	.	.	PUNCT
ap-7583	356	1	(	(	PUNCT
ap-7583	356	2	57	57	NUM
ap-7583	356	3	)	)	PUNCT
ap-7583	356	4	•	•	NOUN
ap-7583	356	5	for	for	ADP
ap-7583	356	6	a	a	DET
ap-7583	356	7	second	second	ADJ
ap-7583	356	8	-	-	PUNCT
ap-7583	356	9	degree	degree	NOUN
ap-7583	356	10	polynomial	polynomial	ADJ
ap-7583	356	11	solution	solution	NOUN
ap-7583	356	12	,	,	PUNCT
ap-7583	356	13	n	n	NOUN
ap-7583	356	14	=	=	SYM
ap-7583	356	15	2	2	NUM
ap-7583	356	16	,	,	PUNCT
ap-7583	356	17	the	the	DET
ap-7583	356	18	coefficients	coefficient	NOUN
ap-7583	356	19	cj	cj	X
ap-7583	356	20	=	=	SYM
ap-7583	356	21	0	0	NUM
ap-7583	356	22	for	for	ADP
ap-7583	356	23	all	all	DET
ap-7583	356	24	j	j	PROPN
ap-7583	356	25	≥	≥	NUM
ap-7583	356	26	3	3	NUM
ap-7583	356	27	where	where	SCONJ
ap-7583	356	28	k	k	PROPN
ap-7583	356	29	=	=	SYM
ap-7583	356	30	0	0	NUM
ap-7583	356	31	,	,	PUNCT
ap-7583	356	32	1	1	NUM
ap-7583	356	33	,	,	PUNCT
ap-7583	356	34	2	2	NUM
ap-7583	356	35	,	,	PUNCT
ap-7583	356	36	3	3	NUM
ap-7583	356	37	give	give	VERB
ap-7583	356	38	the	the	DET
ap-7583	356	39	four	four	NUM
ap-7583	356	40	linear	linear	ADJ
ap-7583	356	41	equations	equation	NOUN
ap-7583	356	42			PROPN
ap-7583	356	43	ε0;2	ε0;2	PROPN
ap-7583	356	44	c0	c0	PROPN
ap-7583	356	45	+	+	CCONJ
ap-7583	356	46	β0	β0	PROPN
ap-7583	356	47	c1	c1	PROPN
ap-7583	356	48	+	+	CCONJ
ap-7583	356	49	2	2	NUM
ap-7583	356	50	α0	α0	ADJ
ap-7583	356	51	c2	c2	PROPN
ap-7583	356	52	=	=	SYM
ap-7583	356	53	0	0	NUM
ap-7583	356	54	,	,	PUNCT
ap-7583	356	55	ε1;2	ε1;2	NOUN
ap-7583	356	56	c0	c0	NOUN
ap-7583	356	57	+	+	CCONJ
ap-7583	356	58	(	(	PUNCT
ap-7583	356	59	β1	β1	PROPN
ap-7583	356	60	+	+	CCONJ
ap-7583	356	61	ε0;2	ε0;2	PROPN
ap-7583	356	62	)	)	PUNCT
ap-7583	356	63	c1	c1	NOUN
ap-7583	356	64	+	+	CCONJ
ap-7583	356	65	2(α1	2(α1	NUM
ap-7583	356	66	+	+	CCONJ
ap-7583	356	67	β0	β0	ADJ
ap-7583	356	68	)	)	PUNCT
ap-7583	356	69	c2	c2	PROPN
ap-7583	356	70	=	=	SYM
ap-7583	356	71	0	0	PROPN
ap-7583	356	72	,	,	PUNCT
ap-7583	356	73	(	(	PUNCT
ap-7583	356	74	β2	β2	NOUN
ap-7583	356	75	+	+	CCONJ
ap-7583	356	76	ε1;2	ε1;2	PROPN
ap-7583	356	77	)	)	PUNCT
ap-7583	356	78	c1	c1	NOUN
ap-7583	356	79	+	+	CCONJ
ap-7583	356	80	(	(	PUNCT
ap-7583	356	81	2	2	NUM
ap-7583	356	82	α2	α2	ADJ
ap-7583	356	83	+	+	CCONJ
ap-7583	356	84	2	2	NUM
ap-7583	356	85	β1	β1	NOUN
ap-7583	356	86	+	+	CCONJ
ap-7583	356	87	ε0;2	ε0;2	PROPN
ap-7583	356	88	)	)	PUNCT
ap-7583	356	89	c2	c2	PROPN
ap-7583	356	90	=	=	SYM
ap-7583	356	91	0	0	PROPN
ap-7583	356	92	,	,	PUNCT
ap-7583	356	93	(	(	PUNCT
ap-7583	356	94	2α3	2α3	NUM
ap-7583	356	95	+	+	SYM
ap-7583	356	96	2	2	NUM
ap-7583	356	97	β2	β2	NOUN
ap-7583	356	98	+	+	CCONJ
ap-7583	356	99	ε1;2)c2	ε1;2)c2	PROPN
ap-7583	356	100	=	=	SYM
ap-7583	356	101	0	0	X
ap-7583	356	102	.	.	PUNCT
ap-7583	357	1	(	(	PUNCT
ap-7583	357	2	58	58	NUM
ap-7583	357	3	)	)	PUNCT
ap-7583	357	4	the	the	DET
ap-7583	357	5	very	very	ADV
ap-7583	357	6	last	last	ADJ
ap-7583	357	7	equation	equation	NOUN
ap-7583	357	8	of	of	ADP
ap-7583	357	9	(	(	PUNCT
ap-7583	357	10	58	58	NUM
ap-7583	357	11	)	)	PUNCT
ap-7583	357	12	,	,	PUNCT
ap-7583	357	13	correspondent	correspondent	NOUN
ap-7583	357	14	to	to	ADP
ap-7583	357	15	k	k	PROPN
ap-7583	357	16	=	=	SYM
ap-7583	357	17	3	3	NUM
ap-7583	357	18	,	,	PUNCT
ap-7583	357	19	gives	give	VERB
ap-7583	357	20	the	the	DET
ap-7583	357	21	necessary	necessary	ADJ
ap-7583	357	22	condition	condition	NOUN
ap-7583	357	23	ε1;2	ε1;2	NOUN
ap-7583	357	24	=	=	SYM
ap-7583	357	25	−2	−2	NOUN
ap-7583	357	26	α3	α3	NOUN
ap-7583	357	27	−	−	PROPN
ap-7583	357	28	2	2	NUM
ap-7583	357	29	β2	β2	NOUN
ap-7583	357	30	,	,	PUNCT
ap-7583	357	31	(	(	PUNCT
ap-7583	357	32	59	59	NUM
ap-7583	357	33	)	)	PUNCT
ap-7583	357	34	and	and	CCONJ
ap-7583	357	35	for	for	ADP
ap-7583	357	36	k	k	PROPN
ap-7583	357	37	=	=	SYM
ap-7583	357	38	0	0	NUM
ap-7583	357	39	,	,	PUNCT
ap-7583	357	40	1	1	NUM
ap-7583	357	41	,	,	PUNCT
ap-7583	357	42	the	the	DET
ap-7583	357	43	coefficients	coefficient	NOUN
ap-7583	357	44	of	of	ADP
ap-7583	357	45	the	the	DET
ap-7583	357	46	polynomial	polynomial	ADJ
ap-7583	357	47	solution	solution	NOUN
ap-7583	357	48	y(r	y(r	NOUN
ap-7583	357	49	)	)	PUNCT
ap-7583	357	50	=	=	SYM
ap-7583	357	51	1	1	NUM
ap-7583	357	52	+	+	NUM
ap-7583	357	53	c1	c1	NOUN
ap-7583	357	54	r	r	NOUN
ap-7583	357	55	+	+	CCONJ
ap-7583	357	56	c2	c2	PROPN
ap-7583	357	57	r2	r2	PROPN
ap-7583	357	58	read	read	PROPN
ap-7583	357	59	c1	c1	PROPN
ap-7583	357	60	=	=	PUNCT
ap-7583	357	61	∣∣∣∣∣	∣∣∣∣∣	PROPN
ap-7583	357	62	−ε0;2	−ε0;2	NOUN
ap-7583	357	63	2α0	2α0	NUM
ap-7583	357	64	2	2	NUM
ap-7583	357	65	α3	α3	NOUN
ap-7583	357	66	+	+	CCONJ
ap-7583	357	67	2	2	NUM
ap-7583	357	68	β2	β2	NOUN
ap-7583	357	69	2α1	2α1	NUM
ap-7583	357	70	+	+	CCONJ
ap-7583	357	71	2β0	2β0	NUM
ap-7583	357	72	,	,	PUNCT
ap-7583	357	73	∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	357	74	β0	β0	PROPN
ap-7583	357	75	2α0	2α0	NUM
ap-7583	357	76	β1	β1	NOUN
ap-7583	357	77	+	+	CCONJ
ap-7583	357	78	ε0;2	ε0;2	PROPN
ap-7583	357	79	2α1	2α1	NUM
ap-7583	357	80	+	+	CCONJ
ap-7583	357	81	2β0	2β0	NUM
ap-7583	357	82	∣∣∣∣∣	∣∣∣∣∣	PUNCT
ap-7583	357	83	,	,	PUNCT
ap-7583	357	84	c2	c2	PROPN
ap-7583	357	85	=	=	PUNCT
ap-7583	358	1	∣∣∣∣∣	∣∣∣∣∣	PROPN
ap-7583	358	2	β0	β0	PROPN
ap-7583	358	3	−ε0;2	−ε0;2	NOUN
ap-7583	358	4	β1	β1	PROPN
ap-7583	358	5	+	+	CCONJ
ap-7583	358	6	ε0;2	ε0;2	PROPN
ap-7583	358	7	2	2	NUM
ap-7583	358	8	α3	α3	NOUN
ap-7583	358	9	+	+	CCONJ
ap-7583	358	10	2	2	NUM
ap-7583	358	11	β2	β2	NOUN
ap-7583	358	12	∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	358	13	β0	β0	NOUN
ap-7583	358	14	2α0	2α0	NUM
ap-7583	358	15	β1	β1	NOUN
ap-7583	358	16	+	+	CCONJ
ap-7583	358	17	ε0;2	ε0;2	PROPN
ap-7583	358	18	2α1	2α1	NUM
ap-7583	358	19	+	+	CCONJ
ap-7583	358	20	2β0	2β0	NUM
ap-7583	358	21	∣∣∣∣∣	∣∣∣∣∣	ADJ
ap-7583	358	22	.	.	PUNCT
ap-7583	359	1	(	(	PUNCT
ap-7583	359	2	60	60	NUM
ap-7583	359	3	)	)	PUNCT
ap-7583	359	4	175	175	NUM
ap-7583	359	5	nasser	nasser	PROPN
ap-7583	359	6	saad	saad	PROPN
ap-7583	359	7	acta	acta	PROPN
ap-7583	359	8	polytechnica	polytechnica	PROPN
ap-7583	359	9	the	the	DET
ap-7583	359	10	equation	equation	NOUN
ap-7583	359	11	corresponding	correspond	VERB
ap-7583	359	12	to	to	ADP
ap-7583	359	13	k	k	PROPN
ap-7583	359	14	=	=	SYM
ap-7583	359	15	2	2	NUM
ap-7583	359	16	and	and	CCONJ
ap-7583	359	17	n	n	NOUN
ap-7583	359	18	=	=	SYM
ap-7583	359	19	2	2	NUM
ap-7583	359	20	establishes	establish	VERB
ap-7583	359	21	the	the	DET
ap-7583	359	22	sufficient	sufficient	ADJ
ap-7583	359	23	condition∣∣∣∣∣∣∣∣	condition∣∣∣∣∣∣∣∣	ADJ
ap-7583	359	24	εℓ	εℓ	X
ap-7583	359	25	0;2	0;2	NOUN
ap-7583	359	26	β0	β0	NOUN
ap-7583	359	27	2α0	2α0	NUM
ap-7583	359	28	−2	−2	NOUN
ap-7583	359	29	α3	α3	NOUN
ap-7583	359	30	−	−	PROPN
ap-7583	359	31	2	2	NUM
ap-7583	359	32	β2	β2	NOUN
ap-7583	359	33	β1	β1	NOUN
ap-7583	359	34	+	+	CCONJ
ap-7583	359	35	εℓ	εℓ	X
ap-7583	359	36	0;2	0;2	NOUN
ap-7583	359	37	2α1	2α1	NOUN
ap-7583	359	38	+	+	CCONJ
ap-7583	359	39	2β0	2β0	NUM
ap-7583	359	40	0	0	NUM
ap-7583	359	41	β2	β2	NOUN
ap-7583	359	42	−	−	PROPN
ap-7583	359	43	2	2	NUM
ap-7583	359	44	α3	α3	NOUN
ap-7583	359	45	−	−	NOUN
ap-7583	359	46	2	2	NUM
ap-7583	359	47	β2	β2	NOUN
ap-7583	359	48	2	2	NUM
ap-7583	359	49	α2	α2	ADJ
ap-7583	359	50	+	+	CCONJ
ap-7583	359	51	2	2	NUM
ap-7583	359	52	β1	β1	NOUN
ap-7583	359	53	+	+	CCONJ
ap-7583	359	54	εℓ	εℓ	NOUN
ap-7583	359	55	0;2	0;2	NOUN
ap-7583	359	56	∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣	ADP
ap-7583	359	57	=	=	SYM
ap-7583	359	58	0	0	NUM
ap-7583	359	59	,	,	PUNCT
ap-7583	359	60	(	(	PUNCT
ap-7583	359	61	61	61	NUM
ap-7583	359	62	)	)	PUNCT
ap-7583	359	63	where	where	SCONJ
ap-7583	359	64	ℓ	ℓ	NOUN
ap-7583	359	65	=	=	SYM
ap-7583	359	66	1	1	NUM
ap-7583	359	67	,	,	PUNCT
ap-7583	359	68	2	2	NUM
ap-7583	359	69	,	,	PUNCT
ap-7583	359	70	3	3	NUM
ap-7583	359	71	refers	refer	VERB
ap-7583	359	72	to	to	ADP
ap-7583	359	73	the	the	DET
ap-7583	359	74	three	three	NUM
ap-7583	359	75	distinct	distinct	ADJ
ap-7583	359	76	simple	simple	ADJ
ap-7583	359	77	roots	root	NOUN
ap-7583	359	78	εℓ	εℓ	ADP
ap-7583	359	79	0;2	0;2	NOUN
ap-7583	359	80	,	,	PUNCT
ap-7583	359	81	ℓ	ℓ	PROPN
ap-7583	359	82	=	=	SYM
ap-7583	359	83	1	1	NUM
ap-7583	359	84	,	,	PUNCT
ap-7583	359	85	2	2	NUM
ap-7583	359	86	,	,	PUNCT
ap-7583	359	87	3	3	NUM
ap-7583	359	88	,	,	PUNCT
ap-7583	359	89	if	if	SCONJ
ap-7583	359	90	any	any	PRON
ap-7583	359	91	,	,	PUNCT
ap-7583	359	92	of	of	ADP
ap-7583	359	93	the	the	DET
ap-7583	359	94	polynomial	polynomial	NOUN
ap-7583	359	95	generated	generate	VERB
ap-7583	359	96	by	by	ADP
ap-7583	359	97	the	the	DET
ap-7583	359	98	determinant	determinant	ADJ
ap-7583	359	99	(	(	PUNCT
ap-7583	359	100	61	61	NUM
ap-7583	359	101	)	)	PUNCT
ap-7583	359	102	.	.	PUNCT
ap-7583	360	1	hence	hence	ADV
ap-7583	360	2	,	,	PUNCT
ap-7583	360	3	for	for	ADP
ap-7583	360	4	each	each	DET
ap-7583	360	5	index	index	NOUN
ap-7583	360	6	ℓ	ℓ	NOUN
ap-7583	360	7	=	=	SYM
ap-7583	360	8	1	1	NUM
ap-7583	360	9	,	,	PUNCT
ap-7583	360	10	2	2	NUM
ap-7583	360	11	,	,	PUNCT
ap-7583	360	12	3	3	NUM
ap-7583	360	13	,	,	PUNCT
ap-7583	360	14	the	the	DET
ap-7583	360	15	differential	differential	ADJ
ap-7583	360	16	equation	equation	NOUN
ap-7583	360	17	(	(	PUNCT
ap-7583	360	18	α0	α0	ADJ
ap-7583	360	19	+	+	NUM
ap-7583	360	20	α1	α1	PROPN
ap-7583	360	21	r	r	NOUN
ap-7583	360	22	+	+	CCONJ
ap-7583	360	23	α2	α2	ADJ
ap-7583	360	24	r2	r2	NOUN
ap-7583	360	25	+	+	CCONJ
ap-7583	360	26	α3	α3	NOUN
ap-7583	360	27	r3)p′′	r3)p′′	VERB
ap-7583	360	28	2;ℓ	2;ℓ	NUM
ap-7583	360	29	(	(	PUNCT
ap-7583	360	30	r	r	NOUN
ap-7583	360	31	)	)	PUNCT
ap-7583	361	1	+	+	CCONJ
ap-7583	361	2	(	(	PUNCT
ap-7583	361	3	β0	β0	NOUN
ap-7583	361	4	+	+	CCONJ
ap-7583	361	5	β1	β1	PROPN
ap-7583	361	6	r	r	NOUN
ap-7583	361	7	+	+	NUM
ap-7583	361	8	β2	β2	NOUN
ap-7583	361	9	r2)p′	r2)p′	PROPN
ap-7583	361	10	2;ℓ	2;ℓ	NUM
ap-7583	361	11	(	(	PUNCT
ap-7583	361	12	r	r	NOUN
ap-7583	361	13	)	)	PUNCT
ap-7583	361	14	+	+	CCONJ
ap-7583	361	15	(	(	PUNCT
ap-7583	361	16	εℓ	εℓ	X
ap-7583	361	17	0;2	0;2	NOUN
ap-7583	361	18	−	−	PROPN
ap-7583	362	1	(	(	PUNCT
ap-7583	362	2	2α3	2α3	NUM
ap-7583	362	3	+	+	CCONJ
ap-7583	362	4	2β2	2β2	NUM
ap-7583	362	5	)	)	PUNCT
ap-7583	362	6	r	r	NOUN
ap-7583	362	7	)	)	PUNCT
ap-7583	362	8	p2;ℓ	p2;ℓ	PROPN
ap-7583	362	9	(	(	PUNCT
ap-7583	362	10	r	r	NOUN
ap-7583	362	11	)	)	PUNCT
ap-7583	362	12	=	=	SYM
ap-7583	362	13	0	0	NUM
ap-7583	362	14	,	,	PUNCT
ap-7583	362	15	(	(	PUNCT
ap-7583	362	16	62	62	NUM
ap-7583	362	17	)	)	PUNCT
ap-7583	362	18	has	have	VERB
ap-7583	362	19	the	the	DET
ap-7583	362	20	polynomial	polynomial	ADJ
ap-7583	362	21	solution	solution	NOUN
ap-7583	362	22	(	(	PUNCT
ap-7583	362	23	for	for	ADP
ap-7583	362	24	ℓ	ℓ	NOUN
ap-7583	362	25	=	=	SYM
ap-7583	362	26	1	1	NUM
ap-7583	362	27	,	,	PUNCT
ap-7583	362	28	2	2	NUM
ap-7583	362	29	,	,	PUNCT
ap-7583	362	30	3	3	NUM
ap-7583	362	31	.	.	PUNCT
ap-7583	362	32	)	)	PUNCT
ap-7583	363	1	p2;ℓ	p2;ℓ	PROPN
ap-7583	363	2	(	(	PUNCT
ap-7583	363	3	r	r	NOUN
ap-7583	363	4	)	)	PUNCT
ap-7583	363	5	=	=	SYM
ap-7583	363	6	1	1	NUM
ap-7583	363	7	+	+	CCONJ
ap-7583	363	8	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ap-7583	363	9	−εℓ	−εℓ	PROPN
ap-7583	363	10	0;2	0;2	NOUN
ap-7583	363	11	2α0	2α0	NUM
ap-7583	363	12	2	2	NUM
ap-7583	363	13	α3	α3	NOUN
ap-7583	363	14	+	+	CCONJ
ap-7583	363	15	2	2	NUM
ap-7583	363	16	β2	β2	NOUN
ap-7583	363	17	2α1	2α1	NUM
ap-7583	363	18	+	+	CCONJ
ap-7583	363	19	2β0	2β0	NUM
ap-7583	363	20	∣∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	363	21	β0	β0	NOUN
ap-7583	363	22	2α0	2α0	NUM
ap-7583	363	23	β1	β1	NOUN
ap-7583	363	24	+	+	CCONJ
ap-7583	363	25	εℓ	εℓ	NOUN
ap-7583	363	26	0;2	0;2	NOUN
ap-7583	363	27	2α1	2α1	NOUN
ap-7583	363	28	+	+	CCONJ
ap-7583	363	29	2β0	2β0	NUM
ap-7583	363	30	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ap-7583	363	31	r	r	NOUN
ap-7583	363	32	+	+	CCONJ
ap-7583	363	33	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ap-7583	363	34	β0	β0	PROPN
ap-7583	363	35	−εℓ	−εℓ	PROPN
ap-7583	363	36	0;2	0;2	NOUN
ap-7583	363	37	β1	β1	PROPN
ap-7583	363	38	+	+	CCONJ
ap-7583	363	39	εℓ	εℓ	X
ap-7583	363	40	0;2	0;2	NOUN
ap-7583	363	41	2	2	NUM
ap-7583	363	42	α3	α3	NOUN
ap-7583	363	43	+	+	CCONJ
ap-7583	363	44	2	2	NUM
ap-7583	363	45	β2	β2	NOUN
ap-7583	363	46	∣∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	363	47	β0	β0	NOUN
ap-7583	363	48	2α0	2α0	NUM
ap-7583	363	49	β1	β1	NOUN
ap-7583	363	50	+	+	CCONJ
ap-7583	363	51	εℓ	εℓ	NOUN
ap-7583	363	52	0;2	0;2	NOUN
ap-7583	363	53	2α1	2α1	NOUN
ap-7583	363	54	+	+	CCONJ
ap-7583	363	55	2β0	2β0	NUM
ap-7583	363	56	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ap-7583	363	57	r2	r2	PROPN
ap-7583	363	58	,	,	PUNCT
ap-7583	363	59	(	(	PUNCT
ap-7583	363	60	63	63	NUM
ap-7583	363	61	)	)	PUNCT
ap-7583	363	62	the	the	DET
ap-7583	363	63	above	above	ADJ
ap-7583	363	64	constructive	constructive	ADJ
ap-7583	363	65	approach	approach	NOUN
ap-7583	363	66	can	can	AUX
ap-7583	363	67	be	be	AUX
ap-7583	363	68	continued	continue	VERB
ap-7583	363	69	to	to	PART
ap-7583	363	70	generate	generate	VERB
ap-7583	363	71	higher	high	ADJ
ap-7583	363	72	-	-	PUNCT
ap-7583	363	73	order	order	NOUN
ap-7583	363	74	polynomial	polynomial	ADJ
ap-7583	363	75	solutions	solution	NOUN
ap-7583	363	76	of	of	ADP
ap-7583	363	77	an	an	DET
ap-7583	363	78	arbitrary	arbitrary	ADJ
ap-7583	363	79	degree	degree	NOUN
ap-7583	363	80	.	.	PUNCT
ap-7583	364	1	theorem	theorem	ADJ
ap-7583	364	2	3.2	3.2	NUM
ap-7583	364	3	.	.	PUNCT
ap-7583	365	1	suppose	suppose	VERB
ap-7583	365	2	the	the	DET
ap-7583	365	3	polynomial	polynomial	NOUN
ap-7583	365	4	in	in	ADP
ap-7583	365	5	εℓ	εℓ	NOUN
ap-7583	365	6	0;n	0;n	NOUN
ap-7583	365	7	generated	generate	VERB
ap-7583	365	8	by	by	ADP
ap-7583	365	9	the	the	DET
ap-7583	365	10	determinant	determinant	ADJ
ap-7583	365	11	(	(	PUNCT
ap-7583	365	12	52	52	NUM
ap-7583	365	13	)	)	PUNCT
ap-7583	365	14	has	have	VERB
ap-7583	365	15	n	n	PROPN
ap-7583	365	16	+	+	CCONJ
ap-7583	365	17	1	1	NUM
ap-7583	365	18	distinct	distinct	ADJ
ap-7583	365	19	real	real	ADJ
ap-7583	365	20	roots	root	NOUN
ap-7583	365	21	arranged	arrange	VERB
ap-7583	365	22	in	in	ADP
ap-7583	365	23	ascending	ascend	VERB
ap-7583	365	24	order	order	NOUN
ap-7583	365	25	ε0	ε0	NOUN
ap-7583	365	26	0;n	0;n	NOUN
ap-7583	365	27	<	<	X
ap-7583	365	28	ε1	ε1	VERB
ap-7583	365	29	0;n	0;n	NOUN
ap-7583	365	30	<	<	X
ap-7583	365	31	ε2	ε2	NOUN
ap-7583	365	32	0;n	0;n	NOUN
ap-7583	365	33	<	<	X
ap-7583	365	34	·	·	PUNCT
ap-7583	365	35	·	·	PUNCT
ap-7583	365	36	·	·	PUNCT
ap-7583	365	37	<	<	X
ap-7583	365	38	εn	εn	ADP
ap-7583	365	39	0;n	0;n	NOUN
ap-7583	365	40	,	,	PUNCT
ap-7583	365	41	then	then	ADV
ap-7583	365	42	,	,	PUNCT
ap-7583	365	43	the	the	DET
ap-7583	365	44	eigenvalue	eigenvalue	PROPN
ap-7583	365	45	problem	problem	NOUN
ap-7583	365	46	(	(	PUNCT
ap-7583	365	47	α0	α0	ADJ
ap-7583	365	48	+	+	NUM
ap-7583	365	49	α1	α1	PROPN
ap-7583	365	50	r	r	NOUN
ap-7583	365	51	+	+	CCONJ
ap-7583	365	52	α2	α2	ADJ
ap-7583	365	53	r2	r2	PROPN
ap-7583	365	54	+	+	CCONJ
ap-7583	365	55	α3	α3	PROPN
ap-7583	365	56	r3	r3	PROPN
ap-7583	365	57	)	)	PUNCT
ap-7583	365	58	d2pn;ℓ	d2pn;ℓ	NOUN
ap-7583	365	59	dr2	dr2	PROPN
ap-7583	365	60	+	+	CCONJ
ap-7583	365	61	(	(	PUNCT
ap-7583	365	62	β0	β0	NOUN
ap-7583	365	63	+	+	CCONJ
ap-7583	365	64	β1	β1	PROPN
ap-7583	365	65	r	r	NOUN
ap-7583	365	66	+	+	CCONJ
ap-7583	365	67	β2	β2	NOUN
ap-7583	365	68	r2	r2	NOUN
ap-7583	365	69	)	)	PUNCT
ap-7583	365	70	dpn;ℓ	dpn;ℓ	NOUN
ap-7583	365	71	dr	dr	PROPN
ap-7583	365	72	−	−	PROPN
ap-7583	365	73	n	n	CCONJ
ap-7583	365	74	(	(	PUNCT
ap-7583	365	75	(	(	PUNCT
ap-7583	365	76	n	n	CCONJ
ap-7583	365	77	−	−	PROPN
ap-7583	365	78	1	1	NUM
ap-7583	365	79	)	)	PUNCT
ap-7583	365	80	α3	α3	NOUN
ap-7583	365	81	+	+	CCONJ
ap-7583	365	82	β2	β2	NOUN
ap-7583	365	83	)	)	PUNCT
ap-7583	365	84	r	r	NOUN
ap-7583	365	85	pn;ℓ	pn;ℓ	NOUN
ap-7583	365	86	=	=	SYM
ap-7583	365	87	−εℓ	−εℓ	NOUN
ap-7583	365	88	0;n	0;n	NUM
ap-7583	365	89	pn;ℓ	pn;ℓ	NOUN
ap-7583	365	90	,	,	PUNCT
ap-7583	365	91	(	(	PUNCT
ap-7583	365	92	64	64	NUM
ap-7583	365	93	)	)	PUNCT
ap-7583	365	94	has	have	VERB
ap-7583	365	95	a	a	DET
ap-7583	365	96	polynomial	polynomial	ADJ
ap-7583	365	97	solution	solution	NOUN
ap-7583	365	98	of	of	ADP
ap-7583	365	99	the	the	DET
ap-7583	365	100	degree	degree	NOUN
ap-7583	365	101	n	n	CCONJ
ap-7583	365	102	,	,	PUNCT
ap-7583	365	103	for	for	ADP
ap-7583	365	104	ℓ	ℓ	PROPN
ap-7583	365	105	=	=	SYM
ap-7583	365	106	1	1	NUM
ap-7583	365	107	,	,	PUNCT
ap-7583	365	108	2	2	NUM
ap-7583	365	109	,	,	PUNCT
ap-7583	365	110	.	.	PUNCT
ap-7583	365	111	.	.	PUNCT
ap-7583	366	1	.	.	PUNCT
ap-7583	367	1	,	,	PUNCT
ap-7583	367	2	n	n	PROPN
ap-7583	367	3	+	+	NOUN
ap-7583	367	4	1	1	X
ap-7583	367	5	.	.	PUNCT
ap-7583	368	1	this	this	DET
ap-7583	368	2	theorem	theorem	NOUN
ap-7583	368	3	is	be	AUX
ap-7583	368	4	illustrated	illustrate	VERB
ap-7583	368	5	by	by	ADP
ap-7583	368	6	figure	figure	NOUN
ap-7583	368	7	1	1	NUM
ap-7583	368	8	,	,	PUNCT
ap-7583	368	9	for	for	ADP
ap-7583	368	10	n	n	NOUN
ap-7583	368	11	=	=	SYM
ap-7583	368	12	0	0	NUM
ap-7583	368	13	,	,	PUNCT
ap-7583	368	14	1	1	NUM
ap-7583	368	15	,	,	PUNCT
ap-7583	368	16	2	2	NUM
ap-7583	368	17	,	,	PUNCT
ap-7583	368	18	3	3	NUM
ap-7583	368	19	,	,	PUNCT
ap-7583	368	20	n	n	NOUN
ap-7583	368	21	=	=	SYM
ap-7583	368	22	0	0	NUM
ap-7583	369	1	ε1	ε1	VERB
ap-7583	369	2	0;0	0;0	NUM
ap-7583	369	3	p0;1(r	p0;1(r	NOUN
ap-7583	369	4	)	)	PUNCT
ap-7583	369	5	n	n	NOUN
ap-7583	369	6	=	=	SYM
ap-7583	369	7	1	1	NUM
ap-7583	369	8	ε1	ε1	VERB
ap-7583	369	9	0;1	0;1	NOUN
ap-7583	369	10	p1;1(r	p1;1(r	NOUN
ap-7583	369	11	)	)	PUNCT
ap-7583	369	12	ε2	ε2	ADJ
ap-7583	369	13	0;1	0;1	X
ap-7583	369	14	p1;2(r	p1;2(r	NOUN
ap-7583	369	15	)	)	PUNCT
ap-7583	369	16	n	n	NOUN
ap-7583	369	17	=	=	SYM
ap-7583	369	18	2	2	NUM
ap-7583	369	19	ε1	ε1	VERB
ap-7583	369	20	0;2	0;2	NOUN
ap-7583	369	21	p2;1(r	p2;1(r	ADJ
ap-7583	369	22	)	)	PUNCT
ap-7583	369	23	ε2	ε2	ADJ
ap-7583	369	24	0;2	0;2	NOUN
ap-7583	369	25	p2;2(r	p2;2(r	NOUN
ap-7583	369	26	)	)	PUNCT
ap-7583	369	27	ε3	ε3	PROPN
ap-7583	369	28	0;2	0;2	NOUN
ap-7583	369	29	p2;3(r	p2;3(r	NOUN
ap-7583	369	30	)	)	PUNCT
ap-7583	369	31	n	n	NOUN
ap-7583	369	32	=	=	SYM
ap-7583	369	33	3	3	NUM
ap-7583	369	34	ε1	ε1	VERB
ap-7583	369	35	0;3	0;3	NOUN
ap-7583	369	36	p3;1(r	p3;1(r	NOUN
ap-7583	369	37	)	)	PUNCT
ap-7583	369	38	ε2	ε2	ADJ
ap-7583	369	39	0;3	0;3	NOUN
ap-7583	369	40	p3;2(r	p3;2(r	NOUN
ap-7583	369	41	)	)	PUNCT
ap-7583	369	42	ε3	ε3	VERB
ap-7583	369	43	0;3	0;3	VERB
ap-7583	369	44	p3;3(r	p3;3(r	NOUN
ap-7583	369	45	)	)	PUNCT
ap-7583	369	46	ε4	ε4	NOUN
ap-7583	369	47	0;3	0;3	NOUN
ap-7583	369	48	p3;4(r	p3;4(r	NOUN
ap-7583	369	49	)	)	PUNCT
ap-7583	369	50	·	·	PUNCT
ap-7583	369	51	·	·	PUNCT
ap-7583	369	52	·	·	PUNCT
ap-7583	369	53	·	·	PUNCT
ap-7583	369	54	·	·	PUNCT
ap-7583	369	55	·	·	PUNCT
ap-7583	369	56	·	·	PUNCT
ap-7583	369	57	·	·	PUNCT
ap-7583	369	58	·	·	PUNCT
ap-7583	369	59	figure	figure	NOUN
ap-7583	369	60	1	1	NUM
ap-7583	369	61	.	.	PUNCT
ap-7583	370	1	a	a	DET
ap-7583	370	2	graphical	graphical	ADJ
ap-7583	370	3	representation	representation	NOUN
ap-7583	370	4	of	of	ADP
ap-7583	370	5	theorem	theorem	ADJ
ap-7583	370	6	3.2	3.2	NUM
ap-7583	370	7	open	open	ADJ
ap-7583	370	8	problem	problem	NOUN
ap-7583	370	9	:	:	PUNCT
ap-7583	370	10	it	it	PRON
ap-7583	370	11	is	be	AUX
ap-7583	370	12	an	an	DET
ap-7583	370	13	open	open	ADJ
ap-7583	370	14	question	question	NOUN
ap-7583	370	15	to	to	PART
ap-7583	370	16	establish	establish	VERB
ap-7583	370	17	the	the	DET
ap-7583	370	18	condition(s	condition(s	NOUN
ap-7583	370	19	)	)	PUNCT
ap-7583	370	20	on	on	ADP
ap-7583	370	21	the	the	DET
ap-7583	370	22	parameters	parameter	NOUN
ap-7583	370	23	so	so	SCONJ
ap-7583	370	24	that	that	SCONJ
ap-7583	370	25	the	the	DET
ap-7583	370	26	polynomial	polynomial	NOUN
ap-7583	370	27	generated	generate	VERB
ap-7583	370	28	by	by	ADP
ap-7583	370	29	the	the	DET
ap-7583	370	30	determinant	determinant	ADJ
ap-7583	370	31	(	(	PUNCT
ap-7583	370	32	32	32	NUM
ap-7583	370	33	)	)	PUNCT
ap-7583	370	34	has	have	VERB
ap-7583	370	35	simple	simple	ADJ
ap-7583	370	36	and	and	CCONJ
ap-7583	370	37	real	real	ADJ
ap-7583	370	38	distinct	distinct	ADJ
ap-7583	370	39	roots	root	NOUN
ap-7583	370	40	.	.	PUNCT
ap-7583	371	1	4	4	X
ap-7583	371	2	.	.	X
ap-7583	371	3	the	the	DET
ap-7583	371	4	solutions	solution	NOUN
ap-7583	371	5	in	in	ADP
ap-7583	371	6	the	the	DET
ap-7583	371	7	neighbourhood	neighbourhood	NOUN
ap-7583	371	8	of	of	ADP
ap-7583	371	9	a	a	DET
ap-7583	371	10	singular	singular	ADJ
ap-7583	371	11	point	point	NOUN
ap-7583	371	12	4.1	4.1	NUM
ap-7583	371	13	.	.	PUNCT
ap-7583	372	1	series	series	NOUN
ap-7583	372	2	solution	solution	NOUN
ap-7583	372	3	and	and	CCONJ
ap-7583	372	4	infinite	infinite	ADJ
ap-7583	372	5	sequence	sequence	NOUN
ap-7583	372	6	of	of	ADP
ap-7583	372	7	orthogonal	orthogonal	ADJ
ap-7583	372	8	polynomials	polynomial	NOUN
ap-7583	372	9	{	{	PUNCT
ap-7583	372	10	pk(ε0)}∞	pk(ε0)}∞	NOUN
ap-7583	372	11	k=0	k=0	PROPN
ap-7583	372	12	as	as	SCONJ
ap-7583	372	13	mentioned	mention	VERB
ap-7583	372	14	earlier	early	ADV
ap-7583	372	15	,	,	PUNCT
ap-7583	372	16	if	if	SCONJ
ap-7583	372	17	α0	α0	ADJ
ap-7583	372	18	=	=	SYM
ap-7583	372	19	0	0	NUM
ap-7583	372	20	,	,	PUNCT
ap-7583	372	21	there	there	PRON
ap-7583	372	22	are	be	VERB
ap-7583	372	23	seven	seven	NUM
ap-7583	372	24	subclasses	subclass	NOUN
ap-7583	372	25	characterized	characterize	VERB
ap-7583	372	26	by	by	ADP
ap-7583	372	27	the	the	DET
ap-7583	372	28	equation	equation	NOUN
ap-7583	372	29	r	r	NOUN
ap-7583	372	30	(	(	PUNCT
ap-7583	372	31	α1	α1	PROPN
ap-7583	372	32	+	+	CCONJ
ap-7583	372	33	α2	α2	ADJ
ap-7583	372	34	r	r	NOUN
ap-7583	372	35	+	+	NOUN
ap-7583	372	36	α3r2	α3r2	X
ap-7583	372	37	)	)	PUNCT
ap-7583	372	38	y′′	y′′	NOUN
ap-7583	372	39	+	+	CCONJ
ap-7583	372	40	(	(	PUNCT
ap-7583	372	41	β0	β0	NOUN
ap-7583	372	42	+	+	CCONJ
ap-7583	372	43	β1	β1	PROPN
ap-7583	372	44	r	r	NOUN
ap-7583	372	45	+	+	CCONJ
ap-7583	372	46	β2	β2	NOUN
ap-7583	372	47	r2	r2	NOUN
ap-7583	372	48	)	)	PUNCT
ap-7583	372	49	y′	y′	PUNCT
ap-7583	373	1	+	+	CCONJ
ap-7583	373	2	(	(	PUNCT
ap-7583	373	3	ε0	ε0	PROPN
ap-7583	373	4	+	+	CCONJ
ap-7583	373	5	ε1	ε1	PROPN
ap-7583	373	6	r	r	NOUN
ap-7583	373	7	)	)	PUNCT
ap-7583	373	8	y	y	PROPN
ap-7583	373	9	=	=	NOUN
ap-7583	373	10	0	0	PROPN
ap-7583	373	11	.	.	PUNCT
ap-7583	374	1	(	(	PUNCT
ap-7583	374	2	65	65	NUM
ap-7583	374	3	)	)	PUNCT
ap-7583	374	4	176	176	NUM
ap-7583	374	5	vol	vol	NOUN
ap-7583	374	6	.	.	PUNCT
ap-7583	375	1	62	62	NUM
ap-7583	375	2	no	no	INTJ
ap-7583	375	3	.	.	PUNCT
ap-7583	376	1	1/2022	1/2022	NUM
ap-7583	376	2	on	on	ADP
ap-7583	376	3	generalized	generalized	ADJ
ap-7583	376	4	heun	heun	NOUN
ap-7583	376	5	equation	equation	NOUN
ap-7583	376	6	with	with	ADP
ap-7583	376	7	some	some	DET
ap-7583	376	8	mathematical	mathematical	NOUN
ap-7583	376	9	.	.	PUNCT
ap-7583	376	10	.	.	PUNCT
ap-7583	376	11	.	.	PUNCT
ap-7583	377	1	the	the	DET
ap-7583	377	2	classification	classification	NOUN
ap-7583	377	3	of	of	ADP
ap-7583	377	4	these	these	DET
ap-7583	377	5	seven	seven	NUM
ap-7583	377	6	equations	equation	NOUN
ap-7583	377	7	along	along	ADP
ap-7583	377	8	with	with	ADP
ap-7583	377	9	their	their	PRON
ap-7583	377	10	singularities	singularity	NOUN
ap-7583	377	11	and	and	CCONJ
ap-7583	377	12	the	the	DET
ap-7583	377	13	associated	associate	VERB
ap-7583	377	14	domains	domain	NOUN
ap-7583	377	15	are	be	AUX
ap-7583	377	16	summarized	summarize	VERB
ap-7583	377	17	in	in	ADP
ap-7583	377	18	table	table	NOUN
ap-7583	377	19	3	3	NUM
ap-7583	377	20	.	.	PUNCT
ap-7583	377	21	from	from	ADP
ap-7583	377	22	this	this	DET
ap-7583	377	23	table	table	NOUN
ap-7583	377	24	,	,	PUNCT
ap-7583	377	25	it	it	PRON
ap-7583	377	26	is	be	AUX
ap-7583	377	27	noted	note	VERB
ap-7583	377	28	that	that	SCONJ
ap-7583	377	29	if	if	SCONJ
ap-7583	377	30	α1	α1	PROPN
ap-7583	377	31	̸=	̸=	PROPN
ap-7583	377	32	0	0	NUM
ap-7583	377	33	,	,	PUNCT
ap-7583	377	34	there	there	PRON
ap-7583	377	35	are	be	VERB
ap-7583	377	36	four	four	NUM
ap-7583	377	37	subclasses	subclass	NOUN
ap-7583	377	38	where	where	SCONJ
ap-7583	377	39	the	the	DET
ap-7583	377	40	point	point	NOUN
ap-7583	377	41	r	r	NOUN
ap-7583	377	42	=	=	SYM
ap-7583	377	43	0	0	NUM
ap-7583	377	44	is	be	AUX
ap-7583	377	45	a	a	DET
ap-7583	377	46	regular	regular	ADJ
ap-7583	377	47	singular	singular	ADJ
ap-7583	377	48	point	point	NOUN
ap-7583	377	49	,	,	PUNCT
ap-7583	377	50	while	while	SCONJ
ap-7583	377	51	if	if	SCONJ
ap-7583	377	52	α1	α1	PROPN
ap-7583	377	53	=	=	SYM
ap-7583	377	54	0	0	NUM
ap-7583	377	55	,	,	PUNCT
ap-7583	377	56	the	the	DET
ap-7583	377	57	condition	condition	NOUN
ap-7583	377	58	β0	β0	PROPN
ap-7583	377	59	=	=	NOUN
ap-7583	377	60	0	0	NUM
ap-7583	377	61	is	be	AUX
ap-7583	377	62	necessary	necessary	ADJ
ap-7583	377	63	to	to	PART
ap-7583	377	64	ensure	ensure	VERB
ap-7583	377	65	that	that	SCONJ
ap-7583	377	66	r	r	NOUN
ap-7583	377	67	=	=	SYM
ap-7583	377	68	0	0	NUM
ap-7583	377	69	is	be	AUX
ap-7583	377	70	a	a	DET
ap-7583	377	71	regular	regular	ADJ
ap-7583	377	72	singular	singular	ADJ
ap-7583	377	73	point	point	NOUN
ap-7583	377	74	for	for	ADP
ap-7583	377	75	two	two	NUM
ap-7583	377	76	additional	additional	ADJ
ap-7583	377	77	subclasses	subclass	NOUN
ap-7583	377	78	and	and	CCONJ
ap-7583	377	79	the	the	DET
ap-7583	377	80	last	last	ADJ
ap-7583	377	81	equation	equation	NOUN
ap-7583	377	82	is	be	AUX
ap-7583	377	83	a	a	DET
ap-7583	377	84	class	class	NOUN
ap-7583	377	85	where	where	SCONJ
ap-7583	377	86	r	r	NOUN
ap-7583	377	87	=	=	SYM
ap-7583	377	88	0	0	NUM
ap-7583	377	89	is	be	AUX
ap-7583	377	90	irregular	irregular	ADJ
ap-7583	377	91	singular	singular	ADJ
ap-7583	377	92	point	point	NOUN
ap-7583	377	93	unless	unless	SCONJ
ap-7583	377	94	we	we	PRON
ap-7583	377	95	reduce	reduce	VERB
ap-7583	377	96	to	to	PART
ap-7583	377	97	euler	euler	VERB
ap-7583	377	98	’s	’s	PART
ap-7583	377	99	type	type	NOUN
ap-7583	377	100	(	(	PUNCT
ap-7583	377	101	α1	α1	NOUN
ap-7583	377	102	=	=	SYM
ap-7583	377	103	α2	α2	PROPN
ap-7583	377	104	=	=	SYM
ap-7583	377	105	β1	β1	PROPN
ap-7583	377	106	=	=	PUNCT
ap-7583	377	107	β0	β0	PROPN
ap-7583	377	108	=	=	PROPN
ap-7583	377	109	ε0	ε0	PROPN
ap-7583	377	110	=	=	NOUN
ap-7583	377	111	0	0	NUM
ap-7583	377	112	)	)	PUNCT
ap-7583	377	113	.	.	PUNCT
ap-7583	378	1	in	in	ADP
ap-7583	378	2	the	the	DET
ap-7583	378	3	neighbourhood	neighbourhood	NOUN
ap-7583	378	4	of	of	ADP
ap-7583	378	5	the	the	DET
ap-7583	378	6	regular	regular	ADJ
ap-7583	378	7	singular	singular	ADJ
ap-7583	378	8	point	point	NOUN
ap-7583	378	9	r	r	NOUN
ap-7583	378	10	=	=	SYM
ap-7583	378	11	0	0	NUM
ap-7583	378	12	,	,	PUNCT
ap-7583	378	13	the	the	DET
ap-7583	378	14	formal	formal	ADJ
ap-7583	378	15	series	series	NOUN
ap-7583	378	16	solution	solution	NOUN
ap-7583	378	17	y(r	y(r	NOUN
ap-7583	378	18	)	)	PUNCT
ap-7583	378	19	=	=	SYM
ap-7583	378	20	rs	rs	NOUN
ap-7583	378	21	∑∞	∑∞	NOUN
ap-7583	378	22	k=0	k=0	PROPN
ap-7583	378	23	ckrk	ckrk	NOUN
ap-7583	378	24	is	be	AUX
ap-7583	378	25	then	then	ADV
ap-7583	378	26	valid	valid	ADJ
ap-7583	378	27	within	within	ADP
ap-7583	378	28	the	the	DET
ap-7583	378	29	interval	interval	NOUN
ap-7583	378	30	(	(	PUNCT
ap-7583	378	31	0	0	NUM
ap-7583	378	32	,	,	PUNCT
ap-7583	378	33	ζ	ζ	NOUN
ap-7583	378	34	)	)	PUNCT
ap-7583	378	35	where	where	SCONJ
ap-7583	378	36	ζ	ζ	NOUN
ap-7583	378	37	is	be	AUX
ap-7583	378	38	the	the	DET
ap-7583	378	39	nearest	near	ADJ
ap-7583	378	40	singular	singular	ADJ
ap-7583	378	41	point	point	NOUN
ap-7583	378	42	obtained	obtain	VERB
ap-7583	378	43	via	via	ADP
ap-7583	378	44	the	the	DET
ap-7583	378	45	roots	root	NOUN
ap-7583	378	46	of	of	ADP
ap-7583	378	47	the	the	DET
ap-7583	378	48	quadratic	quadratic	ADJ
ap-7583	378	49	equation	equation	NOUN
ap-7583	378	50	α1	α1	PROPN
ap-7583	378	51	+	+	CCONJ
ap-7583	378	52	α2	α2	ADJ
ap-7583	378	53	r	r	NOUN
ap-7583	378	54	+	+	CCONJ
ap-7583	378	55	α3r2	α3r2	CCONJ
ap-7583	378	56	=	=	SYM
ap-7583	378	57	0	0	PROPN
ap-7583	378	58	.	.	PUNCT
ap-7583	379	1	here	here	ADV
ap-7583	379	2	,	,	PUNCT
ap-7583	379	3	s	s	VERB
ap-7583	379	4	are	be	AUX
ap-7583	379	5	the	the	DET
ap-7583	379	6	roots	root	NOUN
ap-7583	379	7	of	of	ADP
ap-7583	379	8	the	the	DET
ap-7583	379	9	indicial	indicial	ADJ
ap-7583	379	10	equation	equation	NOUN
ap-7583	379	11	α1	α1	PROPN
ap-7583	379	12	s(s	s(s	PROPN
ap-7583	379	13	−	−	PROPN
ap-7583	379	14	1	1	NUM
ap-7583	379	15	)	)	PUNCT
ap-7583	380	1	+	+	CCONJ
ap-7583	380	2	β0	β0	PROPN
ap-7583	380	3	s	s	PART
ap-7583	380	4	=	=	NOUN
ap-7583	380	5	0	0	NUM
ap-7583	380	6	,	,	PUNCT
ap-7583	380	7	i.e.	i.e.	X
ap-7583	380	8	s1	s1	NOUN
ap-7583	380	9	=	=	SYM
ap-7583	380	10	0	0	NUM
ap-7583	380	11	and	and	CCONJ
ap-7583	380	12	s	s	X
ap-7583	380	13	=	=	SYM
ap-7583	380	14	1	1	NUM
ap-7583	380	15	−	−	PROPN
ap-7583	380	16	β0	β0	PROPN
ap-7583	380	17	/	/	SYM
ap-7583	380	18	α1	α1	PROPN
ap-7583	380	19	.	.	PUNCT
ap-7583	381	1	using	use	VERB
ap-7583	381	2	frobenius	frobenius	ADJ
ap-7583	381	3	method	method	NOUN
ap-7583	381	4	,	,	PUNCT
ap-7583	381	5	it	it	PRON
ap-7583	381	6	is	be	AUX
ap-7583	381	7	straightforward	straightforward	ADJ
ap-7583	381	8	to	to	PART
ap-7583	381	9	show	show	VERB
ap-7583	381	10	that	that	SCONJ
ap-7583	381	11	the	the	DET
ap-7583	381	12	coefficients	coefficient	NOUN
ap-7583	381	13	{	{	PUNCT
ap-7583	381	14	ck}∞	ck}∞	X
ap-7583	381	15	k=0	k=0	PROPN
ap-7583	381	16	satisfy	satisfy	VERB
ap-7583	381	17	the	the	DET
ap-7583	381	18	three	three	NUM
ap-7583	381	19	-	-	PUNCT
ap-7583	381	20	term	term	NOUN
ap-7583	381	21	recurrence	recurrence	NOUN
ap-7583	381	22	relation	relation	NOUN
ap-7583	381	23	(	(	PUNCT
ap-7583	381	24	k	k	PROPN
ap-7583	381	25	+	+	SYM
ap-7583	381	26	s	s	PART
ap-7583	381	27	+	+	NUM
ap-7583	381	28	1	1	NUM
ap-7583	381	29	)	)	PUNCT
ap-7583	381	30	(	(	PUNCT
ap-7583	381	31	α1(k	α1(k	PUNCT
ap-7583	381	32	+	+	NUM
ap-7583	381	33	s	s	X
ap-7583	381	34	)	)	PUNCT
ap-7583	382	1	+	+	CCONJ
ap-7583	382	2	β0	β0	NOUN
ap-7583	382	3	)	)	PUNCT
ap-7583	382	4	ck+1	ck+1	PUNCT
ap-7583	383	1	+	+	CCONJ
ap-7583	383	2	(	(	PUNCT
ap-7583	383	3	(	(	PUNCT
ap-7583	383	4	k	k	X
ap-7583	383	5	+	+	CCONJ
ap-7583	383	6	s)[α2	s)[α2	PROPN
ap-7583	383	7	(	(	PUNCT
ap-7583	383	8	k	k	PROPN
ap-7583	383	9	+	+	SYM
ap-7583	383	10	s	s	VERB
ap-7583	383	11	−	−	NOUN
ap-7583	383	12	1	1	NUM
ap-7583	383	13	)	)	PUNCT
ap-7583	383	14	+	+	CCONJ
ap-7583	384	1	β1	β1	NOUN
ap-7583	384	2	]	]	PUNCT
ap-7583	385	1	+	+	CCONJ
ap-7583	385	2	ε0	ε0	PROPN
ap-7583	385	3	)	)	PUNCT
ap-7583	385	4	ck	ck	PROPN
ap-7583	386	1	+	+	CCONJ
ap-7583	386	2	(	(	PUNCT
ap-7583	386	3	(	(	PUNCT
ap-7583	386	4	k	k	X
ap-7583	386	5	+	+	SYM
ap-7583	386	6	s	s	VERB
ap-7583	386	7	−	−	PROPN
ap-7583	386	8	1)[α3(k	1)[α3(k	NOUN
ap-7583	387	1	+	+	SYM
ap-7583	387	2	s	s	NOUN
ap-7583	387	3	−	−	NOUN
ap-7583	387	4	2	2	NUM
ap-7583	387	5	)	)	PUNCT
ap-7583	387	6	+	+	CCONJ
ap-7583	387	7	β2	β2	VERB
ap-7583	387	8	]	]	X
ap-7583	388	1	+	+	CCONJ
ap-7583	388	2	ε1	ε1	PROPN
ap-7583	388	3	)	)	PUNCT
ap-7583	388	4	ck−1	ck−1	NOUN
ap-7583	388	5	=	=	SYM
ap-7583	388	6	0	0	NUM
ap-7583	388	7	,	,	PUNCT
ap-7583	388	8	(	(	PUNCT
ap-7583	388	9	66	66	NUM
ap-7583	388	10	)	)	PUNCT
ap-7583	388	11	where	where	SCONJ
ap-7583	388	12	k	k	NOUN
ap-7583	388	13	=	=	SYM
ap-7583	388	14	1	1	NUM
ap-7583	388	15	,	,	PUNCT
ap-7583	388	16	2	2	NUM
ap-7583	388	17	,	,	PUNCT
ap-7583	388	18	.	.	PUNCT
ap-7583	388	19	.	.	PUNCT
ap-7583	388	20	.	.	PUNCT
ap-7583	388	21	.	.	PUNCT
ap-7583	389	1	for	for	ADP
ap-7583	389	2			NUM
ap-7583	389	3	c−1	c−1	PROPN
ap-7583	389	4	=	=	SYM
ap-7583	389	5	0	0	PROPN
ap-7583	389	6	,	,	PUNCT
ap-7583	389	7	c0	c0	NOUN
ap-7583	389	8	=	=	SYM
ap-7583	389	9	1	1	NUM
ap-7583	389	10	,	,	PUNCT
ap-7583	389	11	c1	c1	NOUN
ap-7583	389	12	=	=	SYM
ap-7583	389	13	−s(α2	−s(α2	PROPN
ap-7583	389	14	(	(	PUNCT
ap-7583	389	15	s	s	NOUN
ap-7583	389	16	−	−	PROPN
ap-7583	389	17	1	1	NUM
ap-7583	389	18	)	)	PUNCT
ap-7583	389	19	+	+	CCONJ
ap-7583	389	20	β1	β1	NOUN
ap-7583	389	21	)	)	PUNCT
ap-7583	389	22	+	+	CCONJ
ap-7583	389	23	ε0	ε0	PROPN
ap-7583	389	24	(	(	PUNCT
ap-7583	389	25	α1	α1	PROPN
ap-7583	389	26	s	s	PART
ap-7583	389	27	+	+	X
ap-7583	389	28	β0)(s	β0)(s	NOUN
ap-7583	389	29	+	+	CCONJ
ap-7583	389	30	1	1	X
ap-7583	389	31	)	)	PUNCT
ap-7583	389	32	=	=	NOUN
ap-7583	389	33	−	−	PROPN
ap-7583	389	34	p1;s(ε0	p1;s(ε0	NOUN
ap-7583	389	35	)	)	PUNCT
ap-7583	389	36	α1	α1	PROPN
ap-7583	389	37	(	(	PUNCT
ap-7583	389	38	s	s	PROPN
ap-7583	389	39	+	+	CCONJ
ap-7583	389	40	β0	β0	ADJ
ap-7583	389	41	α1	α1	PROPN
ap-7583	389	42	)	)	PUNCT
ap-7583	389	43	(	(	PUNCT
ap-7583	389	44	s	s	VERB
ap-7583	389	45	+	+	NOUN
ap-7583	389	46	1	1	NUM
ap-7583	389	47	)	)	PUNCT
ap-7583	389	48	,	,	PUNCT
ap-7583	389	49	this	this	DET
ap-7583	389	50	equation	equation	NOUN
ap-7583	389	51	can	can	AUX
ap-7583	389	52	be	be	AUX
ap-7583	389	53	written	write	VERB
ap-7583	389	54	as	as	ADP
ap-7583	389	55	ck+2	ck+2	NOUN
ap-7583	389	56	=	=	SYM
ap-7583	389	57	λ0(k	λ0(k	X
ap-7583	389	58	)	)	PUNCT
ap-7583	389	59	ck+1	ck+1	PUNCT
ap-7583	389	60	+	+	CCONJ
ap-7583	389	61	s0(k	s0(k	X
ap-7583	389	62	)	)	PUNCT
ap-7583	389	63	ck	ck	PROPN
ap-7583	389	64	,	,	PUNCT
ap-7583	389	65	where	where	SCONJ
ap-7583	389	66			PROPN
ap-7583	389	67	λ0(k	λ0(k	ADJ
ap-7583	389	68	)	)	PUNCT
ap-7583	389	69	=	=	SYM
ap-7583	390	1	−	−	PROPN
ap-7583	390	2	(	(	PUNCT
ap-7583	390	3	α2	α2	PROPN
ap-7583	390	4	(	(	PUNCT
ap-7583	390	5	k	k	PROPN
ap-7583	390	6	+	+	PROPN
ap-7583	390	7	s	s	X
ap-7583	390	8	)	)	PUNCT
ap-7583	390	9	+	+	CCONJ
ap-7583	390	10	β1	β1	NOUN
ap-7583	390	11	)	)	PUNCT
ap-7583	390	12	(	(	PUNCT
ap-7583	391	1	k	k	X
ap-7583	391	2	+	+	SYM
ap-7583	391	3	s	s	PART
ap-7583	391	4	+	+	NUM
ap-7583	391	5	1	1	NUM
ap-7583	391	6	)	)	PUNCT
ap-7583	391	7	+	+	CCONJ
ap-7583	391	8	ε0	ε0	PROPN
ap-7583	391	9	(	(	PUNCT
ap-7583	391	10	α1(k	α1(k	NOUN
ap-7583	391	11	+	+	SYM
ap-7583	391	12	s	s	PART
ap-7583	391	13	+	+	NUM
ap-7583	391	14	1	1	NUM
ap-7583	391	15	)	)	PUNCT
ap-7583	391	16	+	+	CCONJ
ap-7583	391	17	β0	β0	X
ap-7583	391	18	)	)	PUNCT
ap-7583	391	19	(	(	PUNCT
ap-7583	392	1	k	k	X
ap-7583	392	2	+	+	SYM
ap-7583	392	3	s	s	PART
ap-7583	392	4	+	+	NUM
ap-7583	392	5	2	2	NUM
ap-7583	392	6	)	)	PUNCT
ap-7583	392	7	,	,	PUNCT
ap-7583	392	8	s0(k	s0(k	X
ap-7583	392	9	)	)	PUNCT
ap-7583	392	10	=	=	SYM
ap-7583	393	1	−	−	PROPN
ap-7583	393	2	(	(	PUNCT
ap-7583	393	3	α3(k	α3(k	NUM
ap-7583	394	1	+	+	SYM
ap-7583	394	2	s	s	NOUN
ap-7583	394	3	−	−	PROPN
ap-7583	394	4	1	1	NUM
ap-7583	394	5	)	)	PUNCT
ap-7583	394	6	+	+	CCONJ
ap-7583	394	7	β2	β2	VERB
ap-7583	394	8	)	)	PUNCT
ap-7583	394	9	(	(	PUNCT
ap-7583	394	10	k	k	PROPN
ap-7583	394	11	+	+	PROPN
ap-7583	394	12	s	s	X
ap-7583	394	13	)	)	PUNCT
ap-7583	395	1	+	+	CCONJ
ap-7583	395	2	ε1	ε1	PROPN
ap-7583	395	3	(	(	PUNCT
ap-7583	395	4	α1(k	α1(k	NOUN
ap-7583	395	5	+	+	SYM
ap-7583	395	6	s	s	PART
ap-7583	395	7	+	+	NUM
ap-7583	395	8	1	1	NUM
ap-7583	395	9	)	)	PUNCT
ap-7583	395	10	+	+	CCONJ
ap-7583	395	11	β0	β0	X
ap-7583	395	12	)	)	PUNCT
ap-7583	395	13	(	(	PUNCT
ap-7583	395	14	k	k	X
ap-7583	395	15	+	+	SYM
ap-7583	395	16	s	s	PART
ap-7583	395	17	+	+	NUM
ap-7583	395	18	2	2	NUM
ap-7583	395	19	)	)	PUNCT
ap-7583	395	20	,	,	PUNCT
ap-7583	395	21	from	from	ADP
ap-7583	395	22	this	this	DET
ap-7583	395	23	equation	equation	NOUN
ap-7583	395	24	,	,	PUNCT
ap-7583	395	25	we	we	PRON
ap-7583	395	26	note	note	VERB
ap-7583	395	27	that	that	SCONJ
ap-7583	395	28	ck+3	ck+3	NOUN
ap-7583	395	29	=	=	SYM
ap-7583	395	30	λ1(k	λ1(k	NOUN
ap-7583	395	31	)	)	PUNCT
ap-7583	395	32	ck+1	ck+1	PUNCT
ap-7583	395	33	+	+	CCONJ
ap-7583	395	34	s1(k	s1(k	NOUN
ap-7583	395	35	)	)	PUNCT
ap-7583	395	36	ck	ck	PROPN
ap-7583	395	37	,	,	PUNCT
ap-7583	395	38			PUNCT
ap-7583	395	39	λ1(k	λ1(k	NOUN
ap-7583	395	40	)	)	PUNCT
ap-7583	395	41	=	=	PUNCT
ap-7583	396	1	λ0(k	λ0(k	X
ap-7583	396	2	+	+	PUNCT
ap-7583	396	3	1	1	X
ap-7583	396	4	)	)	PUNCT
ap-7583	396	5	λ0(k	λ0(k	PUNCT
ap-7583	396	6	)	)	PUNCT
ap-7583	397	1	+	+	CCONJ
ap-7583	397	2	s0(k	s0(k	X
ap-7583	397	3	+	+	CCONJ
ap-7583	397	4	1	1	X
ap-7583	397	5	)	)	PUNCT
ap-7583	397	6	s1(k	s1(k	NOUN
ap-7583	397	7	)	)	PUNCT
ap-7583	397	8	=	=	VERB
ap-7583	398	1	λ0(k	λ0(k	X
ap-7583	398	2	+	+	PUNCT
ap-7583	398	3	1	1	X
ap-7583	398	4	)	)	PUNCT
ap-7583	398	5	s0(k	s0(k	NOUN
ap-7583	398	6	)	)	PUNCT
ap-7583	398	7	,	,	PUNCT
ap-7583	398	8	ck+4	ck+4	PROPN
ap-7583	398	9	=	=	SYM
ap-7583	398	10	λ2(k	λ2(k	PROPN
ap-7583	398	11	)	)	PUNCT
ap-7583	398	12	ck+1	ck+1	PUNCT
ap-7583	398	13	+	+	CCONJ
ap-7583	398	14	s2(k	s2(k	NOUN
ap-7583	398	15	)	)	PUNCT
ap-7583	398	16	ck	ck	PROPN
ap-7583	398	17	,	,	PUNCT
ap-7583	398	18			PUNCT
ap-7583	398	19	λ2(k	λ2(k	PROPN
ap-7583	398	20	)	)	PUNCT
ap-7583	398	21	=	=	PUNCT
ap-7583	399	1	λ1(k	λ1(k	X
ap-7583	399	2	+	+	NOUN
ap-7583	399	3	1)λ0(k	1)λ0(k	NUM
ap-7583	399	4	)	)	PUNCT
ap-7583	400	1	+	+	CCONJ
ap-7583	400	2	s1(k	s1(k	X
ap-7583	400	3	+	+	NOUN
ap-7583	400	4	1	1	NUM
ap-7583	400	5	)	)	PUNCT
ap-7583	400	6	s2(k	s2(k	NOUN
ap-7583	400	7	)	)	PUNCT
ap-7583	400	8	=	=	PUNCT
ap-7583	401	1	λ1(k	λ1(k	X
ap-7583	401	2	+	+	NUM
ap-7583	401	3	1)s0(k	1)s0(k	NUM
ap-7583	401	4	)	)	PUNCT
ap-7583	401	5	,	,	PUNCT
ap-7583	401	6	ck+5	ck+5	NOUN
ap-7583	401	7	=	=	SYM
ap-7583	401	8	λ3(k	λ3(k	NOUN
ap-7583	401	9	)	)	PUNCT
ap-7583	401	10	ck+1	ck+1	PUNCT
ap-7583	402	1	+	+	CCONJ
ap-7583	402	2	s3(k	s3(k	NOUN
ap-7583	402	3	)	)	PUNCT
ap-7583	402	4	ck	ck	PROPN
ap-7583	402	5	,	,	PUNCT
ap-7583	402	6			PUNCT
ap-7583	402	7	λ3(k	λ3(k	NOUN
ap-7583	402	8	)	)	PUNCT
ap-7583	403	1	=	=	SYM
ap-7583	404	1	λ2(k	λ2(k	PROPN
ap-7583	404	2	+	+	NUM
ap-7583	404	3	1)λ0(k	1)λ0(k	NUM
ap-7583	404	4	)	)	PUNCT
ap-7583	405	1	+	+	CCONJ
ap-7583	405	2	s2(k	s2(k	X
ap-7583	405	3	+	+	CCONJ
ap-7583	405	4	1	1	NUM
ap-7583	405	5	)	)	PUNCT
ap-7583	405	6	s3(k	s3(k	NOUN
ap-7583	405	7	)	)	PUNCT
ap-7583	405	8	=	=	SYM
ap-7583	406	1	λ2(k	λ2(k	PROPN
ap-7583	406	2	+	+	NUM
ap-7583	406	3	1)s0(k	1)s0(k	NUM
ap-7583	406	4	)	)	PUNCT
ap-7583	406	5	,	,	PUNCT
ap-7583	406	6	and	and	CCONJ
ap-7583	406	7	in	in	ADP
ap-7583	406	8	general	general	PROPN
ap-7583	406	9	ck+m	ck+m	PROPN
ap-7583	406	10	=	=	SYM
ap-7583	406	11	λm−2(k	λm−2(k	PROPN
ap-7583	406	12	)	)	PUNCT
ap-7583	406	13	ck+1	ck+1	VERB
ap-7583	406	14	+	+	CCONJ
ap-7583	406	15	sm−2(k	sm−2(k	NOUN
ap-7583	406	16	)	)	PUNCT
ap-7583	406	17	ck	ck	PROPN
ap-7583	406	18	,	,	PUNCT
ap-7583	406	19			PUNCT
ap-7583	406	20	λm(k	λm(k	X
ap-7583	406	21	)	)	PUNCT
ap-7583	406	22	=	=	PUNCT
ap-7583	407	1	λm−1(k	λm−1(k	NOUN
ap-7583	407	2	+	+	CCONJ
ap-7583	407	3	1)λ0(k	1)λ0(k	NUM
ap-7583	407	4	)	)	PUNCT
ap-7583	408	1	+	+	NUM
ap-7583	408	2	sm−1(k	sm−1(k	NOUN
ap-7583	408	3	+	+	CCONJ
ap-7583	408	4	1	1	NUM
ap-7583	408	5	)	)	PUNCT
ap-7583	408	6	sm(k	sm(k	NOUN
ap-7583	408	7	)	)	PUNCT
ap-7583	409	1	=	=	PUNCT
ap-7583	410	1	λm−1(k	λm−1(k	NOUN
ap-7583	410	2	+	+	CCONJ
ap-7583	410	3	1)s0(k	1)s0(k	NUM
ap-7583	410	4	)	)	PUNCT
ap-7583	410	5	,	,	PUNCT
ap-7583	410	6	and	and	CCONJ
ap-7583	410	7	therefore	therefore	ADV
ap-7583	410	8	c2	c2	PROPN
ap-7583	410	9	=	=	PUNCT
ap-7583	410	10	(	(	PUNCT
ap-7583	410	11	s	s	X
ap-7583	410	12	+	+	X
ap-7583	410	13	1)(α2	1)(α2	NUM
ap-7583	410	14	s	s	PART
ap-7583	410	15	+	+	CCONJ
ap-7583	410	16	β1	β1	NOUN
ap-7583	410	17	)	)	PUNCT
ap-7583	411	1	+	+	CCONJ
ap-7583	411	2	ε0	ε0	PROPN
ap-7583	411	3	(	(	PUNCT
ap-7583	411	4	α1(s	α1(s	NUM
ap-7583	411	5	+	+	NOUN
ap-7583	411	6	1	1	NUM
ap-7583	411	7	)	)	PUNCT
ap-7583	411	8	+	+	CCONJ
ap-7583	411	9	β0)(s	β0)(s	NOUN
ap-7583	412	1	+	+	CCONJ
ap-7583	412	2	2	2	X
ap-7583	412	3	)	)	PUNCT
ap-7583	412	4	(	(	PUNCT
ap-7583	412	5	s(α2	s(α2	X
ap-7583	412	6	(	(	PUNCT
ap-7583	412	7	s	s	VERB
ap-7583	412	8	−	−	NOUN
ap-7583	412	9	1	1	NUM
ap-7583	412	10	)	)	PUNCT
ap-7583	412	11	+	+	CCONJ
ap-7583	412	12	β1	β1	NOUN
ap-7583	412	13	)	)	PUNCT
ap-7583	413	1	+	+	CCONJ
ap-7583	413	2	ε0	ε0	PROPN
ap-7583	413	3	(	(	PUNCT
ap-7583	413	4	α1	α1	PROPN
ap-7583	413	5	s	s	PART
ap-7583	413	6	+	+	X
ap-7583	413	7	β0)(s	β0)(s	PROPN
ap-7583	413	8	+	+	CCONJ
ap-7583	413	9	1	1	NUM
ap-7583	413	10	)	)	PUNCT
ap-7583	413	11	)	)	PUNCT
ap-7583	414	1	−	−	PROPN
ap-7583	415	1	s(α3(s	s(α3(s	PROPN
ap-7583	415	2	−	−	PROPN
ap-7583	415	3	1	1	NUM
ap-7583	415	4	)	)	PUNCT
ap-7583	415	5	+	+	CCONJ
ap-7583	415	6	β2	β2	VERB
ap-7583	415	7	)	)	PUNCT
ap-7583	415	8	+	+	CCONJ
ap-7583	415	9	ε1	ε1	X
ap-7583	415	10	(	(	PUNCT
ap-7583	415	11	α1(s	α1(s	X
ap-7583	415	12	+	+	NOUN
ap-7583	415	13	1	1	NUM
ap-7583	415	14	)	)	PUNCT
ap-7583	416	1	+	+	CCONJ
ap-7583	416	2	β0)(s	β0)(s	NOUN
ap-7583	417	1	+	+	CCONJ
ap-7583	417	2	2	2	X
ap-7583	417	3	)	)	PUNCT
ap-7583	417	4	=	=	SYM
ap-7583	417	5	p2;s(ε0	p2;s(ε0	X
ap-7583	417	6	)	)	PUNCT
ap-7583	417	7	α2	α2	NOUN
ap-7583	417	8	1	1	NUM
ap-7583	417	9	(	(	PUNCT
ap-7583	417	10	s	s	PROPN
ap-7583	417	11	+	+	CCONJ
ap-7583	417	12	β0	β0	ADJ
ap-7583	417	13	α1	α1	PROPN
ap-7583	417	14	)	)	PUNCT
ap-7583	417	15	2	2	NUM
ap-7583	417	16	(	(	PUNCT
ap-7583	417	17	s	s	X
ap-7583	417	18	+	+	X
ap-7583	417	19	1)2	1)2	NUM
ap-7583	417	20	.	.	PUNCT
ap-7583	418	1	(	(	PUNCT
ap-7583	418	2	67	67	NUM
ap-7583	418	3	)	)	PUNCT
ap-7583	418	4	177	177	NUM
ap-7583	418	5	nasser	nasser	PROPN
ap-7583	418	6	saad	saad	PROPN
ap-7583	418	7	acta	acta	PROPN
ap-7583	418	8	polytechnica	polytechnica	PROPN
ap-7583	418	9	de	de	PROPN
ap-7583	418	10	α3	α3	ADJ
ap-7583	418	11	α2	α2	ADJ
ap-7583	418	12	α1	α1	PROPN
ap-7583	418	13	condition	condition	NOUN
ap-7583	418	14	roots	root	NOUN
ap-7583	418	15	of	of	ADP
ap-7583	418	16	lpc	lpc	PROPN
ap-7583	418	17	domain	domain	NOUN
ap-7583	418	18	definition	definition	NOUN
ap-7583	418	19	i	i	PRON
ap-7583	418	20	α3	α3	VERB
ap-7583	418	21	α2	α2	ADJ
ap-7583	418	22	α1	α1	PROPN
ap-7583	418	23	a2	a2	PROPN
ap-7583	418	24	2	2	NUM
ap-7583	418	25	−	−	NOUN
ap-7583	418	26	4a1a3	4a1a3	NUM
ap-7583	418	27	>	>	X
ap-7583	418	28	0	0	PUNCT
ap-7583	419	1	r	r	NOUN
ap-7583	419	2	=	=	SYM
ap-7583	419	3	0	0	NUM
ap-7583	419	4	,	,	PUNCT
ap-7583	419	5	ξ+	ξ+	NUM
ap-7583	419	6	̸=	̸=	PROPN
ap-7583	419	7	ξ−	ξ−	PROPN
ap-7583	419	8	r	r	NOUN
ap-7583	419	9	∈	∈	PROPN
ap-7583	419	10	(	(	PUNCT
ap-7583	419	11	0	0	NUM
ap-7583	419	12	,	,	PUNCT
ap-7583	419	13	min	min	NOUN
ap-7583	419	14	ξ±	ξ±	PROPN
ap-7583	419	15	)	)	PUNCT
ap-7583	420	1	if	if	SCONJ
ap-7583	420	2	ξ±	ξ±	NUM
ap-7583	420	3	>	>	SYM
ap-7583	420	4	0	0	PUNCT
ap-7583	420	5	r	r	NOUN
ap-7583	420	6	∈	∈	PROPN
ap-7583	420	7	(	(	PUNCT
ap-7583	420	8	max	max	PROPN
ap-7583	420	9	ξ±	ξ±	PROPN
ap-7583	420	10	,	,	PUNCT
ap-7583	420	11	0	0	NUM
ap-7583	420	12	)	)	PUNCT
ap-7583	420	13	if	if	SCONJ
ap-7583	420	14	ξ±	ξ±	ADV
ap-7583	420	15	<	<	X
ap-7583	420	16	0	0	NUM
ap-7583	420	17	r	r	NOUN
ap-7583	420	18	∈	∈	PROPN
ap-7583	420	19	(	(	PUNCT
ap-7583	420	20	0	0	NUM
ap-7583	420	21	,	,	PUNCT
ap-7583	420	22	ξ+	ξ+	NUM
ap-7583	420	23	)	)	PUNCT
ap-7583	420	24	if	if	SCONJ
ap-7583	420	25	ξ−	ξ−	PROPN
ap-7583	420	26	<	<	X
ap-7583	420	27	0	0	X
ap-7583	420	28	<	<	X
ap-7583	420	29	ξ+	ξ+	PROPN
ap-7583	420	30	,	,	PUNCT
ap-7583	420	31	|ξ−|	|ξ−|	NOUN
ap-7583	420	32	>	>	X
ap-7583	420	33	ξ+	ξ+	PRON
ap-7583	420	34	r	r	NOUN
ap-7583	420	35	∈	∈	PROPN
ap-7583	420	36	(	(	PUNCT
ap-7583	420	37	ξ−	ξ−	PROPN
ap-7583	420	38	,	,	PUNCT
ap-7583	420	39	0	0	NUM
ap-7583	420	40	)	)	PUNCT
ap-7583	420	41	if	if	SCONJ
ap-7583	420	42	ξ−	ξ−	PROPN
ap-7583	420	43	<	<	X
ap-7583	420	44	0	0	X
ap-7583	420	45	<	<	X
ap-7583	420	46	ξ+	ξ+	PROPN
ap-7583	420	47	,	,	PUNCT
ap-7583	420	48	|ξ−|	|ξ−|	NOUN
ap-7583	420	49	<	<	X
ap-7583	420	50	ξ+	ξ+	X
ap-7583	420	51	a2	a2	PROPN
ap-7583	420	52	2	2	NUM
ap-7583	420	53	−	−	NOUN
ap-7583	420	54	4a1a3	4a1a3	NUM
ap-7583	420	55	=	=	SYM
ap-7583	420	56	0	0	NUM
ap-7583	420	57	r	r	NOUN
ap-7583	420	58	=	=	SYM
ap-7583	420	59	0	0	NUM
ap-7583	420	60	,	,	PUNCT
ap-7583	420	61	ξ+	ξ+	PRON
ap-7583	420	62	=	=	PUNCT
ap-7583	420	63	ξ−	ξ−	PROPN
ap-7583	420	64	=	=	PUNCT
ap-7583	421	1	ξ	ξ	PRON
ap-7583	421	2	r	r	NOUN
ap-7583	421	3	∈	∈	PROPN
ap-7583	421	4	(	(	PUNCT
ap-7583	421	5	0	0	NUM
ap-7583	421	6	,	,	PUNCT
ap-7583	421	7	ξ	ξ	NOUN
ap-7583	421	8	)	)	PUNCT
ap-7583	421	9	differential	differential	ADJ
ap-7583	421	10	equation	equation	NOUN
ap-7583	421	11	:	:	PUNCT
ap-7583	422	1	r	r	NOUN
ap-7583	422	2	(	(	PUNCT
ap-7583	422	3	ξ1	ξ1	PROPN
ap-7583	422	4	−	−	PROPN
ap-7583	422	5	r)(ξ2	r)(ξ2	NOUN
ap-7583	422	6	−	−	NOUN
ap-7583	422	7	r	r	NOUN
ap-7583	422	8	)	)	PUNCT
ap-7583	422	9	y′′	y′′	NOUN
ap-7583	422	10	+	+	CCONJ
ap-7583	422	11	(	(	PUNCT
ap-7583	422	12	β0	β0	NOUN
ap-7583	422	13	+	+	CCONJ
ap-7583	422	14	β1	β1	PROPN
ap-7583	422	15	r	r	NOUN
ap-7583	422	16	+	+	CCONJ
ap-7583	422	17	β2	β2	NOUN
ap-7583	422	18	r2	r2	NOUN
ap-7583	422	19	)	)	PUNCT
ap-7583	422	20	y′	y′	PUNCT
ap-7583	423	1	+	+	CCONJ
ap-7583	423	2	(	(	PUNCT
ap-7583	423	3	ε0	ε0	PROPN
ap-7583	423	4	+	+	CCONJ
ap-7583	423	5	ε1	ε1	PROPN
ap-7583	423	6	r	r	NOUN
ap-7583	423	7	)	)	PUNCT
ap-7583	423	8	y	y	NOUN
ap-7583	423	9	=	=	SYM
ap-7583	423	10	0	0	NUM
ap-7583	423	11	roots	root	NOUN
ap-7583	423	12	:	:	PUNCT
ap-7583	423	13	r	r	NOUN
ap-7583	423	14	=	=	SYM
ap-7583	423	15	0	0	NUM
ap-7583	423	16	;	;	PUNCT
ap-7583	423	17	r	r	NOUN
ap-7583	423	18	=	=	SYM
ap-7583	423	19	ξ±	ξ±	PROPN
ap-7583	423	20	≡	≡	PROPN
ap-7583	423	21	(	(	PUNCT
ap-7583	423	22	−α2	−α2	PROPN
ap-7583	423	23	±	±	NUM
ap-7583	423	24	√	√	ADV
ap-7583	423	25	α2	α2	ADJ
ap-7583	423	26	2	2	NUM
ap-7583	423	27	−	−	NUM
ap-7583	423	28	4α1α3)/(2α3	4α1α3)/(2α3	NUM
ap-7583	423	29	)	)	PUNCT
ap-7583	423	30	singularity	singularity	NOUN
ap-7583	423	31	:	:	PUNCT
ap-7583	423	32	r	r	NOUN
ap-7583	423	33	=	=	SYM
ap-7583	423	34	0	0	NUM
ap-7583	423	35	,	,	PUNCT
ap-7583	423	36	ξ±	ξ±	NUM
ap-7583	423	37	,	,	PUNCT
ap-7583	423	38	∞	∞	PROPN
ap-7583	423	39	:	:	PUNCT
ap-7583	423	40	regular	regular	ADJ
ap-7583	423	41	differential	differential	ADJ
ap-7583	423	42	equation	equation	NOUN
ap-7583	423	43	:	:	PUNCT
ap-7583	423	44	r	r	NOUN
ap-7583	423	45	(	(	PUNCT
ap-7583	423	46	ξ	ξ	PROPN
ap-7583	423	47	−	−	PROPN
ap-7583	423	48	r)2	r)2	NOUN
ap-7583	423	49	y′′	y′′	NOUN
ap-7583	423	50	+	+	CCONJ
ap-7583	423	51	(	(	PUNCT
ap-7583	423	52	β0	β0	NOUN
ap-7583	423	53	+	+	CCONJ
ap-7583	423	54	β1	β1	PROPN
ap-7583	423	55	r	r	NOUN
ap-7583	423	56	+	+	CCONJ
ap-7583	423	57	β2	β2	NOUN
ap-7583	423	58	r2	r2	NOUN
ap-7583	423	59	)	)	PUNCT
ap-7583	423	60	y′	y′	PUNCT
ap-7583	424	1	+	+	CCONJ
ap-7583	424	2	(	(	PUNCT
ap-7583	424	3	ε0	ε0	PROPN
ap-7583	424	4	+	+	CCONJ
ap-7583	424	5	ε1	ε1	PROPN
ap-7583	424	6	r	r	NOUN
ap-7583	424	7	)	)	PUNCT
ap-7583	424	8	y	y	NOUN
ap-7583	424	9	=	=	SYM
ap-7583	424	10	0	0	NUM
ap-7583	424	11	roots	root	NOUN
ap-7583	424	12	:	:	PUNCT
ap-7583	424	13	r	r	NOUN
ap-7583	424	14	=	=	SYM
ap-7583	424	15	0	0	NUM
ap-7583	424	16	;	;	PUNCT
ap-7583	424	17	r	r	NOUN
ap-7583	424	18	=	=	SYM
ap-7583	424	19	ξ	ξ	X
ap-7583	424	20	≡	≡	PROPN
ap-7583	424	21	−α2/(2α3	−α2/(2α3	NOUN
ap-7583	424	22	)	)	PUNCT
ap-7583	424	23	singularity	singularity	NOUN
ap-7583	424	24	:	:	PUNCT
ap-7583	424	25	r	r	NOUN
ap-7583	424	26	=	=	SYM
ap-7583	424	27	0	0	NUM
ap-7583	424	28	,	,	PUNCT
ap-7583	424	29	ξ	ξ	NOUN
ap-7583	424	30	:	:	PUNCT
ap-7583	424	31	regular	regular	ADJ
ap-7583	424	32	;	;	PUNCT
ap-7583	424	33	r	r	NOUN
ap-7583	424	34	=	=	SYM
ap-7583	424	35	∞	∞	NOUN
ap-7583	424	36	:	:	PUNCT
ap-7583	424	37	irregular	irregular	ADJ
ap-7583	424	38	ii	ii	PROPN
ap-7583	424	39	0	0	NUM
ap-7583	424	40	α2	α2	ADJ
ap-7583	424	41	α1	α1	PROPN
ap-7583	424	42	r	r	NOUN
ap-7583	424	43	=	=	SYM
ap-7583	424	44	0	0	NUM
ap-7583	424	45	,	,	PUNCT
ap-7583	424	46	r	r	NOUN
ap-7583	424	47	=	=	SYM
ap-7583	424	48	−α1	−α1	PROPN
ap-7583	424	49	/	/	SYM
ap-7583	424	50	α2	α2	PROPN
ap-7583	424	51	r	r	NOUN
ap-7583	424	52	∈	∈	PROPN
ap-7583	424	53	(	(	PUNCT
ap-7583	424	54	0	0	NUM
ap-7583	424	55	,	,	PUNCT
ap-7583	424	56	−α1	−α1	PROPN
ap-7583	424	57	/	/	SYM
ap-7583	424	58	α2	α2	ADJ
ap-7583	424	59	)	)	PUNCT
ap-7583	425	1	if	if	SCONJ
ap-7583	425	2	α1	α1	PROPN
ap-7583	425	3	/	/	SYM
ap-7583	425	4	α2	α2	PROPN
ap-7583	425	5	<	<	X
ap-7583	425	6	0	0	PUNCT
ap-7583	425	7	r	r	NOUN
ap-7583	425	8	∈	∈	PROPN
ap-7583	425	9	(	(	PUNCT
ap-7583	425	10	−α1	−α1	PROPN
ap-7583	425	11	/	/	SYM
ap-7583	425	12	α2	α2	ADJ
ap-7583	425	13	,	,	PUNCT
ap-7583	425	14	0	0	NUM
ap-7583	425	15	)	)	PUNCT
ap-7583	425	16	if	if	SCONJ
ap-7583	425	17	α1	α1	PROPN
ap-7583	425	18	/	/	SYM
ap-7583	425	19	α2	α2	PROPN
ap-7583	425	20	>	>	SYM
ap-7583	425	21	0	0	NUM
ap-7583	425	22	differential	differential	ADJ
ap-7583	425	23	equation	equation	NOUN
ap-7583	425	24	:	:	PUNCT
ap-7583	425	25	r(α1	r(α1	NOUN
ap-7583	425	26	+	+	CCONJ
ap-7583	425	27	α2	α2	ADJ
ap-7583	425	28	r	r	NOUN
ap-7583	425	29	)	)	PUNCT
ap-7583	425	30	y′′	y′′	NOUN
ap-7583	425	31	+	+	CCONJ
ap-7583	425	32	(	(	PUNCT
ap-7583	425	33	β0	β0	NOUN
ap-7583	425	34	+	+	CCONJ
ap-7583	425	35	β1	β1	PROPN
ap-7583	425	36	r	r	NOUN
ap-7583	425	37	+	+	CCONJ
ap-7583	425	38	β2	β2	NOUN
ap-7583	425	39	r2	r2	NOUN
ap-7583	425	40	)	)	PUNCT
ap-7583	425	41	y′	y′	PUNCT
ap-7583	426	1	+	+	CCONJ
ap-7583	426	2	(	(	PUNCT
ap-7583	426	3	ε0	ε0	PROPN
ap-7583	426	4	+	+	CCONJ
ap-7583	426	5	ε1	ε1	PROPN
ap-7583	426	6	r	r	NOUN
ap-7583	426	7	)	)	PUNCT
ap-7583	426	8	y	y	NOUN
ap-7583	426	9	=	=	SYM
ap-7583	426	10	0	0	NUM
ap-7583	426	11	singularity	singularity	NOUN
ap-7583	426	12	:	:	PUNCT
ap-7583	426	13	r	r	NOUN
ap-7583	426	14	=	=	SYM
ap-7583	426	15	0	0	NUM
ap-7583	426	16	,	,	PUNCT
ap-7583	426	17	−α1	−α1	PROPN
ap-7583	426	18	/	/	SYM
ap-7583	426	19	α2	α2	ADJ
ap-7583	426	20	:	:	PUNCT
ap-7583	426	21	regular	regular	ADJ
ap-7583	426	22	;	;	PUNCT
ap-7583	426	23	r	r	NOUN
ap-7583	426	24	=	=	SYM
ap-7583	426	25	∞	∞	NOUN
ap-7583	426	26	:	:	PUNCT
ap-7583	426	27	irregular	irregular	ADJ
ap-7583	426	28	iii	iii	NUM
ap-7583	426	29	α3	α3	NOUN
ap-7583	426	30	0	0	NUM
ap-7583	426	31	α1	α1	PROPN
ap-7583	426	32	a1a3	a1a3	PUNCT
ap-7583	426	33	<	<	X
ap-7583	426	34	0	0	NUM
ap-7583	426	35	r	r	NOUN
ap-7583	426	36	=	=	SYM
ap-7583	426	37	0	0	NUM
ap-7583	426	38	,	,	PUNCT
ap-7583	426	39	±	±	NOUN
ap-7583	426	40	√	√	NOUN
ap-7583	426	41	−α1	−α1	PROPN
ap-7583	426	42	/	/	SYM
ap-7583	426	43	α3	α3	NOUN
ap-7583	426	44	r	r	NOUN
ap-7583	426	45	∈	∈	PROPN
ap-7583	426	46	(	(	PUNCT
ap-7583	426	47	0	0	NUM
ap-7583	426	48	,	,	PUNCT
ap-7583	426	49	√	√	ADV
ap-7583	426	50	−α1	−α1	PROPN
ap-7583	426	51	/	/	SYM
ap-7583	426	52	α3	α3	NOUN
ap-7583	426	53	)	)	PUNCT
ap-7583	427	1	α1α3	α1α3	NOUN
ap-7583	427	2	>	>	NOUN
ap-7583	427	3	0	0	PUNCT
ap-7583	427	4	r	r	NOUN
ap-7583	427	5	=	=	SYM
ap-7583	427	6	0	0	NUM
ap-7583	427	7	,	,	PUNCT
ap-7583	427	8	r	r	NOUN
ap-7583	427	9	∈	∈	PROPN
ap-7583	427	10	(	(	PUNCT
ap-7583	427	11	0	0	NUM
ap-7583	427	12	,	,	PUNCT
ap-7583	427	13	∞	∞	NUM
ap-7583	427	14	)	)	PUNCT
ap-7583	427	15	differential	differential	NOUN
ap-7583	427	16	equation	equation	NOUN
ap-7583	427	17	:	:	PUNCT
ap-7583	428	1	r	r	NOUN
ap-7583	428	2	(	(	PUNCT
ap-7583	428	3	α3r2	α3r2	CCONJ
ap-7583	428	4	+	+	NUM
ap-7583	428	5	α1	α1	NOUN
ap-7583	428	6	)	)	PUNCT
ap-7583	428	7	y′′	y′′	NOUN
ap-7583	428	8	+	+	CCONJ
ap-7583	428	9	(	(	PUNCT
ap-7583	428	10	β0	β0	NOUN
ap-7583	428	11	+	+	CCONJ
ap-7583	428	12	β1	β1	PROPN
ap-7583	428	13	r	r	NOUN
ap-7583	428	14	+	+	CCONJ
ap-7583	428	15	β2	β2	NOUN
ap-7583	428	16	r2	r2	NOUN
ap-7583	428	17	)	)	PUNCT
ap-7583	428	18	y′	y′	PUNCT
ap-7583	429	1	+	+	CCONJ
ap-7583	429	2	(	(	PUNCT
ap-7583	429	3	ε0	ε0	PROPN
ap-7583	429	4	+	+	CCONJ
ap-7583	429	5	ε1	ε1	PROPN
ap-7583	429	6	r	r	NOUN
ap-7583	429	7	)	)	PUNCT
ap-7583	429	8	y	y	NOUN
ap-7583	429	9	=	=	SYM
ap-7583	429	10	0	0	PROPN
ap-7583	429	11	,	,	PUNCT
ap-7583	429	12	α3α1	α3α1	PROPN
ap-7583	429	13	<	<	X
ap-7583	429	14	0	0	NUM
ap-7583	429	15	singularity	singularity	NOUN
ap-7583	429	16	:	:	PUNCT
ap-7583	429	17	r	r	NOUN
ap-7583	429	18	=	=	SYM
ap-7583	429	19	0	0	NUM
ap-7583	429	20	,	,	PUNCT
ap-7583	429	21	±	±	NOUN
ap-7583	429	22	√	√	NOUN
ap-7583	429	23	−α1	−α1	PROPN
ap-7583	429	24	/	/	SYM
ap-7583	429	25	α3	α3	NOUN
ap-7583	429	26	,	,	PUNCT
ap-7583	429	27	∞	∞	PROPN
ap-7583	429	28	:	:	PUNCT
ap-7583	429	29	regular	regular	ADJ
ap-7583	429	30	differential	differential	ADJ
ap-7583	429	31	equation	equation	NOUN
ap-7583	429	32	:	:	PUNCT
ap-7583	430	1	r	r	NOUN
ap-7583	430	2	(	(	PUNCT
ap-7583	430	3	α3r2	α3r2	CCONJ
ap-7583	430	4	+	+	NUM
ap-7583	430	5	α1	α1	NOUN
ap-7583	430	6	)	)	PUNCT
ap-7583	430	7	y′′	y′′	NOUN
ap-7583	430	8	+	+	CCONJ
ap-7583	430	9	(	(	PUNCT
ap-7583	430	10	β0	β0	NOUN
ap-7583	430	11	+	+	CCONJ
ap-7583	430	12	β1	β1	PROPN
ap-7583	430	13	r	r	NOUN
ap-7583	430	14	+	+	CCONJ
ap-7583	430	15	β2	β2	NOUN
ap-7583	430	16	r2	r2	NOUN
ap-7583	430	17	)	)	PUNCT
ap-7583	430	18	y′	y′	PUNCT
ap-7583	431	1	+	+	CCONJ
ap-7583	431	2	(	(	PUNCT
ap-7583	431	3	ε0	ε0	PROPN
ap-7583	431	4	+	+	CCONJ
ap-7583	431	5	ε1	ε1	PROPN
ap-7583	431	6	r	r	NOUN
ap-7583	431	7	)	)	PUNCT
ap-7583	431	8	y	y	NOUN
ap-7583	431	9	=	=	SYM
ap-7583	431	10	0	0	PROPN
ap-7583	431	11	,	,	PUNCT
ap-7583	431	12	α3α1	α3α1	PROPN
ap-7583	431	13	>	>	X
ap-7583	431	14	0	0	NUM
ap-7583	431	15	singularity	singularity	NOUN
ap-7583	431	16	:	:	PUNCT
ap-7583	431	17	r	r	NOUN
ap-7583	431	18	=	=	SYM
ap-7583	431	19	0	0	NUM
ap-7583	431	20	:	:	PUNCT
ap-7583	431	21	regular	regular	ADJ
ap-7583	431	22	;	;	PUNCT
ap-7583	431	23	r	r	NOUN
ap-7583	431	24	=	=	SYM
ap-7583	431	25	∞	∞	NOUN
ap-7583	431	26	:	:	PUNCT
ap-7583	431	27	irregular	irregular	ADJ
ap-7583	431	28	iv	iv	NUM
ap-7583	431	29	α3	α3	ADJ
ap-7583	431	30	α2	α2	ADJ
ap-7583	431	31	0	0	NUM
ap-7583	432	1	β0	β0	NOUN
ap-7583	432	2	=	=	NOUN
ap-7583	432	3	0	0	NUM
ap-7583	432	4	r	r	NOUN
ap-7583	432	5	=	=	SYM
ap-7583	432	6	0	0	NUM
ap-7583	432	7	,	,	PUNCT
ap-7583	432	8	−α2	−α2	PROPN
ap-7583	432	9	/	/	SYM
ap-7583	432	10	α3	α3	NOUN
ap-7583	432	11	r	r	NOUN
ap-7583	432	12	∈	∈	PROPN
ap-7583	432	13	(	(	PUNCT
ap-7583	432	14	0	0	NUM
ap-7583	432	15	,	,	PUNCT
ap-7583	432	16	−α2	−α2	PROPN
ap-7583	432	17	/	/	SYM
ap-7583	432	18	α3	α3	NOUN
ap-7583	432	19	)	)	PUNCT
ap-7583	432	20	if	if	SCONJ
ap-7583	432	21	α2	α2	VERB
ap-7583	432	22	/	/	SYM
ap-7583	432	23	α3	α3	NOUN
ap-7583	432	24	<	<	X
ap-7583	432	25	0	0	NUM
ap-7583	432	26	r	r	NOUN
ap-7583	432	27	∈	∈	PROPN
ap-7583	432	28	(	(	PUNCT
ap-7583	432	29	−α2	−α2	PROPN
ap-7583	432	30	/	/	SYM
ap-7583	432	31	α3	α3	PROPN
ap-7583	432	32	,	,	PUNCT
ap-7583	432	33	0	0	NUM
ap-7583	432	34	)	)	PUNCT
ap-7583	432	35	if	if	SCONJ
ap-7583	432	36	α2	α2	VERB
ap-7583	432	37	/	/	SYM
ap-7583	432	38	α3	α3	NOUN
ap-7583	432	39	>	>	SYM
ap-7583	432	40	0	0	NUM
ap-7583	432	41	differential	differential	ADJ
ap-7583	432	42	equation	equation	NOUN
ap-7583	432	43	:	:	PUNCT
ap-7583	432	44	r2(α3r	r2(α3r	PROPN
ap-7583	432	45	+	+	CCONJ
ap-7583	432	46	α2	α2	ADJ
ap-7583	432	47	)	)	PUNCT
ap-7583	432	48	y′′	y′′	NOUN
ap-7583	432	49	+	+	NUM
ap-7583	432	50	r(β1	r(β1	NOUN
ap-7583	432	51	+	+	CCONJ
ap-7583	432	52	β2	β2	NOUN
ap-7583	432	53	r	r	NOUN
ap-7583	432	54	)	)	PUNCT
ap-7583	432	55	y′	y′	PUNCT
ap-7583	433	1	+	+	CCONJ
ap-7583	433	2	(	(	PUNCT
ap-7583	433	3	ε0	ε0	PROPN
ap-7583	433	4	+	+	CCONJ
ap-7583	433	5	ε1	ε1	PROPN
ap-7583	433	6	r	r	NOUN
ap-7583	433	7	)	)	PUNCT
ap-7583	433	8	y	y	NOUN
ap-7583	433	9	=	=	SYM
ap-7583	433	10	0	0	NUM
ap-7583	433	11	singularity	singularity	NOUN
ap-7583	433	12	:	:	PUNCT
ap-7583	433	13	r	r	NOUN
ap-7583	433	14	=	=	SYM
ap-7583	433	15	0	0	NUM
ap-7583	433	16	,	,	PUNCT
ap-7583	433	17	−α2	−α2	PROPN
ap-7583	433	18	/	/	SYM
ap-7583	433	19	α3	α3	NOUN
ap-7583	433	20	:	:	PUNCT
ap-7583	433	21	regular	regular	ADJ
ap-7583	433	22	;	;	PUNCT
ap-7583	433	23	r	r	NOUN
ap-7583	433	24	=	=	SYM
ap-7583	433	25	∞	∞	NOUN
ap-7583	433	26	:	:	PUNCT
ap-7583	433	27	irregular	irregular	ADJ
ap-7583	433	28	v	v	ADP
ap-7583	433	29	0	0	NUM
ap-7583	433	30	0	0	NUM
ap-7583	433	31	α1	α1	PROPN
ap-7583	433	32	r	r	NOUN
ap-7583	433	33	=	=	SYM
ap-7583	433	34	0	0	NUM
ap-7583	433	35	r	r	NOUN
ap-7583	433	36	∈	∈	PROPN
ap-7583	433	37	(	(	PUNCT
ap-7583	433	38	0	0	NUM
ap-7583	433	39	,	,	PUNCT
ap-7583	433	40	∞	∞	NUM
ap-7583	433	41	)	)	PUNCT
ap-7583	433	42	differential	differential	NOUN
ap-7583	433	43	equation	equation	NOUN
ap-7583	433	44	:	:	PUNCT
ap-7583	433	45	α1	α1	PROPN
ap-7583	433	46	r	r	NOUN
ap-7583	433	47	y′′	y′′	NOUN
ap-7583	433	48	+	+	CCONJ
ap-7583	433	49	(	(	PUNCT
ap-7583	433	50	β0	β0	NOUN
ap-7583	433	51	+	+	CCONJ
ap-7583	433	52	β1	β1	PROPN
ap-7583	433	53	r	r	NOUN
ap-7583	433	54	+	+	CCONJ
ap-7583	433	55	β2	β2	NOUN
ap-7583	433	56	r2	r2	NOUN
ap-7583	433	57	)	)	PUNCT
ap-7583	433	58	y′	y′	PUNCT
ap-7583	434	1	+	+	CCONJ
ap-7583	434	2	(	(	PUNCT
ap-7583	434	3	ε0	ε0	PROPN
ap-7583	434	4	+	+	CCONJ
ap-7583	434	5	ε1	ε1	PROPN
ap-7583	434	6	r	r	NOUN
ap-7583	434	7	)	)	PUNCT
ap-7583	434	8	y	y	NOUN
ap-7583	434	9	=	=	SYM
ap-7583	434	10	0	0	NUM
ap-7583	434	11	singularity	singularity	NOUN
ap-7583	434	12	:	:	PUNCT
ap-7583	434	13	r	r	NOUN
ap-7583	434	14	=	=	SYM
ap-7583	434	15	0	0	NUM
ap-7583	434	16	:	:	PUNCT
ap-7583	434	17	regular	regular	ADJ
ap-7583	434	18	;	;	PUNCT
ap-7583	434	19	r	r	NOUN
ap-7583	434	20	=	=	SYM
ap-7583	434	21	∞	∞	NOUN
ap-7583	434	22	:	:	PUNCT
ap-7583	434	23	irregular	irregular	ADJ
ap-7583	434	24	vi	vi	NOUN
ap-7583	434	25	0	0	NUM
ap-7583	434	26	α2	α2	NOUN
ap-7583	434	27	0	0	NUM
ap-7583	434	28	β0	β0	NOUN
ap-7583	434	29	=	=	NOUN
ap-7583	434	30	0	0	NUM
ap-7583	434	31	r	r	NOUN
ap-7583	434	32	=	=	SYM
ap-7583	434	33	0	0	NUM
ap-7583	434	34	r	r	NOUN
ap-7583	434	35	∈	∈	PROPN
ap-7583	434	36	(	(	PUNCT
ap-7583	434	37	0	0	NUM
ap-7583	434	38	,	,	PUNCT
ap-7583	434	39	∞	∞	NUM
ap-7583	434	40	)	)	PUNCT
ap-7583	434	41	differential	differential	NOUN
ap-7583	434	42	equation	equation	NOUN
ap-7583	434	43	:	:	PUNCT
ap-7583	434	44	α2	α2	ADJ
ap-7583	434	45	r2	r2	PROPN
ap-7583	434	46	y′′	y′′	PROPN
ap-7583	434	47	+	+	CCONJ
ap-7583	434	48	r(β1	r(β1	NOUN
ap-7583	434	49	+	+	CCONJ
ap-7583	434	50	β2	β2	NOUN
ap-7583	434	51	r	r	NOUN
ap-7583	434	52	)	)	PUNCT
ap-7583	434	53	y′	y′	PUNCT
ap-7583	435	1	+	+	CCONJ
ap-7583	435	2	(	(	PUNCT
ap-7583	435	3	ε0	ε0	PROPN
ap-7583	435	4	+	+	CCONJ
ap-7583	435	5	ε1	ε1	PROPN
ap-7583	435	6	r	r	NOUN
ap-7583	435	7	)	)	PUNCT
ap-7583	435	8	y	y	NOUN
ap-7583	435	9	=	=	SYM
ap-7583	435	10	0	0	NUM
ap-7583	435	11	singularity	singularity	NOUN
ap-7583	435	12	:	:	PUNCT
ap-7583	435	13	r	r	NOUN
ap-7583	435	14	=	=	SYM
ap-7583	435	15	0	0	NUM
ap-7583	435	16	:	:	PUNCT
ap-7583	435	17	regular	regular	ADJ
ap-7583	435	18	;	;	PUNCT
ap-7583	435	19	r	r	NOUN
ap-7583	435	20	=	=	SYM
ap-7583	435	21	∞	∞	NOUN
ap-7583	435	22	:	:	PUNCT
ap-7583	435	23	irregular	irregular	ADJ
ap-7583	435	24	vii	vii	PROPN
ap-7583	435	25	α3	α3	PROPN
ap-7583	435	26	0	0	NUM
ap-7583	435	27	0	0	NUM
ap-7583	435	28	r	r	NOUN
ap-7583	435	29	=	=	SYM
ap-7583	435	30	0	0	NUM
ap-7583	435	31	r	r	NOUN
ap-7583	435	32	∈	∈	PROPN
ap-7583	435	33	(	(	PUNCT
ap-7583	435	34	0	0	NUM
ap-7583	435	35	,	,	PUNCT
ap-7583	435	36	∞	∞	NUM
ap-7583	435	37	)	)	PUNCT
ap-7583	435	38	differential	differential	NOUN
ap-7583	435	39	equation	equation	NOUN
ap-7583	435	40	:	:	PUNCT
ap-7583	435	41	α3	α3	PROPN
ap-7583	435	42	r3	r3	PROPN
ap-7583	435	43	y′′	y′′	PROPN
ap-7583	435	44	+	+	CCONJ
ap-7583	435	45	(	(	PUNCT
ap-7583	435	46	β0	β0	NOUN
ap-7583	435	47	+	+	CCONJ
ap-7583	435	48	β1	β1	PROPN
ap-7583	435	49	r	r	NOUN
ap-7583	435	50	+	+	CCONJ
ap-7583	435	51	β2	β2	NOUN
ap-7583	435	52	r2	r2	NOUN
ap-7583	435	53	)	)	PUNCT
ap-7583	435	54	y′	y′	PUNCT
ap-7583	436	1	+	+	CCONJ
ap-7583	436	2	(	(	PUNCT
ap-7583	436	3	ε0	ε0	PROPN
ap-7583	436	4	+	+	CCONJ
ap-7583	436	5	ε1	ε1	PROPN
ap-7583	436	6	r	r	NOUN
ap-7583	436	7	)	)	PUNCT
ap-7583	436	8	y	y	NOUN
ap-7583	436	9	=	=	SYM
ap-7583	436	10	0	0	NUM
ap-7583	436	11	singulaity	singulaity	NOUN
ap-7583	436	12	:	:	PUNCT
ap-7583	436	13	r	r	NOUN
ap-7583	436	14	=	=	SYM
ap-7583	436	15	0	0	NUM
ap-7583	436	16	,	,	PUNCT
ap-7583	436	17	∞	∞	PROPN
ap-7583	436	18	:	:	PUNCT
ap-7583	436	19	irregular	irregular	ADJ
ap-7583	436	20	table	table	NOUN
ap-7583	436	21	3	3	X
ap-7583	436	22	.	.	PUNCT
ap-7583	436	23	tabulating	tabulate	VERB
ap-7583	436	24	the	the	DET
ap-7583	436	25	seven	seven	NUM
ap-7583	436	26	different	different	ADJ
ap-7583	436	27	types	type	NOUN
ap-7583	436	28	of	of	ADP
ap-7583	436	29	differential	differential	ADJ
ap-7583	436	30	equations	equation	NOUN
ap-7583	436	31	,	,	PUNCT
ap-7583	436	32	which	which	PRON
ap-7583	436	33	apply	apply	VERB
ap-7583	436	34	to	to	ADP
ap-7583	436	35	theorem	theorem	VERB
ap-7583	436	36	4.1	4.1	NUM
ap-7583	436	37	.	.	NOUN
ap-7583	436	38	178	178	NUM
ap-7583	436	39	vol	vol	NOUN
ap-7583	436	40	.	.	PUNCT
ap-7583	437	1	62	62	NUM
ap-7583	437	2	no	no	INTJ
ap-7583	437	3	.	.	PUNCT
ap-7583	438	1	1/2022	1/2022	NUM
ap-7583	438	2	on	on	ADP
ap-7583	438	3	generalized	generalized	ADJ
ap-7583	438	4	heun	heun	NOUN
ap-7583	438	5	equation	equation	NOUN
ap-7583	438	6	with	with	ADP
ap-7583	438	7	some	some	DET
ap-7583	438	8	mathematical	mathematical	NOUN
ap-7583	438	9	.	.	PUNCT
ap-7583	438	10	.	.	PUNCT
ap-7583	439	1	.	.	PUNCT
ap-7583	440	1	initiated	initiate	VERB
ap-7583	440	2	with	with	ADP
ap-7583	440	3	p2,s(ε	p2,s(ε	NOUN
ap-7583	440	4	)	)	PUNCT
ap-7583	440	5	=	=	PUNCT
ap-7583	440	6	(	(	PUNCT
ap-7583	440	7	(	(	PUNCT
ap-7583	440	8	s	s	X
ap-7583	440	9	+	+	X
ap-7583	440	10	1)(α2	1)(α2	NUM
ap-7583	440	11	s	s	PART
ap-7583	440	12	+	+	CCONJ
ap-7583	440	13	β1	β1	NOUN
ap-7583	440	14	)	)	PUNCT
ap-7583	441	1	+	+	NUM
ap-7583	441	2	ε0)p1;s(ε0	ε0)p1;s(ε0	NOUN
ap-7583	441	3	)	)	PUNCT
ap-7583	441	4	−	−	PROPN
ap-7583	442	1	(	(	PUNCT
ap-7583	442	2	α1	α1	PROPN
ap-7583	442	3	s	s	PART
ap-7583	442	4	+	+	X
ap-7583	442	5	β0)(s	β0)(s	PROPN
ap-7583	443	1	+	+	CCONJ
ap-7583	443	2	1)(s(α3(s	1)(s(α3(s	NUM
ap-7583	443	3	−	−	NOUN
ap-7583	443	4	1	1	NUM
ap-7583	443	5	)	)	PUNCT
ap-7583	443	6	+	+	CCONJ
ap-7583	443	7	β2	β2	VERB
ap-7583	443	8	)	)	PUNCT
ap-7583	443	9	+	+	SYM
ap-7583	444	1	ε1	ε1	PROPN
ap-7583	444	2	)	)	PUNCT
ap-7583	444	3	continuing	continue	VERB
ap-7583	444	4	with	with	ADP
ap-7583	444	5	this	this	DET
ap-7583	444	6	process	process	NOUN
ap-7583	444	7	,	,	PUNCT
ap-7583	444	8	it	it	PRON
ap-7583	444	9	is	be	AUX
ap-7583	444	10	straightforward	straightforward	ADJ
ap-7583	444	11	to	to	PART
ap-7583	444	12	conclude	conclude	VERB
ap-7583	444	13	that	that	SCONJ
ap-7583	444	14	the	the	DET
ap-7583	444	15	series	series	NOUN
ap-7583	444	16	solution	solution	NOUN
ap-7583	444	17	can	can	AUX
ap-7583	444	18	be	be	AUX
ap-7583	444	19	written	write	VERB
ap-7583	444	20	as	as	ADP
ap-7583	444	21	y(r	y(r	NOUN
ap-7583	444	22	)	)	PUNCT
ap-7583	444	23	=	=	PUNCT
ap-7583	445	1	rs	r	NOUN
ap-7583	446	1	∞∑	∞∑	PRON
ap-7583	446	2	k=0	k=0	PROPN
ap-7583	446	3	ck	ck	ADP
ap-7583	446	4	rk	rk	NOUN
ap-7583	446	5	=	=	NOUN
ap-7583	446	6	∞∑	∞∑	NUM
ap-7583	446	7	k=0	k=0	PROPN
ap-7583	446	8	(	(	PUNCT
ap-7583	446	9	−1)k	−1)k	PROPN
ap-7583	446	10	pk;s(ε0	pk;s(ε0	NOUN
ap-7583	446	11	)	)	PUNCT
ap-7583	446	12	αk	αk	CCONJ
ap-7583	446	13	1	1	NUM
ap-7583	446	14	(	(	PUNCT
ap-7583	446	15	β0	β0	PROPN
ap-7583	446	16	α1	α1	PROPN
ap-7583	446	17	+	+	SYM
ap-7583	446	18	s	s	NOUN
ap-7583	446	19	)	)	PUNCT
ap-7583	447	1	k	k	NOUN
ap-7583	447	2	(	(	PUNCT
ap-7583	447	3	1	1	NUM
ap-7583	447	4	+	+	NUM
ap-7583	447	5	s)k	s)k	NOUN
ap-7583	447	6	rk+s	rk+s	PROPN
ap-7583	447	7	,	,	PUNCT
ap-7583	447	8	(	(	PUNCT
ap-7583	447	9	68	68	NUM
ap-7583	447	10	)	)	PUNCT
ap-7583	447	11	where	where	SCONJ
ap-7583	447	12	the	the	DET
ap-7583	447	13	k	k	ADJ
ap-7583	447	14	-	-	PUNCT
ap-7583	447	15	degree	degree	NOUN
ap-7583	447	16	polynomials	polynomial	NOUN
ap-7583	447	17	of	of	ADP
ap-7583	447	18	the	the	DET
ap-7583	447	19	parameter	parameter	NOUN
ap-7583	447	20	ε0	ε0	PROPN
ap-7583	447	21	,	,	PUNCT
ap-7583	447	22	namely	namely	ADV
ap-7583	447	23	{	{	PUNCT
ap-7583	447	24	pk;s(ε0)}∞	pk;s(ε0)}∞	ADJ
ap-7583	447	25	k=0	k=0	X
ap-7583	447	26	,	,	PUNCT
ap-7583	447	27	satisfy	satisfy	VERB
ap-7583	447	28	the	the	DET
ap-7583	447	29	following	follow	VERB
ap-7583	447	30	three	three	NUM
ap-7583	447	31	-	-	PUNCT
ap-7583	447	32	term	term	NOUN
ap-7583	447	33	recurrence	recurrence	NOUN
ap-7583	447	34	relation	relation	NOUN
ap-7583	447	35	:	:	PUNCT
ap-7583	447	36	pk+1;s(ε0	pk+1;s(ε0	NOUN
ap-7583	447	37	)	)	PUNCT
ap-7583	447	38	=	=	SYM
ap-7583	447	39	(	(	PUNCT
ap-7583	447	40	(	(	PUNCT
ap-7583	447	41	k	k	X
ap-7583	447	42	+	+	PROPN
ap-7583	447	43	s	s	X
ap-7583	447	44	)	)	PUNCT
ap-7583	447	45	[	[	PUNCT
ap-7583	447	46	(	(	PUNCT
ap-7583	447	47	k	k	X
ap-7583	447	48	+	+	SYM
ap-7583	447	49	s	s	VERB
ap-7583	447	50	−	−	PROPN
ap-7583	447	51	1)α2	1)α2	PRON
ap-7583	447	52	+	+	CCONJ
ap-7583	447	53	β1	β1	X
ap-7583	447	54	]	]	PUNCT
ap-7583	447	55	+	+	CCONJ
ap-7583	447	56	ε0	ε0	ADJ
ap-7583	447	57	)	)	PUNCT
ap-7583	447	58	pk;s(ε0	pk;s(ε0	NOUN
ap-7583	447	59	)	)	PUNCT
ap-7583	447	60	−	−	PROPN
ap-7583	448	1	(	(	PUNCT
ap-7583	448	2	k	k	PROPN
ap-7583	448	3	+	+	PROPN
ap-7583	448	4	s	s	X
ap-7583	448	5	)	)	PUNCT
ap-7583	448	6	(	(	PUNCT
ap-7583	448	7	(	(	PUNCT
ap-7583	448	8	k	k	X
ap-7583	448	9	+	+	SYM
ap-7583	448	10	s	s	VERB
ap-7583	448	11	−	−	PROPN
ap-7583	448	12	1)α1	1)α1	NUM
ap-7583	448	13	+	+	CCONJ
ap-7583	448	14	β0	β0	NOUN
ap-7583	448	15	)	)	PUNCT
ap-7583	448	16	(	(	PUNCT
ap-7583	448	17	(	(	PUNCT
ap-7583	448	18	k	k	X
ap-7583	448	19	+	+	SYM
ap-7583	448	20	s	s	VERB
ap-7583	448	21	−	−	PROPN
ap-7583	448	22	1	1	NUM
ap-7583	448	23	)	)	PUNCT
ap-7583	448	24	×	×	NOUN
ap-7583	448	25	[	[	PUNCT
ap-7583	448	26	(	(	PUNCT
ap-7583	448	27	k	k	X
ap-7583	448	28	+	+	SYM
ap-7583	448	29	s	s	VERB
ap-7583	448	30	−	−	PROPN
ap-7583	448	31	2)α3	2)α3	NUM
ap-7583	448	32	+	+	CCONJ
ap-7583	448	33	β2	β2	NOUN
ap-7583	448	34	]	]	PUNCT
ap-7583	449	1	+	+	CCONJ
ap-7583	449	2	ε1	ε1	PROPN
ap-7583	449	3	)	)	PUNCT
ap-7583	449	4	pk−1;s(ε0	pk−1;s(ε0	PROPN
ap-7583	449	5	)	)	PUNCT
ap-7583	449	6	,	,	PUNCT
ap-7583	449	7	(	(	PUNCT
ap-7583	449	8	69	69	NUM
ap-7583	449	9	)	)	PUNCT
ap-7583	449	10	initiated	initiate	VERB
ap-7583	449	11	with	with	ADP
ap-7583	449	12	p−1;s(ε0	p−1;s(ε0	NOUN
ap-7583	449	13	)	)	PUNCT
ap-7583	450	1	=	=	SYM
ap-7583	450	2	0	0	NUM
ap-7583	450	3	and	and	CCONJ
ap-7583	450	4	p0;s(ε0	p0;s(ε0	NOUN
ap-7583	450	5	)	)	PUNCT
ap-7583	450	6	=	=	SYM
ap-7583	450	7	1	1	X
ap-7583	450	8	.	.	X
ap-7583	450	9	for	for	ADP
ap-7583	450	10	the	the	DET
ap-7583	450	11	classes	class	NOUN
ap-7583	450	12	i	i	PRON
ap-7583	450	13	-	-	PUNCT
ap-7583	450	14	iv	iv	NUM
ap-7583	450	15	in	in	ADP
ap-7583	450	16	table	table	NOUN
ap-7583	450	17	3	3	NUM
ap-7583	450	18	,	,	PUNCT
ap-7583	450	19	including	include	VERB
ap-7583	450	20	,	,	PUNCT
ap-7583	450	21	of	of	ADP
ap-7583	450	22	course	course	NOUN
ap-7583	450	23	,	,	PUNCT
ap-7583	450	24	the	the	DET
ap-7583	450	25	classical	classical	ADJ
ap-7583	450	26	heun	heun	NOUN
ap-7583	450	27	equation	equation	NOUN
ap-7583	450	28	,	,	PUNCT
ap-7583	450	29	r	r	NOUN
ap-7583	450	30	=	=	SYM
ap-7583	450	31	0	0	NUM
ap-7583	450	32	is	be	AUX
ap-7583	450	33	a	a	DET
ap-7583	450	34	regular	regular	ADJ
ap-7583	450	35	singular	singular	ADJ
ap-7583	450	36	point	point	NOUN
ap-7583	450	37	with	with	ADP
ap-7583	450	38	one	one	NUM
ap-7583	450	39	of	of	ADP
ap-7583	450	40	the	the	DET
ap-7583	450	41	exponents	exponent	NOUN
ap-7583	450	42	of	of	ADP
ap-7583	450	43	singularities	singularity	NOUN
ap-7583	450	44	being	be	AUX
ap-7583	450	45	s	s	NOUN
ap-7583	450	46	=	=	NOUN
ap-7583	450	47	0	0	NUM
ap-7583	450	48	,	,	PUNCT
ap-7583	450	49	in	in	ADP
ap-7583	450	50	which	which	DET
ap-7583	450	51	case	case	NOUN
ap-7583	450	52	,	,	PUNCT
ap-7583	450	53	the	the	DET
ap-7583	450	54	coefficients	coefficient	NOUN
ap-7583	450	55	{	{	PUNCT
ap-7583	450	56	ck}∞	ck}∞	X
ap-7583	450	57	k=0	k=0	PROPN
ap-7583	450	58	of	of	ADP
ap-7583	450	59	the	the	DET
ap-7583	450	60	series	series	NOUN
ap-7583	450	61	solution	solution	NOUN
ap-7583	450	62	y(r	y(r	NOUN
ap-7583	450	63	)	)	PUNCT
ap-7583	450	64	=	=	NOUN
ap-7583	450	65	∑∞	∑∞	NOUN
ap-7583	450	66	k=0	k=0	PROPN
ap-7583	450	67	ckrk	ckrk	NOUN
ap-7583	450	68	satisfy	satisfy	VERB
ap-7583	450	69	the	the	DET
ap-7583	450	70	three	three	NUM
ap-7583	450	71	-	-	PUNCT
ap-7583	450	72	term	term	NOUN
ap-7583	450	73	recurrence	recurrence	NOUN
ap-7583	450	74	relation	relation	NOUN
ap-7583	450	75	(	(	PUNCT
ap-7583	450	76	(	(	PUNCT
ap-7583	450	77	k	k	X
ap-7583	450	78	+	+	PROPN
ap-7583	450	79	1)(k	1)(k	NUM
ap-7583	450	80	α1	α1	PROPN
ap-7583	450	81	+	+	CCONJ
ap-7583	450	82	β0	β0	NOUN
ap-7583	450	83	)	)	PUNCT
ap-7583	450	84	)	)	PUNCT
ap-7583	450	85	ck+1	ck+1	PUNCT
ap-7583	451	1	+	+	CCONJ
ap-7583	451	2	(	(	PUNCT
ap-7583	451	3	k	k	X
ap-7583	451	4	(	(	PUNCT
ap-7583	451	5	(	(	PUNCT
ap-7583	451	6	k	k	X
ap-7583	451	7	−	−	PROPN
ap-7583	451	8	1)α2	1)α2	PROPN
ap-7583	451	9	+	+	CCONJ
ap-7583	451	10	β1	β1	PROPN
ap-7583	451	11	)	)	PUNCT
ap-7583	452	1	+	+	CCONJ
ap-7583	452	2	ε0	ε0	PROPN
ap-7583	452	3	)	)	PUNCT
ap-7583	452	4	ck	ck	PROPN
ap-7583	453	1	+	+	CCONJ
ap-7583	453	2	(	(	PUNCT
ap-7583	453	3	(	(	PUNCT
ap-7583	453	4	k	k	X
ap-7583	453	5	−	−	PROPN
ap-7583	453	6	1	1	NUM
ap-7583	453	7	)	)	PUNCT
ap-7583	453	8	(	(	PUNCT
ap-7583	453	9	(	(	PUNCT
ap-7583	453	10	k	k	X
ap-7583	453	11	−	−	PROPN
ap-7583	453	12	2)α3	2)α3	PROPN
ap-7583	453	13	+	+	CCONJ
ap-7583	453	14	β2	β2	NOUN
ap-7583	453	15	)	)	PUNCT
ap-7583	454	1	+	+	CCONJ
ap-7583	454	2	ε1	ε1	PROPN
ap-7583	454	3	)	)	PUNCT
ap-7583	454	4	ck−1	ck−1	NOUN
ap-7583	454	5	=	=	SYM
ap-7583	454	6	0	0	NUM
ap-7583	454	7	,	,	PUNCT
ap-7583	454	8	(	(	PUNCT
ap-7583	454	9	70	70	NUM
ap-7583	454	10	)	)	PUNCT
ap-7583	454	11	and	and	CCONJ
ap-7583	454	12	we	we	PRON
ap-7583	454	13	have	have	VERB
ap-7583	454	14	the	the	DET
ap-7583	454	15	following	follow	VERB
ap-7583	454	16	general	general	ADJ
ap-7583	454	17	result	result	NOUN
ap-7583	454	18	concerning	concern	VERB
ap-7583	454	19	the	the	DET
ap-7583	454	20	series	series	NOUN
ap-7583	454	21	solutions	solution	NOUN
ap-7583	454	22	of	of	ADP
ap-7583	454	23	the	the	DET
ap-7583	454	24	equation	equation	NOUN
ap-7583	454	25	(	(	PUNCT
ap-7583	454	26	65	65	NUM
ap-7583	454	27	):	):	PUNCT
ap-7583	454	28	theorem	theorem	NOUN
ap-7583	454	29	4.1	4.1	NUM
ap-7583	454	30	.	.	PUNCT
ap-7583	455	1	in	in	ADP
ap-7583	455	2	the	the	DET
ap-7583	455	3	neighbourhood	neighbourhood	NOUN
ap-7583	455	4	of	of	ADP
ap-7583	455	5	the	the	DET
ap-7583	455	6	regular	regular	ADJ
ap-7583	455	7	singular	singular	ADJ
ap-7583	455	8	point	point	NOUN
ap-7583	455	9	r	r	NOUN
ap-7583	455	10	=	=	SYM
ap-7583	455	11	0	0	NUM
ap-7583	455	12	,	,	PUNCT
ap-7583	455	13	the	the	DET
ap-7583	455	14	series	series	NOUN
ap-7583	455	15	solution	solution	NOUN
ap-7583	455	16	y(r	y(r	NOUN
ap-7583	455	17	)	)	PUNCT
ap-7583	455	18	=	=	NOUN
ap-7583	455	19	∑∞	∑∞	NOUN
ap-7583	455	20	k=0	k=0	PROPN
ap-7583	455	21	ckrk	ckrk	NOUN
ap-7583	455	22	of	of	ADP
ap-7583	455	23	the	the	DET
ap-7583	455	24	differential	differential	ADJ
ap-7583	455	25	equation	equation	NOUN
ap-7583	455	26	(	(	PUNCT
ap-7583	455	27	65	65	NUM
ap-7583	455	28	)	)	PUNCT
ap-7583	455	29	,	,	PUNCT
ap-7583	455	30	with	with	ADP
ap-7583	455	31	α1	α1	PROPN
ap-7583	455	32	̸=	̸=	PROPN
ap-7583	455	33	0	0	NUM
ap-7583	455	34	,	,	PUNCT
ap-7583	455	35	is	be	AUX
ap-7583	455	36	explicitly	explicitly	ADV
ap-7583	455	37	given	give	VERB
ap-7583	455	38	by	by	ADP
ap-7583	455	39	y(r	y(r	NOUN
ap-7583	455	40	)	)	PUNCT
ap-7583	455	41	=	=	NOUN
ap-7583	456	1	∞∑	∞∑	NUM
ap-7583	456	2	k=0	k=0	PROPN
ap-7583	456	3	(	(	PUNCT
ap-7583	456	4	−1)k	−1)k	NOUN
ap-7583	456	5	pk(ε0	pk(ε0	NOUN
ap-7583	456	6	)	)	PUNCT
ap-7583	456	7	k	k	X
ap-7583	456	8	!	!	PUNCT
ap-7583	457	1	αk	αk	CCONJ
ap-7583	457	2	1	1	NUM
ap-7583	457	3	(	(	PUNCT
ap-7583	457	4	β0	β0	PROPN
ap-7583	457	5	α1	α1	PROPN
ap-7583	457	6	)	)	PUNCT
ap-7583	458	1	k	k	PROPN
ap-7583	458	2	rk	rk	NOUN
ap-7583	458	3	,	,	PUNCT
ap-7583	458	4	(	(	PUNCT
ap-7583	458	5	71	71	NUM
ap-7583	458	6	)	)	PUNCT
ap-7583	458	7	where	where	SCONJ
ap-7583	458	8	the	the	DET
ap-7583	458	9	infinite	infinite	ADJ
ap-7583	458	10	sequence	sequence	NOUN
ap-7583	458	11	{	{	PUNCT
ap-7583	458	12	pk(ε0)}∞	pk(ε0)}∞	NOUN
ap-7583	458	13	k=0	k=0	PROPN
ap-7583	458	14	is	be	AUX
ap-7583	458	15	evaluated	evaluate	VERB
ap-7583	458	16	using	use	VERB
ap-7583	458	17	the	the	DET
ap-7583	458	18	three	three	NUM
ap-7583	458	19	-	-	PUNCT
ap-7583	458	20	term	term	NOUN
ap-7583	458	21	recurrence	recurrence	NOUN
ap-7583	458	22	relation	relation	NOUN
ap-7583	458	23	pk+1(ε0	pk+1(ε0	NOUN
ap-7583	458	24	)	)	PUNCT
ap-7583	458	25	=	=	SYM
ap-7583	458	26	(	(	PUNCT
ap-7583	458	27	k(k	k(k	ADV
ap-7583	458	28	−	−	PROPN
ap-7583	458	29	1)α2	1)α2	PROPN
ap-7583	458	30	+	+	NUM
ap-7583	458	31	kβ1	kβ1	NOUN
ap-7583	458	32	+	+	X
ap-7583	458	33	ε0	ε0	NOUN
ap-7583	458	34	)	)	PUNCT
ap-7583	458	35	pk(ε0	pk(ε0	NOUN
ap-7583	458	36	)	)	PUNCT
ap-7583	458	37	−	−	PROPN
ap-7583	459	1	k	k	NOUN
ap-7583	459	2	(	(	PUNCT
ap-7583	459	3	(	(	PUNCT
ap-7583	459	4	k	k	X
ap-7583	459	5	−	−	PROPN
ap-7583	459	6	1)α1	1)α1	NUM
ap-7583	459	7	+	+	CCONJ
ap-7583	459	8	β0	β0	ADJ
ap-7583	459	9	)	)	PUNCT
ap-7583	459	10	×	×	NOUN
ap-7583	459	11	(	(	PUNCT
ap-7583	459	12	(	(	PUNCT
ap-7583	459	13	k	k	PROPN
ap-7583	460	1	−	−	PROPN
ap-7583	460	2	1)(k	1)(k	NUM
ap-7583	460	3	−	−	PROPN
ap-7583	460	4	2)α3	2)α3	NUM
ap-7583	460	5	+	+	CCONJ
ap-7583	460	6	(	(	PUNCT
ap-7583	460	7	k	k	PROPN
ap-7583	460	8	−	−	PROPN
ap-7583	460	9	1)β2	1)β2	NUM
ap-7583	460	10	+	+	NUM
ap-7583	460	11	ε1	ε1	PROPN
ap-7583	460	12	)	)	PUNCT
ap-7583	460	13	pk−1(ε0	pk−1(ε0	PROPN
ap-7583	460	14	)	)	PUNCT
ap-7583	460	15	,	,	PUNCT
ap-7583	460	16	(	(	PUNCT
ap-7583	460	17	72	72	NUM
ap-7583	460	18	)	)	PUNCT
ap-7583	461	1	where	where	SCONJ
ap-7583	461	2	p−1(ε0	p−1(ε0	NOUN
ap-7583	461	3	)	)	PUNCT
ap-7583	461	4	=	=	SYM
ap-7583	461	5	0	0	NUM
ap-7583	461	6	,	,	PUNCT
ap-7583	461	7	and	and	CCONJ
ap-7583	461	8	p0(ε0	p0(ε0	NOUN
ap-7583	461	9	)	)	PUNCT
ap-7583	461	10	=	=	SYM
ap-7583	462	1	1	1	X
ap-7583	462	2	.	.	PUNCT
ap-7583	463	1	here	here	ADV
ap-7583	463	2	,	,	PUNCT
ap-7583	463	3	(	(	PUNCT
ap-7583	463	4	α)n	α)n	ADJ
ap-7583	463	5	refers	refer	VERB
ap-7583	463	6	to	to	ADP
ap-7583	463	7	the	the	DET
ap-7583	463	8	pochhammer	pochhammer	NOUN
ap-7583	463	9	symbol	symbol	NOUN
ap-7583	463	10	(	(	PUNCT
ap-7583	463	11	α)n	α)n	ADJ
ap-7583	463	12	=	=	PUNCT
ap-7583	463	13	α(α	α(α	PROPN
ap-7583	463	14	+	+	NOUN
ap-7583	463	15	1	1	NUM
ap-7583	463	16	)	)	PUNCT
ap-7583	463	17	·	·	PUNCT
ap-7583	463	18	·	·	PUNCT
ap-7583	463	19	·	·	PUNCT
ap-7583	463	20	(	(	PUNCT
ap-7583	463	21	α	α	NOUN
ap-7583	463	22	−	−	NOUN
ap-7583	463	23	n	n	NOUN
ap-7583	463	24	+	+	NOUN
ap-7583	463	25	1	1	NUM
ap-7583	463	26	)	)	PUNCT
ap-7583	463	27	=	=	VERB
ap-7583	464	1	γ(α	γ(α	PROPN
ap-7583	464	2	+	+	CCONJ
ap-7583	464	3	n)/γ(α	n)/γ(α	PROPN
ap-7583	464	4	)	)	PUNCT
ap-7583	464	5	which	which	PRON
ap-7583	464	6	is	be	AUX
ap-7583	464	7	defined	define	VERB
ap-7583	464	8	in	in	ADP
ap-7583	464	9	terms	term	NOUN
ap-7583	464	10	of	of	ADP
ap-7583	464	11	gamma	gamma	NOUN
ap-7583	464	12	functions	function	NOUN
ap-7583	464	13	and	and	CCONJ
ap-7583	464	14	satisfies	satisfy	VERB
ap-7583	464	15	the	the	DET
ap-7583	464	16	identity	identity	NOUN
ap-7583	464	17	(	(	PUNCT
ap-7583	464	18	−n)k	−n)k	ADJ
ap-7583	464	19	=	=	SYM
ap-7583	464	20	0	0	NUM
ap-7583	465	1	for	for	ADP
ap-7583	465	2	any	any	DET
ap-7583	465	3	positive	positive	ADJ
ap-7583	465	4	integers	integer	NOUN
ap-7583	465	5	k	k	X
ap-7583	465	6	≥	≥	PROPN
ap-7583	465	7	n	n	PROPN
ap-7583	465	8	+	+	NUM
ap-7583	465	9	1	1	NUM
ap-7583	465	10	.	.	X
ap-7583	465	11	equation	equation	NOUN
ap-7583	465	12	(	(	PUNCT
ap-7583	465	13	72	72	NUM
ap-7583	465	14	)	)	PUNCT
ap-7583	465	15	in	in	ADP
ap-7583	465	16	theorem	theorem	NOUN
ap-7583	465	17	follows	follow	VERB
ap-7583	465	18	directly	directly	ADV
ap-7583	465	19	by	by	ADP
ap-7583	465	20	substituting	substitute	VERB
ap-7583	465	21	the	the	DET
ap-7583	465	22	coefficients	coefficient	NOUN
ap-7583	465	23	of	of	ADP
ap-7583	465	24	(	(	PUNCT
ap-7583	465	25	71	71	NUM
ap-7583	465	26	)	)	PUNCT
ap-7583	465	27	in	in	ADP
ap-7583	465	28	the	the	DET
ap-7583	465	29	recurrence	recurrence	NOUN
ap-7583	465	30	relation	relation	NOUN
ap-7583	465	31	(	(	PUNCT
ap-7583	465	32	65	65	NUM
ap-7583	465	33	)	)	PUNCT
ap-7583	465	34	and	and	CCONJ
ap-7583	465	35	eliminates	eliminate	VERB
ap-7583	465	36	the	the	DET
ap-7583	465	37	common	common	ADJ
ap-7583	465	38	terms	term	NOUN
ap-7583	465	39	.	.	PUNCT
ap-7583	466	1	corollary	corollary	ADJ
ap-7583	466	2	4.2	4.2	NUM
ap-7583	466	3	.	.	PUNCT
ap-7583	467	1	in	in	ADP
ap-7583	467	2	the	the	DET
ap-7583	467	3	neighbourhood	neighbourhood	NOUN
ap-7583	467	4	of	of	ADP
ap-7583	467	5	the	the	DET
ap-7583	467	6	regular	regular	ADJ
ap-7583	467	7	singular	singular	ADJ
ap-7583	467	8	point	point	NOUN
ap-7583	467	9	r	r	NOUN
ap-7583	467	10	=	=	SYM
ap-7583	467	11	0	0	NUM
ap-7583	467	12	,	,	PUNCT
ap-7583	467	13	the	the	DET
ap-7583	467	14	series	series	NOUN
ap-7583	467	15	solution	solution	NOUN
ap-7583	467	16	y(r	y(r	NOUN
ap-7583	467	17	)	)	PUNCT
ap-7583	467	18	=	=	NOUN
ap-7583	467	19	∑∞	∑∞	NOUN
ap-7583	467	20	k=0	k=0	PROPN
ap-7583	467	21	ckrk	ckrk	NOUN
ap-7583	467	22	of	of	ADP
ap-7583	467	23	the	the	DET
ap-7583	467	24	differential	differential	ADJ
ap-7583	467	25	equation	equation	NOUN
ap-7583	467	26	r	r	NOUN
ap-7583	467	27	(	(	PUNCT
ap-7583	467	28	α1	α1	PROPN
ap-7583	467	29	+	+	CCONJ
ap-7583	467	30	α3r2	α3r2	X
ap-7583	467	31	)	)	PUNCT
ap-7583	467	32	y′′	y′′	NOUN
ap-7583	467	33	+	+	CCONJ
ap-7583	467	34	(	(	PUNCT
ap-7583	467	35	β0	β0	NOUN
ap-7583	467	36	+	+	CCONJ
ap-7583	467	37	β1	β1	PROPN
ap-7583	467	38	r	r	NOUN
ap-7583	467	39	+	+	CCONJ
ap-7583	467	40	β2	β2	NOUN
ap-7583	467	41	r2	r2	NOUN
ap-7583	467	42	)	)	PUNCT
ap-7583	467	43	y′	y′	PUNCT
ap-7583	468	1	+	+	CCONJ
ap-7583	468	2	(	(	PUNCT
ap-7583	468	3	ε0	ε0	PROPN
ap-7583	468	4	+	+	CCONJ
ap-7583	468	5	ε1	ε1	PROPN
ap-7583	468	6	r	r	NOUN
ap-7583	468	7	)	)	PUNCT
ap-7583	468	8	y	y	NOUN
ap-7583	468	9	=	=	SYM
ap-7583	468	10	0	0	PROPN
ap-7583	468	11	,	,	PUNCT
ap-7583	468	12	(	(	PUNCT
ap-7583	468	13	73	73	NUM
ap-7583	468	14	)	)	PUNCT
ap-7583	468	15	is	be	AUX
ap-7583	468	16	given	give	VERB
ap-7583	468	17	,	,	PUNCT
ap-7583	468	18	explicitly	explicitly	ADV
ap-7583	468	19	,	,	PUNCT
ap-7583	468	20	by	by	ADP
ap-7583	468	21	y(r	y(r	NOUN
ap-7583	468	22	)	)	PUNCT
ap-7583	468	23	=	=	NOUN
ap-7583	469	1	∞∑	∞∑	NUM
ap-7583	469	2	k=0	k=0	PROPN
ap-7583	469	3	(	(	PUNCT
ap-7583	469	4	−1)k	−1)k	NOUN
ap-7583	469	5	pk(ε0	pk(ε0	NOUN
ap-7583	469	6	)	)	PUNCT
ap-7583	469	7	k	k	X
ap-7583	469	8	!	!	PUNCT
ap-7583	470	1	αk	αk	CCONJ
ap-7583	470	2	1	1	NUM
ap-7583	470	3	(	(	PUNCT
ap-7583	470	4	β0	β0	PROPN
ap-7583	470	5	α1	α1	PROPN
ap-7583	470	6	)	)	PUNCT
ap-7583	471	1	k	k	X
ap-7583	471	2	rk	rk	NOUN
ap-7583	471	3	,	,	PUNCT
ap-7583	471	4	(	(	PUNCT
ap-7583	471	5	74	74	NUM
ap-7583	471	6	)	)	PUNCT
ap-7583	471	7	where	where	SCONJ
ap-7583	471	8	pk+1(ε0	pk+1(ε0	NOUN
ap-7583	471	9	)	)	PUNCT
ap-7583	471	10	=	=	SYM
ap-7583	471	11	(	(	PUNCT
ap-7583	471	12	k	k	X
ap-7583	471	13	β1	β1	PROPN
ap-7583	471	14	+	+	CCONJ
ap-7583	471	15	ε0)pk(ε0	ε0)pk(ε0	PROPN
ap-7583	471	16	)	)	PUNCT
ap-7583	472	1	−	−	PROPN
ap-7583	472	2	k	k	NOUN
ap-7583	472	3	(	(	PUNCT
ap-7583	472	4	(	(	PUNCT
ap-7583	472	5	k	k	PROPN
ap-7583	472	6	−	−	PROPN
ap-7583	472	7	1)α1	1)α1	NUM
ap-7583	472	8	+	+	CCONJ
ap-7583	472	9	β0	β0	ADJ
ap-7583	472	10	)	)	PUNCT
ap-7583	472	11	×	×	NOUN
ap-7583	472	12	(	(	PUNCT
ap-7583	472	13	(	(	PUNCT
ap-7583	472	14	k	k	PROPN
ap-7583	472	15	−	−	PROPN
ap-7583	472	16	1)(k	1)(k	NUM
ap-7583	472	17	−	−	PROPN
ap-7583	472	18	2)α3	2)α3	NUM
ap-7583	472	19	+	+	CCONJ
ap-7583	472	20	(	(	PUNCT
ap-7583	472	21	k	k	PROPN
ap-7583	472	22	−	−	PROPN
ap-7583	472	23	1)β2	1)β2	NUM
ap-7583	472	24	+	+	NUM
ap-7583	472	25	ε1)pk−1(ε0	ε1)pk−1(ε0	NOUN
ap-7583	472	26	)	)	PUNCT
ap-7583	472	27	,	,	PUNCT
ap-7583	472	28	(	(	PUNCT
ap-7583	472	29	75	75	NUM
ap-7583	472	30	)	)	PUNCT
ap-7583	472	31	initiated	initiate	VERB
ap-7583	472	32	with	with	ADP
ap-7583	472	33	p−1(ε0	p−1(ε0	NOUN
ap-7583	472	34	)	)	PUNCT
ap-7583	472	35	=	=	SYM
ap-7583	472	36	0	0	NUM
ap-7583	472	37	,	,	PUNCT
ap-7583	472	38	p0(ε0	p0(ε0	NOUN
ap-7583	472	39	)	)	PUNCT
ap-7583	472	40	=	=	SYM
ap-7583	472	41	1	1	NUM
ap-7583	472	42	.	.	NUM
ap-7583	472	43	179	179	NUM
ap-7583	472	44	nasser	nasser	PROPN
ap-7583	472	45	saad	saad	PROPN
ap-7583	472	46	acta	acta	PROPN
ap-7583	472	47	polytechnica	polytechnica	PROPN
ap-7583	472	48	corollary	corollary	PROPN
ap-7583	472	49	4.3	4.3	NUM
ap-7583	472	50	.	.	PUNCT
ap-7583	473	1	in	in	ADP
ap-7583	473	2	the	the	DET
ap-7583	473	3	neighbourhood	neighbourhood	NOUN
ap-7583	473	4	of	of	ADP
ap-7583	473	5	the	the	DET
ap-7583	473	6	regular	regular	ADJ
ap-7583	473	7	singular	singular	ADJ
ap-7583	473	8	point	point	NOUN
ap-7583	473	9	r	r	NOUN
ap-7583	473	10	=	=	SYM
ap-7583	473	11	0	0	NUM
ap-7583	473	12	,	,	PUNCT
ap-7583	473	13	the	the	DET
ap-7583	473	14	series	series	NOUN
ap-7583	473	15	solution	solution	NOUN
ap-7583	473	16	y(r	y(r	NOUN
ap-7583	473	17	)	)	PUNCT
ap-7583	473	18	=	=	NOUN
ap-7583	473	19	∑∞	∑∞	NOUN
ap-7583	473	20	k=0	k=0	PROPN
ap-7583	473	21	ckrk	ckrk	NOUN
ap-7583	473	22	of	of	ADP
ap-7583	473	23	the	the	DET
ap-7583	473	24	differential	differential	ADJ
ap-7583	473	25	equation	equation	NOUN
ap-7583	473	26	r	r	NOUN
ap-7583	473	27	(	(	PUNCT
ap-7583	473	28	α1	α1	PROPN
ap-7583	473	29	+	+	CCONJ
ap-7583	473	30	α2	α2	ADJ
ap-7583	473	31	r	r	NOUN
ap-7583	473	32	)	)	PUNCT
ap-7583	473	33	y′′	y′′	NOUN
ap-7583	473	34	+	+	CCONJ
ap-7583	473	35	(	(	PUNCT
ap-7583	473	36	β0	β0	NOUN
ap-7583	473	37	+	+	CCONJ
ap-7583	473	38	β1	β1	PROPN
ap-7583	473	39	r	r	NOUN
ap-7583	473	40	+	+	CCONJ
ap-7583	473	41	β2	β2	NOUN
ap-7583	473	42	r2	r2	NOUN
ap-7583	473	43	)	)	PUNCT
ap-7583	473	44	y′	y′	PUNCT
ap-7583	474	1	+	+	CCONJ
ap-7583	474	2	(	(	PUNCT
ap-7583	474	3	ε0	ε0	PROPN
ap-7583	474	4	+	+	CCONJ
ap-7583	474	5	ε1	ε1	PROPN
ap-7583	474	6	r	r	NOUN
ap-7583	474	7	)	)	PUNCT
ap-7583	474	8	y	y	NOUN
ap-7583	474	9	=	=	SYM
ap-7583	474	10	0	0	PROPN
ap-7583	474	11	,	,	PUNCT
ap-7583	474	12	(	(	PUNCT
ap-7583	474	13	76	76	NUM
ap-7583	474	14	)	)	PUNCT
ap-7583	474	15	is	be	AUX
ap-7583	474	16	given	give	VERB
ap-7583	474	17	,	,	PUNCT
ap-7583	474	18	explicitly	explicitly	ADV
ap-7583	474	19	,	,	PUNCT
ap-7583	474	20	by	by	ADP
ap-7583	474	21	y(r	y(r	NOUN
ap-7583	474	22	)	)	PUNCT
ap-7583	474	23	=	=	NOUN
ap-7583	475	1	∞∑	∞∑	NUM
ap-7583	475	2	k=0	k=0	PROPN
ap-7583	475	3	(	(	PUNCT
ap-7583	475	4	−1)k	−1)k	NOUN
ap-7583	475	5	pk(ε0	pk(ε0	NOUN
ap-7583	475	6	)	)	PUNCT
ap-7583	475	7	k	k	X
ap-7583	475	8	!	!	PUNCT
ap-7583	476	1	αk	αk	CCONJ
ap-7583	476	2	1	1	NUM
ap-7583	476	3	(	(	PUNCT
ap-7583	476	4	β0	β0	PROPN
ap-7583	476	5	α1	α1	PROPN
ap-7583	476	6	)	)	PUNCT
ap-7583	477	1	k	k	X
ap-7583	477	2	rk	rk	NOUN
ap-7583	477	3	,	,	PUNCT
ap-7583	477	4	(	(	PUNCT
ap-7583	477	5	77	77	NUM
ap-7583	477	6	)	)	PUNCT
ap-7583	477	7	where	where	SCONJ
ap-7583	477	8	pk+1(ε0	pk+1(ε0	NOUN
ap-7583	477	9	)	)	PUNCT
ap-7583	477	10	=	=	SYM
ap-7583	477	11	(	(	PUNCT
ap-7583	477	12	k(k	k(k	ADV
ap-7583	477	13	−	−	PROPN
ap-7583	477	14	1)α2	1)α2	PROPN
ap-7583	477	15	+	+	NUM
ap-7583	477	16	kβ1	kβ1	NOUN
ap-7583	477	17	+	+	SYM
ap-7583	477	18	ε0)pk(ε0	ε0)pk(ε0	PROPN
ap-7583	477	19	)	)	PUNCT
ap-7583	478	1	−	−	PROPN
ap-7583	478	2	k((k	k((k	NOUN
ap-7583	479	1	−	−	PROPN
ap-7583	479	2	1)α1	1)α1	NUM
ap-7583	480	1	+	+	CCONJ
ap-7583	480	2	β0)((k	β0)((k	VERB
ap-7583	480	3	−	−	PROPN
ap-7583	480	4	1)β2	1)β2	NUM
ap-7583	480	5	+	+	CCONJ
ap-7583	480	6	ε1)pk−1(ε0	ε1)pk−1(ε0	NOUN
ap-7583	480	7	)	)	PUNCT
ap-7583	480	8	,	,	PUNCT
ap-7583	480	9	(	(	PUNCT
ap-7583	480	10	78	78	X
ap-7583	480	11	)	)	PUNCT
ap-7583	480	12	initiated	initiate	VERB
ap-7583	480	13	with	with	ADP
ap-7583	480	14	p−1(ε0	p−1(ε0	NOUN
ap-7583	480	15	)	)	PUNCT
ap-7583	480	16	=	=	SYM
ap-7583	480	17	0	0	NUM
ap-7583	480	18	,	,	PUNCT
ap-7583	480	19	p0(ε0	p0(ε0	NOUN
ap-7583	480	20	)	)	PUNCT
ap-7583	480	21	=	=	SYM
ap-7583	480	22	1	1	X
ap-7583	480	23	.	.	PUNCT
ap-7583	480	24	corollary	corollary	ADJ
ap-7583	480	25	4.4	4.4	NUM
ap-7583	480	26	.	.	PUNCT
ap-7583	481	1	in	in	ADP
ap-7583	481	2	the	the	DET
ap-7583	481	3	neighbourhood	neighbourhood	NOUN
ap-7583	481	4	of	of	ADP
ap-7583	481	5	the	the	DET
ap-7583	481	6	regular	regular	ADJ
ap-7583	481	7	singular	singular	ADJ
ap-7583	481	8	point	point	NOUN
ap-7583	481	9	r	r	NOUN
ap-7583	481	10	=	=	SYM
ap-7583	481	11	0	0	NUM
ap-7583	481	12	,	,	PUNCT
ap-7583	481	13	the	the	DET
ap-7583	481	14	series	series	NOUN
ap-7583	481	15	solution	solution	NOUN
ap-7583	481	16	y(x	y(x	PROPN
ap-7583	481	17	)	)	PUNCT
ap-7583	481	18	=	=	SYM
ap-7583	481	19	∑∞	∑∞	NOUN
ap-7583	481	20	k=0	k=0	PROPN
ap-7583	481	21	ckrk	ckrk	NOUN
ap-7583	481	22	of	of	ADP
ap-7583	481	23	the	the	DET
ap-7583	481	24	differential	differential	ADJ
ap-7583	481	25	equation	equation	NOUN
ap-7583	481	26	α1	α1	PROPN
ap-7583	481	27	r	r	NOUN
ap-7583	481	28	y′′	y′′	NOUN
ap-7583	481	29	+	+	CCONJ
ap-7583	481	30	(	(	PUNCT
ap-7583	481	31	β0	β0	NOUN
ap-7583	481	32	+	+	CCONJ
ap-7583	481	33	β1	β1	PROPN
ap-7583	481	34	r	r	NOUN
ap-7583	481	35	+	+	CCONJ
ap-7583	481	36	β2	β2	NOUN
ap-7583	481	37	r2	r2	NOUN
ap-7583	481	38	)	)	PUNCT
ap-7583	481	39	y′	y′	PUNCT
ap-7583	482	1	+	+	CCONJ
ap-7583	482	2	(	(	PUNCT
ap-7583	482	3	ε0	ε0	PROPN
ap-7583	482	4	+	+	CCONJ
ap-7583	482	5	ε1	ε1	PROPN
ap-7583	482	6	r	r	NOUN
ap-7583	482	7	)	)	PUNCT
ap-7583	482	8	y	y	NOUN
ap-7583	482	9	=	=	SYM
ap-7583	482	10	0	0	PROPN
ap-7583	482	11	,	,	PUNCT
ap-7583	482	12	(	(	PUNCT
ap-7583	482	13	79	79	NUM
ap-7583	482	14	)	)	PUNCT
ap-7583	482	15	is	be	AUX
ap-7583	482	16	given	give	VERB
ap-7583	482	17	,	,	PUNCT
ap-7583	482	18	explicitly	explicitly	ADV
ap-7583	482	19	,	,	PUNCT
ap-7583	482	20	by	by	ADP
ap-7583	482	21	y(r	y(r	NOUN
ap-7583	482	22	)	)	PUNCT
ap-7583	482	23	=	=	NOUN
ap-7583	483	1	∞∑	∞∑	NUM
ap-7583	483	2	k=0	k=0	PROPN
ap-7583	483	3	(	(	PUNCT
ap-7583	483	4	−1)k	−1)k	NOUN
ap-7583	483	5	pk(ε0	pk(ε0	NOUN
ap-7583	483	6	)	)	PUNCT
ap-7583	483	7	k	k	X
ap-7583	483	8	!	!	PUNCT
ap-7583	484	1	αk	αk	CCONJ
ap-7583	484	2	1	1	NUM
ap-7583	484	3	(	(	PUNCT
ap-7583	484	4	β0	β0	PROPN
ap-7583	484	5	α1	α1	PROPN
ap-7583	484	6	)	)	PUNCT
ap-7583	485	1	k	k	X
ap-7583	485	2	rk	rk	NOUN
ap-7583	485	3	,	,	PUNCT
ap-7583	485	4	(	(	PUNCT
ap-7583	485	5	80	80	NUM
ap-7583	485	6	)	)	PUNCT
ap-7583	485	7	where	where	SCONJ
ap-7583	485	8	pk+1(ε0	pk+1(ε0	NOUN
ap-7583	485	9	)	)	PUNCT
ap-7583	485	10	=	=	SYM
ap-7583	485	11	(	(	PUNCT
ap-7583	485	12	kβ1	kβ1	NOUN
ap-7583	485	13	+	+	SYM
ap-7583	485	14	ε0)pk(ε0	ε0)pk(ε0	PROPN
ap-7583	485	15	)	)	PUNCT
ap-7583	485	16	−	−	PROPN
ap-7583	486	1	k	k	NOUN
ap-7583	486	2	(	(	PUNCT
ap-7583	486	3	(	(	PUNCT
ap-7583	486	4	k	k	NOUN
ap-7583	486	5	−	−	PROPN
ap-7583	486	6	1	1	NUM
ap-7583	486	7	)	)	PUNCT
ap-7583	486	8	α1	α1	PROPN
ap-7583	486	9	+	+	CCONJ
ap-7583	486	10	β0)((k	β0)((k	NOUN
ap-7583	486	11	−	−	PROPN
ap-7583	486	12	1)β2	1)β2	NUM
ap-7583	486	13	+	+	CCONJ
ap-7583	486	14	ε1)pk−1(ε0	ε1)pk−1(ε0	NOUN
ap-7583	486	15	)	)	PUNCT
ap-7583	486	16	,	,	PUNCT
ap-7583	486	17	(	(	PUNCT
ap-7583	486	18	81	81	NUM
ap-7583	486	19	)	)	PUNCT
ap-7583	486	20	initiated	initiate	VERB
ap-7583	486	21	with	with	ADP
ap-7583	486	22	p−1(ε0	p−1(ε0	NOUN
ap-7583	486	23	)	)	PUNCT
ap-7583	486	24	=	=	SYM
ap-7583	486	25	0	0	NUM
ap-7583	486	26	,	,	PUNCT
ap-7583	486	27	p0(ε0	p0(ε0	NOUN
ap-7583	486	28	)	)	PUNCT
ap-7583	486	29	=	=	SYM
ap-7583	486	30	0	0	X
ap-7583	486	31	.	.	PUNCT
ap-7583	486	32	corollary	corollary	ADJ
ap-7583	486	33	4.5	4.5	NUM
ap-7583	486	34	.	.	PUNCT
ap-7583	487	1	in	in	ADP
ap-7583	487	2	the	the	DET
ap-7583	487	3	neighbourhood	neighbourhood	NOUN
ap-7583	487	4	of	of	ADP
ap-7583	487	5	the	the	DET
ap-7583	487	6	regular	regular	ADJ
ap-7583	487	7	singular	singular	ADJ
ap-7583	487	8	point	point	NOUN
ap-7583	487	9	x	x	PUNCT
ap-7583	487	10	=	=	SYM
ap-7583	487	11	0	0	NUM
ap-7583	487	12	,	,	PUNCT
ap-7583	487	13	the	the	DET
ap-7583	487	14	series	series	NOUN
ap-7583	487	15	solution	solution	NOUN
ap-7583	487	16	y(x	y(x	PROPN
ap-7583	487	17	)	)	PUNCT
ap-7583	487	18	=	=	PUNCT
ap-7583	487	19	∑∞	∑∞	NOUN
ap-7583	487	20	k=0	k=0	PROPN
ap-7583	487	21	ckxk	ckxk	ADV
ap-7583	487	22	of	of	ADP
ap-7583	487	23	the	the	DET
ap-7583	487	24	differential	differential	ADJ
ap-7583	487	25	equation	equation	NOUN
ap-7583	487	26	α1r	α1r	NOUN
ap-7583	487	27	y′′	y′′	PROPN
ap-7583	487	28	+	+	CCONJ
ap-7583	488	1	(	(	PUNCT
ap-7583	488	2	β0	β0	PROPN
ap-7583	488	3	+	+	CCONJ
ap-7583	488	4	β2	β2	NOUN
ap-7583	488	5	r2	r2	NOUN
ap-7583	488	6	)	)	PUNCT
ap-7583	488	7	y′	y′	PUNCT
ap-7583	489	1	+	+	CCONJ
ap-7583	489	2	(	(	PUNCT
ap-7583	489	3	ε0	ε0	PROPN
ap-7583	489	4	+	+	CCONJ
ap-7583	489	5	ε1	ε1	PROPN
ap-7583	489	6	r	r	NOUN
ap-7583	489	7	)	)	PUNCT
ap-7583	489	8	y	y	NOUN
ap-7583	489	9	=	=	SYM
ap-7583	489	10	0	0	PROPN
ap-7583	489	11	,	,	PUNCT
ap-7583	489	12	(	(	PUNCT
ap-7583	489	13	82	82	NUM
ap-7583	489	14	)	)	PUNCT
ap-7583	489	15	is	be	AUX
ap-7583	489	16	given	give	VERB
ap-7583	489	17	,	,	PUNCT
ap-7583	489	18	explicitly	explicitly	ADV
ap-7583	489	19	,	,	PUNCT
ap-7583	489	20	by	by	ADP
ap-7583	489	21	y(r	y(r	NOUN
ap-7583	489	22	)	)	PUNCT
ap-7583	489	23	=	=	NOUN
ap-7583	490	1	∞∑	∞∑	NUM
ap-7583	490	2	k=0	k=0	PROPN
ap-7583	490	3	(	(	PUNCT
ap-7583	490	4	−1)k	−1)k	NOUN
ap-7583	490	5	pk(ε0	pk(ε0	NOUN
ap-7583	490	6	)	)	PUNCT
ap-7583	490	7	k	k	X
ap-7583	490	8	!	!	PUNCT
ap-7583	491	1	αk	αk	CCONJ
ap-7583	491	2	1	1	NUM
ap-7583	491	3	(	(	PUNCT
ap-7583	491	4	β0	β0	PROPN
ap-7583	491	5	α1	α1	PROPN
ap-7583	491	6	)	)	PUNCT
ap-7583	492	1	k	k	X
ap-7583	492	2	rk	rk	NOUN
ap-7583	492	3	,	,	PUNCT
ap-7583	492	4	(	(	PUNCT
ap-7583	492	5	83	83	NUM
ap-7583	492	6	)	)	PUNCT
ap-7583	492	7	where	where	SCONJ
ap-7583	492	8	pk+1(ε0	pk+1(ε0	NOUN
ap-7583	492	9	)	)	PUNCT
ap-7583	492	10	=	=	SYM
ap-7583	492	11	ε0pk(ε0	ε0pk(ε0	NOUN
ap-7583	492	12	)	)	PUNCT
ap-7583	492	13	−	−	PROPN
ap-7583	492	14	k((k	k((k	NOUN
ap-7583	492	15	−	−	PROPN
ap-7583	493	1	1)α1	1)α1	NUM
ap-7583	493	2	+	+	CCONJ
ap-7583	493	3	β0)((k	β0)((k	VERB
ap-7583	493	4	−	−	PROPN
ap-7583	493	5	1)β2	1)β2	NUM
ap-7583	493	6	+	+	CCONJ
ap-7583	493	7	ε1)pk−1(ε0	ε1)pk−1(ε0	NOUN
ap-7583	493	8	)	)	PUNCT
ap-7583	493	9	,	,	PUNCT
ap-7583	493	10	(	(	PUNCT
ap-7583	493	11	84	84	NUM
ap-7583	493	12	)	)	PUNCT
ap-7583	493	13	initiated	initiate	VERB
ap-7583	493	14	with	with	ADP
ap-7583	493	15	p−1(ε0	p−1(ε0	NOUN
ap-7583	493	16	)	)	PUNCT
ap-7583	493	17	=	=	SYM
ap-7583	493	18	0	0	NUM
ap-7583	493	19	,	,	PUNCT
ap-7583	493	20	p1(ε0	p1(ε0	NOUN
ap-7583	493	21	)	)	PUNCT
ap-7583	493	22	=	=	SYM
ap-7583	493	23	1	1	X
ap-7583	493	24	.	.	PUNCT
ap-7583	493	25	remark	remark	NOUN
ap-7583	493	26	4.6	4.6	NUM
ap-7583	493	27	.	.	PUNCT
ap-7583	494	1	if	if	SCONJ
ap-7583	494	2	,	,	PUNCT
ap-7583	494	3	in	in	ADP
ap-7583	494	4	addition	addition	NOUN
ap-7583	494	5	to	to	ADP
ap-7583	494	6	α0	α0	ADJ
ap-7583	494	7	=	=	SYM
ap-7583	494	8	0	0	NUM
ap-7583	494	9	,	,	PUNCT
ap-7583	494	10	we	we	PRON
ap-7583	494	11	also	also	ADV
ap-7583	494	12	have	have	VERB
ap-7583	494	13	α1	α1	PROPN
ap-7583	494	14	=	=	SYM
ap-7583	494	15	0	0	NUM
ap-7583	494	16	,	,	PUNCT
ap-7583	494	17	then	then	ADV
ap-7583	494	18	r	r	NOUN
ap-7583	494	19	=	=	SYM
ap-7583	494	20	0	0	NUM
ap-7583	494	21	is	be	AUX
ap-7583	494	22	a	a	DET
ap-7583	494	23	regular	regular	ADJ
ap-7583	494	24	singular	singular	ADJ
ap-7583	494	25	point	point	NOUN
ap-7583	494	26	only	only	ADV
ap-7583	494	27	if	if	SCONJ
ap-7583	494	28	β0	β0	PROPN
ap-7583	494	29	=	=	NOUN
ap-7583	494	30	0	0	PROPN
ap-7583	494	31	,	,	PUNCT
ap-7583	494	32	in	in	ADP
ap-7583	494	33	which	which	DET
ap-7583	494	34	case	case	NOUN
ap-7583	494	35	the	the	DET
ap-7583	494	36	differential	differential	ADJ
ap-7583	494	37	equation	equation	NOUN
ap-7583	494	38	reduces	reduce	VERB
ap-7583	494	39	to	to	ADP
ap-7583	494	40	an	an	DET
ap-7583	494	41	equation	equation	NOUN
ap-7583	494	42	that	that	PRON
ap-7583	494	43	resembles	resemble	VERB
ap-7583	494	44	euler	euler	PROPN
ap-7583	494	45	’s	’s	PART
ap-7583	494	46	equation	equation	NOUN
ap-7583	494	47	,	,	PUNCT
ap-7583	494	48	namely	namely	ADV
ap-7583	494	49	r2	r2	PROPN
ap-7583	494	50	(	(	PUNCT
ap-7583	494	51	α2	α2	PROPN
ap-7583	494	52	+	+	CCONJ
ap-7583	494	53	α3r	α3r	NUM
ap-7583	494	54	)	)	PUNCT
ap-7583	495	1	y′′	y′′	PROPN
ap-7583	495	2	+	+	PUNCT
ap-7583	495	3	r	r	NOUN
ap-7583	495	4	(	(	PUNCT
ap-7583	495	5	β1	β1	PROPN
ap-7583	495	6	+	+	CCONJ
ap-7583	495	7	β2	β2	NOUN
ap-7583	495	8	r	r	NOUN
ap-7583	495	9	)	)	PUNCT
ap-7583	495	10	y′	y′	PUNCT
ap-7583	496	1	+	+	CCONJ
ap-7583	496	2	(	(	PUNCT
ap-7583	496	3	ε0	ε0	PROPN
ap-7583	496	4	+	+	CCONJ
ap-7583	496	5	ε1	ε1	PROPN
ap-7583	496	6	r	r	NOUN
ap-7583	496	7	)	)	PUNCT
ap-7583	496	8	y	y	PROPN
ap-7583	496	9	=	=	PUNCT
ap-7583	496	10	0	0	PROPN
ap-7583	496	11	.	.	PUNCT
ap-7583	496	12	(	(	PUNCT
ap-7583	496	13	85	85	NUM
ap-7583	496	14	)	)	PUNCT
ap-7583	496	15	the	the	DET
ap-7583	496	16	exponents	exponent	NOUN
ap-7583	496	17	of	of	ADP
ap-7583	496	18	the	the	DET
ap-7583	496	19	singularity	singularity	NOUN
ap-7583	496	20	r	r	NOUN
ap-7583	496	21	=	=	SYM
ap-7583	496	22	0	0	NUM
ap-7583	496	23	are	be	AUX
ap-7583	496	24	s±	s±	X
ap-7583	496	25	=	=	X
ap-7583	496	26	(	(	PUNCT
ap-7583	496	27	α2	α2	PROPN
ap-7583	496	28	−	−	PROPN
ap-7583	496	29	β1	β1	PROPN
ap-7583	496	30	±	±	PROPN
ap-7583	496	31	√	√	PROPN
ap-7583	496	32	(	(	PUNCT
ap-7583	496	33	α2	α2	PROPN
ap-7583	496	34	−	−	PROPN
ap-7583	496	35	β1)2	β1)2	NUM
ap-7583	496	36	−	−	PROPN
ap-7583	496	37	4α2ε0	4α2ε0	PROPN
ap-7583	496	38	)	)	PUNCT
ap-7583	496	39	/(2α2	/(2α2	PUNCT
ap-7583	496	40	)	)	PUNCT
ap-7583	496	41	.	.	PUNCT
ap-7583	497	1	from	from	ADP
ap-7583	497	2	the	the	DET
ap-7583	497	3	relation	relation	NOUN
ap-7583	497	4	(	(	PUNCT
ap-7583	497	5	66	66	NUM
ap-7583	497	6	)	)	PUNCT
ap-7583	497	7	,	,	PUNCT
ap-7583	497	8	the	the	DET
ap-7583	497	9	coefficients	coefficient	NOUN
ap-7583	497	10	of	of	ADP
ap-7583	497	11	the	the	DET
ap-7583	497	12	formal	formal	ADJ
ap-7583	497	13	series	series	NOUN
ap-7583	497	14	solution	solution	NOUN
ap-7583	497	15	y(r	y(r	NOUN
ap-7583	497	16	)	)	PUNCT
ap-7583	497	17	=	=	SYM
ap-7583	497	18	rs	rs	NOUN
ap-7583	497	19	∑∞	∑∞	NOUN
ap-7583	497	20	k=0	k=0	PROPN
ap-7583	497	21	ck	ck	INTJ
ap-7583	497	22	rk	rk	PRON
ap-7583	497	23	satisfy	satisfy	VERB
ap-7583	497	24	the	the	DET
ap-7583	497	25	two	two	NUM
ap-7583	497	26	-	-	PUNCT
ap-7583	497	27	term	term	NOUN
ap-7583	497	28	recurrence	recurrence	NOUN
ap-7583	497	29	relation	relation	NOUN
ap-7583	497	30	(	(	PUNCT
ap-7583	497	31	k	k	NOUN
ap-7583	497	32	=	=	SYM
ap-7583	497	33	1	1	NUM
ap-7583	497	34	,	,	PUNCT
ap-7583	497	35	2	2	NUM
ap-7583	497	36	,	,	PUNCT
ap-7583	497	37	.	.	PUNCT
ap-7583	497	38	.	.	PUNCT
ap-7583	498	1	.	.	PUNCT
ap-7583	499	1	,	,	PUNCT
ap-7583	499	2	c0	c0	NOUN
ap-7583	499	3	=	=	PROPN
ap-7583	499	4	1	1	NUM
ap-7583	499	5	)	)	PUNCT
ap-7583	499	6	,	,	PUNCT
ap-7583	499	7	ck	ck	NOUN
ap-7583	499	8	=	=	SYM
ap-7583	500	1	−	−	PROPN
ap-7583	500	2	(	(	PUNCT
ap-7583	500	3	k−1+s±)(k−2+s±)α3+(k−1+s±)β2+ε1	k−1+s±)(k−2+s±)α3+(k−1+s±)β2+ε1	PROPN
ap-7583	500	4	(	(	PUNCT
ap-7583	500	5	k+s±)(k−1+s±)α2+(k+s±)β1+ε0	k+s±)(k−1+s±)α2+(k+s±)β1+ε0	PROPN
ap-7583	500	6	ck−1	ck−1	PROPN
ap-7583	500	7	,	,	PUNCT
ap-7583	500	8	=	=	PUNCT
ap-7583	500	9	k∏	k∏	PROPN
ap-7583	500	10	j=1	j=1	NOUN
ap-7583	500	11	(	(	PUNCT
ap-7583	500	12	−1)j	−1)j	X
ap-7583	500	13	(	(	PUNCT
ap-7583	500	14	j−1+s±)(j−2+s±)α3+(j−1+s±)β2+ε1	j−1+s±)(j−2+s±)α3+(j−1+s±)β2+ε1	PROPN
ap-7583	500	15	(	(	PUNCT
ap-7583	500	16	j+s±)(j−1+s±)α2+(j+s±)β1+ε0	j+s±)(j−1+s±)α2+(j+s±)β1+ε0	NOUN
ap-7583	500	17	,	,	PUNCT
ap-7583	500	18	(	(	PUNCT
ap-7583	500	19	86	86	NUM
ap-7583	500	20	)	)	PUNCT
ap-7583	500	21	that	that	PRON
ap-7583	500	22	allows	allow	VERB
ap-7583	500	23	to	to	PART
ap-7583	500	24	obtain	obtain	VERB
ap-7583	500	25	a	a	DET
ap-7583	500	26	closed	closed	ADJ
ap-7583	500	27	form	form	NOUN
ap-7583	500	28	of	of	ADP
ap-7583	500	29	the	the	DET
ap-7583	500	30	series	series	NOUN
ap-7583	500	31	solution	solution	NOUN
ap-7583	500	32	of	of	ADP
ap-7583	500	33	(	(	PUNCT
ap-7583	500	34	71	71	NUM
ap-7583	500	35	)	)	PUNCT
ap-7583	500	36	in	in	ADP
ap-7583	500	37	terms	term	NOUN
ap-7583	500	38	of	of	ADP
ap-7583	500	39	the	the	DET
ap-7583	500	40	generalized	generalized	ADJ
ap-7583	500	41	hypergeometric	hypergeometric	ADJ
ap-7583	500	42	function	function	NOUN
ap-7583	500	43	as	as	ADP
ap-7583	500	44	y(r	y(r	NOUN
ap-7583	500	45	)	)	PUNCT
ap-7583	501	1	=	=	NOUN
ap-7583	501	2	rs±	rs±	NOUN
ap-7583	502	1	3f2	3f2	NUM
ap-7583	502	2	(	(	PUNCT
ap-7583	502	3	1	1	NUM
ap-7583	502	4	,	,	PUNCT
ap-7583	502	5	s±	s±	PROPN
ap-7583	502	6	−	−	PROPN
ap-7583	502	7	1	1	NUM
ap-7583	502	8	2	2	NUM
ap-7583	502	9	+	+	CCONJ
ap-7583	502	10	β2	β2	NOUN
ap-7583	502	11	2α3	2α3	NUM
ap-7583	502	12	−	−	PROPN
ap-7583	503	1	√	√	NUM
ap-7583	503	2	(	(	PUNCT
ap-7583	503	3	α3−β2)2−4α3ε1	α3−β2)2−4α3ε1	NUM
ap-7583	503	4	2α3	2α3	NUM
ap-7583	503	5	,	,	PUNCT
ap-7583	503	6	s±	s±	PROPN
ap-7583	503	7	+	+	SYM
ap-7583	503	8	1	1	NUM
ap-7583	503	9	2	2	NUM
ap-7583	503	10	+	+	CCONJ
ap-7583	503	11	β2	β2	NOUN
ap-7583	503	12	2α3	2α3	NUM
ap-7583	503	13	−	−	PROPN
ap-7583	504	1	√	√	NUM
ap-7583	504	2	(	(	PUNCT
ap-7583	504	3	α3−β2)2−4α3ε1	α3−β2)2−4α3ε1	NUM
ap-7583	504	4	2α3	2α3	NUM
ap-7583	504	5	;	;	PUNCT
ap-7583	504	6	s±	s±	PROPN
ap-7583	504	7	+	+	SYM
ap-7583	504	8	1	1	NUM
ap-7583	504	9	2	2	NUM
ap-7583	504	10	+	+	CCONJ
ap-7583	504	11	β1	β1	VERB
ap-7583	504	12	2α2	2α2	NUM
ap-7583	504	13	−	−	NOUN
ap-7583	505	1	√	√	PROPN
ap-7583	506	1	(	(	PUNCT
ap-7583	506	2	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	506	3	2α2	2α2	NUM
ap-7583	506	4	,	,	PUNCT
ap-7583	506	5	s±	s±	PROPN
ap-7583	506	6	+	+	SYM
ap-7583	506	7	1	1	NUM
ap-7583	506	8	2	2	NUM
ap-7583	506	9	+	+	CCONJ
ap-7583	506	10	β1	β1	VERB
ap-7583	506	11	2α2	2α2	NUM
ap-7583	507	1	+	+	CCONJ
ap-7583	507	2	√	√	PROPN
ap-7583	507	3	(	(	PUNCT
ap-7583	507	4	α2−β1)2−4α2ε0	α2−β1)2−4α2ε0	PROPN
ap-7583	507	5	2α2	2α2	NUM
ap-7583	507	6	;	;	PUNCT
ap-7583	507	7	−α3	−α3	PROPN
ap-7583	507	8	α2	α2	ADJ
ap-7583	507	9	r	r	NOUN
ap-7583	507	10	)	)	PUNCT
ap-7583	507	11	.	.	PUNCT
ap-7583	508	1	(	(	PUNCT
ap-7583	508	2	87	87	NUM
ap-7583	508	3	)	)	PUNCT
ap-7583	508	4	180	180	NUM
ap-7583	508	5	vol	vol	NOUN
ap-7583	508	6	.	.	PUNCT
ap-7583	509	1	62	62	NUM
ap-7583	509	2	no	no	INTJ
ap-7583	509	3	.	.	PUNCT
ap-7583	510	1	1/2022	1/2022	NUM
ap-7583	510	2	on	on	ADP
ap-7583	510	3	generalized	generalized	ADJ
ap-7583	510	4	heun	heun	NOUN
ap-7583	510	5	equation	equation	NOUN
ap-7583	510	6	with	with	ADP
ap-7583	510	7	some	some	DET
ap-7583	510	8	mathematical	mathematical	NOUN
ap-7583	510	9	.	.	PUNCT
ap-7583	510	10	.	.	PUNCT
ap-7583	510	11	.	.	PUNCT
ap-7583	511	1	4.2	4.2	NUM
ap-7583	511	2	.	.	PUNCT
ap-7583	511	3	polynomial	polynomial	ADJ
ap-7583	511	4	solution	solution	NOUN
ap-7583	511	5	and	and	CCONJ
ap-7583	511	6	finite	finite	ADJ
ap-7583	511	7	sequence	sequence	NOUN
ap-7583	511	8	of	of	ADP
ap-7583	511	9	orthogonal	orthogonal	ADJ
ap-7583	511	10	polynomials	polynomial	NOUN
ap-7583	511	11	theorem	theorem	VERB
ap-7583	511	12	4.7	4.7	NUM
ap-7583	511	13	.	.	PUNCT
ap-7583	512	1	the	the	DET
ap-7583	512	2	necessary	necessary	ADJ
ap-7583	512	3	condition	condition	NOUN
ap-7583	512	4	for	for	ADP
ap-7583	512	5	the	the	DET
ap-7583	512	6	second	second	ADJ
ap-7583	512	7	-	-	PUNCT
ap-7583	512	8	order	order	NOUN
ap-7583	512	9	linear	linear	ADJ
ap-7583	512	10	differential	differential	NOUN
ap-7583	512	11	equation	equation	NOUN
ap-7583	512	12	(	(	PUNCT
ap-7583	512	13	65	65	NUM
ap-7583	512	14	)	)	PUNCT
ap-7583	512	15	to	to	PART
ap-7583	512	16	have	have	VERB
ap-7583	512	17	an	an	DET
ap-7583	512	18	nthdegree	nthdegree	ADJ
ap-7583	512	19	polynomial	polynomial	ADJ
ap-7583	512	20	solution	solution	NOUN
ap-7583	512	21	yn(r	yn(r	NOUN
ap-7583	512	22	)	)	PUNCT
ap-7583	513	1	=	=	SYM
ap-7583	514	1	∑n	∑n	PROPN
ap-7583	514	2	k=0	k=0	PROPN
ap-7583	514	3	ck	ck	PROPN
ap-7583	514	4	rk	rk	NOUN
ap-7583	514	5	,	,	PUNCT
ap-7583	514	6	n	n	PROPN
ap-7583	514	7	=	=	SYM
ap-7583	514	8	0	0	NUM
ap-7583	514	9	,	,	PUNCT
ap-7583	514	10	1	1	NUM
ap-7583	514	11	,	,	PUNCT
ap-7583	514	12	2	2	NUM
ap-7583	514	13	,	,	PUNCT
ap-7583	514	14	.	.	PUNCT
ap-7583	514	15	.	.	PUNCT
ap-7583	515	1	.	.	PUNCT
ap-7583	515	2	,	,	PUNCT
ap-7583	515	3	in	in	ADP
ap-7583	515	4	the	the	DET
ap-7583	515	5	neighbourhood	neighbourhood	NOUN
ap-7583	515	6	of	of	ADP
ap-7583	515	7	the	the	DET
ap-7583	515	8	regular	regular	ADJ
ap-7583	515	9	singular	singular	ADJ
ap-7583	515	10	point	point	NOUN
ap-7583	515	11	r	r	NOUN
ap-7583	515	12	=	=	NOUN
ap-7583	515	13	0	0	NUM
ap-7583	515	14	with	with	ADP
ap-7583	515	15	one	one	NUM
ap-7583	515	16	of	of	ADP
ap-7583	515	17	the	the	DET
ap-7583	515	18	indicial	indicial	ADJ
ap-7583	515	19	equation	equation	NOUN
ap-7583	515	20	exponents	exponent	NOUN
ap-7583	515	21	s	s	PART
ap-7583	515	22	=	=	SYM
ap-7583	515	23	0	0	NUM
ap-7583	515	24	,	,	PUNCT
ap-7583	515	25	is	be	AUX
ap-7583	515	26	ε1;n	ε1;n	NOUN
ap-7583	515	27	=	=	SYM
ap-7583	515	28	−n	−n	ADJ
ap-7583	515	29	(	(	PUNCT
ap-7583	515	30	n	n	CCONJ
ap-7583	515	31	−	−	PROPN
ap-7583	515	32	1	1	NUM
ap-7583	515	33	)	)	PUNCT
ap-7583	515	34	α3	α3	NOUN
ap-7583	515	35	−	−	PROPN
ap-7583	516	1	n	n	DET
ap-7583	516	2	β2	β2	NOUN
ap-7583	516	3	,	,	PUNCT
ap-7583	516	4	n	n	NOUN
ap-7583	516	5	=	=	SYM
ap-7583	516	6	0	0	NUM
ap-7583	516	7	,	,	PUNCT
ap-7583	516	8	1	1	NUM
ap-7583	516	9	,	,	PUNCT
ap-7583	516	10	2	2	NUM
ap-7583	516	11	,	,	PUNCT
ap-7583	516	12	.	.	PUNCT
ap-7583	516	13	.	.	PUNCT
ap-7583	516	14	.	.	PUNCT
ap-7583	517	1	,	,	PUNCT
ap-7583	517	2	(	(	PUNCT
ap-7583	517	3	88	88	NUM
ap-7583	517	4	)	)	PUNCT
ap-7583	517	5	along	along	ADP
ap-7583	517	6	with	with	ADP
ap-7583	517	7	the	the	DET
ap-7583	517	8	sufficient	sufficient	ADJ
ap-7583	517	9	condition	condition	NOUN
ap-7583	517	10	,	,	PUNCT
ap-7583	517	11	relating	relate	VERB
ap-7583	517	12	the	the	DET
ap-7583	517	13	remaining	remain	VERB
ap-7583	517	14	coefficients	coefficient	NOUN
ap-7583	517	15	,	,	PUNCT
ap-7583	517	16	given	give	VERB
ap-7583	517	17	by	by	ADP
ap-7583	517	18	the	the	DET
ap-7583	517	19	vanishing	vanishing	NOUN
ap-7583	517	20	of	of	ADP
ap-7583	517	21	the	the	DET
ap-7583	517	22	tridiagonal	tridiagonal	NOUN
ap-7583	517	23	(	(	PUNCT
ap-7583	517	24	n	n	NOUN
ap-7583	517	25	+	+	CCONJ
ap-7583	517	26	1	1	NUM
ap-7583	517	27	)	)	PUNCT
ap-7583	517	28	×	×	NOUN
ap-7583	517	29	(	(	PUNCT
ap-7583	517	30	n	n	X
ap-7583	517	31	+	+	CCONJ
ap-7583	517	32	1)-determinant	1)-determinant	PROPN
ap-7583	517	33	∆n+1	∆n+1	PROPN
ap-7583	517	34	≡	≡	PROPN
ap-7583	517	35	0	0	PUNCT
ap-7583	518	1	given	give	VERB
ap-7583	518	2	by	by	ADP
ap-7583	518	3	∆n+1	∆n+1	PROPN
ap-7583	518	4	=	=	SYM
ap-7583	518	5	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	518	6	s0	s0	PROPN
ap-7583	518	7	t1	t1	PROPN
ap-7583	518	8	γ1	γ1	PROPN
ap-7583	518	9	s1	s1	PROPN
ap-7583	518	10	t2	t2	PROPN
ap-7583	518	11	γ2	γ2	PROPN
ap-7583	518	12	s2	s2	PROPN
ap-7583	518	13	t3	t3	NOUN
ap-7583	518	14	.	.	PUNCT
ap-7583	518	15	.	.	PUNCT
ap-7583	518	16	.	.	PUNCT
ap-7583	518	17	.	.	PUNCT
ap-7583	518	18	.	.	PUNCT
ap-7583	518	19	.	.	PUNCT
ap-7583	518	20	.	.	PUNCT
ap-7583	518	21	.	.	PUNCT
ap-7583	518	22	.	.	PUNCT
ap-7583	519	1	γn−2	γn−2	PROPN
ap-7583	519	2	sn−2	sn−2	PROPN
ap-7583	519	3	tn−1	tn−1	PROPN
ap-7583	519	4	γn−1	γn−1	PROPN
ap-7583	519	5	sn−1	sn−1	PROPN
ap-7583	519	6	tn	tn	PROPN
ap-7583	519	7	γn	γn	ADP
ap-7583	519	8	sn	sn	PROPN
ap-7583	519	9	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣	PROPN
ap-7583	519	10	,	,	PUNCT
ap-7583	519	11	(	(	PUNCT
ap-7583	519	12	89	89	NUM
ap-7583	519	13	)	)	PUNCT
ap-7583	519	14	where	where	SCONJ
ap-7583	519	15	,	,	PUNCT
ap-7583	519	16	for	for	ADP
ap-7583	519	17	fixed	fixed	ADJ
ap-7583	519	18	n	n	NOUN
ap-7583	519	19	:	:	PUNCT
ap-7583	519	20	ε1;n	ε1;n	NOUN
ap-7583	519	21	=	=	SYM
ap-7583	519	22	−n	−n	ADJ
ap-7583	519	23	(	(	PUNCT
ap-7583	519	24	n	n	CCONJ
ap-7583	519	25	−	−	PROPN
ap-7583	519	26	1	1	NUM
ap-7583	519	27	)	)	PUNCT
ap-7583	519	28	α3	α3	NOUN
ap-7583	519	29	−	−	PROPN
ap-7583	519	30	n	n	PROPN
ap-7583	519	31	β2,	β2,	ADP
ap-7583	519	32	sk	sk	X
ap-7583	519	33	=	=	PROPN
ap-7583	519	34	ε0;n	ε0;n	PROPN
ap-7583	519	35	+	+	CCONJ
ap-7583	519	36	k	k	X
ap-7583	519	37	(	(	PUNCT
ap-7583	519	38	(	(	PUNCT
ap-7583	519	39	k	k	X
ap-7583	519	40	−	−	PROPN
ap-7583	519	41	1)α2	1)α2	PROPN
ap-7583	519	42	+	+	CCONJ
ap-7583	519	43	β1	β1	PROPN
ap-7583	519	44	)	)	PUNCT
ap-7583	519	45	,	,	PUNCT
ap-7583	519	46	tk	tk	PROPN
ap-7583	520	1	=	=	NOUN
ap-7583	520	2	−k	−k	PROPN
ap-7583	520	3	(	(	PUNCT
ap-7583	520	4	(	(	PUNCT
ap-7583	520	5	k	k	X
ap-7583	520	6	−	−	PROPN
ap-7583	521	1	1)α1	1)α1	NUM
ap-7583	522	1	+	+	CCONJ
ap-7583	522	2	β0	β0	PROPN
ap-7583	522	3	)	)	PUNCT
ap-7583	522	4	,	,	PUNCT
ap-7583	522	5	γk	γk	X
ap-7583	522	6	=	=	PUNCT
ap-7583	522	7	−ε1;n	−ε1;n	PROPN
ap-7583	523	1	−	−	PROPN
ap-7583	524	1	(	(	PUNCT
ap-7583	524	2	k	k	NOUN
ap-7583	524	3	−	−	PROPN
ap-7583	524	4	1	1	NUM
ap-7583	524	5	)	)	PUNCT
ap-7583	524	6	(	(	PUNCT
ap-7583	524	7	(	(	PUNCT
ap-7583	524	8	k	k	X
ap-7583	524	9	−	−	PROPN
ap-7583	524	10	2)α3	2)α3	PROPN
ap-7583	524	11	+	+	CCONJ
ap-7583	524	12	β2	β2	NOUN
ap-7583	524	13	)	)	PUNCT
ap-7583	524	14	,	,	PUNCT
ap-7583	524	15	and	and	CCONJ
ap-7583	524	16	all	all	DET
ap-7583	524	17	other	other	ADJ
ap-7583	524	18	entries	entry	NOUN
ap-7583	524	19	are	be	AUX
ap-7583	524	20	zeros	zero	NOUN
ap-7583	524	21	.	.	PUNCT
ap-7583	525	1	in	in	ADP
ap-7583	525	2	this	this	DET
ap-7583	525	3	case	case	NOUN
ap-7583	525	4	,	,	PUNCT
ap-7583	525	5	the	the	DET
ap-7583	525	6	polynomial	polynomial	ADJ
ap-7583	525	7	solutions	solution	NOUN
ap-7583	525	8	are	be	AUX
ap-7583	525	9	given	give	VERB
ap-7583	525	10	explicitly	explicitly	ADV
ap-7583	525	11	by	by	ADP
ap-7583	525	12	yn(r	yn(r	PRON
ap-7583	525	13	)	)	PUNCT
ap-7583	525	14	=	=	SYM
ap-7583	525	15	n∑	n∑	NOUN
ap-7583	525	16	k=0	k=0	PROPN
ap-7583	525	17	(	(	PUNCT
ap-7583	525	18	−1)k	−1)k	PROPN
ap-7583	525	19	pn	pn	PROPN
ap-7583	525	20	k	k	PROPN
ap-7583	525	21	(	(	PUNCT
ap-7583	525	22	ε0;n	ε0;n	PROPN
ap-7583	525	23	)	)	PUNCT
ap-7583	526	1	k	k	NOUN
ap-7583	526	2	!	!	PUNCT
ap-7583	527	1	αk	αk	CCONJ
ap-7583	527	2	1	1	NUM
ap-7583	527	3	(	(	PUNCT
ap-7583	527	4	β0	β0	PROPN
ap-7583	527	5	α1	α1	PROPN
ap-7583	527	6	)	)	PUNCT
ap-7583	528	1	k	k	PROPN
ap-7583	528	2	rk	rk	NOUN
ap-7583	528	3	,	,	PUNCT
ap-7583	528	4	(	(	PUNCT
ap-7583	528	5	90	90	NUM
ap-7583	528	6	)	)	PUNCT
ap-7583	528	7	where	where	SCONJ
ap-7583	528	8	the	the	DET
ap-7583	528	9	finite	finite	ADJ
ap-7583	528	10	orthogonal	orthogonal	ADJ
ap-7583	528	11	sequences	sequence	NOUN
ap-7583	528	12	{	{	PUNCT
ap-7583	528	13	pn	pn	PROPN
ap-7583	528	14	k	k	PROPN
ap-7583	528	15	(	(	PUNCT
ap-7583	528	16	ε0;n)}n	ε0;n)}n	PROPN
ap-7583	528	17	k=0	k=0	PROPN
ap-7583	528	18	are	be	AUX
ap-7583	528	19	evaluated	evaluate	VERB
ap-7583	528	20	using	use	VERB
ap-7583	528	21	the	the	DET
ap-7583	528	22	three	three	NUM
ap-7583	528	23	-	-	PUNCT
ap-7583	528	24	term	term	NOUN
ap-7583	528	25	recurrence	recurrence	NOUN
ap-7583	528	26	relation	relation	NOUN
ap-7583	528	27	pn	pn	PROPN
ap-7583	528	28	k+1(ε0;n	k+1(ε0;n	PROPN
ap-7583	528	29	)	)	PUNCT
ap-7583	528	30	=	=	PUNCT
ap-7583	528	31	(	(	PUNCT
ap-7583	528	32	sk	sk	ADP
ap-7583	528	33	+	+	CCONJ
ap-7583	528	34	ε0;n	ε0;n	NOUN
ap-7583	528	35	)	)	PUNCT
ap-7583	528	36	pn	pn	PROPN
ap-7583	528	37	k	k	PROPN
ap-7583	528	38	(	(	PUNCT
ap-7583	528	39	ε0;n	ε0;n	PROPN
ap-7583	528	40	)	)	PUNCT
ap-7583	528	41	−	−	PROPN
ap-7583	528	42	γktkpn	γktkpn	PROPN
ap-7583	528	43	k−1(ε0;n	k−1(ε0;n	NUM
ap-7583	528	44	)	)	PUNCT
ap-7583	528	45	,	,	PUNCT
ap-7583	528	46	or	or	CCONJ
ap-7583	528	47	,	,	PUNCT
ap-7583	528	48	more	more	ADV
ap-7583	528	49	explicitly	explicitly	ADV
ap-7583	528	50	,	,	PUNCT
ap-7583	528	51	pn	pn	PROPN
ap-7583	528	52	k+1(ε0;n	k+1(ε0;n	PROPN
ap-7583	528	53	)	)	PUNCT
ap-7583	528	54	=	=	PRON
ap-7583	528	55	(	(	PUNCT
ap-7583	528	56	k(k	k(k	ADV
ap-7583	528	57	−	−	PROPN
ap-7583	528	58	1)α2	1)α2	PROPN
ap-7583	528	59	+	+	NUM
ap-7583	528	60	kβ1	kβ1	NOUN
ap-7583	528	61	+	+	CCONJ
ap-7583	528	62	ε0;n	ε0;n	NUM
ap-7583	528	63	)	)	PUNCT
ap-7583	528	64	pn	pn	PROPN
ap-7583	528	65	k	k	PROPN
ap-7583	528	66	(	(	PUNCT
ap-7583	528	67	ε0;n	ε0;n	PROPN
ap-7583	528	68	)	)	PUNCT
ap-7583	529	1	+	+	CCONJ
ap-7583	530	1	k(n	k(n	PROPN
ap-7583	530	2	−	−	PROPN
ap-7583	530	3	k	k	NOUN
ap-7583	531	1	+	+	PROPN
ap-7583	531	2	1	1	X
ap-7583	531	3	)	)	PUNCT
ap-7583	531	4	(	(	PUNCT
ap-7583	531	5	(	(	PUNCT
ap-7583	531	6	k	k	X
ap-7583	531	7	−	−	PROPN
ap-7583	531	8	1)α1	1)α1	NUM
ap-7583	531	9	+	+	CCONJ
ap-7583	531	10	β0	β0	ADJ
ap-7583	531	11	)	)	PUNCT
ap-7583	531	12	×	×	NOUN
ap-7583	531	13	(	(	PUNCT
ap-7583	531	14	β2	β2	VERB
ap-7583	531	15	+	+	CCONJ
ap-7583	531	16	α3(k	α3(k	PROPN
ap-7583	531	17	+	+	CCONJ
ap-7583	531	18	n	n	CCONJ
ap-7583	531	19	−	−	PROPN
ap-7583	531	20	2	2	NUM
ap-7583	531	21	)	)	PUNCT
ap-7583	531	22	)	)	PUNCT
ap-7583	531	23	pn	pn	PROPN
ap-7583	531	24	k−1(ε0;n	k−1(ε0;n	PROPN
ap-7583	531	25	)	)	PUNCT
ap-7583	531	26	,	,	PUNCT
ap-7583	531	27	(	(	PUNCT
ap-7583	531	28	91	91	NUM
ap-7583	531	29	)	)	PUNCT
ap-7583	531	30	where	where	SCONJ
ap-7583	531	31	pn	pn	PROPN
ap-7583	531	32	−1(ε0;n	−1(ε0;n	PROPN
ap-7583	531	33	)	)	PUNCT
ap-7583	531	34	=	=	SYM
ap-7583	531	35	0	0	NUM
ap-7583	531	36	,	,	PUNCT
ap-7583	531	37	and	and	CCONJ
ap-7583	531	38	pn	pn	PROPN
ap-7583	531	39	0	0	SYM
ap-7583	531	40	(	(	PUNCT
ap-7583	531	41	ε0;n	ε0;n	NUM
ap-7583	531	42	)	)	PUNCT
ap-7583	531	43	=	=	SYM
ap-7583	531	44	1	1	NUM
ap-7583	531	45	for	for	ADP
ap-7583	531	46	the	the	DET
ap-7583	531	47	non	non	ADJ
ap-7583	531	48	-	-	ADJ
ap-7583	531	49	negative	negative	ADJ
ap-7583	531	50	integer	integer	NOUN
ap-7583	531	51	n.	n.	NOUN
ap-7583	531	52	expanding	expand	VERB
ap-7583	531	53	∆k+1	∆k+1	NOUN
ap-7583	531	54	with	with	ADP
ap-7583	531	55	respect	respect	NOUN
ap-7583	531	56	to	to	ADP
ap-7583	531	57	the	the	DET
ap-7583	531	58	last	last	ADJ
ap-7583	531	59	column	column	NOUN
ap-7583	531	60	,	,	PUNCT
ap-7583	531	61	it	it	PRON
ap-7583	531	62	is	be	AUX
ap-7583	531	63	clear	clear	ADJ
ap-7583	531	64	that	that	SCONJ
ap-7583	531	65	the	the	DET
ap-7583	531	66	determinant	determinant	ADJ
ap-7583	531	67	(	(	PUNCT
ap-7583	531	68	89	89	NUM
ap-7583	531	69	)	)	PUNCT
ap-7583	531	70	satisfies	satisfy	VERB
ap-7583	531	71	a	a	DET
ap-7583	531	72	three	three	NUM
ap-7583	531	73	-	-	PUNCT
ap-7583	531	74	term	term	NOUN
ap-7583	531	75	recurrence	recurrence	NOUN
ap-7583	531	76	relation	relation	NOUN
ap-7583	531	77	{	{	PUNCT
ap-7583	531	78	∆k+1	∆k+1	PROPN
ap-7583	531	79	=	=	SYM
ap-7583	531	80	(	(	PUNCT
ap-7583	531	81	sk	sk	X
ap-7583	531	82	+	+	CCONJ
ap-7583	531	83	ε0;n	ε0;n	NUM
ap-7583	531	84	)	)	PUNCT
ap-7583	531	85	∆k	∆k	PROPN
ap-7583	532	1	−	−	PROPN
ap-7583	532	2	γk	γk	PROPN
ap-7583	532	3	tk	tk	PROPN
ap-7583	532	4	∆k−1	∆k−1	PROPN
ap-7583	532	5	,	,	PUNCT
ap-7583	532	6	∆0	∆0	NOUN
ap-7583	532	7	=	=	SYM
ap-7583	532	8	1	1	NUM
ap-7583	532	9	,	,	PUNCT
ap-7583	532	10	∆−1	∆−1	PROPN
ap-7583	532	11	=	=	SYM
ap-7583	532	12	0	0	PROPN
ap-7583	532	13	,	,	PUNCT
ap-7583	532	14	k	k	NOUN
ap-7583	532	15	=	=	SYM
ap-7583	532	16	0	0	NUM
ap-7583	532	17	,	,	PUNCT
ap-7583	532	18	1	1	NUM
ap-7583	532	19	,	,	PUNCT
ap-7583	532	20	.	.	PUNCT
ap-7583	532	21	.	.	PUNCT
ap-7583	533	1	.	.	PUNCT
ap-7583	534	1	,	,	PUNCT
ap-7583	534	2	n	n	X
ap-7583	534	3	,	,	PUNCT
ap-7583	534	4	(	(	PUNCT
ap-7583	534	5	92	92	NUM
ap-7583	534	6	)	)	PUNCT
ap-7583	534	7	that	that	PRON
ap-7583	534	8	allow	allow	VERB
ap-7583	534	9	to	to	PART
ap-7583	534	10	compute	compute	VERB
ap-7583	534	11	the	the	DET
ap-7583	534	12	determinant	determinant	ADJ
ap-7583	534	13	∆k	∆k	PROPN
ap-7583	534	14	recursively	recursively	ADV
ap-7583	534	15	in	in	ADP
ap-7583	534	16	terms	term	NOUN
ap-7583	534	17	of	of	ADP
ap-7583	534	18	lower	low	ADJ
ap-7583	534	19	-	-	PUNCT
ap-7583	534	20	order	order	NOUN
ap-7583	534	21	determinants	determinant	NOUN
ap-7583	534	22	.	.	PUNCT
ap-7583	535	1	we	we	PRON
ap-7583	535	2	now	now	ADV
ap-7583	535	3	show	show	VERB
ap-7583	535	4	,	,	PUNCT
ap-7583	535	5	by	by	ADP
ap-7583	535	6	induction	induction	NOUN
ap-7583	535	7	on	on	ADP
ap-7583	535	8	k	k	PROPN
ap-7583	535	9	,	,	PUNCT
ap-7583	535	10	that	that	SCONJ
ap-7583	535	11	∆k+1	∆k+1	NOUN
ap-7583	535	12	=	=	SYM
ap-7583	535	13	pk+1(ε0;n	pk+1(ε0;n	NOUN
ap-7583	535	14	)	)	PUNCT
ap-7583	535	15	.	.	PUNCT
ap-7583	536	1	(	(	PUNCT
ap-7583	536	2	93	93	NUM
ap-7583	536	3	)	)	PUNCT
ap-7583	536	4	for	for	ADP
ap-7583	536	5	k	k	PROPN
ap-7583	536	6	=	=	SYM
ap-7583	536	7	0	0	PROPN
ap-7583	536	8	,	,	PUNCT
ap-7583	536	9	we	we	PRON
ap-7583	536	10	find	find	VERB
ap-7583	536	11	by	by	ADP
ap-7583	536	12	(	(	PUNCT
ap-7583	536	13	89	89	NUM
ap-7583	536	14	)	)	PUNCT
ap-7583	536	15	that	that	PRON
ap-7583	536	16	∆1	∆1	PRON
ap-7583	537	1	=	=	SYM
ap-7583	537	2	(	(	PUNCT
ap-7583	537	3	s0	s0	PROPN
ap-7583	537	4	+	+	CCONJ
ap-7583	537	5	ε0;n	ε0;n	NUM
ap-7583	537	6	)	)	PUNCT
ap-7583	537	7	where	where	SCONJ
ap-7583	537	8	the	the	DET
ap-7583	537	9	right	right	ADJ
ap-7583	537	10	hand	hand	NOUN
ap-7583	537	11	side	side	NOUN
ap-7583	537	12	equals	equal	VERB
ap-7583	537	13	to	to	ADP
ap-7583	537	14	p	p	PROPN
ap-7583	537	15	n	n	PROPN
ap-7583	537	16	1	1	NUM
ap-7583	537	17	(	(	PUNCT
ap-7583	537	18	ε0;n	ε0;n	NOUN
ap-7583	537	19	)	)	PUNCT
ap-7583	537	20	using	use	VERB
ap-7583	537	21	(	(	PUNCT
ap-7583	537	22	91	91	NUM
ap-7583	537	23	)	)	PUNCT
ap-7583	537	24	.	.	PUNCT
ap-7583	538	1	next	next	ADV
ap-7583	538	2	,	,	PUNCT
ap-7583	538	3	suppose	suppose	VERB
ap-7583	538	4	that	that	SCONJ
ap-7583	538	5	∆j	∆j	PROPN
ap-7583	538	6	=	=	SYM
ap-7583	538	7	pj(ε0;n	pj(ε0;n	PROPN
ap-7583	538	8	)	)	PUNCT
ap-7583	538	9	,	,	PUNCT
ap-7583	538	10	for	for	ADP
ap-7583	538	11	j	j	PROPN
ap-7583	538	12	=	=	SYM
ap-7583	538	13	0	0	PROPN
ap-7583	538	14	,	,	PUNCT
ap-7583	538	15	1	1	NUM
ap-7583	538	16	,	,	PUNCT
ap-7583	538	17	2	2	NUM
ap-7583	538	18	,	,	PUNCT
ap-7583	538	19	·	·	PUNCT
ap-7583	538	20	·	·	PUNCT
ap-7583	538	21	·	·	PUNCT
ap-7583	538	22	,	,	PUNCT
ap-7583	538	23	k	k	NOUN
ap-7583	538	24	,	,	PUNCT
ap-7583	538	25	then	then	ADV
ap-7583	538	26	from	from	ADP
ap-7583	538	27	(	(	PUNCT
ap-7583	538	28	91	91	NUM
ap-7583	538	29	)	)	PUNCT
ap-7583	538	30	pn	pn	PROPN
ap-7583	538	31	k+1(ε0;n	k+1(ε0;n	PROPN
ap-7583	538	32	)	)	PUNCT
ap-7583	538	33	=	=	PUNCT
ap-7583	538	34	(	(	PUNCT
ap-7583	538	35	sk	sk	ADP
ap-7583	538	36	+	+	CCONJ
ap-7583	538	37	ε0;n	ε0;n	NOUN
ap-7583	538	38	)	)	PUNCT
ap-7583	538	39	pn	pn	PROPN
ap-7583	538	40	k	k	PROPN
ap-7583	538	41	(	(	PUNCT
ap-7583	538	42	ε0;n	ε0;n	PROPN
ap-7583	538	43	)	)	PUNCT
ap-7583	538	44	−	−	PROPN
ap-7583	538	45	γk	γk	PROPN
ap-7583	538	46	tk	tk	PROPN
ap-7583	538	47	pn	pn	PROPN
ap-7583	538	48	k−1(ε0;n	k−1(ε0;n	PROPN
ap-7583	538	49	)	)	PUNCT
ap-7583	538	50	=	=	PUNCT
ap-7583	538	51	(	(	PUNCT
ap-7583	538	52	sk	sk	ADP
ap-7583	538	53	+	+	CCONJ
ap-7583	538	54	ε0;n	ε0;n	NOUN
ap-7583	538	55	)	)	PUNCT
ap-7583	538	56	∆k	∆k	PROPN
ap-7583	539	1	−	−	PROPN
ap-7583	539	2	γk	γk	PROPN
ap-7583	539	3	tk	tk	PROPN
ap-7583	539	4	∆k−1	∆k−1	PROPN
ap-7583	539	5	=	=	SYM
ap-7583	539	6	∆k+1	∆k+1	NOUN
ap-7583	539	7	and	and	CCONJ
ap-7583	539	8	the	the	DET
ap-7583	539	9	induction	induction	NOUN
ap-7583	539	10	step	step	NOUN
ap-7583	539	11	is	be	AUX
ap-7583	539	12	reached	reach	VERB
ap-7583	539	13	.	.	PUNCT
ap-7583	540	1	these	these	DET
ap-7583	540	2	results	result	NOUN
ap-7583	540	3	can	can	AUX
ap-7583	540	4	be	be	AUX
ap-7583	540	5	represented	represent	VERB
ap-7583	540	6	by	by	ADP
ap-7583	540	7	the	the	DET
ap-7583	540	8	graphical	graphical	ADJ
ap-7583	540	9	representation	representation	NOUN
ap-7583	540	10	(	(	PUNCT
ap-7583	540	11	figure	figure	NOUN
ap-7583	540	12	2	2	NUM
ap-7583	540	13	)	)	PUNCT
ap-7583	540	14	.	.	PUNCT
ap-7583	541	1	some	some	PRON
ap-7583	541	2	of	of	ADP
ap-7583	541	3	the	the	DET
ap-7583	541	4	mathematical	mathematical	ADJ
ap-7583	541	5	properties	property	NOUN
ap-7583	541	6	of	of	ADP
ap-7583	541	7	the	the	DET
ap-7583	541	8	finite	finite	ADJ
ap-7583	541	9	sequence	sequence	NOUN
ap-7583	541	10	of	of	ADP
ap-7583	541	11	polynomials	polynomial	NOUN
ap-7583	541	12	{	{	PUNCT
ap-7583	541	13	pn	pn	NOUN
ap-7583	541	14	k	k	PROPN
ap-7583	541	15	(	(	PUNCT
ap-7583	541	16	ε0;n)}n	ε0;n)}n	PROPN
ap-7583	541	17	k=0	k=0	PROPN
ap-7583	541	18	will	will	AUX
ap-7583	541	19	be	be	AUX
ap-7583	541	20	explored	explore	VERB
ap-7583	541	21	in	in	ADP
ap-7583	541	22	later	later	ADJ
ap-7583	541	23	sections	section	NOUN
ap-7583	541	24	.	.	PUNCT
ap-7583	542	1	181	181	NUM
ap-7583	542	2	nasser	nasser	PROPN
ap-7583	542	3	saad	saad	PROPN
ap-7583	542	4	acta	acta	PROPN
ap-7583	542	5	polytechnica	polytechnica	PROPN
ap-7583	542	6	∆k+1	∆k+1	PROPN
ap-7583	542	7	=	=	SYM
ap-7583	542	8	det	det	PROPN
ap-7583	542	9	s0	s0	PROPN
ap-7583	542	10	t1	t1	PROPN
ap-7583	542	11	0	0	NUM
ap-7583	542	12	...	...	SYM
ap-7583	542	13	0	0	NUM
ap-7583	542	14	γ1	γ1	PROPN
ap-7583	542	15	s1	s1	PROPN
ap-7583	542	16	t2	t2	PROPN
ap-7583	542	17	...	...	PUNCT
ap-7583	542	18	0	0	NUM
ap-7583	542	19	0	0	NUM
ap-7583	542	20	γ2	γ2	PROPN
ap-7583	542	21	s2	s2	NOUN
ap-7583	542	22	...	...	PUNCT
ap-7583	542	23	0	0	PUNCT
ap-7583	542	24	·	·	PUNCT
ap-7583	542	25	·	·	PUNCT
ap-7583	542	26	·	·	PUNCT
ap-7583	542	27	·	·	PUNCT
ap-7583	542	28	·	·	PUNCT
ap-7583	542	29	·	·	PUNCT
ap-7583	542	30	·	·	PUNCT
ap-7583	542	31	·	·	PUNCT
ap-7583	542	32	·	·	PUNCT
ap-7583	542	33	.	.	PUNCT
ap-7583	542	34	.	.	PUNCT
ap-7583	542	35	.	.	PUNCT
ap-7583	543	1	...	...	PUNCT
ap-7583	544	1	0	0	NUM
ap-7583	544	2	0	0	NUM
ap-7583	544	3	0	0	NUM
ap-7583	544	4	·	·	PUNCT
ap-7583	544	5	·	·	PUNCT
ap-7583	544	6	·	·	PUNCT
ap-7583	544	7	sn	sn	INTJ
ap-7583	544	8			NOUN
ap-7583	544	9			PUNCT
ap-7583	545	1	pn	pn	PROPN
ap-7583	545	2	1	1	NUM
ap-7583	545	3	pn	pn	PROPN
ap-7583	545	4	2	2	NUM
ap-7583	545	5	pn	pn	PROPN
ap-7583	545	6	3	3	NUM
ap-7583	545	7	pn	pn	NOUN
ap-7583	545	8	n	n	PRON
ap-7583	545	9	figure	figure	NOUN
ap-7583	545	10	2	2	NUM
ap-7583	545	11	.	.	PUNCT
ap-7583	546	1	a	a	DET
ap-7583	546	2	demonstration	demonstration	NOUN
ap-7583	546	3	of	of	ADP
ap-7583	546	4	how	how	SCONJ
ap-7583	546	5	the	the	DET
ap-7583	546	6	polynomials	polynomial	NOUN
ap-7583	546	7	{	{	PUNCT
ap-7583	546	8	pn	pn	PROPN
ap-7583	546	9	k	k	PROPN
ap-7583	546	10	(	(	PUNCT
ap-7583	546	11	ε0;n)}n	ε0;n)}n	PROPN
ap-7583	546	12	k=0	k=0	PROPN
ap-7583	546	13	may	may	AUX
ap-7583	546	14	be	be	AUX
ap-7583	546	15	obtained	obtain	VERB
ap-7583	546	16	from	from	ADP
ap-7583	546	17	the	the	DET
ap-7583	546	18	(	(	PUNCT
ap-7583	546	19	k	k	PROPN
ap-7583	546	20	+	+	PROPN
ap-7583	546	21	1)-determinant	1)-determinant	ADJ
ap-7583	546	22	∆k+1	∆k+1	NOUN
ap-7583	546	23	for	for	ADP
ap-7583	546	24	k	k	X
ap-7583	546	25	=	=	SYM
ap-7583	546	26	0	0	NUM
ap-7583	546	27	,	,	PUNCT
ap-7583	546	28	1	1	NUM
ap-7583	546	29	,	,	PUNCT
ap-7583	546	30	2	2	NUM
ap-7583	546	31	,	,	PUNCT
ap-7583	546	32	.	.	PUNCT
ap-7583	546	33	.	.	PUNCT
ap-7583	547	1	.	.	PUNCT
ap-7583	548	1	,	,	PUNCT
ap-7583	548	2	n	n	X
ap-7583	548	3	.	.	PUNCT
ap-7583	549	1	remark	remark	PROPN
ap-7583	549	2	4.8	4.8	NUM
ap-7583	549	3	.	.	PUNCT
ap-7583	550	1	for	for	ADP
ap-7583	550	2	α3	α3	NOUN
ap-7583	550	3	+	+	CCONJ
ap-7583	550	4	α2	α2	ADJ
ap-7583	550	5	+	+	CCONJ
ap-7583	550	6	α1	α1	PROPN
ap-7583	550	7	=	=	SYM
ap-7583	550	8	0	0	NUM
ap-7583	550	9	,	,	PUNCT
ap-7583	550	10	the	the	DET
ap-7583	550	11	canonical	canonical	ADJ
ap-7583	550	12	form	form	NOUN
ap-7583	550	13	of	of	ADP
ap-7583	550	14	heun	heun	PROPN
ap-7583	550	15	’s	’s	PART
ap-7583	550	16	equation	equation	NOUN
ap-7583	550	17	can	can	AUX
ap-7583	550	18	be	be	AUX
ap-7583	550	19	deduced	deduce	VERB
ap-7583	550	20	from	from	ADP
ap-7583	550	21	(	(	PUNCT
ap-7583	550	22	65	65	NUM
ap-7583	550	23	)	)	PUNCT
ap-7583	550	24	by	by	ADP
ap-7583	550	25	means	mean	NOUN
ap-7583	550	26	of	of	ADP
ap-7583	550	27	the	the	DET
ap-7583	550	28	following	follow	VERB
ap-7583	550	29	substitutions	substitution	NOUN
ap-7583	550	30	:	:	PUNCT
ap-7583	550	31	y′′(r	y′′(r	NOUN
ap-7583	550	32	)	)	PUNCT
ap-7583	551	1	+	+	CCONJ
ap-7583	551	2			PROPN
ap-7583	551	3	β0+β1+β3	β0+β1+β3	PROPN
ap-7583	551	4	α3−α1	α3−α1	NUM
ap-7583	551	5	r	r	NOUN
ap-7583	551	6	−	−	NOUN
ap-7583	551	7	1	1	NUM
ap-7583	551	8	+	+	CCONJ
ap-7583	551	9	β0	β0	ADJ
ap-7583	551	10	α1	α1	PROPN
ap-7583	551	11	r	r	NOUN
ap-7583	551	12	+	+	CCONJ
ap-7583	551	13	α2	α2	ADJ
ap-7583	551	14	3β0+α1α3β1+α2	3β0+α1α3β1+α2	PRON
ap-7583	551	15	1β2	1β2	NUM
ap-7583	551	16	α1α3(α1−α3	α1α3(α1−α3	NUM
ap-7583	551	17	)	)	PUNCT
ap-7583	551	18	(	(	PUNCT
ap-7583	551	19	r	r	NOUN
ap-7583	551	20	−	−	PROPN
ap-7583	551	21	α1	α1	PROPN
ap-7583	551	22	α3	α3	NOUN
ap-7583	551	23	)	)	PUNCT
ap-7583	551	24			PROPN
ap-7583	551	25	y′(r	y′(r	NOUN
ap-7583	551	26	)	)	PUNCT
ap-7583	552	1	+	+	CCONJ
ap-7583	552	2	ε1	ε1	VERB
ap-7583	552	3	α3	α3	PROPN
ap-7583	552	4	r	r	NOUN
ap-7583	552	5	+	+	CCONJ
ap-7583	552	6	ε0	ε0	PROPN
ap-7583	552	7	α3	α3	NOUN
ap-7583	552	8	r	r	NOUN
ap-7583	552	9	(	(	PUNCT
ap-7583	552	10	r	r	NOUN
ap-7583	552	11	−	−	PROPN
ap-7583	552	12	1	1	NUM
ap-7583	552	13	)	)	PUNCT
ap-7583	552	14	(	(	PUNCT
ap-7583	552	15	r	r	NOUN
ap-7583	552	16	−	−	PROPN
ap-7583	552	17	α1	α1	PROPN
ap-7583	552	18	α3	α3	NOUN
ap-7583	552	19	)	)	PUNCT
ap-7583	552	20	y(r	y(r	PROPN
ap-7583	552	21	)	)	PUNCT
ap-7583	552	22	=	=	SYM
ap-7583	553	1	0	0	X
ap-7583	553	2	.	.	PUNCT
ap-7583	554	1	(	(	PUNCT
ap-7583	554	2	94	94	NUM
ap-7583	554	3	)	)	PUNCT
ap-7583	554	4	or	or	CCONJ
ap-7583	554	5	,	,	PUNCT
ap-7583	554	6	simply	simply	ADV
ap-7583	554	7	in	in	ADP
ap-7583	554	8	the	the	DET
ap-7583	554	9	standard	standard	ADJ
ap-7583	554	10	form	form	NOUN
ap-7583	554	11	as	as	ADP
ap-7583	554	12	y′′(r	y′′(r	NOUN
ap-7583	554	13	)	)	PUNCT
ap-7583	555	1	+	+	CCONJ
ap-7583	555	2	(	(	PUNCT
ap-7583	555	3	γ	γ	X
ap-7583	555	4	r	r	NOUN
ap-7583	555	5	+	+	NUM
ap-7583	555	6	δ	δ	PROPN
ap-7583	555	7	r	r	NOUN
ap-7583	555	8	−	−	PROPN
ap-7583	555	9	1	1	NUM
ap-7583	555	10	+	+	NUM
ap-7583	555	11	ε	ε	PROPN
ap-7583	555	12	r	r	NOUN
ap-7583	555	13	−	−	PROPN
ap-7583	555	14	b	b	PROPN
ap-7583	555	15	)	)	PUNCT
ap-7583	555	16	y′(r	y′(r	NOUN
ap-7583	555	17	)	)	PUNCT
ap-7583	556	1	+	+	CCONJ
ap-7583	556	2	α	α	NUM
ap-7583	556	3	β	β	NOUN
ap-7583	556	4	r	r	NOUN
ap-7583	556	5	−	−	NOUN
ap-7583	556	6	q	q	NOUN
ap-7583	556	7	r(r	r(r	NOUN
ap-7583	556	8	−	−	PROPN
ap-7583	556	9	1)(r	1)(r	NUM
ap-7583	556	10	−	−	PROPN
ap-7583	556	11	b)y(r	b)y(r	NOUN
ap-7583	556	12	)	)	PUNCT
ap-7583	556	13	=	=	SYM
ap-7583	556	14	0	0	NUM
ap-7583	556	15	,	,	PUNCT
ap-7583	556	16	(	(	PUNCT
ap-7583	556	17	95	95	NUM
ap-7583	556	18	)	)	PUNCT
ap-7583	556	19	where	where	SCONJ
ap-7583	556	20	γ	γ	X
ap-7583	556	21	δ	δ	PROPN
ap-7583	556	22	ε	ε	PROPN
ap-7583	556	23	α	α	PROPN
ap-7583	556	24	β	β	X
ap-7583	556	25	q	q	X
ap-7583	556	26	b	b	NOUN
ap-7583	556	27	⇓	⇓	PROPN
ap-7583	556	28	⇓	⇓	PROPN
ap-7583	556	29	⇓	⇓	PROPN
ap-7583	556	30	⇓	⇓	PROPN
ap-7583	556	31	⇓	⇓	PROPN
ap-7583	556	32	⇓	⇓	PROPN
ap-7583	556	33	β0	β0	PROPN
ap-7583	556	34	α1	α1	PROPN
ap-7583	556	35	β2	β2	PROPN
ap-7583	556	36	+	+	CCONJ
ap-7583	556	37	β1	β1	PROPN
ap-7583	557	1	+	+	CCONJ
ap-7583	557	2	β0	β0	PROPN
ap-7583	557	3	α3	α3	PROPN
ap-7583	558	1	−	−	PROPN
ap-7583	558	2	α1	α1	PROPN
ap-7583	558	3	β2α2	β2α2	PUNCT
ap-7583	558	4	1	1	NUM
ap-7583	559	1	+	+	CCONJ
ap-7583	559	2	β1α1α3	β1α1α3	NOUN
ap-7583	560	1	+	+	X
ap-7583	560	2	β0α2	β0α2	CCONJ
ap-7583	560	3	3	3	NUM
ap-7583	560	4	α3α1(α1	α3α1(α1	NOUN
ap-7583	560	5	−	−	PROPN
ap-7583	560	6	α3	α3	NOUN
ap-7583	560	7	)	)	PUNCT
ap-7583	560	8	β2	β2	NOUN
ap-7583	560	9	+	+	CCONJ
ap-7583	560	10	(	(	PUNCT
ap-7583	560	11	n	n	CCONJ
ap-7583	560	12	−	−	PROPN
ap-7583	560	13	1)α3	1)α3	NUM
ap-7583	560	14	α3	α3	NOUN
ap-7583	560	15	−n	−n	NOUN
ap-7583	560	16	−	−	PROPN
ap-7583	560	17	ε0	ε0	PROPN
ap-7583	560	18	α3	α3	PROPN
ap-7583	560	19	α1	α1	PROPN
ap-7583	560	20	α3	α3	PROPN
ap-7583	560	21	⇑	⇑	PROPN
ap-7583	560	22	⇑	⇑	PROPN
ap-7583	560	23	⇑	⇑	PROPN
ap-7583	560	24	⇑	⇑	PROPN
ap-7583	560	25	⇑	⇑	PROPN
ap-7583	560	26	⇑	⇑	PROPN
ap-7583	560	27	γ	γ	PROPN
ap-7583	560	28	δ	δ	PROPN
ap-7583	560	29	ε	ε	PROPN
ap-7583	560	30	β	β	X
ap-7583	560	31	α	α	PROPN
ap-7583	560	32	q	q	PROPN
ap-7583	560	33	b	b	PROPN
ap-7583	560	34	where	where	SCONJ
ap-7583	560	35	,	,	PUNCT
ap-7583	560	36	in	in	ADP
ap-7583	560	37	either	either	DET
ap-7583	560	38	case	case	NOUN
ap-7583	560	39	,	,	PUNCT
ap-7583	560	40	it	it	PRON
ap-7583	560	41	follows	follow	VERB
ap-7583	560	42	γ	γ	PROPN
ap-7583	560	43	+	+	PROPN
ap-7583	560	44	δ	δ	PROPN
ap-7583	560	45	+	+	CCONJ
ap-7583	560	46	ε	ε	PROPN
ap-7583	560	47	=	=	SYM
ap-7583	560	48	α	α	PROPN
ap-7583	561	1	+	+	X
ap-7583	561	2	β	β	X
ap-7583	561	3	+	+	ADP
ap-7583	561	4	1	1	NUM
ap-7583	561	5	that	that	PRON
ap-7583	561	6	ensures	ensure	VERB
ap-7583	561	7	the	the	DET
ap-7583	561	8	regularity	regularity	NOUN
ap-7583	561	9	of	of	ADP
ap-7583	561	10	the	the	DET
ap-7583	561	11	singular	singular	ADJ
ap-7583	561	12	point	point	NOUN
ap-7583	561	13	∞.	∞.	PROPN
ap-7583	561	14	with	with	ADP
ap-7583	561	15	these	these	DET
ap-7583	561	16	parameters	parameter	NOUN
ap-7583	561	17	,	,	PUNCT
ap-7583	561	18	the	the	DET
ap-7583	561	19	sturm	sturm	NOUN
ap-7583	561	20	-	-	PUNCT
ap-7583	561	21	liouville	liouville	NOUN
ap-7583	561	22	form	form	NOUN
ap-7583	561	23	of	of	ADP
ap-7583	561	24	the	the	DET
ap-7583	561	25	differential	differential	ADJ
ap-7583	561	26	equation	equation	NOUN
ap-7583	561	27	(	(	PUNCT
ap-7583	561	28	65	65	NUM
ap-7583	561	29	)	)	PUNCT
ap-7583	561	30	is	be	AUX
ap-7583	561	31	−	−	PROPN
ap-7583	561	32	d	d	X
ap-7583	561	33	dr	dr	PROPN
ap-7583	561	34	(	(	PUNCT
ap-7583	561	35	rγ	rγ	PROPN
ap-7583	561	36	(	(	PUNCT
ap-7583	561	37	r	r	NOUN
ap-7583	561	38	−	−	PROPN
ap-7583	561	39	1)δ	1)δ	NUM
ap-7583	561	40	(	(	PUNCT
ap-7583	561	41	r	r	NOUN
ap-7583	561	42	−	−	PROPN
ap-7583	561	43	b)ε	b)ε	NOUN
ap-7583	561	44	dy	dy	PROPN
ap-7583	561	45	dr	dr	PROPN
ap-7583	561	46	)	)	PUNCT
ap-7583	562	1	+	+	CCONJ
ap-7583	562	2	α	α	X
ap-7583	562	3	β	β	X
ap-7583	562	4	rγ	rγ	X
ap-7583	562	5	(	(	PUNCT
ap-7583	563	1	r	r	NOUN
ap-7583	563	2	−	−	PROPN
ap-7583	563	3	1)δ−1(r	1)δ−1(r	NUM
ap-7583	563	4	−	−	PROPN
ap-7583	563	5	b)ε−1y	b)ε−1y	NOUN
ap-7583	563	6	=	=	PUNCT
ap-7583	563	7	q	q	PUNCT
ap-7583	563	8	rγ−1(r	rγ−1(r	ADJ
ap-7583	563	9	−	−	PROPN
ap-7583	563	10	1)δ−1(r	1)δ−1(r	NUM
ap-7583	563	11	−	−	PROPN
ap-7583	563	12	b)ε−1	b)ε−1	PROPN
ap-7583	563	13	y	y	PROPN
ap-7583	563	14	(	(	PUNCT
ap-7583	563	15	96	96	NUM
ap-7583	563	16	)	)	PUNCT
ap-7583	563	17	where	where	SCONJ
ap-7583	563	18	,	,	PUNCT
ap-7583	563	19	for	for	ADP
ap-7583	563	20	b	b	PROPN
ap-7583	563	21	≥	≥	NUM
ap-7583	563	22	1	1	NUM
ap-7583	563	23	,	,	PUNCT
ap-7583	563	24	γ	γ	X
ap-7583	563	25	≥	≥	NOUN
ap-7583	563	26	0	0	NUM
ap-7583	563	27	,	,	PUNCT
ap-7583	563	28	δ	δ	PROPN
ap-7583	563	29	≥	≥	NUM
ap-7583	563	30	1	1	NUM
ap-7583	563	31	,	,	PUNCT
ap-7583	563	32	r	r	NOUN
ap-7583	563	33	∈	∈	PROPN
ap-7583	563	34	(	(	PUNCT
ap-7583	563	35	0	0	NUM
ap-7583	563	36	,	,	PUNCT
ap-7583	563	37	1	1	NUM
ap-7583	563	38	)	)	PUNCT
ap-7583	563	39	.	.	PUNCT
ap-7583	564	1	corollary	corollary	ADJ
ap-7583	564	2	4.9	4.9	NUM
ap-7583	564	3	.	.	PUNCT
ap-7583	565	1	the	the	DET
ap-7583	565	2	second	second	ADJ
ap-7583	565	3	-	-	PUNCT
ap-7583	565	4	order	order	NOUN
ap-7583	565	5	linear	linear	NOUN
ap-7583	565	6	differential	differential	NOUN
ap-7583	565	7	equation	equation	NOUN
ap-7583	565	8	r2(α3	r2(α3	VERB
ap-7583	565	9	r	r	NOUN
ap-7583	565	10	+	+	NOUN
ap-7583	565	11	α2)y′′(r	α2)y′′(r	NUM
ap-7583	565	12	)	)	PUNCT
ap-7583	566	1	+	+	CCONJ
ap-7583	566	2	r	r	NOUN
ap-7583	566	3	(	(	PUNCT
ap-7583	566	4	β2	β2	NOUN
ap-7583	566	5	r	r	NOUN
ap-7583	566	6	+	+	CCONJ
ap-7583	566	7	β1	β1	PROPN
ap-7583	566	8	)	)	PUNCT
ap-7583	566	9	y′	y′	PUNCT
ap-7583	567	1	+	+	CCONJ
ap-7583	567	2	(	(	PUNCT
ap-7583	567	3	−(n(n	−(n(n	PROPN
ap-7583	567	4	−	−	PROPN
ap-7583	567	5	1	1	NUM
ap-7583	567	6	)	)	PUNCT
ap-7583	567	7	α3	α3	NOUN
ap-7583	567	8	+	+	CCONJ
ap-7583	567	9	n	n	CCONJ
ap-7583	567	10	β2	β2	ADJ
ap-7583	567	11	)	)	PUNCT
ap-7583	568	1	r	r	NOUN
ap-7583	568	2	+	+	CCONJ
ap-7583	568	3	ε0	ε0	PROPN
ap-7583	568	4	)	)	PUNCT
ap-7583	568	5	y	y	PROPN
ap-7583	568	6	=	=	SYM
ap-7583	568	7	0	0	PROPN
ap-7583	568	8	,	,	PUNCT
ap-7583	568	9	(	(	PUNCT
ap-7583	568	10	97	97	NUM
ap-7583	568	11	)	)	PUNCT
ap-7583	568	12	where	where	SCONJ
ap-7583	568	13	r	r	NOUN
ap-7583	568	14	∈	∈	PROPN
ap-7583	568	15	(	(	PUNCT
ap-7583	568	16	−α2	−α2	PROPN
ap-7583	568	17	/	/	SYM
ap-7583	568	18	α3	α3	PROPN
ap-7583	568	19	,	,	PUNCT
ap-7583	568	20	0	0	NUM
ap-7583	568	21	)	)	PUNCT
ap-7583	568	22	if	if	SCONJ
ap-7583	568	23	α2α3	α2α3	ADP
ap-7583	568	24	>	>	X
ap-7583	568	25	0	0	NUM
ap-7583	568	26	or	or	CCONJ
ap-7583	568	27	r	r	NOUN
ap-7583	568	28	∈	∈	PROPN
ap-7583	568	29	(	(	PUNCT
ap-7583	568	30	0	0	NUM
ap-7583	568	31	,	,	PUNCT
ap-7583	568	32	α2	α2	ADJ
ap-7583	568	33	/	/	SYM
ap-7583	568	34	α3	α3	NOUN
ap-7583	568	35	)	)	PUNCT
ap-7583	569	1	if	if	SCONJ
ap-7583	569	2	α2α3	α2α3	ADP
ap-7583	569	3	<	<	X
ap-7583	569	4	0	0	NUM
ap-7583	569	5	,	,	PUNCT
ap-7583	569	6	has	have	VERB
ap-7583	569	7	a	a	DET
ap-7583	569	8	polynomial	polynomial	ADJ
ap-7583	569	9	solution	solution	NOUN
ap-7583	569	10	of	of	ADP
ap-7583	569	11	degree	degree	NOUN
ap-7583	569	12	n	n	CCONJ
ap-7583	569	13	subject	subject	NOUN
ap-7583	569	14	to	to	ADP
ap-7583	569	15			PROPN
ap-7583	569	16	n∏	n∏	PROPN
ap-7583	569	17	k=0	k=0	PROPN
ap-7583	569	18	(	(	PUNCT
ap-7583	569	19	ε0	ε0	PROPN
ap-7583	569	20	+	+	PROPN
ap-7583	569	21	k((k	k((k	NOUN
ap-7583	569	22	−	−	PROPN
ap-7583	570	1	1)α2	1)α2	PROPN
ap-7583	570	2	+	+	CCONJ
ap-7583	570	3	β1	β1	NOUN
ap-7583	570	4	)	)	PUNCT
ap-7583	570	5	=	=	SYM
ap-7583	570	6	0	0	PUNCT
ap-7583	571	1	=	=	AUX
ap-7583	571	2	⇒	⇒	X
ap-7583	571	3	ε0	ε0	NOUN
ap-7583	571	4	=	=	SYM
ap-7583	571	5	−n	−n	PROPN
ap-7583	571	6	(	(	PUNCT
ap-7583	571	7	n	n	CCONJ
ap-7583	571	8	−	−	PROPN
ap-7583	571	9	1	1	X
ap-7583	571	10	)	)	PUNCT
ap-7583	571	11	α2	α2	ADJ
ap-7583	571	12	−	−	PROPN
ap-7583	571	13	n	n	PRON
ap-7583	571	14	β1	β1	PROPN
ap-7583	571	15	,	,	PUNCT
ap-7583	571	16	n	n	NOUN
ap-7583	571	17	=	=	SYM
ap-7583	571	18	0	0	NUM
ap-7583	571	19	,	,	PUNCT
ap-7583	571	20	1	1	NUM
ap-7583	571	21	,	,	PUNCT
ap-7583	571	22	2	2	NUM
ap-7583	571	23	,	,	PUNCT
ap-7583	571	24	·	·	PUNCT
ap-7583	571	25	·	·	PUNCT
ap-7583	571	26	·	·	PUNCT
ap-7583	571	27	.	.	PUNCT
ap-7583	572	1	(	(	PUNCT
ap-7583	572	2	98	98	NUM
ap-7583	572	3	)	)	PUNCT
ap-7583	572	4	182	182	NUM
ap-7583	572	5	vol	vol	NOUN
ap-7583	572	6	.	.	PUNCT
ap-7583	573	1	62	62	NUM
ap-7583	573	2	no	no	INTJ
ap-7583	573	3	.	.	PUNCT
ap-7583	574	1	1/2022	1/2022	NUM
ap-7583	574	2	on	on	ADP
ap-7583	574	3	generalized	generalized	ADJ
ap-7583	574	4	heun	heun	NOUN
ap-7583	574	5	equation	equation	NOUN
ap-7583	574	6	with	with	ADP
ap-7583	574	7	some	some	DET
ap-7583	574	8	mathematical	mathematical	NOUN
ap-7583	574	9	.	.	PUNCT
ap-7583	574	10	.	.	PUNCT
ap-7583	574	11	.	.	PUNCT
ap-7583	575	1	in	in	ADP
ap-7583	575	2	particular	particular	ADJ
ap-7583	575	3	,	,	PUNCT
ap-7583	575	4	the	the	DET
ap-7583	575	5	differential	differential	ADJ
ap-7583	575	6	equation	equation	NOUN
ap-7583	575	7	r2(α3r	r2(α3r	VERB
ap-7583	575	8	+	+	CCONJ
ap-7583	576	1	α2	α2	ADJ
ap-7583	576	2	)	)	PUNCT
ap-7583	576	3	y′′(r	y′′(r	NOUN
ap-7583	576	4	)	)	PUNCT
ap-7583	577	1	+	+	NUM
ap-7583	577	2	r(β2	r(β2	NOUN
ap-7583	577	3	r	r	NOUN
ap-7583	577	4	+	+	CCONJ
ap-7583	577	5	β1	β1	PROPN
ap-7583	577	6	)	)	PUNCT
ap-7583	577	7	y′(r	y′(r	NOUN
ap-7583	577	8	)	)	PUNCT
ap-7583	577	9	−	−	PROPN
ap-7583	578	1	(	(	PUNCT
ap-7583	578	2	(	(	PUNCT
ap-7583	578	3	n	n	X
ap-7583	578	4	(	(	PUNCT
ap-7583	578	5	n	n	CCONJ
ap-7583	578	6	−	−	PROPN
ap-7583	578	7	1	1	NUM
ap-7583	578	8	)	)	PUNCT
ap-7583	578	9	α3	α3	NOUN
ap-7583	578	10	+	+	CCONJ
ap-7583	578	11	n	n	CCONJ
ap-7583	578	12	β2	β2	ADJ
ap-7583	578	13	)	)	PUNCT
ap-7583	578	14	r	r	NOUN
ap-7583	578	15	+	+	NUM
ap-7583	578	16	n	n	CCONJ
ap-7583	578	17	(	(	PUNCT
ap-7583	578	18	n	n	CCONJ
ap-7583	578	19	−	−	PROPN
ap-7583	578	20	1	1	NUM
ap-7583	578	21	)	)	PUNCT
ap-7583	578	22	α2	α2	PROPN
ap-7583	578	23	+	+	CCONJ
ap-7583	578	24	n	n	CCONJ
ap-7583	578	25	β1	β1	PROPN
ap-7583	578	26	)	)	PUNCT
ap-7583	578	27	y(r	y(r	PROPN
ap-7583	578	28	)	)	PUNCT
ap-7583	578	29	=	=	SYM
ap-7583	578	30	0	0	NUM
ap-7583	578	31	,	,	PUNCT
ap-7583	578	32	(	(	PUNCT
ap-7583	578	33	99	99	NUM
ap-7583	578	34	)	)	PUNCT
ap-7583	578	35	has	have	VERB
ap-7583	578	36	the	the	DET
ap-7583	578	37	polynomial	polynomial	ADJ
ap-7583	578	38	solutions	solution	NOUN
ap-7583	578	39	yn(r	yn(r	NUM
ap-7583	578	40	)	)	PUNCT
ap-7583	578	41	=	=	SYM
ap-7583	578	42	rn	rn	PROPN
ap-7583	578	43	,	,	PUNCT
ap-7583	578	44	n	n	NOUN
ap-7583	578	45	=	=	SYM
ap-7583	578	46	0	0	NUM
ap-7583	578	47	,	,	PUNCT
ap-7583	578	48	1	1	NUM
ap-7583	578	49	,	,	PUNCT
ap-7583	578	50	2	2	NUM
ap-7583	578	51	,	,	PUNCT
ap-7583	578	52	.	.	PUNCT
ap-7583	578	53	.	.	PUNCT
ap-7583	578	54	.	.	PUNCT
ap-7583	578	55	.	.	PUNCT
ap-7583	579	1	(	(	PUNCT
ap-7583	579	2	100	100	NUM
ap-7583	579	3	)	)	PUNCT
ap-7583	579	4	proof	proof	NOUN
ap-7583	579	5	.	.	PUNCT
ap-7583	579	6	follows	follow	VERB
ap-7583	579	7	immediately	immediately	ADV
ap-7583	579	8	from	from	ADP
ap-7583	579	9	theorem	theorem	ADJ
ap-7583	579	10	4.7	4.7	NUM
ap-7583	579	11	with	with	ADP
ap-7583	579	12	α1	α1	PROPN
ap-7583	579	13	=	=	SYM
ap-7583	579	14	β0	β0	PROPN
ap-7583	579	15	=	=	NOUN
ap-7583	579	16	0	0	NUM
ap-7583	579	17	.	.	NOUN
ap-7583	579	18	5	5	NUM
ap-7583	579	19	.	.	NOUN
ap-7583	579	20	mathematical	mathematical	ADJ
ap-7583	579	21	properties	property	NOUN
ap-7583	579	22	of	of	ADP
ap-7583	579	23	the	the	DET
ap-7583	579	24	orthogonal	orthogonal	ADJ
ap-7583	579	25	polynomials	polynomial	NOUN
ap-7583	579	26	{	{	PUNCT
ap-7583	579	27	pk(ε0)}∞	pk(ε0)}∞	NOUN
ap-7583	579	28	k=0	k=0	PROPN
ap-7583	579	29	as	as	SCONJ
ap-7583	579	30	pointed	point	VERB
ap-7583	579	31	out	out	ADP
ap-7583	579	32	by	by	ADP
ap-7583	579	33	theorem	theorem	NOUN
ap-7583	579	34	4.1	4.1	NUM
ap-7583	579	35	,	,	PUNCT
ap-7583	579	36	in	in	ADP
ap-7583	579	37	the	the	DET
ap-7583	579	38	neighbourhood	neighbourhood	NOUN
ap-7583	579	39	of	of	ADP
ap-7583	579	40	the	the	DET
ap-7583	579	41	singular	singular	ADJ
ap-7583	579	42	point	point	NOUN
ap-7583	579	43	r	r	NOUN
ap-7583	579	44	=	=	NOUN
ap-7583	579	45	0	0	NUM
ap-7583	579	46	with	with	SCONJ
ap-7583	579	47	an	an	DET
ap-7583	579	48	indicial	indicial	ADJ
ap-7583	579	49	exponent	exponent	NOUN
ap-7583	579	50	root	root	NOUN
ap-7583	579	51	zero	zero	NUM
ap-7583	579	52	,	,	PUNCT
ap-7583	579	53	the	the	DET
ap-7583	579	54	series	series	NOUN
ap-7583	579	55	solution	solution	NOUN
ap-7583	579	56	of	of	ADP
ap-7583	579	57	the	the	DET
ap-7583	579	58	differential	differential	ADJ
ap-7583	579	59	equation	equation	NOUN
ap-7583	579	60	with	with	ADP
ap-7583	579	61	four	four	NUM
ap-7583	579	62	singular	singular	ADJ
ap-7583	579	63	points	point	NOUN
ap-7583	579	64	,	,	PUNCT
ap-7583	579	65	see	see	VERB
ap-7583	579	66	(	(	PUNCT
ap-7583	579	67	65	65	NUM
ap-7583	579	68	)	)	PUNCT
ap-7583	579	69	,	,	PUNCT
ap-7583	579	70	r	r	NOUN
ap-7583	579	71	(	(	PUNCT
ap-7583	579	72	α1	α1	PROPN
ap-7583	579	73	+	+	CCONJ
ap-7583	579	74	α2	α2	ADJ
ap-7583	579	75	r	r	NOUN
ap-7583	579	76	+	+	NOUN
ap-7583	579	77	α3r2	α3r2	X
ap-7583	579	78	)	)	PUNCT
ap-7583	579	79	y′′	y′′	NOUN
ap-7583	579	80	+	+	CCONJ
ap-7583	579	81	(	(	PUNCT
ap-7583	579	82	β0	β0	NOUN
ap-7583	579	83	+	+	CCONJ
ap-7583	579	84	β1	β1	PROPN
ap-7583	579	85	r	r	NOUN
ap-7583	579	86	+	+	CCONJ
ap-7583	579	87	β2	β2	NOUN
ap-7583	579	88	r2	r2	NOUN
ap-7583	579	89	)	)	PUNCT
ap-7583	579	90	y′	y′	PUNCT
ap-7583	580	1	+	+	CCONJ
ap-7583	580	2	(	(	PUNCT
ap-7583	580	3	ε0	ε0	PROPN
ap-7583	580	4	+	+	CCONJ
ap-7583	580	5	ε1	ε1	PROPN
ap-7583	580	6	r	r	NOUN
ap-7583	580	7	)	)	PUNCT
ap-7583	580	8	y	y	PROPN
ap-7583	580	9	=	=	NOUN
ap-7583	580	10	0	0	PROPN
ap-7583	580	11	.	.	PUNCT
ap-7583	581	1	can	can	AUX
ap-7583	581	2	be	be	AUX
ap-7583	581	3	written	write	VERB
ap-7583	581	4	as	as	ADP
ap-7583	581	5	y(r	y(r	NOUN
ap-7583	581	6	)	)	PUNCT
ap-7583	581	7	=	=	PUNCT
ap-7583	582	1	∞∑	∞∑	NUM
ap-7583	582	2	k=0	k=0	PROPN
ap-7583	582	3	(	(	PUNCT
ap-7583	582	4	−1)k	−1)k	NOUN
ap-7583	582	5	pk(ε0	pk(ε0	NOUN
ap-7583	582	6	)	)	PUNCT
ap-7583	582	7	k	k	X
ap-7583	582	8	!	!	PUNCT
ap-7583	583	1	αk	αk	CCONJ
ap-7583	583	2	1	1	NUM
ap-7583	583	3	(	(	PUNCT
ap-7583	583	4	β0	β0	PROPN
ap-7583	583	5	α1	α1	PROPN
ap-7583	583	6	)	)	PUNCT
ap-7583	584	1	k	k	PROPN
ap-7583	584	2	rk	rk	NOUN
ap-7583	584	3	,	,	PUNCT
ap-7583	584	4	(	(	PUNCT
ap-7583	584	5	101	101	NUM
ap-7583	584	6	)	)	PUNCT
ap-7583	584	7	where	where	SCONJ
ap-7583	584	8	the	the	DET
ap-7583	584	9	infinite	infinite	ADJ
ap-7583	584	10	sequence	sequence	NOUN
ap-7583	584	11	of	of	ADP
ap-7583	584	12	polynomials	polynomial	NOUN
ap-7583	584	13	{	{	PUNCT
ap-7583	584	14	pk(ε0)}∞	pk(ε0)}∞	NOUN
ap-7583	584	15	k=0	k=0	PROPN
ap-7583	584	16	in	in	ADP
ap-7583	584	17	the	the	DET
ap-7583	584	18	real	real	ADJ
ap-7583	584	19	variable	variable	ADJ
ap-7583	584	20	ε0	ε0	NOUN
ap-7583	584	21	satisfies	satisfy	VERB
ap-7583	584	22	the	the	DET
ap-7583	584	23	three	three	NUM
ap-7583	584	24	-	-	PUNCT
ap-7583	584	25	term	term	NOUN
ap-7583	584	26	recurrence	recurrence	NOUN
ap-7583	584	27	relation	relation	NOUN
ap-7583	584	28	pk+1(ε0	pk+1(ε0	NOUN
ap-7583	584	29	)	)	PUNCT
ap-7583	584	30	=	=	SYM
ap-7583	584	31	(	(	PUNCT
ap-7583	584	32	ε0	ε0	PROPN
ap-7583	584	33	−	−	PROPN
ap-7583	584	34	ak)pk(ε0	ak)pk(ε0	ADJ
ap-7583	584	35	)	)	PUNCT
ap-7583	584	36	−	−	PROPN
ap-7583	584	37	bkpk−1(ε0	bkpk−1(ε0	NOUN
ap-7583	584	38	)	)	PUNCT
ap-7583	584	39	,	,	PUNCT
ap-7583	584	40	(	(	PUNCT
ap-7583	584	41	102	102	NUM
ap-7583	584	42	)	)	PUNCT
ap-7583	584	43	initiated	initiate	VERB
ap-7583	584	44	with	with	ADP
ap-7583	584	45	p−1(ε0	p−1(ε0	NOUN
ap-7583	584	46	)	)	PUNCT
ap-7583	584	47	=	=	SYM
ap-7583	584	48	0	0	NUM
ap-7583	584	49	,	,	PUNCT
ap-7583	584	50	p0(ε0	p0(ε0	NOUN
ap-7583	584	51	)	)	PUNCT
ap-7583	584	52	=	=	SYM
ap-7583	584	53	1	1	NUM
ap-7583	584	54	,	,	PUNCT
ap-7583	584	55	k	k	NOUN
ap-7583	584	56	=	=	SYM
ap-7583	584	57	1	1	NUM
ap-7583	584	58	,	,	PUNCT
ap-7583	584	59	2	2	NUM
ap-7583	584	60	,	,	PUNCT
ap-7583	584	61	3	3	NUM
ap-7583	584	62	,	,	PUNCT
ap-7583	584	63	·	·	PUNCT
ap-7583	584	64	·	·	PUNCT
ap-7583	584	65	·	·	PUNCT
ap-7583	584	66	.	.	PUNCT
ap-7583	585	1	where	where	SCONJ
ap-7583	585	2	ak	ak	PROPN
ap-7583	585	3	=	=	PROPN
ap-7583	585	4	−k(k	−k(k	PROPN
ap-7583	585	5	−	−	PROPN
ap-7583	586	1	1)α2	1)α2	PRON
ap-7583	586	2	−	−	PROPN
ap-7583	586	3	kβ1	kβ1	NOUN
ap-7583	586	4	,	,	PUNCT
ap-7583	586	5	bk	bk	X
ap-7583	586	6	=	=	SYM
ap-7583	586	7	k	k	X
ap-7583	586	8	(	(	PUNCT
ap-7583	586	9	(	(	PUNCT
ap-7583	586	10	k	k	X
ap-7583	586	11	−	−	PROPN
ap-7583	587	1	1)α1	1)α1	NUM
ap-7583	588	1	+	+	CCONJ
ap-7583	588	2	β0	β0	NOUN
ap-7583	588	3	)	)	PUNCT
ap-7583	588	4	(	(	PUNCT
ap-7583	588	5	(	(	PUNCT
ap-7583	588	6	k	k	X
ap-7583	588	7	−	−	PROPN
ap-7583	588	8	1)((k	1)((k	NOUN
ap-7583	588	9	−	−	PROPN
ap-7583	589	1	2)α3	2)α3	PROPN
ap-7583	589	2	+	+	CCONJ
ap-7583	589	3	β2	β2	VERB
ap-7583	589	4	)	)	PUNCT
ap-7583	589	5	+	+	NUM
ap-7583	589	6	ε1	ε1	PROPN
ap-7583	589	7	)	)	PUNCT
ap-7583	589	8	.	.	PUNCT
ap-7583	590	1	for	for	ADP
ap-7583	590	2	ak	ak	PROPN
ap-7583	590	3	,	,	PUNCT
ap-7583	590	4	bk	bk	ADP
ap-7583	590	5	∈	∈	NOUN
ap-7583	590	6	r	r	NOUN
ap-7583	590	7	and	and	CCONJ
ap-7583	590	8	if	if	SCONJ
ap-7583	590	9	bk	bk	NOUN
ap-7583	590	10	>	>	X
ap-7583	590	11	0	0	NUM
ap-7583	590	12	,	,	PUNCT
ap-7583	590	13	then	then	ADV
ap-7583	590	14	according	accord	VERB
ap-7583	590	15	to	to	ADP
ap-7583	590	16	favard	favard	PROPN
ap-7583	590	17	theorem	theorem	NOUN
ap-7583	590	18	[	[	X
ap-7583	590	19	25	25	NUM
ap-7583	590	20	]	]	PUNCT
ap-7583	590	21	,	,	PUNCT
ap-7583	590	22	see	see	VERB
ap-7583	590	23	also	also	ADV
ap-7583	590	24	[	[	X
ap-7583	590	25	26	26	NUM
ap-7583	590	26	,	,	PUNCT
ap-7583	590	27	theorem	theorem	VERB
ap-7583	590	28	2.14	2.14	NUM
ap-7583	590	29	]	]	PUNCT
ap-7583	590	30	,	,	PUNCT
ap-7583	590	31	there	there	PRON
ap-7583	590	32	exists	exist	VERB
ap-7583	590	33	a	a	DET
ap-7583	590	34	positive	positive	ADJ
ap-7583	590	35	borel	borel	NOUN
ap-7583	590	36	measure	measure	NOUN
ap-7583	590	37	µ	µ	PRON
ap-7583	590	38	such	such	ADJ
ap-7583	590	39	that	that	SCONJ
ap-7583	590	40	{	{	PUNCT
ap-7583	590	41	pk}∞	pk}∞	NOUN
ap-7583	590	42	k=0	k=0	PROPN
ap-7583	590	43	is	be	AUX
ap-7583	590	44	orthogonal	orthogonal	ADJ
ap-7583	590	45	with	with	ADP
ap-7583	590	46	respect	respect	NOUN
ap-7583	590	47	to	to	ADP
ap-7583	590	48	the	the	DET
ap-7583	590	49	inner	inner	ADJ
ap-7583	590	50	product	product	NOUN
ap-7583	590	51	⟨pk	⟨pk	PROPN
ap-7583	590	52	,	,	PUNCT
ap-7583	590	53	pk′⟩	pk′⟩	ADV
ap-7583	590	54	=	=	SYM
ap-7583	590	55	∫	∫	PROPN
ap-7583	590	56	r	r	PROPN
ap-7583	590	57	pk(ε0)pp′(ε0)dµ	pk(ε0)pp′(ε0)dµ	NOUN
ap-7583	590	58	(	(	PUNCT
ap-7583	590	59	103	103	NUM
ap-7583	590	60	)	)	PUNCT
ap-7583	590	61	such	such	ADJ
ap-7583	590	62	that	that	DET
ap-7583	590	63	∫	∫	PROPN
ap-7583	590	64	r	r	NOUN
ap-7583	590	65	pk(ε0)pk′(ε0)dµ	pk(ε0)pk′(ε0)dµ	PROPN
ap-7583	590	66	=	=	SYM
ap-7583	590	67	pkpk′δkk′	pkpk′δkk′	PROPN
ap-7583	590	68	,	,	PUNCT
ap-7583	590	69	∫	∫	PROPN
ap-7583	590	70	r	r	NOUN
ap-7583	590	71	dµ	dµ	PROPN
ap-7583	590	72	=	=	SYM
ap-7583	590	73	1	1	NUM
ap-7583	590	74	,	,	PUNCT
ap-7583	590	75	(	(	PUNCT
ap-7583	590	76	104	104	NUM
ap-7583	590	77	)	)	PUNCT
ap-7583	590	78	where	where	SCONJ
ap-7583	590	79	δkk′	δkk′	PROPN
ap-7583	590	80	is	be	AUX
ap-7583	590	81	the	the	DET
ap-7583	590	82	kronecker	kronecker	NOUN
ap-7583	590	83	symbol	symbol	NOUN
ap-7583	590	84	.	.	PUNCT
ap-7583	591	1	in	in	ADP
ap-7583	591	2	particular,∫	particular,∫	NOUN
ap-7583	591	3	r	r	NOUN
ap-7583	591	4	εk	εk	NOUN
ap-7583	591	5	0pk′(ε0)dµ	0pk′(ε0)dµ	PROPN
ap-7583	591	6	=	=	NOUN
ap-7583	591	7	0	0	NUM
ap-7583	591	8	for	for	ADP
ap-7583	591	9	all	all	PRON
ap-7583	591	10	0	0	NUM
ap-7583	591	11	<	<	X
ap-7583	591	12	k	k	X
ap-7583	591	13	<	<	X
ap-7583	591	14	k′.	k′.	X
ap-7583	591	15	(	(	PUNCT
ap-7583	591	16	105	105	NUM
ap-7583	591	17	)	)	PUNCT
ap-7583	591	18	the	the	DET
ap-7583	591	19	norm	norm	NOUN
ap-7583	591	20	pk	pk	NOUN
ap-7583	591	21	can	can	AUX
ap-7583	591	22	be	be	AUX
ap-7583	591	23	found	find	VERB
ap-7583	591	24	using	use	VERB
ap-7583	591	25	the	the	DET
ap-7583	591	26	recurrence	recurrence	NOUN
ap-7583	591	27	relations	relation	NOUN
ap-7583	591	28	(	(	PUNCT
ap-7583	591	29	102	102	NUM
ap-7583	591	30	)	)	PUNCT
ap-7583	591	31	by	by	ADP
ap-7583	591	32	multiplying	multiply	VERB
ap-7583	591	33	with	with	ADP
ap-7583	591	34	εk−1	εk−1	PROPN
ap-7583	591	35	0	0	PUNCT
ap-7583	591	36	and	and	CCONJ
ap-7583	591	37	taking	take	VERB
ap-7583	591	38	the	the	DET
ap-7583	591	39	integral	integral	ADJ
ap-7583	591	40	over	over	ADP
ap-7583	591	41	ε0	ε0	NOUN
ap-7583	591	42	with	with	ADP
ap-7583	591	43	respect	respect	NOUN
ap-7583	591	44	to	to	ADP
ap-7583	591	45	µ	µ	PROPN
ap-7583	591	46	that	that	SCONJ
ap-7583	591	47	yields	yield	NOUN
ap-7583	591	48	∫	∫	PROPN
ap-7583	591	49	r	r	PROPN
ap-7583	591	50	εk	εk	PROPN
ap-7583	591	51	0pk(ε0)dµ	0pk(ε0)dµ	NOUN
ap-7583	591	52	=	=	PUNCT
ap-7583	592	1	bk	bk	ADP
ap-7583	592	2	∫	∫	PROPN
ap-7583	592	3	r	r	PROPN
ap-7583	592	4	εk−1	εk−1	PROPN
ap-7583	592	5	0	0	PUNCT
ap-7583	593	1	pk−1(ε0)dµ	pk−1(ε0)dµ	PROPN
ap-7583	593	2	=	=	PUNCT
ap-7583	593	3	bkbk−1	bkbk−1	NOUN
ap-7583	593	4	∫	∫	PROPN
ap-7583	593	5	r	r	NOUN
ap-7583	593	6	εk−2	εk−2	PROPN
ap-7583	593	7	0	0	NUM
ap-7583	593	8	pk−2(ε0)dµ	pk−2(ε0)dµ	PROPN
ap-7583	593	9	=	=	SYM
ap-7583	593	10	·	·	PUNCT
ap-7583	593	11	·	·	PUNCT
ap-7583	593	12	·	·	PUNCT
ap-7583	594	1	=	=	PUNCT
ap-7583	594	2			PROPN
ap-7583	594	3	k∏	k∏	PROPN
ap-7583	594	4	j=2	j=2	NOUN
ap-7583	594	5	bj	bj	VERB
ap-7583	594	6	∫	∫	VERB
ap-7583	594	7	r	r	NOUN
ap-7583	594	8	dµ	dµ	PROPN
ap-7583	594	9	(	(	PUNCT
ap-7583	594	10	106	106	NUM
ap-7583	594	11	)	)	PUNCT
ap-7583	594	12	183	183	NUM
ap-7583	594	13	nasser	nasser	PROPN
ap-7583	594	14	saad	saad	PROPN
ap-7583	594	15	acta	acta	PROPN
ap-7583	594	16	polytechnica	polytechnica	PROPN
ap-7583	594	17	and	and	CCONJ
ap-7583	594	18	∫	∫	PROPN
ap-7583	594	19	r	r	NOUN
ap-7583	594	20	pk(ε0)pk′(ε0	pk(ε0)pk′(ε0	NOUN
ap-7583	594	21	)	)	PUNCT
ap-7583	594	22	dµ	dµ	PROPN
ap-7583	595	1	=	=	SYM
ap-7583	595	2	k	k	X
ap-7583	595	3	!	!	PUNCT
ap-7583	596	1	(	(	PUNCT
ap-7583	596	2	α1	α1	PROPN
ap-7583	596	3	α3)k	α3)k	PROPN
ap-7583	596	4	β0ε1	β0ε1	PUNCT
ap-7583	596	5	(	(	PUNCT
ap-7583	596	6	β0	β0	PROPN
ap-7583	596	7	α1	α1	PROPN
ap-7583	596	8	)	)	PUNCT
ap-7583	597	1	k	k	PROPN
ap-7583	597	2	×	×	NOUN
ap-7583	597	3	(	(	PUNCT
ap-7583	597	4	−α3	−α3	PROPN
ap-7583	597	5	+	+	SYM
ap-7583	597	6	β2	β2	NOUN
ap-7583	597	7	−	−	PROPN
ap-7583	597	8	√	√	PROPN
ap-7583	597	9	(	(	PUNCT
ap-7583	597	10	α3	α3	NOUN
ap-7583	597	11	−	−	PROPN
ap-7583	597	12	β2)2	β2)2	PUNCT
ap-7583	597	13	−	−	PROPN
ap-7583	597	14	4α3ε1	4α3ε1	PROPN
ap-7583	597	15	2α3	2α3	NUM
ap-7583	597	16	)	)	PUNCT
ap-7583	598	1	k	k	X
ap-7583	598	2	×	×	NOUN
ap-7583	598	3	(	(	PUNCT
ap-7583	598	4	−α3	−α3	PROPN
ap-7583	598	5	+	+	CCONJ
ap-7583	598	6	β2	β2	NOUN
ap-7583	598	7	+	+	CCONJ
ap-7583	598	8	√	√	PROPN
ap-7583	598	9	(	(	PUNCT
ap-7583	598	10	α3	α3	NOUN
ap-7583	598	11	−	−	PROPN
ap-7583	598	12	β2)2	β2)2	PUNCT
ap-7583	598	13	−	−	PROPN
ap-7583	598	14	4α3ε1	4α3ε1	PROPN
ap-7583	598	15	2α3	2α3	NUM
ap-7583	598	16	)	)	PUNCT
ap-7583	599	1	k	k	PROPN
ap-7583	599	2	δkk′	δkk′	PROPN
ap-7583	599	3	.	.	PUNCT
ap-7583	600	1	(	(	PUNCT
ap-7583	600	2	107	107	NUM
ap-7583	600	3	)	)	PUNCT
ap-7583	600	4	using	use	VERB
ap-7583	600	5	the	the	DET
ap-7583	600	6	recurrence	recurrence	NOUN
ap-7583	600	7	relation	relation	NOUN
ap-7583	600	8	(	(	PUNCT
ap-7583	600	9	102	102	NUM
ap-7583	600	10	)	)	PUNCT
ap-7583	600	11	,	,	PUNCT
ap-7583	600	12	it	it	PRON
ap-7583	600	13	also	also	ADV
ap-7583	600	14	follows	follow	VERB
ap-7583	600	15	that∫	that∫	NOUN
ap-7583	600	16	r	r	NOUN
ap-7583	600	17	ε0[pk(ε0)]2dµ	ε0[pk(ε0)]2dµ	PROPN
ap-7583	600	18	=	=	SYM
ap-7583	600	19	−k((k	−k((k	NOUN
ap-7583	600	20	−	−	PROPN
ap-7583	600	21	1)α2	1)α2	PROPN
ap-7583	601	1	+	+	CCONJ
ap-7583	601	2	β1)k	β1)k	ADJ
ap-7583	601	3	!	!	PUNCT
ap-7583	602	1	(	(	PUNCT
ap-7583	602	2	α1	α1	PROPN
ap-7583	602	3	α3)k	α3)k	PROPN
ap-7583	602	4	β0ε1	β0ε1	PUNCT
ap-7583	602	5	(	(	PUNCT
ap-7583	602	6	β0	β0	PROPN
ap-7583	602	7	α1	α1	PROPN
ap-7583	602	8	)	)	PUNCT
ap-7583	602	9	k	k	PROPN
ap-7583	603	1	(	(	PUNCT
ap-7583	603	2	−α3	−α3	PROPN
ap-7583	603	3	+	+	SYM
ap-7583	603	4	β2	β2	NOUN
ap-7583	603	5	−	−	PROPN
ap-7583	603	6	√	√	PROPN
ap-7583	603	7	(	(	PUNCT
ap-7583	603	8	α3	α3	NOUN
ap-7583	603	9	−	−	PROPN
ap-7583	603	10	β2)2	β2)2	PUNCT
ap-7583	603	11	−	−	PROPN
ap-7583	603	12	4α3ε1	4α3ε1	PROPN
ap-7583	603	13	2α3	2α3	NUM
ap-7583	603	14	)	)	PUNCT
ap-7583	604	1	k	k	X
ap-7583	604	2	×	×	NOUN
ap-7583	604	3	(	(	PUNCT
ap-7583	604	4	−α3	−α3	PROPN
ap-7583	604	5	+	+	CCONJ
ap-7583	604	6	β2	β2	NOUN
ap-7583	604	7	+	+	CCONJ
ap-7583	604	8	√	√	PROPN
ap-7583	604	9	(	(	PUNCT
ap-7583	604	10	α3	α3	NOUN
ap-7583	604	11	−	−	PROPN
ap-7583	604	12	β2)2	β2)2	PUNCT
ap-7583	604	13	−	−	PROPN
ap-7583	604	14	4α3ε1	4α3ε1	PROPN
ap-7583	604	15	2α3	2α3	NUM
ap-7583	604	16	)	)	PUNCT
ap-7583	605	1	k	k	PROPN
ap-7583	605	2	.	.	PUNCT
ap-7583	606	1	(	(	PUNCT
ap-7583	606	2	108	108	NUM
ap-7583	606	3	)	)	PUNCT
ap-7583	606	4	further	far	ADV
ap-7583	606	5	,	,	PUNCT
ap-7583	606	6	for	for	ADP
ap-7583	606	7	k	k	PROPN
ap-7583	606	8	=	=	SYM
ap-7583	606	9	0	0	NUM
ap-7583	606	10	,	,	PUNCT
ap-7583	606	11	1	1	NUM
ap-7583	606	12	,	,	PUNCT
ap-7583	606	13	2	2	NUM
ap-7583	606	14	,	,	PUNCT
ap-7583	606	15	·	·	PUNCT
ap-7583	606	16	·	·	PUNCT
ap-7583	606	17	·	·	PUNCT
ap-7583	606	18	,	,	PUNCT
ap-7583	606	19	∫	∫	PROPN
ap-7583	606	20	r	r	PROPN
ap-7583	606	21	ε0pk+1(ε0)pk(ε0)dµ	ε0pk+1(ε0)pk(ε0)dµ	PROPN
ap-7583	606	22	=	=	PRON
ap-7583	606	23	(	(	PUNCT
ap-7583	606	24	k	k	PROPN
ap-7583	606	25	+	+	PROPN
ap-7583	606	26	1	1	NUM
ap-7583	606	27	)	)	PUNCT
ap-7583	606	28	!	!	PUNCT
ap-7583	607	1	(	(	PUNCT
ap-7583	607	2	α1	α1	PROPN
ap-7583	607	3	α3)k+1	α3)k+1	PROPN
ap-7583	607	4	β0ε1	β0ε1	PUNCT
ap-7583	607	5	×	×	NOUN
ap-7583	607	6	(	(	PUNCT
ap-7583	607	7	β0	β0	PROPN
ap-7583	607	8	α1	α1	PROPN
ap-7583	607	9	)	)	PUNCT
ap-7583	607	10	k+1	k+1	X
ap-7583	607	11	(	(	PUNCT
ap-7583	607	12	−α3	−α3	PROPN
ap-7583	607	13	+	+	SYM
ap-7583	607	14	β2	β2	NOUN
ap-7583	607	15	−	−	PROPN
ap-7583	607	16	√	√	PROPN
ap-7583	607	17	(	(	PUNCT
ap-7583	607	18	α3	α3	NOUN
ap-7583	607	19	−	−	PROPN
ap-7583	607	20	β2)2	β2)2	PUNCT
ap-7583	607	21	−	−	PROPN
ap-7583	607	22	4α3ε1	4α3ε1	PROPN
ap-7583	607	23	2α3	2α3	NUM
ap-7583	607	24	)	)	PUNCT
ap-7583	607	25	k+1	k+1	X
ap-7583	608	1	(	(	PUNCT
ap-7583	608	2	−α3	−α3	PROPN
ap-7583	608	3	+	+	SYM
ap-7583	608	4	β2	β2	NOUN
ap-7583	608	5	+	+	CCONJ
ap-7583	608	6	√	√	PROPN
ap-7583	608	7	(	(	PUNCT
ap-7583	608	8	α3	α3	NOUN
ap-7583	608	9	−	−	PROPN
ap-7583	608	10	β2)2	β2)2	PUNCT
ap-7583	608	11	−	−	PROPN
ap-7583	608	12	4α3ε1	4α3ε1	PROPN
ap-7583	608	13	2α3	2α3	NUM
ap-7583	608	14	)	)	PUNCT
ap-7583	609	1	k+1	k+1	X
ap-7583	609	2	.	.	PUNCT
ap-7583	610	1	(	(	PUNCT
ap-7583	610	2	109	109	NUM
ap-7583	610	3	)	)	PUNCT
ap-7583	610	4	other	other	ADJ
ap-7583	610	5	integrals	integral	NOUN
ap-7583	610	6	can	can	AUX
ap-7583	610	7	be	be	AUX
ap-7583	610	8	evaluated	evaluate	VERB
ap-7583	610	9	similarly	similarly	ADV
ap-7583	610	10	,	,	PUNCT
ap-7583	610	11	for	for	ADP
ap-7583	610	12	example	example	NOUN
ap-7583	610	13	∫	∫	PROPN
ap-7583	610	14	r[ε0pk(ε0)]2dµ	r[ε0pk(ε0)]2dµ	NOUN
ap-7583	610	15	can	can	AUX
ap-7583	610	16	be	be	AUX
ap-7583	610	17	evaluated	evaluate	VERB
ap-7583	610	18	by	by	ADP
ap-7583	610	19	multiplying	multiply	VERB
ap-7583	610	20	(	(	PUNCT
ap-7583	610	21	102	102	NUM
ap-7583	610	22	)	)	PUNCT
ap-7583	610	23	by	by	ADP
ap-7583	610	24	ε0pk(ε0	ε0pk(ε0	PROPN
ap-7583	610	25	)	)	PUNCT
ap-7583	610	26	and	and	CCONJ
ap-7583	610	27	integrate	integrate	VERB
ap-7583	610	28	with	with	ADP
ap-7583	610	29	respect	respect	NOUN
ap-7583	610	30	to	to	ADP
ap-7583	610	31	the	the	DET
ap-7583	610	32	measure	measure	NOUN
ap-7583	610	33	µ	µ	ADV
ap-7583	610	34	using	use	VERB
ap-7583	610	35	(	(	PUNCT
ap-7583	610	36	107	107	NUM
ap-7583	610	37	)	)	PUNCT
ap-7583	610	38	,	,	PUNCT
ap-7583	610	39	(	(	PUNCT
ap-7583	610	40	108	108	NUM
ap-7583	610	41	)	)	PUNCT
ap-7583	610	42	,	,	PUNCT
ap-7583	610	43	and	and	CCONJ
ap-7583	610	44	(	(	PUNCT
ap-7583	610	45	109	109	NUM
ap-7583	610	46	)	)	PUNCT
ap-7583	610	47	and	and	CCONJ
ap-7583	610	48	we	we	PRON
ap-7583	610	49	continue	continue	VERB
ap-7583	610	50	similarly	similarly	ADV
ap-7583	610	51	for	for	ADP
ap-7583	610	52	∫	∫	PROPN
ap-7583	610	53	r	r	NOUN
ap-7583	610	54	εm	εm	NOUN
ap-7583	610	55	0	0	PUNCT
ap-7583	611	1	[	[	X
ap-7583	611	2	pk(ε0)]2dµ	pk(ε0)]2dµ	PROPN
ap-7583	611	3	,	,	PUNCT
ap-7583	611	4	m	m	VERB
ap-7583	611	5	=	=	NOUN
ap-7583	611	6	0	0	NUM
ap-7583	611	7	,	,	PUNCT
ap-7583	611	8	1	1	NUM
ap-7583	611	9	,	,	PUNCT
ap-7583	611	10	2	2	NUM
ap-7583	611	11	,	,	PUNCT
ap-7583	611	12	·	·	PUNCT
ap-7583	611	13	·	·	PUNCT
ap-7583	611	14	·	·	PUNCT
ap-7583	611	15	.	.	PUNCT
ap-7583	612	1	the	the	DET
ap-7583	612	2	recurrence	recurrence	NOUN
ap-7583	612	3	relations	relation	NOUN
ap-7583	612	4	(	(	PUNCT
ap-7583	612	5	102	102	NUM
ap-7583	612	6	)	)	PUNCT
ap-7583	612	7	for	for	ADP
ap-7583	612	8	x	x	X
ap-7583	612	9	=	=	SYM
ap-7583	612	10	ε0	ε0	PROPN
ap-7583	612	11	and	and	CCONJ
ap-7583	612	12	y	y	NOUN
ap-7583	612	13	=	=	SYM
ap-7583	612	14	ε′	ε′	NOUN
ap-7583	612	15	0	0	NUM
ap-7583	612	16	read	read	NOUN
ap-7583	612	17	pk+1(x	pk+1(x	PROPN
ap-7583	612	18	)	)	PUNCT
ap-7583	612	19	=	=	PRON
ap-7583	613	1	(	(	PUNCT
ap-7583	613	2	x	x	X
ap-7583	613	3	−	−	PROPN
ap-7583	613	4	ak	ak	PROPN
ap-7583	613	5	)	)	PUNCT
ap-7583	614	1	pn	pn	PROPN
ap-7583	615	1	k	k	PROPN
ap-7583	616	1	(	(	PUNCT
ap-7583	616	2	x	x	NOUN
ap-7583	616	3	)	)	PUNCT
ap-7583	616	4	−	−	PROPN
ap-7583	616	5	bkpn	bkpn	PROPN
ap-7583	616	6	k−1(x	k−1(x	PROPN
ap-7583	616	7	)	)	PUNCT
ap-7583	616	8	,	,	PUNCT
ap-7583	616	9	pk+1(y	pk+1(y	PROPN
ap-7583	616	10	)	)	PUNCT
ap-7583	617	1	=	=	PRON
ap-7583	617	2	(	(	PUNCT
ap-7583	617	3	y	y	PROPN
ap-7583	617	4	−	−	PROPN
ap-7583	617	5	ak	ak	PROPN
ap-7583	617	6	)	)	PUNCT
ap-7583	617	7	pn	pn	PROPN
ap-7583	617	8	k	k	PROPN
ap-7583	617	9	(	(	PUNCT
ap-7583	617	10	y	y	PROPN
ap-7583	617	11	)	)	PUNCT
ap-7583	617	12	−	−	PROPN
ap-7583	617	13	bkpn	bkpn	PROPN
ap-7583	617	14	k−1(y	k−1(y	PROPN
ap-7583	617	15	)	)	PUNCT
ap-7583	617	16	,	,	PUNCT
ap-7583	617	17	respectively	respectively	ADV
ap-7583	617	18	.	.	PUNCT
ap-7583	618	1	by	by	ADP
ap-7583	618	2	multiplying	multiply	VERB
ap-7583	618	3	the	the	DET
ap-7583	618	4	first	first	ADJ
ap-7583	618	5	by	by	ADP
ap-7583	618	6	pk(y	pk(y	NOUN
ap-7583	618	7	)	)	PUNCT
ap-7583	618	8	and	and	CCONJ
ap-7583	618	9	the	the	DET
ap-7583	618	10	second	second	ADJ
ap-7583	618	11	by	by	ADP
ap-7583	618	12	pk(x	pk(x	NOUN
ap-7583	618	13	)	)	PUNCT
ap-7583	618	14	and	and	CCONJ
ap-7583	618	15	subtracting	subtracting	NOUN
ap-7583	618	16	,	,	PUNCT
ap-7583	618	17	the	the	DET
ap-7583	618	18	resulting	result	VERB
ap-7583	618	19	equation	equation	NOUN
ap-7583	618	20	becomes	become	VERB
ap-7583	618	21	(	(	PUNCT
ap-7583	618	22	x	x	NOUN
ap-7583	618	23	−	−	NOUN
ap-7583	618	24	y)pk(y)pk(x	y)pk(y)pk(x	NOUN
ap-7583	618	25	)	)	PUNCT
ap-7583	618	26	=	=	SYM
ap-7583	618	27	qk+1(x	qk+1(x	PROPN
ap-7583	618	28	,	,	PUNCT
ap-7583	618	29	y	y	PROPN
ap-7583	618	30	)	)	PUNCT
ap-7583	618	31	−	−	NOUN
ap-7583	618	32	bk	bk	VERB
ap-7583	618	33	qk(x	qk(x	NOUN
ap-7583	618	34	,	,	PUNCT
ap-7583	618	35	y	y	NOUN
ap-7583	618	36	)	)	PUNCT
ap-7583	618	37	(	(	PUNCT
ap-7583	618	38	110	110	NUM
ap-7583	618	39	)	)	PUNCT
ap-7583	618	40	where	where	SCONJ
ap-7583	618	41	qk+1(x	qk+1(x	PROPN
ap-7583	618	42	,	,	PUNCT
ap-7583	618	43	y	y	NOUN
ap-7583	618	44	)	)	PUNCT
ap-7583	618	45	=	=	SYM
ap-7583	618	46	pk+1(x)pk(y	pk+1(x)pk(y	PROPN
ap-7583	618	47	)	)	PUNCT
ap-7583	619	1	−	−	PROPN
ap-7583	619	2	pk(x)pk+1(y	pk(x)pk+1(y	PROPN
ap-7583	619	3	)	)	PUNCT
ap-7583	619	4	.	.	PUNCT
ap-7583	620	1	thus	thus	ADV
ap-7583	620	2	,	,	PUNCT
ap-7583	620	3	recursively	recursively	ADV
ap-7583	620	4	over	over	ADP
ap-7583	620	5	k	k	PROPN
ap-7583	620	6	,	,	PUNCT
ap-7583	620	7	we	we	PRON
ap-7583	620	8	have	have	VERB
ap-7583	620	9	(	(	PUNCT
ap-7583	620	10	x	x	X
ap-7583	620	11	−	−	NOUN
ap-7583	620	12	y)pk(x)pk(y	y)pk(x)pk(y	NUM
ap-7583	620	13	)	)	PUNCT
ap-7583	620	14	=	=	SYM
ap-7583	620	15	qk+1(x	qk+1(x	PROPN
ap-7583	620	16	,	,	PUNCT
ap-7583	620	17	y	y	PROPN
ap-7583	620	18	)	)	PUNCT
ap-7583	620	19	−	−	NOUN
ap-7583	620	20	bk	bk	VERB
ap-7583	620	21	qk(x	qk(x	NOUN
ap-7583	620	22	,	,	PUNCT
ap-7583	620	23	y	y	NOUN
ap-7583	620	24	)	)	PUNCT
ap-7583	620	25	(	(	PUNCT
ap-7583	620	26	x	x	X
ap-7583	620	27	−	−	PROPN
ap-7583	620	28	y)pk−1(x)pk−1(y	y)pk−1(x)pk−1(y	PROPN
ap-7583	620	29	)	)	PUNCT
ap-7583	620	30	=	=	PUNCT
ap-7583	621	1	qk(x	qk(x	X
ap-7583	621	2	,	,	PUNCT
ap-7583	621	3	y	y	NOUN
ap-7583	621	4	)	)	PUNCT
ap-7583	621	5	−	−	PROPN
ap-7583	622	1	bk−1	bk−1	ADJ
ap-7583	622	2	qk−1(x	qk−1(x	NOUN
ap-7583	622	3	,	,	PUNCT
ap-7583	622	4	y	y	PROPN
ap-7583	622	5	)	)	PUNCT
ap-7583	622	6	...	...	PUNCT
ap-7583	623	1	(	(	PUNCT
ap-7583	623	2	x	x	X
ap-7583	623	3	−	−	PUNCT
ap-7583	623	4	y)pn	y)pn	PROPN
ap-7583	623	5	0	0	NUM
ap-7583	623	6	(	(	PUNCT
ap-7583	623	7	x)pn	x)pn	PROPN
ap-7583	623	8	0	0	NUM
ap-7583	623	9	(	(	PUNCT
ap-7583	623	10	y	y	NOUN
ap-7583	623	11	)	)	PUNCT
ap-7583	623	12	=	=	SYM
ap-7583	624	1	q1(x	q1(x	PROPN
ap-7583	624	2	,	,	PUNCT
ap-7583	624	3	y	y	PROPN
ap-7583	624	4	)	)	PUNCT
ap-7583	624	5	,	,	PUNCT
ap-7583	624	6	from	from	ADP
ap-7583	624	7	which	which	PRON
ap-7583	624	8	it	it	PRON
ap-7583	624	9	is	be	AUX
ap-7583	624	10	straightforward	straightforward	ADJ
ap-7583	624	11	to	to	PART
ap-7583	624	12	obtain	obtain	VERB
ap-7583	624	13	(	(	PUNCT
ap-7583	624	14	x	x	NOUN
ap-7583	624	15	−	−	PROPN
ap-7583	624	16	y	y	PROPN
ap-7583	624	17	)	)	PUNCT
ap-7583	624	18	[	[	PUNCT
ap-7583	624	19	pk(x)pk(y	pk(x)pk(y	PROPN
ap-7583	624	20	)	)	PUNCT
ap-7583	624	21	+	+	CCONJ
ap-7583	624	22	bkpk−1(x)pk−1(y	bkpk−1(x)pk−1(y	X
ap-7583	624	23	)	)	PUNCT
ap-7583	624	24	+	+	NUM
ap-7583	624	25	bkbk−1pk−2(x)pk−2(y	bkbk−1pk−2(x)pk−2(y	X
ap-7583	624	26	)	)	PUNCT
ap-7583	624	27	+	+	CCONJ
ap-7583	624	28	bkbk−1bk−2pk−3(x)pk−3(y	bkbk−1bk−2pk−3(x)pk−3(y	NOUN
ap-7583	624	29	)	)	PUNCT
ap-7583	624	30	+	+	NUM
ap-7583	624	31	·	·	PUNCT
ap-7583	624	32	·	·	PUNCT
ap-7583	624	33	·	·	PUNCT
ap-7583	625	1	+	+	NUM
ap-7583	625	2	λk+1λkλk−1λk−2	λk+1λkλk−1λk−2	X
ap-7583	625	3	.	.	PUNCT
ap-7583	625	4	.	.	PUNCT
ap-7583	625	5	.	.	PUNCT
ap-7583	626	1	λ2p0(x)p0(y	λ2p0(x)p0(y	PROPN
ap-7583	626	2	)	)	PUNCT
ap-7583	626	3	]	]	PUNCT
ap-7583	627	1	=	=	SYM
ap-7583	627	2	qk+1(ε0	qk+1(ε0	X
ap-7583	627	3	,	,	PUNCT
ap-7583	627	4	y	y	PROPN
ap-7583	627	5	)	)	PUNCT
ap-7583	627	6	.	.	PUNCT
ap-7583	628	1	dividing	divide	VERB
ap-7583	628	2	both	both	DET
ap-7583	628	3	sides	side	NOUN
ap-7583	628	4	by	by	ADP
ap-7583	628	5	(	(	PUNCT
ap-7583	628	6	x	x	X
ap-7583	628	7	−	−	NOUN
ap-7583	628	8	y)bkbk−1bk−2	y)bkbk−1bk−2	NUM
ap-7583	628	9	.	.	PUNCT
ap-7583	628	10	.	.	PUNCT
ap-7583	628	11	.	.	PUNCT
ap-7583	629	1	b2	b2	NOUN
ap-7583	629	2	and	and	CCONJ
ap-7583	629	3	summing	sum	VERB
ap-7583	629	4	over	over	ADP
ap-7583	629	5	k	k	PROPN
ap-7583	629	6	results	result	NOUN
ap-7583	629	7	in	in	ADP
ap-7583	629	8	k∑	k∑	PROPN
ap-7583	629	9	j=0	j=0	PROPN
ap-7583	629	10	pj(x)pj(y	pj(x)pj(y	PROPN
ap-7583	629	11	)	)	PUNCT
ap-7583	629	12	bjbj−1bj−2	bjbj−1bj−2	PROPN
ap-7583	629	13	.	.	PUNCT
ap-7583	629	14	.	.	PUNCT
ap-7583	629	15	.	.	PUNCT
ap-7583	630	1	b2	b2	NOUN
ap-7583	630	2	=	=	SYM
ap-7583	630	3	(	(	PUNCT
ap-7583	630	4	bkbk−1bk−2	bkbk−1bk−2	PROPN
ap-7583	630	5	.	.	PUNCT
ap-7583	630	6	.	.	PUNCT
ap-7583	630	7	.	.	PUNCT
ap-7583	631	1	b2)−1	b2)−1	PROPN
ap-7583	631	2	×	×	PROPN
ap-7583	631	3	pk+1(x)pk(y	pk+1(x)pk(y	PROPN
ap-7583	631	4	)	)	PUNCT
ap-7583	632	1	−	−	PROPN
ap-7583	633	1	pn	pn	PROPN
ap-7583	633	2	k	k	PROPN
ap-7583	633	3	(	(	PUNCT
ap-7583	633	4	x)pk+1(y	x)pk+1(y	PROPN
ap-7583	633	5	)	)	PUNCT
ap-7583	633	6	x	x	X
ap-7583	634	1	−	−	PROPN
ap-7583	634	2	y	y	NOUN
ap-7583	634	3	.	.	PUNCT
ap-7583	635	1	184	184	NUM
ap-7583	635	2	vol	vol	NOUN
ap-7583	635	3	.	.	PUNCT
ap-7583	636	1	62	62	NUM
ap-7583	636	2	no	no	INTJ
ap-7583	636	3	.	.	PUNCT
ap-7583	637	1	1/2022	1/2022	NUM
ap-7583	637	2	on	on	ADP
ap-7583	637	3	generalized	generalized	ADJ
ap-7583	637	4	heun	heun	NOUN
ap-7583	637	5	equation	equation	NOUN
ap-7583	637	6	with	with	ADP
ap-7583	637	7	some	some	DET
ap-7583	637	8	mathematical	mathematical	NOUN
ap-7583	637	9	.	.	PUNCT
ap-7583	637	10	.	.	PUNCT
ap-7583	637	11	.	.	PUNCT
ap-7583	638	1	(	(	PUNCT
ap-7583	638	2	101	101	NUM
ap-7583	638	3	)	)	PUNCT
ap-7583	638	4	then	then	ADV
ap-7583	638	5	follows	follow	VERB
ap-7583	638	6	using	use	VERB
ap-7583	638	7	bkbk−1bk−2	bkbk−1bk−2	PROPN
ap-7583	638	8	.	.	PUNCT
ap-7583	638	9	.	.	PUNCT
ap-7583	638	10	.	.	PUNCT
ap-7583	639	1	b2	b2	NOUN
ap-7583	639	2	=	=	SYM
ap-7583	639	3	k∏	k∏	PROPN
ap-7583	639	4	i=2	i=2	PROPN
ap-7583	639	5	bi	bi	NOUN
ap-7583	639	6	=	=	PROPN
ap-7583	639	7	k	k	PROPN
ap-7583	639	8	!	!	PUNCT
ap-7583	640	1	(	(	PUNCT
ap-7583	640	2	α1	α1	PROPN
ap-7583	640	3	α3)k	α3)k	PROPN
ap-7583	640	4	β0ε1	β0ε1	PUNCT
ap-7583	640	5	(	(	PUNCT
ap-7583	640	6	β0	β0	PROPN
ap-7583	640	7	α1	α1	PROPN
ap-7583	640	8	)	)	PUNCT
ap-7583	641	1	k	k	PROPN
ap-7583	641	2	×	×	NOUN
ap-7583	641	3	(	(	PUNCT
ap-7583	641	4	−α3	−α3	PROPN
ap-7583	641	5	+	+	SYM
ap-7583	641	6	β2	β2	NOUN
ap-7583	641	7	−	−	PROPN
ap-7583	641	8	√	√	PROPN
ap-7583	641	9	(	(	PUNCT
ap-7583	641	10	α3	α3	NOUN
ap-7583	641	11	−	−	PROPN
ap-7583	641	12	β2)2	β2)2	PUNCT
ap-7583	641	13	−	−	PROPN
ap-7583	641	14	4α3ε1	4α3ε1	PROPN
ap-7583	641	15	2α3	2α3	NUM
ap-7583	641	16	)	)	PUNCT
ap-7583	642	1	k	k	X
ap-7583	642	2	×	×	NOUN
ap-7583	642	3	(	(	PUNCT
ap-7583	642	4	−α3	−α3	PROPN
ap-7583	642	5	+	+	CCONJ
ap-7583	642	6	β2	β2	NOUN
ap-7583	642	7	+	+	CCONJ
ap-7583	642	8	√	√	PROPN
ap-7583	642	9	(	(	PUNCT
ap-7583	642	10	α3	α3	NOUN
ap-7583	642	11	−	−	PROPN
ap-7583	642	12	β2)2	β2)2	PUNCT
ap-7583	642	13	−	−	PROPN
ap-7583	642	14	4α3ε1	4α3ε1	PROPN
ap-7583	642	15	2α3	2α3	NUM
ap-7583	642	16	)	)	PUNCT
ap-7583	643	1	k	k	NOUN
ap-7583	643	2	and	and	CCONJ
ap-7583	643	3	finally	finally	ADV
ap-7583	643	4	,	,	PUNCT
ap-7583	643	5	we	we	PRON
ap-7583	643	6	have	have	VERB
ap-7583	643	7	,	,	PUNCT
ap-7583	643	8	for	for	ADP
ap-7583	643	9	k	k	PROPN
ap-7583	643	10	≥	≥	PROPN
ap-7583	643	11	0	0	NUM
ap-7583	643	12	,	,	PUNCT
ap-7583	643	13	christoffel	christoffel	NOUN
ap-7583	643	14	-	-	PUNCT
ap-7583	643	15	darboux	darboux	VERB
ap-7583	643	16	identities	identity	NOUN
ap-7583	643	17	:	:	PUNCT
ap-7583	643	18	k∑	k∑	PROPN
ap-7583	643	19	j=0	j=0	PROPN
ap-7583	643	20	pj(x)pj(y	pj(x)pj(y	NOUN
ap-7583	643	21	)	)	PUNCT
ap-7583	643	22	j	j	PROPN
ap-7583	643	23	!	!	PUNCT
ap-7583	644	1	(	(	PUNCT
ap-7583	644	2	α1	α1	PROPN
ap-7583	644	3	α3)j	α3)j	PROPN
ap-7583	644	4	(	(	PUNCT
ap-7583	644	5	β0	β0	PROPN
ap-7583	644	6	α1	α1	PROPN
ap-7583	644	7	)	)	PUNCT
ap-7583	644	8	j	j	PROPN
ap-7583	644	9	(	(	PUNCT
ap-7583	644	10	ξ+)j	ξ+)j	PROPN
ap-7583	644	11	(	(	PUNCT
ap-7583	644	12	ξ−)j	ξ−)j	PROPN
ap-7583	644	13	=	=	SYM
ap-7583	644	14	pk+1(x)pk(y	pk+1(x)pk(y	PROPN
ap-7583	644	15	)	)	PUNCT
ap-7583	645	1	−	−	PROPN
ap-7583	645	2	pk(x)pk+1(y	pk(x)pk+1(y	PROPN
ap-7583	645	3	)	)	PUNCT
ap-7583	646	1	k	k	X
ap-7583	646	2	!	!	PUNCT
ap-7583	647	1	(	(	PUNCT
ap-7583	647	2	α1	α1	PROPN
ap-7583	647	3	α3)k	α3)k	PROPN
ap-7583	647	4	(	(	PUNCT
ap-7583	647	5	β0	β0	PROPN
ap-7583	647	6	α1	α1	PROPN
ap-7583	647	7	)	)	PUNCT
ap-7583	647	8	k	k	PROPN
ap-7583	648	1	(	(	PUNCT
ap-7583	648	2	ξ+)k	ξ+)k	PROPN
ap-7583	648	3	(	(	PUNCT
ap-7583	648	4	ξ−)k	ξ−)k	X
ap-7583	648	5	(	(	PUNCT
ap-7583	648	6	x	x	PROPN
ap-7583	648	7	−	−	PROPN
ap-7583	648	8	y	y	PROPN
ap-7583	648	9	)	)	PUNCT
ap-7583	648	10	,	,	PUNCT
ap-7583	648	11	(	(	PUNCT
ap-7583	648	12	111	111	NUM
ap-7583	648	13	)	)	PUNCT
ap-7583	648	14	where	where	SCONJ
ap-7583	648	15	ξ±	ξ±	ADV
ap-7583	648	16	=	=	SYM
ap-7583	648	17	−α3	−α3	PROPN
ap-7583	648	18	+	+	CCONJ
ap-7583	648	19	β2	β2	PROPN
ap-7583	648	20	±	±	NOUN
ap-7583	648	21	√	√	PROPN
ap-7583	648	22	(	(	PUNCT
ap-7583	648	23	α3	α3	NOUN
ap-7583	648	24	−	−	PROPN
ap-7583	648	25	β2)2	β2)2	PUNCT
ap-7583	648	26	−	−	PROPN
ap-7583	648	27	4α3ε1	4α3ε1	PROPN
ap-7583	648	28	2α3	2α3	NUM
ap-7583	648	29	and	and	CCONJ
ap-7583	648	30	by	by	ADP
ap-7583	648	31	evaluating	evaluate	VERB
ap-7583	648	32	the	the	DET
ap-7583	648	33	limit	limit	NOUN
ap-7583	648	34	of	of	ADP
ap-7583	648	35	both	both	DET
ap-7583	648	36	sides	side	NOUN
ap-7583	648	37	as	as	ADP
ap-7583	648	38	y	y	PROPN
ap-7583	648	39	→	→	SYM
ap-7583	648	40	x	x	PROPN
ap-7583	648	41	,	,	PUNCT
ap-7583	648	42	its	its	PRON
ap-7583	648	43	confluent	confluent	ADJ
ap-7583	648	44	form	form	NOUN
ap-7583	648	45	k∑	k∑	VERB
ap-7583	648	46	j=0	j=0	PROPN
ap-7583	649	1	[	[	X
ap-7583	649	2	pj(x)]2	pj(x)]2	PROPN
ap-7583	649	3	j	j	PROPN
ap-7583	649	4	!	!	PUNCT
ap-7583	650	1	(	(	PUNCT
ap-7583	650	2	α1	α1	PROPN
ap-7583	650	3	α3)j	α3)j	PROPN
ap-7583	650	4	(	(	PUNCT
ap-7583	650	5	ξ+)j	ξ+)j	PROPN
ap-7583	650	6	(	(	PUNCT
ap-7583	650	7	ξ−)j	ξ−)j	PROPN
ap-7583	650	8	=	=	SYM
ap-7583	651	1	p	p	NOUN
ap-7583	651	2	′	′	NUM
ap-7583	651	3	k+1(x)pk(x	k+1(x)pk(x	PROPN
ap-7583	651	4	)	)	PUNCT
ap-7583	651	5	−	−	PROPN
ap-7583	651	6	p	p	NOUN
ap-7583	651	7	′	′	NUM
ap-7583	651	8	k(x)pk+1(x	k(x)pk+1(x	PROPN
ap-7583	651	9	)	)	PUNCT
ap-7583	652	1	k	k	X
ap-7583	652	2	!	!	PUNCT
ap-7583	653	1	(	(	PUNCT
ap-7583	653	2	α1	α1	PROPN
ap-7583	653	3	α3)k	α3)k	PROPN
ap-7583	653	4	(	(	PUNCT
ap-7583	653	5	β0	β0	PROPN
ap-7583	653	6	α1	α1	PROPN
ap-7583	653	7	)	)	PUNCT
ap-7583	653	8	k	k	PROPN
ap-7583	654	1	(	(	PUNCT
ap-7583	654	2	ξ+)k	ξ+)k	PROPN
ap-7583	654	3	(	(	PUNCT
ap-7583	654	4	ξ−)k	ξ−)k	PROPN
ap-7583	654	5	(	(	PUNCT
ap-7583	654	6	112	112	NUM
ap-7583	654	7	)	)	PUNCT
ap-7583	654	8	follows	follow	VERB
ap-7583	654	9	.	.	PUNCT
ap-7583	655	1	here	here	ADV
ap-7583	655	2	,	,	PUNCT
ap-7583	655	3	the	the	DET
ap-7583	655	4	prime	prime	NOUN
ap-7583	655	5	refers	refer	VERB
ap-7583	655	6	to	to	ADP
ap-7583	655	7	the	the	DET
ap-7583	655	8	derivative	derivative	NOUN
ap-7583	655	9	with	with	ADP
ap-7583	655	10	respect	respect	NOUN
ap-7583	655	11	to	to	ADP
ap-7583	655	12	the	the	DET
ap-7583	655	13	variable	variable	ADJ
ap-7583	655	14	x.	x.	NOUN
ap-7583	655	15	as	as	ADP
ap-7583	655	16	a	a	DET
ap-7583	655	17	direct	direct	ADJ
ap-7583	655	18	consequence	consequence	NOUN
ap-7583	655	19	of	of	ADP
ap-7583	655	20	the	the	DET
ap-7583	655	21	christoffel	christoffel	NOUN
ap-7583	655	22	-	-	PUNCT
ap-7583	655	23	darboux	darboux	VERB
ap-7583	655	24	formula	formula	NOUN
ap-7583	655	25	(	(	PUNCT
ap-7583	655	26	112	112	NUM
ap-7583	655	27	)	)	PUNCT
ap-7583	655	28	,	,	PUNCT
ap-7583	655	29	all	all	DET
ap-7583	655	30	the	the	DET
ap-7583	655	31	zeros	zero	NOUN
ap-7583	655	32	of	of	ADP
ap-7583	655	33	the	the	DET
ap-7583	655	34	n	n	CCONJ
ap-7583	655	35	-	-	PUNCT
ap-7583	655	36	degree	degree	NOUN
ap-7583	655	37	polynomial	polynomial	ADJ
ap-7583	655	38	pn(ε	pn(ε	NOUN
ap-7583	655	39	)	)	PUNCT
ap-7583	655	40	are	be	AUX
ap-7583	655	41	simple	simple	ADJ
ap-7583	655	42	.	.	PUNCT
ap-7583	656	1	to	to	PART
ap-7583	656	2	prove	prove	VERB
ap-7583	656	3	that	that	SCONJ
ap-7583	656	4	they	they	PRON
ap-7583	656	5	are	be	AUX
ap-7583	656	6	also	also	ADV
ap-7583	656	7	real	real	ADJ
ap-7583	656	8	,	,	PUNCT
ap-7583	656	9	we	we	PRON
ap-7583	656	10	note	note	VERB
ap-7583	656	11	that	that	SCONJ
ap-7583	656	12	the	the	DET
ap-7583	656	13	recurrence	recurrence	NOUN
ap-7583	656	14	relation	relation	NOUN
ap-7583	656	15	(	(	PUNCT
ap-7583	656	16	102	102	NUM
ap-7583	656	17	)	)	PUNCT
ap-7583	656	18	can	can	AUX
ap-7583	656	19	be	be	AUX
ap-7583	656	20	written	write	VERB
ap-7583	656	21	in	in	ADP
ap-7583	656	22	a	a	DET
ap-7583	656	23	matrix	matrix	NOUN
ap-7583	656	24	form	form	NOUN
ap-7583	656	25	as	as	ADP
ap-7583	656	26	x	x	X
ap-7583	656	27			ADJ
ap-7583	656	28	p0(x	p0(x	NOUN
ap-7583	656	29	)	)	PUNCT
ap-7583	656	30	p1(x	p1(x	NOUN
ap-7583	656	31	)	)	PUNCT
ap-7583	656	32	p2(x	p2(x	NOUN
ap-7583	656	33	)	)	PUNCT
ap-7583	656	34	...	...	PUNCT
ap-7583	657	1	pk−1(x	pk−1(x	NOUN
ap-7583	657	2	)	)	PUNCT
ap-7583	657	3			PUNCT
ap-7583	658	1	=	=	PRON
ap-7583	658	2			PROPN
ap-7583	658	3	a0	a0	NOUN
ap-7583	658	4	1	1	NUM
ap-7583	658	5	0	0	NUM
ap-7583	658	6	·	·	PUNCT
ap-7583	658	7	·	·	PUNCT
ap-7583	658	8	·	·	PUNCT
ap-7583	658	9	0	0	NUM
ap-7583	658	10	0	0	NUM
ap-7583	658	11	b1	b1	NOUN
ap-7583	658	12	a1	a1	NOUN
ap-7583	658	13	1	1	NUM
ap-7583	658	14	·	·	PUNCT
ap-7583	658	15	·	·	PUNCT
ap-7583	658	16	·	·	PUNCT
ap-7583	658	17	0	0	NUM
ap-7583	658	18	0	0	NUM
ap-7583	658	19	0	0	NUM
ap-7583	658	20	b2	b2	NOUN
ap-7583	658	21	a2	a2	PROPN
ap-7583	658	22	·	·	PUNCT
ap-7583	658	23	·	·	PUNCT
ap-7583	658	24	·	·	PUNCT
ap-7583	658	25	0	0	NUM
ap-7583	658	26	0	0	NUM
ap-7583	658	27	...	...	PUNCT
ap-7583	658	28	...	...	PUNCT
ap-7583	658	29	...	...	PUNCT
ap-7583	658	30	.	.	PUNCT
ap-7583	658	31	.	.	PUNCT
ap-7583	658	32	.	.	PUNCT
ap-7583	659	1	...	...	PUNCT
ap-7583	660	1	...	...	PUNCT
ap-7583	661	1	0	0	NUM
ap-7583	661	2	0	0	NUM
ap-7583	661	3	0	0	NUM
ap-7583	661	4	·	·	PUNCT
ap-7583	661	5	·	·	PUNCT
ap-7583	661	6	·	·	PUNCT
ap-7583	662	1	bk−1	bk−1	VERB
ap-7583	662	2	ak−1	ak−1	ADV
ap-7583	662	3			PRON
ap-7583	662	4			ADJ
ap-7583	662	5	p0(x	p0(x	NOUN
ap-7583	662	6	)	)	PUNCT
ap-7583	662	7	p1(x	p1(x	NOUN
ap-7583	662	8	)	)	PUNCT
ap-7583	662	9	p2(x	p2(x	NOUN
ap-7583	662	10	)	)	PUNCT
ap-7583	662	11	...	...	PUNCT
ap-7583	662	12	pk−1(x	pk−1(x	NOUN
ap-7583	662	13	)	)	PUNCT
ap-7583	662	14			PROPN
ap-7583	663	1	+	+	CCONJ
ap-7583	663	2	pk(x	pk(x	X
ap-7583	663	3	)	)	PUNCT
ap-7583	663	4			NOUN
ap-7583	663	5	0	0	NUM
ap-7583	663	6	0	0	NUM
ap-7583	663	7	0	0	NUM
ap-7583	663	8	...	...	SYM
ap-7583	663	9	1	1	NUM
ap-7583	663	10			NOUN
ap-7583	663	11	(	(	PUNCT
ap-7583	663	12	113	113	NUM
ap-7583	663	13	)	)	PUNCT
ap-7583	663	14	thus	thus	ADV
ap-7583	663	15	,	,	PUNCT
ap-7583	663	16	if	if	SCONJ
ap-7583	663	17	xi	xi	PROPN
ap-7583	663	18	is	be	AUX
ap-7583	663	19	a	a	DET
ap-7583	663	20	zero	zero	NUM
ap-7583	663	21	of	of	ADP
ap-7583	663	22	pk(x	pk(x	NUM
ap-7583	663	23	)	)	PUNCT
ap-7583	663	24	,	,	PUNCT
ap-7583	663	25	it	it	PRON
ap-7583	663	26	is	be	AUX
ap-7583	663	27	an	an	DET
ap-7583	663	28	eigenvalue	eigenvalue	NOUN
ap-7583	663	29	of	of	ADP
ap-7583	663	30	the	the	DET
ap-7583	663	31	given	give	VERB
ap-7583	663	32	tridiagonal	tridiagonal	ADJ
ap-7583	663	33	matrix	matrix	NOUN
ap-7583	663	34	.	.	PUNCT
ap-7583	664	1	since	since	SCONJ
ap-7583	664	2	,	,	PUNCT
ap-7583	664	3	by	by	ADP
ap-7583	664	4	the	the	DET
ap-7583	664	5	hypothesis	hypothesis	NOUN
ap-7583	664	6	of	of	ADP
ap-7583	664	7	(	(	PUNCT
ap-7583	664	8	102	102	NUM
ap-7583	664	9	)	)	PUNCT
ap-7583	664	10	,	,	PUNCT
ap-7583	664	11	bk	bk	VERB
ap-7583	664	12	>	>	X
ap-7583	664	13	0	0	PUNCT
ap-7583	664	14	for	for	ADP
ap-7583	664	15	all	all	DET
ap-7583	664	16	k	k	PROPN
ap-7583	664	17	≥	≥	NUM
ap-7583	664	18	1	1	NUM
ap-7583	664	19	,	,	PUNCT
ap-7583	664	20	the	the	DET
ap-7583	664	21	results	result	NOUN
ap-7583	664	22	of	of	ADP
ap-7583	664	23	arscott	arscott	NOUN
ap-7583	664	24	[	[	X
ap-7583	664	25	24	24	NUM
ap-7583	664	26	]	]	PUNCT
ap-7583	664	27	confirm	confirm	VERB
ap-7583	664	28	that	that	SCONJ
ap-7583	664	29	(	(	PUNCT
ap-7583	664	30	i	i	NOUN
ap-7583	664	31	)	)	PUNCT
ap-7583	664	32	the	the	DET
ap-7583	664	33	zeros	zero	NOUN
ap-7583	664	34	of	of	ADP
ap-7583	664	35	pk−1(x	pk−1(x	NOUN
ap-7583	664	36	)	)	PUNCT
ap-7583	664	37	and	and	CCONJ
ap-7583	664	38	pk(x	pk(x	X
ap-7583	664	39	)	)	PUNCT
ap-7583	664	40	interlace	interlace	NOUN
ap-7583	664	41	–	–	PUNCT
ap-7583	664	42	that	that	ADV
ap-7583	664	43	is	is	ADV
ap-7583	664	44	,	,	PUNCT
ap-7583	664	45	between	between	ADP
ap-7583	664	46	two	two	NUM
ap-7583	664	47	consecutive	consecutive	ADJ
ap-7583	664	48	zeros	zero	NOUN
ap-7583	664	49	of	of	ADP
ap-7583	664	50	either	either	CCONJ
ap-7583	664	51	polynomial	polynomial	ADJ
ap-7583	664	52	lies	lie	VERB
ap-7583	664	53	precisely	precisely	ADV
ap-7583	664	54	one	one	NUM
ap-7583	664	55	zero	zero	NUM
ap-7583	664	56	of	of	ADP
ap-7583	664	57	the	the	DET
ap-7583	664	58	other	other	ADJ
ap-7583	664	59	(	(	PUNCT
ap-7583	664	60	ii	ii	NOUN
ap-7583	664	61	)	)	PUNCT
ap-7583	664	62	at	at	ADP
ap-7583	664	63	the	the	DET
ap-7583	664	64	zeros	zero	NOUN
ap-7583	664	65	of	of	ADP
ap-7583	664	66	pk(x	pk(x	NOUN
ap-7583	664	67	)	)	PUNCT
ap-7583	664	68	the	the	DET
ap-7583	664	69	values	value	NOUN
ap-7583	664	70	of	of	ADP
ap-7583	664	71	pk−1(x	pk−1(x	NOUN
ap-7583	664	72	)	)	PUNCT
ap-7583	664	73	are	be	AUX
ap-7583	664	74	alternately	alternately	ADV
ap-7583	664	75	positive	positive	ADJ
ap-7583	664	76	and	and	CCONJ
ap-7583	664	77	negative	negative	ADJ
ap-7583	664	78	,	,	PUNCT
ap-7583	664	79	(	(	PUNCT
ap-7583	664	80	iii	iii	X
ap-7583	664	81	)	)	PUNCT
ap-7583	664	82	all	all	DET
ap-7583	664	83	the	the	DET
ap-7583	664	84	zeros	zero	NOUN
ap-7583	664	85	of	of	ADP
ap-7583	664	86	pk(x	pk(x	NOUN
ap-7583	664	87	)	)	PUNCT
ap-7583	664	88	–	–	PUNCT
ap-7583	664	89	i.e.	i.e.	X
ap-7583	664	90	all	all	DET
ap-7583	664	91	the	the	DET
ap-7583	664	92	eigenvalues	eigenvalue	NOUN
ap-7583	664	93	of	of	ADP
ap-7583	664	94	tridiagonal	tridiagonal	ADJ
ap-7583	664	95	matrix	matrix	NOUN
ap-7583	664	96	are	be	AUX
ap-7583	664	97	real	real	ADJ
ap-7583	664	98	and	and	CCONJ
ap-7583	664	99	different	different	ADJ
ap-7583	664	100	.	.	PUNCT
ap-7583	665	1	6	6	X
ap-7583	665	2	.	.	X
ap-7583	665	3	mathematical	mathematical	ADJ
ap-7583	665	4	properties	property	NOUN
ap-7583	665	5	of	of	ADP
ap-7583	665	6	the	the	DET
ap-7583	665	7	finite	finite	ADJ
ap-7583	665	8	orthogonal	orthogonal	ADJ
ap-7583	665	9	polynomials	polynomial	NOUN
ap-7583	665	10	{	{	PUNCT
ap-7583	665	11	pn	pn	PROPN
ap-7583	665	12	k	k	PROPN
ap-7583	665	13	(	(	PUNCT
ap-7583	665	14	ε0)}n	ε0)}n	PROPN
ap-7583	665	15	k=0	k=0	PROPN
ap-7583	665	16	in	in	ADP
ap-7583	665	17	this	this	DET
ap-7583	665	18	section	section	NOUN
ap-7583	665	19	,	,	PUNCT
ap-7583	665	20	we	we	PRON
ap-7583	665	21	shall	shall	AUX
ap-7583	665	22	study	study	VERB
ap-7583	665	23	some	some	PRON
ap-7583	665	24	of	of	ADP
ap-7583	665	25	the	the	DET
ap-7583	665	26	mathematical	mathematical	ADJ
ap-7583	665	27	properties	property	NOUN
ap-7583	665	28	of	of	ADP
ap-7583	665	29	the	the	DET
ap-7583	665	30	orthogonal	orthogonal	ADJ
ap-7583	665	31	polynomials	polynomial	NOUN
ap-7583	665	32	{	{	PUNCT
ap-7583	665	33	pn	pn	NOUN
ap-7583	665	34	k	k	PROPN
ap-7583	665	35	(	(	PUNCT
ap-7583	665	36	ε0;n)}n	ε0;n)}n	PROPN
ap-7583	665	37	k=0	k=0	PROPN
ap-7583	665	38	.	.	PUNCT
ap-7583	666	1	first	first	ADV
ap-7583	666	2	,	,	PUNCT
ap-7583	666	3	the	the	DET
ap-7583	666	4	zeros	zero	NOUN
ap-7583	666	5	of	of	ADP
ap-7583	666	6	the	the	DET
ap-7583	666	7	polynomial	polynomial	NOUN
ap-7583	666	8	generated	generate	VERB
ap-7583	666	9	by	by	ADP
ap-7583	666	10	the	the	DET
ap-7583	666	11	aforementioned	aforementione	VERB
ap-7583	666	12	determinant	determinant	NOUN
ap-7583	666	13	are	be	AUX
ap-7583	666	14	all	all	ADV
ap-7583	666	15	simple	simple	ADJ
ap-7583	666	16	.	.	PUNCT
ap-7583	667	1	this	this	DET
ap-7583	667	2	fact	fact	NOUN
ap-7583	667	3	can	can	AUX
ap-7583	667	4	be	be	AUX
ap-7583	667	5	confirmed	confirm	VERB
ap-7583	667	6	by	by	ADP
ap-7583	667	7	establishing	establish	VERB
ap-7583	667	8	the	the	DET
ap-7583	667	9	christoffel	christoffel	NOUN
ap-7583	667	10	-	-	PUNCT
ap-7583	667	11	darboux	darboux	VERB
ap-7583	667	12	formula	formula	NOUN
ap-7583	667	13	.	.	PUNCT
ap-7583	668	1	denote	denote	VERB
ap-7583	668	2	x	x	X
ap-7583	669	1	=	=	PUNCT
ap-7583	669	2	ε0;k	ε0;k	PROPN
ap-7583	669	3	and	and	CCONJ
ap-7583	669	4	y	y	PROPN
ap-7583	669	5	=	=	SYM
ap-7583	669	6	ε0;k′	ε0;k′	PROPN
ap-7583	669	7	,	,	PUNCT
ap-7583	669	8	where	where	SCONJ
ap-7583	669	9	k	k	PROPN
ap-7583	669	10	̸=	̸=	PROPN
ap-7583	669	11	k′	k′	PROPN
ap-7583	669	12	and	and	CCONJ
ap-7583	669	13	k	k	PROPN
ap-7583	669	14	,	,	PUNCT
ap-7583	669	15	k′	k′	PROPN
ap-7583	669	16	=	=	SYM
ap-7583	669	17	0	0	NUM
ap-7583	669	18	,	,	PUNCT
ap-7583	669	19	1	1	NUM
ap-7583	669	20	,	,	PUNCT
ap-7583	669	21	2	2	NUM
ap-7583	669	22	,	,	PUNCT
ap-7583	669	23	·	·	PUNCT
ap-7583	669	24	·	·	PUNCT
ap-7583	669	25	·	·	PUNCT
ap-7583	669	26	,	,	PUNCT
ap-7583	669	27	n	n	CCONJ
ap-7583	669	28	−	−	PROPN
ap-7583	669	29	1	1	NUM
ap-7583	669	30	:	:	PUNCT
ap-7583	669	31	for	for	ADP
ap-7583	669	32	x	x	SYM
ap-7583	669	33	̸=	̸=	PROPN
ap-7583	669	34	y	y	PROPN
ap-7583	669	35	k∑	k∑	PROPN
ap-7583	669	36	j=0	j=0	PROPN
ap-7583	669	37	pn	pn	PROPN
ap-7583	669	38	j	j	PROPN
ap-7583	669	39	(	(	PUNCT
ap-7583	669	40	x)pn	x)pn	PROPN
ap-7583	669	41	j	j	PROPN
ap-7583	669	42	(	(	PUNCT
ap-7583	669	43	y	y	NOUN
ap-7583	669	44	)	)	PUNCT
ap-7583	669	45	j!(α1α3)j(−n)j	j!(α1α3)j(−n)j	PROPN
ap-7583	669	46	(	(	PUNCT
ap-7583	669	47	β0	β0	PROPN
ap-7583	669	48	α1	α1	PROPN
ap-7583	669	49	)	)	PUNCT
ap-7583	669	50	j	j	PROPN
ap-7583	670	1	(	(	PUNCT
ap-7583	670	2	β2	β2	NOUN
ap-7583	670	3	α3	α3	PROPN
ap-7583	670	4	+	+	CCONJ
ap-7583	670	5	n	n	CCONJ
ap-7583	670	6	−	−	PROPN
ap-7583	670	7	1	1	NUM
ap-7583	670	8	)	)	PUNCT
ap-7583	670	9	j	j	NOUN
ap-7583	671	1	=	=	PUNCT
ap-7583	671	2	pn	pn	PROPN
ap-7583	671	3	k+1(x)pn	k+1(x)pn	PROPN
ap-7583	671	4	k	k	PROPN
ap-7583	671	5	(	(	PUNCT
ap-7583	671	6	y	y	NOUN
ap-7583	671	7	)	)	PUNCT
ap-7583	671	8	−	−	PROPN
ap-7583	672	1	pn	pn	PROPN
ap-7583	672	2	k	k	PROPN
ap-7583	672	3	(	(	PUNCT
ap-7583	672	4	x)pn	x)pn	PROPN
ap-7583	672	5	k+1(y	k+1(y	PROPN
ap-7583	672	6	)	)	PUNCT
ap-7583	673	1	k!(α1α3)k(−n)k	k!(α1α3)k(−n)k	PROPN
ap-7583	673	2	(	(	PUNCT
ap-7583	673	3	β0	β0	PROPN
ap-7583	673	4	α1	α1	PROPN
ap-7583	673	5	)	)	PUNCT
ap-7583	674	1	k	k	PROPN
ap-7583	675	1	(	(	PUNCT
ap-7583	675	2	β2	β2	VERB
ap-7583	675	3	α3	α3	NOUN
ap-7583	675	4	+	+	CCONJ
ap-7583	675	5	n	n	CCONJ
ap-7583	675	6	−	−	PROPN
ap-7583	675	7	1	1	NUM
ap-7583	675	8	)	)	PUNCT
ap-7583	675	9	k	k	NOUN
ap-7583	676	1	(	(	PUNCT
ap-7583	676	2	x	x	PROPN
ap-7583	676	3	−	−	PROPN
ap-7583	676	4	y	y	PROPN
ap-7583	676	5	)	)	PUNCT
ap-7583	676	6	,	,	PUNCT
ap-7583	676	7	(	(	PUNCT
ap-7583	676	8	114	114	NUM
ap-7583	676	9	)	)	PUNCT
ap-7583	676	10	while	while	SCONJ
ap-7583	676	11	,	,	PUNCT
ap-7583	676	12	for	for	ADP
ap-7583	676	13	the	the	DET
ap-7583	676	14	limit	limit	NOUN
ap-7583	676	15	y	y	PROPN
ap-7583	676	16	→	→	SYM
ap-7583	676	17	x	x	PROPN
ap-7583	676	18	,	,	PUNCT
ap-7583	676	19	k∑	k∑	VERB
ap-7583	676	20	j=0	j=0	PROPN
ap-7583	676	21	(	(	PUNCT
ap-7583	676	22	pn	pn	PROPN
ap-7583	676	23	j	j	PROPN
ap-7583	676	24	(	(	PUNCT
ap-7583	676	25	x	x	NOUN
ap-7583	676	26	)	)	PUNCT
ap-7583	676	27	)	)	PUNCT
ap-7583	676	28	2	2	NUM
ap-7583	676	29	j!(α1α3)j(−n)j	j!(α1α3)j(−n)j	PROPN
ap-7583	676	30	(	(	PUNCT
ap-7583	676	31	β0	β0	PROPN
ap-7583	676	32	α1	α1	PROPN
ap-7583	676	33	)	)	PUNCT
ap-7583	676	34	j	j	PROPN
ap-7583	676	35	(	(	PUNCT
ap-7583	676	36	β2	β2	NOUN
ap-7583	676	37	α3	α3	PROPN
ap-7583	676	38	+	+	CCONJ
ap-7583	676	39	n	n	CCONJ
ap-7583	676	40	−	−	PROPN
ap-7583	676	41	1	1	NUM
ap-7583	676	42	)	)	PUNCT
ap-7583	676	43	j	j	NOUN
ap-7583	677	1	=	=	PUNCT
ap-7583	678	1	[	[	X
ap-7583	678	2	pn	pn	PROPN
ap-7583	678	3	k+1(x)]′pn	k+1(x)]′pn	PROPN
ap-7583	678	4	k	k	PROPN
ap-7583	678	5	(	(	PUNCT
ap-7583	678	6	x	x	NOUN
ap-7583	678	7	)	)	PUNCT
ap-7583	678	8	−	−	PROPN
ap-7583	679	1	[	[	X
ap-7583	679	2	pn	pn	X
ap-7583	679	3	k	k	X
ap-7583	679	4	(	(	PUNCT
ap-7583	679	5	x)]′pn	x)]′pn	PROPN
ap-7583	679	6	k+1(x	k+1(x	PROPN
ap-7583	679	7	)	)	PUNCT
ap-7583	679	8	k!(α1α3)k(−n)k	k!(α1α3)k(−n)k	PROPN
ap-7583	679	9	(	(	PUNCT
ap-7583	679	10	β0	β0	PROPN
ap-7583	679	11	α1	α1	PROPN
ap-7583	679	12	)	)	PUNCT
ap-7583	680	1	k	k	PROPN
ap-7583	681	1	(	(	PUNCT
ap-7583	681	2	β2	β2	VERB
ap-7583	681	3	α3	α3	NOUN
ap-7583	681	4	+	+	CCONJ
ap-7583	681	5	n	n	CCONJ
ap-7583	681	6	−	−	PROPN
ap-7583	681	7	1	1	NUM
ap-7583	681	8	)	)	PUNCT
ap-7583	681	9	k	k	PROPN
ap-7583	681	10	.	.	PUNCT
ap-7583	682	1	(	(	PUNCT
ap-7583	682	2	115	115	NUM
ap-7583	682	3	)	)	SYM
ap-7583	682	4	185	185	NUM
ap-7583	682	5	nasser	nasser	PROPN
ap-7583	682	6	saad	saad	PROPN
ap-7583	682	7	acta	acta	PROPN
ap-7583	682	8	polytechnica	polytechnica	PROPN
ap-7583	682	9	here	here	ADV
ap-7583	682	10	,	,	PUNCT
ap-7583	682	11	the	the	DET
ap-7583	682	12	prime	prime	NOUN
ap-7583	682	13	refers	refer	VERB
ap-7583	682	14	to	to	ADP
ap-7583	682	15	the	the	DET
ap-7583	682	16	derivative	derivative	NOUN
ap-7583	682	17	with	with	ADP
ap-7583	682	18	respect	respect	NOUN
ap-7583	682	19	to	to	ADP
ap-7583	682	20	the	the	DET
ap-7583	682	21	variable	variable	ADJ
ap-7583	682	22	x.	x.	NOUN
ap-7583	683	1	if	if	SCONJ
ap-7583	683	2	x	x	PRON
ap-7583	683	3	=	=	SYM
ap-7583	683	4	xk	xk	PROPN
ap-7583	683	5	is	be	AUX
ap-7583	683	6	a	a	DET
ap-7583	683	7	zero	zero	NUM
ap-7583	683	8	of	of	ADP
ap-7583	683	9	the	the	DET
ap-7583	683	10	polynomial	polynomial	ADJ
ap-7583	683	11	pn	pn	PROPN
ap-7583	683	12	k	k	PROPN
ap-7583	683	13	(	(	PUNCT
ap-7583	683	14	x	x	X
ap-7583	683	15	)	)	PUNCT
ap-7583	683	16	with	with	ADP
ap-7583	683	17	multiplicity	multiplicity	NOUN
ap-7583	683	18	>	>	X
ap-7583	683	19	1	1	NUM
ap-7583	683	20	,	,	PUNCT
ap-7583	683	21	then	then	ADV
ap-7583	683	22	pn	pn	PROPN
ap-7583	683	23	k	k	PROPN
ap-7583	683	24	(	(	PUNCT
ap-7583	683	25	xk	xk	PROPN
ap-7583	683	26	)	)	PUNCT
ap-7583	683	27	=	=	SYM
ap-7583	683	28	0	0	NUM
ap-7583	683	29	and	and	CCONJ
ap-7583	683	30	(	(	PUNCT
ap-7583	683	31	115	115	NUM
ap-7583	683	32	)	)	PUNCT
ap-7583	683	33	yields	yield	VERB
ap-7583	683	34	the	the	DET
ap-7583	683	35	contradiction	contradiction	NOUN
ap-7583	683	36	0	0	PUNCT
ap-7583	683	37	<	<	X
ap-7583	683	38	k−1∑	k−1∑	PROPN
ap-7583	683	39	j=0	j=0	PROPN
ap-7583	683	40	(	(	PUNCT
ap-7583	683	41	pn	pn	PROPN
ap-7583	683	42	j	j	PROPN
ap-7583	683	43	(	(	PUNCT
ap-7583	683	44	xi	xi	PROPN
ap-7583	683	45	)	)	PUNCT
ap-7583	683	46	)	)	PUNCT
ap-7583	683	47	2	2	NUM
ap-7583	683	48	j!(α1α3)j(−n)j	j!(α1α3)j(−n)j	PROPN
ap-7583	683	49	(	(	PUNCT
ap-7583	683	50	β0	β0	PROPN
ap-7583	683	51	α1	α1	PROPN
ap-7583	683	52	)	)	PUNCT
ap-7583	683	53	j	j	PROPN
ap-7583	684	1	(	(	PUNCT
ap-7583	684	2	β2	β2	NOUN
ap-7583	684	3	α3	α3	PROPN
ap-7583	684	4	+	+	CCONJ
ap-7583	684	5	n	n	CCONJ
ap-7583	684	6	−	−	PROPN
ap-7583	684	7	1	1	NUM
ap-7583	684	8	)	)	PUNCT
ap-7583	684	9	j	j	NOUN
ap-7583	685	1	=	=	SYM
ap-7583	685	2	0	0	PROPN
ap-7583	685	3	,	,	PUNCT
ap-7583	685	4	(	(	PUNCT
ap-7583	685	5	116	116	NUM
ap-7583	685	6	)	)	PUNCT
ap-7583	685	7	and	and	CCONJ
ap-7583	685	8	the	the	DET
ap-7583	685	9	zeros	zero	NOUN
ap-7583	685	10	of	of	ADP
ap-7583	685	11	the	the	DET
ap-7583	685	12	polynomial	polynomial	ADJ
ap-7583	685	13	p	p	PROPN
ap-7583	685	14	n	n	CCONJ
ap-7583	685	15	k	k	PROPN
ap-7583	685	16	(	(	PUNCT
ap-7583	685	17	x	x	NOUN
ap-7583	685	18	)	)	PUNCT
ap-7583	685	19	,	,	PUNCT
ap-7583	686	1	k	k	PROPN
ap-7583	686	2	=	=	SYM
ap-7583	686	3	1	1	NUM
ap-7583	686	4	,	,	PUNCT
ap-7583	686	5	2	2	NUM
ap-7583	686	6	,	,	PUNCT
ap-7583	686	7	·	·	PUNCT
ap-7583	686	8	·	·	PUNCT
ap-7583	686	9	·	·	PUNCT
ap-7583	687	1	n	n	PRON
ap-7583	687	2	are	be	AUX
ap-7583	687	3	distinct	distinct	ADJ
ap-7583	687	4	.	.	PUNCT
ap-7583	688	1	6.1	6.1	NUM
ap-7583	688	2	.	.	PUNCT
ap-7583	688	3	norms	norm	NOUN
ap-7583	688	4	of	of	ADP
ap-7583	688	5	the	the	DET
ap-7583	688	6	orthogonal	orthogonal	ADJ
ap-7583	688	7	polynomials	polynomial	NOUN
ap-7583	688	8	denote	denote	VERB
ap-7583	688	9	ε0;n	ε0;n	NOUN
ap-7583	688	10	=	=	SYM
ap-7583	689	1	x	x	NOUN
ap-7583	689	2	,	,	PUNCT
ap-7583	689	3	the	the	DET
ap-7583	689	4	general	general	ADJ
ap-7583	689	5	theory	theory	NOUN
ap-7583	689	6	of	of	ADP
ap-7583	689	7	orthogonal	orthogonal	ADJ
ap-7583	689	8	polynomials	polynomial	NOUN
ap-7583	689	9	[	[	X
ap-7583	689	10	27	27	NUM
ap-7583	689	11	]	]	PUNCT
ap-7583	689	12	guarantees	guarantee	VERB
ap-7583	689	13	that	that	SCONJ
ap-7583	689	14	the	the	DET
ap-7583	689	15	finite	finite	ADJ
ap-7583	689	16	sequence	sequence	NOUN
ap-7583	689	17	of	of	ADP
ap-7583	689	18	polynomials	polynomial	NOUN
ap-7583	689	19	{	{	PUNCT
ap-7583	689	20	pk(x)}n	pk(x)}n	PROPN
ap-7583	689	21	k=0	k=0	PROPN
ap-7583	689	22	form	form	VERB
ap-7583	689	23	a	a	DET
ap-7583	689	24	set	set	NOUN
ap-7583	689	25	of	of	ADP
ap-7583	689	26	orthogonal	orthogonal	ADJ
ap-7583	689	27	polynomials	polynomial	NOUN
ap-7583	689	28	for	for	ADP
ap-7583	689	29	each	each	DET
ap-7583	689	30	n.	n.	NOUN
ap-7583	689	31	this	this	PRON
ap-7583	689	32	implies	imply	VERB
ap-7583	689	33	the	the	DET
ap-7583	689	34	existence	existence	NOUN
ap-7583	689	35	of	of	ADP
ap-7583	689	36	a	a	DET
ap-7583	689	37	certain	certain	ADJ
ap-7583	689	38	weight	weight	NOUN
ap-7583	689	39	function	function	NOUN
ap-7583	689	40	,	,	PUNCT
ap-7583	689	41	w(x	w(x	PROPN
ap-7583	689	42	)	)	PUNCT
ap-7583	689	43	,	,	PUNCT
ap-7583	689	44	which	which	PRON
ap-7583	689	45	can	can	AUX
ap-7583	689	46	be	be	AUX
ap-7583	689	47	normalized	normalize	VERB
ap-7583	689	48	as∫	as∫	PROPN
ap-7583	689	49	dw	dw	PROPN
ap-7583	689	50	=	=	SYM
ap-7583	689	51	1	1	NUM
ap-7583	689	52	,	,	PUNCT
ap-7583	689	53	(	(	PUNCT
ap-7583	689	54	117	117	NUM
ap-7583	689	55	)	)	PUNCT
ap-7583	689	56	for	for	ADP
ap-7583	689	57	which	which	PRON
ap-7583	689	58	∫	∫	PROPN
ap-7583	689	59	pk(x)pk′(x)dw	pk(x)pk′(x)dw	NOUN
ap-7583	689	60	=	=	SYM
ap-7583	689	61	pk	pk	NOUN
ap-7583	689	62	pk′	pk′	PROPN
ap-7583	689	63	δkk′	δkk′	PROPN
ap-7583	689	64	,	,	PUNCT
ap-7583	689	65	0	0	NUM
ap-7583	689	66	≤	≤	NUM
ap-7583	690	1	k	k	PROPN
ap-7583	690	2	,	,	PUNCT
ap-7583	690	3	k′	k′	PROPN
ap-7583	690	4	≤	≤	ADJ
ap-7583	690	5	n	n	CCONJ
ap-7583	690	6	,	,	PUNCT
ap-7583	690	7	(	(	PUNCT
ap-7583	690	8	118	118	NUM
ap-7583	690	9	)	)	PUNCT
ap-7583	690	10	where	where	SCONJ
ap-7583	690	11	pk	pk	NOUN
ap-7583	690	12	denotes	denote	VERB
ap-7583	690	13	the	the	DET
ap-7583	690	14	norms	norm	NOUN
ap-7583	690	15	of	of	ADP
ap-7583	690	16	polynomials	polynomial	NOUN
ap-7583	690	17	pk(x	pk(x	NOUN
ap-7583	690	18	)	)	PUNCT
ap-7583	690	19	.	.	PUNCT
ap-7583	691	1	these	these	DET
ap-7583	691	2	norms	norm	NOUN
ap-7583	691	3	can	can	AUX
ap-7583	691	4	be	be	AUX
ap-7583	691	5	found	find	VERB
ap-7583	691	6	from	from	ADP
ap-7583	691	7	the	the	DET
ap-7583	691	8	recurrence	recurrence	NOUN
ap-7583	691	9	relations	relation	NOUN
ap-7583	691	10	(	(	PUNCT
ap-7583	691	11	36	36	NUM
ap-7583	691	12	)	)	PUNCT
ap-7583	691	13	by	by	ADP
ap-7583	691	14	multiplying	multiply	VERB
ap-7583	691	15	with	with	ADP
ap-7583	691	16	xk−1w(x	xk−1w(x	PROPN
ap-7583	691	17	)	)	PUNCT
ap-7583	691	18	and	and	CCONJ
ap-7583	691	19	taking	take	VERB
ap-7583	691	20	the	the	DET
ap-7583	691	21	integral	integral	ADJ
ap-7583	691	22	over	over	ADP
ap-7583	691	23	x	x	SYM
ap-7583	691	24	yields	yield	NOUN
ap-7583	691	25	the	the	DET
ap-7583	691	26	recurrence	recurrence	NOUN
ap-7583	691	27	formula∫	formula∫	PROPN
ap-7583	691	28	xk	xk	PROPN
ap-7583	691	29	pn	pn	PROPN
ap-7583	691	30	k	k	PROPN
ap-7583	691	31	(	(	PUNCT
ap-7583	691	32	x	x	X
ap-7583	691	33	)	)	PUNCT
ap-7583	691	34	w(x	w(x	NOUN
ap-7583	691	35	)	)	PUNCT
ap-7583	691	36	dx	dx	PROPN
ap-7583	692	1	=	=	PRON
ap-7583	692	2	−k(n	−k(n	PROPN
ap-7583	692	3	−	−	PROPN
ap-7583	693	1	k	k	PROPN
ap-7583	694	1	+	+	PROPN
ap-7583	694	2	1	1	X
ap-7583	694	3	)	)	PUNCT
ap-7583	694	4	(	(	PUNCT
ap-7583	694	5	(	(	PUNCT
ap-7583	694	6	k	k	X
ap-7583	694	7	−	−	PROPN
ap-7583	694	8	1)α1	1)α1	NUM
ap-7583	694	9	+	+	CCONJ
ap-7583	694	10	β0	β0	ADJ
ap-7583	694	11	)	)	PUNCT
ap-7583	694	12	×	×	NOUN
ap-7583	694	13	(	(	PUNCT
ap-7583	694	14	β2	β2	VERB
ap-7583	694	15	+	+	CCONJ
ap-7583	694	16	α3(k	α3(k	PROPN
ap-7583	694	17	+	+	CCONJ
ap-7583	694	18	n	n	CCONJ
ap-7583	694	19	−	−	PROPN
ap-7583	694	20	2	2	NUM
ap-7583	694	21	)	)	PUNCT
ap-7583	694	22	)	)	PUNCT
ap-7583	695	1	∫	∫	PROPN
ap-7583	696	1	xk−1pn	xk−1pn	NUM
ap-7583	696	2	k−1(x)w(x)dx	k−1(x)w(x)dx	X
ap-7583	696	3	,	,	PUNCT
ap-7583	696	4	(	(	PUNCT
ap-7583	696	5	119	119	NUM
ap-7583	696	6	)	)	PUNCT
ap-7583	696	7	and	and	CCONJ
ap-7583	696	8	thus	thus	ADV
ap-7583	696	9	∫	∫	PROPN
ap-7583	696	10	pn	pn	PROPN
ap-7583	696	11	k	k	PROPN
ap-7583	696	12	(	(	PUNCT
ap-7583	696	13	x	x	PROPN
ap-7583	696	14	)	)	PUNCT
ap-7583	696	15	xk	xk	PROPN
ap-7583	696	16	w(x	w(x	PROPN
ap-7583	696	17	)	)	PUNCT
ap-7583	696	18	dx	dx	PROPN
ap-7583	697	1	=	=	SYM
ap-7583	697	2	k	k	PROPN
ap-7583	697	3	!	!	PUNCT
ap-7583	698	1	(	(	PUNCT
ap-7583	698	2	α1α3)k	α1α3)k	PROPN
ap-7583	698	3	(	(	PUNCT
ap-7583	698	4	−n)k	−n)k	PROPN
ap-7583	698	5	(	(	PUNCT
ap-7583	698	6	β0	β0	PROPN
ap-7583	698	7	α1	α1	PROPN
ap-7583	698	8	)	)	PUNCT
ap-7583	698	9	k	k	PROPN
ap-7583	698	10	(	(	PUNCT
ap-7583	698	11	β2	β2	VERB
ap-7583	698	12	α3	α3	NOUN
ap-7583	698	13	+	+	CCONJ
ap-7583	698	14	n	n	CCONJ
ap-7583	698	15	−	−	PROPN
ap-7583	698	16	1	1	NUM
ap-7583	698	17	)	)	PUNCT
ap-7583	698	18	k	k	PROPN
ap-7583	698	19	.	.	PUNCT
ap-7583	699	1	(	(	PUNCT
ap-7583	699	2	120	120	NUM
ap-7583	699	3	)	)	PUNCT
ap-7583	699	4	from	from	ADP
ap-7583	699	5	which	which	PRON
ap-7583	699	6	it	it	PRON
ap-7583	699	7	follows∫	follows∫	VERB
ap-7583	700	1	[	[	X
ap-7583	700	2	pn	pn	X
ap-7583	700	3	k	k	X
ap-7583	700	4	(	(	PUNCT
ap-7583	700	5	x)]2w(x)dx	x)]2w(x)dx	X
ap-7583	700	6	=	=	SYM
ap-7583	700	7	p2	p2	PROPN
ap-7583	700	8	k	k	NOUN
ap-7583	700	9	=	=	PUNCT
ap-7583	700	10	k	k	PROPN
ap-7583	700	11	!	!	PUNCT
ap-7583	701	1	(	(	PUNCT
ap-7583	701	2	α1α3)k	α1α3)k	PROPN
ap-7583	701	3	(	(	PUNCT
ap-7583	701	4	−n)k	−n)k	PROPN
ap-7583	701	5	(	(	PUNCT
ap-7583	701	6	β0	β0	PROPN
ap-7583	701	7	α1	α1	PROPN
ap-7583	701	8	)	)	PUNCT
ap-7583	701	9	k	k	PROPN
ap-7583	701	10	(	(	PUNCT
ap-7583	701	11	β2	β2	VERB
ap-7583	701	12	α3	α3	NOUN
ap-7583	701	13	+	+	CCONJ
ap-7583	701	14	n	n	CCONJ
ap-7583	701	15	−	−	PROPN
ap-7583	701	16	1	1	NUM
ap-7583	701	17	)	)	PUNCT
ap-7583	701	18	k	k	NOUN
ap-7583	701	19	(	(	PUNCT
ap-7583	701	20	121	121	NUM
ap-7583	701	21	)	)	PUNCT
ap-7583	701	22	for	for	ADP
ap-7583	701	23	all	all	PRON
ap-7583	701	24	0	0	NUM
ap-7583	701	25	≤	≤	NUM
ap-7583	701	26	k	k	X
ap-7583	701	27	≤	≤	PROPN
ap-7583	701	28	n.	n.	NOUN
ap-7583	701	29	because	because	SCONJ
ap-7583	701	30	of	of	ADP
ap-7583	701	31	the	the	DET
ap-7583	701	32	pochhammer	pochhammer	NOUN
ap-7583	701	33	identity	identity	NOUN
ap-7583	701	34	(	(	PUNCT
ap-7583	701	35	−n)k	−n)k	ADJ
ap-7583	701	36	=	=	SYM
ap-7583	701	37	0	0	NUM
ap-7583	701	38	for	for	ADP
ap-7583	701	39	k	k	PROPN
ap-7583	701	40	>	>	PUNCT
ap-7583	701	41	n	n	CCONJ
ap-7583	701	42	,	,	PUNCT
ap-7583	701	43	it	it	PRON
ap-7583	701	44	follows	follow	VERB
ap-7583	701	45	from	from	ADP
ap-7583	701	46	(	(	PUNCT
ap-7583	701	47	71	71	NUM
ap-7583	701	48	)	)	PUNCT
ap-7583	702	1	that	that	SCONJ
ap-7583	702	2	the	the	DET
ap-7583	702	3	norms	norm	NOUN
ap-7583	702	4	of	of	ADP
ap-7583	702	5	all	all	DET
ap-7583	702	6	polynomials	polynomial	NOUN
ap-7583	702	7	pn	pn	PROPN
ap-7583	702	8	k	k	PROPN
ap-7583	702	9	(	(	PUNCT
ap-7583	702	10	x	x	X
ap-7583	702	11	)	)	PUNCT
ap-7583	702	12	with	with	ADP
ap-7583	702	13	k	k	PROPN
ap-7583	702	14	≥	≥	PROPN
ap-7583	702	15	n	n	PROPN
ap-7583	702	16	+	+	CCONJ
ap-7583	702	17	1	1	NUM
ap-7583	702	18	vanish	vanish	NOUN
ap-7583	703	1	.	.	PUNCT
ap-7583	704	1	thus	thus	ADV
ap-7583	704	2	pk	pk	X
ap-7583	704	3	=	=	NOUN
ap-7583	704	4	0	0	PROPN
ap-7583	704	5	,	,	PUNCT
ap-7583	704	6	k	k	X
ap-7583	704	7	≥	≥	PROPN
ap-7583	704	8	n	n	PROPN
ap-7583	704	9	+	+	NUM
ap-7583	704	10	1	1	NUM
ap-7583	704	11	.	.	PUNCT
ap-7583	705	1	(	(	PUNCT
ap-7583	705	2	122	122	NUM
ap-7583	705	3	)	)	PUNCT
ap-7583	705	4	we	we	PRON
ap-7583	705	5	may	may	AUX
ap-7583	705	6	also	also	ADV
ap-7583	705	7	note	note	VERB
ap-7583	705	8	,	,	PUNCT
ap-7583	705	9	using	use	VERB
ap-7583	705	10	the	the	DET
ap-7583	705	11	recurrence	recurrence	NOUN
ap-7583	705	12	relation	relation	NOUN
ap-7583	705	13	,	,	PUNCT
ap-7583	705	14	that∫	that∫	NOUN
ap-7583	705	15	x[pk(x)]2w(x)dx	x[pk(x)]2w(x)dx	PUNCT
ap-7583	706	1	=	=	PUNCT
ap-7583	706	2	−k((k	−k((k	NOUN
ap-7583	706	3	−	−	PROPN
ap-7583	706	4	1)α2	1)α2	PROPN
ap-7583	706	5	+	+	CCONJ
ap-7583	706	6	β1	β1	PROPN
ap-7583	706	7	)	)	PUNCT
ap-7583	707	1	k	k	X
ap-7583	707	2	!	!	PUNCT
ap-7583	708	1	(	(	PUNCT
ap-7583	708	2	α1α3)k	α1α3)k	PROPN
ap-7583	708	3	(	(	PUNCT
ap-7583	708	4	−n)k	−n)k	PROPN
ap-7583	708	5	(	(	PUNCT
ap-7583	708	6	β0	β0	PROPN
ap-7583	708	7	α1	α1	PROPN
ap-7583	708	8	)	)	PUNCT
ap-7583	708	9	k	k	PROPN
ap-7583	708	10	(	(	PUNCT
ap-7583	708	11	β2	β2	VERB
ap-7583	708	12	α3	α3	NOUN
ap-7583	708	13	+	+	CCONJ
ap-7583	708	14	n	n	CCONJ
ap-7583	708	15	−	−	PROPN
ap-7583	708	16	1	1	NUM
ap-7583	708	17	)	)	PUNCT
ap-7583	708	18	k	k	PROPN
ap-7583	708	19	.	.	PUNCT
ap-7583	709	1	(	(	PUNCT
ap-7583	709	2	123	123	NUM
ap-7583	709	3	)	)	PUNCT
ap-7583	709	4	6.2	6.2	NUM
ap-7583	709	5	.	.	PUNCT
ap-7583	710	1	the	the	DET
ap-7583	710	2	zeros	zero	NOUN
ap-7583	710	3	of	of	ADP
ap-7583	710	4	the	the	DET
ap-7583	710	5	polynomials	polynomial	NOUN
ap-7583	710	6	{	{	PUNCT
ap-7583	710	7	pn	pn	NOUN
ap-7583	710	8	k	k	PROPN
ap-7583	710	9	(	(	PUNCT
ap-7583	710	10	ε0;n)}n	ε0;n)}n	PROPN
ap-7583	710	11	k=0	k=0	PUNCT
ap-7583	710	12	one	one	NUM
ap-7583	710	13	of	of	ADP
ap-7583	710	14	the	the	DET
ap-7583	710	15	important	important	ADJ
ap-7583	710	16	properties	property	NOUN
ap-7583	710	17	of	of	ADP
ap-7583	710	18	the	the	DET
ap-7583	710	19	polynomials	polynomial	NOUN
ap-7583	710	20	pn	pn	PROPN
ap-7583	710	21	n+1(ε0;n	n+1(ε0;n	PROPN
ap-7583	710	22	)	)	PUNCT
ap-7583	710	23	concerns	concern	VERB
ap-7583	710	24	their	their	PRON
ap-7583	710	25	zeros	zero	NOUN
ap-7583	710	26	.	.	PUNCT
ap-7583	711	1	an	an	DET
ap-7583	711	2	argument	argument	NOUN
ap-7583	711	3	provided	provide	VERB
ap-7583	711	4	by	by	ADP
ap-7583	711	5	arscott	arscott	PROPN
ap-7583	711	6	[	[	X
ap-7583	711	7	24	24	NUM
ap-7583	711	8	]	]	PUNCT
ap-7583	711	9	proves	prove	VERB
ap-7583	711	10	that	that	SCONJ
ap-7583	711	11	if	if	SCONJ
ap-7583	711	12	the	the	DET
ap-7583	711	13	product	product	NOUN
ap-7583	711	14	(	(	PUNCT
ap-7583	711	15	γk	γk	PROPN
ap-7583	711	16	·	·	PUNCT
ap-7583	711	17	tk	tk	PROPN
ap-7583	711	18	)	)	PUNCT
ap-7583	711	19	>	>	X
ap-7583	711	20	0	0	PUNCT
ap-7583	711	21	for	for	ADP
ap-7583	711	22	all	all	PRON
ap-7583	711	23	k	k	NOUN
ap-7583	711	24	=	=	SYM
ap-7583	711	25	1	1	NUM
ap-7583	711	26	,	,	PUNCT
ap-7583	711	27	2	2	NUM
ap-7583	711	28	,	,	PUNCT
ap-7583	711	29	.	.	PUNCT
ap-7583	711	30	.	.	PUNCT
ap-7583	711	31	.	.	PUNCT
ap-7583	712	1	,	,	PUNCT
ap-7583	712	2	n	n	CCONJ
ap-7583	712	3	,	,	PUNCT
ap-7583	712	4	then	then	ADV
ap-7583	712	5	the	the	DET
ap-7583	712	6	polynomials	polynomial	NOUN
ap-7583	712	7	that	that	PRON
ap-7583	712	8	satisfy	satisfy	VERB
ap-7583	712	9	the	the	DET
ap-7583	712	10	tri	tri	ADJ
ap-7583	712	11	-	-	ADJ
ap-7583	712	12	diagonal	diagonal	ADJ
ap-7583	712	13	determinant	determinant	ADJ
ap-7583	712	14	(	(	PUNCT
ap-7583	712	15	67	67	NUM
ap-7583	712	16	)	)	PUNCT
ap-7583	712	17	are	be	AUX
ap-7583	712	18	real	real	ADJ
ap-7583	712	19	and	and	CCONJ
ap-7583	712	20	simple	simple	ADJ
ap-7583	712	21	.	.	PUNCT
ap-7583	713	1	let	let	VERB
ap-7583	713	2	us	we	PRON
ap-7583	713	3	denote	denote	VERB
ap-7583	713	4	that	that	SCONJ
ap-7583	713	5	the	the	DET
ap-7583	713	6	roots	root	NOUN
ap-7583	713	7	of	of	ADP
ap-7583	713	8	the	the	DET
ap-7583	713	9	polynomials	polynomial	NOUN
ap-7583	713	10	pn	pn	PROPN
ap-7583	713	11	n+1(ε0;n	n+1(ε0;n	PROPN
ap-7583	713	12	)	)	PUNCT
ap-7583	713	13	=	=	NOUN
ap-7583	713	14	0	0	NUM
ap-7583	713	15	by	by	ADP
ap-7583	713	16	εℓ	εℓ	NOUN
ap-7583	713	17	0;n	0;n	NOUN
ap-7583	713	18	,	,	PUNCT
ap-7583	713	19	ℓ	ℓ	PROPN
ap-7583	713	20	=	=	SYM
ap-7583	713	21	0	0	NUM
ap-7583	713	22	,	,	PUNCT
ap-7583	713	23	1	1	NUM
ap-7583	713	24	,	,	PUNCT
ap-7583	713	25	.	.	PUNCT
ap-7583	713	26	.	.	PUNCT
ap-7583	714	1	.	.	PUNCT
ap-7583	715	1	,	,	PUNCT
ap-7583	715	2	n	n	PRON
ap-7583	715	3	such	such	ADJ
ap-7583	715	4	that	that	SCONJ
ap-7583	715	5	pn	pn	PROPN
ap-7583	715	6	n+1(εℓ	n+1(εℓ	PROPN
ap-7583	715	7	0;n	0;n	NUM
ap-7583	715	8	)	)	PUNCT
ap-7583	715	9	=	=	SYM
ap-7583	715	10	0	0	NUM
ap-7583	715	11	,	,	PUNCT
ap-7583	715	12	(	(	PUNCT
ap-7583	715	13	124	124	NUM
ap-7583	715	14	)	)	PUNCT
ap-7583	715	15	where	where	SCONJ
ap-7583	715	16	ε0	ε0	PROPN
ap-7583	715	17	0;n	0;n	NOUN
ap-7583	715	18	<	<	X
ap-7583	715	19	ε1	ε1	VERB
ap-7583	715	20	0;n	0;n	NOUN
ap-7583	715	21	<	<	X
ap-7583	715	22	·	·	PUNCT
ap-7583	715	23	·	·	PUNCT
ap-7583	715	24	·	·	PUNCT
ap-7583	716	1	<	<	X
ap-7583	716	2	εn	εn	ADP
ap-7583	716	3	0;n	0;n	NOUN
ap-7583	716	4	.	.	PUNCT
ap-7583	717	1	186	186	NUM
ap-7583	717	2	vol	vol	NOUN
ap-7583	717	3	.	.	PUNCT
ap-7583	718	1	62	62	NUM
ap-7583	718	2	no	no	INTJ
ap-7583	718	3	.	.	PUNCT
ap-7583	719	1	1/2022	1/2022	NUM
ap-7583	719	2	on	on	ADP
ap-7583	719	3	generalized	generalized	ADJ
ap-7583	719	4	heun	heun	NOUN
ap-7583	719	5	equation	equation	NOUN
ap-7583	719	6	with	with	ADP
ap-7583	719	7	some	some	DET
ap-7583	719	8	mathematical	mathematical	NOUN
ap-7583	719	9	.	.	PUNCT
ap-7583	719	10	.	.	PUNCT
ap-7583	719	11	.	.	PUNCT
ap-7583	720	1	in	in	ADP
ap-7583	720	2	particular	particular	ADJ
ap-7583	720	3	,	,	PUNCT
ap-7583	720	4	since	since	SCONJ
ap-7583	720	5	pn	pn	PROPN
ap-7583	720	6	n+1(ε0;n	n+1(ε0;n	PROPN
ap-7583	720	7	)	)	PUNCT
ap-7583	720	8	is	be	AUX
ap-7583	720	9	of	of	ADP
ap-7583	720	10	degree	degree	NOUN
ap-7583	720	11	n	n	NOUN
ap-7583	720	12	+	+	CCONJ
ap-7583	720	13	1	1	NUM
ap-7583	720	14	and	and	CCONJ
ap-7583	720	15	all	all	DET
ap-7583	720	16	the	the	DET
ap-7583	720	17	roots	root	NOUN
ap-7583	720	18	are	be	AUX
ap-7583	720	19	simple	simple	ADJ
ap-7583	720	20	and	and	CCONJ
ap-7583	720	21	different	different	ADJ
ap-7583	720	22	,	,	PUNCT
ap-7583	720	23	it	it	PRON
ap-7583	720	24	follows	follow	VERB
ap-7583	720	25	that	that	SCONJ
ap-7583	720	26	pn	pn	PROPN
ap-7583	720	27	n+1(ε0;n	n+1(ε0;n	PROPN
ap-7583	720	28	)	)	PUNCT
ap-7583	720	29	=	=	SYM
ap-7583	720	30	n∏	n∏	PROPN
ap-7583	720	31	ℓ=0	ℓ=0	SYM
ap-7583	720	32	(	(	PUNCT
ap-7583	720	33	ε0;n	ε0;n	NOUN
ap-7583	720	34	−	−	PROPN
ap-7583	720	35	εℓ	εℓ	NOUN
ap-7583	720	36	0;n	0;n	NUM
ap-7583	720	37	)	)	PUNCT
ap-7583	720	38	.	.	PUNCT
ap-7583	721	1	(	(	PUNCT
ap-7583	721	2	125	125	NUM
ap-7583	721	3	)	)	PUNCT
ap-7583	721	4	the	the	DET
ap-7583	721	5	‘	'	PUNCT
ap-7583	721	6	discrete	discrete	ADJ
ap-7583	721	7	’	'	PUNCT
ap-7583	721	8	weight	weight	NOUN
ap-7583	721	9	function	function	NOUN
ap-7583	721	10	w	w	NOUN
ap-7583	721	11	can	can	AUX
ap-7583	721	12	be	be	AUX
ap-7583	721	13	computed	compute	VERB
ap-7583	721	14	numerically	numerically	ADV
ap-7583	721	15	[	[	X
ap-7583	721	16	28	28	NUM
ap-7583	721	17	]	]	X
ap-7583	721	18	using	use	VERB
ap-7583	721	19	(	(	PUNCT
ap-7583	721	20	118	118	NUM
ap-7583	721	21	)	)	PUNCT
ap-7583	721	22	,	,	PUNCT
ap-7583	721	23	(	(	PUNCT
ap-7583	721	24	119	119	NUM
ap-7583	721	25	)	)	PUNCT
ap-7583	721	26	and	and	CCONJ
ap-7583	721	27	(	(	PUNCT
ap-7583	721	28	125	125	NUM
ap-7583	721	29	)	)	PUNCT
ap-7583	721	30	for	for	ADP
ap-7583	721	31	the	the	DET
ap-7583	721	32	given	give	VERB
ap-7583	721	33	n.	n.	PROPN
ap-7583	721	34	denote	denote	NOUN
ap-7583	721	35	pℓ(ε0	pℓ(ε0	NOUN
ap-7583	721	36	)	)	PUNCT
ap-7583	721	37	=	=	PUNCT
ap-7583	721	38	pn	pn	PROPN
ap-7583	721	39	ℓ	ℓ	PROPN
ap-7583	721	40	(	(	PUNCT
ap-7583	721	41	ε0	ε0	PROPN
ap-7583	721	42	)	)	PUNCT
ap-7583	721	43	,	,	PUNCT
ap-7583	721	44	and	and	CCONJ
ap-7583	721	45	let	let	VERB
ap-7583	721	46	the	the	DET
ap-7583	721	47	roots	root	NOUN
ap-7583	721	48	of	of	ADP
ap-7583	721	49	pn	pn	PROPN
ap-7583	721	50	n+1(ε0;n	n+1(ε0;n	PROPN
ap-7583	721	51	)	)	PUNCT
ap-7583	722	1	=	=	SYM
ap-7583	722	2	0	0	NUM
ap-7583	722	3	be	be	AUX
ap-7583	722	4	εj	εj	NOUN
ap-7583	722	5	0;n	0;n	NOUN
ap-7583	722	6	arranged	arrange	VERB
ap-7583	722	7	in	in	ADP
ap-7583	722	8	ascending	ascend	VERB
ap-7583	722	9	order	order	NOUN
ap-7583	722	10	for	for	ADP
ap-7583	722	11	j	j	PROPN
ap-7583	722	12	=	=	SYM
ap-7583	722	13	0	0	PROPN
ap-7583	722	14	,	,	PUNCT
ap-7583	722	15	1	1	NUM
ap-7583	722	16	,	,	PUNCT
ap-7583	722	17	2	2	NUM
ap-7583	722	18	,	,	PUNCT
ap-7583	722	19	·	·	PUNCT
ap-7583	722	20	·	·	PUNCT
ap-7583	722	21	·	·	PUNCT
ap-7583	722	22	,	,	PUNCT
ap-7583	722	23	n.	n.	VERB
ap-7583	722	24	the	the	DET
ap-7583	722	25	weights	weight	NOUN
ap-7583	722	26	wj	wj	PROPN
ap-7583	722	27	,	,	PUNCT
ap-7583	722	28	j	j	PROPN
ap-7583	722	29	=	=	SYM
ap-7583	722	30	0	0	NUM
ap-7583	722	31	,	,	PUNCT
ap-7583	722	32	1	1	NUM
ap-7583	722	33	,	,	PUNCT
ap-7583	722	34	·	·	PUNCT
ap-7583	722	35	·	·	PUNCT
ap-7583	722	36	·	·	PUNCT
ap-7583	722	37	,	,	PUNCT
ap-7583	722	38	n	n	CCONJ
ap-7583	722	39	,	,	PUNCT
ap-7583	722	40	for	for	ADP
ap-7583	722	41	the	the	DET
ap-7583	722	42	orthogonal	orthogonal	ADJ
ap-7583	722	43	polynomials	polynomial	NOUN
ap-7583	722	44	{	{	PUNCT
ap-7583	722	45	pn	pn	PROPN
ap-7583	722	46	ℓ	ℓ	PROPN
ap-7583	722	47	(	(	PUNCT
ap-7583	722	48	ε0;n)}n	ε0;n)}n	PROPN
ap-7583	722	49	k=0	k=0	PROPN
ap-7583	722	50	can	can	AUX
ap-7583	722	51	be	be	AUX
ap-7583	722	52	computed	compute	VERB
ap-7583	722	53	by	by	ADP
ap-7583	722	54	solving	solve	VERB
ap-7583	722	55	the	the	DET
ap-7583	722	56	linear	linear	ADJ
ap-7583	722	57	system	system	NOUN
ap-7583	722	58	n∑	n∑	PROPN
ap-7583	722	59	j=0	j=0	PROPN
ap-7583	722	60	wj	wj	PROPN
ap-7583	722	61	pn	pn	PROPN
ap-7583	722	62	ℓ	ℓ	PROPN
ap-7583	722	63	(	(	PUNCT
ap-7583	722	64	εj	εj	NOUN
ap-7583	722	65	0;n	0;n	NUM
ap-7583	722	66	)	)	PUNCT
ap-7583	722	67	=	=	SYM
ap-7583	722	68	0	0	NUM
ap-7583	723	1	(	(	PUNCT
ap-7583	723	2	126	126	NUM
ap-7583	723	3	)	)	PUNCT
ap-7583	723	4	for	for	ADP
ap-7583	723	5	ℓ	ℓ	PROPN
ap-7583	723	6	=	=	SYM
ap-7583	723	7	0	0	NUM
ap-7583	723	8	,	,	PUNCT
ap-7583	723	9	1	1	NUM
ap-7583	723	10	,	,	PUNCT
ap-7583	723	11	·	·	PUNCT
ap-7583	723	12	·	·	PUNCT
ap-7583	723	13	·	·	PUNCT
ap-7583	723	14	,	,	PUNCT
ap-7583	723	15	n.	n.	NOUN
ap-7583	723	16	6.3	6.3	NUM
ap-7583	723	17	.	.	PUNCT
ap-7583	724	1	factorization	factorization	NOUN
ap-7583	724	2	property	property	NOUN
ap-7583	724	3	another	another	DET
ap-7583	724	4	interesting	interesting	ADJ
ap-7583	724	5	property	property	NOUN
ap-7583	724	6	of	of	ADP
ap-7583	724	7	the	the	DET
ap-7583	724	8	polynomials	polynomial	NOUN
ap-7583	724	9	{	{	PUNCT
ap-7583	724	10	pn	pn	PROPN
ap-7583	724	11	k	k	PROPN
ap-7583	724	12	(	(	PUNCT
ap-7583	724	13	x)}n	x)}n	PROPN
ap-7583	724	14	k=0	k=0	PROPN
ap-7583	724	15	,	,	PUNCT
ap-7583	724	16	aside	aside	ADV
ap-7583	724	17	from	from	ADP
ap-7583	724	18	being	be	AUX
ap-7583	724	19	an	an	DET
ap-7583	724	20	orthogonal	orthogonal	ADJ
ap-7583	724	21	sequence	sequence	NOUN
ap-7583	724	22	,	,	PUNCT
ap-7583	724	23	is	be	AUX
ap-7583	724	24	that	that	SCONJ
ap-7583	724	25	when	when	SCONJ
ap-7583	724	26	the	the	DET
ap-7583	724	27	parameter	parameter	NOUN
ap-7583	724	28	n	n	PRON
ap-7583	724	29	takes	take	VERB
ap-7583	724	30	positive	positive	ADJ
ap-7583	724	31	integer	integer	NOUN
ap-7583	724	32	values	value	NOUN
ap-7583	724	33	,	,	PUNCT
ap-7583	724	34	the	the	DET
ap-7583	724	35	polynomials	polynomial	NOUN
ap-7583	724	36	exhibit	exhibit	VERB
ap-7583	724	37	a	a	DET
ap-7583	724	38	factorization	factorization	NOUN
ap-7583	724	39	property	property	NOUN
ap-7583	724	40	.	.	PUNCT
ap-7583	725	1	clearly	clearly	ADV
ap-7583	725	2	,	,	PUNCT
ap-7583	725	3	the	the	DET
ap-7583	725	4	factorization	factorization	NOUN
ap-7583	725	5	occurs	occur	VERB
ap-7583	725	6	because	because	SCONJ
ap-7583	725	7	the	the	DET
ap-7583	725	8	third	third	ADJ
ap-7583	725	9	term	term	NOUN
ap-7583	725	10	in	in	ADP
ap-7583	725	11	the	the	DET
ap-7583	725	12	recursion	recursion	NOUN
ap-7583	725	13	relation	relation	NOUN
ap-7583	725	14	(	(	PUNCT
ap-7583	725	15	36	36	NUM
ap-7583	725	16	)	)	PUNCT
ap-7583	725	17	vanishes	vanish	VERB
ap-7583	725	18	when	when	SCONJ
ap-7583	725	19	k	k	PROPN
ap-7583	725	20	=	=	PUNCT
ap-7583	725	21	n	n	PROPN
ap-7583	725	22	+	+	NOUN
ap-7583	725	23	1	1	NUM
ap-7583	725	24	,	,	PUNCT
ap-7583	725	25	so	so	SCONJ
ap-7583	725	26	that	that	SCONJ
ap-7583	725	27	all	all	DET
ap-7583	725	28	subsequent	subsequent	ADJ
ap-7583	725	29	polynomials	polynomial	NOUN
ap-7583	725	30	have	have	VERB
ap-7583	725	31	a	a	DET
ap-7583	725	32	common	common	ADJ
ap-7583	725	33	factor	factor	NOUN
ap-7583	725	34	pn	pn	PROPN
ap-7583	725	35	n+1(ζ	n+1(ζ	PROPN
ap-7583	725	36	)	)	PUNCT
ap-7583	725	37	called	call	VERB
ap-7583	725	38	a	a	DET
ap-7583	725	39	critical	critical	ADJ
ap-7583	725	40	polynomial	polynomial	NOUN
ap-7583	725	41	.	.	PUNCT
ap-7583	726	1	indeed	indeed	ADV
ap-7583	726	2	,	,	PUNCT
ap-7583	726	3	all	all	DET
ap-7583	726	4	the	the	DET
ap-7583	726	5	polynomials	polynomial	NOUN
ap-7583	726	6	pn	pn	PROPN
ap-7583	726	7	k+n+1(x	k+n+1(x	PROPN
ap-7583	726	8	)	)	PUNCT
ap-7583	727	1	,	,	PUNCT
ap-7583	727	2	beyond	beyond	ADP
ap-7583	727	3	the	the	DET
ap-7583	727	4	critical	critical	ADJ
ap-7583	727	5	polynomial	polynomial	NOUN
ap-7583	727	6	pn	pn	PROPN
ap-7583	727	7	n+1(x	n+1(x	PROPN
ap-7583	727	8	)	)	PUNCT
ap-7583	727	9	are	be	AUX
ap-7583	727	10	factored	factor	VERB
ap-7583	727	11	into	into	ADP
ap-7583	727	12	the	the	DET
ap-7583	727	13	product	product	NOUN
ap-7583	727	14	pn	pn	PROPN
ap-7583	727	15	k+n+1(x	k+n+1(x	NOUN
ap-7583	727	16	)	)	PUNCT
ap-7583	728	1	=	=	PUNCT
ap-7583	728	2	qn	qn	PROPN
ap-7583	728	3	k	k	PROPN
ap-7583	728	4	(	(	PUNCT
ap-7583	728	5	x	x	X
ap-7583	728	6	)	)	PUNCT
ap-7583	728	7	pn	pn	PROPN
ap-7583	728	8	n+1(x	n+1(x	PROPN
ap-7583	728	9	)	)	PUNCT
ap-7583	728	10	,	,	PUNCT
ap-7583	728	11	k	k	PROPN
ap-7583	729	1	=	=	SYM
ap-7583	729	2	0	0	NUM
ap-7583	729	3	,	,	PUNCT
ap-7583	729	4	1	1	NUM
ap-7583	729	5	,	,	PUNCT
ap-7583	729	6	.	.	PUNCT
ap-7583	729	7	.	.	PUNCT
ap-7583	729	8	.	.	PUNCT
ap-7583	730	1	,	,	PUNCT
ap-7583	730	2	(	(	PUNCT
ap-7583	730	3	127	127	NUM
ap-7583	730	4	)	)	PUNCT
ap-7583	730	5	where	where	SCONJ
ap-7583	730	6	the	the	DET
ap-7583	730	7	sequence	sequence	NOUN
ap-7583	730	8	{	{	PUNCT
ap-7583	730	9	qn	qn	NOUN
ap-7583	730	10	k	k	PROPN
ap-7583	730	11	(	(	PUNCT
ap-7583	730	12	x	x	NOUN
ap-7583	730	13	)	)	PUNCT
ap-7583	730	14	}	}	PUNCT
ap-7583	730	15	are	be	AUX
ap-7583	730	16	polynomials	polynomial	NOUN
ap-7583	730	17	of	of	ADP
ap-7583	730	18	degree	degree	NOUN
ap-7583	730	19	k	k	NOUN
ap-7583	730	20	=	=	SYM
ap-7583	730	21	0	0	NUM
ap-7583	730	22	,	,	PUNCT
ap-7583	730	23	1	1	NUM
ap-7583	730	24	,	,	PUNCT
ap-7583	730	25	.	.	PUNCT
ap-7583	730	26	.	.	PUNCT
ap-7583	730	27	.	.	PUNCT
ap-7583	730	28	.	.	PUNCT
ap-7583	731	1	interestingly	interestingly	ADV
ap-7583	731	2	,	,	PUNCT
ap-7583	731	3	the	the	DET
ap-7583	731	4	quotient	quotient	NOUN
ap-7583	731	5	polynomials	polynomial	VERB
ap-7583	731	6	{	{	PUNCT
ap-7583	731	7	qn	qn	NOUN
ap-7583	731	8	k	k	PROPN
ap-7583	731	9	(	(	PUNCT
ap-7583	731	10	x)}∞	x)}∞	ADJ
ap-7583	731	11	k=0	k=0	PROPN
ap-7583	731	12	form	form	VERB
ap-7583	731	13	an	an	DET
ap-7583	731	14	infinite	infinite	ADJ
ap-7583	731	15	sequence	sequence	NOUN
ap-7583	731	16	of	of	ADP
ap-7583	731	17	orthogonal	orthogonal	ADJ
ap-7583	731	18	polynomials	polynomial	NOUN
ap-7583	731	19	.	.	PUNCT
ap-7583	732	1	to	to	PART
ap-7583	732	2	prove	prove	VERB
ap-7583	732	3	this	this	DET
ap-7583	732	4	claim	claim	NOUN
ap-7583	732	5	,	,	PUNCT
ap-7583	732	6	we	we	PRON
ap-7583	732	7	substitute	substitute	VERB
ap-7583	732	8	(	(	PUNCT
ap-7583	732	9	128	128	NUM
ap-7583	732	10	)	)	PUNCT
ap-7583	732	11	into	into	ADP
ap-7583	732	12	(	(	PUNCT
ap-7583	732	13	36	36	NUM
ap-7583	732	14	)	)	PUNCT
ap-7583	732	15	and	and	CCONJ
ap-7583	732	16	re	re	NOUN
ap-7583	732	17	-	-	NOUN
ap-7583	732	18	index	index	VERB
ap-7583	732	19	the	the	DET
ap-7583	732	20	polynomials	polynomial	NOUN
ap-7583	732	21	to	to	PART
ap-7583	732	22	eliminate	eliminate	VERB
ap-7583	732	23	the	the	DET
ap-7583	732	24	common	common	ADJ
ap-7583	732	25	factor	factor	NOUN
ap-7583	732	26	pn	pn	PROPN
ap-7583	732	27	n+1(ζ	n+1(ζ	PROPN
ap-7583	732	28	)	)	PUNCT
ap-7583	732	29	from	from	ADP
ap-7583	732	30	both	both	DET
ap-7583	732	31	sides	side	NOUN
ap-7583	732	32	.	.	PUNCT
ap-7583	733	1	the	the	DET
ap-7583	733	2	recurrence	recurrence	NOUN
ap-7583	733	3	relation	relation	NOUN
ap-7583	733	4	(	(	PUNCT
ap-7583	733	5	36	36	NUM
ap-7583	733	6	)	)	PUNCT
ap-7583	733	7	then	then	ADV
ap-7583	733	8	reduces	reduce	VERB
ap-7583	733	9	to	to	ADP
ap-7583	733	10	a	a	DET
ap-7583	733	11	three	three	NUM
ap-7583	733	12	-	-	PUNCT
ap-7583	733	13	term	term	NOUN
ap-7583	733	14	recurrence	recurrence	NOUN
ap-7583	733	15	relation	relation	NOUN
ap-7583	733	16	for	for	ADP
ap-7583	733	17	the	the	DET
ap-7583	733	18	polynomials	polynomial	NOUN
ap-7583	733	19	{	{	PUNCT
ap-7583	733	20	qn	qn	NOUN
ap-7583	733	21	k	k	PROPN
ap-7583	733	22	(	(	PUNCT
ap-7583	733	23	ζ)}k≥0	ζ)}k≥0	PROPN
ap-7583	733	24	that	that	PRON
ap-7583	733	25	reads	read	VERB
ap-7583	733	26	qn	qn	PROPN
ap-7583	733	27	k	k	PROPN
ap-7583	733	28	(	(	PUNCT
ap-7583	733	29	x	x	X
ap-7583	733	30	)	)	PUNCT
ap-7583	733	31	=	=	SYM
ap-7583	734	1	(	(	PUNCT
ap-7583	734	2	(	(	PUNCT
ap-7583	734	3	k	k	X
ap-7583	734	4	+	+	NOUN
ap-7583	734	5	n)(k	n)(k	PUNCT
ap-7583	734	6	+	+	CCONJ
ap-7583	734	7	n	n	CCONJ
ap-7583	734	8	−	−	PROPN
ap-7583	734	9	1)α2	1)α2	NUM
ap-7583	735	1	+	+	CCONJ
ap-7583	735	2	(	(	PUNCT
ap-7583	735	3	k	k	X
ap-7583	735	4	+	+	CCONJ
ap-7583	735	5	n)β1	n)β1	PROPN
ap-7583	735	6	+	+	NOUN
ap-7583	735	7	x	x	X
ap-7583	735	8	)	)	PUNCT
ap-7583	735	9	qn	qn	PROPN
ap-7583	735	10	k−1(x	k−1(x	NOUN
ap-7583	735	11	)	)	PUNCT
ap-7583	735	12	−	−	PROPN
ap-7583	736	1	(	(	PUNCT
ap-7583	736	2	k	k	X
ap-7583	736	3	+	+	CCONJ
ap-7583	736	4	n)(k	n)(k	NOUN
ap-7583	736	5	−	−	ADP
ap-7583	736	6	1	1	NUM
ap-7583	736	7	)	)	PUNCT
ap-7583	736	8	(	(	PUNCT
ap-7583	736	9	(	(	PUNCT
ap-7583	736	10	k	k	X
ap-7583	736	11	+	+	CCONJ
ap-7583	736	12	n	n	CCONJ
ap-7583	736	13	−	−	PROPN
ap-7583	736	14	1)α1	1)α1	NUM
ap-7583	736	15	+	+	CCONJ
ap-7583	736	16	β0	β0	ADJ
ap-7583	736	17	)	)	PUNCT
ap-7583	736	18	×	×	NOUN
ap-7583	736	19	(	(	PUNCT
ap-7583	736	20	β2	β2	VERB
ap-7583	736	21	+	+	CCONJ
ap-7583	737	1	α3(k	α3(k	NUM
ap-7583	738	1	+	+	CCONJ
ap-7583	738	2	2n	2n	NUM
ap-7583	738	3	−	−	NOUN
ap-7583	738	4	2	2	NUM
ap-7583	738	5	)	)	PUNCT
ap-7583	738	6	)	)	PUNCT
ap-7583	739	1	qn	qn	PROPN
ap-7583	739	2	k−2(x	k−2(x	PROPN
ap-7583	739	3	)	)	PUNCT
ap-7583	739	4	,	,	PUNCT
ap-7583	739	5	(	(	PUNCT
ap-7583	739	6	128	128	NUM
ap-7583	739	7	)	)	PUNCT
ap-7583	740	1	where	where	SCONJ
ap-7583	740	2	qn	qn	PROPN
ap-7583	740	3	−1(ζ	−1(ζ	PROPN
ap-7583	740	4	)	)	PUNCT
ap-7583	740	5	=	=	SYM
ap-7583	740	6	0	0	NUM
ap-7583	740	7	,	,	PUNCT
ap-7583	740	8	and	and	CCONJ
ap-7583	740	9	qn	qn	NOUN
ap-7583	740	10	0	0	NUM
ap-7583	740	11	(	(	PUNCT
ap-7583	740	12	ζ	ζ	NOUN
ap-7583	740	13	)	)	PUNCT
ap-7583	740	14	=	=	SYM
ap-7583	740	15	1	1	X
ap-7583	740	16	.	.	PUNCT
ap-7583	740	17	hence	hence	ADV
ap-7583	740	18	,	,	PUNCT
ap-7583	740	19	the	the	DET
ap-7583	740	20	quotient	quotient	NOUN
ap-7583	740	21	polynomials	polynomial	VERB
ap-7583	740	22	qn	qn	PROPN
ap-7583	740	23	k	k	PROPN
ap-7583	740	24	(	(	PUNCT
ap-7583	740	25	ζ	ζ	NOUN
ap-7583	740	26	)	)	PUNCT
ap-7583	740	27	also	also	ADV
ap-7583	740	28	form	form	VERB
ap-7583	740	29	a	a	DET
ap-7583	740	30	new	new	ADJ
ap-7583	740	31	sequence	sequence	NOUN
ap-7583	740	32	of	of	ADP
ap-7583	740	33	orthogonal	orthogonal	ADJ
ap-7583	740	34	polynomials	polynomial	NOUN
ap-7583	740	35	for	for	ADP
ap-7583	740	36	each	each	DET
ap-7583	740	37	value	value	NOUN
ap-7583	740	38	of	of	ADP
ap-7583	740	39	n.	n.	NOUN
ap-7583	740	40	for	for	ADP
ap-7583	740	41	example	example	NOUN
ap-7583	740	42	,	,	PUNCT
ap-7583	740	43	if	if	SCONJ
ap-7583	740	44	n	n	NOUN
ap-7583	740	45	=	=	SYM
ap-7583	740	46	2	2	NUM
ap-7583	740	47	,	,	PUNCT
ap-7583	740	48	the	the	DET
ap-7583	740	49	critical	critical	ADJ
ap-7583	740	50	polynomial	polynomial	NOUN
ap-7583	740	51	is	be	AUX
ap-7583	740	52	p2	p2	PROPN
ap-7583	740	53	3	3	NUM
ap-7583	740	54	(	(	PUNCT
ap-7583	740	55	x	x	NOUN
ap-7583	740	56	)	)	PUNCT
ap-7583	741	1	=	=	SYM
ap-7583	741	2	x3	x3	VERB
ap-7583	741	3	+	+	CCONJ
ap-7583	741	4	(	(	PUNCT
ap-7583	741	5	2α2	2α2	NUM
ap-7583	742	1	+	+	CCONJ
ap-7583	742	2	3β1)x2	3β1)x2	NUM
ap-7583	742	3	+	+	SYM
ap-7583	742	4	2	2	NUM
ap-7583	742	5	(	(	PUNCT
ap-7583	742	6	(	(	PUNCT
ap-7583	742	7	3α3	3α3	NUM
ap-7583	742	8	+	+	CCONJ
ap-7583	742	9	2β2)β0	2β2)β0	NUM
ap-7583	742	10	+	+	CCONJ
ap-7583	742	11	β1(α2	β1(α2	PRON
ap-7583	742	12	+	+	CCONJ
ap-7583	742	13	β1	β1	NOUN
ap-7583	742	14	)	)	PUNCT
ap-7583	743	1	+	+	CCONJ
ap-7583	743	2	α1(2α3	α1(2α3	NUM
ap-7583	743	3	+	+	CCONJ
ap-7583	743	4	β2	β2	VERB
ap-7583	743	5	)	)	PUNCT
ap-7583	743	6	)	)	PUNCT
ap-7583	744	1	x	x	PUNCT
ap-7583	745	1	+	+	NUM
ap-7583	745	2	4β0(α2	4β0(α2	NUM
ap-7583	745	3	+	+	CCONJ
ap-7583	745	4	β1)(α3	β1)(α3	NOUN
ap-7583	745	5	+	+	CCONJ
ap-7583	745	6	β2	β2	VERB
ap-7583	745	7	)	)	PUNCT
ap-7583	745	8	.	.	PUNCT
ap-7583	746	1	(	(	PUNCT
ap-7583	746	2	129	129	NUM
ap-7583	746	3	)	)	PUNCT
ap-7583	746	4	and	and	CCONJ
ap-7583	746	5	p2	p2	PROPN
ap-7583	746	6	4	4	NUM
ap-7583	746	7	(	(	PUNCT
ap-7583	746	8	x	x	NOUN
ap-7583	746	9	)	)	PUNCT
ap-7583	746	10	=	=	SYM
ap-7583	747	1	(	(	PUNCT
ap-7583	747	2	x	x	SYM
ap-7583	747	3	+	+	PUNCT
ap-7583	747	4	6α2	6α2	NUM
ap-7583	747	5	+	+	CCONJ
ap-7583	747	6	3β1	3β1	NUM
ap-7583	747	7	)	)	PUNCT
ap-7583	747	8	p2	p2	PROPN
ap-7583	747	9	3	3	NUM
ap-7583	747	10	(	(	PUNCT
ap-7583	747	11	x	x	NOUN
ap-7583	747	12	)	)	PUNCT
ap-7583	747	13	,	,	PUNCT
ap-7583	747	14	p2	p2	X
ap-7583	747	15	5	5	NUM
ap-7583	747	16	(	(	PUNCT
ap-7583	747	17	x	x	NOUN
ap-7583	747	18	)	)	PUNCT
ap-7583	747	19	=	=	SYM
ap-7583	748	1	(	(	PUNCT
ap-7583	748	2	x2	x2	SYM
ap-7583	748	3	+	+	CCONJ
ap-7583	748	4	(	(	PUNCT
ap-7583	748	5	18α2	18α2	NUM
ap-7583	748	6	+	+	CCONJ
ap-7583	748	7	7β1)x	7β1)x	NUM
ap-7583	748	8	−	−	NOUN
ap-7583	748	9	4	4	NUM
ap-7583	748	10	(	(	PUNCT
ap-7583	748	11	(	(	PUNCT
ap-7583	748	12	3α1	3α1	NUM
ap-7583	748	13	+	+	CCONJ
ap-7583	748	14	β0)(4α3	β0)(4α3	ADJ
ap-7583	748	15	+	+	CCONJ
ap-7583	748	16	β2	β2	ADJ
ap-7583	748	17	)	)	PUNCT
ap-7583	748	18	−	−	PROPN
ap-7583	748	19	3(2α2	3(2α2	NOUN
ap-7583	748	20	+	+	CCONJ
ap-7583	748	21	β1)(3α2	β1)(3α2	NOUN
ap-7583	748	22	+	+	CCONJ
ap-7583	748	23	β1	β1	NOUN
ap-7583	748	24	)	)	PUNCT
ap-7583	748	25	)	)	PUNCT
ap-7583	748	26	)	)	PUNCT
ap-7583	749	1	p2	p2	PROPN
ap-7583	749	2	3	3	NUM
ap-7583	749	3	(	(	PUNCT
ap-7583	749	4	x	x	NOUN
ap-7583	749	5	)	)	PUNCT
ap-7583	749	6	,	,	PUNCT
ap-7583	749	7	p2	p2	PROPN
ap-7583	749	8	6	6	NUM
ap-7583	749	9	(	(	PUNCT
ap-7583	749	10	x	x	NOUN
ap-7583	749	11	)	)	PUNCT
ap-7583	749	12	=	=	SYM
ap-7583	750	1	(	(	PUNCT
ap-7583	750	2	x3	x3	VERB
ap-7583	750	3	+	+	CCONJ
ap-7583	750	4	(	(	PUNCT
ap-7583	750	5	38α2	38α2	NUM
ap-7583	750	6	+	+	NUM
ap-7583	750	7	12β1)x2	12β1)x2	NUM
ap-7583	751	1	+	+	CCONJ
ap-7583	751	2	(	(	PUNCT
ap-7583	751	3	432α2	432α2	NUM
ap-7583	751	4	2	2	NUM
ap-7583	751	5	+	+	CCONJ
ap-7583	751	6	290α2β1	290α2β1	NUM
ap-7583	752	1	+	+	CCONJ
ap-7583	752	2	47β2	47β2	NUM
ap-7583	752	3	1	1	NUM
ap-7583	752	4	−	−	NOUN
ap-7583	752	5	2(124α1α3	2(124α1α3	PRON
ap-7583	753	1	+	+	PUNCT
ap-7583	753	2	33α3β0	33α3β0	NOUN
ap-7583	753	3	+	+	CCONJ
ap-7583	753	4	26α1β2	26α1β2	NUM
ap-7583	753	5	+	+	CCONJ
ap-7583	753	6	7β0β2	7β0β2	NOUN
ap-7583	753	7	)	)	PUNCT
ap-7583	753	8	)	)	PUNCT
ap-7583	754	1	x	x	X
ap-7583	755	1	−	−	PROPN
ap-7583	755	2	10	10	NUM
ap-7583	755	3	(	(	PUNCT
ap-7583	755	4	β1(84α1α3	β1(84α1α3	NOUN
ap-7583	755	5	+	+	NUM
ap-7583	755	6	23α3β0	23α3β0	NUM
ap-7583	755	7	−	−	NOUN
ap-7583	755	8	6β2	6β2	NUM
ap-7583	755	9	1	1	NUM
ap-7583	756	1	+	+	CCONJ
ap-7583	756	2	18α1β2	18α1β2	NOUN
ap-7583	756	3	+	+	ADJ
ap-7583	756	4	5β0β2	5β0β2	NUM
ap-7583	756	5	)	)	PUNCT
ap-7583	757	1	+	+	CCONJ
ap-7583	757	2	2α2	2α2	NUM
ap-7583	757	3	(	(	PUNCT
ap-7583	757	4	31α3β0	31α3β0	NUM
ap-7583	757	5	−	−	PROPN
ap-7583	757	6	27β2	27β2	NUM
ap-7583	757	7	1	1	NUM
ap-7583	757	8	+	+	NUM
ap-7583	757	9	7β0β2	7β0β2	NUM
ap-7583	758	1	+	+	CCONJ
ap-7583	758	2	12α1(9α3	12α1(9α3	NUM
ap-7583	758	3	+	+	NUM
ap-7583	758	4	2β2	2β2	NUM
ap-7583	758	5	)	)	PUNCT
ap-7583	758	6	)	)	PUNCT
ap-7583	759	1	−	−	PROPN
ap-7583	759	2	144α3	144α3	NUM
ap-7583	759	3	2	2	NUM
ap-7583	759	4	−	−	NUM
ap-7583	759	5	156α2	156α2	NUM
ap-7583	759	6	2β1	2β1	NUM
ap-7583	759	7	)	)	PUNCT
ap-7583	759	8	)	)	PUNCT
ap-7583	760	1	p2	p2	PROPN
ap-7583	760	2	3	3	NUM
ap-7583	760	3	(	(	PUNCT
ap-7583	760	4	x	x	NOUN
ap-7583	760	5	)	)	PUNCT
ap-7583	760	6	,	,	PUNCT
ap-7583	760	7	...	...	PUNCT
ap-7583	760	8	from	from	ADP
ap-7583	760	9	which	which	PRON
ap-7583	760	10	we	we	PRON
ap-7583	760	11	have	have	VERB
ap-7583	760	12	q0(x	q0(x	NOUN
ap-7583	760	13	)	)	PUNCT
ap-7583	760	14	=	=	SYM
ap-7583	760	15	1	1	NUM
ap-7583	760	16	,	,	PUNCT
ap-7583	760	17	q1(x	q1(x	NOUN
ap-7583	760	18	)	)	PUNCT
ap-7583	760	19	=	=	SYM
ap-7583	761	1	x	x	PUNCT
ap-7583	762	1	+	+	NUM
ap-7583	762	2	6α2	6α2	NUM
ap-7583	762	3	+	+	CCONJ
ap-7583	762	4	3β1	3β1	NUM
ap-7583	762	5	,	,	PUNCT
ap-7583	762	6	q2(x	q2(x	X
ap-7583	762	7	)	)	PUNCT
ap-7583	762	8	=	=	SYM
ap-7583	763	1	x2	x2	PROPN
ap-7583	764	1	+	+	CCONJ
ap-7583	764	2	(	(	PUNCT
ap-7583	764	3	18α2	18α2	NUM
ap-7583	764	4	+	+	CCONJ
ap-7583	764	5	7β1)x	7β1)x	NUM
ap-7583	764	6	−	−	NOUN
ap-7583	764	7	4	4	NUM
ap-7583	764	8	(	(	PUNCT
ap-7583	764	9	(	(	PUNCT
ap-7583	764	10	3α1	3α1	NUM
ap-7583	764	11	+	+	CCONJ
ap-7583	764	12	β0)(4α3	β0)(4α3	ADJ
ap-7583	764	13	+	+	CCONJ
ap-7583	764	14	β2	β2	ADJ
ap-7583	764	15	)	)	PUNCT
ap-7583	764	16	−	−	PROPN
ap-7583	765	1	3(2α2	3(2α2	NOUN
ap-7583	765	2	+	+	CCONJ
ap-7583	765	3	β1)(3α2	β1)(3α2	NOUN
ap-7583	765	4	+	+	CCONJ
ap-7583	765	5	β1	β1	NOUN
ap-7583	765	6	)	)	PUNCT
ap-7583	765	7	)	)	PUNCT
ap-7583	765	8	,	,	PUNCT
ap-7583	765	9	q3(x	q3(x	X
ap-7583	765	10	)	)	PUNCT
ap-7583	765	11	=	=	PUNCT
ap-7583	766	1	x3	x3	VERB
ap-7583	766	2	+	+	CCONJ
ap-7583	766	3	(	(	PUNCT
ap-7583	766	4	38α2	38α2	NUM
ap-7583	766	5	+	+	NUM
ap-7583	766	6	12β1)x2	12β1)x2	NUM
ap-7583	767	1	+	+	CCONJ
ap-7583	767	2	(	(	PUNCT
ap-7583	767	3	432α2	432α2	NUM
ap-7583	767	4	2	2	NUM
ap-7583	767	5	+	+	CCONJ
ap-7583	767	6	290α2β1	290α2β1	NUM
ap-7583	768	1	+	+	CCONJ
ap-7583	768	2	47β2	47β2	NUM
ap-7583	768	3	1	1	NUM
ap-7583	768	4	−	−	NOUN
ap-7583	768	5	2(124α1α3	2(124α1α3	PRON
ap-7583	769	1	+	+	PUNCT
ap-7583	769	2	33α3β0	33α3β0	NOUN
ap-7583	769	3	+	+	CCONJ
ap-7583	769	4	26α1β2	26α1β2	NUM
ap-7583	769	5	+	+	CCONJ
ap-7583	769	6	7β0β2	7β0β2	NOUN
ap-7583	769	7	)	)	PUNCT
ap-7583	769	8	)	)	PUNCT
ap-7583	770	1	x	x	X
ap-7583	771	1	−	−	PROPN
ap-7583	771	2	10	10	NUM
ap-7583	771	3	(	(	PUNCT
ap-7583	771	4	β1(84α1α3	β1(84α1α3	NOUN
ap-7583	771	5	+	+	NUM
ap-7583	771	6	23α3β0	23α3β0	NUM
ap-7583	771	7	−	−	NOUN
ap-7583	771	8	6β2	6β2	NUM
ap-7583	771	9	1	1	NUM
ap-7583	772	1	+	+	CCONJ
ap-7583	772	2	18α1β2	18α1β2	NOUN
ap-7583	772	3	+	+	ADJ
ap-7583	772	4	5β0β2	5β0β2	NUM
ap-7583	772	5	)	)	PUNCT
ap-7583	773	1	+	+	CCONJ
ap-7583	773	2	2α2	2α2	NUM
ap-7583	773	3	(	(	PUNCT
ap-7583	773	4	31α3β0	31α3β0	NUM
ap-7583	773	5	−	−	PROPN
ap-7583	773	6	27β2	27β2	NUM
ap-7583	773	7	1	1	NUM
ap-7583	773	8	+	+	NUM
ap-7583	773	9	7β0β2	7β0β2	NUM
ap-7583	774	1	+	+	CCONJ
ap-7583	774	2	12α1(9α3	12α1(9α3	NUM
ap-7583	774	3	+	+	NUM
ap-7583	774	4	2β2	2β2	NUM
ap-7583	774	5	)	)	PUNCT
ap-7583	774	6	)	)	PUNCT
ap-7583	775	1	−	−	PROPN
ap-7583	775	2	144α3	144α3	NUM
ap-7583	775	3	2	2	NUM
ap-7583	775	4	−	−	NUM
ap-7583	775	5	156α2	156α2	NUM
ap-7583	775	6	2β1	2β1	NUM
ap-7583	775	7	)	)	PUNCT
ap-7583	775	8	,	,	PUNCT
ap-7583	775	9	...	...	PUNCT
ap-7583	775	10	187	187	NUM
ap-7583	775	11	nasser	nasser	PROPN
ap-7583	775	12	saad	saad	PROPN
ap-7583	775	13	acta	acta	PROPN
ap-7583	775	14	polytechnica	polytechnica	PROPN
ap-7583	775	15	and	and	CCONJ
ap-7583	775	16	so	so	ADV
ap-7583	775	17	on	on	ADV
ap-7583	775	18	.	.	PUNCT
ap-7583	776	1	the	the	DET
ap-7583	776	2	christoffel	christoffel	NOUN
ap-7583	776	3	-	-	PUNCT
ap-7583	776	4	darboux	darboux	VERB
ap-7583	776	5	formula	formula	NOUN
ap-7583	776	6	for	for	ADP
ap-7583	776	7	this	this	DET
ap-7583	776	8	infinite	infinite	ADJ
ap-7583	776	9	sequence	sequence	NOUN
ap-7583	776	10	of	of	ADP
ap-7583	776	11	orthogonal	orthogonal	ADJ
ap-7583	776	12	polynomials	polynomial	NOUN
ap-7583	776	13	reads	read	VERB
ap-7583	776	14	k∑	k∑	VERB
ap-7583	776	15	j=0	j=0	PROPN
ap-7583	776	16	qn	qn	PROPN
ap-7583	776	17	j	j	PROPN
ap-7583	777	1	(	(	PUNCT
ap-7583	777	2	x)qn	x)qn	PROPN
ap-7583	777	3	j	j	PROPN
ap-7583	777	4	(	(	PUNCT
ap-7583	777	5	y	y	PROPN
ap-7583	777	6	)	)	PUNCT
ap-7583	777	7	j	j	PROPN
ap-7583	777	8	!	!	PUNCT
ap-7583	778	1	(	(	PUNCT
ap-7583	778	2	α1α3)j(n	α1α3)j(n	PROPN
ap-7583	778	3	+	+	NUM
ap-7583	778	4	2)j	2)j	PROPN
ap-7583	778	5	(	(	PUNCT
ap-7583	778	6	β0	β0	PROPN
ap-7583	778	7	α1	α1	PROPN
ap-7583	778	8	+	+	CCONJ
ap-7583	778	9	n	n	PROPN
ap-7583	778	10	+	+	CCONJ
ap-7583	778	11	1	1	NUM
ap-7583	778	12	)	)	PUNCT
ap-7583	778	13	j	j	NOUN
ap-7583	778	14	(	(	PUNCT
ap-7583	778	15	β2	β2	NOUN
ap-7583	778	16	α3	α3	PROPN
ap-7583	778	17	+	+	CCONJ
ap-7583	778	18	2n	2n	NUM
ap-7583	778	19	)	)	PUNCT
ap-7583	779	1	j	j	PROPN
ap-7583	780	1	=	=	PUNCT
ap-7583	780	2	qn	qn	PROPN
ap-7583	781	1	k+1(x)qn	k+1(x)qn	PROPN
ap-7583	781	2	k	k	PROPN
ap-7583	781	3	(	(	PUNCT
ap-7583	781	4	y	y	NOUN
ap-7583	781	5	)	)	PUNCT
ap-7583	781	6	−	−	PROPN
ap-7583	782	1	qn	qn	INTJ
ap-7583	782	2	k	k	PROPN
ap-7583	782	3	(	(	PUNCT
ap-7583	782	4	x)qn	x)qn	PROPN
ap-7583	782	5	k+1(y	k+1(y	PROPN
ap-7583	782	6	)	)	PUNCT
ap-7583	783	1	k	k	X
ap-7583	783	2	!	!	PUNCT
ap-7583	784	1	(	(	PUNCT
ap-7583	784	2	α1α3)k(n	α1α3)k(n	NUM
ap-7583	784	3	+	+	NUM
ap-7583	784	4	2)k	2)k	NOUN
ap-7583	784	5	(	(	PUNCT
ap-7583	784	6	β0	β0	PROPN
ap-7583	784	7	α1	α1	PROPN
ap-7583	784	8	+	+	CCONJ
ap-7583	784	9	n	n	PROPN
ap-7583	784	10	+	+	CCONJ
ap-7583	784	11	1	1	NUM
ap-7583	784	12	)	)	PUNCT
ap-7583	784	13	k	k	NOUN
ap-7583	785	1	(	(	PUNCT
ap-7583	785	2	β2	β2	NOUN
ap-7583	785	3	α3	α3	PROPN
ap-7583	785	4	+	+	CCONJ
ap-7583	785	5	2n	2n	NUM
ap-7583	785	6	)	)	PUNCT
ap-7583	786	1	k	k	NOUN
ap-7583	786	2	(	(	PUNCT
ap-7583	786	3	x	x	PROPN
ap-7583	786	4	−	−	PROPN
ap-7583	786	5	y	y	PROPN
ap-7583	786	6	)	)	PUNCT
ap-7583	786	7	,	,	PUNCT
ap-7583	786	8	(	(	PUNCT
ap-7583	786	9	130	130	NUM
ap-7583	786	10	)	)	PUNCT
ap-7583	786	11	and	and	CCONJ
ap-7583	786	12	as	as	ADP
ap-7583	786	13	y	y	PROPN
ap-7583	786	14	→	→	SYM
ap-7583	786	15	x	x	PROPN
ap-7583	786	16	k∑	k∑	VERB
ap-7583	786	17	j=0	j=0	PROPN
ap-7583	786	18	(	(	PUNCT
ap-7583	786	19	qn	qn	PROPN
ap-7583	786	20	j	j	PROPN
ap-7583	786	21	(	(	PUNCT
ap-7583	786	22	x	x	NOUN
ap-7583	786	23	)	)	PUNCT
ap-7583	786	24	)	)	PUNCT
ap-7583	786	25	2	2	NUM
ap-7583	786	26	j	j	NOUN
ap-7583	786	27	!	!	PUNCT
ap-7583	787	1	(	(	PUNCT
ap-7583	787	2	α1α3)j(n	α1α3)j(n	PROPN
ap-7583	787	3	+	+	NUM
ap-7583	787	4	2)j	2)j	PROPN
ap-7583	787	5	(	(	PUNCT
ap-7583	787	6	β0	β0	PROPN
ap-7583	787	7	α1	α1	PROPN
ap-7583	787	8	+	+	CCONJ
ap-7583	787	9	n	n	PROPN
ap-7583	787	10	+	+	CCONJ
ap-7583	787	11	1	1	NUM
ap-7583	787	12	)	)	PUNCT
ap-7583	787	13	j	j	NOUN
ap-7583	787	14	(	(	PUNCT
ap-7583	787	15	β2	β2	NOUN
ap-7583	787	16	α3	α3	PROPN
ap-7583	787	17	+	+	CCONJ
ap-7583	787	18	2n	2n	NUM
ap-7583	787	19	)	)	PUNCT
ap-7583	788	1	j	j	X
ap-7583	789	1	=	=	PUNCT
ap-7583	790	1	[	[	X
ap-7583	790	2	qn	qn	INTJ
ap-7583	790	3	k+1(x)]′qn	k+1(x)]′qn	PROPN
ap-7583	790	4	k	k	X
ap-7583	790	5	(	(	PUNCT
ap-7583	790	6	x	x	NOUN
ap-7583	790	7	)	)	PUNCT
ap-7583	790	8	−	−	PROPN
ap-7583	791	1	[	[	X
ap-7583	791	2	qn	qn	X
ap-7583	791	3	k	k	X
ap-7583	791	4	(	(	PUNCT
ap-7583	791	5	x)]′qn	x)]′qn	PROPN
ap-7583	791	6	k+1(x	k+1(x	PROPN
ap-7583	791	7	)	)	PUNCT
ap-7583	791	8	k	k	X
ap-7583	791	9	!	!	PUNCT
ap-7583	792	1	(	(	PUNCT
ap-7583	792	2	α1α3)k(n	α1α3)k(n	NUM
ap-7583	792	3	+	+	NUM
ap-7583	792	4	2)k	2)k	NOUN
ap-7583	792	5	(	(	PUNCT
ap-7583	792	6	β0	β0	PROPN
ap-7583	792	7	α1	α1	PROPN
ap-7583	792	8	+	+	CCONJ
ap-7583	792	9	n	n	PROPN
ap-7583	792	10	+	+	CCONJ
ap-7583	792	11	1	1	NUM
ap-7583	792	12	)	)	PUNCT
ap-7583	792	13	k	k	NOUN
ap-7583	793	1	(	(	PUNCT
ap-7583	793	2	β2	β2	NOUN
ap-7583	793	3	α3	α3	PROPN
ap-7583	793	4	+	+	CCONJ
ap-7583	793	5	2n	2n	NUM
ap-7583	793	6	)	)	PUNCT
ap-7583	794	1	k	k	X
ap-7583	794	2	.	.	PUNCT
ap-7583	795	1	(	(	PUNCT
ap-7583	795	2	131	131	X
ap-7583	795	3	)	)	PUNCT
ap-7583	795	4	theorem	theorem	VERB
ap-7583	795	5	6.1	6.1	NUM
ap-7583	795	6	.	.	PUNCT
ap-7583	796	1	the	the	DET
ap-7583	796	2	norms	norm	NOUN
ap-7583	796	3	of	of	ADP
ap-7583	796	4	all	all	DET
ap-7583	796	5	polynomials	polynomial	NOUN
ap-7583	796	6	qn	qn	PROPN
ap-7583	796	7	k	k	PROPN
ap-7583	796	8	(	(	PUNCT
ap-7583	796	9	ξ	ξ	NOUN
ap-7583	796	10	)	)	PUNCT
ap-7583	796	11	are	be	AUX
ap-7583	796	12	given	give	VERB
ap-7583	796	13	by	by	ADP
ap-7583	796	14	gq	gq	PROPN
ap-7583	796	15	k	k	PROPN
ap-7583	796	16	=	=	PUNCT
ap-7583	796	17	k	k	PROPN
ap-7583	796	18	!	!	PUNCT
ap-7583	797	1	(	(	PUNCT
ap-7583	797	2	α1α3)k(n	α1α3)k(n	NUM
ap-7583	797	3	+	+	NUM
ap-7583	797	4	2)k	2)k	NOUN
ap-7583	797	5	(	(	PUNCT
ap-7583	797	6	β0	β0	PROPN
ap-7583	797	7	α1	α1	PROPN
ap-7583	797	8	+	+	CCONJ
ap-7583	797	9	n	n	PROPN
ap-7583	797	10	+	+	CCONJ
ap-7583	797	11	1	1	NUM
ap-7583	797	12	)	)	PUNCT
ap-7583	797	13	k	k	NOUN
ap-7583	798	1	(	(	PUNCT
ap-7583	798	2	β2	β2	NOUN
ap-7583	798	3	α3	α3	PROPN
ap-7583	798	4	+	+	CCONJ
ap-7583	798	5	2n	2n	NUM
ap-7583	798	6	)	)	PUNCT
ap-7583	799	1	k	k	X
ap-7583	799	2	.	.	PUNCT
ap-7583	800	1	(	(	PUNCT
ap-7583	800	2	132	132	NUM
ap-7583	800	3	)	)	PUNCT
ap-7583	800	4	proof	proof	NOUN
ap-7583	800	5	.	.	PUNCT
ap-7583	801	1	the	the	DET
ap-7583	801	2	proof	proof	NOUN
ap-7583	801	3	follows	follow	VERB
ap-7583	801	4	by	by	ADP
ap-7583	801	5	multiplying	multiply	VERB
ap-7583	801	6	the	the	DET
ap-7583	801	7	recurrence	recurrence	NOUN
ap-7583	801	8	relation	relation	NOUN
ap-7583	801	9	(	(	PUNCT
ap-7583	801	10	128	128	NUM
ap-7583	801	11	)	)	PUNCT
ap-7583	801	12	by	by	ADP
ap-7583	801	13	xk−2ρ(x	xk−2ρ(x	PROPN
ap-7583	801	14	)	)	PUNCT
ap-7583	801	15	,	,	PUNCT
ap-7583	801	16	with	with	ADP
ap-7583	801	17	the	the	DET
ap-7583	801	18	normalized	normalize	VERB
ap-7583	801	19	weight	weight	NOUN
ap-7583	801	20	function	function	NOUN
ap-7583	801	21	∫	∫	PROPN
ap-7583	801	22	ρ(x)dx	ρ(x)dx	ADP
ap-7583	801	23	=	=	SYM
ap-7583	801	24	1	1	NUM
ap-7583	801	25	,	,	PUNCT
ap-7583	801	26	and	and	CCONJ
ap-7583	801	27	integrating	integrate	VERB
ap-7583	801	28	over	over	ADP
ap-7583	801	29	x.	x.	NOUN
ap-7583	802	1	this	this	DET
ap-7583	802	2	procedure	procedure	NOUN
ap-7583	802	3	yields	yield	VERB
ap-7583	802	4	a	a	DET
ap-7583	802	5	two	two	NUM
ap-7583	802	6	-	-	PUNCT
ap-7583	802	7	term	term	NOUN
ap-7583	802	8	recurrence	recurrence	NOUN
ap-7583	802	9	relation	relation	NOUN
ap-7583	802	10	gq	gq	PROPN
ap-7583	803	1	k	k	PROPN
ap-7583	803	2	=	=	PUNCT
ap-7583	803	3	k	k	PROPN
ap-7583	803	4	(	(	PUNCT
ap-7583	803	5	k	k	PROPN
ap-7583	803	6	+	+	CCONJ
ap-7583	803	7	n	n	PROPN
ap-7583	803	8	+	+	NOUN
ap-7583	803	9	1	1	NUM
ap-7583	803	10	)	)	PUNCT
ap-7583	803	11	(	(	PUNCT
ap-7583	803	12	(	(	PUNCT
ap-7583	803	13	k	k	X
ap-7583	803	14	+	+	CCONJ
ap-7583	803	15	n)α1	n)α1	PROPN
ap-7583	803	16	+	+	CCONJ
ap-7583	803	17	β0	β0	NOUN
ap-7583	803	18	)	)	PUNCT
ap-7583	803	19	(	(	PUNCT
ap-7583	803	20	β2	β2	VERB
ap-7583	803	21	+	+	CCONJ
ap-7583	803	22	α3(k	α3(k	NUM
ap-7583	803	23	+	+	CCONJ
ap-7583	803	24	2n	2n	NUM
ap-7583	803	25	−	−	NOUN
ap-7583	803	26	1	1	NUM
ap-7583	803	27	)	)	PUNCT
ap-7583	803	28	)	)	PUNCT
ap-7583	803	29	)	)	PUNCT
ap-7583	804	1	gq	gq	PROPN
ap-7583	804	2	k−1	k−1	PROPN
ap-7583	804	3	,	,	PUNCT
ap-7583	804	4	where	where	SCONJ
ap-7583	804	5	gq	gq	VERB
ap-7583	804	6	k	k	PROPN
ap-7583	804	7	=	=	SYM
ap-7583	804	8	∫	∫	PROPN
ap-7583	804	9	|qn	|qn	NUM
ap-7583	805	1	k	k	X
ap-7583	805	2	(	(	PUNCT
ap-7583	805	3	x)|2ρ(z)dz	x)|2ρ(z)dz	PROPN
ap-7583	805	4	=	=	SYM
ap-7583	805	5	∫	∫	PROPN
ap-7583	806	1	xkqn	xkqn	PROPN
ap-7583	806	2	k	k	PROPN
ap-7583	806	3	(	(	PUNCT
ap-7583	806	4	x)ρ(x)dx	x)ρ(x)dx	PROPN
ap-7583	806	5	with	with	ADP
ap-7583	806	6	a	a	DET
ap-7583	806	7	solution	solution	NOUN
ap-7583	806	8	given	give	VERB
ap-7583	806	9	by	by	ADP
ap-7583	806	10	(	(	PUNCT
ap-7583	806	11	132	132	NUM
ap-7583	806	12	)	)	PUNCT
ap-7583	806	13	.	.	PUNCT
ap-7583	807	1	we	we	PRON
ap-7583	807	2	see	see	VERB
ap-7583	807	3	that	that	SCONJ
ap-7583	807	4	,	,	PUNCT
ap-7583	807	5	in	in	ADP
ap-7583	807	6	general	general	ADJ
ap-7583	807	7	,	,	PUNCT
ap-7583	807	8	the	the	DET
ap-7583	807	9	norm	norm	NOUN
ap-7583	807	10	of	of	ADP
ap-7583	807	11	the	the	DET
ap-7583	807	12	polynomials	polynomial	NOUN
ap-7583	808	1	qn	qn	PROPN
ap-7583	808	2	k	k	PROPN
ap-7583	808	3	(	(	PUNCT
ap-7583	808	4	x	x	X
ap-7583	808	5	)	)	PUNCT
ap-7583	808	6	does	do	AUX
ap-7583	808	7	not	not	PART
ap-7583	808	8	vanish	vanish	VERB
ap-7583	808	9	.	.	PUNCT
ap-7583	809	1	acknowledgements	acknowledgement	NOUN
ap-7583	809	2	partial	partial	ADJ
ap-7583	809	3	financial	financial	ADJ
ap-7583	809	4	support	support	NOUN
ap-7583	809	5	of	of	ADP
ap-7583	809	6	this	this	DET
ap-7583	809	7	work	work	NOUN
ap-7583	809	8	under	under	ADP
ap-7583	809	9	grant	grant	NOUN
ap-7583	809	10	number	number	NOUN
ap-7583	809	11	gp249507	gp249507	NOUN
ap-7583	809	12	from	from	ADP
ap-7583	809	13	the	the	DET
ap-7583	809	14	natural	natural	ADJ
ap-7583	809	15	sciences	science	NOUN
ap-7583	809	16	and	and	CCONJ
ap-7583	809	17	engineering	engineering	NOUN
ap-7583	809	18	research	research	NOUN
ap-7583	809	19	council	council	PROPN
ap-7583	809	20	of	of	ADP
ap-7583	809	21	canada	canada	PROPN
ap-7583	809	22	is	be	AUX
ap-7583	809	23	gratefully	gratefully	ADV
ap-7583	809	24	acknowledged	acknowledge	VERB
ap-7583	809	25	.	.	PUNCT
ap-7583	810	1	references	reference	NOUN
ap-7583	810	2	[	[	X
ap-7583	810	3	1	1	NUM
ap-7583	810	4	]	]	PUNCT
ap-7583	810	5	a.	a.	PROPN
ap-7583	810	6	f.	f.	PROPN
ap-7583	810	7	nikiforov	nikiforov	PROPN
ap-7583	810	8	,	,	PUNCT
ap-7583	810	9	v.	v.	PROPN
ap-7583	810	10	b.	b.	PROPN
ap-7583	810	11	uvarov	uvarov	PROPN
ap-7583	810	12	.	.	PUNCT
ap-7583	811	1	special	special	ADJ
ap-7583	811	2	functions	function	NOUN
ap-7583	811	3	of	of	ADP
ap-7583	811	4	mathematical	mathematical	ADJ
ap-7583	811	5	physics	physics	NOUN
ap-7583	811	6	.	.	PUNCT
ap-7583	812	1	birkhäuser	birkhäuser	PROPN
ap-7583	812	2	verlag	verlag	PROPN
ap-7583	812	3	,	,	PUNCT
ap-7583	812	4	basel	basel	PROPN
ap-7583	812	5	,	,	PUNCT
ap-7583	812	6	1988	1988	NUM
ap-7583	812	7	.	.	PUNCT
ap-7583	813	1	https://doi.org/10.1007/978-1-4757-1595-8	https://doi.org/10.1007/978-1-4757-1595-8	PROPN
ap-7583	813	2	.	.	PUNCT
ap-7583	814	1	[	[	X
ap-7583	814	2	2	2	X
ap-7583	814	3	]	]	PUNCT
ap-7583	814	4	e.	e.	PROPN
ap-7583	814	5	j.	j.	PROPN
ap-7583	814	6	routh	routh	PROPN
ap-7583	814	7	.	.	PUNCT
ap-7583	815	1	on	on	ADP
ap-7583	815	2	some	some	DET
ap-7583	815	3	properties	property	NOUN
ap-7583	815	4	of	of	ADP
ap-7583	815	5	certain	certain	ADJ
ap-7583	815	6	solutions	solution	NOUN
ap-7583	815	7	of	of	ADP
ap-7583	815	8	a	a	DET
ap-7583	815	9	differential	differential	ADJ
ap-7583	815	10	equation	equation	NOUN
ap-7583	815	11	of	of	ADP
ap-7583	815	12	the	the	DET
ap-7583	815	13	second	second	ADJ
ap-7583	815	14	order	order	NOUN
ap-7583	815	15	.	.	PUNCT
ap-7583	816	1	proceedings	proceeding	NOUN
ap-7583	816	2	of	of	ADP
ap-7583	816	3	the	the	DET
ap-7583	816	4	london	london	PROPN
ap-7583	816	5	mathematical	mathematical	ADJ
ap-7583	816	6	society	society	NOUN
ap-7583	816	7	16:245–262	16:245–262	PROPN
ap-7583	816	8	,	,	PUNCT
ap-7583	816	9	1884/85	1884/85	NUM
ap-7583	816	10	.	.	PUNCT
ap-7583	817	1	https://doi.org/10.1112/plms/s1-16.1.245	https://doi.org/10.1112/plms/s1-16.1.245	PROPN
ap-7583	817	2	.	.	PUNCT
ap-7583	818	1	[	[	X
ap-7583	818	2	3	3	X
ap-7583	818	3	]	]	X
ap-7583	818	4	n.	n.	NOUN
ap-7583	818	5	saad	saad	PROPN
ap-7583	818	6	,	,	PUNCT
ap-7583	818	7	r.	r.	PROPN
ap-7583	818	8	l.	l.	PROPN
ap-7583	818	9	hall	hall	PROPN
ap-7583	818	10	,	,	PUNCT
ap-7583	818	11	h.	h.	PROPN
ap-7583	818	12	ciftci	ciftci	PROPN
ap-7583	818	13	.	.	PUNCT
ap-7583	818	14	criterion	criterion	NOUN
ap-7583	818	15	for	for	ADP
ap-7583	818	16	polynomial	polynomial	ADJ
ap-7583	818	17	solutions	solution	NOUN
ap-7583	818	18	to	to	ADP
ap-7583	818	19	a	a	DET
ap-7583	818	20	class	class	NOUN
ap-7583	818	21	of	of	ADP
ap-7583	818	22	linear	linear	PROPN
ap-7583	818	23	differential	differential	ADJ
ap-7583	818	24	equations	equation	NOUN
ap-7583	818	25	of	of	ADP
ap-7583	818	26	second	second	ADJ
ap-7583	818	27	order	order	NOUN
ap-7583	818	28	.	.	PUNCT
ap-7583	819	1	journal	journal	PROPN
ap-7583	819	2	of	of	ADP
ap-7583	819	3	physics	physics	PROPN
ap-7583	819	4	a	a	PRON
ap-7583	819	5	:	:	PUNCT
ap-7583	819	6	mathematical	mathematical	ADJ
ap-7583	819	7	and	and	CCONJ
ap-7583	819	8	general	general	ADJ
ap-7583	819	9	39(43):13445–13454	39(43):13445–13454	NUM
ap-7583	819	10	,	,	PUNCT
ap-7583	819	11	2006	2006	NUM
ap-7583	819	12	.	.	PUNCT
ap-7583	820	1	https://doi.org/10.1088/0305-4470/39/43/004	https://doi.org/10.1088/0305-4470/39/43/004	PROPN
ap-7583	820	2	.	.	PUNCT
ap-7583	821	1	[	[	X
ap-7583	821	2	4	4	NUM
ap-7583	821	3	]	]	X
ap-7583	821	4	n.	n.	NOUN
ap-7583	821	5	saad	saad	PROPN
ap-7583	821	6	,	,	PUNCT
ap-7583	821	7	r.	r.	PROPN
ap-7583	821	8	l.	l.	PROPN
ap-7583	821	9	hall	hall	PROPN
ap-7583	821	10	,	,	PUNCT
ap-7583	821	11	v.	v.	PROPN
ap-7583	821	12	a.	a.	PROPN
ap-7583	821	13	trenton	trenton	PROPN
ap-7583	821	14	.	.	PUNCT
ap-7583	822	1	polynomial	polynomial	ADJ
ap-7583	822	2	solutions	solution	NOUN
ap-7583	822	3	for	for	ADP
ap-7583	822	4	a	a	DET
ap-7583	822	5	class	class	NOUN
ap-7583	822	6	of	of	ADP
ap-7583	822	7	second	second	ADJ
ap-7583	822	8	-	-	PUNCT
ap-7583	822	9	order	order	NOUN
ap-7583	822	10	linear	linear	ADJ
ap-7583	822	11	differential	differential	NOUN
ap-7583	822	12	equations	equation	NOUN
ap-7583	822	13	.	.	PUNCT
ap-7583	823	1	applied	apply	VERB
ap-7583	823	2	mathematics	mathematic	NOUN
ap-7583	823	3	and	and	CCONJ
ap-7583	823	4	computation	computation	NOUN
ap-7583	823	5	226:615–634	226:615–634	NUM
ap-7583	823	6	,	,	PUNCT
ap-7583	823	7	2014	2014	NUM
ap-7583	823	8	.	.	PUNCT
ap-7583	824	1	https://doi.org/10.1016/j.amc.2013.10.056	https://doi.org/10.1016/j.amc.2013.10.056	PROPN
ap-7583	824	2	.	.	PUNCT
ap-7583	825	1	[	[	X
ap-7583	825	2	5	5	NUM
ap-7583	825	3	]	]	PUNCT
ap-7583	825	4	a.	a.	NOUN
ap-7583	825	5	ronveaux	ronveaux	NOUN
ap-7583	825	6	(	(	PUNCT
ap-7583	825	7	ed	ed	NOUN
ap-7583	825	8	.	.	PUNCT
ap-7583	825	9	)	)	PUNCT
ap-7583	825	10	.	.	PUNCT
ap-7583	826	1	heun	heun	PROPN
ap-7583	826	2	’s	’s	PART
ap-7583	826	3	differential	differential	ADJ
ap-7583	826	4	equations	equation	NOUN
ap-7583	826	5	.	.	PUNCT
ap-7583	827	1	oxford	oxford	PROPN
ap-7583	827	2	university	university	PROPN
ap-7583	827	3	press	press	NOUN
ap-7583	827	4	,	,	PUNCT
ap-7583	827	5	new	new	PROPN
ap-7583	827	6	york	york	PROPN
ap-7583	827	7	,	,	PUNCT
ap-7583	827	8	1995	1995	NUM
ap-7583	827	9	.	.	PUNCT
ap-7583	828	1	[	[	X
ap-7583	828	2	6	6	NUM
ap-7583	828	3	]	]	PUNCT
ap-7583	828	4	s.	s.	PROPN
ap-7583	828	5	y.	y.	PROPN
ap-7583	828	6	slavyanov	slavyanov	PROPN
ap-7583	828	7	,	,	PUNCT
ap-7583	828	8	w.	w.	PROPN
ap-7583	828	9	lay	lay	PROPN
ap-7583	828	10	.	.	PUNCT
ap-7583	829	1	special	special	ADJ
ap-7583	829	2	functions	function	NOUN
ap-7583	829	3	:	:	PUNCT
ap-7583	829	4	a	a	DET
ap-7583	829	5	unified	unified	ADJ
ap-7583	829	6	theory	theory	NOUN
ap-7583	829	7	based	base	VERB
ap-7583	829	8	on	on	ADP
ap-7583	829	9	singularities	singularity	NOUN
ap-7583	829	10	.	.	PUNCT
ap-7583	830	1	oxford	oxford	PROPN
ap-7583	830	2	university	university	PROPN
ap-7583	830	3	press	press	NOUN
ap-7583	830	4	,	,	PUNCT
ap-7583	830	5	oxford	oxford	PROPN
ap-7583	830	6	,	,	PUNCT
ap-7583	830	7	2000	2000	NUM
ap-7583	830	8	.	.	PUNCT
ap-7583	831	1	isbn	isbn	ADJ
ap-7583	831	2	0	0	NUM
ap-7583	831	3	-	-	SYM
ap-7583	831	4	19	19	NUM
ap-7583	831	5	-	-	PUNCT
ap-7583	831	6	850573	850573	NUM
ap-7583	831	7	-	-	SYM
ap-7583	831	8	6	6	NUM
ap-7583	831	9	.	.	PUNCT
ap-7583	832	1	[	[	X
ap-7583	832	2	7	7	NUM
ap-7583	832	3	]	]	PUNCT
ap-7583	832	4	a.	a.	NOUN
ap-7583	832	5	decarreau	decarreau	PROPN
ap-7583	832	6	,	,	PUNCT
ap-7583	832	7	m.-c	m.-c	PROPN
ap-7583	832	8	.	.	PUNCT
ap-7583	833	1	dumont	dumont	PROPN
ap-7583	833	2	-	-	PUNCT
ap-7583	833	3	lepage	lepage	NOUN
ap-7583	833	4	,	,	PUNCT
ap-7583	833	5	p.	p.	PROPN
ap-7583	833	6	maroni	maroni	PROPN
ap-7583	833	7	,	,	PUNCT
ap-7583	833	8	et	et	PROPN
ap-7583	833	9	al	al	PROPN
ap-7583	833	10	.	.	PROPN
ap-7583	833	11	formes	formes	PROPN
ap-7583	833	12	canoniques	canoniques	PROPN
ap-7583	833	13	des	des	PROPN
ap-7583	833	14	équations	équations	PROPN
ap-7583	833	15	confluentes	confluente	VERB
ap-7583	833	16	de	de	X
ap-7583	833	17	l’équation	l’équation	PROPN
ap-7583	833	18	de	de	X
ap-7583	833	19	heun	heun	NOUN
ap-7583	833	20	.	.	PUNCT
ap-7583	834	1	annales	annales	PROPN
ap-7583	834	2	de	de	X
ap-7583	834	3	la	la	X
ap-7583	834	4	societé	societé	VERB
ap-7583	834	5	scientifique	scientifique	X
ap-7583	834	6	de	de	X
ap-7583	834	7	bruxelles	bruxelles	PROPN
ap-7583	834	8	série	série	PROPN
ap-7583	834	9	i	i	PROPN
ap-7583	834	10	sciences	science	VERB
ap-7583	834	11	mathématiques	mathématique	NOUN
ap-7583	834	12	,	,	PUNCT
ap-7583	834	13	astronomiques	astronomique	VERB
ap-7583	834	14	et	et	NOUN
ap-7583	834	15	physiques	physique	NOUN
ap-7583	834	16	92(1	92(1	ADV
ap-7583	834	17	-	-	PUNCT
ap-7583	834	18	2):53–78	2):53–78	NUM
ap-7583	834	19	,	,	PUNCT
ap-7583	834	20	1978	1978	NUM
ap-7583	834	21	.	.	PUNCT
ap-7583	835	1	[	[	X
ap-7583	835	2	8	8	NUM
ap-7583	835	3	]	]	PUNCT
ap-7583	835	4	a.	a.	NOUN
ap-7583	835	5	decarreau	decarreau	PROPN
ap-7583	835	6	,	,	PUNCT
ap-7583	835	7	p.	p.	PROPN
ap-7583	835	8	maroni	maroni	PROPN
ap-7583	835	9	,	,	PUNCT
ap-7583	836	1	a.	a.	PROPN
ap-7583	836	2	robert	robert	PROPN
ap-7583	836	3	.	.	PUNCT
ap-7583	836	4	sur	sur	PROPN
ap-7583	836	5	les	les	PROPN
ap-7583	836	6	équations	équations	PROPN
ap-7583	836	7	confluentes	confluente	VERB
ap-7583	836	8	de	de	X
ap-7583	836	9	l’équation	l’équation	PROPN
ap-7583	836	10	de	de	X
ap-7583	836	11	heun	heun	NOUN
ap-7583	836	12	.	.	PUNCT
ap-7583	837	1	annales	annales	PROPN
ap-7583	837	2	de	de	X
ap-7583	837	3	la	la	X
ap-7583	837	4	societé	societé	VERB
ap-7583	837	5	scientifique	scientifique	X
ap-7583	837	6	de	de	X
ap-7583	837	7	bruxelles	bruxelles	PROPN
ap-7583	837	8	série	série	PROPN
ap-7583	837	9	i	i	PROPN
ap-7583	837	10	sciences	science	VERB
ap-7583	837	11	mathématiques	mathématique	NOUN
ap-7583	837	12	,	,	PUNCT
ap-7583	837	13	astronomiques	astronomique	VERB
ap-7583	837	14	et	et	NOUN
ap-7583	837	15	physiques	physique	NOUN
ap-7583	837	16	92(3):151–189	92(3):151–189	NUM
ap-7583	837	17	,	,	PUNCT
ap-7583	837	18	1978	1978	NUM
ap-7583	837	19	.	.	PUNCT
ap-7583	838	1	[	[	X
ap-7583	838	2	9	9	NUM
ap-7583	838	3	]	]	PUNCT
ap-7583	838	4	k.	k.	PROPN
ap-7583	838	5	heun	heun	PROPN
ap-7583	838	6	.	.	PUNCT
ap-7583	839	1	zur	zur	PROPN
ap-7583	840	1	theorie	theorie	PROPN
ap-7583	840	2	der	der	NOUN
ap-7583	840	3	riemann’schen	riemann’schen	VERB
ap-7583	840	4	functionen	functionen	PROPN
ap-7583	840	5	zweiter	zweiter	PROPN
ap-7583	840	6	ordnung	ordnung	PROPN
ap-7583	840	7	mit	mit	PROPN
ap-7583	840	8	vier	vier	NOUN
ap-7583	840	9	verzweigungspunkten	verzweigungspunkten	ADJ
ap-7583	840	10	.	.	PUNCT
ap-7583	841	1	mathematische	mathematische	NOUN
ap-7583	841	2	annalen	annalen	VERB
ap-7583	841	3	33(2):161–179	33(2):161–179	PROPN
ap-7583	841	4	,	,	PUNCT
ap-7583	841	5	1888	1888	NUM
ap-7583	841	6	.	.	PUNCT
ap-7583	842	1	https://doi.org/10.1007/bf01443849	https://doi.org/10.1007/bf01443849	X
ap-7583	842	2	.	.	PUNCT
ap-7583	843	1	[	[	X
ap-7583	843	2	10	10	NUM
ap-7583	843	3	]	]	X
ap-7583	843	4	f.	f.	PROPN
ap-7583	843	5	beukers	beukers	PROPN
ap-7583	843	6	,	,	PUNCT
ap-7583	843	7	a.	a.	PROPN
ap-7583	843	8	van	van	PROPN
ap-7583	843	9	der	der	PROPN
ap-7583	843	10	waall	waall	PROPN
ap-7583	843	11	.	.	PUNCT
ap-7583	844	1	lamé	lamé	NOUN
ap-7583	844	2	equations	equation	NOUN
ap-7583	844	3	with	with	ADP
ap-7583	844	4	algebraic	algebraic	ADJ
ap-7583	844	5	solutions	solution	NOUN
ap-7583	844	6	.	.	PUNCT
ap-7583	845	1	journal	journal	PROPN
ap-7583	845	2	of	of	ADP
ap-7583	845	3	differential	differential	ADJ
ap-7583	845	4	equations	equation	NOUN
ap-7583	845	5	197(1):1–25	197(1):1–25	NUM
ap-7583	845	6	,	,	PUNCT
ap-7583	845	7	2004	2004	NUM
ap-7583	845	8	.	.	PUNCT
ap-7583	846	1	https://doi.org/10.1016/j.jde.2003.10.017	https://doi.org/10.1016/j.jde.2003.10.017	X
ap-7583	846	2	.	.	PUNCT
ap-7583	847	1	[	[	X
ap-7583	847	2	11	11	NUM
ap-7583	847	3	]	]	PUNCT
ap-7583	847	4	a.	a.	NOUN
ap-7583	847	5	turbiner	turbiner	NOUN
ap-7583	847	6	.	.	PUNCT
ap-7583	848	1	on	on	ADP
ap-7583	848	2	polynomial	polynomial	ADJ
ap-7583	848	3	solutions	solution	NOUN
ap-7583	848	4	of	of	ADP
ap-7583	848	5	differential	differential	ADJ
ap-7583	848	6	equations	equation	NOUN
ap-7583	848	7	.	.	PUNCT
ap-7583	849	1	journal	journal	PROPN
ap-7583	849	2	of	of	ADP
ap-7583	849	3	mathematical	mathematical	ADJ
ap-7583	849	4	physics	physics	NOUN
ap-7583	849	5	33(12):3989–3993	33(12):3989–3993	NUM
ap-7583	849	6	,	,	PUNCT
ap-7583	849	7	1992	1992	NUM
ap-7583	849	8	.	.	PUNCT
ap-7583	850	1	https://doi.org/10.1063/1.529848	https://doi.org/10.1063/1.529848	X
ap-7583	850	2	.	.	PUNCT
ap-7583	851	1	[	[	X
ap-7583	851	2	12	12	NUM
ap-7583	851	3	]	]	X
ap-7583	851	4	r.	r.	PROPN
ap-7583	851	5	v.	v.	PROPN
ap-7583	851	6	craster	craster	PROPN
ap-7583	851	7	,	,	PUNCT
ap-7583	851	8	v.	v.	PROPN
ap-7583	851	9	h.	h.	PROPN
ap-7583	851	10	hoàng	hoàng	PROPN
ap-7583	851	11	.	.	PUNCT
ap-7583	852	1	applications	application	NOUN
ap-7583	852	2	of	of	ADP
ap-7583	852	3	fuchsian	fuchsian	ADJ
ap-7583	852	4	differential	differential	ADJ
ap-7583	852	5	equations	equation	NOUN
ap-7583	852	6	to	to	PART
ap-7583	852	7	free	free	VERB
ap-7583	852	8	boundary	boundary	ADJ
ap-7583	852	9	problems	problem	NOUN
ap-7583	852	10	.	.	PUNCT
ap-7583	853	1	proceedings	proceeding	NOUN
ap-7583	853	2	of	of	ADP
ap-7583	853	3	the	the	DET
ap-7583	853	4	royal	royal	ADJ
ap-7583	853	5	society	society	NOUN
ap-7583	853	6	a	a	DET
ap-7583	853	7	mathematical	mathematical	ADJ
ap-7583	853	8	,	,	PUNCT
ap-7583	853	9	physical	physical	ADJ
ap-7583	853	10	and	and	CCONJ
ap-7583	853	11	engineering	engineering	NOUN
ap-7583	853	12	sciences	science	NOUN
ap-7583	853	13	454(1972):1241–1252	454(1972):1241–1252	NUM
ap-7583	853	14	,	,	PUNCT
ap-7583	853	15	1998	1998	NUM
ap-7583	853	16	.	.	PUNCT
ap-7583	854	1	https://doi.org/10.1098/rspa.1998.0204	https://doi.org/10.1098/rspa.1998.0204	PROPN
ap-7583	854	2	.	.	PROPN
ap-7583	854	3	188	188	NUM
ap-7583	854	4	https://doi.org/10.1007/978-1-4757-1595-8	https://doi.org/10.1007/978-1-4757-1595-8	PROPN
ap-7583	854	5	https://doi.org/10.1112/plms/s1-16.1.245	https://doi.org/10.1112/plms/s1-16.1.245	VERB
ap-7583	854	6	https://doi.org/10.1088/0305-4470/39/43/004	https://doi.org/10.1088/0305-4470/39/43/004	PROPN
ap-7583	854	7	https://doi.org/10.1016/j.amc.2013.10.056	https://doi.org/10.1016/j.amc.2013.10.056	PROPN
ap-7583	854	8	https://doi.org/10.1007/bf01443849	https://doi.org/10.1007/bf01443849	PROPN
ap-7583	854	9	https://doi.org/10.1016/j.jde.2003.10.017	https://doi.org/10.1016/j.jde.2003.10.017	ADP
ap-7583	854	10	https://doi.org/10.1063/1.529848	https://doi.org/10.1063/1.529848	PROPN
ap-7583	854	11	https://doi.org/10.1098/rspa.1998.0204	https://doi.org/10.1098/rspa.1998.0204	PROPN
ap-7583	854	12	vol	vol	NOUN
ap-7583	854	13	.	.	PUNCT
ap-7583	855	1	62	62	NUM
ap-7583	855	2	no	no	INTJ
ap-7583	855	3	.	.	PUNCT
ap-7583	856	1	1/2022	1/2022	NUM
ap-7583	856	2	on	on	ADP
ap-7583	856	3	generalized	generalized	ADJ
ap-7583	856	4	heun	heun	NOUN
ap-7583	856	5	equation	equation	NOUN
ap-7583	856	6	with	with	ADP
ap-7583	856	7	some	some	DET
ap-7583	856	8	mathematical	mathematical	NOUN
ap-7583	856	9	.	.	PUNCT
ap-7583	856	10	.	.	PUNCT
ap-7583	856	11	.	.	PUNCT
ap-7583	857	1	[	[	X
ap-7583	857	2	13	13	NUM
ap-7583	857	3	]	]	PUNCT
ap-7583	858	1	p.	p.	NOUN
ap-7583	858	2	p.	p.	NOUN
ap-7583	858	3	fiziev	fiziev	PROPN
ap-7583	858	4	.	.	PUNCT
ap-7583	859	1	the	the	DET
ap-7583	859	2	heun	heun	NOUN
ap-7583	859	3	functions	function	NOUN
ap-7583	859	4	as	as	ADP
ap-7583	859	5	a	a	DET
ap-7583	859	6	modern	modern	ADJ
ap-7583	859	7	powerful	powerful	ADJ
ap-7583	859	8	tool	tool	NOUN
ap-7583	859	9	for	for	ADP
ap-7583	859	10	research	research	NOUN
ap-7583	859	11	in	in	ADP
ap-7583	859	12	different	different	ADJ
ap-7583	859	13	scientific	scientific	ADJ
ap-7583	859	14	domains	domain	NOUN
ap-7583	859	15	,	,	PUNCT
ap-7583	859	16	2015	2015	NUM
ap-7583	859	17	.	.	PUNCT
ap-7583	860	1	arxiv:1512.04025v1	arxiv:1512.04025v1	NOUN
ap-7583	860	2	.	.	PUNCT
ap-7583	861	1	[	[	X
ap-7583	861	2	14	14	NUM
ap-7583	861	3	]	]	X
ap-7583	861	4	mathematical	mathematical	ADJ
ap-7583	861	5	physics	physics	NOUN
ap-7583	861	6	.	.	PUNCT
ap-7583	862	1	in	in	ADP
ap-7583	862	2	u.	u.	PROPN
ap-7583	862	3	camcı	camcı	PROPN
ap-7583	862	4	,	,	PUNCT
ap-7583	862	5	i.	i.	PROPN
ap-7583	862	6	semiz	semiz	PROPN
ap-7583	862	7	(	(	PUNCT
ap-7583	862	8	eds	ed	NOUN
ap-7583	862	9	.	.	PUNCT
ap-7583	862	10	)	)	PUNCT
ap-7583	862	11	,	,	PUNCT
ap-7583	862	12	proceedings	proceeding	NOUN
ap-7583	862	13	of	of	ADP
ap-7583	862	14	the	the	DET
ap-7583	862	15	13th	13th	ADJ
ap-7583	862	16	regional	regional	ADJ
ap-7583	862	17	conference	conference	NOUN
ap-7583	862	18	held	hold	VERB
ap-7583	862	19	in	in	ADP
ap-7583	862	20	antalya	antalya	PROPN
ap-7583	862	21	,	,	PUNCT
ap-7583	862	22	october	october	PROPN
ap-7583	862	23	27–31	27–31	PROPN
ap-7583	862	24	,	,	PUNCT
ap-7583	862	25	2010	2010	NUM
ap-7583	862	26	,	,	PUNCT
ap-7583	862	27	pp	pp	ADJ
ap-7583	862	28	.	.	PUNCT
ap-7583	863	1	23–39	23–39	NUM
ap-7583	863	2	.	.	PUNCT
ap-7583	864	1	world	world	NOUN
ap-7583	864	2	scientific	scientific	PROPN
ap-7583	864	3	publishing	publishing	PROPN
ap-7583	864	4	co.	co.	PROPN
ap-7583	864	5	pte	pte	PROPN
ap-7583	864	6	.	.	PROPN
ap-7583	864	7	ltd	ltd	PROPN
ap-7583	864	8	.	.	PROPN
ap-7583	864	9	,	,	PUNCT
ap-7583	864	10	hackensack	hackensack	PROPN
ap-7583	864	11	,	,	PUNCT
ap-7583	864	12	nj	nj	PROPN
ap-7583	864	13	,	,	PUNCT
ap-7583	864	14	2013	2013	NUM
ap-7583	864	15	.	.	PUNCT
ap-7583	865	1	https://doi.org/10.1142/8566	https://doi.org/10.1142/8566	NOUN
ap-7583	865	2	.	.	PUNCT
ap-7583	866	1	[	[	X
ap-7583	866	2	15	15	NUM
ap-7583	866	3	]	]	X
ap-7583	866	4	h.	h.	PROPN
ap-7583	866	5	ciftci	ciftci	PROPN
ap-7583	866	6	,	,	PUNCT
ap-7583	866	7	r.	r.	PROPN
ap-7583	866	8	l.	l.	PROPN
ap-7583	866	9	hall	hall	PROPN
ap-7583	866	10	,	,	PUNCT
ap-7583	866	11	n.	n.	PROPN
ap-7583	866	12	saad	saad	PROPN
ap-7583	866	13	,	,	PUNCT
ap-7583	866	14	e.	e.	PROPN
ap-7583	866	15	dogu	dogu	PROPN
ap-7583	866	16	.	.	PUNCT
ap-7583	867	1	physical	physical	ADJ
ap-7583	867	2	applications	application	NOUN
ap-7583	867	3	of	of	ADP
ap-7583	867	4	second	second	ADJ
ap-7583	867	5	-	-	PUNCT
ap-7583	867	6	order	order	NOUN
ap-7583	867	7	linear	linear	NOUN
ap-7583	867	8	differential	differential	NOUN
ap-7583	867	9	equations	equation	NOUN
ap-7583	867	10	that	that	PRON
ap-7583	867	11	admit	admit	VERB
ap-7583	867	12	polynomial	polynomial	ADJ
ap-7583	867	13	solutions	solution	NOUN
ap-7583	867	14	.	.	PUNCT
ap-7583	868	1	journal	journal	PROPN
ap-7583	868	2	of	of	ADP
ap-7583	868	3	physics	physics	PROPN
ap-7583	868	4	a	a	PRON
ap-7583	868	5	:	:	PUNCT
ap-7583	868	6	mathematical	mathematical	ADJ
ap-7583	868	7	and	and	CCONJ
ap-7583	868	8	theoretical	theoretical	ADJ
ap-7583	868	9	43(41):415206	43(41):415206	NUM
ap-7583	868	10	,	,	PUNCT
ap-7583	868	11	14	14	NUM
ap-7583	868	12	,	,	PUNCT
ap-7583	868	13	2010	2010	NUM
ap-7583	868	14	.	.	PUNCT
ap-7583	869	1	https://doi.org/10.1088/1751-8113/43/41/415206	https://doi.org/10.1088/1751-8113/43/41/415206	NOUN
ap-7583	869	2	.	.	PUNCT
ap-7583	870	1	[	[	X
ap-7583	870	2	16	16	NUM
ap-7583	870	3	]	]	PUNCT
ap-7583	870	4	y.-z	y.-z	NOUN
ap-7583	870	5	.	.	PUNCT
ap-7583	871	1	zhang	zhang	PROPN
ap-7583	871	2	.	.	PUNCT
ap-7583	872	1	exact	exact	ADJ
ap-7583	872	2	polynomial	polynomial	ADJ
ap-7583	872	3	solutions	solution	NOUN
ap-7583	872	4	of	of	ADP
ap-7583	872	5	second	second	ADJ
ap-7583	872	6	order	order	NOUN
ap-7583	872	7	differential	differential	ADJ
ap-7583	872	8	equations	equation	NOUN
ap-7583	872	9	and	and	CCONJ
ap-7583	872	10	their	their	PRON
ap-7583	872	11	applications	application	NOUN
ap-7583	872	12	.	.	PUNCT
ap-7583	873	1	journal	journal	PROPN
ap-7583	873	2	of	of	ADP
ap-7583	873	3	physics	physics	PROPN
ap-7583	873	4	a	a	PRON
ap-7583	873	5	:	:	PUNCT
ap-7583	873	6	mathematical	mathematical	ADJ
ap-7583	873	7	and	and	CCONJ
ap-7583	873	8	theoretical	theoretical	ADJ
ap-7583	873	9	45(6):065206	45(6):065206	PROPN
ap-7583	873	10	,	,	PUNCT
ap-7583	873	11	2012	2012	NUM
ap-7583	873	12	.	.	PUNCT
ap-7583	874	1	https://doi.org/10.1088/1751-8113/45/6/065206	https://doi.org/10.1088/1751-8113/45/6/065206	NOUN
ap-7583	874	2	.	.	PUNCT
ap-7583	875	1	[	[	X
ap-7583	875	2	17	17	NUM
ap-7583	875	3	]	]	PUNCT
ap-7583	875	4	b.-h	b.-h	NOUN
ap-7583	875	5	.	.	PUNCT
ap-7583	876	1	chen	chen	PROPN
ap-7583	876	2	,	,	PUNCT
ap-7583	876	3	y.	y.	PROPN
ap-7583	876	4	wu	wu	PROPN
ap-7583	876	5	,	,	PUNCT
ap-7583	876	6	q.-t	q.-t	NOUN
ap-7583	876	7	.	.	PUNCT
ap-7583	877	1	xie	xie	PROPN
ap-7583	877	2	.	.	PUNCT
ap-7583	877	3	heun	heun	PROPN
ap-7583	877	4	functions	function	NOUN
ap-7583	877	5	and	and	CCONJ
ap-7583	877	6	quasi	quasi	ADJ
ap-7583	877	7	-	-	ADJ
ap-7583	877	8	exactly	exactly	ADV
ap-7583	877	9	solvable	solvable	ADJ
ap-7583	877	10	double	double	ADJ
ap-7583	877	11	-	-	PUNCT
ap-7583	877	12	well	well	NOUN
ap-7583	877	13	potentials	potential	NOUN
ap-7583	877	14	.	.	PUNCT
ap-7583	878	1	journal	journal	PROPN
ap-7583	878	2	of	of	ADP
ap-7583	878	3	physics	physics	PROPN
ap-7583	878	4	a	a	PRON
ap-7583	878	5	:	:	PUNCT
ap-7583	878	6	mathematical	mathematical	ADJ
ap-7583	878	7	and	and	CCONJ
ap-7583	878	8	theoretical	theoretical	ADJ
ap-7583	878	9	46(3):035301	46(3):035301	PROPN
ap-7583	878	10	,	,	PUNCT
ap-7583	878	11	2013	2013	NUM
ap-7583	878	12	.	.	PUNCT
ap-7583	879	1	https://doi.org/10.1088/1751-8113/46/3/035301	https://doi.org/10.1088/1751-8113/46/3/035301	NOUN
ap-7583	879	2	.	.	PUNCT
ap-7583	880	1	[	[	X
ap-7583	880	2	18	18	NUM
ap-7583	880	3	]	]	X
ap-7583	880	4	f.	f.	PROPN
ap-7583	880	5	caruso	caruso	PROPN
ap-7583	880	6	,	,	PUNCT
ap-7583	880	7	j.	j.	PROPN
ap-7583	880	8	martins	martins	PROPN
ap-7583	880	9	,	,	PUNCT
ap-7583	880	10	v.	v.	PROPN
ap-7583	880	11	oguri	oguri	PROPN
ap-7583	880	12	.	.	PUNCT
ap-7583	881	1	solving	solve	VERB
ap-7583	881	2	a	a	DET
ap-7583	881	3	two	two	NUM
ap-7583	881	4	-	-	PUNCT
ap-7583	881	5	electron	electron	NOUN
ap-7583	881	6	quantum	quantum	NOUN
ap-7583	881	7	dot	dot	NOUN
ap-7583	881	8	model	model	NOUN
ap-7583	881	9	in	in	ADP
ap-7583	881	10	terms	term	NOUN
ap-7583	881	11	of	of	ADP
ap-7583	881	12	polynomial	polynomial	ADJ
ap-7583	881	13	solutions	solution	NOUN
ap-7583	881	14	of	of	ADP
ap-7583	881	15	a	a	DET
ap-7583	881	16	biconfluent	biconfluent	ADJ
ap-7583	881	17	heun	heun	NOUN
ap-7583	881	18	equation	equation	NOUN
ap-7583	881	19	.	.	PUNCT
ap-7583	882	1	annals	annal	NOUN
ap-7583	882	2	of	of	ADP
ap-7583	882	3	physics	physics	NOUN
ap-7583	882	4	347:130–140	347:130–140	NUM
ap-7583	882	5	,	,	PUNCT
ap-7583	882	6	2014	2014	NUM
ap-7583	882	7	.	.	PUNCT
ap-7583	883	1	https://doi.org/10.1016/j.aop.2014.04.023	https://doi.org/10.1016/j.aop.2014.04.023	NUM
ap-7583	883	2	.	.	PUNCT
ap-7583	884	1	[	[	X
ap-7583	884	2	19	19	NUM
ap-7583	884	3	]	]	X
ap-7583	884	4	a.	a.	NOUN
ap-7583	884	5	v.	v.	PROPN
ap-7583	884	6	turbiner	turbiner	NOUN
ap-7583	884	7	.	.	PUNCT
ap-7583	885	1	one	one	NUM
ap-7583	885	2	-	-	PUNCT
ap-7583	885	3	dimensional	dimensional	ADJ
ap-7583	885	4	quasi	quasi	ADJ
ap-7583	885	5	-	-	ADJ
ap-7583	885	6	exactly	exactly	ADV
ap-7583	885	7	solvable	solvable	ADJ
ap-7583	885	8	schrödinger	schrödinger	ADJ
ap-7583	885	9	equations	equation	NOUN
ap-7583	885	10	.	.	PUNCT
ap-7583	886	1	physics	physics	NOUN
ap-7583	886	2	reports	report	VERB
ap-7583	886	3	a	a	DET
ap-7583	886	4	review	review	NOUN
ap-7583	886	5	section	section	NOUN
ap-7583	886	6	of	of	ADP
ap-7583	886	7	physics	physics	NOUN
ap-7583	886	8	letters	letter	NOUN
ap-7583	886	9	642:1–71	642:1–71	NUM
ap-7583	886	10	,	,	PUNCT
ap-7583	886	11	2016	2016	NUM
ap-7583	886	12	.	.	PUNCT
ap-7583	887	1	https://doi.org/10.1016/j.physrep.2016.06.002	https://doi.org/10.1016/j.physrep.2016.06.002	NOUN
ap-7583	887	2	.	.	PUNCT
ap-7583	888	1	[	[	X
ap-7583	888	2	20	20	NUM
ap-7583	888	3	]	]	PUNCT
ap-7583	888	4	h.	h.	PROPN
ap-7583	888	5	karayer	karayer	PROPN
ap-7583	888	6	,	,	PUNCT
ap-7583	888	7	d.	d.	PROPN
ap-7583	888	8	demirhan	demirhan	PROPN
ap-7583	888	9	,	,	PUNCT
ap-7583	888	10	f.	f.	PROPN
ap-7583	888	11	büyükkılıç	büyükkılıç	PROPN
ap-7583	888	12	.	.	PUNCT
ap-7583	889	1	solution	solution	NOUN
ap-7583	889	2	of	of	ADP
ap-7583	889	3	schrödinger	schrödinger	ADJ
ap-7583	889	4	equation	equation	NOUN
ap-7583	889	5	for	for	ADP
ap-7583	889	6	two	two	NUM
ap-7583	889	7	different	different	ADJ
ap-7583	889	8	potentials	potential	NOUN
ap-7583	889	9	using	use	VERB
ap-7583	889	10	extended	extended	ADJ
ap-7583	889	11	nikiforov	nikiforov	NOUN
ap-7583	889	12	-	-	PUNCT
ap-7583	889	13	uvarov	uvarov	ADJ
ap-7583	889	14	method	method	NOUN
ap-7583	889	15	and	and	CCONJ
ap-7583	889	16	polynomial	polynomial	ADJ
ap-7583	889	17	solutions	solution	NOUN
ap-7583	889	18	of	of	ADP
ap-7583	889	19	biconfluent	biconfluent	ADJ
ap-7583	889	20	heun	heun	NOUN
ap-7583	889	21	equation	equation	NOUN
ap-7583	889	22	.	.	PUNCT
ap-7583	890	1	journal	journal	PROPN
ap-7583	890	2	of	of	ADP
ap-7583	890	3	mathematical	mathematical	ADJ
ap-7583	890	4	physics	physics	NOUN
ap-7583	890	5	59(5):053501	59(5):053501	NUM
ap-7583	890	6	,	,	PUNCT
ap-7583	890	7	2018	2018	NUM
ap-7583	890	8	.	.	PUNCT
ap-7583	891	1	https://doi.org/10.1063/1.5022008	https://doi.org/10.1063/1.5022008	NOUN
ap-7583	891	2	.	.	PUNCT
ap-7583	892	1	[	[	X
ap-7583	892	2	21	21	NUM
ap-7583	892	3	]	]	X
ap-7583	892	4	h.	h.	PROPN
ap-7583	892	5	ciftci	ciftci	PROPN
ap-7583	892	6	,	,	PUNCT
ap-7583	892	7	r.	r.	PROPN
ap-7583	892	8	l.	l.	PROPN
ap-7583	892	9	hall	hall	PROPN
ap-7583	892	10	,	,	PUNCT
ap-7583	892	11	n.	n.	PROPN
ap-7583	892	12	saad	saad	PROPN
ap-7583	892	13	.	.	PUNCT
ap-7583	893	1	asymptotic	asymptotic	ADJ
ap-7583	893	2	iteration	iteration	NOUN
ap-7583	893	3	method	method	NOUN
ap-7583	893	4	for	for	ADP
ap-7583	893	5	eigenvalue	eigenvalue	NOUN
ap-7583	893	6	problems	problem	NOUN
ap-7583	893	7	.	.	PUNCT
ap-7583	894	1	journal	journal	PROPN
ap-7583	894	2	of	of	ADP
ap-7583	894	3	physics	physics	PROPN
ap-7583	894	4	a	a	PRON
ap-7583	894	5	:	:	PUNCT
ap-7583	894	6	mathematical	mathematical	ADJ
ap-7583	894	7	and	and	CCONJ
ap-7583	894	8	general	general	ADJ
ap-7583	894	9	36(47):11807–11816	36(47):11807–11816	NUM
ap-7583	894	10	,	,	PUNCT
ap-7583	894	11	2003	2003	NUM
ap-7583	894	12	.	.	PUNCT
ap-7583	895	1	https://doi.org/10.1088/0305-4470/36/47/008	https://doi.org/10.1088/0305-4470/36/47/008	NOUN
ap-7583	895	2	.	.	PUNCT
ap-7583	896	1	[	[	X
ap-7583	896	2	22	22	NUM
ap-7583	896	3	]	]	X
ap-7583	896	4	h.	h.	PROPN
ap-7583	896	5	scheffé	scheffé	PROPN
ap-7583	896	6	.	.	PUNCT
ap-7583	897	1	linear	linear	ADJ
ap-7583	897	2	differential	differential	ADJ
ap-7583	897	3	equations	equation	NOUN
ap-7583	897	4	with	with	ADP
ap-7583	897	5	two	two	NUM
ap-7583	897	6	-	-	PUNCT
ap-7583	897	7	term	term	NOUN
ap-7583	897	8	recurrence	recurrence	NOUN
ap-7583	897	9	formulas	formula	NOUN
ap-7583	897	10	.	.	PUNCT
ap-7583	898	1	journal	journal	NOUN
ap-7583	898	2	of	of	ADP
ap-7583	898	3	mathematics	mathematics	PROPN
ap-7583	898	4	and	and	CCONJ
ap-7583	898	5	physics	physics	NOUN
ap-7583	898	6	21(1	21(1	PROPN
ap-7583	898	7	-	-	SYM
ap-7583	898	8	4):240–249	4):240–249	NUM
ap-7583	898	9	,	,	PUNCT
ap-7583	898	10	1942	1942	NUM
ap-7583	898	11	.	.	PUNCT
ap-7583	899	1	https://doi.org/10.1002/sapm1942211240	https://doi.org/10.1002/sapm1942211240	NOUN
ap-7583	899	2	.	.	PUNCT
ap-7583	900	1	[	[	X
ap-7583	900	2	23	23	NUM
ap-7583	900	3	]	]	PUNCT
ap-7583	900	4	r.	r.	PROPN
ap-7583	900	5	s.	s.	PROPN
ap-7583	900	6	irving	irving	PROPN
ap-7583	900	7	.	.	PUNCT
ap-7583	901	1	integers	integer	NOUN
ap-7583	901	2	,	,	PUNCT
ap-7583	901	3	polynomials	polynomial	NOUN
ap-7583	901	4	,	,	PUNCT
ap-7583	901	5	and	and	CCONJ
ap-7583	901	6	rings	ring	NOUN
ap-7583	901	7	.	.	PUNCT
ap-7583	902	1	springer	springer	NOUN
ap-7583	902	2	-	-	PUNCT
ap-7583	902	3	verlag	verlag	PROPN
ap-7583	902	4	,	,	PUNCT
ap-7583	902	5	new	new	PROPN
ap-7583	902	6	york	york	PROPN
ap-7583	902	7	,	,	PUNCT
ap-7583	902	8	2004	2004	NUM
ap-7583	902	9	.	.	PUNCT
ap-7583	903	1	https://doi.org/10.1007/b97633	https://doi.org/10.1007/b97633	X
ap-7583	903	2	.	.	PUNCT
ap-7583	904	1	[	[	X
ap-7583	904	2	24	24	NUM
ap-7583	904	3	]	]	PUNCT
ap-7583	904	4	f.	f.	PROPN
ap-7583	904	5	m.	m.	PROPN
ap-7583	904	6	arscott	arscott	PROPN
ap-7583	904	7	.	.	PUNCT
ap-7583	905	1	latent	latent	PROPN
ap-7583	905	2	roots	root	NOUN
ap-7583	905	3	of	of	ADP
ap-7583	905	4	tri	tri	ADJ
ap-7583	905	5	-	-	ADJ
ap-7583	905	6	diagonal	diagonal	ADJ
ap-7583	905	7	matrices	matrix	NOUN
ap-7583	905	8	.	.	PUNCT
ap-7583	906	1	edinburgh	edinburgh	PROPN
ap-7583	906	2	mathematical	mathematical	PROPN
ap-7583	906	3	notes	note	VERB
ap-7583	906	4	44:5–7	44:5–7	PROPN
ap-7583	906	5	,	,	PUNCT
ap-7583	906	6	1961	1961	NUM
ap-7583	906	7	.	.	PUNCT
ap-7583	907	1	https://doi.org/10.1017/s095018430000330x	https://doi.org/10.1017/s095018430000330x	NOUN
ap-7583	907	2	.	.	PUNCT
ap-7583	908	1	[	[	X
ap-7583	908	2	25	25	NUM
ap-7583	908	3	]	]	PUNCT
ap-7583	908	4	j.	j.	PROPN
ap-7583	908	5	favard	favard	PROPN
ap-7583	908	6	.	.	PUNCT
ap-7583	909	1	sur	sur	PROPN
ap-7583	909	2	les	les	PROPN
ap-7583	909	3	polynômes	polynôme	NOUN
ap-7583	909	4	de	de	ADP
ap-7583	909	5	tchebicheff	tchebicheff	PROPN
ap-7583	909	6	.	.	PUNCT
ap-7583	910	1	comptes	compte	VERB
ap-7583	910	2	rendus	rendus	PROPN
ap-7583	910	3	hebdomadaires	hebdomadaires	PROPN
ap-7583	910	4	des	des	PROPN
ap-7583	910	5	séances	séances	PROPN
ap-7583	910	6	de	de	X
ap-7583	910	7	l’académie	l’académie	PROPN
ap-7583	910	8	des	des	PROPN
ap-7583	910	9	sciences	sciences	PROPN
ap-7583	910	10	,	,	PUNCT
ap-7583	910	11	paris	paris	PROPN
ap-7583	910	12	200:2052–2053	200:2052–2053	NUM
ap-7583	910	13	,	,	PUNCT
ap-7583	910	14	1935	1935	NUM
ap-7583	910	15	.	.	PUNCT
ap-7583	911	1	[	[	X
ap-7583	911	2	26	26	NUM
ap-7583	911	3	]	]	X
ap-7583	911	4	f.	f.	PROPN
ap-7583	911	5	marcellán	marcellán	PROPN
ap-7583	911	6	,	,	PUNCT
ap-7583	911	7	r.	r.	PROPN
ap-7583	911	8	álvarez	álvarez	PROPN
ap-7583	911	9	nodarse	nodarse	PROPN
ap-7583	911	10	.	.	PUNCT
ap-7583	912	1	on	on	ADP
ap-7583	912	2	the	the	DET
ap-7583	912	3	“	"	PUNCT
ap-7583	912	4	favard	favard	NOUN
ap-7583	912	5	theorem	theorem	NOUN
ap-7583	912	6	”	"	PUNCT
ap-7583	912	7	and	and	CCONJ
ap-7583	912	8	its	its	PRON
ap-7583	912	9	extensions	extension	NOUN
ap-7583	912	10	.	.	PUNCT
ap-7583	913	1	journal	journal	NOUN
ap-7583	913	2	of	of	ADP
ap-7583	913	3	computational	computational	ADJ
ap-7583	913	4	and	and	CCONJ
ap-7583	913	5	applied	apply	VERB
ap-7583	913	6	mathematics	mathematic	NOUN
ap-7583	913	7	127(1	127(1	NUM
ap-7583	913	8	-	-	PUNCT
ap-7583	913	9	2):231–254	2):231–254	NUM
ap-7583	913	10	,	,	PUNCT
ap-7583	913	11	2001	2001	NUM
ap-7583	913	12	.	.	PUNCT
ap-7583	914	1	https://doi.org/10.1016/s0377-0427(00)00497-0	https://doi.org/10.1016/s0377-0427(00)00497-0	PROPN
ap-7583	914	2	.	.	PUNCT
ap-7583	915	1	[	[	X
ap-7583	915	2	27	27	NUM
ap-7583	915	3	]	]	PUNCT
ap-7583	915	4	m.	m.	PROPN
ap-7583	915	5	e.	e.	PROPN
ap-7583	915	6	h.	h.	PROPN
ap-7583	915	7	ismail	ismail	PROPN
ap-7583	915	8	.	.	PUNCT
ap-7583	916	1	classical	classical	ADJ
ap-7583	916	2	and	and	CCONJ
ap-7583	916	3	quantum	quantum	ADJ
ap-7583	916	4	orthogonal	orthogonal	ADJ
ap-7583	916	5	polynomials	polynomial	NOUN
ap-7583	916	6	in	in	ADP
ap-7583	916	7	one	one	NUM
ap-7583	916	8	variable	variable	NOUN
ap-7583	916	9	.	.	PUNCT
ap-7583	917	1	cambridge	cambridge	PROPN
ap-7583	917	2	university	university	PROPN
ap-7583	917	3	press	press	PROPN
ap-7583	917	4	,	,	PUNCT
ap-7583	917	5	cambridge	cambridge	PROPN
ap-7583	917	6	,	,	PUNCT
ap-7583	917	7	2009	2009	NUM
ap-7583	917	8	.	.	PUNCT
ap-7583	918	1	[	[	X
ap-7583	918	2	28	28	NUM
ap-7583	918	3	]	]	X
ap-7583	918	4	a.	a.	NOUN
ap-7583	918	5	krajewska	krajewska	PROPN
ap-7583	918	6	,	,	PUNCT
ap-7583	918	7	a.	a.	NOUN
ap-7583	918	8	ushveridze	ushveridze	PROPN
ap-7583	918	9	,	,	PUNCT
ap-7583	918	10	z.	z.	PROPN
ap-7583	918	11	walczak	walczak	PROPN
ap-7583	918	12	.	.	PUNCT
ap-7583	919	1	bender	bender	NOUN
ap-7583	919	2	-	-	PUNCT
ap-7583	919	3	dunne	dunne	PROPN
ap-7583	919	4	orthogonal	orthogonal	ADJ
ap-7583	919	5	polynomials	polynomial	NOUN
ap-7583	919	6	general	general	ADJ
ap-7583	919	7	theory	theory	NOUN
ap-7583	919	8	.	.	PUNCT
ap-7583	920	1	modern	modern	ADJ
ap-7583	920	2	physics	physics	NOUN
ap-7583	920	3	letters	letter	NOUN
ap-7583	920	4	a	a	DET
ap-7583	920	5	12(16):1131–1144	12(16):1131–1144	NUM
ap-7583	920	6	,	,	PUNCT
ap-7583	920	7	1997	1997	NUM
ap-7583	920	8	.	.	PUNCT
ap-7583	921	1	https://doi.org/10.1142/s0217732397001163	https://doi.org/10.1142/s0217732397001163	NUM
ap-7583	921	2	.	.	PUNCT
ap-7583	922	1	189	189	NUM
ap-7583	922	2	http://arxiv.org/abs/1512.04025v1	http://arxiv.org/abs/1512.04025v1	NUM
ap-7583	922	3	https://doi.org/10.1142/8566	https://doi.org/10.1142/8566	NOUN
ap-7583	922	4	https://doi.org/10.1088/1751-8113/43/41/415206	https://doi.org/10.1088/1751-8113/43/41/415206	PROPN
ap-7583	922	5	https://doi.org/10.1088/1751-8113/45/6/065206	https://doi.org/10.1088/1751-8113/45/6/065206	PROPN
ap-7583	922	6	https://doi.org/10.1088/1751-8113/46/3/035301	https://doi.org/10.1088/1751-8113/46/3/035301	PROPN
ap-7583	922	7	https://doi.org/10.1016/j.aop.2014.04.023	https://doi.org/10.1016/j.aop.2014.04.023	PROPN
ap-7583	922	8	https://doi.org/10.1016/j.physrep.2016.06.002	https://doi.org/10.1016/j.physrep.2016.06.002	NOUN
ap-7583	922	9	https://doi.org/10.1063/1.5022008	https://doi.org/10.1063/1.5022008	PROPN
ap-7583	922	10	https://doi.org/10.1088/0305-4470/36/47/008	https://doi.org/10.1088/0305-4470/36/47/008	NOUN
ap-7583	922	11	https://doi.org/10.1002/sapm1942211240	https://doi.org/10.1002/sapm1942211240	NOUN
ap-7583	922	12	https://doi.org/10.1007/b97633	https://doi.org/10.1007/b97633	PROPN
ap-7583	922	13	https://doi.org/10.1017/s095018430000330x	https://doi.org/10.1017/s095018430000330x	PROPN
ap-7583	922	14	https://doi.org/10.1016/s0377-0427(00)00497-0	https://doi.org/10.1016/s0377-0427(00)00497-0	NOUN
ap-7583	922	15	https://doi.org/10.1142/s0217732397001163	https://doi.org/10.1142/s0217732397001163	NUM
ap-7583	922	16	acta	acta	PROPN
ap-7583	922	17	polytechnica	polytechnica	PROPN
ap-7583	922	18	62(1):165–189	62(1):165–189	PROPN
ap-7583	922	19	,	,	PUNCT
ap-7583	922	20	2022	2022	NUM
ap-7583	922	21	1	1	NUM
ap-7583	922	22	introduction	introduction	NOUN
ap-7583	922	23	2	2	NUM
ap-7583	922	24	elementary	elementary	ADJ
ap-7583	922	25	observations	observation	NOUN
ap-7583	922	26	3	3	NUM
ap-7583	922	27	the	the	DET
ap-7583	922	28	solutions	solution	NOUN
ap-7583	922	29	in	in	ADP
ap-7583	922	30	the	the	DET
ap-7583	922	31	neighbourhood	neighbourhood	NOUN
ap-7583	922	32	of	of	ADP
ap-7583	922	33	an	an	DET
ap-7583	922	34	ordinary	ordinary	ADJ
ap-7583	922	35	point	point	NOUN
ap-7583	922	36	3.1	3.1	NUM
ap-7583	922	37	series	series	NOUN
ap-7583	922	38	solutions	solution	NOUN
ap-7583	922	39	3.2	3.2	NUM
ap-7583	922	40	polynomial	polynomial	ADJ
ap-7583	922	41	solutions	solution	NOUN
ap-7583	922	42	4	4	NUM
ap-7583	922	43	the	the	DET
ap-7583	922	44	solutions	solution	NOUN
ap-7583	922	45	in	in	ADP
ap-7583	922	46	the	the	DET
ap-7583	922	47	neighbourhood	neighbourhood	NOUN
ap-7583	922	48	of	of	ADP
ap-7583	922	49	a	a	DET
ap-7583	922	50	singular	singular	ADJ
ap-7583	922	51	point	point	NOUN
ap-7583	922	52	4.1	4.1	NUM
ap-7583	922	53	series	series	NOUN
ap-7583	922	54	solution	solution	NOUN
ap-7583	922	55	and	and	CCONJ
ap-7583	922	56	infinite	infinite	ADJ
ap-7583	922	57	sequence	sequence	NOUN
ap-7583	922	58	of	of	ADP
ap-7583	922	59	orthogonal	orthogonal	ADJ
ap-7583	922	60	polynomials	polynomial	NOUN
ap-7583	922	61	{	{	PUNCT
ap-7583	922	62	pk	pk	NOUN
ap-7583	922	63	(	(	PUNCT
ap-7583	922	64	0)}k=0	0)}k=0	PROPN
ap-7583	922	65	4.2	4.2	NUM
ap-7583	922	66	polynomial	polynomial	ADJ
ap-7583	922	67	solution	solution	NOUN
ap-7583	922	68	and	and	CCONJ
ap-7583	922	69	finite	finite	ADJ
ap-7583	922	70	sequence	sequence	NOUN
ap-7583	922	71	of	of	ADP
ap-7583	922	72	orthogonal	orthogonal	ADJ
ap-7583	922	73	polynomials	polynomial	NOUN
ap-7583	922	74	5	5	NUM
ap-7583	922	75	mathematical	mathematical	ADJ
ap-7583	922	76	properties	property	NOUN
ap-7583	922	77	of	of	ADP
ap-7583	922	78	the	the	DET
ap-7583	922	79	orthogonal	orthogonal	ADJ
ap-7583	922	80	polynomials	polynomial	NOUN
ap-7583	922	81	{	{	PUNCT
ap-7583	922	82	pk	pk	NOUN
ap-7583	922	83	(	(	PUNCT
ap-7583	922	84	0)}k=0	0)}k=0	PROPN
ap-7583	922	85	6	6	NUM
ap-7583	922	86	mathematical	mathematical	ADJ
ap-7583	922	87	properties	property	NOUN
ap-7583	922	88	of	of	ADP
ap-7583	922	89	the	the	DET
ap-7583	922	90	finite	finite	ADJ
ap-7583	922	91	orthogonal	orthogonal	ADJ
ap-7583	922	92	polynomials	polynomial	NOUN
ap-7583	922	93	{	{	PUNCT
ap-7583	922	94	pkn(0)}k=0n	pkn(0)}k=0n	PROPN
ap-7583	922	95	6.1	6.1	NUM
ap-7583	922	96	norms	norm	NOUN
ap-7583	922	97	of	of	ADP
ap-7583	922	98	the	the	DET
ap-7583	922	99	orthogonal	orthogonal	ADJ
ap-7583	922	100	polynomials	polynomial	NOUN
ap-7583	922	101	6.2	6.2	NUM
ap-7583	922	102	the	the	DET
ap-7583	922	103	zeros	zero	NOUN
ap-7583	922	104	of	of	ADP
ap-7583	922	105	the	the	DET
ap-7583	922	106	polynomials	polynomial	NOUN
ap-7583	922	107	{	{	PUNCT
ap-7583	922	108	pkn(0;n)}k=0n	pkn(0;n)}k=0n	NUM
ap-7583	922	109	6.3	6.3	NUM
ap-7583	922	110	factorization	factorization	NOUN
ap-7583	922	111	property	property	NOUN
ap-7583	922	112	acknowledgements	acknowledgement	NOUN
ap-7583	922	113	references	reference	NOUN