id	sid	tid	token	lemma	pos
ejpam-5886	1	1	european	european	PROPN
ejpam-5886	1	2	journal	journal	PROPN
ejpam-5886	1	3	of	of	ADP
ejpam-5886	1	4	pure	pure	ADJ
ejpam-5886	1	5	and	and	CCONJ
ejpam-5886	1	6	applied	applied	ADJ
ejpam-5886	1	7	mathematics	mathematic	NOUN
ejpam-5886	1	8	2025	2025	NUM
ejpam-5886	1	9	,	,	PUNCT
ejpam-5886	1	10	vol	vol	NOUN
ejpam-5886	1	11	.	.	PROPN
ejpam-5886	1	12	18	18	NUM
ejpam-5886	1	13	,	,	PUNCT
ejpam-5886	1	14	issue	issue	NOUN
ejpam-5886	1	15	2	2	NUM
ejpam-5886	1	16	,	,	PUNCT
ejpam-5886	1	17	article	article	NOUN
ejpam-5886	1	18	number	number	NOUN
ejpam-5886	1	19	5886	5886	NUM
ejpam-5886	1	20	issn	issn	PROPN
ejpam-5886	1	21	1307	1307	NUM
ejpam-5886	1	22	-	-	SYM
ejpam-5886	1	23	5543	5543	NUM
ejpam-5886	1	24	–	–	PUNCT
ejpam-5886	1	25	ejpam.com	ejpam.com	X
ejpam-5886	1	26	published	publish	VERB
ejpam-5886	1	27	by	by	ADP
ejpam-5886	1	28	new	new	PROPN
ejpam-5886	1	29	york	york	PROPN
ejpam-5886	1	30	business	business	PROPN
ejpam-5886	1	31	global	global	ADJ
ejpam-5886	1	32	certain	certain	ADJ
ejpam-5886	1	33	properties	property	NOUN
ejpam-5886	1	34	of	of	ADP
ejpam-5886	1	35	∆h	∆h	PROPN
ejpam-5886	1	36	legendre	legendre	NOUN
ejpam-5886	1	37	polynomials	polynomial	NOUN
ejpam-5886	1	38	and	and	CCONJ
ejpam-5886	1	39	applications	application	NOUN
ejpam-5886	1	40	in	in	ADP
ejpam-5886	1	41	computer	computer	NOUN
ejpam-5886	1	42	modelling	model	VERB
ejpam-5886	1	43	shahid	shahid	PROPN
ejpam-5886	1	44	ahmad	ahmad	PROPN
ejpam-5886	1	45	wani1,∗	wani1,∗	PROPN
ejpam-5886	1	46	,	,	PUNCT
ejpam-5886	1	47	waseem	waseem	PROPN
ejpam-5886	1	48	ahmad	ahmad	PROPN
ejpam-5886	1	49	khan2	khan2	PROPN
ejpam-5886	1	50	,	,	PUNCT
ejpam-5886	1	51	taghreed	taghreed	NOUN
ejpam-5886	1	52	alqurashi3	alqurashi3	NOUN
ejpam-5886	1	53	,	,	PUNCT
ejpam-5886	1	54	javid	javid	PROPN
ejpam-5886	1	55	gani	gani	PROPN
ejpam-5886	1	56	dar1	dar1	PROPN
ejpam-5886	1	57	,	,	PUNCT
ejpam-5886	1	58	dxion	dxion	NOUN
ejpam-5886	1	59	salcedo4	salcedo4	NOUN
ejpam-5886	1	60	1	1	NUM
ejpam-5886	1	61	symbiosis	symbiosis	NOUN
ejpam-5886	1	62	institute	institute	NOUN
ejpam-5886	1	63	of	of	ADP
ejpam-5886	1	64	technology	technology	PROPN
ejpam-5886	1	65	,	,	PUNCT
ejpam-5886	1	66	pune	pune	NOUN
ejpam-5886	1	67	campus	campus	NOUN
ejpam-5886	1	68	,	,	PUNCT
ejpam-5886	1	69	symbiosis	symbiosis	NOUN
ejpam-5886	1	70	international	international	ADJ
ejpam-5886	1	71	(	(	PUNCT
ejpam-5886	1	72	deemed	deem	VERB
ejpam-5886	1	73	)	)	PUNCT
ejpam-5886	1	74	university	university	NOUN
ejpam-5886	1	75	,	,	PUNCT
ejpam-5886	1	76	pune	pune	NOUN
ejpam-5886	1	77	,	,	PUNCT
ejpam-5886	1	78	india	india	PROPN
ejpam-5886	1	79	2	2	NUM
ejpam-5886	1	80	department	department	NOUN
ejpam-5886	1	81	of	of	ADP
ejpam-5886	1	82	electrical	electrical	ADJ
ejpam-5886	1	83	engineering	engineering	NOUN
ejpam-5886	1	84	,	,	PUNCT
ejpam-5886	1	85	prince	prince	PROPN
ejpam-5886	1	86	mohammad	mohammad	PROPN
ejpam-5886	1	87	bin	bin	PROPN
ejpam-5886	1	88	fahd	fahd	PROPN
ejpam-5886	1	89	university	university	PROPN
ejpam-5886	1	90	,	,	PUNCT
ejpam-5886	1	91	p.o	p.o	PROPN
ejpam-5886	1	92	box	box	PROPN
ejpam-5886	1	93	1664	1664	NUM
ejpam-5886	1	94	,	,	PUNCT
ejpam-5886	1	95	al	al	PROPN
ejpam-5886	1	96	khobar	khobar	PROPN
ejpam-5886	1	97	31952	31952	NUM
ejpam-5886	1	98	,	,	PUNCT
ejpam-5886	1	99	saudi	saudi	PROPN
ejpam-5886	1	100	arabia	arabia	PROPN
ejpam-5886	1	101	3	3	NUM
ejpam-5886	1	102	mathematics	mathematics	PROPN
ejpam-5886	1	103	department	department	NOUN
ejpam-5886	1	104	,	,	PUNCT
ejpam-5886	1	105	faculty	faculty	NOUN
ejpam-5886	1	106	of	of	ADP
ejpam-5886	1	107	science	science	NOUN
ejpam-5886	1	108	,	,	PUNCT
ejpam-5886	1	109	al	al	PROPN
ejpam-5886	1	110	-	-	PUNCT
ejpam-5886	1	111	baha	baha	PROPN
ejpam-5886	1	112	university	university	PROPN
ejpam-5886	1	113	,	,	PUNCT
ejpam-5886	1	114	65779	65779	NUM
ejpam-5886	1	115	-	-	SYM
ejpam-5886	1	116	7738	7738	NUM
ejpam-5886	1	117	,	,	PUNCT
ejpam-5886	1	118	albaha	albaha	NOUN
ejpam-5886	1	119	city	city	NOUN
ejpam-5886	1	120	,	,	PUNCT
ejpam-5886	1	121	kingdom	kingdom	NOUN
ejpam-5886	1	122	of	of	ADP
ejpam-5886	1	123	saudi	saudi	PROPN
ejpam-5886	1	124	arabia	arabia	PROPN
ejpam-5886	1	125	4	4	NUM
ejpam-5886	1	126	computer	computer	NOUN
ejpam-5886	1	127	science	science	NOUN
ejpam-5886	1	128	and	and	CCONJ
ejpam-5886	1	129	electronics	electronic	NOUN
ejpam-5886	1	130	development	development	NOUN
ejpam-5886	1	131	,	,	PUNCT
ejpam-5886	1	132	universidad	universidad	PROPN
ejpam-5886	1	133	de	de	PROPN
ejpam-5886	1	134	la	la	PROPN
ejpam-5886	1	135	costa	costa	PROPN
ejpam-5886	1	136	,	,	PUNCT
ejpam-5886	1	137	barranquilla	barranquilla	PROPN
ejpam-5886	1	138	,	,	PUNCT
ejpam-5886	1	139	colombia	colombia	PROPN
ejpam-5886	1	140	abstract	abstract	NOUN
ejpam-5886	1	141	.	.	PUNCT
ejpam-5886	2	1	this	this	DET
ejpam-5886	2	2	study	study	NOUN
ejpam-5886	2	3	investigates	investigate	VERB
ejpam-5886	2	4	the	the	DET
ejpam-5886	2	5	development	development	NOUN
ejpam-5886	2	6	of	of	ADP
ejpam-5886	2	7	polynomials	polynomial	NOUN
ejpam-5886	2	8	,	,	PUNCT
ejpam-5886	2	9	with	with	ADP
ejpam-5886	2	10	a	a	DET
ejpam-5886	2	11	particular	particular	ADJ
ejpam-5886	2	12	focus	focus	NOUN
ejpam-5886	2	13	on	on	ADP
ejpam-5886	2	14	the	the	DET
ejpam-5886	2	15	unique	unique	ADJ
ejpam-5886	2	16	∆h	∆h	PROPN
ejpam-5886	2	17	legendre	legendre	NOUN
ejpam-5886	2	18	polynomials	polynomial	NOUN
ejpam-5886	2	19	.	.	PUNCT
ejpam-5886	3	1	explicit	explicit	ADJ
ejpam-5886	3	2	formulas	formula	NOUN
ejpam-5886	3	3	for	for	ADP
ejpam-5886	3	4	these	these	DET
ejpam-5886	3	5	polynomials	polynomial	NOUN
ejpam-5886	3	6	are	be	AUX
ejpam-5886	3	7	derived	derive	VERB
ejpam-5886	3	8	,	,	PUNCT
ejpam-5886	3	9	along	along	ADP
ejpam-5886	3	10	with	with	ADP
ejpam-5886	3	11	summation	summation	NOUN
ejpam-5886	3	12	formulae	formulae	NOUN
ejpam-5886	3	13	that	that	PRON
ejpam-5886	3	14	provide	provide	VERB
ejpam-5886	3	15	further	further	ADJ
ejpam-5886	3	16	structural	structural	ADJ
ejpam-5886	3	17	insights	insight	NOUN
ejpam-5886	3	18	.	.	PUNCT
ejpam-5886	4	1	additionally	additionally	ADV
ejpam-5886	4	2	,	,	PUNCT
ejpam-5886	4	3	the	the	DET
ejpam-5886	4	4	monomiality	monomiality	NOUN
ejpam-5886	4	5	principle	principle	NOUN
ejpam-5886	4	6	is	be	AUX
ejpam-5886	4	7	established	establish	VERB
ejpam-5886	4	8	,	,	PUNCT
ejpam-5886	4	9	reinforcing	reinforce	VERB
ejpam-5886	4	10	the	the	DET
ejpam-5886	4	11	algebraic	algebraic	ADJ
ejpam-5886	4	12	framework	framework	NOUN
ejpam-5886	4	13	of	of	ADP
ejpam-5886	4	14	these	these	DET
ejpam-5886	4	15	polynomials	polynomial	NOUN
ejpam-5886	4	16	.	.	PUNCT
ejpam-5886	5	1	symmetric	symmetric	ADJ
ejpam-5886	5	2	identities	identity	NOUN
ejpam-5886	5	3	are	be	AUX
ejpam-5886	5	4	also	also	ADV
ejpam-5886	5	5	formulated	formulate	VERB
ejpam-5886	5	6	,	,	PUNCT
ejpam-5886	5	7	highlighting	highlight	VERB
ejpam-5886	5	8	their	their	PRON
ejpam-5886	5	9	fundamental	fundamental	ADJ
ejpam-5886	5	10	properties	property	NOUN
ejpam-5886	5	11	.	.	PUNCT
ejpam-5886	6	1	the	the	DET
ejpam-5886	6	2	study	study	NOUN
ejpam-5886	6	3	concludes	conclude	VERB
ejpam-5886	6	4	with	with	ADP
ejpam-5886	6	5	remarks	remark	NOUN
ejpam-5886	6	6	summarizing	summarize	VERB
ejpam-5886	6	7	the	the	DET
ejpam-5886	6	8	key	key	ADJ
ejpam-5886	6	9	findings	finding	NOUN
ejpam-5886	6	10	and	and	CCONJ
ejpam-5886	6	11	potential	potential	ADJ
ejpam-5886	6	12	directions	direction	NOUN
ejpam-5886	6	13	for	for	ADP
ejpam-5886	6	14	future	future	ADJ
ejpam-5886	6	15	research	research	NOUN
ejpam-5886	6	16	.	.	PUNCT
ejpam-5886	7	1	2020	2020	NUM
ejpam-5886	7	2	mathematics	mathematic	NOUN
ejpam-5886	7	3	subject	subject	NOUN
ejpam-5886	7	4	classifications	classification	NOUN
ejpam-5886	7	5	:	:	PUNCT
ejpam-5886	7	6	33e20	33e20	NUM
ejpam-5886	7	7	,	,	PUNCT
ejpam-5886	7	8	33b10	33b10	NUM
ejpam-5886	7	9	,	,	PUNCT
ejpam-5886	7	10	33e30	33e30	NUM
ejpam-5886	7	11	,	,	PUNCT
ejpam-5886	7	12	11t23	11t23	DET
ejpam-5886	7	13	key	key	ADJ
ejpam-5886	7	14	words	word	NOUN
ejpam-5886	7	15	and	and	CCONJ
ejpam-5886	7	16	phrases	phrase	NOUN
ejpam-5886	7	17	:	:	PUNCT
ejpam-5886	7	18	∆h	∆h	NUM
ejpam-5886	7	19	sequences	sequence	NOUN
ejpam-5886	7	20	,	,	PUNCT
ejpam-5886	7	21	monomiality	monomiality	NOUN
ejpam-5886	7	22	principle	principle	NOUN
ejpam-5886	7	23	,	,	PUNCT
ejpam-5886	7	24	explicit	explicit	ADJ
ejpam-5886	7	25	forms	form	NOUN
ejpam-5886	7	26	,	,	PUNCT
ejpam-5886	7	27	symmetric	symmetric	ADJ
ejpam-5886	7	28	identities	identity	NOUN
ejpam-5886	7	29	1	1	NUM
ejpam-5886	7	30	.	.	PUNCT
ejpam-5886	8	1	introduction	introduction	NOUN
ejpam-5886	8	2	and	and	CCONJ
ejpam-5886	8	3	preliminaries	preliminary	NOUN
ejpam-5886	8	4	many	many	ADJ
ejpam-5886	8	5	fields	field	NOUN
ejpam-5886	8	6	,	,	PUNCT
ejpam-5886	8	7	such	such	ADJ
ejpam-5886	8	8	as	as	ADP
ejpam-5886	8	9	statistical	statistical	ADJ
ejpam-5886	8	10	mechanics	mechanic	NOUN
ejpam-5886	8	11	and	and	CCONJ
ejpam-5886	8	12	quantum	quantum	NOUN
ejpam-5886	8	13	mechanics	mechanic	NOUN
ejpam-5886	8	14	,	,	PUNCT
ejpam-5886	8	15	have	have	AUX
ejpam-5886	8	16	used	use	VERB
ejpam-5886	8	17	specific	specific	ADJ
ejpam-5886	8	18	polynomials	polynomial	NOUN
ejpam-5886	8	19	to	to	PART
ejpam-5886	8	20	represent	represent	VERB
ejpam-5886	8	21	and	and	CCONJ
ejpam-5886	8	22	characterize	characterize	VERB
ejpam-5886	8	23	the	the	DET
ejpam-5886	8	24	behavior	behavior	NOUN
ejpam-5886	8	25	of	of	ADP
ejpam-5886	8	26	complex	complex	ADJ
ejpam-5886	8	27	systems	system	NOUN
ejpam-5886	8	28	.	.	PUNCT
ejpam-5886	9	1	there	there	PRON
ejpam-5886	9	2	are	be	VERB
ejpam-5886	9	3	several	several	ADJ
ejpam-5886	9	4	other	other	ADJ
ejpam-5886	9	5	fields	field	NOUN
ejpam-5886	9	6	,	,	PUNCT
ejpam-5886	9	7	including	include	VERB
ejpam-5886	9	8	statistics	statistic	NOUN
ejpam-5886	9	9	and	and	CCONJ
ejpam-5886	9	10	quantum	quantum	NOUN
ejpam-5886	9	11	mechanics	mechanic	NOUN
ejpam-5886	9	12	,	,	PUNCT
ejpam-5886	9	13	where	where	SCONJ
ejpam-5886	9	14	complex	complex	ADJ
ejpam-5886	9	15	systems	system	NOUN
ejpam-5886	9	16	have	have	AUX
ejpam-5886	9	17	been	be	AUX
ejpam-5886	9	18	described	describe	VERB
ejpam-5886	9	19	and	and	CCONJ
ejpam-5886	9	20	analyzed	analyze	VERB
ejpam-5886	9	21	using	use	VERB
ejpam-5886	9	22	these	these	DET
ejpam-5886	9	23	special	special	ADJ
ejpam-5886	9	24	polynomials	polynomial	NOUN
ejpam-5886	9	25	.	.	PUNCT
ejpam-5886	10	1	in	in	ADP
ejpam-5886	10	2	a	a	DET
ejpam-5886	10	3	number	number	NOUN
ejpam-5886	10	4	of	of	ADP
ejpam-5886	10	5	mathematical	mathematical	ADJ
ejpam-5886	10	6	fields	field	NOUN
ejpam-5886	10	7	,	,	PUNCT
ejpam-5886	10	8	including	include	VERB
ejpam-5886	10	9	combinatorics	combinatoric	NOUN
ejpam-5886	10	10	,	,	PUNCT
ejpam-5886	10	11	entropy	entropy	NOUN
ejpam-5886	10	12	,	,	PUNCT
ejpam-5886	10	13	and	and	CCONJ
ejpam-5886	10	14	algebraic	algebraic	ADJ
ejpam-5886	10	15	combinatorics	combinatoric	NOUN
ejpam-5886	10	16	,	,	PUNCT
ejpam-5886	10	17	polynomial	polynomial	ADJ
ejpam-5886	10	18	sequences	sequence	NOUN
ejpam-5886	10	19	are	be	AUX
ejpam-5886	10	20	essential	essential	ADJ
ejpam-5886	10	21	.	.	PUNCT
ejpam-5886	11	1	in	in	ADP
ejpam-5886	11	2	approximation	approximation	NOUN
ejpam-5886	11	3	theory	theory	NOUN
ejpam-5886	11	4	and	and	CCONJ
ejpam-5886	11	5	physics	physics	PROPN
ejpam-5886	11	6	,	,	PUNCT
ejpam-5886	11	7	legendre	legendre	PROPN
ejpam-5886	11	8	polynomials	polynomial	NOUN
ejpam-5886	11	9	,	,	PUNCT
ejpam-5886	11	10	∗corresponding	∗corresponde	VERB
ejpam-5886	11	11	author	author	NOUN
ejpam-5886	11	12	.	.	PUNCT
ejpam-5886	12	1	doi	doi	NOUN
ejpam-5886	12	2	:	:	PUNCT
ejpam-5886	12	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5886	https://doi.org/10.29020/nybg.ejpam.v18i2.5886	PROPN
ejpam-5886	12	4	email	email	NOUN
ejpam-5886	12	5	addresses	address	NOUN
ejpam-5886	12	6	:	:	PUNCT
ejpam-5886	12	7	shahidwani177@gmail.com	shahidwani177@gmail.com	X
ejpam-5886	12	8	(	(	PUNCT
ejpam-5886	12	9	s.	s.	PROPN
ejpam-5886	12	10	a.	a.	PROPN
ejpam-5886	12	11	wani	wani	PROPN
ejpam-5886	12	12	)	)	PUNCT
ejpam-5886	12	13	,	,	PUNCT
ejpam-5886	12	14	wkhan1@pmu.edu.sa	wkhan1@pmu.edu.sa	PROPN
ejpam-5886	12	15	(	(	PUNCT
ejpam-5886	12	16	w.	w.	PROPN
ejpam-5886	12	17	a.	a.	PROPN
ejpam-5886	12	18	khan	khan	PROPN
ejpam-5886	12	19	)	)	PUNCT
ejpam-5886	12	20	,	,	PUNCT
ejpam-5886	12	21	talqorashi@bu.edu.sa	talqorashi@bu.edu.sa	PROPN
ejpam-5886	12	22	(	(	PUNCT
ejpam-5886	12	23	t.	t.	PROPN
ejpam-5886	12	24	alqurashi	alqurashi	PROPN
ejpam-5886	12	25	)	)	PUNCT
ejpam-5886	12	26	,	,	PUNCT
ejpam-5886	12	27	javid.dar@sitpune.edu.in	javid.dar@sitpune.edu.in	NOUN
ejpam-5886	12	28	(	(	PUNCT
ejpam-5886	12	29	j.	j.	PROPN
ejpam-5886	12	30	g.	g.	PROPN
ejpam-5886	12	31	dar	dar	PROPN
ejpam-5886	12	32	)	)	PUNCT
ejpam-5886	12	33	,	,	PUNCT
ejpam-5886	12	34	dsalcedo2@cuc.edu.co	dsalcedo2@cuc.edu.co	PROPN
ejpam-5886	12	35	(	(	PUNCT
ejpam-5886	12	36	d.	d.	PROPN
ejpam-5886	12	37	salcedo	salcedo	PROPN
ejpam-5886	12	38	)	)	PUNCT
ejpam-5886	12	39	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5886	13	1	1	1	NUM
ejpam-5886	13	2	copyright	copyright	NOUN
ejpam-5886	13	3	:	:	PUNCT
ejpam-5886	13	4	©	©	PROPN
ejpam-5886	13	5	2025	2025	NUM
ejpam-5886	13	6	the	the	DET
ejpam-5886	13	7	author(s	author(s	NOUN
ejpam-5886	13	8	)	)	PUNCT
ejpam-5886	13	9	.	.	PUNCT
ejpam-5886	14	1	(	(	PUNCT
ejpam-5886	14	2	cc	cc	NOUN
ejpam-5886	14	3	by	by	ADP
ejpam-5886	14	4	-	-	PUNCT
ejpam-5886	14	5	nc	nc	PROPN
ejpam-5886	14	6	4.0	4.0	NUM
ejpam-5886	14	7	)	)	PUNCT
ejpam-5886	14	8	s.	s.	PROPN
ejpam-5886	14	9	a.	a.	PROPN
ejpam-5886	14	10	wani	wani	PROPN
ejpam-5886	14	11	et	et	PROPN
ejpam-5886	14	12	al	al	PROPN
ejpam-5886	14	13	.	.	PUNCT
ejpam-5886	14	14	/	/	SYM
ejpam-5886	14	15	eur	eur	PROPN
ejpam-5886	14	16	.	.	PUNCT
ejpam-5886	15	1	j.	j.	PROPN
ejpam-5886	15	2	pure	pure	PROPN
ejpam-5886	15	3	appl	appl	PROPN
ejpam-5886	15	4	.	.	PROPN
ejpam-5886	15	5	math	math	PROPN
ejpam-5886	15	6	,	,	PUNCT
ejpam-5886	15	7	18	18	NUM
ejpam-5886	15	8	(	(	PUNCT
ejpam-5886	15	9	2	2	NUM
ejpam-5886	15	10	)	)	PUNCT
ejpam-5886	15	11	(	(	PUNCT
ejpam-5886	15	12	2025	2025	NUM
ejpam-5886	15	13	)	)	PUNCT
ejpam-5886	15	14	,	,	PUNCT
ejpam-5886	15	15	5886	5886	NUM
ejpam-5886	15	16	2	2	NUM
ejpam-5886	15	17	of	of	ADP
ejpam-5886	15	18	16	16	NUM
ejpam-5886	15	19	named	name	VERB
ejpam-5886	15	20	after	after	ADP
ejpam-5886	15	21	the	the	DET
ejpam-5886	15	22	french	french	ADJ
ejpam-5886	15	23	mathematician	mathematician	NOUN
ejpam-5886	15	24	edmond	edmond	PROPN
ejpam-5886	15	25	legendre	legendre	PROPN
ejpam-5886	15	26	,	,	PUNCT
ejpam-5886	15	27	were	be	AUX
ejpam-5886	15	28	introduced	introduce	VERB
ejpam-5886	15	29	in	in	ADP
ejpam-5886	15	30	the	the	DET
ejpam-5886	15	31	19th	19th	ADJ
ejpam-5886	15	32	century	century	NOUN
ejpam-5886	15	33	.	.	PUNCT
ejpam-5886	16	1	these	these	DET
ejpam-5886	16	2	polynomials	polynomial	NOUN
ejpam-5886	16	3	arise	arise	VERB
ejpam-5886	16	4	as	as	ADP
ejpam-5886	16	5	solutions	solution	NOUN
ejpam-5886	16	6	to	to	ADP
ejpam-5886	16	7	the	the	DET
ejpam-5886	16	8	second	second	ADJ
ejpam-5886	16	9	-	-	PUNCT
ejpam-5886	16	10	order	order	NOUN
ejpam-5886	16	11	legendre	legendre	NOUN
ejpam-5886	16	12	differential	differential	PROPN
ejpam-5886	16	13	equation	equation	NOUN
ejpam-5886	16	14	and	and	CCONJ
ejpam-5886	16	15	are	be	AUX
ejpam-5886	16	16	defined	define	VERB
ejpam-5886	16	17	on	on	ADP
ejpam-5886	16	18	the	the	DET
ejpam-5886	16	19	interval	interval	NOUN
ejpam-5886	16	20	[	[	X
ejpam-5886	16	21	0,+∞	0,+∞	NUM
ejpam-5886	16	22	)	)	PUNCT
ejpam-5886	16	23	,	,	PUNCT
ejpam-5886	16	24	commonly	commonly	ADV
ejpam-5886	16	25	denoted	denote	VERB
ejpam-5886	16	26	by	by	ADP
ejpam-5886	16	27	sn(u	sn(u	NOUN
ejpam-5886	16	28	)	)	PUNCT
ejpam-5886	16	29	,	,	PUNCT
ejpam-5886	16	30	where	where	SCONJ
ejpam-5886	16	31	n	n	PRON
ejpam-5886	16	32	represents	represent	VERB
ejpam-5886	16	33	the	the	DET
ejpam-5886	16	34	degree	degree	NOUN
ejpam-5886	16	35	.	.	PUNCT
ejpam-5886	17	1	legendre	legendre	PROPN
ejpam-5886	17	2	polynomials	polynomial	NOUN
ejpam-5886	17	3	exhibit	exhibit	VERB
ejpam-5886	17	4	orthogonality	orthogonality	NOUN
ejpam-5886	17	5	with	with	ADP
ejpam-5886	17	6	respect	respect	NOUN
ejpam-5886	17	7	to	to	ADP
ejpam-5886	17	8	the	the	DET
ejpam-5886	17	9	weight	weight	NOUN
ejpam-5886	17	10	function	function	NOUN
ejpam-5886	17	11	e−u	e−u	ADV
ejpam-5886	17	12	on	on	ADP
ejpam-5886	17	13	[	[	X
ejpam-5886	17	14	0,+∞	0,+∞	NUM
ejpam-5886	17	15	)	)	PUNCT
ejpam-5886	17	16	,	,	PUNCT
ejpam-5886	17	17	ensuring	ensure	VERB
ejpam-5886	17	18	that	that	SCONJ
ejpam-5886	17	19	the	the	DET
ejpam-5886	17	20	weighted	weight	VERB
ejpam-5886	17	21	integral	integral	ADJ
ejpam-5886	17	22	of	of	ADP
ejpam-5886	17	23	two	two	NUM
ejpam-5886	17	24	polynomials	polynomial	NOUN
ejpam-5886	17	25	with	with	ADP
ejpam-5886	17	26	different	different	ADJ
ejpam-5886	17	27	degrees	degree	NOUN
ejpam-5886	17	28	is	be	AUX
ejpam-5886	17	29	zero	zero	NUM
ejpam-5886	17	30	.	.	PUNCT
ejpam-5886	18	1	this	this	DET
ejpam-5886	18	2	orthogonality	orthogonality	NOUN
ejpam-5886	18	3	and	and	CCONJ
ejpam-5886	18	4	their	their	PRON
ejpam-5886	18	5	recurrence	recurrence	NOUN
ejpam-5886	18	6	relation	relation	NOUN
ejpam-5886	18	7	enable	enable	VERB
ejpam-5886	18	8	efficient	efficient	ADJ
ejpam-5886	18	9	computation	computation	NOUN
ejpam-5886	18	10	of	of	ADP
ejpam-5886	18	11	higher	high	ADJ
ejpam-5886	18	12	-	-	PUNCT
ejpam-5886	18	13	degree	degree	NOUN
ejpam-5886	18	14	polynomials	polynomial	NOUN
ejpam-5886	18	15	from	from	ADP
ejpam-5886	18	16	lower	low	ADJ
ejpam-5886	18	17	-	-	PUNCT
ejpam-5886	18	18	degree	degree	NOUN
ejpam-5886	18	19	ones	one	NOUN
ejpam-5886	18	20	.	.	PUNCT
ejpam-5886	19	1	additionally	additionally	ADV
ejpam-5886	19	2	,	,	PUNCT
ejpam-5886	19	3	they	they	PRON
ejpam-5886	19	4	allow	allow	VERB
ejpam-5886	19	5	functions	function	NOUN
ejpam-5886	19	6	to	to	PART
ejpam-5886	19	7	be	be	AUX
ejpam-5886	19	8	expressed	express	VERB
ejpam-5886	19	9	as	as	ADP
ejpam-5886	19	10	series	series	NOUN
ejpam-5886	19	11	expansions	expansion	NOUN
ejpam-5886	19	12	using	use	VERB
ejpam-5886	19	13	their	their	PRON
ejpam-5886	19	14	generating	generate	VERB
ejpam-5886	19	15	function	function	NOUN
ejpam-5886	19	16	,	,	PUNCT
ejpam-5886	19	17	simplifying	simplify	VERB
ejpam-5886	19	18	the	the	DET
ejpam-5886	19	19	derivation	derivation	NOUN
ejpam-5886	19	20	of	of	ADP
ejpam-5886	19	21	closed	closed	ADJ
ejpam-5886	19	22	-	-	PUNCT
ejpam-5886	19	23	form	form	NOUN
ejpam-5886	19	24	solutions	solution	NOUN
ejpam-5886	19	25	for	for	ADP
ejpam-5886	19	26	certain	certain	ADJ
ejpam-5886	19	27	differential	differential	ADJ
ejpam-5886	19	28	equations	equation	NOUN
ejpam-5886	19	29	.	.	PUNCT
ejpam-5886	20	1	widely	widely	ADV
ejpam-5886	20	2	applied	apply	VERB
ejpam-5886	20	3	in	in	ADP
ejpam-5886	20	4	mathematics	mathematic	NOUN
ejpam-5886	20	5	,	,	PUNCT
ejpam-5886	20	6	physics	physics	NOUN
ejpam-5886	20	7	,	,	PUNCT
ejpam-5886	20	8	and	and	CCONJ
ejpam-5886	20	9	engineering	engineering	NOUN
ejpam-5886	20	10	,	,	PUNCT
ejpam-5886	20	11	legendre	legendre	PROPN
ejpam-5886	20	12	polynomials	polynomial	NOUN
ejpam-5886	20	13	are	be	AUX
ejpam-5886	20	14	integral	integral	ADJ
ejpam-5886	20	15	to	to	ADP
ejpam-5886	20	16	solving	solve	VERB
ejpam-5886	20	17	the	the	DET
ejpam-5886	20	18	schrödinger	schrödinger	NOUN
ejpam-5886	20	19	equation	equation	NOUN
ejpam-5886	20	20	for	for	ADP
ejpam-5886	20	21	spherically	spherically	NOUN
ejpam-5886	20	22	symmetric	symmetric	ADJ
ejpam-5886	20	23	quantum	quantum	NOUN
ejpam-5886	20	24	systems	system	NOUN
ejpam-5886	20	25	,	,	PUNCT
ejpam-5886	20	26	including	include	VERB
ejpam-5886	20	27	the	the	DET
ejpam-5886	20	28	hydrogen	hydrogen	NOUN
ejpam-5886	20	29	atom	atom	NOUN
ejpam-5886	20	30	.	.	PUNCT
ejpam-5886	21	1	they	they	PRON
ejpam-5886	21	2	also	also	ADV
ejpam-5886	21	3	play	play	VERB
ejpam-5886	21	4	a	a	DET
ejpam-5886	21	5	crucial	crucial	ADJ
ejpam-5886	21	6	role	role	NOUN
ejpam-5886	21	7	in	in	ADP
ejpam-5886	21	8	problems	problem	NOUN
ejpam-5886	21	9	involving	involve	VERB
ejpam-5886	21	10	diffusion	diffusion	NOUN
ejpam-5886	21	11	,	,	PUNCT
ejpam-5886	21	12	wave	wave	NOUN
ejpam-5886	21	13	propagation	propagation	NOUN
ejpam-5886	21	14	,	,	PUNCT
ejpam-5886	21	15	and	and	CCONJ
ejpam-5886	21	16	heat	heat	NOUN
ejpam-5886	21	17	conduction	conduction	NOUN
ejpam-5886	21	18	.	.	PUNCT
ejpam-5886	22	1	much	much	ADJ
ejpam-5886	22	2	recent	recent	ADJ
ejpam-5886	22	3	work	work	NOUN
ejpam-5886	22	4	has	have	AUX
ejpam-5886	22	5	focused	focus	VERB
ejpam-5886	22	6	on	on	ADP
ejpam-5886	22	7	two	two	NUM
ejpam-5886	22	8	-	-	PUNCT
ejpam-5886	22	9	variable	variable	NOUN
ejpam-5886	22	10	special	special	ADJ
ejpam-5886	22	11	polynomials	polynomial	NOUN
ejpam-5886	22	12	in	in	ADP
ejpam-5886	22	13	mathematical	mathematical	ADJ
ejpam-5886	22	14	physics	physics	NOUN
ejpam-5886	22	15	.	.	PUNCT
ejpam-5886	23	1	there	there	PRON
ejpam-5886	23	2	are	be	VERB
ejpam-5886	23	3	characteristics	characteristic	NOUN
ejpam-5886	23	4	of	of	ADP
ejpam-5886	23	5	a	a	DET
ejpam-5886	23	6	class	class	NOUN
ejpam-5886	23	7	of	of	ADP
ejpam-5886	23	8	polynomials	polynomial	NOUN
ejpam-5886	23	9	called	call	VERB
ejpam-5886	23	10	two	two	NUM
ejpam-5886	23	11	-	-	PUNCT
ejpam-5886	23	12	variable	variable	NOUN
ejpam-5886	23	13	special	special	ADJ
ejpam-5886	23	14	polynomials	polynomial	NOUN
ejpam-5886	23	15	,	,	PUNCT
ejpam-5886	23	16	such	such	ADJ
ejpam-5886	23	17	as	as	ADP
ejpam-5886	23	18	[	[	X
ejpam-5886	23	19	1–10	1–10	NOUN
ejpam-5886	23	20	]	]	PUNCT
ejpam-5886	23	21	.	.	PUNCT
ejpam-5886	24	1	they	they	PRON
ejpam-5886	24	2	are	be	AUX
ejpam-5886	24	3	widely	widely	ADV
ejpam-5886	24	4	studied	study	VERB
ejpam-5886	24	5	in	in	ADP
ejpam-5886	24	6	the	the	DET
ejpam-5886	24	7	subject	subject	NOUN
ejpam-5886	24	8	of	of	ADP
ejpam-5886	24	9	algebraic	algebraic	ADJ
ejpam-5886	24	10	geometry	geometry	NOUN
ejpam-5886	24	11	and	and	CCONJ
ejpam-5886	24	12	have	have	VERB
ejpam-5886	24	13	many	many	ADJ
ejpam-5886	24	14	applications	application	NOUN
ejpam-5886	24	15	in	in	ADP
ejpam-5886	24	16	mathematics	mathematic	NOUN
ejpam-5886	24	17	and	and	CCONJ
ejpam-5886	24	18	other	other	ADJ
ejpam-5886	24	19	domains	domain	NOUN
ejpam-5886	24	20	.	.	PUNCT
ejpam-5886	25	1	notable	notable	ADJ
ejpam-5886	25	2	instances	instance	NOUN
ejpam-5886	25	3	of	of	ADP
ejpam-5886	25	4	two	two	NUM
ejpam-5886	25	5	-	-	PUNCT
ejpam-5886	25	6	variable	variable	NOUN
ejpam-5886	25	7	special	special	ADJ
ejpam-5886	25	8	polynomials	polynomial	NOUN
ejpam-5886	25	9	include	include	VERB
ejpam-5886	25	10	bivariate	bivariate	ADJ
ejpam-5886	25	11	chebyshev	chebyshev	NOUN
ejpam-5886	25	12	,	,	PUNCT
ejpam-5886	25	13	hermite	hermite	ADJ
ejpam-5886	25	14	,	,	PUNCT
ejpam-5886	25	15	laguerre	laguerre	NOUN
ejpam-5886	25	16	,	,	PUNCT
ejpam-5886	25	17	and	and	CCONJ
ejpam-5886	25	18	legendre	legendre	PROPN
ejpam-5886	25	19	polynomials	polynomial	NOUN
ejpam-5886	25	20	,	,	PUNCT
ejpam-5886	25	21	etcetra	etcetra	PROPN
ejpam-5886	25	22	.	.	PUNCT
ejpam-5886	26	1	approximation	approximation	NOUN
ejpam-5886	26	2	theory	theory	NOUN
ejpam-5886	26	3	,	,	PUNCT
ejpam-5886	26	4	numerical	numerical	ADJ
ejpam-5886	26	5	analysis	analysis	NOUN
ejpam-5886	26	6	,	,	PUNCT
ejpam-5886	26	7	and	and	CCONJ
ejpam-5886	26	8	signal	signal	NOUN
ejpam-5886	26	9	processing	processing	NOUN
ejpam-5886	26	10	all	all	PRON
ejpam-5886	26	11	make	make	VERB
ejpam-5886	26	12	extensive	extensive	ADJ
ejpam-5886	26	13	use	use	NOUN
ejpam-5886	26	14	of	of	ADP
ejpam-5886	26	15	them	they	PRON
ejpam-5886	26	16	.	.	PUNCT
ejpam-5886	27	1	bivariate	bivariate	ADJ
ejpam-5886	27	2	legendre	legendre	PROPN
ejpam-5886	27	3	polynomials	polynomial	NOUN
ejpam-5886	27	4	are	be	AUX
ejpam-5886	27	5	legendre	legendre	NOUN
ejpam-5886	27	6	polynomials	polynomial	NOUN
ejpam-5886	27	7	extended	extend	VERB
ejpam-5886	27	8	to	to	ADP
ejpam-5886	27	9	two	two	NUM
ejpam-5886	27	10	variables	variable	NOUN
ejpam-5886	27	11	.	.	PUNCT
ejpam-5886	28	1	they	they	PRON
ejpam-5886	28	2	are	be	AUX
ejpam-5886	28	3	useful	useful	ADJ
ejpam-5886	28	4	in	in	ADP
ejpam-5886	28	5	quantum	quantum	ADJ
ejpam-5886	28	6	mechanics	mechanic	NOUN
ejpam-5886	28	7	,	,	PUNCT
ejpam-5886	28	8	potential	potential	ADJ
ejpam-5886	28	9	theory	theory	NOUN
ejpam-5886	28	10	,	,	PUNCT
ejpam-5886	28	11	and	and	CCONJ
ejpam-5886	28	12	random	random	ADJ
ejpam-5886	28	13	matrix	matrix	NOUN
ejpam-5886	28	14	theory	theory	NOUN
ejpam-5886	28	15	,	,	PUNCT
ejpam-5886	28	16	and	and	CCONJ
ejpam-5886	28	17	they	they	PRON
ejpam-5886	28	18	satisfy	satisfy	VERB
ejpam-5886	28	19	a	a	DET
ejpam-5886	28	20	bivariate	bivariate	ADJ
ejpam-5886	28	21	counterpart	counterpart	NOUN
ejpam-5886	28	22	of	of	ADP
ejpam-5886	28	23	the	the	DET
ejpam-5886	28	24	legendre	legendre	PROPN
ejpam-5886	28	25	differential	differential	PROPN
ejpam-5886	28	26	equation	equation	NOUN
ejpam-5886	28	27	.	.	PUNCT
ejpam-5886	29	1	the	the	DET
ejpam-5886	29	2	behavior	behavior	NOUN
ejpam-5886	29	3	of	of	ADP
ejpam-5886	29	4	two	two	NUM
ejpam-5886	29	5	-	-	PUNCT
ejpam-5886	29	6	degree	degree	NOUN
ejpam-5886	29	7	-	-	PUNCT
ejpam-5886	29	8	of	of	ADP
ejpam-5886	29	9	-	-	PUNCT
ejpam-5886	29	10	freedom	freedom	NOUN
ejpam-5886	29	11	systems	system	NOUN
ejpam-5886	29	12	can	can	AUX
ejpam-5886	29	13	be	be	AUX
ejpam-5886	29	14	studied	study	VERB
ejpam-5886	29	15	with	with	ADP
ejpam-5886	29	16	the	the	DET
ejpam-5886	29	17	help	help	NOUN
ejpam-5886	29	18	of	of	ADP
ejpam-5886	29	19	bivariate	bivariate	ADJ
ejpam-5886	29	20	legendre	legendre	NOUN
ejpam-5886	29	21	polynomials	polynomial	NOUN
ejpam-5886	29	22	.	.	PUNCT
ejpam-5886	30	1	these	these	DET
ejpam-5886	30	2	polynomials	polynomial	NOUN
ejpam-5886	30	3	are	be	AUX
ejpam-5886	30	4	widely	widely	ADV
ejpam-5886	30	5	studied	study	VERB
ejpam-5886	30	6	in	in	ADP
ejpam-5886	30	7	mathematical	mathematical	ADJ
ejpam-5886	30	8	physics	physics	NOUN
ejpam-5886	30	9	,	,	PUNCT
ejpam-5886	30	10	probability	probability	NOUN
ejpam-5886	30	11	theory	theory	NOUN
ejpam-5886	30	12	,	,	PUNCT
ejpam-5886	30	13	and	and	CCONJ
ejpam-5886	30	14	approximation	approximation	NOUN
ejpam-5886	30	15	theory	theory	NOUN
ejpam-5886	30	16	because	because	SCONJ
ejpam-5886	30	17	they	they	PRON
ejpam-5886	30	18	meet	meet	VERB
ejpam-5886	30	19	a	a	DET
ejpam-5886	30	20	specific	specific	ADJ
ejpam-5886	30	21	orthogonality	orthogonality	NOUN
ejpam-5886	30	22	condition	condition	NOUN
ejpam-5886	30	23	concerning	concern	VERB
ejpam-5886	30	24	a	a	DET
ejpam-5886	30	25	weight	weight	NOUN
ejpam-5886	30	26	function	function	NOUN
ejpam-5886	30	27	.	.	PUNCT
ejpam-5886	31	1	these	these	DET
ejpam-5886	31	2	two	two	NUM
ejpam-5886	31	3	-	-	PUNCT
ejpam-5886	31	4	variable	variable	NOUN
ejpam-5886	31	5	special	special	ADJ
ejpam-5886	31	6	polynomials	polynomial	NOUN
ejpam-5886	31	7	are	be	AUX
ejpam-5886	31	8	important	important	ADJ
ejpam-5886	31	9	because	because	SCONJ
ejpam-5886	31	10	they	they	PRON
ejpam-5886	31	11	can	can	AUX
ejpam-5886	31	12	be	be	AUX
ejpam-5886	31	13	used	use	VERB
ejpam-5886	31	14	to	to	PART
ejpam-5886	31	15	solve	solve	VERB
ejpam-5886	31	16	problems	problem	NOUN
ejpam-5886	31	17	in	in	ADP
ejpam-5886	31	18	a	a	DET
ejpam-5886	31	19	variety	variety	NOUN
ejpam-5886	31	20	of	of	ADP
ejpam-5886	31	21	mathematical	mathematical	ADJ
ejpam-5886	31	22	and	and	CCONJ
ejpam-5886	31	23	scientific	scientific	ADJ
ejpam-5886	31	24	fields	field	NOUN
ejpam-5886	31	25	,	,	PUNCT
ejpam-5886	31	26	offer	offer	VERB
ejpam-5886	31	27	a	a	DET
ejpam-5886	31	28	rich	rich	ADJ
ejpam-5886	31	29	framework	framework	NOUN
ejpam-5886	31	30	for	for	ADP
ejpam-5886	31	31	expressing	express	VERB
ejpam-5886	31	32	and	and	CCONJ
ejpam-5886	31	33	analyzing	analyze	VERB
ejpam-5886	31	34	multivariate	multivariate	NOUN
ejpam-5886	31	35	functions	function	NOUN
ejpam-5886	31	36	,	,	PUNCT
ejpam-5886	31	37	and	and	CCONJ
ejpam-5886	31	38	have	have	VERB
ejpam-5886	31	39	particular	particular	ADJ
ejpam-5886	31	40	characteristics	characteristic	NOUN
ejpam-5886	31	41	that	that	PRON
ejpam-5886	31	42	make	make	VERB
ejpam-5886	31	43	them	they	PRON
ejpam-5886	31	44	appropriate	appropriate	ADJ
ejpam-5886	31	45	for	for	ADP
ejpam-5886	31	46	particular	particular	ADJ
ejpam-5886	31	47	applications	application	NOUN
ejpam-5886	31	48	.	.	PUNCT
ejpam-5886	32	1	two	two	NUM
ejpam-5886	32	2	-	-	PUNCT
ejpam-5886	32	3	variable	variable	NOUN
ejpam-5886	32	4	legendre	legendre	NOUN
ejpam-5886	32	5	polynomials	polynomial	NOUN
ejpam-5886	32	6	,	,	PUNCT
ejpam-5886	32	7	denoted	denote	VERB
ejpam-5886	32	8	as	as	ADP
ejpam-5886	32	9	sn(u	sn(u	NOUN
ejpam-5886	32	10	,	,	PUNCT
ejpam-5886	32	11	v	v	NOUN
ejpam-5886	32	12	)	)	PUNCT
ejpam-5886	33	1	[	[	X
ejpam-5886	33	2	11	11	NUM
ejpam-5886	33	3	]	]	PUNCT
ejpam-5886	33	4	,	,	PUNCT
ejpam-5886	33	5	hold	hold	VERB
ejpam-5886	33	6	significant	significant	ADJ
ejpam-5886	33	7	mathematical	mathematical	ADJ
ejpam-5886	33	8	value	value	NOUN
ejpam-5886	33	9	and	and	CCONJ
ejpam-5886	33	10	serve	serve	VERB
ejpam-5886	33	11	as	as	ADP
ejpam-5886	33	12	essential	essential	ADJ
ejpam-5886	33	13	tools	tool	NOUN
ejpam-5886	33	14	in	in	ADP
ejpam-5886	33	15	physics	physics	NOUN
ejpam-5886	33	16	due	due	ADP
ejpam-5886	33	17	to	to	ADP
ejpam-5886	33	18	their	their	PRON
ejpam-5886	33	19	broad	broad	ADJ
ejpam-5886	33	20	applications	application	NOUN
ejpam-5886	33	21	.	.	PUNCT
ejpam-5886	34	1	these	these	DET
ejpam-5886	34	2	polynomials	polynomial	NOUN
ejpam-5886	34	3	provide	provide	VERB
ejpam-5886	34	4	a	a	DET
ejpam-5886	34	5	powerful	powerful	ADJ
ejpam-5886	34	6	framework	framework	NOUN
ejpam-5886	34	7	for	for	ADP
ejpam-5886	34	8	analyzing	analyze	VERB
ejpam-5886	34	9	solutions	solution	NOUN
ejpam-5886	34	10	to	to	ADP
ejpam-5886	34	11	various	various	ADJ
ejpam-5886	34	12	partial	partial	ADJ
ejpam-5886	34	13	differential	differential	ADJ
ejpam-5886	34	14	equations	equation	NOUN
ejpam-5886	34	15	commonly	commonly	ADV
ejpam-5886	34	16	arising	arise	VERB
ejpam-5886	34	17	in	in	ADP
ejpam-5886	34	18	physical	physical	ADJ
ejpam-5886	34	19	contexts	contexts	NOUN
ejpam-5886	34	20	.	.	PUNCT
ejpam-5886	35	1	the	the	DET
ejpam-5886	35	2	2	2	NUM
ejpam-5886	35	3	-	-	PUNCT
ejpam-5886	35	4	variable	variable	NOUN
ejpam-5886	35	5	legendre	legendre	NOUN
ejpam-5886	35	6	polynomials	polynomial	NOUN
ejpam-5886	35	7	(	(	PUNCT
ejpam-5886	35	8	2vlep	2vlep	NUM
ejpam-5886	35	9	)	)	PUNCT
ejpam-5886	35	10	sn(u	sn(u	NOUN
ejpam-5886	35	11	,	,	PUNCT
ejpam-5886	35	12	v	v	NOUN
ejpam-5886	35	13	)	)	PUNCT
ejpam-5886	35	14	are	be	AUX
ejpam-5886	35	15	defined	define	VERB
ejpam-5886	35	16	through	through	ADP
ejpam-5886	35	17	the	the	DET
ejpam-5886	35	18	following	follow	VERB
ejpam-5886	35	19	generating	generate	VERB
ejpam-5886	35	20	function	function	NOUN
ejpam-5886	35	21	:	:	PUNCT
ejpam-5886	36	1	evtj0(2	evtj0(2	X
ejpam-5886	36	2	t	t	NOUN
ejpam-5886	36	3	√	√	PROPN
ejpam-5886	36	4	−u	−u	PROPN
ejpam-5886	36	5	)	)	PUNCT
ejpam-5886	36	6	=	=	PUNCT
ejpam-5886	37	1	∞∑	∞∑	PRON
ejpam-5886	37	2	n=0	n=0	NUM
ejpam-5886	37	3	sn(u	sn(u	NOUN
ejpam-5886	37	4	,	,	PUNCT
ejpam-5886	37	5	v	v	NOUN
ejpam-5886	37	6	)	)	PUNCT
ejpam-5886	37	7	tn	tn	PROPN
ejpam-5886	37	8	n	n	CCONJ
ejpam-5886	37	9	!	!	PROPN
ejpam-5886	37	10	,	,	PUNCT
ejpam-5886	37	11	(	(	PUNCT
ejpam-5886	37	12	1	1	X
ejpam-5886	37	13	)	)	PUNCT
ejpam-5886	37	14	s.	s.	PROPN
ejpam-5886	37	15	a.	a.	PROPN
ejpam-5886	37	16	wani	wani	PROPN
ejpam-5886	37	17	et	et	PROPN
ejpam-5886	37	18	al	al	PROPN
ejpam-5886	37	19	.	.	PUNCT
ejpam-5886	37	20	/	/	SYM
ejpam-5886	37	21	eur	eur	PROPN
ejpam-5886	37	22	.	.	PUNCT
ejpam-5886	38	1	j.	j.	PROPN
ejpam-5886	38	2	pure	pure	PROPN
ejpam-5886	38	3	appl	appl	PROPN
ejpam-5886	38	4	.	.	PROPN
ejpam-5886	38	5	math	math	PROPN
ejpam-5886	38	6	,	,	PUNCT
ejpam-5886	38	7	18	18	NUM
ejpam-5886	38	8	(	(	PUNCT
ejpam-5886	38	9	2	2	NUM
ejpam-5886	38	10	)	)	PUNCT
ejpam-5886	38	11	(	(	PUNCT
ejpam-5886	38	12	2025	2025	NUM
ejpam-5886	38	13	)	)	PUNCT
ejpam-5886	38	14	,	,	PUNCT
ejpam-5886	38	15	5886	5886	NUM
ejpam-5886	38	16	3	3	NUM
ejpam-5886	38	17	of	of	ADP
ejpam-5886	38	18	16	16	NUM
ejpam-5886	38	19	where	where	SCONJ
ejpam-5886	38	20	j0(ut	j0(ut	PROPN
ejpam-5886	38	21	)	)	PUNCT
ejpam-5886	38	22	is	be	AUX
ejpam-5886	38	23	the	the	DET
ejpam-5886	38	24	0th	0th	ADJ
ejpam-5886	38	25	order	order	NOUN
ejpam-5886	38	26	ordinary	ordinary	ADJ
ejpam-5886	38	27	bessel	bessel	ADJ
ejpam-5886	38	28	function	function	NOUN
ejpam-5886	38	29	of	of	ADP
ejpam-5886	38	30	first	first	ADJ
ejpam-5886	38	31	kind	kind	NOUN
ejpam-5886	39	1	[	[	X
ejpam-5886	39	2	12	12	NUM
ejpam-5886	39	3	]	]	PUNCT
ejpam-5886	39	4	defined	define	VERB
ejpam-5886	39	5	by	by	ADP
ejpam-5886	39	6	jn(2	jn(2	PROPN
ejpam-5886	39	7	√	√	NUM
ejpam-5886	39	8	u	u	NOUN
ejpam-5886	39	9	)	)	PUNCT
ejpam-5886	39	10	=	=	SYM
ejpam-5886	40	1	∞∑	∞∑	NUM
ejpam-5886	40	2	k=0	k=0	PROPN
ejpam-5886	40	3	(	(	PUNCT
ejpam-5886	40	4	−1)k	−1)k	PROPN
ejpam-5886	40	5	(	(	PUNCT
ejpam-5886	40	6	√	√	NUM
ejpam-5886	40	7	u	u	NOUN
ejpam-5886	40	8	)	)	PUNCT
ejpam-5886	40	9	n+2k	n+2k	PROPN
ejpam-5886	40	10	k	k	NOUN
ejpam-5886	40	11	!	!	PUNCT
ejpam-5886	40	12	(	(	PUNCT
ejpam-5886	40	13	n+	n+	X
ejpam-5886	40	14	k	k	X
ejpam-5886	40	15	)	)	PUNCT
ejpam-5886	40	16	!	!	PUNCT
ejpam-5886	40	17	.	.	PUNCT
ejpam-5886	41	1	(	(	PUNCT
ejpam-5886	41	2	2	2	X
ejpam-5886	41	3	)	)	PUNCT
ejpam-5886	41	4	we	we	PRON
ejpam-5886	41	5	also	also	ADV
ejpam-5886	41	6	note	note	VERB
ejpam-5886	41	7	that	that	SCONJ
ejpam-5886	41	8	exp(−αd−1	exp(−αd−1	X
ejpam-5886	41	9	u	u	NOUN
ejpam-5886	41	10	)	)	PUNCT
ejpam-5886	41	11	=	=	PUNCT
ejpam-5886	41	12	j0(2	j0(2	PROPN
ejpam-5886	41	13	√	√	NUM
ejpam-5886	41	14	αu	αu	NOUN
ejpam-5886	41	15	)	)	PUNCT
ejpam-5886	41	16	,	,	PUNCT
ejpam-5886	41	17	d−n	d−n	PUNCT
ejpam-5886	41	18	u	u	NOUN
ejpam-5886	41	19	{	{	PUNCT
ejpam-5886	41	20	1	1	NUM
ejpam-5886	41	21	}	}	PUNCT
ejpam-5886	41	22	:	:	PUNCT
ejpam-5886	41	23	=	=	SYM
ejpam-5886	41	24	un	un	PROPN
ejpam-5886	41	25	n	n	X
ejpam-5886	41	26	!	!	PUNCT
ejpam-5886	42	1	(	(	PUNCT
ejpam-5886	42	2	3	3	X
ejpam-5886	42	3	)	)	PUNCT
ejpam-5886	42	4	is	be	AUX
ejpam-5886	42	5	the	the	DET
ejpam-5886	42	6	inverse	inverse	ADJ
ejpam-5886	42	7	derivative	derivative	ADJ
ejpam-5886	42	8	operator	operator	NOUN
ejpam-5886	42	9	.	.	PUNCT
ejpam-5886	43	1	or	or	CCONJ
ejpam-5886	43	2	,	,	PUNCT
ejpam-5886	43	3	alternatively	alternatively	ADV
ejpam-5886	43	4	by	by	ADP
ejpam-5886	43	5	evtc0(−ut2	evtc0(−ut2	NOUN
ejpam-5886	43	6	)	)	PUNCT
ejpam-5886	43	7	=	=	PUNCT
ejpam-5886	44	1	∞∑	∞∑	PRON
ejpam-5886	44	2	n=0	n=0	NUM
ejpam-5886	44	3	sn(u	sn(u	NOUN
ejpam-5886	44	4	,	,	PUNCT
ejpam-5886	44	5	v	v	NOUN
ejpam-5886	44	6	)	)	PUNCT
ejpam-5886	44	7	tn	tn	PROPN
ejpam-5886	44	8	n	n	CCONJ
ejpam-5886	44	9	!	!	PROPN
ejpam-5886	44	10	,	,	PUNCT
ejpam-5886	44	11	(	(	PUNCT
ejpam-5886	44	12	4	4	X
ejpam-5886	44	13	)	)	PUNCT
ejpam-5886	44	14	where	where	SCONJ
ejpam-5886	44	15	c0(ut	c0(ut	PROPN
ejpam-5886	44	16	)	)	PUNCT
ejpam-5886	44	17	is	be	AUX
ejpam-5886	44	18	the	the	DET
ejpam-5886	44	19	0th	0th	ADJ
ejpam-5886	44	20	order	order	NOUN
ejpam-5886	44	21	tricommi	tricommi	NOUN
ejpam-5886	44	22	function	function	NOUN
ejpam-5886	44	23	of	of	ADP
ejpam-5886	44	24	first	first	ADJ
ejpam-5886	44	25	kind	kind	NOUN
ejpam-5886	45	1	[	[	X
ejpam-5886	45	2	12	12	NUM
ejpam-5886	45	3	]	]	PUNCT
ejpam-5886	45	4	with	with	ADP
ejpam-5886	45	5	c0(−ut2	c0(−ut2	PROPN
ejpam-5886	45	6	)	)	PUNCT
ejpam-5886	45	7	=	=	SYM
ejpam-5886	45	8	ed	ed	NOUN
ejpam-5886	45	9	−1	−1	NOUN
ejpam-5886	45	10	u	u	PROPN
ejpam-5886	45	11	t2	t2	PROPN
ejpam-5886	45	12	.	.	PUNCT
ejpam-5886	46	1	(	(	PUNCT
ejpam-5886	46	2	5	5	NUM
ejpam-5886	46	3	)	)	PUNCT
ejpam-5886	46	4	thus	thus	ADV
ejpam-5886	46	5	,	,	PUNCT
ejpam-5886	46	6	in	in	ADP
ejpam-5886	46	7	view	view	NOUN
ejpam-5886	46	8	of	of	ADP
ejpam-5886	46	9	equation	equation	NOUN
ejpam-5886	46	10	(	(	PUNCT
ejpam-5886	46	11	3	3	NUM
ejpam-5886	46	12	)	)	PUNCT
ejpam-5886	46	13	or	or	CCONJ
ejpam-5886	46	14	(	(	PUNCT
ejpam-5886	46	15	5	5	NUM
ejpam-5886	46	16	)	)	PUNCT
ejpam-5886	46	17	,	,	PUNCT
ejpam-5886	46	18	the	the	DET
ejpam-5886	46	19	generating	generate	VERB
ejpam-5886	46	20	expression	expression	NOUN
ejpam-5886	46	21	for	for	ADP
ejpam-5886	46	22	legendre	legendre	PROPN
ejpam-5886	46	23	polynomials	polynomial	NOUN
ejpam-5886	46	24	can	can	AUX
ejpam-5886	46	25	be	be	AUX
ejpam-5886	46	26	casted	cast	VERB
ejpam-5886	46	27	as	as	ADP
ejpam-5886	46	28	:	:	PUNCT
ejpam-5886	46	29	evt	evt	PROPN
ejpam-5886	46	30	ed	ed	PROPN
ejpam-5886	46	31	−1	−1	NOUN
ejpam-5886	46	32	u	u	NOUN
ejpam-5886	46	33	t2	t2	NOUN
ejpam-5886	46	34	=	=	PUNCT
ejpam-5886	46	35	∞∑	∞∑	PRON
ejpam-5886	46	36	n=0	n=0	NUM
ejpam-5886	46	37	sn(u	sn(u	NOUN
ejpam-5886	46	38	,	,	PUNCT
ejpam-5886	46	39	v	v	NOUN
ejpam-5886	46	40	)	)	PUNCT
ejpam-5886	46	41	tn	tn	PROPN
ejpam-5886	46	42	n	n	NUM
ejpam-5886	46	43	!	!	PUNCT
ejpam-5886	46	44	.	.	PUNCT
ejpam-5886	47	1	(	(	PUNCT
ejpam-5886	47	2	6	6	NUM
ejpam-5886	47	3	)	)	PUNCT
ejpam-5886	47	4	in	in	ADP
ejpam-5886	47	5	recent	recent	ADJ
ejpam-5886	47	6	years	year	NOUN
ejpam-5886	47	7	,	,	PUNCT
ejpam-5886	47	8	mathematicians	mathematician	NOUN
ejpam-5886	47	9	have	have	AUX
ejpam-5886	47	10	shown	show	VERB
ejpam-5886	47	11	a	a	DET
ejpam-5886	47	12	growing	grow	VERB
ejpam-5886	47	13	interest	interest	NOUN
ejpam-5886	47	14	in	in	ADP
ejpam-5886	47	15	the	the	DET
ejpam-5886	47	16	study	study	NOUN
ejpam-5886	47	17	of	of	ADP
ejpam-5886	47	18	∆h	∆h	PROPN
ejpam-5886	47	19	forms	form	NOUN
ejpam-5886	47	20	of	of	ADP
ejpam-5886	47	21	special	special	ADJ
ejpam-5886	47	22	polynomials	polynomial	NOUN
ejpam-5886	47	23	,	,	PUNCT
ejpam-5886	47	24	motivated	motivate	VERB
ejpam-5886	47	25	by	by	ADP
ejpam-5886	47	26	their	their	PRON
ejpam-5886	47	27	analytical	analytical	ADJ
ejpam-5886	47	28	significance	significance	NOUN
ejpam-5886	47	29	and	and	CCONJ
ejpam-5886	47	30	computational	computational	ADJ
ejpam-5886	47	31	utility	utility	NOUN
ejpam-5886	47	32	.	.	PUNCT
ejpam-5886	48	1	several	several	ADJ
ejpam-5886	48	2	generalizations	generalization	NOUN
ejpam-5886	48	3	of	of	ADP
ejpam-5886	48	4	these	these	DET
ejpam-5886	48	5	polynomials	polynomial	NOUN
ejpam-5886	48	6	have	have	AUX
ejpam-5886	48	7	been	be	AUX
ejpam-5886	48	8	explored	explore	VERB
ejpam-5886	48	9	in	in	ADP
ejpam-5886	48	10	works	work	NOUN
ejpam-5886	48	11	such	such	ADJ
ejpam-5886	48	12	as	as	ADP
ejpam-5886	48	13	[	[	X
ejpam-5886	48	14	13–18	13–18	NUM
ejpam-5886	48	15	]	]	PUNCT
ejpam-5886	48	16	.	.	PUNCT
ejpam-5886	49	1	expanding	expand	VERB
ejpam-5886	49	2	upon	upon	SCONJ
ejpam-5886	49	3	these	these	DET
ejpam-5886	49	4	studies	study	NOUN
ejpam-5886	49	5	,	,	PUNCT
ejpam-5886	49	6	the	the	DET
ejpam-5886	49	7	classical	classical	ADJ
ejpam-5886	49	8	finite	finite	ADJ
ejpam-5886	49	9	difference	difference	NOUN
ejpam-5886	49	10	operator	operator	NOUN
ejpam-5886	49	11	∆h	∆h	PROPN
ejpam-5886	49	12	has	have	AUX
ejpam-5886	49	13	been	be	AUX
ejpam-5886	49	14	employed	employ	VERB
ejpam-5886	49	15	to	to	PART
ejpam-5886	49	16	introduce	introduce	VERB
ejpam-5886	49	17	a	a	DET
ejpam-5886	49	18	novel	novel	ADJ
ejpam-5886	49	19	class	class	NOUN
ejpam-5886	49	20	known	know	VERB
ejpam-5886	49	21	as	as	ADP
ejpam-5886	49	22	∆h	∆h	PROPN
ejpam-5886	49	23	special	special	ADJ
ejpam-5886	49	24	polynomials	polynomial	NOUN
ejpam-5886	49	25	,	,	PUNCT
ejpam-5886	49	26	as	as	SCONJ
ejpam-5886	49	27	discussed	discuss	VERB
ejpam-5886	49	28	in	in	ADP
ejpam-5886	49	29	[	[	X
ejpam-5886	49	30	19–23	19–23	NUM
ejpam-5886	49	31	]	]	PUNCT
ejpam-5886	49	32	.	.	PUNCT
ejpam-5886	50	1	these	these	DET
ejpam-5886	50	2	polynomials	polynomial	NOUN
ejpam-5886	50	3	are	be	AUX
ejpam-5886	50	4	not	not	PART
ejpam-5886	50	5	only	only	ADV
ejpam-5886	50	6	significant	significant	ADJ
ejpam-5886	50	7	in	in	ADP
ejpam-5886	50	8	pure	pure	ADJ
ejpam-5886	50	9	mathematical	mathematical	ADJ
ejpam-5886	50	10	theory	theory	NOUN
ejpam-5886	50	11	but	but	CCONJ
ejpam-5886	50	12	also	also	ADV
ejpam-5886	50	13	hold	hold	VERB
ejpam-5886	50	14	substantial	substantial	ADJ
ejpam-5886	50	15	applications	application	NOUN
ejpam-5886	50	16	in	in	ADP
ejpam-5886	50	17	computational	computational	ADJ
ejpam-5886	50	18	modeling	modeling	NOUN
ejpam-5886	50	19	.	.	PUNCT
ejpam-5886	51	1	in	in	ADP
ejpam-5886	51	2	numerical	numerical	ADJ
ejpam-5886	51	3	analysis	analysis	NOUN
ejpam-5886	51	4	,	,	PUNCT
ejpam-5886	51	5	∆h	∆h	PROPN
ejpam-5886	51	6	special	special	ADJ
ejpam-5886	51	7	polynomials	polynomial	NOUN
ejpam-5886	51	8	serve	serve	VERB
ejpam-5886	51	9	as	as	ADP
ejpam-5886	51	10	essential	essential	ADJ
ejpam-5886	51	11	tools	tool	NOUN
ejpam-5886	51	12	for	for	ADP
ejpam-5886	51	13	approximating	approximate	VERB
ejpam-5886	51	14	solutions	solution	NOUN
ejpam-5886	51	15	to	to	PART
ejpam-5886	51	16	differential	differential	VERB
ejpam-5886	51	17	and	and	CCONJ
ejpam-5886	51	18	integral	integral	ADJ
ejpam-5886	51	19	equations	equation	NOUN
ejpam-5886	51	20	,	,	PUNCT
ejpam-5886	51	21	particularly	particularly	ADV
ejpam-5886	51	22	in	in	ADP
ejpam-5886	51	23	discretized	discretized	ADJ
ejpam-5886	51	24	frameworks	framework	NOUN
ejpam-5886	51	25	.	.	PUNCT
ejpam-5886	52	1	in	in	ADP
ejpam-5886	52	2	physics	physics	NOUN
ejpam-5886	52	3	and	and	CCONJ
ejpam-5886	52	4	engineering	engineering	NOUN
ejpam-5886	52	5	,	,	PUNCT
ejpam-5886	52	6	they	they	PRON
ejpam-5886	52	7	provide	provide	VERB
ejpam-5886	52	8	efficient	efficient	ADJ
ejpam-5886	52	9	computational	computational	ADJ
ejpam-5886	52	10	schemes	scheme	NOUN
ejpam-5886	52	11	for	for	ADP
ejpam-5886	52	12	modeling	model	VERB
ejpam-5886	52	13	wave	wave	NOUN
ejpam-5886	52	14	propagation	propagation	NOUN
ejpam-5886	52	15	,	,	PUNCT
ejpam-5886	52	16	heat	heat	NOUN
ejpam-5886	52	17	conduction	conduction	NOUN
ejpam-5886	52	18	,	,	PUNCT
ejpam-5886	52	19	and	and	CCONJ
ejpam-5886	52	20	fluid	fluid	ADJ
ejpam-5886	52	21	dynamics	dynamic	NOUN
ejpam-5886	52	22	.	.	PUNCT
ejpam-5886	53	1	their	their	PRON
ejpam-5886	53	2	structured	structured	ADJ
ejpam-5886	53	3	recurrence	recurrence	NOUN
ejpam-5886	53	4	relations	relation	NOUN
ejpam-5886	53	5	and	and	CCONJ
ejpam-5886	53	6	explicit	explicit	ADJ
ejpam-5886	53	7	forms	form	NOUN
ejpam-5886	53	8	allow	allow	VERB
ejpam-5886	53	9	for	for	ADP
ejpam-5886	53	10	the	the	DET
ejpam-5886	53	11	development	development	NOUN
ejpam-5886	53	12	of	of	ADP
ejpam-5886	53	13	stable	stable	ADJ
ejpam-5886	53	14	numerical	numerical	ADJ
ejpam-5886	53	15	algorithms	algorithm	NOUN
ejpam-5886	53	16	,	,	PUNCT
ejpam-5886	53	17	making	make	VERB
ejpam-5886	53	18	them	they	PRON
ejpam-5886	53	19	valuable	valuable	ADJ
ejpam-5886	53	20	in	in	ADP
ejpam-5886	53	21	computer	computer	NOUN
ejpam-5886	53	22	simulations	simulation	NOUN
ejpam-5886	53	23	.	.	PUNCT
ejpam-5886	54	1	additionally	additionally	ADV
ejpam-5886	54	2	,	,	PUNCT
ejpam-5886	54	3	in	in	ADP
ejpam-5886	54	4	statistical	statistical	ADJ
ejpam-5886	54	5	computing	computing	NOUN
ejpam-5886	54	6	,	,	PUNCT
ejpam-5886	54	7	these	these	DET
ejpam-5886	54	8	polynomials	polynomial	NOUN
ejpam-5886	54	9	facilitate	facilitate	VERB
ejpam-5886	54	10	the	the	DET
ejpam-5886	54	11	design	design	NOUN
ejpam-5886	54	12	of	of	ADP
ejpam-5886	54	13	discrete	discrete	ADJ
ejpam-5886	54	14	probability	probability	NOUN
ejpam-5886	54	15	distributions	distribution	NOUN
ejpam-5886	54	16	and	and	CCONJ
ejpam-5886	54	17	contribute	contribute	VERB
ejpam-5886	54	18	to	to	ADP
ejpam-5886	54	19	data	datum	NOUN
ejpam-5886	54	20	interpolation	interpolation	NOUN
ejpam-5886	54	21	techniques	technique	NOUN
ejpam-5886	54	22	,	,	PUNCT
ejpam-5886	54	23	enhancing	enhance	VERB
ejpam-5886	54	24	predictive	predictive	ADJ
ejpam-5886	54	25	modeling	modeling	NOUN
ejpam-5886	54	26	.	.	PUNCT
ejpam-5886	55	1	their	their	PRON
ejpam-5886	55	2	role	role	NOUN
ejpam-5886	55	3	in	in	ADP
ejpam-5886	55	4	symbolic	symbolic	ADJ
ejpam-5886	55	5	computation	computation	NOUN
ejpam-5886	55	6	further	far	ADV
ejpam-5886	55	7	extends	extend	VERB
ejpam-5886	55	8	to	to	ADP
ejpam-5886	55	9	software	software	NOUN
ejpam-5886	55	10	implementations	implementation	NOUN
ejpam-5886	55	11	for	for	ADP
ejpam-5886	55	12	solving	solve	VERB
ejpam-5886	55	13	partial	partial	ADJ
ejpam-5886	55	14	difference	difference	NOUN
ejpam-5886	55	15	equations	equation	NOUN
ejpam-5886	55	16	in	in	ADP
ejpam-5886	55	17	image	image	NOUN
ejpam-5886	55	18	processing	processing	NOUN
ejpam-5886	55	19	and	and	CCONJ
ejpam-5886	55	20	digital	digital	ADJ
ejpam-5886	55	21	signal	signal	NOUN
ejpam-5886	55	22	analysis	analysis	NOUN
ejpam-5886	55	23	.	.	PUNCT
ejpam-5886	56	1	given	give	VERB
ejpam-5886	56	2	their	their	PRON
ejpam-5886	56	3	computational	computational	ADJ
ejpam-5886	56	4	efficiency	efficiency	NOUN
ejpam-5886	56	5	and	and	CCONJ
ejpam-5886	56	6	adaptability	adaptability	NOUN
ejpam-5886	56	7	,	,	PUNCT
ejpam-5886	56	8	∆h	∆h	PROPN
ejpam-5886	56	9	special	special	ADJ
ejpam-5886	56	10	polynomials	polynomial	NOUN
ejpam-5886	56	11	are	be	AUX
ejpam-5886	56	12	emerging	emerge	VERB
ejpam-5886	56	13	as	as	ADP
ejpam-5886	56	14	powerful	powerful	ADJ
ejpam-5886	56	15	mathematical	mathematical	ADJ
ejpam-5886	56	16	tools	tool	NOUN
ejpam-5886	56	17	with	with	ADP
ejpam-5886	56	18	extensive	extensive	ADJ
ejpam-5886	56	19	applications	application	NOUN
ejpam-5886	56	20	in	in	ADP
ejpam-5886	56	21	computeraided	computeraide	VERB
ejpam-5886	56	22	modeling	modeling	NOUN
ejpam-5886	56	23	and	and	CCONJ
ejpam-5886	56	24	simulations	simulation	NOUN
ejpam-5886	56	25	.	.	PUNCT
ejpam-5886	57	1	in	in	ADP
ejpam-5886	57	2	symbolic	symbolic	ADJ
ejpam-5886	57	3	computation	computation	NOUN
ejpam-5886	57	4	and	and	CCONJ
ejpam-5886	57	5	computer	computer	NOUN
ejpam-5886	57	6	algebra	algebra	NOUN
ejpam-5886	57	7	systems	system	NOUN
ejpam-5886	57	8	(	(	PUNCT
ejpam-5886	57	9	cas	cas	PROPN
ejpam-5886	57	10	)	)	PUNCT
ejpam-5886	57	11	,	,	PUNCT
ejpam-5886	57	12	these	these	PRON
ejpam-5886	57	13	polynomials	polynomial	NOUN
ejpam-5886	57	14	aid	aid	VERB
ejpam-5886	57	15	in	in	ADP
ejpam-5886	57	16	the	the	DET
ejpam-5886	57	17	efficient	efficient	ADJ
ejpam-5886	57	18	manipulation	manipulation	NOUN
ejpam-5886	57	19	of	of	ADP
ejpam-5886	57	20	large	large	ADJ
ejpam-5886	57	21	-	-	PUNCT
ejpam-5886	57	22	scale	scale	NOUN
ejpam-5886	57	23	algebraic	algebraic	PROPN
ejpam-5886	57	24	s.	s.	PROPN
ejpam-5886	57	25	a.	a.	PROPN
ejpam-5886	57	26	wani	wani	PROPN
ejpam-5886	57	27	et	et	PROPN
ejpam-5886	57	28	al	al	PROPN
ejpam-5886	57	29	.	.	PUNCT
ejpam-5886	57	30	/	/	SYM
ejpam-5886	57	31	eur	eur	PROPN
ejpam-5886	57	32	.	.	PUNCT
ejpam-5886	58	1	j.	j.	PROPN
ejpam-5886	58	2	pure	pure	PROPN
ejpam-5886	58	3	appl	appl	PROPN
ejpam-5886	58	4	.	.	PROPN
ejpam-5886	58	5	math	math	PROPN
ejpam-5886	58	6	,	,	PUNCT
ejpam-5886	58	7	18	18	NUM
ejpam-5886	58	8	(	(	PUNCT
ejpam-5886	58	9	2	2	NUM
ejpam-5886	58	10	)	)	PUNCT
ejpam-5886	58	11	(	(	PUNCT
ejpam-5886	58	12	2025	2025	NUM
ejpam-5886	58	13	)	)	PUNCT
ejpam-5886	58	14	,	,	PUNCT
ejpam-5886	58	15	5886	5886	NUM
ejpam-5886	58	16	4	4	NUM
ejpam-5886	58	17	of	of	ADP
ejpam-5886	58	18	16	16	NUM
ejpam-5886	58	19	expressions	expression	NOUN
ejpam-5886	58	20	.	.	PUNCT
ejpam-5886	59	1	their	their	PRON
ejpam-5886	59	2	structured	structured	ADJ
ejpam-5886	59	3	operational	operational	ADJ
ejpam-5886	59	4	rules	rule	NOUN
ejpam-5886	59	5	allow	allow	VERB
ejpam-5886	59	6	for	for	ADP
ejpam-5886	59	7	automated	automate	VERB
ejpam-5886	59	8	simplifications	simplification	NOUN
ejpam-5886	59	9	and	and	CCONJ
ejpam-5886	59	10	symbolic	symbolic	ADJ
ejpam-5886	59	11	differentiation	differentiation	NOUN
ejpam-5886	59	12	,	,	PUNCT
ejpam-5886	59	13	making	make	VERB
ejpam-5886	59	14	them	they	PRON
ejpam-5886	59	15	valuable	valuable	ADJ
ejpam-5886	59	16	in	in	ADP
ejpam-5886	59	17	software	software	NOUN
ejpam-5886	59	18	implementations	implementation	NOUN
ejpam-5886	59	19	for	for	ADP
ejpam-5886	59	20	solving	solve	VERB
ejpam-5886	59	21	discrete	discrete	ADJ
ejpam-5886	59	22	versions	version	NOUN
ejpam-5886	59	23	of	of	ADP
ejpam-5886	59	24	differential	differential	ADJ
ejpam-5886	59	25	equations	equation	NOUN
ejpam-5886	59	26	,	,	PUNCT
ejpam-5886	59	27	such	such	ADJ
ejpam-5886	59	28	as	as	ADP
ejpam-5886	59	29	in	in	ADP
ejpam-5886	59	30	image	image	NOUN
ejpam-5886	59	31	processing	processing	NOUN
ejpam-5886	59	32	,	,	PUNCT
ejpam-5886	59	33	digital	digital	ADJ
ejpam-5886	59	34	signal	signal	NOUN
ejpam-5886	59	35	analysis	analysis	NOUN
ejpam-5886	59	36	,	,	PUNCT
ejpam-5886	59	37	and	and	CCONJ
ejpam-5886	59	38	network	network	NOUN
ejpam-5886	59	39	modelling	modelling	NOUN
ejpam-5886	59	40	.	.	PUNCT
ejpam-5886	60	1	furthermore	furthermore	ADV
ejpam-5886	60	2	,	,	PUNCT
ejpam-5886	60	3	in	in	ADP
ejpam-5886	60	4	graph	graph	NOUN
ejpam-5886	60	5	theory	theory	NOUN
ejpam-5886	60	6	and	and	CCONJ
ejpam-5886	60	7	combinatorial	combinatorial	ADJ
ejpam-5886	60	8	optimization	optimization	NOUN
ejpam-5886	60	9	,	,	PUNCT
ejpam-5886	60	10	∆h	∆h	PROPN
ejpam-5886	60	11	polynomials	polynomial	NOUN
ejpam-5886	60	12	provide	provide	VERB
ejpam-5886	60	13	analytical	analytical	ADJ
ejpam-5886	60	14	tools	tool	NOUN
ejpam-5886	60	15	for	for	ADP
ejpam-5886	60	16	studying	study	VERB
ejpam-5886	60	17	discrete	discrete	ADJ
ejpam-5886	60	18	structures	structure	NOUN
ejpam-5886	60	19	and	and	CCONJ
ejpam-5886	60	20	optimizing	optimize	VERB
ejpam-5886	60	21	network	network	NOUN
ejpam-5886	60	22	algorithms	algorithm	NOUN
ejpam-5886	60	23	.	.	PUNCT
ejpam-5886	61	1	their	their	PRON
ejpam-5886	61	2	application	application	NOUN
ejpam-5886	61	3	extends	extend	VERB
ejpam-5886	61	4	to	to	PART
ejpam-5886	61	5	shortest	short	ADJ
ejpam-5886	61	6	path	path	NOUN
ejpam-5886	61	7	problems	problem	NOUN
ejpam-5886	61	8	,	,	PUNCT
ejpam-5886	61	9	spanning	span	VERB
ejpam-5886	61	10	tree	tree	NOUN
ejpam-5886	61	11	calculations	calculation	NOUN
ejpam-5886	61	12	,	,	PUNCT
ejpam-5886	61	13	and	and	CCONJ
ejpam-5886	61	14	network	network	NOUN
ejpam-5886	61	15	flow	flow	NOUN
ejpam-5886	61	16	optimizations	optimization	NOUN
ejpam-5886	61	17	,	,	PUNCT
ejpam-5886	61	18	which	which	PRON
ejpam-5886	61	19	are	be	AUX
ejpam-5886	61	20	crucial	crucial	ADJ
ejpam-5886	61	21	in	in	ADP
ejpam-5886	61	22	computer	computer	NOUN
ejpam-5886	61	23	science	science	NOUN
ejpam-5886	61	24	and	and	CCONJ
ejpam-5886	61	25	logistics	logistic	NOUN
ejpam-5886	61	26	.	.	PUNCT
ejpam-5886	62	1	“	"	PUNCT
ejpam-5886	62	2	these	these	DET
ejpam-5886	62	3	∆h	∆h	NUM
ejpam-5886	62	4	-	-	PUNCT
ejpam-5886	62	5	appell	appell	NOUN
ejpam-5886	62	6	polynomial	polynomial	NOUN
ejpam-5886	62	7	are	be	AUX
ejpam-5886	62	8	represented	represent	VERB
ejpam-5886	62	9	as	as	ADP
ejpam-5886	62	10	:	:	PUNCT
ejpam-5886	62	11	a[h	a[h	X
ejpam-5886	62	12	]	]	PUNCT
ejpam-5886	62	13	n	n	CCONJ
ejpam-5886	62	14	(	(	PUNCT
ejpam-5886	62	15	u	u	NOUN
ejpam-5886	62	16	)	)	PUNCT
ejpam-5886	62	17	:	:	PUNCT
ejpam-5886	62	18	=	=	NOUN
ejpam-5886	62	19	an(u	an(u	ADJ
ejpam-5886	62	20	)	)	PUNCT
ejpam-5886	62	21	,	,	PUNCT
ejpam-5886	62	22	n	n	PROPN
ejpam-5886	62	23	∈	∈	PROPN
ejpam-5886	62	24	n0	n0	X
ejpam-5886	62	25	(	(	PUNCT
ejpam-5886	62	26	7	7	NUM
ejpam-5886	62	27	)	)	PUNCT
ejpam-5886	62	28	and	and	CCONJ
ejpam-5886	62	29	defined	define	VERB
ejpam-5886	62	30	by	by	ADP
ejpam-5886	62	31	a[h	a[h	X
ejpam-5886	62	32	]	]	PUNCT
ejpam-5886	62	33	n	n	CCONJ
ejpam-5886	62	34	(	(	PUNCT
ejpam-5886	62	35	u	u	NOUN
ejpam-5886	62	36	)	)	PUNCT
ejpam-5886	62	37	=	=	SYM
ejpam-5886	62	38	nhan−1(u	nhan−1(u	NOUN
ejpam-5886	62	39	)	)	PUNCT
ejpam-5886	62	40	,	,	PUNCT
ejpam-5886	62	41	n	n	PROPN
ejpam-5886	62	42	∈	∈	PROPN
ejpam-5886	62	43	n	n	CCONJ
ejpam-5886	62	44	,	,	PUNCT
ejpam-5886	62	45	(	(	PUNCT
ejpam-5886	62	46	8)	8)	NUM
ejpam-5886	62	47	where	where	SCONJ
ejpam-5886	62	48	∆h	∆h	PROPN
ejpam-5886	62	49	is	be	AUX
ejpam-5886	62	50	the	the	DET
ejpam-5886	62	51	finite	finite	ADJ
ejpam-5886	62	52	difference	difference	NOUN
ejpam-5886	62	53	operator	operator	NOUN
ejpam-5886	62	54	:	:	PUNCT
ejpam-5886	62	55	∆h	∆h	NOUN
ejpam-5886	62	56	h[h](u	h[h](u	NOUN
ejpam-5886	62	57	)	)	PUNCT
ejpam-5886	62	58	=	=	SYM
ejpam-5886	62	59	h(u+	h(u+	NOUN
ejpam-5886	62	60	h)−h(u	h)−h(u	NOUN
ejpam-5886	62	61	)	)	PUNCT
ejpam-5886	62	62	”	"	PUNCT
ejpam-5886	62	63	.	.	PUNCT
ejpam-5886	63	1	(	(	PUNCT
ejpam-5886	63	2	9	9	X
ejpam-5886	63	3	)	)	PUNCT
ejpam-5886	63	4	the	the	DET
ejpam-5886	63	5	∆h	∆h	NUM
ejpam-5886	63	6	-	-	PUNCT
ejpam-5886	63	7	appell	appell	NOUN
ejpam-5886	63	8	polynomials	polynomial	NOUN
ejpam-5886	63	9	an(u	an(u	ADV
ejpam-5886	63	10	)	)	PUNCT
ejpam-5886	63	11	are	be	AUX
ejpam-5886	63	12	specified	specify	VERB
ejpam-5886	63	13	by	by	ADP
ejpam-5886	63	14	the	the	DET
ejpam-5886	63	15	following	follow	VERB
ejpam-5886	63	16	generating	generate	VERB
ejpam-5886	63	17	function	function	NOUN
ejpam-5886	63	18	[	[	X
ejpam-5886	63	19	19	19	NUM
ejpam-5886	63	20	]	]	X
ejpam-5886	63	21	:	:	PUNCT
ejpam-5886	63	22	γ(t)(1	γ(t)(1	PROPN
ejpam-5886	64	1	+	+	CCONJ
ejpam-5886	64	2	ht	ht	X
ejpam-5886	64	3	)	)	PUNCT
ejpam-5886	64	4	u	u	NOUN
ejpam-5886	64	5	h	h	NOUN
ejpam-5886	64	6	=	=	PUNCT
ejpam-5886	65	1	∞∑	∞∑	PRON
ejpam-5886	65	2	n=0	n=0	ADV
ejpam-5886	65	3	a[h	a[h	ADJ
ejpam-5886	65	4	]	]	PUNCT
ejpam-5886	65	5	n	n	CCONJ
ejpam-5886	65	6	(	(	PUNCT
ejpam-5886	65	7	u	u	NOUN
ejpam-5886	65	8	)	)	PUNCT
ejpam-5886	65	9	tn	tn	PROPN
ejpam-5886	65	10	n	n	PROPN
ejpam-5886	65	11	!	!	PROPN
ejpam-5886	65	12	,	,	PUNCT
ejpam-5886	65	13	(	(	PUNCT
ejpam-5886	65	14	10	10	NUM
ejpam-5886	65	15	)	)	PUNCT
ejpam-5886	65	16	where	where	SCONJ
ejpam-5886	65	17	γ(t	γ(t	NOUN
ejpam-5886	65	18	)	)	PUNCT
ejpam-5886	65	19	=	=	PUNCT
ejpam-5886	66	1	∞∑	∞∑	PRON
ejpam-5886	66	2	n=0	n=0	NUM
ejpam-5886	66	3	γn	γn	NOUN
ejpam-5886	66	4	,	,	PUNCT
ejpam-5886	66	5	h	h	PROPN
ejpam-5886	66	6	tn	tn	PROPN
ejpam-5886	66	7	n	n	PROPN
ejpam-5886	66	8	!	!	PROPN
ejpam-5886	66	9	,	,	PUNCT
ejpam-5886	66	10	γ0,h	γ0,h	PROPN
ejpam-5886	66	11	̸=	̸=	PROPN
ejpam-5886	66	12	0	0	NUM
ejpam-5886	66	13	.	.	PUNCT
ejpam-5886	67	1	(	(	PUNCT
ejpam-5886	67	2	11	11	NUM
ejpam-5886	67	3	)	)	PUNCT
ejpam-5886	67	4	motivated	motivate	VERB
ejpam-5886	67	5	by	by	ADP
ejpam-5886	67	6	costabile	costabile	NOUN
ejpam-5886	68	1	[	[	X
ejpam-5886	68	2	19	19	NUM
ejpam-5886	68	3	]	]	X
ejpam-5886	68	4	,	,	PUNCT
ejpam-5886	68	5	here	here	ADV
ejpam-5886	68	6	we	we	PRON
ejpam-5886	68	7	introduced	introduce	VERB
ejpam-5886	68	8	the	the	DET
ejpam-5886	68	9	two	two	NUM
ejpam-5886	68	10	variable	variable	ADJ
ejpam-5886	68	11	∆h	∆h	PROPN
ejpam-5886	68	12	legendre	legendre	NOUN
ejpam-5886	68	13	polynomials	polynomial	NOUN
ejpam-5886	68	14	:	:	PUNCT
ejpam-5886	68	15	(	(	PUNCT
ejpam-5886	68	16	1	1	X
ejpam-5886	68	17	+	+	NUM
ejpam-5886	68	18	ht	ht	PROPN
ejpam-5886	68	19	)	)	PUNCT
ejpam-5886	68	20	v	v	NOUN
ejpam-5886	68	21	h	h	NOUN
ejpam-5886	68	22	(	(	PUNCT
ejpam-5886	68	23	1	1	NUM
ejpam-5886	68	24	+	+	NUM
ejpam-5886	68	25	ht2	ht2	NOUN
ejpam-5886	68	26	)	)	PUNCT
ejpam-5886	69	1	d−1	d−1	PROPN
ejpam-5886	69	2	u	u	NOUN
ejpam-5886	69	3	h	h	NOUN
ejpam-5886	69	4	=	=	SYM
ejpam-5886	69	5	s[h]n	s[h]n	PROPN
ejpam-5886	69	6	(	(	PUNCT
ejpam-5886	69	7	u	u	NOUN
ejpam-5886	69	8	,	,	PUNCT
ejpam-5886	69	9	v	v	NOUN
ejpam-5886	69	10	)	)	PUNCT
ejpam-5886	69	11	tn	tn	PROPN
ejpam-5886	69	12	n	n	CCONJ
ejpam-5886	69	13	!	!	PUNCT
ejpam-5886	70	1	(	(	PUNCT
ejpam-5886	70	2	12	12	NUM
ejpam-5886	70	3	)	)	PUNCT
ejpam-5886	70	4	through	through	ADP
ejpam-5886	70	5	the	the	DET
ejpam-5886	70	6	generating	generate	VERB
ejpam-5886	70	7	function	function	NOUN
ejpam-5886	70	8	concept	concept	NOUN
ejpam-5886	70	9	.	.	PUNCT
ejpam-5886	71	1	the	the	DET
ejpam-5886	71	2	article	article	NOUN
ejpam-5886	71	3	is	be	AUX
ejpam-5886	71	4	organized	organize	VERB
ejpam-5886	71	5	to	to	PART
ejpam-5886	71	6	give	give	VERB
ejpam-5886	71	7	readers	reader	NOUN
ejpam-5886	71	8	a	a	DET
ejpam-5886	71	9	thorough	thorough	ADJ
ejpam-5886	71	10	grasp	grasp	NOUN
ejpam-5886	71	11	of	of	ADP
ejpam-5886	71	12	the	the	DET
ejpam-5886	71	13	∆h	∆h	PROPN
ejpam-5886	71	14	legendre	legendre	NOUN
ejpam-5886	71	15	polynomials	polynomial	NOUN
ejpam-5886	71	16	,	,	PUNCT
ejpam-5886	71	17	with	with	ADP
ejpam-5886	71	18	section	section	NOUN
ejpam-5886	71	19	2	2	NUM
ejpam-5886	71	20	outlining	outline	VERB
ejpam-5886	71	21	how	how	SCONJ
ejpam-5886	71	22	they	they	PRON
ejpam-5886	71	23	are	be	AUX
ejpam-5886	71	24	generated	generate	VERB
ejpam-5886	71	25	,	,	PUNCT
ejpam-5886	71	26	how	how	SCONJ
ejpam-5886	71	27	they	they	PRON
ejpam-5886	71	28	recur	recur	VERB
ejpam-5886	71	29	,	,	PUNCT
ejpam-5886	71	30	and	and	CCONJ
ejpam-5886	71	31	how	how	SCONJ
ejpam-5886	71	32	they	they	PRON
ejpam-5886	71	33	are	be	AUX
ejpam-5886	71	34	evaluated	evaluate	VERB
ejpam-5886	71	35	.	.	PUNCT
ejpam-5886	72	1	effective	effective	ADJ
ejpam-5886	72	2	calculation	calculation	NOUN
ejpam-5886	72	3	is	be	AUX
ejpam-5886	72	4	made	make	VERB
ejpam-5886	72	5	possible	possible	ADJ
ejpam-5886	72	6	by	by	ADP
ejpam-5886	72	7	the	the	DET
ejpam-5886	72	8	formulas	formula	NOUN
ejpam-5886	72	9	provided	provide	VERB
ejpam-5886	72	10	in	in	ADP
ejpam-5886	72	11	section	section	NOUN
ejpam-5886	72	12	3	3	NUM
ejpam-5886	72	13	for	for	ADP
ejpam-5886	72	14	evaluating	evaluate	VERB
ejpam-5886	72	15	or	or	CCONJ
ejpam-5886	72	16	adding	add	VERB
ejpam-5886	72	17	up	up	ADP
ejpam-5886	72	18	these	these	DET
ejpam-5886	72	19	polynomials	polynomial	NOUN
ejpam-5886	72	20	under	under	ADP
ejpam-5886	72	21	particular	particular	ADJ
ejpam-5886	72	22	circumstances	circumstance	NOUN
ejpam-5886	72	23	.	.	PUNCT
ejpam-5886	73	1	by	by	ADP
ejpam-5886	73	2	analyzing	analyze	VERB
ejpam-5886	73	3	the	the	DET
ejpam-5886	73	4	behavior	behavior	NOUN
ejpam-5886	73	5	of	of	ADP
ejpam-5886	73	6	∆h	∆h	PROPN
ejpam-5886	73	7	legendre	legendre	PROPN
ejpam-5886	73	8	polynomials	polynomial	NOUN
ejpam-5886	73	9	under	under	ADP
ejpam-5886	73	10	different	different	ADJ
ejpam-5886	73	11	operations	operation	NOUN
ejpam-5886	73	12	and	and	CCONJ
ejpam-5886	73	13	determining	determine	VERB
ejpam-5886	73	14	their	their	PRON
ejpam-5886	73	15	determinant	determinant	ADJ
ejpam-5886	73	16	form	form	NOUN
ejpam-5886	73	17	,	,	PUNCT
ejpam-5886	73	18	section	section	NOUN
ejpam-5886	73	19	4	4	NUM
ejpam-5886	73	20	explores	explore	VERB
ejpam-5886	73	21	the	the	DET
ejpam-5886	73	22	momomiality	momomiality	NOUN
ejpam-5886	73	23	principle	principle	NOUN
ejpam-5886	73	24	.	.	PUNCT
ejpam-5886	74	1	in	in	ADP
ejpam-5886	74	2	section	section	NOUN
ejpam-5886	74	3	5	5	NUM
ejpam-5886	74	4	,	,	PUNCT
ejpam-5886	74	5	symmetric	symmetric	ADJ
ejpam-5886	74	6	identities	identity	NOUN
ejpam-5886	74	7	are	be	AUX
ejpam-5886	74	8	obtained	obtain	VERB
ejpam-5886	74	9	for	for	ADP
ejpam-5886	74	10	these	these	DET
ejpam-5886	74	11	polynomials	polynomial	NOUN
ejpam-5886	74	12	.	.	PUNCT
ejpam-5886	75	1	by	by	ADP
ejpam-5886	75	2	summarizing	summarize	VERB
ejpam-5886	75	3	the	the	DET
ejpam-5886	75	4	results	result	NOUN
ejpam-5886	75	5	and	and	CCONJ
ejpam-5886	75	6	going	go	VERB
ejpam-5886	75	7	over	over	ADP
ejpam-5886	75	8	applications	application	NOUN
ejpam-5886	75	9	,	,	PUNCT
ejpam-5886	75	10	consequences	consequence	NOUN
ejpam-5886	75	11	,	,	PUNCT
ejpam-5886	75	12	and	and	CCONJ
ejpam-5886	75	13	possible	possible	ADJ
ejpam-5886	75	14	future	future	ADJ
ejpam-5886	75	15	research	research	NOUN
ejpam-5886	75	16	areas	area	NOUN
ejpam-5886	75	17	related	relate	VERB
ejpam-5886	75	18	to	to	ADP
ejpam-5886	75	19	∆h	∆h	PROPN
ejpam-5886	75	20	legendre	legendre	NOUN
ejpam-5886	75	21	polynomials	polynomial	NOUN
ejpam-5886	75	22	,	,	PUNCT
ejpam-5886	75	23	the	the	DET
ejpam-5886	75	24	conclusion	conclusion	NOUN
ejpam-5886	75	25	enhances	enhance	VERB
ejpam-5886	75	26	understanding	understanding	NOUN
ejpam-5886	75	27	of	of	ADP
ejpam-5886	75	28	their	their	PRON
ejpam-5886	75	29	behavior	behavior	NOUN
ejpam-5886	75	30	and	and	CCONJ
ejpam-5886	75	31	usefulness	usefulness	NOUN
ejpam-5886	75	32	in	in	ADP
ejpam-5886	75	33	a	a	DET
ejpam-5886	75	34	variety	variety	NOUN
ejpam-5886	75	35	of	of	ADP
ejpam-5886	75	36	mathematical	mathematical	ADJ
ejpam-5886	75	37	contexts	contexts	NOUN
ejpam-5886	75	38	.	.	PUNCT
ejpam-5886	76	1	s.	s.	PROPN
ejpam-5886	76	2	a.	a.	PROPN
ejpam-5886	76	3	wani	wani	PROPN
ejpam-5886	76	4	et	et	PROPN
ejpam-5886	76	5	al	al	PROPN
ejpam-5886	76	6	.	.	PUNCT
ejpam-5886	76	7	/	/	SYM
ejpam-5886	76	8	eur	eur	PROPN
ejpam-5886	76	9	.	.	PUNCT
ejpam-5886	77	1	j.	j.	PROPN
ejpam-5886	77	2	pure	pure	PROPN
ejpam-5886	77	3	appl	appl	PROPN
ejpam-5886	77	4	.	.	PROPN
ejpam-5886	77	5	math	math	PROPN
ejpam-5886	77	6	,	,	PUNCT
ejpam-5886	77	7	18	18	NUM
ejpam-5886	77	8	(	(	PUNCT
ejpam-5886	77	9	2	2	NUM
ejpam-5886	77	10	)	)	PUNCT
ejpam-5886	77	11	(	(	PUNCT
ejpam-5886	77	12	2025	2025	NUM
ejpam-5886	77	13	)	)	PUNCT
ejpam-5886	77	14	,	,	PUNCT
ejpam-5886	77	15	5886	5886	NUM
ejpam-5886	77	16	5	5	NUM
ejpam-5886	77	17	of	of	ADP
ejpam-5886	77	18	16	16	NUM
ejpam-5886	77	19	2	2	NUM
ejpam-5886	77	20	.	.	PUNCT
ejpam-5886	78	1	two	two	NUM
ejpam-5886	78	2	variable	variable	ADJ
ejpam-5886	78	3	∆h	∆h	PROPN
ejpam-5886	78	4	legendre	legendre	NOUN
ejpam-5886	78	5	polynomials	polynomial	VERB
ejpam-5886	78	6	this	this	DET
ejpam-5886	78	7	section	section	NOUN
ejpam-5886	78	8	explores	explore	VERB
ejpam-5886	78	9	a	a	DET
ejpam-5886	78	10	new	new	ADJ
ejpam-5886	78	11	class	class	NOUN
ejpam-5886	78	12	of	of	ADP
ejpam-5886	78	13	two	two	NUM
ejpam-5886	78	14	-	-	PUNCT
ejpam-5886	78	15	variable	variable	NOUN
ejpam-5886	78	16	∆h	∆h	PROPN
ejpam-5886	78	17	legendre	legendre	NOUN
ejpam-5886	78	18	polynomials	polynomial	NOUN
ejpam-5886	78	19	,	,	PUNCT
ejpam-5886	78	20	establishing	establish	VERB
ejpam-5886	78	21	their	their	PRON
ejpam-5886	78	22	fundamental	fundamental	ADJ
ejpam-5886	78	23	properties	property	NOUN
ejpam-5886	78	24	and	and	CCONJ
ejpam-5886	78	25	expanding	expand	VERB
ejpam-5886	78	26	the	the	DET
ejpam-5886	78	27	scope	scope	NOUN
ejpam-5886	78	28	of	of	ADP
ejpam-5886	78	29	polynomial	polynomial	ADJ
ejpam-5886	78	30	theory	theory	NOUN
ejpam-5886	78	31	.	.	PUNCT
ejpam-5886	79	1	the	the	DET
ejpam-5886	79	2	development	development	NOUN
ejpam-5886	79	3	of	of	ADP
ejpam-5886	79	4	their	their	PRON
ejpam-5886	79	5	generating	generate	VERB
ejpam-5886	79	6	function	function	NOUN
ejpam-5886	79	7	,	,	PUNCT
ejpam-5886	79	8	s[h]ω	s[h]ω	PROPN
ejpam-5886	79	9	(	(	PUNCT
ejpam-5886	79	10	u	u	NOUN
ejpam-5886	79	11	,	,	PUNCT
ejpam-5886	79	12	v	v	NOUN
ejpam-5886	79	13	)	)	PUNCT
ejpam-5886	79	14	,	,	PUNCT
ejpam-5886	79	15	provides	provide	VERB
ejpam-5886	79	16	deeper	deep	ADJ
ejpam-5886	79	17	insight	insight	NOUN
ejpam-5886	79	18	into	into	ADP
ejpam-5886	79	19	their	their	PRON
ejpam-5886	79	20	structure	structure	NOUN
ejpam-5886	79	21	and	and	CCONJ
ejpam-5886	79	22	behavior	behavior	NOUN
ejpam-5886	79	23	,	,	PUNCT
ejpam-5886	79	24	which	which	PRON
ejpam-5886	79	25	is	be	AUX
ejpam-5886	79	26	crucial	crucial	ADJ
ejpam-5886	79	27	for	for	ADP
ejpam-5886	79	28	applications	application	NOUN
ejpam-5886	79	29	in	in	ADP
ejpam-5886	79	30	combinatorics	combinatoric	NOUN
ejpam-5886	79	31	,	,	PUNCT
ejpam-5886	79	32	analysis	analysis	NOUN
ejpam-5886	79	33	,	,	PUNCT
ejpam-5886	79	34	and	and	CCONJ
ejpam-5886	79	35	mathematical	mathematical	ADJ
ejpam-5886	79	36	physics	physics	NOUN
ejpam-5886	79	37	.	.	PUNCT
ejpam-5886	80	1	by	by	ADP
ejpam-5886	80	2	linking	link	VERB
ejpam-5886	80	3	these	these	DET
ejpam-5886	80	4	polynomials	polynomial	NOUN
ejpam-5886	80	5	to	to	ADP
ejpam-5886	80	6	their	their	PRON
ejpam-5886	80	7	generating	generate	VERB
ejpam-5886	80	8	function	function	NOUN
ejpam-5886	80	9	,	,	PUNCT
ejpam-5886	80	10	this	this	DET
ejpam-5886	80	11	study	study	NOUN
ejpam-5886	80	12	enhances	enhance	VERB
ejpam-5886	80	13	the	the	DET
ejpam-5886	80	14	understanding	understanding	NOUN
ejpam-5886	80	15	of	of	ADP
ejpam-5886	80	16	polynomial	polynomial	ADJ
ejpam-5886	80	17	families	family	NOUN
ejpam-5886	80	18	and	and	CCONJ
ejpam-5886	80	19	their	their	PRON
ejpam-5886	80	20	applications	application	NOUN
ejpam-5886	80	21	.	.	PUNCT
ejpam-5886	81	1	the	the	DET
ejpam-5886	81	2	findings	finding	NOUN
ejpam-5886	81	3	offer	offer	VERB
ejpam-5886	81	4	valuable	valuable	ADJ
ejpam-5886	81	5	perspectives	perspective	NOUN
ejpam-5886	81	6	on	on	ADP
ejpam-5886	81	7	their	their	PRON
ejpam-5886	81	8	special	special	ADJ
ejpam-5886	81	9	properties	property	NOUN
ejpam-5886	81	10	,	,	PUNCT
ejpam-5886	81	11	paving	pave	VERB
ejpam-5886	81	12	the	the	DET
ejpam-5886	81	13	way	way	NOUN
ejpam-5886	81	14	for	for	ADP
ejpam-5886	81	15	further	further	ADJ
ejpam-5886	81	16	research	research	NOUN
ejpam-5886	81	17	in	in	ADP
ejpam-5886	81	18	mathematical	mathematical	ADJ
ejpam-5886	81	19	and	and	CCONJ
ejpam-5886	81	20	scientific	scientific	ADJ
ejpam-5886	81	21	domains	domain	NOUN
ejpam-5886	81	22	.	.	PUNCT
ejpam-5886	82	1	first	first	ADV
ejpam-5886	82	2	,	,	PUNCT
ejpam-5886	82	3	we	we	PRON
ejpam-5886	82	4	derive	derive	VERB
ejpam-5886	82	5	the	the	DET
ejpam-5886	82	6	generating	generate	VERB
ejpam-5886	82	7	function	function	NOUN
ejpam-5886	82	8	for	for	ADP
ejpam-5886	82	9	these	these	DET
ejpam-5886	82	10	∆h	∆h	PROPN
ejpam-5886	82	11	legendre	legendre	PROPN
ejpam-5886	82	12	polynomials	polynomial	NOUN
ejpam-5886	82	13	s[h]n	s[h]n	PROPN
ejpam-5886	82	14	(	(	PUNCT
ejpam-5886	82	15	u	u	NOUN
ejpam-5886	82	16	,	,	PUNCT
ejpam-5886	82	17	v	v	NOUN
ejpam-5886	82	18	)	)	PUNCT
ejpam-5886	82	19	by	by	ADP
ejpam-5886	82	20	proving	prove	VERB
ejpam-5886	82	21	the	the	DET
ejpam-5886	82	22	following	follow	VERB
ejpam-5886	82	23	result	result	NOUN
ejpam-5886	82	24	:	:	PUNCT
ejpam-5886	82	25	theorem	theorem	NOUN
ejpam-5886	82	26	1	1	NUM
ejpam-5886	82	27	.	.	PUNCT
ejpam-5886	83	1	for	for	ADP
ejpam-5886	83	2	the	the	DET
ejpam-5886	83	3	two	two	NUM
ejpam-5886	83	4	variable	variable	ADJ
ejpam-5886	83	5	∆h	∆h	PROPN
ejpam-5886	83	6	legendre	legendre	NOUN
ejpam-5886	83	7	polynomials	polynomial	NOUN
ejpam-5886	83	8	s[h]n	s[h]n	PROPN
ejpam-5886	83	9	(	(	PUNCT
ejpam-5886	83	10	u	u	NOUN
ejpam-5886	83	11	,	,	PUNCT
ejpam-5886	83	12	v	v	NOUN
ejpam-5886	83	13	)	)	PUNCT
ejpam-5886	83	14	,	,	PUNCT
ejpam-5886	83	15	the	the	DET
ejpam-5886	83	16	succeeding	succeed	VERB
ejpam-5886	83	17	generating	generating	NOUN
ejpam-5886	83	18	relation	relation	NOUN
ejpam-5886	83	19	holds	hold	VERB
ejpam-5886	83	20	true	true	ADJ
ejpam-5886	83	21	:	:	PUNCT
ejpam-5886	83	22	(	(	PUNCT
ejpam-5886	83	23	1	1	X
ejpam-5886	83	24	+	+	NUM
ejpam-5886	83	25	ht	ht	PROPN
ejpam-5886	83	26	)	)	PUNCT
ejpam-5886	83	27	v	v	NOUN
ejpam-5886	83	28	h	h	NOUN
ejpam-5886	83	29	(	(	PUNCT
ejpam-5886	83	30	1	1	NUM
ejpam-5886	83	31	+	+	NUM
ejpam-5886	83	32	ht2	ht2	NOUN
ejpam-5886	83	33	)	)	PUNCT
ejpam-5886	84	1	d−1	d−1	PROPN
ejpam-5886	84	2	u	u	NOUN
ejpam-5886	84	3	h	h	NOUN
ejpam-5886	84	4	=	=	PUNCT
ejpam-5886	85	1	∞∑	∞∑	PRON
ejpam-5886	85	2	n=0	n=0	NUM
ejpam-5886	85	3	s[h]n	s[h]n	NOUN
ejpam-5886	85	4	(	(	PUNCT
ejpam-5886	85	5	u	u	NOUN
ejpam-5886	85	6	,	,	PUNCT
ejpam-5886	85	7	v	v	NOUN
ejpam-5886	85	8	)	)	PUNCT
ejpam-5886	85	9	tn	tn	PROPN
ejpam-5886	85	10	n	n	CCONJ
ejpam-5886	85	11	!	!	PROPN
ejpam-5886	85	12	,	,	PUNCT
ejpam-5886	85	13	(	(	PUNCT
ejpam-5886	85	14	13	13	NUM
ejpam-5886	85	15	)	)	PUNCT
ejpam-5886	85	16	or	or	CCONJ
ejpam-5886	85	17	equivalently	equivalently	ADV
ejpam-5886	85	18	(	(	PUNCT
ejpam-5886	85	19	1	1	NUM
ejpam-5886	85	20	+	+	NUM
ejpam-5886	85	21	ht	ht	PROPN
ejpam-5886	85	22	)	)	PUNCT
ejpam-5886	85	23	v	v	NOUN
ejpam-5886	85	24	hc0	hc0	NOUN
ejpam-5886	85	25	(	(	PUNCT
ejpam-5886	85	26	−u	−u	NOUN
ejpam-5886	85	27	h	h	NOUN
ejpam-5886	85	28	log(1	log(1	NOUN
ejpam-5886	85	29	+	+	CCONJ
ejpam-5886	85	30	ht2	ht2	NOUN
ejpam-5886	85	31	)	)	PUNCT
ejpam-5886	85	32	)	)	PUNCT
ejpam-5886	86	1	=	=	PUNCT
ejpam-5886	87	1	∞∑	∞∑	PRON
ejpam-5886	87	2	n=0	n=0	NUM
ejpam-5886	87	3	s[h]n	s[h]n	NOUN
ejpam-5886	87	4	(	(	PUNCT
ejpam-5886	87	5	u	u	NOUN
ejpam-5886	87	6	,	,	PUNCT
ejpam-5886	87	7	v	v	NOUN
ejpam-5886	87	8	)	)	PUNCT
ejpam-5886	87	9	tn	tn	PROPN
ejpam-5886	87	10	n	n	NUM
ejpam-5886	87	11	!	!	PUNCT
ejpam-5886	87	12	.	.	PUNCT
ejpam-5886	88	1	(	(	PUNCT
ejpam-5886	88	2	14	14	NUM
ejpam-5886	88	3	)	)	PUNCT
ejpam-5886	88	4	proof	proof	NOUN
ejpam-5886	88	5	.	.	PUNCT
ejpam-5886	89	1	expanding	expand	VERB
ejpam-5886	89	2	(	(	PUNCT
ejpam-5886	89	3	1	1	NUM
ejpam-5886	89	4	+	+	NUM
ejpam-5886	89	5	ht	ht	PROPN
ejpam-5886	89	6	)	)	PUNCT
ejpam-5886	89	7	v	v	NOUN
ejpam-5886	89	8	h	h	NOUN
ejpam-5886	89	9	(	(	PUNCT
ejpam-5886	89	10	1	1	NUM
ejpam-5886	89	11	+	+	NUM
ejpam-5886	89	12	ht2	ht2	NOUN
ejpam-5886	89	13	)	)	PUNCT
ejpam-5886	90	1	d−1	d−1	PROPN
ejpam-5886	90	2	u	u	PROPN
ejpam-5886	90	3	h	h	NOUN
ejpam-5886	90	4	around	around	ADP
ejpam-5886	90	5	u	u	NOUN
ejpam-5886	91	1	=	=	X
ejpam-5886	91	2	v	v	NOUN
ejpam-5886	91	3	=	=	SYM
ejpam-5886	91	4	0	0	NUM
ejpam-5886	91	5	using	use	VERB
ejpam-5886	91	6	a	a	DET
ejpam-5886	91	7	newton	newton	PROPN
ejpam-5886	91	8	series	series	NOUN
ejpam-5886	91	9	for	for	ADP
ejpam-5886	91	10	finite	finite	ADJ
ejpam-5886	91	11	differences	difference	NOUN
ejpam-5886	91	12	and	and	CCONJ
ejpam-5886	91	13	analyzing	analyze	VERB
ejpam-5886	91	14	the	the	DET
ejpam-5886	91	15	product	product	NOUN
ejpam-5886	91	16	’s	’s	PART
ejpam-5886	91	17	development	development	NOUN
ejpam-5886	91	18	with	with	ADP
ejpam-5886	91	19	respect	respect	NOUN
ejpam-5886	91	20	to	to	ADP
ejpam-5886	91	21	t	t	PROPN
ejpam-5886	91	22	’s	’s	PART
ejpam-5886	91	23	powers	power	NOUN
ejpam-5886	92	1	,	,	PUNCT
ejpam-5886	92	2	we	we	PRON
ejpam-5886	92	3	identify	identify	VERB
ejpam-5886	92	4	the	the	DET
ejpam-5886	92	5	polynomials	polynomial	NOUN
ejpam-5886	92	6	s[h]n	s[h]n	NOUN
ejpam-5886	92	7	(	(	PUNCT
ejpam-5886	92	8	u	u	NOUN
ejpam-5886	92	9	,	,	PUNCT
ejpam-5886	92	10	v	v	NOUN
ejpam-5886	92	11	)	)	PUNCT
ejpam-5886	92	12	as	as	ADP
ejpam-5886	92	13	coefficients	coefficient	NOUN
ejpam-5886	92	14	of	of	ADP
ejpam-5886	92	15	tn	tn	NOUN
ejpam-5886	92	16	n	n	ADP
ejpam-5886	92	17	!	!	PUNCT
ejpam-5886	92	18	.	.	PUNCT
ejpam-5886	93	1	these	these	DET
ejpam-5886	93	2	coefficients	coefficient	NOUN
ejpam-5886	93	3	,	,	PUNCT
ejpam-5886	93	4	given	give	VERB
ejpam-5886	93	5	in	in	ADP
ejpam-5886	93	6	equation	equation	NOUN
ejpam-5886	93	7	(	(	PUNCT
ejpam-5886	93	8	13	13	NUM
ejpam-5886	93	9	)	)	PUNCT
ejpam-5886	93	10	,	,	PUNCT
ejpam-5886	93	11	represent	represent	VERB
ejpam-5886	93	12	the	the	DET
ejpam-5886	93	13	generating	generate	VERB
ejpam-5886	93	14	function	function	NOUN
ejpam-5886	93	15	of	of	ADP
ejpam-5886	93	16	the	the	DET
ejpam-5886	93	17	two	two	NUM
ejpam-5886	93	18	-	-	PUNCT
ejpam-5886	93	19	variable	variable	NOUN
ejpam-5886	93	20	∆h	∆h	PROPN
ejpam-5886	93	21	-	-	PUNCT
ejpam-5886	93	22	legendre	legendre	NOUN
ejpam-5886	93	23	polynomials	polynomials	PROPN
ejpam-5886	93	24	s[h]n	s[h]n	PROPN
ejpam-5886	93	25	(	(	PUNCT
ejpam-5886	93	26	u	u	NOUN
ejpam-5886	93	27	,	,	PUNCT
ejpam-5886	93	28	v	v	NOUN
ejpam-5886	93	29	)	)	PUNCT
ejpam-5886	93	30	.	.	PUNCT
ejpam-5886	94	1	theorem	theorem	NOUN
ejpam-5886	94	2	2	2	NUM
ejpam-5886	94	3	.	.	X
ejpam-5886	94	4	for	for	ADP
ejpam-5886	94	5	the	the	DET
ejpam-5886	94	6	two	two	NUM
ejpam-5886	94	7	variable	variable	ADJ
ejpam-5886	94	8	∆h	∆h	PROPN
ejpam-5886	94	9	legendre	legendre	NOUN
ejpam-5886	94	10	polynomials	polynomial	NOUN
ejpam-5886	94	11	s[h]n	s[h]n	PROPN
ejpam-5886	94	12	(	(	PUNCT
ejpam-5886	94	13	u	u	NOUN
ejpam-5886	94	14	,	,	PUNCT
ejpam-5886	94	15	v	v	NOUN
ejpam-5886	94	16	)	)	PUNCT
ejpam-5886	94	17	,	,	PUNCT
ejpam-5886	94	18	the	the	DET
ejpam-5886	94	19	succeeding	succeed	VERB
ejpam-5886	94	20	relations	relation	NOUN
ejpam-5886	94	21	hold	hold	VERB
ejpam-5886	94	22	true	true	ADJ
ejpam-5886	94	23	:	:	PUNCT
ejpam-5886	94	24	v∆h	v∆h	ADJ
ejpam-5886	94	25	h	h	NOUN
ejpam-5886	94	26	s[h]n	s[h]n	PROPN
ejpam-5886	94	27	(	(	PUNCT
ejpam-5886	94	28	u	u	NOUN
ejpam-5886	94	29	,	,	PUNCT
ejpam-5886	94	30	v	v	NOUN
ejpam-5886	94	31	)	)	PUNCT
ejpam-5886	94	32	=	=	SYM
ejpam-5886	94	33	n	n	PRON
ejpam-5886	94	34	s[h]n−1(u	s[h]n−1(u	NOUN
ejpam-5886	94	35	,	,	PUNCT
ejpam-5886	94	36	v	v	NOUN
ejpam-5886	94	37	)	)	PUNCT
ejpam-5886	94	38	u∆h	u∆h	NOUN
ejpam-5886	94	39	h	h	NOUN
ejpam-5886	94	40	s[h]n	s[h]n	NOUN
ejpam-5886	94	41	(	(	PUNCT
ejpam-5886	94	42	u	u	NOUN
ejpam-5886	94	43	,	,	PUNCT
ejpam-5886	94	44	v	v	NOUN
ejpam-5886	94	45	)	)	PUNCT
ejpam-5886	95	1	=	=	SYM
ejpam-5886	95	2	n(n−	n(n−	ADJ
ejpam-5886	95	3	1	1	NUM
ejpam-5886	95	4	)	)	PUNCT
ejpam-5886	95	5	s[h]n−2(u	s[h]n−2(u	PROPN
ejpam-5886	95	6	,	,	PUNCT
ejpam-5886	95	7	v	v	NOUN
ejpam-5886	95	8	)	)	PUNCT
ejpam-5886	95	9	,	,	PUNCT
ejpam-5886	95	10	d−1	d−1	PROPN
ejpam-5886	95	11	u	u	PROPN
ejpam-5886	95	12	→	→	SYM
ejpam-5886	95	13	u.	u.	PROPN
ejpam-5886	95	14	(	(	PUNCT
ejpam-5886	95	15	15	15	NUM
ejpam-5886	95	16	)	)	PUNCT
ejpam-5886	95	17	proof	proof	NOUN
ejpam-5886	95	18	.	.	PUNCT
ejpam-5886	96	1	by	by	ADP
ejpam-5886	96	2	differentiating	differentiate	VERB
ejpam-5886	96	3	(	(	PUNCT
ejpam-5886	96	4	13	13	NUM
ejpam-5886	96	5	)	)	PUNCT
ejpam-5886	96	6	w.r.t	w.r.t	NOUN
ejpam-5886	96	7	.	.	PUNCT
ejpam-5886	97	1	v	v	NOUN
ejpam-5886	97	2	by	by	ADP
ejpam-5886	97	3	taking	take	VERB
ejpam-5886	97	4	into	into	ADP
ejpam-5886	97	5	consideration	consideration	NOUN
ejpam-5886	97	6	of	of	ADP
ejpam-5886	97	7	expression	expression	NOUN
ejpam-5886	97	8	(	(	PUNCT
ejpam-5886	97	9	5	5	NUM
ejpam-5886	97	10	)	)	PUNCT
ejpam-5886	97	11	,	,	PUNCT
ejpam-5886	97	12	we	we	PRON
ejpam-5886	97	13	have	have	AUX
ejpam-5886	97	14	v∆h(1	v∆h(1	ADV
ejpam-5886	97	15	+	+	NUM
ejpam-5886	97	16	ht	ht	PROPN
ejpam-5886	97	17	)	)	PUNCT
ejpam-5886	97	18	v	v	NOUN
ejpam-5886	97	19	h	h	NOUN
ejpam-5886	98	1	(	(	PUNCT
ejpam-5886	98	2	1	1	NUM
ejpam-5886	98	3	+	+	NUM
ejpam-5886	98	4	ht2	ht2	NOUN
ejpam-5886	98	5	)	)	PUNCT
ejpam-5886	99	1	d−1	d−1	PROPN
ejpam-5886	99	2	u	u	NOUN
ejpam-5886	99	3	h	h	NOUN
ejpam-5886	99	4	=	=	PUNCT
ejpam-5886	99	5	(	(	PUNCT
ejpam-5886	99	6	1	1	NUM
ejpam-5886	99	7	+	+	NUM
ejpam-5886	99	8	ht	ht	PROPN
ejpam-5886	99	9	)	)	PUNCT
ejpam-5886	99	10	v+h	v+h	NUM
ejpam-5886	99	11	h	h	NOUN
ejpam-5886	99	12	(	(	PUNCT
ejpam-5886	99	13	1	1	NUM
ejpam-5886	99	14	+	+	NUM
ejpam-5886	99	15	ht2	ht2	NOUN
ejpam-5886	99	16	)	)	PUNCT
ejpam-5886	100	1	d−1	d−1	PROPN
ejpam-5886	100	2	u	u	NOUN
ejpam-5886	100	3	h	h	NOUN
ejpam-5886	100	4	−	−	PROPN
ejpam-5886	100	5	(	(	PUNCT
ejpam-5886	100	6	1	1	NUM
ejpam-5886	100	7	+	+	NUM
ejpam-5886	100	8	ht	ht	PROPN
ejpam-5886	100	9	)	)	PUNCT
ejpam-5886	100	10	v	v	NOUN
ejpam-5886	100	11	h	h	NOUN
ejpam-5886	100	12	(	(	PUNCT
ejpam-5886	100	13	1	1	NUM
ejpam-5886	100	14	+	+	NUM
ejpam-5886	100	15	ht2	ht2	NOUN
ejpam-5886	100	16	)	)	PUNCT
ejpam-5886	101	1	d−1	d−1	PROPN
ejpam-5886	101	2	u	u	NOUN
ejpam-5886	101	3	h	h	NOUN
ejpam-5886	101	4	=	=	PUNCT
ejpam-5886	101	5	(	(	PUNCT
ejpam-5886	101	6	1	1	NUM
ejpam-5886	101	7	+	+	NUM
ejpam-5886	101	8	ht−	ht−	NUM
ejpam-5886	101	9	1)(1	1)(1	NUM
ejpam-5886	101	10	+	+	NUM
ejpam-5886	101	11	ht	ht	PROPN
ejpam-5886	101	12	)	)	PUNCT
ejpam-5886	101	13	v	v	NOUN
ejpam-5886	101	14	h	h	NOUN
ejpam-5886	101	15	(	(	PUNCT
ejpam-5886	101	16	1	1	NUM
ejpam-5886	101	17	+	+	NUM
ejpam-5886	101	18	ht2	ht2	NOUN
ejpam-5886	101	19	)	)	PUNCT
ejpam-5886	102	1	d−1	d−1	PROPN
ejpam-5886	102	2	u	u	PROPN
ejpam-5886	102	3	h	h	PROPN
ejpam-5886	102	4	s.	s.	PROPN
ejpam-5886	102	5	a.	a.	PROPN
ejpam-5886	102	6	wani	wani	PROPN
ejpam-5886	102	7	et	et	PROPN
ejpam-5886	102	8	al	al	PROPN
ejpam-5886	102	9	.	.	PUNCT
ejpam-5886	102	10	/	/	SYM
ejpam-5886	102	11	eur	eur	PROPN
ejpam-5886	102	12	.	.	PUNCT
ejpam-5886	103	1	j.	j.	PROPN
ejpam-5886	103	2	pure	pure	PROPN
ejpam-5886	103	3	appl	appl	PROPN
ejpam-5886	103	4	.	.	PROPN
ejpam-5886	103	5	math	math	PROPN
ejpam-5886	103	6	,	,	PUNCT
ejpam-5886	103	7	18	18	NUM
ejpam-5886	103	8	(	(	PUNCT
ejpam-5886	103	9	2	2	NUM
ejpam-5886	103	10	)	)	PUNCT
ejpam-5886	103	11	(	(	PUNCT
ejpam-5886	103	12	2025	2025	NUM
ejpam-5886	103	13	)	)	PUNCT
ejpam-5886	103	14	,	,	PUNCT
ejpam-5886	103	15	5886	5886	NUM
ejpam-5886	103	16	6	6	NUM
ejpam-5886	103	17	of	of	ADP
ejpam-5886	103	18	16	16	NUM
ejpam-5886	103	19	=	=	SYM
ejpam-5886	103	20	ht	ht	X
ejpam-5886	103	21	(	(	PUNCT
ejpam-5886	103	22	1	1	NUM
ejpam-5886	103	23	+	+	NUM
ejpam-5886	103	24	ht	ht	PROPN
ejpam-5886	103	25	)	)	PUNCT
ejpam-5886	103	26	v	v	NOUN
ejpam-5886	103	27	h	h	NOUN
ejpam-5886	103	28	(	(	PUNCT
ejpam-5886	103	29	1	1	NUM
ejpam-5886	103	30	+	+	NUM
ejpam-5886	103	31	ht2	ht2	NOUN
ejpam-5886	103	32	)	)	PUNCT
ejpam-5886	104	1	d−1	d−1	PROPN
ejpam-5886	104	2	u	u	PROPN
ejpam-5886	104	3	h	h	NOUN
ejpam-5886	104	4	.	.	PUNCT
ejpam-5886	105	1	(	(	PUNCT
ejpam-5886	105	2	16	16	NUM
ejpam-5886	105	3	)	)	PUNCT
ejpam-5886	105	4	by	by	ADP
ejpam-5886	105	5	substituting	substitute	VERB
ejpam-5886	105	6	r.h.s	r.h.s	NOUN
ejpam-5886	105	7	.	.	PUNCT
ejpam-5886	105	8	of	of	ADP
ejpam-5886	105	9	expression	expression	NOUN
ejpam-5886	105	10	(	(	PUNCT
ejpam-5886	105	11	13	13	NUM
ejpam-5886	105	12	)	)	PUNCT
ejpam-5886	105	13	in	in	ADP
ejpam-5886	105	14	(	(	PUNCT
ejpam-5886	105	15	16	16	NUM
ejpam-5886	105	16	)	)	PUNCT
ejpam-5886	105	17	,	,	PUNCT
ejpam-5886	105	18	we	we	PRON
ejpam-5886	105	19	find	find	VERB
ejpam-5886	105	20	v∆h	v∆h	ADJ
ejpam-5886	105	21	∞∑	∞∑	ADJ
ejpam-5886	105	22	n=0	n=0	ADJ
ejpam-5886	105	23	s[h]n	s[h]n	NOUN
ejpam-5886	105	24	(	(	PUNCT
ejpam-5886	105	25	u	u	NOUN
ejpam-5886	105	26	,	,	PUNCT
ejpam-5886	105	27	v	v	NOUN
ejpam-5886	105	28	)	)	PUNCT
ejpam-5886	105	29	tn	tn	NOUN
ejpam-5886	105	30	n	n	NOUN
ejpam-5886	105	31	!	!	PUNCT
ejpam-5886	106	1	=	=	NOUN
ejpam-5886	106	2	h	h	PROPN
ejpam-5886	107	1	∞∑	∞∑	ADJ
ejpam-5886	107	2	n=0	n=0	PROPN
ejpam-5886	107	3	s[h]n	s[h]n	NOUN
ejpam-5886	107	4	(	(	PUNCT
ejpam-5886	107	5	u	u	NOUN
ejpam-5886	107	6	,	,	PUNCT
ejpam-5886	107	7	v	v	NOUN
ejpam-5886	107	8	)	)	PUNCT
ejpam-5886	107	9	tn+1	tn+1	NOUN
ejpam-5886	107	10	n	n	X
ejpam-5886	107	11	!	!	PUNCT
ejpam-5886	107	12	.	.	PUNCT
ejpam-5886	108	1	(	(	PUNCT
ejpam-5886	108	2	17	17	NUM
ejpam-5886	108	3	)	)	PUNCT
ejpam-5886	108	4	substituting	substitute	VERB
ejpam-5886	108	5	n	n	NOUN
ejpam-5886	108	6	→	→	SYM
ejpam-5886	108	7	n	n	CCONJ
ejpam-5886	108	8	−	−	PROPN
ejpam-5886	108	9	1	1	NUM
ejpam-5886	108	10	into	into	ADP
ejpam-5886	108	11	the	the	DET
ejpam-5886	108	12	right	right	ADJ
ejpam-5886	108	13	-	-	PUNCT
ejpam-5886	108	14	hand	hand	NOUN
ejpam-5886	108	15	side	side	NOUN
ejpam-5886	108	16	of	of	ADP
ejpam-5886	108	17	(	(	PUNCT
ejpam-5886	108	18	16	16	NUM
ejpam-5886	108	19	)	)	PUNCT
ejpam-5886	108	20	and	and	CCONJ
ejpam-5886	108	21	equating	equate	VERB
ejpam-5886	108	22	the	the	DET
ejpam-5886	108	23	coefficients	coefficient	NOUN
ejpam-5886	108	24	of	of	ADP
ejpam-5886	108	25	identical	identical	ADJ
ejpam-5886	108	26	powers	power	NOUN
ejpam-5886	108	27	of	of	ADP
ejpam-5886	108	28	t	t	PROPN
ejpam-5886	108	29	in	in	ADP
ejpam-5886	108	30	the	the	DET
ejpam-5886	108	31	resulting	result	VERB
ejpam-5886	108	32	expression	expression	NOUN
ejpam-5886	108	33	leads	lead	VERB
ejpam-5886	108	34	to	to	ADP
ejpam-5886	108	35	the	the	DET
ejpam-5886	108	36	derivation	derivation	NOUN
ejpam-5886	108	37	of	of	ADP
ejpam-5886	108	38	(	(	PUNCT
ejpam-5886	108	39	15	15	NUM
ejpam-5886	108	40	)	)	PUNCT
ejpam-5886	108	41	.	.	PUNCT
ejpam-5886	109	1	we	we	PRON
ejpam-5886	109	2	now	now	ADV
ejpam-5886	109	3	derive	derive	VERB
ejpam-5886	109	4	the	the	DET
ejpam-5886	109	5	explicit	explicit	ADJ
ejpam-5886	109	6	form	form	NOUN
ejpam-5886	109	7	of	of	ADP
ejpam-5886	109	8	the	the	DET
ejpam-5886	109	9	two	two	NUM
ejpam-5886	109	10	-	-	PUNCT
ejpam-5886	109	11	variable	variable	NOUN
ejpam-5886	109	12	∆h	∆h	PROPN
ejpam-5886	109	13	legendre	legendre	NOUN
ejpam-5886	109	14	polynomials	polynomial	NOUN
ejpam-5886	109	15	,	,	PUNCT
ejpam-5886	109	16	s[h]n	s[h]n	PROPN
ejpam-5886	109	17	(	(	PUNCT
ejpam-5886	109	18	u	u	NOUN
ejpam-5886	109	19	,	,	PUNCT
ejpam-5886	109	20	v	v	NOUN
ejpam-5886	109	21	)	)	PUNCT
ejpam-5886	109	22	,	,	PUNCT
ejpam-5886	109	23	as	as	SCONJ
ejpam-5886	109	24	stated	state	VERB
ejpam-5886	109	25	in	in	ADP
ejpam-5886	109	26	the	the	DET
ejpam-5886	109	27	following	following	NOUN
ejpam-5886	109	28	theorem	theorem	NOUN
ejpam-5886	109	29	:	:	PUNCT
ejpam-5886	109	30	theorem	theorem	NOUN
ejpam-5886	109	31	3	3	NUM
ejpam-5886	109	32	.	.	PUNCT
ejpam-5886	110	1	the	the	DET
ejpam-5886	110	2	two	two	NUM
ejpam-5886	110	3	-	-	PUNCT
ejpam-5886	110	4	variable	variable	NOUN
ejpam-5886	110	5	∆h	∆h	PROPN
ejpam-5886	110	6	legendre	legendre	NOUN
ejpam-5886	110	7	polynomials	polynomial	NOUN
ejpam-5886	110	8	s[h]n	s[h]n	PROPN
ejpam-5886	110	9	(	(	PUNCT
ejpam-5886	110	10	u	u	NOUN
ejpam-5886	110	11	,	,	PUNCT
ejpam-5886	110	12	v	v	NOUN
ejpam-5886	110	13	)	)	PUNCT
ejpam-5886	110	14	satisfy	satisfy	VERB
ejpam-5886	110	15	the	the	DET
ejpam-5886	110	16	following	follow	VERB
ejpam-5886	110	17	relations	relation	NOUN
ejpam-5886	110	18	:	:	PUNCT
ejpam-5886	110	19	s[h]n	s[h]n	PROPN
ejpam-5886	110	20	(	(	PUNCT
ejpam-5886	110	21	u	u	NOUN
ejpam-5886	110	22	,	,	PUNCT
ejpam-5886	110	23	v	v	NOUN
ejpam-5886	110	24	)	)	PUNCT
ejpam-5886	110	25	=	=	SYM
ejpam-5886	110	26	v	v	NOUN
ejpam-5886	110	27	h∑	h∑	NOUN
ejpam-5886	110	28	d=0	d=0	PROPN
ejpam-5886	110	29	(	(	PUNCT
ejpam-5886	110	30	n	n	NOUN
ejpam-5886	110	31	d	d	NOUN
ejpam-5886	110	32	)	)	PUNCT
ejpam-5886	110	33	(	(	PUNCT
ejpam-5886	110	34	v	v	NUM
ejpam-5886	110	35	h	h	NOUN
ejpam-5886	110	36	d	d	NOUN
ejpam-5886	110	37	)	)	PUNCT
ejpam-5886	110	38	hd	hd	PROPN
ejpam-5886	110	39	s[h]n−d(u	s[h]n−d(u	PROPN
ejpam-5886	110	40	)	)	PUNCT
ejpam-5886	110	41	.	.	PUNCT
ejpam-5886	111	1	(	(	PUNCT
ejpam-5886	111	2	18	18	NUM
ejpam-5886	111	3	)	)	PUNCT
ejpam-5886	111	4	proof	proof	NOUN
ejpam-5886	111	5	.	.	PUNCT
ejpam-5886	112	1	expanding	expand	VERB
ejpam-5886	112	2	generating	generating	NOUN
ejpam-5886	112	3	relation	relation	NOUN
ejpam-5886	112	4	(	(	PUNCT
ejpam-5886	112	5	13	13	NUM
ejpam-5886	112	6	)	)	PUNCT
ejpam-5886	112	7	in	in	ADP
ejpam-5886	112	8	the	the	DET
ejpam-5886	112	9	given	give	VERB
ejpam-5886	112	10	manner	manner	NOUN
ejpam-5886	112	11	:	:	PUNCT
ejpam-5886	112	12	(	(	PUNCT
ejpam-5886	112	13	1	1	X
ejpam-5886	112	14	+	+	NUM
ejpam-5886	112	15	ht	ht	PROPN
ejpam-5886	112	16	)	)	PUNCT
ejpam-5886	112	17	v	v	NOUN
ejpam-5886	112	18	h	h	NOUN
ejpam-5886	112	19	(	(	PUNCT
ejpam-5886	112	20	1	1	NUM
ejpam-5886	112	21	+	+	NUM
ejpam-5886	112	22	ht2	ht2	NOUN
ejpam-5886	112	23	)	)	PUNCT
ejpam-5886	113	1	d−1	d−1	PROPN
ejpam-5886	113	2	u	u	NOUN
ejpam-5886	113	3	h	h	NOUN
ejpam-5886	113	4	=	=	NOUN
ejpam-5886	113	5	v	v	PROPN
ejpam-5886	113	6	h∑	h∑	AUX
ejpam-5886	113	7	d=0	d=0	PROPN
ejpam-5886	113	8	(	(	PUNCT
ejpam-5886	113	9	v	v	NOUN
ejpam-5886	113	10	h	h	NOUN
ejpam-5886	113	11	d	d	NOUN
ejpam-5886	113	12	)	)	PUNCT
ejpam-5886	113	13	(	(	PUNCT
ejpam-5886	113	14	ht)d	ht)d	PROPN
ejpam-5886	113	15	d	d	X
ejpam-5886	113	16	!	!	PUNCT
ejpam-5886	114	1	∞∑	∞∑	NUM
ejpam-5886	114	2	n=0	n=0	ADJ
ejpam-5886	114	3	s[h]n	s[h]n	NOUN
ejpam-5886	114	4	(	(	PUNCT
ejpam-5886	114	5	u	u	NOUN
ejpam-5886	114	6	,	,	PUNCT
ejpam-5886	114	7	0	0	NUM
ejpam-5886	114	8	)	)	PUNCT
ejpam-5886	114	9	tn	tn	PROPN
ejpam-5886	114	10	n	n	PROPN
ejpam-5886	114	11	!	!	PUNCT
ejpam-5886	114	12	(	(	PUNCT
ejpam-5886	114	13	19	19	NUM
ejpam-5886	114	14	)	)	PUNCT
ejpam-5886	114	15	which	which	PRON
ejpam-5886	114	16	can	can	AUX
ejpam-5886	114	17	further	far	ADV
ejpam-5886	114	18	be	be	AUX
ejpam-5886	114	19	written	write	VERB
ejpam-5886	114	20	as	as	ADP
ejpam-5886	114	21	s[h]n	s[h]n	PROPN
ejpam-5886	114	22	(	(	PUNCT
ejpam-5886	114	23	u	u	NOUN
ejpam-5886	114	24	,	,	PUNCT
ejpam-5886	114	25	v	v	NOUN
ejpam-5886	114	26	)	)	PUNCT
ejpam-5886	114	27	tn	tn	NOUN
ejpam-5886	114	28	n	n	NOUN
ejpam-5886	114	29	!	!	PUNCT
ejpam-5886	114	30	=	=	NOUN
ejpam-5886	115	1	∞∑	∞∑	DET
ejpam-5886	115	2	n=0	n=0	NUM
ejpam-5886	115	3	v	v	NOUN
ejpam-5886	115	4	h∑	h∑	AUX
ejpam-5886	115	5	d=0	d=0	PROPN
ejpam-5886	115	6	(	(	PUNCT
ejpam-5886	115	7	v	v	NOUN
ejpam-5886	115	8	h	h	NOUN
ejpam-5886	115	9	d	d	NOUN
ejpam-5886	115	10	)	)	PUNCT
ejpam-5886	115	11	hd	hd	VERB
ejpam-5886	115	12	s[h]n	s[h]n	PROPN
ejpam-5886	115	13	(	(	PUNCT
ejpam-5886	115	14	u	u	NOUN
ejpam-5886	115	15	,	,	PUNCT
ejpam-5886	115	16	0	0	NUM
ejpam-5886	115	17	)	)	PUNCT
ejpam-5886	115	18	tn+d	tn+d	NUM
ejpam-5886	115	19	n	n	CCONJ
ejpam-5886	115	20	!	!	PUNCT
ejpam-5886	116	1	d	d	X
ejpam-5886	116	2	!	!	PUNCT
ejpam-5886	116	3	.	.	PUNCT
ejpam-5886	117	1	(	(	PUNCT
ejpam-5886	117	2	20	20	NUM
ejpam-5886	117	3	)	)	PUNCT
ejpam-5886	117	4	by	by	ADP
ejpam-5886	117	5	replacing	replace	VERB
ejpam-5886	117	6	n	n	PRON
ejpam-5886	117	7	→	→	PUNCT
ejpam-5886	117	8	n−	n−	NOUN
ejpam-5886	117	9	d	d	PROPN
ejpam-5886	117	10	in	in	ADP
ejpam-5886	117	11	the	the	DET
ejpam-5886	117	12	r.h.s	r.h.s	NOUN
ejpam-5886	117	13	.	.	PUNCT
ejpam-5886	117	14	of	of	ADP
ejpam-5886	117	15	previous	previous	ADJ
ejpam-5886	117	16	expression	expression	NOUN
ejpam-5886	117	17	,	,	PUNCT
ejpam-5886	117	18	it	it	PRON
ejpam-5886	117	19	follows	follow	VERB
ejpam-5886	117	20	that	that	SCONJ
ejpam-5886	117	21	s[h]n	s[h]n	PROPN
ejpam-5886	117	22	(	(	PUNCT
ejpam-5886	117	23	u	u	NOUN
ejpam-5886	117	24	,	,	PUNCT
ejpam-5886	117	25	v	v	NOUN
ejpam-5886	117	26	)	)	PUNCT
ejpam-5886	117	27	tn	tn	NOUN
ejpam-5886	117	28	n	n	NOUN
ejpam-5886	117	29	!	!	PUNCT
ejpam-5886	118	1	=	=	NOUN
ejpam-5886	119	1	∞∑	∞∑	DET
ejpam-5886	119	2	n=0	n=0	NUM
ejpam-5886	119	3	v	v	NOUN
ejpam-5886	119	4	h∑	h∑	AUX
ejpam-5886	119	5	d=0	d=0	PROPN
ejpam-5886	119	6	(	(	PUNCT
ejpam-5886	119	7	v	v	NOUN
ejpam-5886	119	8	h	h	NOUN
ejpam-5886	119	9	d	d	NOUN
ejpam-5886	119	10	)	)	PUNCT
ejpam-5886	119	11	hd	hd	VERB
ejpam-5886	119	12	s[h]n	s[h]n	PROPN
ejpam-5886	119	13	(	(	PUNCT
ejpam-5886	119	14	u	u	NOUN
ejpam-5886	119	15	,	,	PUNCT
ejpam-5886	119	16	0	0	NUM
ejpam-5886	119	17	)	)	PUNCT
ejpam-5886	119	18	tn	tn	NOUN
ejpam-5886	119	19	(	(	PUNCT
ejpam-5886	119	20	n−	n−	NOUN
ejpam-5886	119	21	d	d	NOUN
ejpam-5886	119	22	)	)	PUNCT
ejpam-5886	119	23	!	!	PUNCT
ejpam-5886	120	1	d	d	X
ejpam-5886	120	2	!	!	PUNCT
ejpam-5886	120	3	.	.	PUNCT
ejpam-5886	121	1	(	(	PUNCT
ejpam-5886	121	2	21	21	NUM
ejpam-5886	121	3	)	)	PUNCT
ejpam-5886	121	4	on	on	ADP
ejpam-5886	121	5	multiplying	multiply	VERB
ejpam-5886	121	6	and	and	CCONJ
ejpam-5886	121	7	dividing	dividing	NOUN
ejpam-5886	121	8	by	by	ADP
ejpam-5886	121	9	n	n	CCONJ
ejpam-5886	121	10	!	!	PUNCT
ejpam-5886	122	1	in	in	ADP
ejpam-5886	122	2	the	the	DET
ejpam-5886	122	3	r.h.s	r.h.s	NOUN
ejpam-5886	122	4	.	.	PUNCT
ejpam-5886	123	1	of	of	ADP
ejpam-5886	123	2	previous	previous	ADJ
ejpam-5886	123	3	expression	expression	NOUN
ejpam-5886	123	4	(	(	PUNCT
ejpam-5886	123	5	21	21	NUM
ejpam-5886	123	6	)	)	PUNCT
ejpam-5886	123	7	and	and	CCONJ
ejpam-5886	123	8	comparing	compare	VERB
ejpam-5886	123	9	the	the	DET
ejpam-5886	123	10	coeffecients	coeffecient	NOUN
ejpam-5886	123	11	of	of	ADP
ejpam-5886	123	12	same	same	ADJ
ejpam-5886	123	13	exponents	exponent	NOUN
ejpam-5886	123	14	of	of	ADP
ejpam-5886	123	15	t	t	PROPN
ejpam-5886	123	16	on	on	ADP
ejpam-5886	123	17	both	both	DET
ejpam-5886	123	18	sides	side	NOUN
ejpam-5886	123	19	,	,	PUNCT
ejpam-5886	123	20	assertion	assertion	NOUN
ejpam-5886	123	21	(	(	PUNCT
ejpam-5886	123	22	18	18	NUM
ejpam-5886	123	23	)	)	PUNCT
ejpam-5886	123	24	is	be	AUX
ejpam-5886	123	25	deduced	deduce	VERB
ejpam-5886	123	26	.	.	PUNCT
ejpam-5886	124	1	s.	s.	PROPN
ejpam-5886	124	2	a.	a.	PROPN
ejpam-5886	124	3	wani	wani	PROPN
ejpam-5886	124	4	et	et	PROPN
ejpam-5886	124	5	al	al	PROPN
ejpam-5886	124	6	.	.	PUNCT
ejpam-5886	124	7	/	/	SYM
ejpam-5886	124	8	eur	eur	PROPN
ejpam-5886	124	9	.	.	PUNCT
ejpam-5886	125	1	j.	j.	PROPN
ejpam-5886	125	2	pure	pure	PROPN
ejpam-5886	125	3	appl	appl	PROPN
ejpam-5886	125	4	.	.	PROPN
ejpam-5886	125	5	math	math	PROPN
ejpam-5886	125	6	,	,	PUNCT
ejpam-5886	125	7	18	18	NUM
ejpam-5886	125	8	(	(	PUNCT
ejpam-5886	125	9	2	2	NUM
ejpam-5886	125	10	)	)	PUNCT
ejpam-5886	125	11	(	(	PUNCT
ejpam-5886	125	12	2025	2025	NUM
ejpam-5886	125	13	)	)	PUNCT
ejpam-5886	125	14	,	,	PUNCT
ejpam-5886	125	15	5886	5886	NUM
ejpam-5886	125	16	7	7	NUM
ejpam-5886	125	17	of	of	ADP
ejpam-5886	125	18	16	16	NUM
ejpam-5886	125	19	3	3	NUM
ejpam-5886	125	20	.	.	PUNCT
ejpam-5886	125	21	summation	summation	NOUN
ejpam-5886	125	22	formulae	formulae	VERB
ejpam-5886	125	23	this	this	DET
ejpam-5886	125	24	section	section	NOUN
ejpam-5886	125	25	presents	present	VERB
ejpam-5886	125	26	summation	summation	NOUN
ejpam-5886	125	27	formulae	formulae	ADJ
ejpam-5886	125	28	,	,	PUNCT
ejpam-5886	125	29	essential	essential	ADJ
ejpam-5886	125	30	tools	tool	NOUN
ejpam-5886	125	31	in	in	ADP
ejpam-5886	125	32	mathematical	mathematical	ADJ
ejpam-5886	125	33	analysis	analysis	NOUN
ejpam-5886	125	34	for	for	ADP
ejpam-5886	125	35	understanding	understand	VERB
ejpam-5886	125	36	polynomial	polynomial	ADJ
ejpam-5886	125	37	structures	structure	NOUN
ejpam-5886	125	38	in	in	ADP
ejpam-5886	125	39	two	two	NUM
ejpam-5886	125	40	variables	variable	NOUN
ejpam-5886	125	41	.	.	PUNCT
ejpam-5886	126	1	these	these	DET
ejpam-5886	126	2	formulas	formula	NOUN
ejpam-5886	126	3	systematically	systematically	ADV
ejpam-5886	126	4	compute	compute	VERB
ejpam-5886	126	5	sums	sum	NOUN
ejpam-5886	126	6	of	of	ADP
ejpam-5886	126	7	specific	specific	ADJ
ejpam-5886	126	8	polynomials	polynomial	NOUN
ejpam-5886	126	9	,	,	PUNCT
ejpam-5886	126	10	revealing	reveal	VERB
ejpam-5886	126	11	hidden	hide	VERB
ejpam-5886	126	12	symmetries	symmetry	NOUN
ejpam-5886	126	13	and	and	CCONJ
ejpam-5886	126	14	interrelationships	interrelationship	NOUN
ejpam-5886	126	15	between	between	ADP
ejpam-5886	126	16	variables	variable	NOUN
ejpam-5886	126	17	.	.	PUNCT
ejpam-5886	127	1	their	their	PRON
ejpam-5886	127	2	applications	application	NOUN
ejpam-5886	127	3	extend	extend	VERB
ejpam-5886	127	4	to	to	ADP
ejpam-5886	127	5	mathematical	mathematical	ADJ
ejpam-5886	127	6	physics	physics	NOUN
ejpam-5886	127	7	,	,	PUNCT
ejpam-5886	127	8	probability	probability	NOUN
ejpam-5886	127	9	theory	theory	NOUN
ejpam-5886	127	10	,	,	PUNCT
ejpam-5886	127	11	and	and	CCONJ
ejpam-5886	127	12	combinatorics	combinatoric	NOUN
ejpam-5886	127	13	,	,	PUNCT
ejpam-5886	127	14	aiding	aid	VERB
ejpam-5886	127	15	in	in	ADP
ejpam-5886	127	16	the	the	DET
ejpam-5886	127	17	development	development	NOUN
ejpam-5886	127	18	of	of	ADP
ejpam-5886	127	19	efficient	efficient	ADJ
ejpam-5886	127	20	computational	computational	ADJ
ejpam-5886	127	21	techniques	technique	NOUN
ejpam-5886	127	22	.	.	PUNCT
ejpam-5886	128	1	summation	summation	NOUN
ejpam-5886	128	2	equations	equation	NOUN
ejpam-5886	128	3	serve	serve	VERB
ejpam-5886	128	4	as	as	ADP
ejpam-5886	128	5	fundamental	fundamental	ADJ
ejpam-5886	128	6	building	building	NOUN
ejpam-5886	128	7	blocks	block	NOUN
ejpam-5886	128	8	for	for	ADP
ejpam-5886	128	9	advancing	advance	VERB
ejpam-5886	128	10	mathematical	mathematical	ADJ
ejpam-5886	128	11	theory	theory	NOUN
ejpam-5886	128	12	and	and	CCONJ
ejpam-5886	128	13	its	its	PRON
ejpam-5886	128	14	real	real	ADJ
ejpam-5886	128	15	-	-	PUNCT
ejpam-5886	128	16	world	world	NOUN
ejpam-5886	128	17	applications	application	NOUN
ejpam-5886	128	18	.	.	PUNCT
ejpam-5886	129	1	below	below	ADV
ejpam-5886	129	2	,	,	PUNCT
ejpam-5886	129	3	we	we	PRON
ejpam-5886	129	4	establish	establish	VERB
ejpam-5886	129	5	these	these	DET
ejpam-5886	129	6	summation	summation	NOUN
ejpam-5886	129	7	formulae	formulae	NOUN
ejpam-5886	129	8	by	by	ADP
ejpam-5886	129	9	proving	prove	VERB
ejpam-5886	129	10	the	the	DET
ejpam-5886	129	11	following	follow	VERB
ejpam-5886	129	12	key	key	ADJ
ejpam-5886	129	13	results	result	NOUN
ejpam-5886	129	14	:	:	PUNCT
ejpam-5886	129	15	theorem	theorem	VERB
ejpam-5886	129	16	4	4	NUM
ejpam-5886	129	17	.	.	PUNCT
ejpam-5886	129	18	for	for	ADP
ejpam-5886	129	19	n	n	PRON
ejpam-5886	129	20	≥	≥	NOUN
ejpam-5886	129	21	0	0	NUM
ejpam-5886	129	22	,	,	PUNCT
ejpam-5886	129	23	we	we	PRON
ejpam-5886	129	24	have	have	VERB
ejpam-5886	129	25	s[h]n	s[h]n	NOUN
ejpam-5886	129	26	(	(	PUNCT
ejpam-5886	129	27	v	v	NOUN
ejpam-5886	129	28	+	+	CCONJ
ejpam-5886	129	29	1	1	NUM
ejpam-5886	129	30	,	,	PUNCT
ejpam-5886	129	31	u	u	NOUN
ejpam-5886	129	32	)	)	PUNCT
ejpam-5886	129	33	=	=	SYM
ejpam-5886	130	1	n∑	n∑	PROPN
ejpam-5886	130	2	m=0	m=0	PROPN
ejpam-5886	130	3	(	(	PUNCT
ejpam-5886	130	4	n	n	NOUN
ejpam-5886	130	5	m	m	VERB
ejpam-5886	130	6	)	)	PUNCT
ejpam-5886	130	7	(	(	PUNCT
ejpam-5886	130	8	−1	−1	NOUN
ejpam-5886	130	9	h	h	NOUN
ejpam-5886	130	10	)	)	PUNCT
ejpam-5886	130	11	m	m	VERB
ejpam-5886	130	12	(	(	PUNCT
ejpam-5886	130	13	−h)ms[h]n−m(u	−h)ms[h]n−m(u	NOUN
ejpam-5886	130	14	,	,	PUNCT
ejpam-5886	130	15	v	v	NOUN
ejpam-5886	130	16	)	)	PUNCT
ejpam-5886	130	17	.	.	PUNCT
ejpam-5886	131	1	(	(	PUNCT
ejpam-5886	131	2	22	22	NUM
ejpam-5886	131	3	)	)	PUNCT
ejpam-5886	131	4	proof	proof	NOUN
ejpam-5886	131	5	.	.	PUNCT
ejpam-5886	132	1	from	from	ADP
ejpam-5886	132	2	(	(	PUNCT
ejpam-5886	132	3	13	13	NUM
ejpam-5886	132	4	)	)	PUNCT
ejpam-5886	132	5	,	,	PUNCT
ejpam-5886	132	6	we	we	PRON
ejpam-5886	132	7	have	have	VERB
ejpam-5886	132	8	∞∑	∞∑	NUM
ejpam-5886	132	9	n=0	n=0	ADJ
ejpam-5886	132	10	s[h]n	s[h]n	NOUN
ejpam-5886	132	11	(	(	PUNCT
ejpam-5886	132	12	v	v	NOUN
ejpam-5886	132	13	+	+	CCONJ
ejpam-5886	132	14	1	1	NUM
ejpam-5886	132	15	,	,	PUNCT
ejpam-5886	132	16	u	u	NOUN
ejpam-5886	132	17	)	)	PUNCT
ejpam-5886	132	18	tn	tn	PROPN
ejpam-5886	132	19	n	n	PROPN
ejpam-5886	132	20	!	!	PUNCT
ejpam-5886	133	1	−	−	PROPN
ejpam-5886	134	1	∞∑	∞∑	PRON
ejpam-5886	134	2	n=0	n=0	PROPN
ejpam-5886	134	3	s[h]n	s[h]n	NOUN
ejpam-5886	134	4	(	(	PUNCT
ejpam-5886	134	5	u	u	NOUN
ejpam-5886	134	6	,	,	PUNCT
ejpam-5886	134	7	v	v	NOUN
ejpam-5886	134	8	)	)	PUNCT
ejpam-5886	134	9	tn	tn	NOUN
ejpam-5886	134	10	n	n	NOUN
ejpam-5886	134	11	!	!	PUNCT
ejpam-5886	134	12	=	=	PUNCT
ejpam-5886	135	1	(	(	PUNCT
ejpam-5886	135	2	1	1	NUM
ejpam-5886	135	3	+	+	NUM
ejpam-5886	135	4	ht	ht	PROPN
ejpam-5886	135	5	)	)	PUNCT
ejpam-5886	135	6	v	v	NOUN
ejpam-5886	135	7	h	h	NOUN
ejpam-5886	135	8	(	(	PUNCT
ejpam-5886	135	9	1	1	NUM
ejpam-5886	135	10	+	+	NUM
ejpam-5886	135	11	ht2	ht2	NOUN
ejpam-5886	135	12	)	)	PUNCT
ejpam-5886	136	1	d−1	d−1	PROPN
ejpam-5886	136	2	u	u	PROPN
ejpam-5886	136	3	h	h	NOUN
ejpam-5886	136	4	(	(	PUNCT
ejpam-5886	136	5	(	(	PUNCT
ejpam-5886	136	6	1	1	NUM
ejpam-5886	136	7	+	+	NUM
ejpam-5886	136	8	ht	ht	X
ejpam-5886	136	9	)	)	PUNCT
ejpam-5886	136	10	1	1	NUM
ejpam-5886	136	11	h	h	NOUN
ejpam-5886	136	12	−	−	NOUN
ejpam-5886	136	13	1	1	NUM
ejpam-5886	136	14	)	)	PUNCT
ejpam-5886	136	15	=	=	NOUN
ejpam-5886	137	1	∞∑	∞∑	PRON
ejpam-5886	137	2	n=0	n=0	NUM
ejpam-5886	137	3	s[h]n	s[h]n	NOUN
ejpam-5886	137	4	(	(	PUNCT
ejpam-5886	137	5	u	u	NOUN
ejpam-5886	137	6	,	,	PUNCT
ejpam-5886	137	7	v	v	NOUN
ejpam-5886	137	8	)	)	PUNCT
ejpam-5886	137	9	tn	tn	PROPN
ejpam-5886	137	10	n	n	CCONJ
ejpam-5886	137	11	!	!	PUNCT
ejpam-5886	138	1	(	(	PUNCT
ejpam-5886	138	2	∞∑	∞∑	NUM
ejpam-5886	138	3	m=0	m=0	PROPN
ejpam-5886	138	4	(	(	PUNCT
ejpam-5886	138	5	−1	−1	NOUN
ejpam-5886	138	6	h	h	NOUN
ejpam-5886	138	7	)	)	PUNCT
ejpam-5886	138	8	m	m	VERB
ejpam-5886	138	9	(	(	PUNCT
ejpam-5886	138	10	−h)m	−h)m	ADJ
ejpam-5886	138	11	tm	tm	PROPN
ejpam-5886	138	12	m	m	PROPN
ejpam-5886	138	13	!	!	PUNCT
ejpam-5886	139	1	−	−	NOUN
ejpam-5886	139	2	1	1	NUM
ejpam-5886	139	3	)	)	PUNCT
ejpam-5886	139	4	=	=	NOUN
ejpam-5886	140	1	∞∑	∞∑	NUM
ejpam-5886	140	2	n=0	n=0	NUM
ejpam-5886	140	3	(	(	PUNCT
ejpam-5886	140	4	n∑	n∑	PROPN
ejpam-5886	140	5	m=0	m=0	PROPN
ejpam-5886	140	6	(	(	PUNCT
ejpam-5886	140	7	n	n	NOUN
ejpam-5886	140	8	m	m	VERB
ejpam-5886	140	9	)	)	PUNCT
ejpam-5886	140	10	(	(	PUNCT
ejpam-5886	140	11	−1	−1	NOUN
ejpam-5886	140	12	h	h	NOUN
ejpam-5886	140	13	)	)	PUNCT
ejpam-5886	140	14	m	m	VERB
ejpam-5886	140	15	(	(	PUNCT
ejpam-5886	140	16	−h)ms[h]n−m(u	−h)ms[h]n−m(u	NOUN
ejpam-5886	140	17	,	,	PUNCT
ejpam-5886	140	18	v	v	NOUN
ejpam-5886	140	19	)	)	PUNCT
ejpam-5886	140	20	)	)	PUNCT
ejpam-5886	140	21	tn	tn	PROPN
ejpam-5886	140	22	n	n	PROPN
ejpam-5886	140	23	!	!	PUNCT
ejpam-5886	141	1	−	−	PROPN
ejpam-5886	142	1	∞∑	∞∑	DET
ejpam-5886	142	2	n=0	n=0	PROPN
ejpam-5886	142	3	s[h]n	s[h]n	NOUN
ejpam-5886	142	4	(	(	PUNCT
ejpam-5886	142	5	u	u	NOUN
ejpam-5886	142	6	,	,	PUNCT
ejpam-5886	142	7	v	v	NOUN
ejpam-5886	142	8	)	)	PUNCT
ejpam-5886	142	9	tn	tn	PROPN
ejpam-5886	142	10	n	n	NUM
ejpam-5886	142	11	!	!	PUNCT
ejpam-5886	142	12	.	.	PUNCT
ejpam-5886	143	1	(	(	PUNCT
ejpam-5886	143	2	23	23	NUM
ejpam-5886	143	3	)	)	PUNCT
ejpam-5886	143	4	comparing	compare	VERB
ejpam-5886	143	5	the	the	DET
ejpam-5886	143	6	coefficients	coefficient	NOUN
ejpam-5886	143	7	of	of	ADP
ejpam-5886	143	8	t	t	PROPN
ejpam-5886	143	9	,	,	PUNCT
ejpam-5886	143	10	we	we	PRON
ejpam-5886	143	11	obtain	obtain	VERB
ejpam-5886	143	12	(	(	PUNCT
ejpam-5886	143	13	22	22	NUM
ejpam-5886	143	14	)	)	PUNCT
ejpam-5886	143	15	.	.	PUNCT
ejpam-5886	144	1	theorem	theorem	ADJ
ejpam-5886	144	2	5	5	NUM
ejpam-5886	144	3	.	.	PUNCT
ejpam-5886	144	4	for	for	ADP
ejpam-5886	144	5	n	n	PRON
ejpam-5886	144	6	≥	≥	NOUN
ejpam-5886	144	7	0	0	NUM
ejpam-5886	144	8	,	,	PUNCT
ejpam-5886	144	9	we	we	PRON
ejpam-5886	144	10	have	have	VERB
ejpam-5886	144	11	s[h]n	s[h]n	NOUN
ejpam-5886	144	12	(	(	PUNCT
ejpam-5886	144	13	u	u	NOUN
ejpam-5886	144	14	,	,	PUNCT
ejpam-5886	144	15	v	v	NOUN
ejpam-5886	144	16	)	)	PUNCT
ejpam-5886	144	17	=	=	PUNCT
ejpam-5886	145	1	[	[	X
ejpam-5886	145	2	n	n	X
ejpam-5886	145	3	2	2	NUM
ejpam-5886	145	4	]	]	PUNCT
ejpam-5886	145	5	∑	∑	PUNCT
ejpam-5886	145	6	j=0	j=0	PROPN
ejpam-5886	145	7	(	(	PUNCT
ejpam-5886	145	8	−v	−v	NOUN
ejpam-5886	145	9	h	h	NOUN
ejpam-5886	145	10	)	)	PUNCT
ejpam-5886	145	11	n−2j	n−2j	PROPN
ejpam-5886	145	12	(	(	PUNCT
ejpam-5886	145	13	−h)n−j	−h)n−j	PROPN
ejpam-5886	145	14	(	(	PUNCT
ejpam-5886	145	15	−u	−u	PROPN
ejpam-5886	145	16	h	h	PROPN
ejpam-5886	145	17	)	)	PUNCT
ejpam-5886	145	18	j	j	PROPN
ejpam-5886	145	19	(	(	PUNCT
ejpam-5886	145	20	−1)j	−1)j	X
ejpam-5886	145	21	n	n	CCONJ
ejpam-5886	145	22	!	!	PUNCT
ejpam-5886	146	1	(	(	PUNCT
ejpam-5886	146	2	n−	n−	NOUN
ejpam-5886	146	3	2j)!(j!)2	2j)!(j!)2	NUM
ejpam-5886	146	4	.	.	PUNCT
ejpam-5886	147	1	(	(	PUNCT
ejpam-5886	147	2	24	24	NUM
ejpam-5886	147	3	)	)	PUNCT
ejpam-5886	147	4	proof	proof	NOUN
ejpam-5886	147	5	.	.	PUNCT
ejpam-5886	148	1	from	from	ADP
ejpam-5886	148	2	(	(	PUNCT
ejpam-5886	148	3	13	13	NUM
ejpam-5886	148	4	)	)	PUNCT
ejpam-5886	148	5	,	,	PUNCT
ejpam-5886	148	6	we	we	PRON
ejpam-5886	148	7	have	have	VERB
ejpam-5886	148	8	∞∑	∞∑	NUM
ejpam-5886	148	9	n=0	n=0	ADJ
ejpam-5886	148	10	s[h]n	s[h]n	NOUN
ejpam-5886	148	11	(	(	PUNCT
ejpam-5886	148	12	u	u	NOUN
ejpam-5886	148	13	,	,	PUNCT
ejpam-5886	148	14	v	v	NOUN
ejpam-5886	148	15	)	)	PUNCT
ejpam-5886	148	16	tn	tn	NOUN
ejpam-5886	148	17	n	n	NOUN
ejpam-5886	148	18	!	!	PUNCT
ejpam-5886	149	1	=	=	PUNCT
ejpam-5886	149	2	(	(	PUNCT
ejpam-5886	149	3	1	1	NUM
ejpam-5886	149	4	+	+	NUM
ejpam-5886	149	5	ht	ht	PROPN
ejpam-5886	149	6	)	)	PUNCT
ejpam-5886	149	7	v	v	NOUN
ejpam-5886	149	8	h	h	NOUN
ejpam-5886	149	9	(	(	PUNCT
ejpam-5886	149	10	1	1	NUM
ejpam-5886	149	11	+	+	NUM
ejpam-5886	149	12	ht2	ht2	NOUN
ejpam-5886	149	13	)	)	PUNCT
ejpam-5886	150	1	d−1	d−1	PROPN
ejpam-5886	150	2	u	u	NOUN
ejpam-5886	150	3	h	h	NOUN
ejpam-5886	150	4	=	=	PUNCT
ejpam-5886	150	5	∞∑	∞∑	NUM
ejpam-5886	150	6	n=0	n=0	NUM
ejpam-5886	150	7	(	(	PUNCT
ejpam-5886	150	8	−v	−v	NOUN
ejpam-5886	150	9	h	h	NOUN
ejpam-5886	150	10	)	)	PUNCT
ejpam-5886	150	11	n	n	CCONJ
ejpam-5886	150	12	(	(	PUNCT
ejpam-5886	150	13	−h)n	−h)n	VERB
ejpam-5886	150	14	tn	tn	PROPN
ejpam-5886	150	15	n	n	CCONJ
ejpam-5886	150	16	!	!	PUNCT
ejpam-5886	151	1	∞∑	∞∑	NUM
ejpam-5886	151	2	j=0	j=0	PROPN
ejpam-5886	151	3	(	(	PUNCT
ejpam-5886	151	4	−u	−u	PROPN
ejpam-5886	151	5	h	h	PROPN
ejpam-5886	151	6	)	)	PUNCT
ejpam-5886	151	7	j	j	PROPN
ejpam-5886	151	8	(	(	PUNCT
ejpam-5886	151	9	−1)j(−h)j	−1)j(−h)j	PROPN
ejpam-5886	151	10	t2j	t2j	NUM
ejpam-5886	151	11	j!j	j!j	PROPN
ejpam-5886	151	12	!	!	PUNCT
ejpam-5886	151	13	=	=	NOUN
ejpam-5886	152	1	∞∑	∞∑	PRON
ejpam-5886	152	2	n=0	n=0	NUM
ejpam-5886	153	1	[	[	X
ejpam-5886	153	2	n	n	X
ejpam-5886	153	3	2	2	NUM
ejpam-5886	153	4	]	]	PUNCT
ejpam-5886	153	5	∑	∑	PUNCT
ejpam-5886	153	6	j=0	j=0	PROPN
ejpam-5886	153	7	(	(	PUNCT
ejpam-5886	153	8	−v	−v	NOUN
ejpam-5886	153	9	h	h	NOUN
ejpam-5886	153	10	)	)	PUNCT
ejpam-5886	153	11	n−2j	n−2j	PROPN
ejpam-5886	153	12	(	(	PUNCT
ejpam-5886	153	13	−h)n−j	−h)n−j	PROPN
ejpam-5886	153	14	(	(	PUNCT
ejpam-5886	153	15	−u	−u	PROPN
ejpam-5886	153	16	h	h	PROPN
ejpam-5886	153	17	)	)	PUNCT
ejpam-5886	153	18	j	j	PROPN
ejpam-5886	153	19	(	(	PUNCT
ejpam-5886	153	20	−1)j	−1)j	X
ejpam-5886	153	21	tn	tn	PROPN
ejpam-5886	153	22	(	(	PUNCT
ejpam-5886	153	23	n−	n−	NOUN
ejpam-5886	153	24	2j)!(j!)2	2j)!(j!)2	NUM
ejpam-5886	153	25	.	.	PUNCT
ejpam-5886	154	1	(	(	PUNCT
ejpam-5886	154	2	25	25	NUM
ejpam-5886	154	3	)	)	PUNCT
ejpam-5886	154	4	s.	s.	PROPN
ejpam-5886	154	5	a.	a.	PROPN
ejpam-5886	154	6	wani	wani	PROPN
ejpam-5886	154	7	et	et	PROPN
ejpam-5886	154	8	al	al	PROPN
ejpam-5886	154	9	.	.	PUNCT
ejpam-5886	154	10	/	/	SYM
ejpam-5886	154	11	eur	eur	PROPN
ejpam-5886	154	12	.	.	PUNCT
ejpam-5886	155	1	j.	j.	PROPN
ejpam-5886	155	2	pure	pure	PROPN
ejpam-5886	155	3	appl	appl	PROPN
ejpam-5886	155	4	.	.	PROPN
ejpam-5886	155	5	math	math	PROPN
ejpam-5886	155	6	,	,	PUNCT
ejpam-5886	155	7	18	18	NUM
ejpam-5886	155	8	(	(	PUNCT
ejpam-5886	155	9	2	2	NUM
ejpam-5886	155	10	)	)	PUNCT
ejpam-5886	155	11	(	(	PUNCT
ejpam-5886	155	12	2025	2025	NUM
ejpam-5886	155	13	)	)	PUNCT
ejpam-5886	155	14	,	,	PUNCT
ejpam-5886	155	15	5886	5886	NUM
ejpam-5886	155	16	8	8	NUM
ejpam-5886	155	17	of	of	ADP
ejpam-5886	155	18	16	16	NUM
ejpam-5886	155	19	comparing	compare	VERB
ejpam-5886	155	20	the	the	DET
ejpam-5886	155	21	coefficients	coefficient	NOUN
ejpam-5886	155	22	of	of	ADP
ejpam-5886	155	23	t	t	PROPN
ejpam-5886	155	24	,	,	PUNCT
ejpam-5886	155	25	we	we	PRON
ejpam-5886	155	26	obtain	obtain	VERB
ejpam-5886	155	27	(	(	PUNCT
ejpam-5886	155	28	24	24	NUM
ejpam-5886	155	29	)	)	PUNCT
ejpam-5886	155	30	.	.	PUNCT
ejpam-5886	156	1	now	now	ADV
ejpam-5886	156	2	,	,	PUNCT
ejpam-5886	156	3	we	we	PRON
ejpam-5886	156	4	investigate	investigate	VERB
ejpam-5886	156	5	the	the	DET
ejpam-5886	156	6	connection	connection	NOUN
ejpam-5886	156	7	between	between	ADP
ejpam-5886	156	8	the	the	DET
ejpam-5886	156	9	stirling	stirling	NOUN
ejpam-5886	156	10	numbers	number	NOUN
ejpam-5886	156	11	of	of	ADP
ejpam-5886	156	12	the	the	DET
ejpam-5886	156	13	first	first	ADJ
ejpam-5886	156	14	kind	kind	NOUN
ejpam-5886	156	15	and	and	CCONJ
ejpam-5886	156	16	2	2	NUM
ejpam-5886	156	17	-	-	PUNCT
ejpam-5886	156	18	variable	variable	ADJ
ejpam-5886	156	19	∆h	∆h	PROPN
ejpam-5886	156	20	legendre	legendre	NOUN
ejpam-5886	156	21	polynomials	polynomial	NOUN
ejpam-5886	156	22	.	.	PUNCT
ejpam-5886	157	1	[	[	X
ejpam-5886	157	2	log(1	log(1	NOUN
ejpam-5886	157	3	+	+	X
ejpam-5886	157	4	t)]k	t)]k	ADV
ejpam-5886	157	5	k	k	NOUN
ejpam-5886	157	6	!	!	PUNCT
ejpam-5886	157	7	=	=	PUNCT
ejpam-5886	158	1	∞∑	∞∑	NUM
ejpam-5886	158	2	i	i	PRON
ejpam-5886	158	3	=	=	PROPN
ejpam-5886	158	4	k	k	PROPN
ejpam-5886	158	5	s1(i	s1(i	PROPN
ejpam-5886	158	6	,	,	PUNCT
ejpam-5886	158	7	k	k	NOUN
ejpam-5886	158	8	)	)	PUNCT
ejpam-5886	158	9	ti	ti	NOUN
ejpam-5886	158	10	i	i	PRON
ejpam-5886	158	11	!	!	PUNCT
ejpam-5886	159	1	,	,	PUNCT
ejpam-5886	160	1	|	|	ADV
ejpam-5886	160	2	t	t	NOUN
ejpam-5886	160	3	|	|	ADV
ejpam-5886	160	4	<	<	X
ejpam-5886	160	5	1	1	NUM
ejpam-5886	160	6	.	.	PUNCT
ejpam-5886	161	1	(	(	PUNCT
ejpam-5886	161	2	26	26	NUM
ejpam-5886	161	3	)	)	PUNCT
ejpam-5886	161	4	from	from	ADP
ejpam-5886	161	5	the	the	DET
ejpam-5886	161	6	above	above	ADJ
ejpam-5886	161	7	definition	definition	NOUN
ejpam-5886	161	8	,	,	PUNCT
ejpam-5886	161	9	we	we	PRON
ejpam-5886	161	10	have	have	VERB
ejpam-5886	161	11	(	(	PUNCT
ejpam-5886	161	12	v)i	v)i	X
ejpam-5886	161	13	=	=	SYM
ejpam-5886	161	14	i∑	i∑	ADJ
ejpam-5886	161	15	k=0	k=0	PROPN
ejpam-5886	161	16	(	(	PUNCT
ejpam-5886	161	17	−1)i−ks1(i	−1)i−ks1(i	PROPN
ejpam-5886	161	18	,	,	PUNCT
ejpam-5886	161	19	k)v	k)v	X
ejpam-5886	161	20	k.	k.	PROPN
ejpam-5886	162	1	(	(	PUNCT
ejpam-5886	162	2	27	27	NUM
ejpam-5886	162	3	)	)	PUNCT
ejpam-5886	162	4	theorem	theorem	VERB
ejpam-5886	162	5	6	6	NUM
ejpam-5886	162	6	.	.	PUNCT
ejpam-5886	162	7	for	for	ADP
ejpam-5886	162	8	n	n	PRON
ejpam-5886	162	9	≥	≥	NOUN
ejpam-5886	162	10	0	0	NUM
ejpam-5886	162	11	,	,	PUNCT
ejpam-5886	162	12	we	we	PRON
ejpam-5886	162	13	have	have	AUX
ejpam-5886	162	14	s[h]n	s[h]n	NOUN
ejpam-5886	162	15	(	(	PUNCT
ejpam-5886	162	16	u	u	NOUN
ejpam-5886	162	17	,	,	PUNCT
ejpam-5886	162	18	v	v	NOUN
ejpam-5886	162	19	)	)	PUNCT
ejpam-5886	163	1	=	=	SYM
ejpam-5886	163	2	n∑	n∑	PROPN
ejpam-5886	163	3	m=0	m=0	PROPN
ejpam-5886	163	4	(	(	PUNCT
ejpam-5886	163	5	n	n	NOUN
ejpam-5886	163	6	m	m	PROPN
ejpam-5886	163	7	)	)	PUNCT
ejpam-5886	163	8	s[h]n	s[h]n	PROPN
ejpam-5886	163	9	(	(	PUNCT
ejpam-5886	163	10	u	u	NOUN
ejpam-5886	163	11	,	,	PUNCT
ejpam-5886	163	12	0	0	NUM
ejpam-5886	163	13	)	)	PUNCT
ejpam-5886	163	14	m∑	m∑	CCONJ
ejpam-5886	163	15	j=0	j=0	PROPN
ejpam-5886	163	16	xjs1(m	xjs1(m	NUM
ejpam-5886	163	17	,	,	PUNCT
ejpam-5886	163	18	j)hm−j	j)hm−j	X
ejpam-5886	163	19	.	.	PUNCT
ejpam-5886	164	1	(	(	PUNCT
ejpam-5886	164	2	28	28	NUM
ejpam-5886	164	3	)	)	PUNCT
ejpam-5886	164	4	proof	proof	NOUN
ejpam-5886	164	5	.	.	PUNCT
ejpam-5886	165	1	from	from	ADP
ejpam-5886	165	2	(	(	PUNCT
ejpam-5886	165	3	13	13	NUM
ejpam-5886	165	4	)	)	PUNCT
ejpam-5886	165	5	,	,	PUNCT
ejpam-5886	165	6	we	we	PRON
ejpam-5886	165	7	have	have	VERB
ejpam-5886	165	8	∞∑	∞∑	NUM
ejpam-5886	165	9	n=0	n=0	ADJ
ejpam-5886	165	10	s[h]n	s[h]n	NOUN
ejpam-5886	165	11	(	(	PUNCT
ejpam-5886	165	12	u	u	NOUN
ejpam-5886	165	13	,	,	PUNCT
ejpam-5886	165	14	v	v	NOUN
ejpam-5886	165	15	)	)	PUNCT
ejpam-5886	165	16	tn	tn	NOUN
ejpam-5886	165	17	n	n	NOUN
ejpam-5886	165	18	!	!	PUNCT
ejpam-5886	166	1	=	=	PUNCT
ejpam-5886	167	1	e	e	X
ejpam-5886	167	2	v	v	NUM
ejpam-5886	167	3	h	h	NOUN
ejpam-5886	167	4	log(1+ht)(1	log(1+ht)(1	NOUN
ejpam-5886	167	5	+	+	CCONJ
ejpam-5886	167	6	ht2	ht2	NOUN
ejpam-5886	167	7	)	)	PUNCT
ejpam-5886	168	1	d−1	d−1	PROPN
ejpam-5886	168	2	u	u	NOUN
ejpam-5886	168	3	h	h	NOUN
ejpam-5886	168	4	=	=	PUNCT
ejpam-5886	168	5	(	(	PUNCT
ejpam-5886	168	6	1	1	NUM
ejpam-5886	168	7	+	+	NUM
ejpam-5886	168	8	ht2	ht2	NOUN
ejpam-5886	168	9	)	)	PUNCT
ejpam-5886	169	1	d−1	d−1	PROPN
ejpam-5886	169	2	u	u	PROPN
ejpam-5886	169	3	h	h	NOUN
ejpam-5886	169	4	∞∑	∞∑	PROPN
ejpam-5886	169	5	j=0	j=0	PROPN
ejpam-5886	169	6	(	(	PUNCT
ejpam-5886	169	7	v	v	NOUN
ejpam-5886	169	8	h	h	NOUN
ejpam-5886	169	9	)	)	PUNCT
ejpam-5886	169	10	j	j	NOUN
ejpam-5886	170	1	[	[	X
ejpam-5886	170	2	log(1	log(1	NOUN
ejpam-5886	170	3	+	+	CCONJ
ejpam-5886	170	4	ht)]j	ht)]j	PROPN
ejpam-5886	170	5	j	j	PROPN
ejpam-5886	170	6	!	!	PUNCT
ejpam-5886	170	7	=	=	PUNCT
ejpam-5886	171	1	∞∑	∞∑	DET
ejpam-5886	171	2	n=0	n=0	NUM
ejpam-5886	171	3	s[h]n	s[h]n	NOUN
ejpam-5886	171	4	(	(	PUNCT
ejpam-5886	171	5	u	u	NOUN
ejpam-5886	171	6	,	,	PUNCT
ejpam-5886	171	7	0	0	NUM
ejpam-5886	171	8	)	)	PUNCT
ejpam-5886	171	9	tn	tn	NOUN
ejpam-5886	171	10	n	n	CCONJ
ejpam-5886	171	11	!	!	PUNCT
ejpam-5886	172	1	∞∑	∞∑	PRON
ejpam-5886	172	2	m=0	m=0	PROPN
ejpam-5886	172	3	m∑	m∑	CCONJ
ejpam-5886	172	4	j=0	j=0	PROPN
ejpam-5886	172	5	(	(	PUNCT
ejpam-5886	172	6	v	v	NUM
ejpam-5886	172	7	h	h	NOUN
ejpam-5886	172	8	)	)	PUNCT
ejpam-5886	172	9	j	j	PROPN
ejpam-5886	172	10	s1(m	s1(m	PROPN
ejpam-5886	172	11	,	,	PUNCT
ejpam-5886	172	12	j)hm	j)hm	PROPN
ejpam-5886	172	13	tm	tm	NOUN
ejpam-5886	172	14	m	m	PROPN
ejpam-5886	172	15	!	!	PUNCT
ejpam-5886	172	16	=	=	NOUN
ejpam-5886	173	1	∞∑	∞∑	PRON
ejpam-5886	173	2	n=0	n=0	PUNCT
ejpam-5886	173	3			PROPN
ejpam-5886	173	4	n∑	n∑	PROPN
ejpam-5886	173	5	m=0	m=0	PROPN
ejpam-5886	173	6	(	(	PUNCT
ejpam-5886	173	7	n	n	NOUN
ejpam-5886	173	8	m	m	VERB
ejpam-5886	173	9	)	)	PUNCT
ejpam-5886	173	10	s[h]n−m(u	s[h]n−m(u	PROPN
ejpam-5886	173	11	,	,	PUNCT
ejpam-5886	173	12	0	0	NUM
ejpam-5886	173	13	)	)	PUNCT
ejpam-5886	173	14	m∑	m∑	CCONJ
ejpam-5886	173	15	j=0	j=0	PROPN
ejpam-5886	173	16	(	(	PUNCT
ejpam-5886	173	17	v	v	NUM
ejpam-5886	173	18	h	h	NOUN
ejpam-5886	173	19	)	)	PUNCT
ejpam-5886	173	20	j	j	PROPN
ejpam-5886	173	21	s1(m	s1(m	PROPN
ejpam-5886	173	22	,	,	PUNCT
ejpam-5886	173	23	j)hm	j)hm	PROPN
ejpam-5886	173	24			PROPN
ejpam-5886	173	25	tn	tn	PROPN
ejpam-5886	173	26	n	n	X
ejpam-5886	173	27	!	!	PUNCT
ejpam-5886	173	28	.	.	PUNCT
ejpam-5886	174	1	(	(	PUNCT
ejpam-5886	174	2	29	29	NUM
ejpam-5886	174	3	)	)	PUNCT
ejpam-5886	174	4	comparing	compare	VERB
ejpam-5886	174	5	the	the	DET
ejpam-5886	174	6	coefficients	coefficient	NOUN
ejpam-5886	174	7	of	of	ADP
ejpam-5886	174	8	t	t	PROPN
ejpam-5886	174	9	,	,	PUNCT
ejpam-5886	174	10	we	we	PRON
ejpam-5886	174	11	obtain	obtain	VERB
ejpam-5886	174	12	(	(	PUNCT
ejpam-5886	174	13	28	28	NUM
ejpam-5886	174	14	)	)	PUNCT
ejpam-5886	174	15	.	.	PUNCT
ejpam-5886	175	1	theorem	theorem	VERB
ejpam-5886	175	2	7	7	NUM
ejpam-5886	175	3	.	.	NOUN
ejpam-5886	175	4	for	for	ADP
ejpam-5886	175	5	n	n	PRON
ejpam-5886	175	6	≥	≥	NOUN
ejpam-5886	175	7	0	0	NUM
ejpam-5886	175	8	,	,	PUNCT
ejpam-5886	175	9	we	we	PRON
ejpam-5886	175	10	have	have	VERB
ejpam-5886	175	11	s[h]n	s[h]n	NOUN
ejpam-5886	175	12	(	(	PUNCT
ejpam-5886	175	13	u	u	NOUN
ejpam-5886	175	14	,	,	PUNCT
ejpam-5886	175	15	v	v	NOUN
ejpam-5886	175	16	)	)	PUNCT
ejpam-5886	176	1	=	=	SYM
ejpam-5886	176	2	n∑	n∑	PROPN
ejpam-5886	176	3	l=0	l=0	PROPN
ejpam-5886	176	4	n−l∑	n−l∑	X
ejpam-5886	176	5	m=0	m=0	PROPN
ejpam-5886	176	6	n	n	CCONJ
ejpam-5886	176	7	!	!	PUNCT
ejpam-5886	177	1	(	(	PUNCT
ejpam-5886	177	2	n−m−	n−m−	NUM
ejpam-5886	177	3	l)!(m+	l)!(m+	NOUN
ejpam-5886	177	4	l	l	NOUN
ejpam-5886	177	5	)	)	PUNCT
ejpam-5886	177	6	!	!	PUNCT
ejpam-5886	178	1	hms[h]n−m−1(0	hms[h]n−m−1(0	PROPN
ejpam-5886	178	2	,	,	PUNCT
ejpam-5886	178	3	u)s1(m+	u)s1(m+	VERB
ejpam-5886	178	4	l	l	NOUN
ejpam-5886	178	5	,	,	PUNCT
ejpam-5886	178	6	l)vl	l)vl	PROPN
ejpam-5886	178	7	.	.	PUNCT
ejpam-5886	179	1	(	(	PUNCT
ejpam-5886	179	2	30	30	NUM
ejpam-5886	179	3	)	)	PUNCT
ejpam-5886	179	4	proof	proof	NOUN
ejpam-5886	179	5	.	.	PUNCT
ejpam-5886	180	1	from	from	ADP
ejpam-5886	180	2	(	(	PUNCT
ejpam-5886	180	3	13	13	NUM
ejpam-5886	180	4	)	)	PUNCT
ejpam-5886	180	5	,	,	PUNCT
ejpam-5886	180	6	we	we	PRON
ejpam-5886	180	7	have	have	VERB
ejpam-5886	180	8	∞∑	∞∑	NUM
ejpam-5886	180	9	n=0	n=0	ADJ
ejpam-5886	180	10	s[h]n	s[h]n	NOUN
ejpam-5886	180	11	(	(	PUNCT
ejpam-5886	180	12	u	u	NOUN
ejpam-5886	180	13	,	,	PUNCT
ejpam-5886	180	14	v	v	NOUN
ejpam-5886	180	15	)	)	PUNCT
ejpam-5886	180	16	tn	tn	NOUN
ejpam-5886	180	17	n	n	NOUN
ejpam-5886	180	18	!	!	PUNCT
ejpam-5886	181	1	=	=	PUNCT
ejpam-5886	181	2	(	(	PUNCT
ejpam-5886	181	3	1	1	NUM
ejpam-5886	181	4	+	+	NUM
ejpam-5886	181	5	ht	ht	PROPN
ejpam-5886	181	6	)	)	PUNCT
ejpam-5886	181	7	v	v	NOUN
ejpam-5886	181	8	h	h	NOUN
ejpam-5886	181	9	(	(	PUNCT
ejpam-5886	181	10	1	1	NUM
ejpam-5886	181	11	+	+	NUM
ejpam-5886	181	12	ht2	ht2	NOUN
ejpam-5886	181	13	)	)	PUNCT
ejpam-5886	182	1	d−1	d−1	PROPN
ejpam-5886	182	2	u	u	PROPN
ejpam-5886	182	3	h	h	PROPN
ejpam-5886	182	4	s.	s.	PROPN
ejpam-5886	182	5	a.	a.	PROPN
ejpam-5886	182	6	wani	wani	PROPN
ejpam-5886	182	7	et	et	PROPN
ejpam-5886	182	8	al	al	PROPN
ejpam-5886	182	9	.	.	PUNCT
ejpam-5886	182	10	/	/	SYM
ejpam-5886	182	11	eur	eur	PROPN
ejpam-5886	182	12	.	.	PUNCT
ejpam-5886	183	1	j.	j.	PROPN
ejpam-5886	183	2	pure	pure	PROPN
ejpam-5886	183	3	appl	appl	PROPN
ejpam-5886	183	4	.	.	PROPN
ejpam-5886	183	5	math	math	PROPN
ejpam-5886	183	6	,	,	PUNCT
ejpam-5886	183	7	18	18	NUM
ejpam-5886	183	8	(	(	PUNCT
ejpam-5886	183	9	2	2	NUM
ejpam-5886	183	10	)	)	PUNCT
ejpam-5886	183	11	(	(	PUNCT
ejpam-5886	183	12	2025	2025	NUM
ejpam-5886	183	13	)	)	PUNCT
ejpam-5886	183	14	,	,	PUNCT
ejpam-5886	183	15	5886	5886	NUM
ejpam-5886	183	16	9	9	NUM
ejpam-5886	183	17	of	of	ADP
ejpam-5886	183	18	16	16	NUM
ejpam-5886	183	19	=	=	SYM
ejpam-5886	183	20	∞∑	∞∑	PRON
ejpam-5886	183	21	n=0	n=0	NUM
ejpam-5886	183	22	s[h]n	s[h]n	NOUN
ejpam-5886	183	23	(	(	PUNCT
ejpam-5886	183	24	0	0	NUM
ejpam-5886	183	25	,	,	PUNCT
ejpam-5886	183	26	u	u	NOUN
ejpam-5886	183	27	)	)	PUNCT
ejpam-5886	183	28	tn	tn	PROPN
ejpam-5886	183	29	n	n	CCONJ
ejpam-5886	183	30	!	!	PUNCT
ejpam-5886	184	1	∞∑	∞∑	ADJ
ejpam-5886	184	2	m=0	m=0	PROPN
ejpam-5886	184	3	(	(	PUNCT
ejpam-5886	184	4	−v	−v	NOUN
ejpam-5886	184	5	h	h	NOUN
ejpam-5886	184	6	)	)	PUNCT
ejpam-5886	184	7	m	m	VERB
ejpam-5886	184	8	(	(	PUNCT
ejpam-5886	184	9	−h)m	−h)m	ADJ
ejpam-5886	184	10	tm	tm	PROPN
ejpam-5886	184	11	m	m	PROPN
ejpam-5886	184	12	!	!	PUNCT
ejpam-5886	185	1	=	=	NOUN
ejpam-5886	186	1	∞∑	∞∑	PRON
ejpam-5886	186	2	n=0	n=0	NUM
ejpam-5886	186	3	(	(	PUNCT
ejpam-5886	186	4	n∑	n∑	PROPN
ejpam-5886	186	5	m=0	m=0	PROPN
ejpam-5886	186	6	(	(	PUNCT
ejpam-5886	186	7	n	n	NOUN
ejpam-5886	186	8	m	m	VERB
ejpam-5886	186	9	)	)	PUNCT
ejpam-5886	186	10	(	(	PUNCT
ejpam-5886	186	11	−v	−v	NOUN
ejpam-5886	186	12	h	h	PROPN
ejpam-5886	186	13	)	)	PUNCT
ejpam-5886	186	14	m	m	VERB
ejpam-5886	186	15	(	(	PUNCT
ejpam-5886	186	16	−h)ms[h]n−m(0	−h)ms[h]n−m(0	NOUN
ejpam-5886	186	17	,	,	PUNCT
ejpam-5886	186	18	u	u	NOUN
ejpam-5886	186	19	)	)	PUNCT
ejpam-5886	186	20	)	)	PUNCT
ejpam-5886	186	21	tn	tn	PROPN
ejpam-5886	186	22	n	n	PROPN
ejpam-5886	186	23	!	!	PUNCT
ejpam-5886	186	24	.	.	PUNCT
ejpam-5886	187	1	(	(	PUNCT
ejpam-5886	187	2	31	31	NUM
ejpam-5886	187	3	)	)	PUNCT
ejpam-5886	187	4	comparing	compare	VERB
ejpam-5886	187	5	the	the	DET
ejpam-5886	187	6	coefficients	coefficient	NOUN
ejpam-5886	187	7	of	of	ADP
ejpam-5886	187	8	t	t	PROPN
ejpam-5886	187	9	,	,	PUNCT
ejpam-5886	187	10	we	we	PRON
ejpam-5886	187	11	get	get	VERB
ejpam-5886	187	12	s[h]n	s[h]n	NOUN
ejpam-5886	187	13	(	(	PUNCT
ejpam-5886	187	14	u	u	NOUN
ejpam-5886	187	15	,	,	PUNCT
ejpam-5886	187	16	v	v	NOUN
ejpam-5886	187	17	)	)	PUNCT
ejpam-5886	188	1	=	=	SYM
ejpam-5886	188	2	n∑	n∑	PROPN
ejpam-5886	188	3	m=0	m=0	PROPN
ejpam-5886	188	4	(	(	PUNCT
ejpam-5886	188	5	n	n	NOUN
ejpam-5886	188	6	m	m	VERB
ejpam-5886	188	7	)	)	PUNCT
ejpam-5886	188	8	(	(	PUNCT
ejpam-5886	188	9	−u	−u	PROPN
ejpam-5886	188	10	h	h	NOUN
ejpam-5886	188	11	)	)	PUNCT
ejpam-5886	188	12	m	m	VERB
ejpam-5886	188	13	(	(	PUNCT
ejpam-5886	188	14	−h)ms[h]n−m(0	−h)ms[h]n−m(0	X
ejpam-5886	188	15	,	,	PUNCT
ejpam-5886	188	16	v	v	NOUN
ejpam-5886	188	17	)	)	PUNCT
ejpam-5886	188	18	.	.	PUNCT
ejpam-5886	189	1	(	(	PUNCT
ejpam-5886	189	2	32	32	NUM
ejpam-5886	189	3	)	)	PUNCT
ejpam-5886	189	4	using	use	VERB
ejpam-5886	189	5	the	the	DET
ejpam-5886	189	6	above	above	ADJ
ejpam-5886	189	7	equality	equality	NOUN
ejpam-5886	189	8	(	(	PUNCT
ejpam-5886	189	9	3.5	3.5	NUM
ejpam-5886	189	10	)	)	PUNCT
ejpam-5886	189	11	,	,	PUNCT
ejpam-5886	189	12	we	we	PRON
ejpam-5886	189	13	get	get	VERB
ejpam-5886	189	14	s[h]n	s[h]n	NOUN
ejpam-5886	189	15	(	(	PUNCT
ejpam-5886	189	16	u	u	NOUN
ejpam-5886	189	17	,	,	PUNCT
ejpam-5886	189	18	v	v	NOUN
ejpam-5886	189	19	)	)	PUNCT
ejpam-5886	189	20	=	=	SYM
ejpam-5886	190	1	(	(	PUNCT
ejpam-5886	190	2	n∑	n∑	INTJ
ejpam-5886	190	3	m=0	m=0	PROPN
ejpam-5886	190	4	(	(	PUNCT
ejpam-5886	190	5	n	n	NOUN
ejpam-5886	190	6	m	m	VERB
ejpam-5886	190	7	)	)	PUNCT
ejpam-5886	190	8	(	(	PUNCT
ejpam-5886	190	9	−h)ms[h]n−m(0	−h)ms[h]n−m(0	X
ejpam-5886	190	10	,	,	PUNCT
ejpam-5886	190	11	u	u	NOUN
ejpam-5886	190	12	)	)	PUNCT
ejpam-5886	190	13	)	)	PUNCT
ejpam-5886	190	14	(	(	PUNCT
ejpam-5886	190	15	m∑	m∑	CCONJ
ejpam-5886	190	16	l=0	l=0	PROPN
ejpam-5886	190	17	(	(	PUNCT
ejpam-5886	190	18	−1)m−ls1(m	−1)m−ls1(m	ADJ
ejpam-5886	190	19	,	,	PUNCT
ejpam-5886	190	20	l)(−h)−lvl	l)(−h)−lvl	NOUN
ejpam-5886	190	21	)	)	PUNCT
ejpam-5886	190	22	=	=	PUNCT
ejpam-5886	191	1	n∑	n∑	PROPN
ejpam-5886	191	2	l=0	l=0	PROPN
ejpam-5886	191	3	n∑	n∑	PROPN
ejpam-5886	191	4	m	m	PROPN
ejpam-5886	191	5	=	=	PROPN
ejpam-5886	191	6	l	l	NOUN
ejpam-5886	191	7	n	n	X
ejpam-5886	191	8	!	!	PUNCT
ejpam-5886	192	1	(	(	PUNCT
ejpam-5886	192	2	n−m)!m	n−m)!m	PROPN
ejpam-5886	192	3	!	!	PUNCT
ejpam-5886	193	1	(	(	PUNCT
ejpam-5886	193	2	−h)m−ls[h]n−m(0	−h)m−ls[h]n−m(0	PROPN
ejpam-5886	193	3	,	,	PUNCT
ejpam-5886	193	4	u)(−1)m−ls1(m	u)(−1)m−ls1(m	PROPN
ejpam-5886	193	5	,	,	PUNCT
ejpam-5886	193	6	l)vl	l)vl	PROPN
ejpam-5886	193	7	=	=	SYM
ejpam-5886	193	8	n∑	n∑	PROPN
ejpam-5886	193	9	l=0	l=0	PROPN
ejpam-5886	193	10	n−l∑	n−l∑	X
ejpam-5886	193	11	m=0	m=0	PROPN
ejpam-5886	193	12	n	n	CCONJ
ejpam-5886	193	13	!	!	PUNCT
ejpam-5886	194	1	(	(	PUNCT
ejpam-5886	194	2	n−m−	n−m−	NUM
ejpam-5886	194	3	l)!(m+	l)!(m+	NOUN
ejpam-5886	194	4	l	l	NOUN
ejpam-5886	194	5	)	)	PUNCT
ejpam-5886	194	6	!	!	PUNCT
ejpam-5886	195	1	(	(	PUNCT
ejpam-5886	195	2	−h)ms[h]n−m−1(0	−h)ms[h]n−m−1(0	X
ejpam-5886	195	3	,	,	PUNCT
ejpam-5886	195	4	u)(−1)ms1(m+	u)(−1)ms1(m+	ADJ
ejpam-5886	195	5	l	l	NOUN
ejpam-5886	195	6	,	,	PUNCT
ejpam-5886	195	7	l)vl	l)vl	PROPN
ejpam-5886	195	8	.	.	PUNCT
ejpam-5886	196	1	(	(	PUNCT
ejpam-5886	196	2	33	33	NUM
ejpam-5886	196	3	)	)	PUNCT
ejpam-5886	196	4	the	the	DET
ejpam-5886	196	5	complete	complete	ADJ
ejpam-5886	196	6	proof	proof	NOUN
ejpam-5886	196	7	of	of	ADP
ejpam-5886	196	8	the	the	DET
ejpam-5886	196	9	theorem	theorem	PROPN
ejpam-5886	196	10	.	.	PUNCT
ejpam-5886	196	11	theorem	theorem	ADJ
ejpam-5886	196	12	8	8	NUM
ejpam-5886	196	13	.	.	PUNCT
ejpam-5886	196	14	for	for	ADP
ejpam-5886	196	15	n	n	PRON
ejpam-5886	196	16	≥	≥	NOUN
ejpam-5886	196	17	0	0	NUM
ejpam-5886	196	18	,	,	PUNCT
ejpam-5886	196	19	we	we	PRON
ejpam-5886	196	20	have	have	VERB
ejpam-5886	196	21	s[h]n	s[h]n	NOUN
ejpam-5886	196	22	(	(	PUNCT
ejpam-5886	196	23	v	v	NOUN
ejpam-5886	196	24	+	+	CCONJ
ejpam-5886	196	25	w	w	PROPN
ejpam-5886	196	26	,	,	PUNCT
ejpam-5886	196	27	u	u	NOUN
ejpam-5886	196	28	)	)	PUNCT
ejpam-5886	197	1	=	=	SYM
ejpam-5886	197	2	n∑	n∑	PROPN
ejpam-5886	197	3	l=0	l=0	PROPN
ejpam-5886	197	4	n−l∑	n−l∑	X
ejpam-5886	197	5	m=0	m=0	PROPN
ejpam-5886	197	6	n	n	CCONJ
ejpam-5886	197	7	!	!	PUNCT
ejpam-5886	198	1	(	(	PUNCT
ejpam-5886	198	2	n−m−	n−m−	NUM
ejpam-5886	198	3	l)!(m+	l)!(m+	NOUN
ejpam-5886	198	4	l	l	NOUN
ejpam-5886	198	5	)	)	PUNCT
ejpam-5886	198	6	!	!	PUNCT
ejpam-5886	199	1	hms[h]n−m−l(u	hms[h]n−m−l(u	PROPN
ejpam-5886	199	2	,	,	PUNCT
ejpam-5886	199	3	v)s1(m+	v)s1(m+	ADV
ejpam-5886	199	4	l	l	NOUN
ejpam-5886	199	5	,	,	PUNCT
ejpam-5886	199	6	l)wl	l)wl	PROPN
ejpam-5886	199	7	.	.	PROPN
ejpam-5886	200	1	(	(	PUNCT
ejpam-5886	200	2	34	34	NUM
ejpam-5886	200	3	)	)	PUNCT
ejpam-5886	200	4	proof	proof	NOUN
ejpam-5886	200	5	.	.	PUNCT
ejpam-5886	201	1	taking	take	VERB
ejpam-5886	201	2	v	v	PRON
ejpam-5886	201	3	+	+	CCONJ
ejpam-5886	201	4	w	w	NOUN
ejpam-5886	201	5	instead	instead	ADV
ejpam-5886	201	6	of	of	ADP
ejpam-5886	201	7	u	u	NOUN
ejpam-5886	201	8	in	in	ADP
ejpam-5886	201	9	(	(	PUNCT
ejpam-5886	201	10	13	13	NUM
ejpam-5886	201	11	)	)	PUNCT
ejpam-5886	201	12	,	,	PUNCT
ejpam-5886	201	13	we	we	PRON
ejpam-5886	201	14	have	have	VERB
ejpam-5886	201	15	∞∑	∞∑	NUM
ejpam-5886	201	16	n=0	n=0	ADJ
ejpam-5886	201	17	s[h]n	s[h]n	NOUN
ejpam-5886	201	18	(	(	PUNCT
ejpam-5886	201	19	v	v	NOUN
ejpam-5886	201	20	+	+	CCONJ
ejpam-5886	201	21	w	w	PROPN
ejpam-5886	201	22	,	,	PUNCT
ejpam-5886	201	23	v	v	NOUN
ejpam-5886	201	24	)	)	PUNCT
ejpam-5886	201	25	tn	tn	NOUN
ejpam-5886	201	26	n	n	NOUN
ejpam-5886	201	27	!	!	PUNCT
ejpam-5886	202	1	=	=	PUNCT
ejpam-5886	202	2	(	(	PUNCT
ejpam-5886	202	3	1	1	NUM
ejpam-5886	202	4	+	+	NUM
ejpam-5886	202	5	ht	ht	X
ejpam-5886	202	6	)	)	PUNCT
ejpam-5886	202	7	v+w	v+w	NUM
ejpam-5886	202	8	h	h	NOUN
ejpam-5886	202	9	(	(	PUNCT
ejpam-5886	202	10	1	1	NUM
ejpam-5886	202	11	+	+	NUM
ejpam-5886	202	12	ht2	ht2	NOUN
ejpam-5886	202	13	)	)	PUNCT
ejpam-5886	203	1	d−1	d−1	PROPN
ejpam-5886	203	2	u	u	NOUN
ejpam-5886	203	3	h	h	NOUN
ejpam-5886	203	4	=	=	PUNCT
ejpam-5886	203	5	(	(	PUNCT
ejpam-5886	203	6	∞∑	∞∑	PROPN
ejpam-5886	203	7	n=0	n=0	PROPN
ejpam-5886	203	8	s[h]n	s[h]n	NOUN
ejpam-5886	203	9	(	(	PUNCT
ejpam-5886	203	10	u	u	NOUN
ejpam-5886	203	11	,	,	PUNCT
ejpam-5886	203	12	v	v	NOUN
ejpam-5886	203	13	)	)	PUNCT
ejpam-5886	203	14	tn	tn	PROPN
ejpam-5886	203	15	n	n	CCONJ
ejpam-5886	203	16	!	!	PUNCT
ejpam-5886	203	17	)	)	PUNCT
ejpam-5886	204	1	(	(	PUNCT
ejpam-5886	204	2	∞∑	∞∑	NUM
ejpam-5886	204	3	m=0	m=0	PROPN
ejpam-5886	204	4	(	(	PUNCT
ejpam-5886	204	5	−w	−w	ADV
ejpam-5886	204	6	h	h	NOUN
ejpam-5886	204	7	)	)	PUNCT
ejpam-5886	204	8	m	m	VERB
ejpam-5886	204	9	(	(	PUNCT
ejpam-5886	204	10	−h)m	−h)m	ADJ
ejpam-5886	204	11	tm	tm	PROPN
ejpam-5886	204	12	m	m	PROPN
ejpam-5886	204	13	!	!	PUNCT
ejpam-5886	204	14	)	)	PUNCT
ejpam-5886	204	15	.	.	PUNCT
ejpam-5886	205	1	(	(	PUNCT
ejpam-5886	205	2	35	35	NUM
ejpam-5886	205	3	)	)	PUNCT
ejpam-5886	205	4	using	use	VERB
ejpam-5886	205	5	the	the	DET
ejpam-5886	205	6	cauchy	cauchy	ADJ
ejpam-5886	205	7	rule	rule	NOUN
ejpam-5886	205	8	and	and	CCONJ
ejpam-5886	205	9	after	after	ADP
ejpam-5886	205	10	comparing	compare	VERB
ejpam-5886	205	11	the	the	DET
ejpam-5886	205	12	coefficients	coefficient	NOUN
ejpam-5886	205	13	of	of	ADP
ejpam-5886	205	14	t	t	PROPN
ejpam-5886	205	15	on	on	ADP
ejpam-5886	205	16	both	both	DET
ejpam-5886	205	17	sides	side	NOUN
ejpam-5886	205	18	of	of	ADP
ejpam-5886	205	19	the	the	DET
ejpam-5886	205	20	resulting	result	VERB
ejpam-5886	205	21	equation	equation	NOUN
ejpam-5886	205	22	,	,	PUNCT
ejpam-5886	205	23	we	we	PRON
ejpam-5886	205	24	have	have	VERB
ejpam-5886	205	25	s[h]n	s[h]n	NOUN
ejpam-5886	205	26	(	(	PUNCT
ejpam-5886	205	27	v	v	NOUN
ejpam-5886	205	28	+	+	CCONJ
ejpam-5886	205	29	w	w	PROPN
ejpam-5886	205	30	,	,	PUNCT
ejpam-5886	205	31	u	u	NOUN
ejpam-5886	205	32	)	)	PUNCT
ejpam-5886	206	1	=	=	SYM
ejpam-5886	206	2	n∑	n∑	PROPN
ejpam-5886	206	3	m=0	m=0	PROPN
ejpam-5886	206	4	(	(	PUNCT
ejpam-5886	206	5	n	n	NOUN
ejpam-5886	206	6	m	m	VERB
ejpam-5886	206	7	)	)	PUNCT
ejpam-5886	206	8	(	(	PUNCT
ejpam-5886	206	9	−w	−w	ADV
ejpam-5886	206	10	h	h	NOUN
ejpam-5886	206	11	)	)	PUNCT
ejpam-5886	206	12	m	m	VERB
ejpam-5886	206	13	(	(	PUNCT
ejpam-5886	206	14	−h)ms[h]n−m(u	−h)ms[h]n−m(u	NOUN
ejpam-5886	206	15	,	,	PUNCT
ejpam-5886	206	16	v	v	NOUN
ejpam-5886	206	17	)	)	PUNCT
ejpam-5886	206	18	.	.	PUNCT
ejpam-5886	207	1	(	(	PUNCT
ejpam-5886	207	2	36	36	NUM
ejpam-5886	207	3	)	)	PUNCT
ejpam-5886	207	4	then	then	ADV
ejpam-5886	207	5	using	use	VERB
ejpam-5886	207	6	(	(	PUNCT
ejpam-5886	207	7	25	25	NUM
ejpam-5886	207	8	)	)	PUNCT
ejpam-5886	207	9	for	for	ADP
ejpam-5886	207	10	(	(	PUNCT
ejpam-5886	207	11	−w	−w	ADV
ejpam-5886	207	12	h	h	NOUN
ejpam-5886	207	13	)	)	PUNCT
ejpam-5886	208	1	m	m	VERB
ejpam-5886	208	2	,	,	PUNCT
ejpam-5886	208	3	we	we	PRON
ejpam-5886	208	4	have	have	AUX
ejpam-5886	208	5	obtain	obtain	VERB
ejpam-5886	208	6	(	(	PUNCT
ejpam-5886	208	7	34	34	NUM
ejpam-5886	208	8	)	)	PUNCT
ejpam-5886	208	9	.	.	PUNCT
ejpam-5886	209	1	s.	s.	PROPN
ejpam-5886	209	2	a.	a.	PROPN
ejpam-5886	209	3	wani	wani	PROPN
ejpam-5886	209	4	et	et	PROPN
ejpam-5886	209	5	al	al	PROPN
ejpam-5886	209	6	.	.	PUNCT
ejpam-5886	209	7	/	/	SYM
ejpam-5886	209	8	eur	eur	PROPN
ejpam-5886	209	9	.	.	PUNCT
ejpam-5886	210	1	j.	j.	PROPN
ejpam-5886	210	2	pure	pure	PROPN
ejpam-5886	210	3	appl	appl	PROPN
ejpam-5886	210	4	.	.	PROPN
ejpam-5886	210	5	math	math	PROPN
ejpam-5886	210	6	,	,	PUNCT
ejpam-5886	210	7	18	18	NUM
ejpam-5886	210	8	(	(	PUNCT
ejpam-5886	210	9	2	2	NUM
ejpam-5886	210	10	)	)	PUNCT
ejpam-5886	210	11	(	(	PUNCT
ejpam-5886	210	12	2025	2025	NUM
ejpam-5886	210	13	)	)	PUNCT
ejpam-5886	210	14	,	,	PUNCT
ejpam-5886	210	15	5886	5886	NUM
ejpam-5886	210	16	10	10	NUM
ejpam-5886	210	17	of	of	ADP
ejpam-5886	210	18	16	16	NUM
ejpam-5886	210	19	theorem	theorem	NOUN
ejpam-5886	210	20	9	9	NUM
ejpam-5886	210	21	.	.	PUNCT
ejpam-5886	210	22	for	for	ADP
ejpam-5886	210	23	n	n	PRON
ejpam-5886	210	24	≥	≥	NOUN
ejpam-5886	210	25	0	0	NUM
ejpam-5886	210	26	,	,	PUNCT
ejpam-5886	210	27	we	we	PRON
ejpam-5886	210	28	have	have	VERB
ejpam-5886	210	29	s[h]n	s[h]n	NOUN
ejpam-5886	210	30	(	(	PUNCT
ejpam-5886	210	31	u	u	NOUN
ejpam-5886	210	32	,	,	PUNCT
ejpam-5886	210	33	v	v	NOUN
ejpam-5886	210	34	)	)	PUNCT
ejpam-5886	210	35	=	=	SYM
ejpam-5886	211	1	n∑	n∑	PROPN
ejpam-5886	211	2	m=0	m=0	PROPN
ejpam-5886	211	3	m∑	m∑	CCONJ
ejpam-5886	211	4	j=0	j=0	PROPN
ejpam-5886	211	5	(	(	PUNCT
ejpam-5886	211	6	n	n	NOUN
ejpam-5886	211	7	m	m	VERB
ejpam-5886	211	8	)	)	PUNCT
ejpam-5886	212	1	(	(	PUNCT
ejpam-5886	212	2	z	z	NOUN
ejpam-5886	212	3	−	−	PROPN
ejpam-5886	212	4	v)js1(m	v)js1(m	PROPN
ejpam-5886	212	5	,	,	PUNCT
ejpam-5886	212	6	j)hm−js[h]n−m(0	j)hm−js[h]n−m(0	NOUN
ejpam-5886	212	7	,	,	PUNCT
ejpam-5886	212	8	u	u	NOUN
ejpam-5886	212	9	)	)	PUNCT
ejpam-5886	212	10	.	.	PUNCT
ejpam-5886	213	1	(	(	PUNCT
ejpam-5886	213	2	37	37	NUM
ejpam-5886	213	3	)	)	PUNCT
ejpam-5886	213	4	proof	proof	NOUN
ejpam-5886	213	5	.	.	PUNCT
ejpam-5886	214	1	from	from	ADP
ejpam-5886	214	2	(	(	PUNCT
ejpam-5886	214	3	13	13	NUM
ejpam-5886	214	4	)	)	PUNCT
ejpam-5886	214	5	,	,	PUNCT
ejpam-5886	214	6	we	we	PRON
ejpam-5886	214	7	have	have	VERB
ejpam-5886	214	8	(	(	PUNCT
ejpam-5886	214	9	1	1	NUM
ejpam-5886	214	10	+	+	NUM
ejpam-5886	214	11	ht2	ht2	NOUN
ejpam-5886	214	12	)	)	PUNCT
ejpam-5886	215	1	d−1	d−1	PROPN
ejpam-5886	215	2	u	u	NOUN
ejpam-5886	215	3	h	h	NOUN
ejpam-5886	215	4	=	=	PUNCT
ejpam-5886	215	5	e−	e−	PROPN
ejpam-5886	215	6	v	v	X
ejpam-5886	215	7	h	h	NOUN
ejpam-5886	215	8	log(1+ht	log(1+ht	NOUN
ejpam-5886	215	9	)	)	PUNCT
ejpam-5886	216	1	∞∑	∞∑	DET
ejpam-5886	216	2	n=0	n=0	NUM
ejpam-5886	216	3	s[h]n	s[h]n	NOUN
ejpam-5886	216	4	(	(	PUNCT
ejpam-5886	216	5	u	u	NOUN
ejpam-5886	216	6	,	,	PUNCT
ejpam-5886	216	7	v	v	NOUN
ejpam-5886	216	8	)	)	PUNCT
ejpam-5886	216	9	tn	tn	PROPN
ejpam-5886	216	10	n	n	NUM
ejpam-5886	216	11	!	!	PUNCT
ejpam-5886	216	12	.	.	PUNCT
ejpam-5886	217	1	(	(	PUNCT
ejpam-5886	217	2	38	38	NUM
ejpam-5886	217	3	)	)	PUNCT
ejpam-5886	217	4	replacing	replace	VERB
ejpam-5886	217	5	v	v	NOUN
ejpam-5886	217	6	by	by	ADP
ejpam-5886	217	7	z	z	NOUN
ejpam-5886	217	8	and	and	CCONJ
ejpam-5886	217	9	comparing	compare	VERB
ejpam-5886	217	10	the	the	DET
ejpam-5886	217	11	resulting	result	VERB
ejpam-5886	217	12	equations	equation	NOUN
ejpam-5886	217	13	,	,	PUNCT
ejpam-5886	217	14	we	we	PRON
ejpam-5886	217	15	get	get	VERB
ejpam-5886	217	16	e	e	NOUN
ejpam-5886	217	17	z−v	z−v	NUM
ejpam-5886	217	18	h	h	NOUN
ejpam-5886	217	19	log(1+ht)(1	log(1+ht)(1	NOUN
ejpam-5886	217	20	+	+	CCONJ
ejpam-5886	217	21	ht2	ht2	NOUN
ejpam-5886	217	22	)	)	PUNCT
ejpam-5886	218	1	d−1	d−1	PROPN
ejpam-5886	218	2	u	u	NOUN
ejpam-5886	218	3	h	h	NOUN
ejpam-5886	218	4	=	=	PUNCT
ejpam-5886	219	1	∞∑	∞∑	PRON
ejpam-5886	219	2	n=0	n=0	NUM
ejpam-5886	219	3	s[h]n	s[h]n	NOUN
ejpam-5886	219	4	(	(	PUNCT
ejpam-5886	219	5	u	u	NOUN
ejpam-5886	219	6	,	,	PUNCT
ejpam-5886	219	7	v	v	NOUN
ejpam-5886	219	8	)	)	PUNCT
ejpam-5886	219	9	tn	tn	PROPN
ejpam-5886	219	10	n	n	NUM
ejpam-5886	219	11	!	!	PUNCT
ejpam-5886	219	12	.	.	PUNCT
ejpam-5886	220	1	(	(	PUNCT
ejpam-5886	220	2	39	39	NUM
ejpam-5886	220	3	)	)	PUNCT
ejpam-5886	220	4	finally	finally	ADV
ejpam-5886	220	5	,	,	PUNCT
ejpam-5886	220	6	expanding	expand	VERB
ejpam-5886	220	7	the	the	DET
ejpam-5886	220	8	exponential	exponential	ADJ
ejpam-5886	220	9	function	function	NOUN
ejpam-5886	220	10	and	and	CCONJ
ejpam-5886	220	11	then	then	ADV
ejpam-5886	220	12	comparing	compare	VERB
ejpam-5886	220	13	the	the	DET
ejpam-5886	220	14	coefficients	coefficient	NOUN
ejpam-5886	220	15	of	of	ADP
ejpam-5886	220	16	equal	equal	ADJ
ejpam-5886	220	17	powers	power	NOUN
ejpam-5886	220	18	of	of	ADP
ejpam-5886	220	19	t	t	PROPN
ejpam-5886	220	20	,	,	PUNCT
ejpam-5886	220	21	we	we	PRON
ejpam-5886	220	22	come	come	VERB
ejpam-5886	220	23	to	to	ADP
ejpam-5886	220	24	assertion	assertion	NOUN
ejpam-5886	220	25	(	(	PUNCT
ejpam-5886	220	26	39	39	NUM
ejpam-5886	220	27	)	)	PUNCT
ejpam-5886	220	28	of	of	ADP
ejpam-5886	220	29	theorem	theorem	ADJ
ejpam-5886	220	30	3.8	3.8	NUM
ejpam-5886	220	31	.	.	PUNCT
ejpam-5886	221	1	remark	remark	PROPN
ejpam-5886	221	2	1	1	NUM
ejpam-5886	221	3	.	.	PUNCT
ejpam-5886	222	1	taking	take	VERB
ejpam-5886	222	2	v	v	NOUN
ejpam-5886	222	3	=	=	SYM
ejpam-5886	222	4	0	0	NUM
ejpam-5886	222	5	in	in	ADP
ejpam-5886	222	6	theorem	theorem	NOUN
ejpam-5886	222	7	3.8	3.8	NUM
ejpam-5886	222	8	,	,	PUNCT
ejpam-5886	222	9	we	we	PRON
ejpam-5886	222	10	immediately	immediately	ADV
ejpam-5886	222	11	deduce	deduce	VERB
ejpam-5886	222	12	the	the	DET
ejpam-5886	222	13	following	follow	VERB
ejpam-5886	222	14	consequence	consequence	NOUN
ejpam-5886	222	15	of	of	ADP
ejpam-5886	222	16	theorem	theorem	ADJ
ejpam-5886	222	17	3.8	3.8	NUM
ejpam-5886	222	18	:	:	PUNCT
ejpam-5886	222	19	s[h]n	s[h]n	PROPN
ejpam-5886	222	20	(	(	PUNCT
ejpam-5886	222	21	0	0	NUM
ejpam-5886	222	22	,	,	PUNCT
ejpam-5886	222	23	u	u	NOUN
ejpam-5886	222	24	)	)	PUNCT
ejpam-5886	222	25	=	=	SYM
ejpam-5886	222	26	n∑	n∑	PROPN
ejpam-5886	222	27	m=0	m=0	PROPN
ejpam-5886	222	28	m∑	m∑	CCONJ
ejpam-5886	222	29	j=0	j=0	PROPN
ejpam-5886	222	30	(	(	PUNCT
ejpam-5886	222	31	n	n	X
ejpam-5886	222	32	m	m	PROPN
ejpam-5886	222	33	)	)	PUNCT
ejpam-5886	222	34	zjs1(m	zjs1(m	NOUN
ejpam-5886	222	35	,	,	PUNCT
ejpam-5886	222	36	j)hm−js[h]n−m(0	j)hm−js[h]n−m(0	NOUN
ejpam-5886	222	37	,	,	PUNCT
ejpam-5886	222	38	u	u	NOUN
ejpam-5886	222	39	)	)	PUNCT
ejpam-5886	222	40	.	.	PUNCT
ejpam-5886	223	1	4	4	X
ejpam-5886	223	2	.	.	X
ejpam-5886	223	3	monomiality	monomiality	NOUN
ejpam-5886	223	4	principle	principle	NOUN
ejpam-5886	223	5	the	the	DET
ejpam-5886	223	6	monomiality	monomiality	NOUN
ejpam-5886	223	7	principle	principle	NOUN
ejpam-5886	223	8	serves	serve	VERB
ejpam-5886	223	9	as	as	ADP
ejpam-5886	223	10	a	a	DET
ejpam-5886	223	11	fundamental	fundamental	ADJ
ejpam-5886	223	12	framework	framework	NOUN
ejpam-5886	223	13	for	for	ADP
ejpam-5886	223	14	understanding	understanding	NOUN
ejpam-5886	223	15	and	and	CCONJ
ejpam-5886	223	16	manipulating	manipulate	VERB
ejpam-5886	223	17	polynomial	polynomial	ADJ
ejpam-5886	223	18	expressions	expression	NOUN
ejpam-5886	223	19	.	.	PUNCT
ejpam-5886	224	1	it	it	PRON
ejpam-5886	224	2	states	state	VERB
ejpam-5886	224	3	that	that	SCONJ
ejpam-5886	224	4	any	any	DET
ejpam-5886	224	5	polynomial	polynomial	NOUN
ejpam-5886	224	6	can	can	AUX
ejpam-5886	224	7	be	be	AUX
ejpam-5886	224	8	uniquely	uniquely	ADV
ejpam-5886	224	9	expressed	express	VERB
ejpam-5886	224	10	as	as	ADP
ejpam-5886	224	11	a	a	DET
ejpam-5886	224	12	linear	linear	ADJ
ejpam-5886	224	13	combination	combination	NOUN
ejpam-5886	224	14	of	of	ADP
ejpam-5886	224	15	monomials	monomial	NOUN
ejpam-5886	224	16	,	,	PUNCT
ejpam-5886	224	17	simplifying	simplify	VERB
ejpam-5886	224	18	their	their	PRON
ejpam-5886	224	19	structure	structure	NOUN
ejpam-5886	224	20	and	and	CCONJ
ejpam-5886	224	21	facilitating	facilitate	VERB
ejpam-5886	224	22	mathematical	mathematical	ADJ
ejpam-5886	224	23	analysis	analysis	NOUN
ejpam-5886	224	24	.	.	PUNCT
ejpam-5886	225	1	this	this	DET
ejpam-5886	225	2	principle	principle	ADJ
ejpam-5886	225	3	aids	aid	NOUN
ejpam-5886	225	4	in	in	ADP
ejpam-5886	225	5	extracting	extract	VERB
ejpam-5886	225	6	key	key	ADJ
ejpam-5886	225	7	properties	property	NOUN
ejpam-5886	225	8	such	such	ADJ
ejpam-5886	225	9	as	as	ADP
ejpam-5886	225	10	degree	degree	NOUN
ejpam-5886	225	11	,	,	PUNCT
ejpam-5886	225	12	leading	leading	ADJ
ejpam-5886	225	13	coefficient	coefficient	NOUN
ejpam-5886	225	14	,	,	PUNCT
ejpam-5886	225	15	and	and	CCONJ
ejpam-5886	225	16	roots	root	NOUN
ejpam-5886	225	17	,	,	PUNCT
ejpam-5886	225	18	enabling	enable	VERB
ejpam-5886	225	19	the	the	DET
ejpam-5886	225	20	development	development	NOUN
ejpam-5886	225	21	of	of	ADP
ejpam-5886	225	22	advanced	advanced	ADJ
ejpam-5886	225	23	mathematical	mathematical	ADJ
ejpam-5886	225	24	techniques	technique	NOUN
ejpam-5886	225	25	and	and	CCONJ
ejpam-5886	225	26	algorithms	algorithm	NOUN
ejpam-5886	225	27	.	.	PUNCT
ejpam-5886	226	1	beyond	beyond	ADP
ejpam-5886	226	2	its	its	PRON
ejpam-5886	226	3	theoretical	theoretical	ADJ
ejpam-5886	226	4	significance	significance	NOUN
ejpam-5886	226	5	,	,	PUNCT
ejpam-5886	226	6	the	the	DET
ejpam-5886	226	7	monomiality	monomiality	NOUN
ejpam-5886	226	8	principle	principle	NOUN
ejpam-5886	226	9	plays	play	VERB
ejpam-5886	226	10	a	a	DET
ejpam-5886	226	11	crucial	crucial	ADJ
ejpam-5886	226	12	role	role	NOUN
ejpam-5886	226	13	in	in	ADP
ejpam-5886	226	14	various	various	ADJ
ejpam-5886	226	15	scientific	scientific	ADJ
ejpam-5886	226	16	and	and	CCONJ
ejpam-5886	226	17	engineering	engineering	NOUN
ejpam-5886	226	18	applications	application	NOUN
ejpam-5886	226	19	.	.	PUNCT
ejpam-5886	227	1	in	in	ADP
ejpam-5886	227	2	computational	computational	ADJ
ejpam-5886	227	3	mathematics	mathematic	NOUN
ejpam-5886	227	4	,	,	PUNCT
ejpam-5886	227	5	it	it	PRON
ejpam-5886	227	6	ensures	ensure	VERB
ejpam-5886	227	7	the	the	DET
ejpam-5886	227	8	efficiency	efficiency	NOUN
ejpam-5886	227	9	of	of	ADP
ejpam-5886	227	10	numerical	numerical	ADJ
ejpam-5886	227	11	integration	integration	NOUN
ejpam-5886	227	12	,	,	PUNCT
ejpam-5886	227	13	approximation	approximation	NOUN
ejpam-5886	227	14	,	,	PUNCT
ejpam-5886	227	15	and	and	CCONJ
ejpam-5886	227	16	interpolation	interpolation	NOUN
ejpam-5886	227	17	methods	method	NOUN
ejpam-5886	227	18	.	.	PUNCT
ejpam-5886	228	1	similarly	similarly	ADV
ejpam-5886	228	2	,	,	PUNCT
ejpam-5886	228	3	in	in	ADP
ejpam-5886	228	4	fields	field	NOUN
ejpam-5886	228	5	like	like	ADP
ejpam-5886	228	6	image	image	NOUN
ejpam-5886	228	7	analysis	analysis	NOUN
ejpam-5886	228	8	,	,	PUNCT
ejpam-5886	228	9	control	control	NOUN
ejpam-5886	228	10	theory	theory	NOUN
ejpam-5886	228	11	,	,	PUNCT
ejpam-5886	228	12	and	and	CCONJ
ejpam-5886	228	13	signal	signal	NOUN
ejpam-5886	228	14	processing	processing	NOUN
ejpam-5886	228	15	,	,	PUNCT
ejpam-5886	228	16	it	it	PRON
ejpam-5886	228	17	provides	provide	VERB
ejpam-5886	228	18	a	a	DET
ejpam-5886	228	19	structured	structured	ADJ
ejpam-5886	228	20	approach	approach	NOUN
ejpam-5886	228	21	for	for	ADP
ejpam-5886	228	22	modeling	model	VERB
ejpam-5886	228	23	complex	complex	ADJ
ejpam-5886	228	24	systems	system	NOUN
ejpam-5886	228	25	.	.	PUNCT
ejpam-5886	229	1	its	its	PRON
ejpam-5886	229	2	relevance	relevance	NOUN
ejpam-5886	229	3	extends	extend	VERB
ejpam-5886	229	4	to	to	ADP
ejpam-5886	229	5	physics	physics	PROPN
ejpam-5886	229	6	,	,	PUNCT
ejpam-5886	229	7	where	where	SCONJ
ejpam-5886	229	8	polynomials	polynomial	NOUN
ejpam-5886	229	9	describe	describe	VERB
ejpam-5886	229	10	fundamental	fundamental	ADJ
ejpam-5886	229	11	laws	law	NOUN
ejpam-5886	229	12	and	and	CCONJ
ejpam-5886	229	13	phenomena	phenomenon	NOUN
ejpam-5886	229	14	.	.	PUNCT
ejpam-5886	230	1	the	the	DET
ejpam-5886	230	2	study	study	NOUN
ejpam-5886	230	3	of	of	ADP
ejpam-5886	230	4	hybrid	hybrid	ADJ
ejpam-5886	230	5	special	special	ADJ
ejpam-5886	230	6	polynomials	polynomial	NOUN
ejpam-5886	230	7	has	have	AUX
ejpam-5886	230	8	further	far	ADV
ejpam-5886	230	9	explored	explore	VERB
ejpam-5886	230	10	monomiality	monomiality	NOUN
ejpam-5886	230	11	and	and	CCONJ
ejpam-5886	230	12	its	its	PRON
ejpam-5886	230	13	operational	operational	ADJ
ejpam-5886	230	14	principles	principle	NOUN
ejpam-5886	230	15	.	.	PUNCT
ejpam-5886	231	1	originally	originally	ADV
ejpam-5886	231	2	introduced	introduce	VERB
ejpam-5886	231	3	by	by	ADP
ejpam-5886	231	4	steffenson	steffenson	NOUN
ejpam-5886	231	5	in	in	ADP
ejpam-5886	231	6	1941	1941	NUM
ejpam-5886	231	7	with	with	ADP
ejpam-5886	231	8	the	the	DET
ejpam-5886	231	9	concept	concept	NOUN
ejpam-5886	231	10	of	of	ADP
ejpam-5886	231	11	poweroids	poweroids	PROPN
ejpam-5886	231	12	,	,	PUNCT
ejpam-5886	231	13	the	the	DET
ejpam-5886	231	14	principle	principle	NOUN
ejpam-5886	231	15	was	be	AUX
ejpam-5886	231	16	later	later	ADV
ejpam-5886	231	17	expanded	expand	VERB
ejpam-5886	231	18	by	by	ADP
ejpam-5886	231	19	dattoli	dattoli	NOUN
ejpam-5886	231	20	.	.	PUNCT
ejpam-5886	232	1	in	in	ADP
ejpam-5886	232	2	this	this	DET
ejpam-5886	232	3	framework	framework	NOUN
ejpam-5886	232	4	,	,	PUNCT
ejpam-5886	232	5	multiplicative	multiplicative	ADJ
ejpam-5886	232	6	and	and	CCONJ
ejpam-5886	232	7	derivative	derivative	ADJ
ejpam-5886	232	8	operators	operator	NOUN
ejpam-5886	232	9	ĵ	ĵ	PROPN
ejpam-5886	232	10	and	and	CCONJ
ejpam-5886	232	11	k̂	k̂	X
ejpam-5886	232	12	play	play	VERB
ejpam-5886	232	13	a	a	DET
ejpam-5886	232	14	crucial	crucial	ADJ
ejpam-5886	232	15	role	role	NOUN
ejpam-5886	232	16	in	in	ADP
ejpam-5886	232	17	defining	define	VERB
ejpam-5886	232	18	polynomial	polynomial	ADJ
ejpam-5886	232	19	sequences	sequence	NOUN
ejpam-5886	232	20	{	{	PUNCT
ejpam-5886	232	21	gk(u1)}k∈n	gk(u1)}k∈n	ADV
ejpam-5886	232	22	,	,	PUNCT
ejpam-5886	232	23	reinforcing	reinforce	VERB
ejpam-5886	232	24	its	its	PRON
ejpam-5886	232	25	mathematical	mathematical	ADJ
ejpam-5886	232	26	and	and	CCONJ
ejpam-5886	232	27	practical	practical	ADJ
ejpam-5886	232	28	significance	significance	NOUN
ejpam-5886	232	29	.	.	PUNCT
ejpam-5886	233	1	s.	s.	PROPN
ejpam-5886	233	2	a.	a.	PROPN
ejpam-5886	233	3	wani	wani	PROPN
ejpam-5886	233	4	et	et	PROPN
ejpam-5886	233	5	al	al	PROPN
ejpam-5886	233	6	.	.	PUNCT
ejpam-5886	233	7	/	/	SYM
ejpam-5886	233	8	eur	eur	PROPN
ejpam-5886	233	9	.	.	PUNCT
ejpam-5886	234	1	j.	j.	PROPN
ejpam-5886	234	2	pure	pure	PROPN
ejpam-5886	234	3	appl	appl	PROPN
ejpam-5886	234	4	.	.	PROPN
ejpam-5886	234	5	math	math	PROPN
ejpam-5886	234	6	,	,	PUNCT
ejpam-5886	234	7	18	18	NUM
ejpam-5886	234	8	(	(	PUNCT
ejpam-5886	234	9	2	2	NUM
ejpam-5886	234	10	)	)	PUNCT
ejpam-5886	234	11	(	(	PUNCT
ejpam-5886	234	12	2025	2025	NUM
ejpam-5886	234	13	)	)	PUNCT
ejpam-5886	234	14	,	,	PUNCT
ejpam-5886	234	15	5886	5886	NUM
ejpam-5886	234	16	11	11	NUM
ejpam-5886	234	17	of	of	ADP
ejpam-5886	234	18	16	16	NUM
ejpam-5886	234	19	the	the	DET
ejpam-5886	234	20	following	follow	VERB
ejpam-5886	234	21	expressions	expression	NOUN
ejpam-5886	234	22	are	be	AUX
ejpam-5886	234	23	satisfied	satisfied	ADJ
ejpam-5886	234	24	by	by	ADP
ejpam-5886	234	25	these	these	DET
ejpam-5886	234	26	operators	operator	NOUN
ejpam-5886	234	27	:	:	PUNCT
ejpam-5886	234	28	gk+1(u1	gk+1(u1	X
ejpam-5886	234	29	)	)	PUNCT
ejpam-5886	235	1	=	=	VERB
ejpam-5886	235	2	ĵ	ĵ	X
ejpam-5886	235	3	{	{	PUNCT
ejpam-5886	235	4	gk(u1	gk(u1	NOUN
ejpam-5886	235	5	)	)	PUNCT
ejpam-5886	235	6	}	}	PUNCT
ejpam-5886	235	7	(	(	PUNCT
ejpam-5886	235	8	40	40	NUM
ejpam-5886	235	9	)	)	PUNCT
ejpam-5886	235	10	and	and	CCONJ
ejpam-5886	235	11	k	k	PROPN
ejpam-5886	235	12	gk−1(u1	gk−1(u1	PROPN
ejpam-5886	235	13	)	)	PUNCT
ejpam-5886	236	1	=	=	SYM
ejpam-5886	236	2	k̂{gk(u1	k̂{gk(u1	NOUN
ejpam-5886	236	3	)	)	PUNCT
ejpam-5886	236	4	}	}	PUNCT
ejpam-5886	236	5	.	.	PUNCT
ejpam-5886	237	1	(	(	PUNCT
ejpam-5886	237	2	41	41	NUM
ejpam-5886	237	3	)	)	PUNCT
ejpam-5886	237	4	therefore	therefore	ADV
ejpam-5886	237	5	,	,	PUNCT
ejpam-5886	237	6	a	a	DET
ejpam-5886	237	7	quasi	quasi	ADJ
ejpam-5886	237	8	-	-	ADJ
ejpam-5886	237	9	monomial	monomial	ADJ
ejpam-5886	237	10	domain	domain	NOUN
ejpam-5886	237	11	is	be	AUX
ejpam-5886	237	12	produced	produce	VERB
ejpam-5886	237	13	when	when	SCONJ
ejpam-5886	237	14	the	the	DET
ejpam-5886	237	15	polynomial	polynomial	ADJ
ejpam-5886	237	16	set	set	NOUN
ejpam-5886	237	17	gk(u1)m∈n	gk(u1)m∈n	PROPN
ejpam-5886	237	18	is	be	AUX
ejpam-5886	237	19	subjected	subject	VERB
ejpam-5886	237	20	to	to	ADP
ejpam-5886	237	21	multiplicative	multiplicative	ADJ
ejpam-5886	237	22	and	and	CCONJ
ejpam-5886	237	23	derivative	derivative	ADJ
ejpam-5886	237	24	operations	operation	NOUN
ejpam-5886	237	25	.	.	PUNCT
ejpam-5886	238	1	for	for	ADP
ejpam-5886	238	2	this	this	DET
ejpam-5886	238	3	quasi	quasi	NOUN
ejpam-5886	238	4	-	-	NOUN
ejpam-5886	238	5	monomial	monomial	ADJ
ejpam-5886	238	6	,	,	PUNCT
ejpam-5886	238	7	it	it	PRON
ejpam-5886	238	8	is	be	AUX
ejpam-5886	238	9	essential	essential	ADJ
ejpam-5886	238	10	to	to	PART
ejpam-5886	238	11	adhere	adhere	VERB
ejpam-5886	238	12	to	to	ADP
ejpam-5886	238	13	the	the	DET
ejpam-5886	238	14	following	follow	VERB
ejpam-5886	238	15	formula	formula	NOUN
ejpam-5886	238	16	:	:	PUNCT
ejpam-5886	239	1	[	[	X
ejpam-5886	239	2	k̂	k̂	X
ejpam-5886	239	3	,	,	PUNCT
ejpam-5886	239	4	ĵ	ĵ	X
ejpam-5886	239	5	]	]	PUNCT
ejpam-5886	239	6	=	=	PUNCT
ejpam-5886	239	7	k̂ĵ	k̂ĵ	X
ejpam-5886	239	8	−	−	PROPN
ejpam-5886	239	9	ĵ	ĵ	X
ejpam-5886	239	10	k̂	k̂	PROPN
ejpam-5886	240	1	=	=	SYM
ejpam-5886	240	2	1̂.	1̂.	NUM
ejpam-5886	240	3	(	(	PUNCT
ejpam-5886	240	4	42	42	NUM
ejpam-5886	240	5	)	)	PUNCT
ejpam-5886	240	6	it	it	PRON
ejpam-5886	240	7	consequently	consequently	ADV
ejpam-5886	240	8	displays	display	VERB
ejpam-5886	240	9	a	a	DET
ejpam-5886	240	10	weyl	weyl	VERB
ejpam-5886	240	11	group	group	NOUN
ejpam-5886	240	12	structure	structure	NOUN
ejpam-5886	240	13	.	.	PUNCT
ejpam-5886	241	1	it	it	PRON
ejpam-5886	241	2	is	be	AUX
ejpam-5886	241	3	possible	possible	ADJ
ejpam-5886	241	4	to	to	PART
ejpam-5886	241	5	determine	determine	VERB
ejpam-5886	241	6	the	the	DET
ejpam-5886	241	7	significance	significance	NOUN
ejpam-5886	241	8	of	of	ADP
ejpam-5886	241	9	the	the	DET
ejpam-5886	241	10	underlined	underline	VERB
ejpam-5886	241	11	set	set	VERB
ejpam-5886	241	12	by	by	ADP
ejpam-5886	241	13	using	use	VERB
ejpam-5886	241	14	the	the	DET
ejpam-5886	241	15	significance	significance	NOUN
ejpam-5886	241	16	and	and	CCONJ
ejpam-5886	241	17	application	application	NOUN
ejpam-5886	241	18	of	of	ADP
ejpam-5886	241	19	the	the	DET
ejpam-5886	241	20	operators	operator	NOUN
ejpam-5886	241	21	ĵ	ĵ	PROPN
ejpam-5886	241	22	and	and	CCONJ
ejpam-5886	241	23	k̂	k̂	NUM
ejpam-5886	241	24	,	,	PUNCT
ejpam-5886	241	25	assuming	assume	VERB
ejpam-5886	241	26	that	that	SCONJ
ejpam-5886	241	27	the	the	DET
ejpam-5886	241	28	set	set	NOUN
ejpam-5886	241	29	{	{	PUNCT
ejpam-5886	241	30	gk(u1)}k∈n	gk(u1)}k∈n	ADV
ejpam-5886	241	31	is	be	AUX
ejpam-5886	241	32	quasimonomial	quasimonomial	ADJ
ejpam-5886	241	33	.	.	PUNCT
ejpam-5886	242	1	as	as	ADP
ejpam-5886	242	2	a	a	DET
ejpam-5886	242	3	result	result	NOUN
ejpam-5886	242	4	,	,	PUNCT
ejpam-5886	242	5	the	the	DET
ejpam-5886	242	6	following	following	ADJ
ejpam-5886	242	7	axioms	axiom	NOUN
ejpam-5886	242	8	hold	hold	VERB
ejpam-5886	242	9	:	:	PUNCT
ejpam-5886	242	10	(	(	PUNCT
ejpam-5886	242	11	i	i	NOUN
ejpam-5886	242	12	)	)	PUNCT
ejpam-5886	242	13	gk(u1	gk(u1	NOUN
ejpam-5886	242	14	)	)	PUNCT
ejpam-5886	242	15	gives	give	VERB
ejpam-5886	242	16	differential	differential	ADJ
ejpam-5886	242	17	equation	equation	NOUN
ejpam-5886	242	18	ĵ	ĵ	SCONJ
ejpam-5886	242	19	k̂{gk(u1	k̂{gk(u1	NOUN
ejpam-5886	242	20	)	)	PUNCT
ejpam-5886	242	21	}	}	PUNCT
ejpam-5886	243	1	=	=	SYM
ejpam-5886	243	2	k	k	X
ejpam-5886	243	3	gk(u1	gk(u1	NOUN
ejpam-5886	243	4	)	)	PUNCT
ejpam-5886	243	5	,	,	PUNCT
ejpam-5886	243	6	(	(	PUNCT
ejpam-5886	243	7	43	43	NUM
ejpam-5886	243	8	)	)	PUNCT
ejpam-5886	243	9	provided	provide	VERB
ejpam-5886	243	10	ĵ	ĵ	PROPN
ejpam-5886	243	11	and	and	CCONJ
ejpam-5886	243	12	k̂	k̂	PROPN
ejpam-5886	243	13	exhibits	exhibit	VERB
ejpam-5886	243	14	differential	differential	ADJ
ejpam-5886	243	15	traits	trait	NOUN
ejpam-5886	243	16	.	.	PUNCT
ejpam-5886	244	1	(	(	PUNCT
ejpam-5886	244	2	ii	ii	NOUN
ejpam-5886	244	3	)	)	PUNCT
ejpam-5886	244	4	the	the	DET
ejpam-5886	244	5	expression	expression	NOUN
ejpam-5886	244	6	gk(u1	gk(u1	NOUN
ejpam-5886	244	7	)	)	PUNCT
ejpam-5886	244	8	=	=	SYM
ejpam-5886	245	1	ĵ	ĵ	PROPN
ejpam-5886	245	2	k	k	X
ejpam-5886	245	3	{	{	PUNCT
ejpam-5886	245	4	1	1	NUM
ejpam-5886	245	5	}	}	PUNCT
ejpam-5886	245	6	,	,	PUNCT
ejpam-5886	245	7	(	(	PUNCT
ejpam-5886	245	8	44	44	NUM
ejpam-5886	245	9	)	)	PUNCT
ejpam-5886	245	10	gives	give	VERB
ejpam-5886	245	11	the	the	DET
ejpam-5886	245	12	explicit	explicit	ADJ
ejpam-5886	245	13	form	form	NOUN
ejpam-5886	245	14	,	,	PUNCT
ejpam-5886	245	15	with	with	ADP
ejpam-5886	245	16	g0(u1	g0(u1	NOUN
ejpam-5886	245	17	)	)	PUNCT
ejpam-5886	245	18	=	=	SYM
ejpam-5886	245	19	1	1	X
ejpam-5886	245	20	.	.	PUNCT
ejpam-5886	245	21	(	(	PUNCT
ejpam-5886	245	22	iii	iii	NOUN
ejpam-5886	245	23	)	)	PUNCT
ejpam-5886	245	24	further	far	ADV
ejpam-5886	245	25	,	,	PUNCT
ejpam-5886	245	26	the	the	DET
ejpam-5886	245	27	expression	expression	NOUN
ejpam-5886	245	28	ewĵ	ewĵ	PROPN
ejpam-5886	245	29	{	{	PUNCT
ejpam-5886	245	30	1	1	NUM
ejpam-5886	245	31	}	}	PUNCT
ejpam-5886	245	32	=	=	NOUN
ejpam-5886	246	1	∞∑	∞∑	NUM
ejpam-5886	246	2	k=0	k=0	PUNCT
ejpam-5886	246	3	gk(u1	gk(u1	X
ejpam-5886	246	4	)	)	PUNCT
ejpam-5886	246	5	wk	wk	X
ejpam-5886	247	1	k	k	X
ejpam-5886	247	2	!	!	PROPN
ejpam-5886	247	3	,	,	PUNCT
ejpam-5886	247	4	|w|	|w|	VERB
ejpam-5886	247	5	<	<	X
ejpam-5886	247	6	∞	∞	PROPN
ejpam-5886	247	7	,	,	PUNCT
ejpam-5886	247	8	(	(	PUNCT
ejpam-5886	247	9	45	45	NUM
ejpam-5886	247	10	)	)	PUNCT
ejpam-5886	247	11	demonstrates	demonstrate	VERB
ejpam-5886	247	12	generating	generate	VERB
ejpam-5886	247	13	expression	expression	NOUN
ejpam-5886	247	14	behavior	behavior	NOUN
ejpam-5886	247	15	and	and	CCONJ
ejpam-5886	247	16	is	be	AUX
ejpam-5886	247	17	obtained	obtain	VERB
ejpam-5886	247	18	by	by	ADP
ejpam-5886	247	19	applying	apply	VERB
ejpam-5886	247	20	identity	identity	NOUN
ejpam-5886	247	21	(	(	PUNCT
ejpam-5886	247	22	44	44	NUM
ejpam-5886	247	23	)	)	PUNCT
ejpam-5886	247	24	.	.	PUNCT
ejpam-5886	248	1	these	these	DET
ejpam-5886	248	2	methods	method	NOUN
ejpam-5886	248	3	,	,	PUNCT
ejpam-5886	248	4	rooted	root	VERB
ejpam-5886	248	5	in	in	ADP
ejpam-5886	248	6	quantum	quantum	ADJ
ejpam-5886	248	7	mechanics	mechanic	NOUN
ejpam-5886	248	8	,	,	PUNCT
ejpam-5886	248	9	mathematical	mathematical	ADJ
ejpam-5886	248	10	physics	physics	NOUN
ejpam-5886	248	11	,	,	PUNCT
ejpam-5886	248	12	and	and	CCONJ
ejpam-5886	248	13	classical	classical	ADJ
ejpam-5886	248	14	optics	optic	NOUN
ejpam-5886	248	15	,	,	PUNCT
ejpam-5886	248	16	remain	remain	VERB
ejpam-5886	248	17	highly	highly	ADV
ejpam-5886	248	18	relevant	relevant	ADJ
ejpam-5886	248	19	in	in	ADP
ejpam-5886	248	20	modern	modern	ADJ
ejpam-5886	248	21	research	research	NOUN
ejpam-5886	248	22	.	.	PUNCT
ejpam-5886	249	1	they	they	PRON
ejpam-5886	249	2	serve	serve	VERB
ejpam-5886	249	3	as	as	ADP
ejpam-5886	249	4	reliable	reliable	ADJ
ejpam-5886	249	5	tools	tool	NOUN
ejpam-5886	249	6	for	for	ADP
ejpam-5886	249	7	analyzing	analyze	VERB
ejpam-5886	249	8	complex	complex	ADJ
ejpam-5886	249	9	phenomena	phenomenon	NOUN
ejpam-5886	249	10	across	across	ADP
ejpam-5886	249	11	various	various	ADJ
ejpam-5886	249	12	fields	field	NOUN
ejpam-5886	249	13	,	,	PUNCT
ejpam-5886	249	14	reinforcing	reinforce	VERB
ejpam-5886	249	15	our	our	PRON
ejpam-5886	249	16	understanding	understanding	NOUN
ejpam-5886	249	17	of	of	ADP
ejpam-5886	249	18	intricate	intricate	ADJ
ejpam-5886	249	19	systems	system	NOUN
ejpam-5886	249	20	.	.	PUNCT
ejpam-5886	250	1	recognizing	recognize	VERB
ejpam-5886	250	2	their	their	PRON
ejpam-5886	250	3	significance	significance	NOUN
ejpam-5886	250	4	,	,	PUNCT
ejpam-5886	250	5	we	we	PRON
ejpam-5886	250	6	validate	validate	VERB
ejpam-5886	250	7	the	the	DET
ejpam-5886	250	8	concept	concept	NOUN
ejpam-5886	250	9	of	of	ADP
ejpam-5886	250	10	monomiality	monomiality	NOUN
ejpam-5886	250	11	for	for	ADP
ejpam-5886	250	12	the	the	DET
ejpam-5886	250	13	∆h	∆h	PROPN
ejpam-5886	250	14	legendre	legendre	NOUN
ejpam-5886	250	15	polynomials	polynomial	NOUN
ejpam-5886	250	16	,	,	PUNCT
ejpam-5886	250	17	denoted	denote	VERB
ejpam-5886	250	18	as	as	ADP
ejpam-5886	250	19	s[h]n	s[h]n	PROPN
ejpam-5886	250	20	(	(	PUNCT
ejpam-5886	250	21	u	u	NOUN
ejpam-5886	250	22	,	,	PUNCT
ejpam-5886	250	23	v	v	NOUN
ejpam-5886	250	24	)	)	PUNCT
ejpam-5886	250	25	.	.	PUNCT
ejpam-5886	251	1	these	these	DET
ejpam-5886	251	2	polynomials	polynomial	NOUN
ejpam-5886	251	3	form	form	VERB
ejpam-5886	251	4	a	a	DET
ejpam-5886	251	5	crucial	crucial	ADJ
ejpam-5886	251	6	mathematical	mathematical	ADJ
ejpam-5886	251	7	framework	framework	NOUN
ejpam-5886	251	8	for	for	ADP
ejpam-5886	251	9	modeling	model	VERB
ejpam-5886	251	10	diverse	diverse	ADJ
ejpam-5886	251	11	phenomena	phenomenon	NOUN
ejpam-5886	251	12	.	.	PUNCT
ejpam-5886	252	1	by	by	ADP
ejpam-5886	252	2	establishing	establish	VERB
ejpam-5886	252	3	their	their	PRON
ejpam-5886	252	4	monomiality	monomiality	NOUN
ejpam-5886	252	5	,	,	PUNCT
ejpam-5886	252	6	we	we	PRON
ejpam-5886	252	7	aim	aim	VERB
ejpam-5886	252	8	to	to	PART
ejpam-5886	252	9	highlight	highlight	VERB
ejpam-5886	252	10	their	their	PRON
ejpam-5886	252	11	fundamental	fundamental	ADJ
ejpam-5886	252	12	properties	property	NOUN
ejpam-5886	252	13	and	and	CCONJ
ejpam-5886	252	14	applications	application	NOUN
ejpam-5886	252	15	.	.	PUNCT
ejpam-5886	253	1	this	this	DET
ejpam-5886	253	2	section	section	NOUN
ejpam-5886	253	3	presents	present	VERB
ejpam-5886	253	4	our	our	PRON
ejpam-5886	253	5	validation	validation	NOUN
ejpam-5886	253	6	results	result	NOUN
ejpam-5886	253	7	,	,	PUNCT
ejpam-5886	253	8	confirming	confirm	VERB
ejpam-5886	253	9	the	the	DET
ejpam-5886	253	10	integrity	integrity	NOUN
ejpam-5886	253	11	and	and	CCONJ
ejpam-5886	253	12	utility	utility	NOUN
ejpam-5886	253	13	of	of	ADP
ejpam-5886	253	14	∆h	∆h	PROPN
ejpam-5886	253	15	legendre	legendre	NOUN
ejpam-5886	253	16	polynomials	polynomial	NOUN
ejpam-5886	253	17	.	.	PUNCT
ejpam-5886	254	1	through	through	ADP
ejpam-5886	254	2	rigorous	rigorous	ADJ
ejpam-5886	254	3	analysis	analysis	NOUN
ejpam-5886	254	4	,	,	PUNCT
ejpam-5886	254	5	we	we	PRON
ejpam-5886	254	6	affirm	affirm	VERB
ejpam-5886	254	7	their	their	PRON
ejpam-5886	254	8	relevance	relevance	NOUN
ejpam-5886	254	9	for	for	ADP
ejpam-5886	254	10	both	both	CCONJ
ejpam-5886	254	11	theoretical	theoretical	ADJ
ejpam-5886	254	12	and	and	CCONJ
ejpam-5886	254	13	applied	applied	ADJ
ejpam-5886	254	14	research	research	NOUN
ejpam-5886	254	15	,	,	PUNCT
ejpam-5886	254	16	thereby	thereby	ADV
ejpam-5886	254	17	substantiating	substantiate	VERB
ejpam-5886	254	18	the	the	DET
ejpam-5886	254	19	monomiality	monomiality	NOUN
ejpam-5886	254	20	principle	principle	NOUN
ejpam-5886	254	21	for	for	ADP
ejpam-5886	254	22	s[h]n	s[h]n	PROPN
ejpam-5886	254	23	(	(	PUNCT
ejpam-5886	254	24	u	u	NOUN
ejpam-5886	254	25	,	,	PUNCT
ejpam-5886	254	26	v	v	NOUN
ejpam-5886	254	27	)	)	PUNCT
ejpam-5886	254	28	.	.	PUNCT
ejpam-5886	255	1	s.	s.	PROPN
ejpam-5886	255	2	a.	a.	PROPN
ejpam-5886	255	3	wani	wani	PROPN
ejpam-5886	255	4	et	et	PROPN
ejpam-5886	255	5	al	al	PROPN
ejpam-5886	255	6	.	.	PUNCT
ejpam-5886	255	7	/	/	SYM
ejpam-5886	255	8	eur	eur	PROPN
ejpam-5886	255	9	.	.	PUNCT
ejpam-5886	256	1	j.	j.	PROPN
ejpam-5886	256	2	pure	pure	PROPN
ejpam-5886	256	3	appl	appl	PROPN
ejpam-5886	256	4	.	.	PROPN
ejpam-5886	256	5	math	math	PROPN
ejpam-5886	256	6	,	,	PUNCT
ejpam-5886	256	7	18	18	NUM
ejpam-5886	256	8	(	(	PUNCT
ejpam-5886	256	9	2	2	NUM
ejpam-5886	256	10	)	)	PUNCT
ejpam-5886	256	11	(	(	PUNCT
ejpam-5886	256	12	2025	2025	NUM
ejpam-5886	256	13	)	)	PUNCT
ejpam-5886	256	14	,	,	PUNCT
ejpam-5886	256	15	5886	5886	NUM
ejpam-5886	256	16	12	12	NUM
ejpam-5886	256	17	of	of	ADP
ejpam-5886	256	18	16	16	NUM
ejpam-5886	256	19	theorem	theorem	VERB
ejpam-5886	256	20	10	10	NUM
ejpam-5886	256	21	.	.	PUNCT
ejpam-5886	257	1	the	the	DET
ejpam-5886	257	2	∆h	∆h	PROPN
ejpam-5886	257	3	legendre	legendre	PROPN
ejpam-5886	257	4	polynomials	polynomial	NOUN
ejpam-5886	257	5	s[h]n	s[h]n	PROPN
ejpam-5886	257	6	(	(	PUNCT
ejpam-5886	257	7	u	u	NOUN
ejpam-5886	257	8	,	,	PUNCT
ejpam-5886	257	9	v	v	NOUN
ejpam-5886	257	10	)	)	PUNCT
ejpam-5886	257	11	satisfy	satisfy	NOUN
ejpam-5886	257	12	the	the	DET
ejpam-5886	257	13	succeeding	succeed	VERB
ejpam-5886	257	14	multiplicative	multiplicative	ADJ
ejpam-5886	257	15	and	and	CCONJ
ejpam-5886	257	16	derivative	derivative	ADJ
ejpam-5886	257	17	operators	operator	NOUN
ejpam-5886	257	18	:	:	PUNCT
ejpam-5886	258	1	m̂sa	m̂sa	PROPN
ejpam-5886	258	2	=	=	PRON
ejpam-5886	258	3	(	(	PUNCT
ejpam-5886	258	4	v	v	NOUN
ejpam-5886	258	5	1	1	NUM
ejpam-5886	258	6	+	+	CCONJ
ejpam-5886	258	7	v∆h	v∆h	ADJ
ejpam-5886	258	8	+	+	CCONJ
ejpam-5886	258	9	2	2	NUM
ejpam-5886	258	10	d−1	d−1	PROPN
ejpam-5886	258	11	u	u	PROPN
ejpam-5886	258	12	v∆h	v∆h	PROPN
ejpam-5886	258	13	h+	h+	X
ejpam-5886	258	14	v∆h	v∆h	PROPN
ejpam-5886	258	15	2	2	NUM
ejpam-5886	258	16	)	)	PUNCT
ejpam-5886	258	17	(	(	PUNCT
ejpam-5886	258	18	46	46	NUM
ejpam-5886	258	19	)	)	PUNCT
ejpam-5886	258	20	and	and	CCONJ
ejpam-5886	258	21	d̂s	d̂s	NOUN
ejpam-5886	258	22	=	=	SYM
ejpam-5886	258	23	v∆h	v∆h	ADJ
ejpam-5886	258	24	h	h	NOUN
ejpam-5886	258	25	.	.	PUNCT
ejpam-5886	259	1	(	(	PUNCT
ejpam-5886	259	2	47	47	NUM
ejpam-5886	259	3	)	)	PUNCT
ejpam-5886	259	4	proof	proof	NOUN
ejpam-5886	259	5	.	.	PUNCT
ejpam-5886	260	1	in	in	ADP
ejpam-5886	260	2	consideration	consideration	NOUN
ejpam-5886	260	3	of	of	ADP
ejpam-5886	260	4	expression	expression	NOUN
ejpam-5886	260	5	(	(	PUNCT
ejpam-5886	260	6	5	5	NUM
ejpam-5886	260	7	)	)	PUNCT
ejpam-5886	260	8	,	,	PUNCT
ejpam-5886	260	9	taking	take	VERB
ejpam-5886	260	10	derivatives	derivative	NOUN
ejpam-5886	260	11	w.r.t	w.r.t	VERB
ejpam-5886	260	12	.	.	PUNCT
ejpam-5886	261	1	v	v	NOUN
ejpam-5886	261	2	of	of	ADP
ejpam-5886	261	3	expression	expression	NOUN
ejpam-5886	261	4	(	(	PUNCT
ejpam-5886	261	5	13	13	NUM
ejpam-5886	261	6	)	)	PUNCT
ejpam-5886	261	7	,	,	PUNCT
ejpam-5886	261	8	we	we	PRON
ejpam-5886	261	9	have	have	VERB
ejpam-5886	261	10	v∆h	v∆h	ADJ
ejpam-5886	261	11	{	{	PUNCT
ejpam-5886	261	12	(	(	PUNCT
ejpam-5886	261	13	1	1	NUM
ejpam-5886	261	14	+	+	NUM
ejpam-5886	261	15	ht	ht	PROPN
ejpam-5886	261	16	)	)	PUNCT
ejpam-5886	261	17	v	v	NOUN
ejpam-5886	261	18	h	h	NOUN
ejpam-5886	261	19	(	(	PUNCT
ejpam-5886	261	20	1	1	NUM
ejpam-5886	261	21	+	+	NUM
ejpam-5886	261	22	ht2	ht2	NOUN
ejpam-5886	261	23	)	)	PUNCT
ejpam-5886	262	1	d−1	d−1	PROPN
ejpam-5886	262	2	u	u	NOUN
ejpam-5886	262	3	h	h	NOUN
ejpam-5886	262	4	}	}	PUNCT
ejpam-5886	262	5	=	=	SYM
ejpam-5886	262	6	(	(	PUNCT
ejpam-5886	262	7	1	1	NUM
ejpam-5886	262	8	+	+	NUM
ejpam-5886	262	9	ht	ht	PROPN
ejpam-5886	262	10	)	)	PUNCT
ejpam-5886	262	11	v+h	v+h	NUM
ejpam-5886	262	12	h	h	NOUN
ejpam-5886	262	13	(	(	PUNCT
ejpam-5886	262	14	1	1	NUM
ejpam-5886	262	15	+	+	NUM
ejpam-5886	262	16	ht2	ht2	NOUN
ejpam-5886	262	17	)	)	PUNCT
ejpam-5886	263	1	d−1	d−1	PROPN
ejpam-5886	263	2	u	u	NOUN
ejpam-5886	263	3	h	h	NOUN
ejpam-5886	263	4	−	−	PROPN
ejpam-5886	263	5	(	(	PUNCT
ejpam-5886	263	6	1	1	NUM
ejpam-5886	263	7	+	+	NUM
ejpam-5886	263	8	ht	ht	PROPN
ejpam-5886	263	9	)	)	PUNCT
ejpam-5886	263	10	v	v	NOUN
ejpam-5886	263	11	h	h	NOUN
ejpam-5886	263	12	(	(	PUNCT
ejpam-5886	263	13	1	1	NUM
ejpam-5886	263	14	+	+	NUM
ejpam-5886	263	15	ht2	ht2	NOUN
ejpam-5886	263	16	)	)	PUNCT
ejpam-5886	264	1	d−1	d−1	PROPN
ejpam-5886	264	2	u	u	NOUN
ejpam-5886	264	3	h	h	NOUN
ejpam-5886	264	4	=	=	PUNCT
ejpam-5886	264	5	(	(	PUNCT
ejpam-5886	264	6	1	1	NUM
ejpam-5886	264	7	+	+	NUM
ejpam-5886	264	8	ht−	ht−	NUM
ejpam-5886	264	9	1)(1	1)(1	NUM
ejpam-5886	264	10	+	+	NUM
ejpam-5886	264	11	ht	ht	PROPN
ejpam-5886	264	12	)	)	PUNCT
ejpam-5886	264	13	v	v	NOUN
ejpam-5886	264	14	h	h	NOUN
ejpam-5886	264	15	(	(	PUNCT
ejpam-5886	264	16	1	1	NUM
ejpam-5886	264	17	+	+	NUM
ejpam-5886	264	18	ht2	ht2	NOUN
ejpam-5886	264	19	)	)	PUNCT
ejpam-5886	265	1	d−1	d−1	PROPN
ejpam-5886	265	2	u	u	NOUN
ejpam-5886	265	3	h	h	NOUN
ejpam-5886	265	4	=	=	SYM
ejpam-5886	265	5	ht	ht	PROPN
ejpam-5886	265	6	(	(	PUNCT
ejpam-5886	265	7	1	1	NUM
ejpam-5886	265	8	+	+	NUM
ejpam-5886	265	9	ht	ht	PROPN
ejpam-5886	265	10	)	)	PUNCT
ejpam-5886	265	11	v	v	NOUN
ejpam-5886	265	12	h	h	NOUN
ejpam-5886	265	13	(	(	PUNCT
ejpam-5886	265	14	1	1	NUM
ejpam-5886	265	15	+	+	NUM
ejpam-5886	265	16	ht2	ht2	NOUN
ejpam-5886	265	17	)	)	PUNCT
ejpam-5886	266	1	d−1	d−1	PROPN
ejpam-5886	266	2	u	u	PROPN
ejpam-5886	266	3	h	h	NOUN
ejpam-5886	266	4	,	,	PUNCT
ejpam-5886	266	5	(	(	PUNCT
ejpam-5886	266	6	48	48	NUM
ejpam-5886	266	7	)	)	PUNCT
ejpam-5886	266	8	thus	thus	ADV
ejpam-5886	266	9	,	,	PUNCT
ejpam-5886	266	10	we	we	PRON
ejpam-5886	266	11	have	have	VERB
ejpam-5886	266	12	v∆h	v∆h	ADJ
ejpam-5886	266	13	h	h	NOUN
ejpam-5886	266	14	[	[	PUNCT
ejpam-5886	266	15	(	(	PUNCT
ejpam-5886	266	16	1	1	NUM
ejpam-5886	266	17	+	+	NUM
ejpam-5886	266	18	ht	ht	PROPN
ejpam-5886	266	19	)	)	PUNCT
ejpam-5886	266	20	v	v	NOUN
ejpam-5886	266	21	h	h	NOUN
ejpam-5886	266	22	(	(	PUNCT
ejpam-5886	266	23	1	1	NUM
ejpam-5886	266	24	+	+	NUM
ejpam-5886	266	25	ht2	ht2	NOUN
ejpam-5886	266	26	)	)	PUNCT
ejpam-5886	267	1	d−1	d−1	PROPN
ejpam-5886	267	2	u	u	NOUN
ejpam-5886	267	3	h	h	NOUN
ejpam-5886	267	4	]	]	X
ejpam-5886	267	5	=	=	SYM
ejpam-5886	267	6	t	t	X
ejpam-5886	267	7	[	[	PUNCT
ejpam-5886	267	8	(	(	PUNCT
ejpam-5886	267	9	1	1	NUM
ejpam-5886	267	10	+	+	NUM
ejpam-5886	267	11	ht	ht	PROPN
ejpam-5886	267	12	)	)	PUNCT
ejpam-5886	267	13	v	v	NOUN
ejpam-5886	267	14	h	h	NOUN
ejpam-5886	267	15	(	(	PUNCT
ejpam-5886	267	16	1	1	NUM
ejpam-5886	267	17	+	+	NUM
ejpam-5886	267	18	ht2	ht2	NOUN
ejpam-5886	267	19	)	)	PUNCT
ejpam-5886	268	1	d−1	d−1	PROPN
ejpam-5886	268	2	u	u	NOUN
ejpam-5886	268	3	h	h	NOUN
ejpam-5886	268	4	]	]	PUNCT
ejpam-5886	268	5	,	,	PUNCT
ejpam-5886	268	6	(	(	PUNCT
ejpam-5886	268	7	49	49	NUM
ejpam-5886	268	8	)	)	PUNCT
ejpam-5886	268	9	which	which	PRON
ejpam-5886	268	10	gives	give	VERB
ejpam-5886	268	11	the	the	DET
ejpam-5886	268	12	identity	identity	NOUN
ejpam-5886	268	13	v∆h	v∆h	ADJ
ejpam-5886	268	14	h	h	NOUN
ejpam-5886	268	15	[	[	PUNCT
ejpam-5886	268	16	s[h]n	s[h]n	PROPN
ejpam-5886	268	17	(	(	PUNCT
ejpam-5886	268	18	u	u	NOUN
ejpam-5886	268	19	,	,	PUNCT
ejpam-5886	268	20	v	v	NOUN
ejpam-5886	268	21	)	)	PUNCT
ejpam-5886	268	22	]	]	PUNCT
ejpam-5886	269	1	=	=	SYM
ejpam-5886	269	2	t	t	X
ejpam-5886	269	3	[	[	PUNCT
ejpam-5886	269	4	s[h]n	s[h]n	PROPN
ejpam-5886	269	5	(	(	PUNCT
ejpam-5886	269	6	u	u	NOUN
ejpam-5886	269	7	,	,	PUNCT
ejpam-5886	269	8	v	v	NOUN
ejpam-5886	269	9	)	)	PUNCT
ejpam-5886	269	10	]	]	PUNCT
ejpam-5886	269	11	.	.	PUNCT
ejpam-5886	270	1	(	(	PUNCT
ejpam-5886	270	2	50	50	NUM
ejpam-5886	270	3	)	)	PUNCT
ejpam-5886	270	4	now	now	ADV
ejpam-5886	270	5	,	,	PUNCT
ejpam-5886	270	6	differentiating	differentiate	VERB
ejpam-5886	270	7	expression	expression	NOUN
ejpam-5886	270	8	(	(	PUNCT
ejpam-5886	270	9	13	13	NUM
ejpam-5886	270	10	)	)	PUNCT
ejpam-5886	270	11	w.r.t	w.r.t	NOUN
ejpam-5886	270	12	.	.	PUNCT
ejpam-5886	271	1	t	t	PROPN
ejpam-5886	271	2	,	,	PUNCT
ejpam-5886	271	3	we	we	PRON
ejpam-5886	271	4	have	have	VERB
ejpam-5886	271	5	∂	∂	NUM
ejpam-5886	271	6	∂t	∂t	PROPN
ejpam-5886	271	7	{	{	PUNCT
ejpam-5886	271	8	(	(	PUNCT
ejpam-5886	271	9	1	1	NUM
ejpam-5886	271	10	+	+	NUM
ejpam-5886	271	11	ht	ht	PROPN
ejpam-5886	271	12	)	)	PUNCT
ejpam-5886	271	13	v	v	NOUN
ejpam-5886	271	14	h	h	NOUN
ejpam-5886	271	15	(	(	PUNCT
ejpam-5886	271	16	1	1	NUM
ejpam-5886	271	17	+	+	NUM
ejpam-5886	271	18	ht2	ht2	NOUN
ejpam-5886	271	19	)	)	PUNCT
ejpam-5886	272	1	d−1	d−1	PROPN
ejpam-5886	272	2	u	u	NOUN
ejpam-5886	272	3	h	h	NOUN
ejpam-5886	272	4	}	}	PUNCT
ejpam-5886	272	5	=	=	SYM
ejpam-5886	272	6	∂	∂	NUM
ejpam-5886	272	7	∂t	∂t	PROPN
ejpam-5886	272	8	{	{	PUNCT
ejpam-5886	272	9	∞∑	∞∑	PROPN
ejpam-5886	272	10	n=0	n=0	PROPN
ejpam-5886	272	11	s[h]n	s[h]n	NOUN
ejpam-5886	272	12	(	(	PUNCT
ejpam-5886	272	13	u	u	NOUN
ejpam-5886	272	14	,	,	PUNCT
ejpam-5886	272	15	v	v	NOUN
ejpam-5886	272	16	)	)	PUNCT
ejpam-5886	272	17	tn	tn	PROPN
ejpam-5886	272	18	n	n	PROPN
ejpam-5886	272	19	!	!	PUNCT
ejpam-5886	272	20	}	}	PUNCT
ejpam-5886	272	21	,	,	PUNCT
ejpam-5886	272	22	(	(	PUNCT
ejpam-5886	272	23	51	51	NUM
ejpam-5886	272	24	)	)	PUNCT
ejpam-5886	272	25	(	(	PUNCT
ejpam-5886	272	26	v	v	NOUN
ejpam-5886	272	27	1	1	NUM
ejpam-5886	272	28	+	+	NUM
ejpam-5886	272	29	ht	ht	PROPN
ejpam-5886	272	30	+	+	CCONJ
ejpam-5886	272	31	2	2	NUM
ejpam-5886	272	32	d−1	d−1	PROPN
ejpam-5886	272	33	u	u	NOUN
ejpam-5886	272	34	t	t	PROPN
ejpam-5886	272	35	1	1	NUM
ejpam-5886	272	36	+	+	CCONJ
ejpam-5886	272	37	ht2	ht2	NOUN
ejpam-5886	272	38	)	)	PUNCT
ejpam-5886	272	39	{	{	PUNCT
ejpam-5886	272	40	∞∑	∞∑	PRON
ejpam-5886	272	41	n=0	n=0	NUM
ejpam-5886	272	42	s[h]n	s[h]n	NOUN
ejpam-5886	272	43	(	(	PUNCT
ejpam-5886	272	44	u	u	NOUN
ejpam-5886	272	45	,	,	PUNCT
ejpam-5886	272	46	v	v	NOUN
ejpam-5886	272	47	)	)	PUNCT
ejpam-5886	272	48	tn	tn	PROPN
ejpam-5886	272	49	n	n	NOUN
ejpam-5886	272	50	!	!	PUNCT
ejpam-5886	272	51	}	}	PUNCT
ejpam-5886	272	52	=	=	PUNCT
ejpam-5886	273	1	∞∑	∞∑	NUM
ejpam-5886	273	2	n=0	n=0	NUM
ejpam-5886	273	3	n	n	PRON
ejpam-5886	273	4	s[h]n	s[h]n	NOUN
ejpam-5886	273	5	(	(	PUNCT
ejpam-5886	273	6	u	u	NOUN
ejpam-5886	273	7	,	,	PUNCT
ejpam-5886	273	8	v	v	NOUN
ejpam-5886	273	9	)	)	PUNCT
ejpam-5886	273	10	tn−1	tn−1	PROPN
ejpam-5886	273	11	n	n	CCONJ
ejpam-5886	273	12	!	!	PUNCT
ejpam-5886	273	13	.	.	PUNCT
ejpam-5886	274	1	(	(	PUNCT
ejpam-5886	274	2	52	52	NUM
ejpam-5886	274	3	)	)	PUNCT
ejpam-5886	274	4	which	which	PRON
ejpam-5886	274	5	on	on	ADP
ejpam-5886	274	6	usage	usage	NOUN
ejpam-5886	274	7	of	of	ADP
ejpam-5886	274	8	identity	identity	NOUN
ejpam-5886	274	9	expression	expression	NOUN
ejpam-5886	274	10	(	(	PUNCT
ejpam-5886	274	11	50	50	NUM
ejpam-5886	274	12	)	)	PUNCT
ejpam-5886	274	13	and	and	CCONJ
ejpam-5886	274	14	replacing	replace	VERB
ejpam-5886	274	15	n	n	PRON
ejpam-5886	274	16	→	→	SYM
ejpam-5886	274	17	n+1	n+1	PROPN
ejpam-5886	274	18	in	in	ADP
ejpam-5886	274	19	the	the	DET
ejpam-5886	274	20	r.h.s	r.h.s	NOUN
ejpam-5886	274	21	.	.	PUNCT
ejpam-5886	274	22	of	of	ADP
ejpam-5886	274	23	previous	previous	ADJ
ejpam-5886	274	24	expression	expression	NOUN
ejpam-5886	274	25	(	(	PUNCT
ejpam-5886	274	26	52	52	NUM
ejpam-5886	274	27	)	)	PUNCT
ejpam-5886	274	28	,	,	PUNCT
ejpam-5886	274	29	assertion	assertion	NOUN
ejpam-5886	274	30	(	(	PUNCT
ejpam-5886	274	31	46	46	NUM
ejpam-5886	274	32	)	)	PUNCT
ejpam-5886	274	33	is	be	AUX
ejpam-5886	274	34	established	establish	VERB
ejpam-5886	274	35	.	.	PUNCT
ejpam-5886	275	1	s.	s.	PROPN
ejpam-5886	275	2	a.	a.	PROPN
ejpam-5886	275	3	wani	wani	PROPN
ejpam-5886	275	4	et	et	PROPN
ejpam-5886	275	5	al	al	PROPN
ejpam-5886	275	6	.	.	PUNCT
ejpam-5886	275	7	/	/	SYM
ejpam-5886	275	8	eur	eur	PROPN
ejpam-5886	275	9	.	.	PUNCT
ejpam-5886	276	1	j.	j.	PROPN
ejpam-5886	276	2	pure	pure	PROPN
ejpam-5886	276	3	appl	appl	PROPN
ejpam-5886	276	4	.	.	PROPN
ejpam-5886	276	5	math	math	PROPN
ejpam-5886	276	6	,	,	PUNCT
ejpam-5886	276	7	18	18	NUM
ejpam-5886	276	8	(	(	PUNCT
ejpam-5886	276	9	2	2	NUM
ejpam-5886	276	10	)	)	PUNCT
ejpam-5886	276	11	(	(	PUNCT
ejpam-5886	276	12	2025	2025	NUM
ejpam-5886	276	13	)	)	PUNCT
ejpam-5886	276	14	,	,	PUNCT
ejpam-5886	276	15	5886	5886	NUM
ejpam-5886	276	16	13	13	NUM
ejpam-5886	276	17	of	of	ADP
ejpam-5886	276	18	16	16	NUM
ejpam-5886	276	19	further	far	ADV
ejpam-5886	276	20	,	,	PUNCT
ejpam-5886	276	21	in	in	ADP
ejpam-5886	276	22	view	view	NOUN
ejpam-5886	276	23	of	of	ADP
ejpam-5886	276	24	identity	identity	NOUN
ejpam-5886	276	25	expression	expression	NOUN
ejpam-5886	276	26	(	(	PUNCT
ejpam-5886	276	27	50	50	NUM
ejpam-5886	276	28	)	)	PUNCT
ejpam-5886	277	1	,	,	PUNCT
ejpam-5886	277	2	we	we	PRON
ejpam-5886	277	3	have	have	VERB
ejpam-5886	277	4	v∆h	v∆h	ADJ
ejpam-5886	277	5	h	h	NOUN
ejpam-5886	277	6	[	[	PUNCT
ejpam-5886	277	7	s[h]n	s[h]n	PROPN
ejpam-5886	277	8	(	(	PUNCT
ejpam-5886	277	9	u	u	NOUN
ejpam-5886	277	10	,	,	PUNCT
ejpam-5886	277	11	v	v	NOUN
ejpam-5886	277	12	)	)	PUNCT
ejpam-5886	277	13	]	]	PUNCT
ejpam-5886	278	1	=	=	PUNCT
ejpam-5886	278	2	[	[	PUNCT
ejpam-5886	278	3	n	n	X
ejpam-5886	278	4	s[h]n−1(u	s[h]n−1(u	NOUN
ejpam-5886	278	5	,	,	PUNCT
ejpam-5886	278	6	v	v	NOUN
ejpam-5886	278	7	)	)	PUNCT
ejpam-5886	278	8	]	]	PUNCT
ejpam-5886	278	9	,	,	PUNCT
ejpam-5886	278	10	(	(	PUNCT
ejpam-5886	278	11	53	53	NUM
ejpam-5886	278	12	)	)	PUNCT
ejpam-5886	278	13	which	which	PRON
ejpam-5886	278	14	gives	give	VERB
ejpam-5886	278	15	expression	expression	NOUN
ejpam-5886	278	16	for	for	ADP
ejpam-5886	278	17	the	the	DET
ejpam-5886	278	18	derivative	derivative	ADJ
ejpam-5886	278	19	operator	operator	NOUN
ejpam-5886	278	20	(	(	PUNCT
ejpam-5886	278	21	47	47	NUM
ejpam-5886	278	22	)	)	PUNCT
ejpam-5886	278	23	.	.	PUNCT
ejpam-5886	279	1	next	next	ADV
ejpam-5886	279	2	,	,	PUNCT
ejpam-5886	279	3	we	we	PRON
ejpam-5886	279	4	deduce	deduce	VERB
ejpam-5886	279	5	the	the	DET
ejpam-5886	279	6	differential	differential	ADJ
ejpam-5886	279	7	equation	equation	NOUN
ejpam-5886	279	8	for	for	ADP
ejpam-5886	279	9	the	the	DET
ejpam-5886	279	10	∆h	∆h	PROPN
ejpam-5886	279	11	legendre	legendre	PROPN
ejpam-5886	279	12	polynomials	polynomial	NOUN
ejpam-5886	279	13	s[h]n	s[h]n	PROPN
ejpam-5886	279	14	(	(	PUNCT
ejpam-5886	279	15	u	u	NOUN
ejpam-5886	279	16	,	,	PUNCT
ejpam-5886	279	17	v	v	NOUN
ejpam-5886	279	18	)	)	PUNCT
ejpam-5886	279	19	by	by	ADP
ejpam-5886	279	20	demonstrating	demonstrate	VERB
ejpam-5886	279	21	the	the	DET
ejpam-5886	279	22	succeeding	succeed	VERB
ejpam-5886	279	23	result	result	NOUN
ejpam-5886	279	24	:	:	PUNCT
ejpam-5886	279	25	theorem	theorem	VERB
ejpam-5886	279	26	11	11	NUM
ejpam-5886	279	27	.	.	PUNCT
ejpam-5886	280	1	the	the	DET
ejpam-5886	280	2	∆h	∆h	PROPN
ejpam-5886	280	3	legendre	legendre	PROPN
ejpam-5886	280	4	polynomials	polynomial	NOUN
ejpam-5886	280	5	s[h]n	s[h]n	PROPN
ejpam-5886	280	6	(	(	PUNCT
ejpam-5886	280	7	u	u	NOUN
ejpam-5886	280	8	,	,	PUNCT
ejpam-5886	280	9	v	v	NOUN
ejpam-5886	280	10	)	)	PUNCT
ejpam-5886	280	11	satisfy	satisfy	VERB
ejpam-5886	280	12	the	the	DET
ejpam-5886	280	13	differential	differential	ADJ
ejpam-5886	280	14	equation	equation	NOUN
ejpam-5886	280	15	:(	:(	PUNCT
ejpam-5886	280	16	v	v	ADP
ejpam-5886	280	17	1	1	NUM
ejpam-5886	280	18	+	+	CCONJ
ejpam-5886	280	19	v∆h	v∆h	ADJ
ejpam-5886	280	20	+	+	CCONJ
ejpam-5886	280	21	2	2	NUM
ejpam-5886	280	22	d−1	d−1	PROPN
ejpam-5886	280	23	u	u	PROPN
ejpam-5886	280	24	v∆h	v∆h	PROPN
ejpam-5886	280	25	h+	h+	X
ejpam-5886	280	26	v∆h	v∆h	ADJ
ejpam-5886	280	27	2	2	NUM
ejpam-5886	280	28	−	−	PROPN
ejpam-5886	280	29	nh	nh	PROPN
ejpam-5886	280	30	v∆h	v∆h	PROPN
ejpam-5886	280	31	)	)	PUNCT
ejpam-5886	280	32	s[h]n	s[h]n	PROPN
ejpam-5886	280	33	(	(	PUNCT
ejpam-5886	280	34	u	u	NOUN
ejpam-5886	280	35	,	,	PUNCT
ejpam-5886	280	36	v	v	NOUN
ejpam-5886	280	37	)	)	PUNCT
ejpam-5886	280	38	=	=	SYM
ejpam-5886	281	1	0	0	X
ejpam-5886	281	2	.	.	PUNCT
ejpam-5886	282	1	(	(	PUNCT
ejpam-5886	282	2	54	54	NUM
ejpam-5886	282	3	)	)	PUNCT
ejpam-5886	282	4	proof	proof	NOUN
ejpam-5886	282	5	.	.	PUNCT
ejpam-5886	283	1	inserting	insert	VERB
ejpam-5886	283	2	expression	expression	NOUN
ejpam-5886	283	3	(	(	PUNCT
ejpam-5886	283	4	46	46	NUM
ejpam-5886	283	5	)	)	PUNCT
ejpam-5886	283	6	and	and	CCONJ
ejpam-5886	283	7	(	(	PUNCT
ejpam-5886	283	8	47	47	NUM
ejpam-5886	283	9	)	)	PUNCT
ejpam-5886	283	10	in	in	ADP
ejpam-5886	283	11	the	the	DET
ejpam-5886	283	12	expression	expression	NOUN
ejpam-5886	283	13	(	(	PUNCT
ejpam-5886	283	14	43	43	NUM
ejpam-5886	283	15	)	)	PUNCT
ejpam-5886	283	16	,	,	PUNCT
ejpam-5886	283	17	the	the	DET
ejpam-5886	283	18	assertion	assertion	NOUN
ejpam-5886	283	19	(	(	PUNCT
ejpam-5886	283	20	54	54	NUM
ejpam-5886	283	21	)	)	PUNCT
ejpam-5886	283	22	is	be	AUX
ejpam-5886	283	23	proved	prove	VERB
ejpam-5886	283	24	.	.	PUNCT
ejpam-5886	284	1	5	5	X
ejpam-5886	284	2	.	.	X
ejpam-5886	284	3	symmetric	symmetric	ADJ
ejpam-5886	284	4	identities	identity	NOUN
ejpam-5886	284	5	the	the	DET
ejpam-5886	284	6	two	two	NUM
ejpam-5886	284	7	-	-	PUNCT
ejpam-5886	284	8	variable	variable	NOUN
ejpam-5886	284	9	∆h	∆h	PROPN
ejpam-5886	284	10	special	special	ADJ
ejpam-5886	284	11	polynomials	polynomial	NOUN
ejpam-5886	284	12	have	have	VERB
ejpam-5886	284	13	symmetric	symmetric	ADJ
ejpam-5886	284	14	identities	identity	NOUN
ejpam-5886	284	15	that	that	PRON
ejpam-5886	284	16	we	we	PRON
ejpam-5886	284	17	examine	examine	VERB
ejpam-5886	284	18	in	in	ADP
ejpam-5886	284	19	this	this	DET
ejpam-5886	284	20	section	section	NOUN
ejpam-5886	284	21	.	.	PUNCT
ejpam-5886	285	1	these	these	DET
ejpam-5886	285	2	identities	identity	NOUN
ejpam-5886	285	3	reveal	reveal	VERB
ejpam-5886	285	4	fascinating	fascinating	ADJ
ejpam-5886	285	5	connections	connection	NOUN
ejpam-5886	285	6	between	between	ADP
ejpam-5886	285	7	the	the	DET
ejpam-5886	285	8	polynomials	polynomial	NOUN
ejpam-5886	285	9	’	'	PUNCT
ejpam-5886	285	10	variables	variable	NOUN
ejpam-5886	285	11	and	and	CCONJ
ejpam-5886	285	12	coefficients	coefficient	NOUN
ejpam-5886	285	13	,	,	PUNCT
ejpam-5886	285	14	illuminating	illuminate	VERB
ejpam-5886	285	15	their	their	PRON
ejpam-5886	285	16	fundamental	fundamental	ADJ
ejpam-5886	285	17	symmetrical	symmetrical	ADJ
ejpam-5886	285	18	characteristics	characteristic	NOUN
ejpam-5886	285	19	.	.	PUNCT
ejpam-5886	286	1	by	by	ADP
ejpam-5886	286	2	examining	examine	VERB
ejpam-5886	286	3	the	the	DET
ejpam-5886	286	4	behavior	behavior	NOUN
ejpam-5886	286	5	of	of	ADP
ejpam-5886	286	6	the	the	DET
ejpam-5886	286	7	polynomials	polynomial	NOUN
ejpam-5886	286	8	when	when	SCONJ
ejpam-5886	286	9	the	the	DET
ejpam-5886	286	10	variables	variable	NOUN
ejpam-5886	286	11	or	or	CCONJ
ejpam-5886	286	12	coefficients	coefficient	NOUN
ejpam-5886	286	13	are	be	AUX
ejpam-5886	286	14	changed	change	VERB
ejpam-5886	286	15	,	,	PUNCT
ejpam-5886	286	16	we	we	PRON
ejpam-5886	286	17	can	can	AUX
ejpam-5886	286	18	find	find	VERB
ejpam-5886	286	19	deep	deep	ADJ
ejpam-5886	286	20	relationships	relationship	NOUN
ejpam-5886	286	21	that	that	PRON
ejpam-5886	286	22	go	go	VERB
ejpam-5886	286	23	beyond	beyond	ADP
ejpam-5886	286	24	their	their	PRON
ejpam-5886	286	25	original	original	ADJ
ejpam-5886	286	26	definitions	definition	NOUN
ejpam-5886	286	27	.	.	PUNCT
ejpam-5886	287	1	in	in	ADP
ejpam-5886	287	2	addition	addition	NOUN
ejpam-5886	287	3	to	to	ADP
ejpam-5886	287	4	helping	help	VERB
ejpam-5886	287	5	us	we	PRON
ejpam-5886	287	6	comprehend	comprehend	VERB
ejpam-5886	287	7	the	the	DET
ejpam-5886	287	8	polynomials	polynomial	NOUN
ejpam-5886	287	9	better	well	ADV
ejpam-5886	287	10	,	,	PUNCT
ejpam-5886	287	11	these	these	DET
ejpam-5886	287	12	symmetric	symmetric	ADJ
ejpam-5886	287	13	identities	identity	NOUN
ejpam-5886	287	14	provide	provide	VERB
ejpam-5886	287	15	important	important	ADJ
ejpam-5886	287	16	new	new	ADJ
ejpam-5886	287	17	information	information	NOUN
ejpam-5886	287	18	on	on	ADP
ejpam-5886	287	19	more	more	ADJ
ejpam-5886	287	20	general	general	ADJ
ejpam-5886	287	21	mathematical	mathematical	ADJ
ejpam-5886	287	22	occurrences	occurrence	NOUN
ejpam-5886	287	23	and	and	CCONJ
ejpam-5886	287	24	structures	structure	NOUN
ejpam-5886	287	25	.	.	PUNCT
ejpam-5886	288	1	we	we	PRON
ejpam-5886	288	2	provide	provide	VERB
ejpam-5886	288	3	a	a	DET
ejpam-5886	288	4	thorough	thorough	ADJ
ejpam-5886	288	5	framework	framework	NOUN
ejpam-5886	288	6	for	for	ADP
ejpam-5886	288	7	comprehending	comprehend	VERB
ejpam-5886	288	8	and	and	CCONJ
ejpam-5886	288	9	taking	take	VERB
ejpam-5886	288	10	use	use	NOUN
ejpam-5886	288	11	of	of	ADP
ejpam-5886	288	12	the	the	DET
ejpam-5886	288	13	symmetrical	symmetrical	ADJ
ejpam-5886	288	14	characteristics	characteristic	NOUN
ejpam-5886	288	15	of	of	ADP
ejpam-5886	288	16	these	these	DET
ejpam-5886	288	17	two	two	NUM
ejpam-5886	288	18	-	-	PUNCT
ejpam-5886	288	19	variable	variable	NOUN
ejpam-5886	288	20	special	special	ADJ
ejpam-5886	288	21	polynomials	polynomial	NOUN
ejpam-5886	288	22	by	by	ADP
ejpam-5886	288	23	methodical	methodical	ADJ
ejpam-5886	288	24	investigation	investigation	NOUN
ejpam-5886	288	25	and	and	CCONJ
ejpam-5886	288	26	exacting	exact	VERB
ejpam-5886	288	27	derivation	derivation	NOUN
ejpam-5886	288	28	,	,	PUNCT
ejpam-5886	288	29	opening	open	VERB
ejpam-5886	288	30	the	the	DET
ejpam-5886	288	31	door	door	NOUN
ejpam-5886	288	32	for	for	ADP
ejpam-5886	288	33	future	future	ADJ
ejpam-5886	288	34	developments	development	NOUN
ejpam-5886	288	35	in	in	ADP
ejpam-5886	288	36	theoretical	theoretical	ADJ
ejpam-5886	288	37	analysis	analysis	NOUN
ejpam-5886	288	38	and	and	CCONJ
ejpam-5886	288	39	real	real	ADJ
ejpam-5886	288	40	-	-	PUNCT
ejpam-5886	288	41	world	world	NOUN
ejpam-5886	288	42	applications	application	NOUN
ejpam-5886	288	43	.	.	PUNCT
ejpam-5886	289	1	consequently	consequently	ADV
ejpam-5886	289	2	,	,	PUNCT
ejpam-5886	289	3	we	we	PRON
ejpam-5886	289	4	discover	discover	VERB
ejpam-5886	289	5	a	a	DET
ejpam-5886	289	6	few	few	ADJ
ejpam-5886	289	7	of	of	ADP
ejpam-5886	289	8	the	the	DET
ejpam-5886	289	9	legenedre	legenedre	NOUN
ejpam-5886	289	10	polynomials	polynomial	VERB
ejpam-5886	289	11	formulae	formulae	VERB
ejpam-5886	289	12	and	and	CCONJ
ejpam-5886	289	13	characteristics	characteristic	NOUN
ejpam-5886	289	14	.	.	PUNCT
ejpam-5886	290	1	theorem	theorem	NOUN
ejpam-5886	290	2	12	12	NUM
ejpam-5886	290	3	.	.	PUNCT
ejpam-5886	291	1	for	for	ADP
ejpam-5886	291	2	a	a	DET
ejpam-5886	291	3	̸=	̸=	PROPN
ejpam-5886	291	4	b	b	PROPN
ejpam-5886	291	5	,	,	PUNCT
ejpam-5886	291	6	a	a	PRON
ejpam-5886	291	7	,	,	PUNCT
ejpam-5886	291	8	b	b	X
ejpam-5886	291	9	>	>	X
ejpam-5886	291	10	0	0	PUNCT
ejpam-5886	291	11	and	and	CCONJ
ejpam-5886	291	12	u1	u1	NOUN
ejpam-5886	291	13	,	,	PUNCT
ejpam-5886	291	14	u2	u2	NOUN
ejpam-5886	291	15	,	,	PUNCT
ejpam-5886	291	16	v1	v1	NOUN
ejpam-5886	291	17	,	,	PUNCT
ejpam-5886	291	18	v2	v2	PROPN
ejpam-5886	291	19	∈	∈	PROPN
ejpam-5886	291	20	c	c	X
ejpam-5886	291	21	,	,	PUNCT
ejpam-5886	291	22	we	we	PRON
ejpam-5886	291	23	have	have	VERB
ejpam-5886	291	24	n∑	n∑	PROPN
ejpam-5886	291	25	m=0	m=0	PROPN
ejpam-5886	291	26	(	(	PUNCT
ejpam-5886	291	27	n	n	NOUN
ejpam-5886	291	28	m	m	PROPN
ejpam-5886	291	29	)	)	PUNCT
ejpam-5886	291	30	an−mbms[h]n−m(au1	an−mbms[h]n−m(au1	PROPN
ejpam-5886	291	31	,	,	PUNCT
ejpam-5886	291	32	av1)s[h]m	av1)s[h]m	NOUN
ejpam-5886	291	33	(	(	PUNCT
ejpam-5886	291	34	bu2	bu2	PROPN
ejpam-5886	291	35	,	,	PUNCT
ejpam-5886	291	36	bv2	bv2	NOUN
ejpam-5886	291	37	)	)	PUNCT
ejpam-5886	291	38	=	=	SYM
ejpam-5886	292	1	n∑	n∑	PROPN
ejpam-5886	292	2	m=0	m=0	PROPN
ejpam-5886	292	3	(	(	PUNCT
ejpam-5886	292	4	n	n	NOUN
ejpam-5886	292	5	m	m	NOUN
ejpam-5886	292	6	)	)	PUNCT
ejpam-5886	292	7	anbn−ms[h]n−m(au2	anbn−ms[h]n−m(au2	ADJ
ejpam-5886	292	8	,	,	PUNCT
ejpam-5886	292	9	av2)s[h]m	av2)s[h]m	NOUN
ejpam-5886	292	10	(	(	PUNCT
ejpam-5886	292	11	bu1	bu1	PROPN
ejpam-5886	292	12	,	,	PUNCT
ejpam-5886	292	13	bv1	bv1	NOUN
ejpam-5886	292	14	)	)	PUNCT
ejpam-5886	292	15	.	.	PUNCT
ejpam-5886	293	1	(	(	PUNCT
ejpam-5886	293	2	55	55	NUM
ejpam-5886	293	3	)	)	PUNCT
ejpam-5886	293	4	proof	proof	NOUN
ejpam-5886	293	5	.	.	PUNCT
ejpam-5886	294	1	let	let	VERB
ejpam-5886	294	2	a(t	a(t	VERB
ejpam-5886	294	3	)	)	PUNCT
ejpam-5886	294	4	=	=	PUNCT
ejpam-5886	295	1	(	(	PUNCT
ejpam-5886	295	2	1	1	NUM
ejpam-5886	295	3	+	+	NUM
ejpam-5886	295	4	ht	ht	PROPN
ejpam-5886	295	5	)	)	PUNCT
ejpam-5886	295	6	ab(v1+v2	ab(v1+v2	PROPN
ejpam-5886	295	7	)	)	PUNCT
ejpam-5886	295	8	h	h	PROPN
ejpam-5886	295	9	c0	c0	NOUN
ejpam-5886	295	10	(	(	PUNCT
ejpam-5886	295	11	−abu1	−abu1	NOUN
ejpam-5886	295	12	h	h	NOUN
ejpam-5886	295	13	log(1	log(1	NOUN
ejpam-5886	295	14	+	+	CCONJ
ejpam-5886	296	1	ht2	ht2	NOUN
ejpam-5886	296	2	)	)	PUNCT
ejpam-5886	296	3	)	)	PUNCT
ejpam-5886	297	1	c0	c0	NOUN
ejpam-5886	297	2	(	(	PUNCT
ejpam-5886	297	3	−abu2	−abu2	PROPN
ejpam-5886	297	4	h	h	NOUN
ejpam-5886	297	5	log(1	log(1	NOUN
ejpam-5886	297	6	+	+	CCONJ
ejpam-5886	297	7	ht2	ht2	NOUN
ejpam-5886	297	8	)	)	PUNCT
ejpam-5886	297	9	)	)	PUNCT
ejpam-5886	298	1	s.	s.	PROPN
ejpam-5886	298	2	a.	a.	PROPN
ejpam-5886	298	3	wani	wani	PROPN
ejpam-5886	298	4	et	et	PROPN
ejpam-5886	298	5	al	al	PROPN
ejpam-5886	298	6	.	.	PUNCT
ejpam-5886	298	7	/	/	SYM
ejpam-5886	298	8	eur	eur	PROPN
ejpam-5886	298	9	.	.	PUNCT
ejpam-5886	299	1	j.	j.	PROPN
ejpam-5886	299	2	pure	pure	PROPN
ejpam-5886	299	3	appl	appl	PROPN
ejpam-5886	299	4	.	.	PROPN
ejpam-5886	299	5	math	math	PROPN
ejpam-5886	299	6	,	,	PUNCT
ejpam-5886	299	7	18	18	NUM
ejpam-5886	299	8	(	(	PUNCT
ejpam-5886	299	9	2	2	NUM
ejpam-5886	299	10	)	)	PUNCT
ejpam-5886	299	11	(	(	PUNCT
ejpam-5886	299	12	2025	2025	NUM
ejpam-5886	299	13	)	)	PUNCT
ejpam-5886	299	14	,	,	PUNCT
ejpam-5886	299	15	5886	5886	NUM
ejpam-5886	299	16	14	14	NUM
ejpam-5886	299	17	of	of	ADP
ejpam-5886	299	18	16	16	NUM
ejpam-5886	299	19	=	=	SYM
ejpam-5886	299	20	∞∑	∞∑	NUM
ejpam-5886	299	21	n=0	n=0	NUM
ejpam-5886	299	22	s[h]n	s[h]n	NOUN
ejpam-5886	299	23	(	(	PUNCT
ejpam-5886	299	24	bu1	bu1	PROPN
ejpam-5886	299	25	,	,	PUNCT
ejpam-5886	299	26	bv1	bv1	NOUN
ejpam-5886	299	27	)	)	PUNCT
ejpam-5886	299	28	(	(	PUNCT
ejpam-5886	299	29	at)n	at)n	PROPN
ejpam-5886	299	30	n	n	CCONJ
ejpam-5886	299	31	!	!	PUNCT
ejpam-5886	300	1	∞∑	∞∑	NUM
ejpam-5886	300	2	m=0	m=0	PROPN
ejpam-5886	300	3	s[h]n	s[h]n	PROPN
ejpam-5886	300	4	(	(	PUNCT
ejpam-5886	300	5	au2	au2	NOUN
ejpam-5886	300	6	,	,	PUNCT
ejpam-5886	300	7	av2	av2	PROPN
ejpam-5886	300	8	)	)	PUNCT
ejpam-5886	300	9	(	(	PUNCT
ejpam-5886	300	10	bt)m	bt)m	PROPN
ejpam-5886	300	11	m	m	PROPN
ejpam-5886	300	12	!	!	PUNCT
ejpam-5886	300	13	=	=	NOUN
ejpam-5886	301	1	∞∑	∞∑	PRON
ejpam-5886	301	2	n=0	n=0	NUM
ejpam-5886	301	3	(	(	PUNCT
ejpam-5886	301	4	n∑	n∑	PROPN
ejpam-5886	301	5	m=0	m=0	PROPN
ejpam-5886	301	6	(	(	PUNCT
ejpam-5886	301	7	n	n	NOUN
ejpam-5886	301	8	m	m	PROPN
ejpam-5886	301	9	)	)	PUNCT
ejpam-5886	301	10	an−mbms[h]n−m(au1	an−mbms[h]n−m(au1	PROPN
ejpam-5886	301	11	,	,	PUNCT
ejpam-5886	301	12	av1)s[h]m	av1)s[h]m	NOUN
ejpam-5886	301	13	(	(	PUNCT
ejpam-5886	301	14	bu2	bu2	PROPN
ejpam-5886	301	15	,	,	PUNCT
ejpam-5886	301	16	bv2	bv2	NOUN
ejpam-5886	301	17	)	)	PUNCT
ejpam-5886	301	18	)	)	PUNCT
ejpam-5886	301	19	tn	tn	PROPN
ejpam-5886	301	20	n	n	PROPN
ejpam-5886	301	21	!	!	PUNCT
ejpam-5886	301	22	.	.	PUNCT
ejpam-5886	302	1	(	(	PUNCT
ejpam-5886	302	2	56	56	NUM
ejpam-5886	302	3	)	)	PUNCT
ejpam-5886	302	4	similarly	similarly	ADV
ejpam-5886	302	5	,	,	PUNCT
ejpam-5886	302	6	we	we	PRON
ejpam-5886	302	7	have	have	AUX
ejpam-5886	302	8	a(t	a(t	VERB
ejpam-5886	302	9	)	)	PUNCT
ejpam-5886	303	1	=	=	NOUN
ejpam-5886	304	1	∞∑	∞∑	NUM
ejpam-5886	304	2	n=0	n=0	NUM
ejpam-5886	304	3	(	(	PUNCT
ejpam-5886	304	4	n∑	n∑	PROPN
ejpam-5886	304	5	m=0	m=0	PROPN
ejpam-5886	304	6	(	(	PUNCT
ejpam-5886	304	7	n	n	NOUN
ejpam-5886	304	8	m	m	NOUN
ejpam-5886	304	9	)	)	PUNCT
ejpam-5886	305	1	anbn−ms[h]n−m(au2	anbn−ms[h]n−m(au2	ADJ
ejpam-5886	305	2	,	,	PUNCT
ejpam-5886	305	3	av2)s[h]m	av2)s[h]m	NOUN
ejpam-5886	305	4	(	(	PUNCT
ejpam-5886	305	5	bu1	bu1	PROPN
ejpam-5886	305	6	,	,	PUNCT
ejpam-5886	305	7	bv1	bv1	NOUN
ejpam-5886	305	8	)	)	PUNCT
ejpam-5886	305	9	)	)	PUNCT
ejpam-5886	305	10	tn	tn	PROPN
ejpam-5886	306	1	n	n	PROPN
ejpam-5886	306	2	!	!	PUNCT
ejpam-5886	306	3	.	.	PUNCT
ejpam-5886	307	1	(	(	PUNCT
ejpam-5886	307	2	57	57	X
ejpam-5886	307	3	)	)	PUNCT
ejpam-5886	307	4	comapring	comapre	VERB
ejpam-5886	307	5	the	the	DET
ejpam-5886	307	6	coefficients	coefficient	NOUN
ejpam-5886	307	7	of	of	ADP
ejpam-5886	307	8	t	t	PROPN
ejpam-5886	307	9	on	on	ADP
ejpam-5886	307	10	both	both	DET
ejpam-5886	307	11	sides	side	NOUN
ejpam-5886	307	12	of	of	ADP
ejpam-5886	307	13	last	last	ADJ
ejpam-5886	307	14	equations	equation	NOUN
ejpam-5886	307	15	,	,	PUNCT
ejpam-5886	307	16	we	we	PRON
ejpam-5886	307	17	get	get	VERB
ejpam-5886	307	18	(	(	PUNCT
ejpam-5886	307	19	55	55	NUM
ejpam-5886	307	20	)	)	PUNCT
ejpam-5886	307	21	.	.	PUNCT
ejpam-5886	308	1	theorem	theorem	VERB
ejpam-5886	308	2	13	13	NUM
ejpam-5886	308	3	.	.	PUNCT
ejpam-5886	309	1	for	for	ADP
ejpam-5886	309	2	a	a	DET
ejpam-5886	309	3	̸=	̸=	PROPN
ejpam-5886	309	4	b	b	PROPN
ejpam-5886	309	5	,	,	PUNCT
ejpam-5886	309	6	a	a	PRON
ejpam-5886	309	7	,	,	PUNCT
ejpam-5886	309	8	b	b	X
ejpam-5886	309	9	>	>	X
ejpam-5886	309	10	0	0	PUNCT
ejpam-5886	309	11	and	and	CCONJ
ejpam-5886	309	12	u1	u1	NOUN
ejpam-5886	309	13	,	,	PUNCT
ejpam-5886	309	14	u2	u2	PROPN
ejpam-5886	309	15	,	,	PUNCT
ejpam-5886	309	16	v	v	NOUN
ejpam-5886	309	17	∈	∈	ADJ
ejpam-5886	309	18	c	c	NOUN
ejpam-5886	309	19	,	,	PUNCT
ejpam-5886	309	20	we	we	PRON
ejpam-5886	309	21	have	have	VERB
ejpam-5886	309	22	n∑	n∑	PROPN
ejpam-5886	309	23	k=0	k=0	PROPN
ejpam-5886	309	24	k∑	k∑	PROPN
ejpam-5886	310	1	m=0	m=0	PROPN
ejpam-5886	310	2	(	(	PUNCT
ejpam-5886	310	3	n	n	NOUN
ejpam-5886	310	4	k	k	NOUN
ejpam-5886	310	5	)	)	PUNCT
ejpam-5886	310	6	(	(	PUNCT
ejpam-5886	310	7	k	k	NOUN
ejpam-5886	310	8	m	m	VERB
ejpam-5886	310	9	)	)	PUNCT
ejpam-5886	311	1	an−mbm+1βn−k(h)s	an−mbm+1βn−k(h)s	VERB
ejpam-5886	312	1	[	[	X
ejpam-5886	312	2	h	h	X
ejpam-5886	312	3	]	]	X
ejpam-5886	312	4	n−m(bu	n−m(bu	PROPN
ejpam-5886	312	5	,	,	PUNCT
ejpam-5886	312	6	bv)σm(a−	bv)σm(a−	PROPN
ejpam-5886	312	7	1;h	1;h	NUM
ejpam-5886	312	8	)	)	PUNCT
ejpam-5886	312	9	=	=	SYM
ejpam-5886	313	1	n∑	n∑	PROPN
ejpam-5886	313	2	k=0	k=0	PROPN
ejpam-5886	313	3	k∑	k∑	PROPN
ejpam-5886	313	4	m=0	m=0	PROPN
ejpam-5886	314	1	(	(	PUNCT
ejpam-5886	314	2	n	n	NOUN
ejpam-5886	314	3	k	k	NOUN
ejpam-5886	314	4	)	)	PUNCT
ejpam-5886	315	1	(	(	PUNCT
ejpam-5886	315	2	k	k	NOUN
ejpam-5886	315	3	m	m	AUX
ejpam-5886	315	4	)	)	PUNCT
ejpam-5886	315	5	bn−mam+1βn−k(h)s	bn−mam+1βn−k(h)s	PROPN
ejpam-5886	316	1	[	[	X
ejpam-5886	316	2	h	h	X
ejpam-5886	316	3	]	]	X
ejpam-5886	316	4	n−m(au	n−m(au	PROPN
ejpam-5886	316	5	,	,	PUNCT
ejpam-5886	316	6	av)σm(b−	av)σm(b−	VERB
ejpam-5886	316	7	1;h	1;h	NUM
ejpam-5886	316	8	)	)	PUNCT
ejpam-5886	316	9	.	.	PUNCT
ejpam-5886	317	1	(	(	PUNCT
ejpam-5886	317	2	58	58	X
ejpam-5886	317	3	)	)	PUNCT
ejpam-5886	317	4	proof	proof	NOUN
ejpam-5886	317	5	.	.	PUNCT
ejpam-5886	318	1	consider	consider	VERB
ejpam-5886	318	2	b(t	b(t	NOUN
ejpam-5886	318	3	)	)	PUNCT
ejpam-5886	319	1	=	=	SYM
ejpam-5886	319	2	abt(1	abt(1	NOUN
ejpam-5886	320	1	+	+	CCONJ
ejpam-5886	320	2	ht	ht	X
ejpam-5886	320	3	)	)	PUNCT
ejpam-5886	320	4	abv	abv	PROPN
ejpam-5886	320	5	h	h	PROPN
ejpam-5886	320	6	c0	c0	PROPN
ejpam-5886	320	7	(	(	PUNCT
ejpam-5886	320	8	−abu	−abu	NOUN
ejpam-5886	320	9	h	h	NOUN
ejpam-5886	320	10	log(1	log(1	NOUN
ejpam-5886	320	11	+	+	CCONJ
ejpam-5886	320	12	ht2	ht2	NOUN
ejpam-5886	320	13	)	)	PUNCT
ejpam-5886	320	14	)	)	PUNCT
ejpam-5886	321	1	(	(	PUNCT
ejpam-5886	321	2	(	(	PUNCT
ejpam-5886	321	3	1	1	NUM
ejpam-5886	321	4	+	+	NUM
ejpam-5886	321	5	ht	ht	PROPN
ejpam-5886	321	6	)	)	PUNCT
ejpam-5886	321	7	ab	ab	PROPN
ejpam-5886	321	8	h	h	NOUN
ejpam-5886	321	9	−	−	PROPN
ejpam-5886	321	10	1	1	NUM
ejpam-5886	321	11	)	)	PUNCT
ejpam-5886	321	12	(	(	PUNCT
ejpam-5886	321	13	(	(	PUNCT
ejpam-5886	321	14	1	1	NUM
ejpam-5886	321	15	+	+	NUM
ejpam-5886	321	16	ht	ht	PROPN
ejpam-5886	321	17	)	)	PUNCT
ejpam-5886	321	18	a	a	DET
ejpam-5886	321	19	h	h	NOUN
ejpam-5886	321	20	−	−	PROPN
ejpam-5886	321	21	1)((1	1)((1	PROPN
ejpam-5886	322	1	+	+	CCONJ
ejpam-5886	322	2	ht	ht	PROPN
ejpam-5886	322	3	)	)	PUNCT
ejpam-5886	322	4	b	b	PROPN
ejpam-5886	322	5	h	h	NOUN
ejpam-5886	322	6	−	−	NOUN
ejpam-5886	322	7	1	1	NUM
ejpam-5886	322	8	)	)	PUNCT
ejpam-5886	322	9	=	=	VERB
ejpam-5886	322	10	abt	abt	INTJ
ejpam-5886	322	11	(	(	PUNCT
ejpam-5886	322	12	(	(	PUNCT
ejpam-5886	322	13	1	1	NUM
ejpam-5886	322	14	+	+	NUM
ejpam-5886	322	15	ht	ht	PROPN
ejpam-5886	322	16	)	)	PUNCT
ejpam-5886	322	17	a	a	DET
ejpam-5886	322	18	h	h	NOUN
ejpam-5886	322	19	−	−	NOUN
ejpam-5886	322	20	1	1	NUM
ejpam-5886	322	21	)	)	PUNCT
ejpam-5886	322	22	(	(	PUNCT
ejpam-5886	322	23	1	1	NUM
ejpam-5886	322	24	+	+	NUM
ejpam-5886	322	25	ht	ht	X
ejpam-5886	322	26	)	)	PUNCT
ejpam-5886	322	27	abv	abv	PROPN
ejpam-5886	322	28	h	h	PROPN
ejpam-5886	322	29	c0	c0	PROPN
ejpam-5886	322	30	(	(	PUNCT
ejpam-5886	322	31	−abu	−abu	NOUN
ejpam-5886	322	32	h	h	NOUN
ejpam-5886	322	33	log(1	log(1	NOUN
ejpam-5886	323	1	+	+	CCONJ
ejpam-5886	324	1	ht2	ht2	NOUN
ejpam-5886	324	2	)	)	PUNCT
ejpam-5886	324	3	)	)	PUNCT
ejpam-5886	325	1	(	(	PUNCT
ejpam-5886	325	2	(	(	PUNCT
ejpam-5886	325	3	1	1	NUM
ejpam-5886	325	4	+	+	NUM
ejpam-5886	325	5	ht	ht	PROPN
ejpam-5886	325	6	)	)	PUNCT
ejpam-5886	325	7	ab	ab	PROPN
ejpam-5886	325	8	h	h	NOUN
ejpam-5886	325	9	−	−	PROPN
ejpam-5886	325	10	1	1	NUM
ejpam-5886	325	11	)	)	PUNCT
ejpam-5886	325	12	(	(	PUNCT
ejpam-5886	325	13	(	(	PUNCT
ejpam-5886	325	14	1	1	NUM
ejpam-5886	325	15	+	+	NUM
ejpam-5886	325	16	ht	ht	PROPN
ejpam-5886	325	17	)	)	PUNCT
ejpam-5886	325	18	b	b	PROPN
ejpam-5886	325	19	h	h	NOUN
ejpam-5886	325	20	−	−	NOUN
ejpam-5886	325	21	1	1	NUM
ejpam-5886	325	22	)	)	PUNCT
ejpam-5886	325	23	=	=	SYM
ejpam-5886	326	1	b	b	PROPN
ejpam-5886	326	2	∞∑	∞∑	PROPN
ejpam-5886	326	3	n=0	n=0	NUM
ejpam-5886	326	4	βn(h	βn(h	NUM
ejpam-5886	326	5	)	)	PUNCT
ejpam-5886	326	6	(	(	PUNCT
ejpam-5886	326	7	at)n	at)n	PROPN
ejpam-5886	326	8	n	n	CCONJ
ejpam-5886	326	9	!	!	PUNCT
ejpam-5886	327	1	∞∑	∞∑	NUM
ejpam-5886	327	2	k=0	k=0	PROPN
ejpam-5886	327	3	s[h]k	s[h]k	PROPN
ejpam-5886	327	4	(	(	PUNCT
ejpam-5886	327	5	bu	bu	PROPN
ejpam-5886	327	6	,	,	PUNCT
ejpam-5886	327	7	bv	bv	PROPN
ejpam-5886	327	8	)	)	PUNCT
ejpam-5886	327	9	(	(	PUNCT
ejpam-5886	327	10	at)k	at)k	PROPN
ejpam-5886	327	11	k	k	X
ejpam-5886	327	12	!	!	PUNCT
ejpam-5886	328	1	∞∑	∞∑	NUM
ejpam-5886	328	2	m=0	m=0	PROPN
ejpam-5886	328	3	σm(a−	σm(a−	PROPN
ejpam-5886	328	4	1;h	1;h	NUM
ejpam-5886	328	5	)	)	PUNCT
ejpam-5886	328	6	(	(	PUNCT
ejpam-5886	328	7	bt)m	bt)m	PROPN
ejpam-5886	328	8	m	m	PROPN
ejpam-5886	328	9	!	!	PUNCT
ejpam-5886	329	1	=	=	PUNCT
ejpam-5886	329	2	b	b	X
ejpam-5886	330	1	∞∑	∞∑	PRON
ejpam-5886	330	2	n=0	n=0	NUM
ejpam-5886	330	3	βn(h	βn(h	NUM
ejpam-5886	330	4	)	)	PUNCT
ejpam-5886	330	5	(	(	PUNCT
ejpam-5886	330	6	at)n	at)n	PROPN
ejpam-5886	330	7	n	n	CCONJ
ejpam-5886	330	8	!	!	PUNCT
ejpam-5886	331	1	∞∑	∞∑	NUM
ejpam-5886	331	2	k=0	k=0	PROPN
ejpam-5886	331	3	k∑	k∑	PROPN
ejpam-5886	331	4	m=0	m=0	PROPN
ejpam-5886	332	1	(	(	PUNCT
ejpam-5886	332	2	k	k	PROPN
ejpam-5886	332	3	m	m	PROPN
ejpam-5886	332	4	)	)	PUNCT
ejpam-5886	332	5	ak−mbms[h]k−m(bu	ak−mbms[h]k−m(bu	PROPN
ejpam-5886	332	6	,	,	PUNCT
ejpam-5886	332	7	bv)σm(a−	bv)σm(a−	PROPN
ejpam-5886	332	8	1;h	1;h	NUM
ejpam-5886	332	9	)	)	PUNCT
ejpam-5886	332	10	tk	tk	PROPN
ejpam-5886	333	1	k	k	NOUN
ejpam-5886	333	2	!	!	PUNCT
ejpam-5886	333	3	=	=	PUNCT
ejpam-5886	334	1	∞∑	∞∑	PRON
ejpam-5886	334	2	n=0	n=0	NUM
ejpam-5886	334	3	(	(	PUNCT
ejpam-5886	334	4	n∑	n∑	PROPN
ejpam-5886	334	5	k=0	k=0	PROPN
ejpam-5886	334	6	k∑	k∑	PROPN
ejpam-5886	335	1	m=0	m=0	PROPN
ejpam-5886	335	2	(	(	PUNCT
ejpam-5886	335	3	n	n	NOUN
ejpam-5886	335	4	k	k	NOUN
ejpam-5886	335	5	)	)	PUNCT
ejpam-5886	335	6	(	(	PUNCT
ejpam-5886	335	7	k	k	NOUN
ejpam-5886	335	8	m	m	VERB
ejpam-5886	335	9	)	)	PUNCT
ejpam-5886	336	1	an−mbm+1βn−k(h)s	an−mbm+1βn−k(h)s	VERB
ejpam-5886	337	1	[	[	X
ejpam-5886	337	2	h	h	X
ejpam-5886	337	3	]	]	X
ejpam-5886	337	4	n−m(bu	n−m(bu	PROPN
ejpam-5886	337	5	,	,	PUNCT
ejpam-5886	337	6	bv)σm(a−	bv)σm(a−	PROPN
ejpam-5886	337	7	1;h	1;h	NUM
ejpam-5886	337	8	)	)	PUNCT
ejpam-5886	337	9	)	)	PUNCT
ejpam-5886	337	10	tn	tn	PROPN
ejpam-5886	338	1	n	n	PROPN
ejpam-5886	338	2	!	!	PUNCT
ejpam-5886	338	3	.	.	PUNCT
ejpam-5886	339	1	(	(	PUNCT
ejpam-5886	339	2	59	59	NUM
ejpam-5886	339	3	)	)	PUNCT
ejpam-5886	339	4	similarly	similarly	ADV
ejpam-5886	339	5	,	,	PUNCT
ejpam-5886	339	6	we	we	PRON
ejpam-5886	339	7	have	have	VERB
ejpam-5886	339	8	b(t	b(t	VERB
ejpam-5886	339	9	)	)	PUNCT
ejpam-5886	339	10	=	=	PUNCT
ejpam-5886	340	1	∞∑	∞∑	NUM
ejpam-5886	340	2	n=0	n=0	NUM
ejpam-5886	340	3	(	(	PUNCT
ejpam-5886	340	4	n∑	n∑	PROPN
ejpam-5886	340	5	k=0	k=0	PROPN
ejpam-5886	340	6	k∑	k∑	PROPN
ejpam-5886	341	1	m=0	m=0	PROPN
ejpam-5886	341	2	(	(	PUNCT
ejpam-5886	341	3	n	n	NOUN
ejpam-5886	341	4	k	k	NOUN
ejpam-5886	341	5	)	)	PUNCT
ejpam-5886	341	6	(	(	PUNCT
ejpam-5886	341	7	k	k	NOUN
ejpam-5886	341	8	m	m	AUX
ejpam-5886	341	9	)	)	PUNCT
ejpam-5886	341	10	bn−mam+1βn−k(h)s	bn−mam+1βn−k(h)s	PROPN
ejpam-5886	342	1	[	[	X
ejpam-5886	342	2	h	h	X
ejpam-5886	342	3	]	]	X
ejpam-5886	342	4	n−m(au	n−m(au	PROPN
ejpam-5886	342	5	,	,	PUNCT
ejpam-5886	342	6	av)σm(b−	av)σm(b−	VERB
ejpam-5886	342	7	1;h	1;h	NUM
ejpam-5886	342	8	)	)	PUNCT
ejpam-5886	342	9	)	)	PUNCT
ejpam-5886	342	10	tn	tn	PROPN
ejpam-5886	343	1	n	n	PROPN
ejpam-5886	343	2	!	!	PUNCT
ejpam-5886	343	3	.	.	PUNCT
ejpam-5886	344	1	(	(	PUNCT
ejpam-5886	344	2	60	60	X
ejpam-5886	344	3	)	)	PUNCT
ejpam-5886	344	4	comparing	compare	VERB
ejpam-5886	344	5	the	the	DET
ejpam-5886	344	6	both	both	DET
ejpam-5886	344	7	sides	side	NOUN
ejpam-5886	344	8	of	of	ADP
ejpam-5886	344	9	the	the	DET
ejpam-5886	344	10	above	above	ADJ
ejpam-5886	344	11	equations	equation	NOUN
ejpam-5886	344	12	,	,	PUNCT
ejpam-5886	344	13	we	we	PRON
ejpam-5886	344	14	get	get	VERB
ejpam-5886	344	15	(	(	PUNCT
ejpam-5886	344	16	58	58	NUM
ejpam-5886	344	17	)	)	PUNCT
ejpam-5886	344	18	.	.	PUNCT
ejpam-5886	345	1	s.	s.	PROPN
ejpam-5886	345	2	a.	a.	PROPN
ejpam-5886	345	3	wani	wani	PROPN
ejpam-5886	345	4	et	et	PROPN
ejpam-5886	345	5	al	al	PROPN
ejpam-5886	345	6	.	.	PUNCT
ejpam-5886	345	7	/	/	SYM
ejpam-5886	345	8	eur	eur	PROPN
ejpam-5886	345	9	.	.	PUNCT
ejpam-5886	346	1	j.	j.	PROPN
ejpam-5886	346	2	pure	pure	PROPN
ejpam-5886	346	3	appl	appl	PROPN
ejpam-5886	346	4	.	.	PROPN
ejpam-5886	346	5	math	math	PROPN
ejpam-5886	346	6	,	,	PUNCT
ejpam-5886	346	7	18	18	NUM
ejpam-5886	346	8	(	(	PUNCT
ejpam-5886	346	9	2	2	NUM
ejpam-5886	346	10	)	)	PUNCT
ejpam-5886	346	11	(	(	PUNCT
ejpam-5886	346	12	2025	2025	NUM
ejpam-5886	346	13	)	)	PUNCT
ejpam-5886	346	14	,	,	PUNCT
ejpam-5886	346	15	5886	5886	NUM
ejpam-5886	346	16	15	15	NUM
ejpam-5886	346	17	of	of	ADP
ejpam-5886	346	18	16	16	NUM
ejpam-5886	346	19	6	6	NUM
ejpam-5886	346	20	.	.	PUNCT
ejpam-5886	346	21	conclusion	conclusion	VERB
ejpam-5886	346	22	the	the	DET
ejpam-5886	346	23	introduction	introduction	NOUN
ejpam-5886	346	24	and	and	CCONJ
ejpam-5886	346	25	study	study	NOUN
ejpam-5886	346	26	of	of	ADP
ejpam-5886	346	27	∆h	∆h	PROPN
ejpam-5886	346	28	legendre	legendre	PROPN
ejpam-5886	346	29	polynomials	polynomial	NOUN
ejpam-5886	346	30	mark	mark	VERB
ejpam-5886	346	31	a	a	DET
ejpam-5886	346	32	significant	significant	ADJ
ejpam-5886	346	33	advancement	advancement	NOUN
ejpam-5886	346	34	in	in	ADP
ejpam-5886	346	35	polynomial	polynomial	ADJ
ejpam-5886	346	36	theory	theory	NOUN
ejpam-5886	346	37	,	,	PUNCT
ejpam-5886	346	38	particularly	particularly	ADV
ejpam-5886	346	39	in	in	ADP
ejpam-5886	346	40	quantum	quantum	ADJ
ejpam-5886	346	41	mechanics	mechanic	NOUN
ejpam-5886	346	42	and	and	CCONJ
ejpam-5886	346	43	entropy	entropy	NOUN
ejpam-5886	346	44	modeling	modeling	NOUN
ejpam-5886	346	45	.	.	PUNCT
ejpam-5886	347	1	by	by	ADP
ejpam-5886	347	2	incorporating	incorporate	VERB
ejpam-5886	347	3	the	the	DET
ejpam-5886	347	4	monomiality	monomiality	NOUN
ejpam-5886	347	5	principle	principle	NOUN
ejpam-5886	347	6	with	with	ADP
ejpam-5886	347	7	operational	operational	ADJ
ejpam-5886	347	8	norms	norm	NOUN
ejpam-5886	347	9	,	,	PUNCT
ejpam-5886	347	10	these	these	DET
ejpam-5886	347	11	polynomials	polynomial	NOUN
ejpam-5886	347	12	offer	offer	VERB
ejpam-5886	347	13	new	new	ADJ
ejpam-5886	347	14	insights	insight	NOUN
ejpam-5886	347	15	into	into	ADP
ejpam-5886	347	16	unexplored	unexplored	ADJ
ejpam-5886	347	17	mathematical	mathematical	ADJ
ejpam-5886	347	18	domains	domain	NOUN
ejpam-5886	347	19	.	.	PUNCT
ejpam-5886	348	1	this	this	DET
ejpam-5886	348	2	study	study	NOUN
ejpam-5886	348	3	not	not	PART
ejpam-5886	348	4	only	only	ADV
ejpam-5886	348	5	provides	provide	VERB
ejpam-5886	348	6	explicit	explicit	ADJ
ejpam-5886	348	7	formulas	formula	NOUN
ejpam-5886	348	8	and	and	CCONJ
ejpam-5886	348	9	fundamental	fundamental	ADJ
ejpam-5886	348	10	properties	property	NOUN
ejpam-5886	348	11	of	of	ADP
ejpam-5886	348	12	∆h	∆h	PROPN
ejpam-5886	348	13	legendre	legendre	NOUN
ejpam-5886	348	14	polynomials	polynomial	NOUN
ejpam-5886	348	15	but	but	CCONJ
ejpam-5886	348	16	also	also	ADV
ejpam-5886	348	17	establishes	establish	VERB
ejpam-5886	348	18	connections	connection	NOUN
ejpam-5886	348	19	with	with	ADP
ejpam-5886	348	20	other	other	ADJ
ejpam-5886	348	21	well	well	ADV
ejpam-5886	348	22	-	-	PUNCT
ejpam-5886	348	23	known	know	VERB
ejpam-5886	348	24	polynomial	polynomial	ADJ
ejpam-5886	348	25	classes	class	NOUN
ejpam-5886	348	26	,	,	PUNCT
ejpam-5886	348	27	enriching	enrich	VERB
ejpam-5886	348	28	mathematical	mathematical	ADJ
ejpam-5886	348	29	understanding	understanding	NOUN
ejpam-5886	348	30	.	.	PUNCT
ejpam-5886	349	1	future	future	ADJ
ejpam-5886	349	2	research	research	NOUN
ejpam-5886	349	3	could	could	AUX
ejpam-5886	349	4	further	far	ADV
ejpam-5886	349	5	explore	explore	VERB
ejpam-5886	349	6	their	their	PRON
ejpam-5886	349	7	algebraic	algebraic	ADJ
ejpam-5886	349	8	structure	structure	NOUN
ejpam-5886	349	9	and	and	CCONJ
ejpam-5886	349	10	applications	application	NOUN
ejpam-5886	349	11	in	in	ADP
ejpam-5886	349	12	mathematical	mathematical	ADJ
ejpam-5886	349	13	physics	physics	NOUN
ejpam-5886	349	14	and	and	CCONJ
ejpam-5886	349	15	quantum	quantum	NOUN
ejpam-5886	349	16	mechanics	mechanic	NOUN
ejpam-5886	349	17	.	.	PUNCT
ejpam-5886	350	1	additionally	additionally	ADV
ejpam-5886	350	2	,	,	PUNCT
ejpam-5886	350	3	bridging	bridge	VERB
ejpam-5886	350	4	the	the	DET
ejpam-5886	350	5	gap	gap	NOUN
ejpam-5886	350	6	between	between	ADP
ejpam-5886	350	7	theory	theory	NOUN
ejpam-5886	350	8	and	and	CCONJ
ejpam-5886	350	9	practical	practical	ADJ
ejpam-5886	350	10	applications	application	NOUN
ejpam-5886	350	11	in	in	ADP
ejpam-5886	350	12	computational	computational	ADJ
ejpam-5886	350	13	science	science	NOUN
ejpam-5886	350	14	,	,	PUNCT
ejpam-5886	350	15	information	information	NOUN
ejpam-5886	350	16	theory	theory	NOUN
ejpam-5886	350	17	,	,	PUNCT
ejpam-5886	350	18	and	and	CCONJ
ejpam-5886	350	19	statistical	statistical	ADJ
ejpam-5886	350	20	mechanics	mechanic	NOUN
ejpam-5886	350	21	is	be	AUX
ejpam-5886	350	22	crucial	crucial	ADJ
ejpam-5886	350	23	.	.	PUNCT
ejpam-5886	351	1	interdisciplinary	interdisciplinary	ADJ
ejpam-5886	351	2	collaboration	collaboration	NOUN
ejpam-5886	351	3	may	may	AUX
ejpam-5886	351	4	unlock	unlock	VERB
ejpam-5886	351	5	the	the	DET
ejpam-5886	351	6	full	full	ADJ
ejpam-5886	351	7	potential	potential	NOUN
ejpam-5886	351	8	of	of	ADP
ejpam-5886	351	9	∆h	∆h	PROPN
ejpam-5886	351	10	legendre	legendre	NOUN
ejpam-5886	351	11	polynomials	polynomial	NOUN
ejpam-5886	351	12	,	,	PUNCT
ejpam-5886	351	13	paving	pave	VERB
ejpam-5886	351	14	the	the	DET
ejpam-5886	351	15	way	way	NOUN
ejpam-5886	351	16	for	for	ADP
ejpam-5886	351	17	new	new	ADJ
ejpam-5886	351	18	developments	development	NOUN
ejpam-5886	351	19	in	in	ADP
ejpam-5886	351	20	various	various	ADJ
ejpam-5886	351	21	scientific	scientific	ADJ
ejpam-5886	351	22	fields	field	NOUN
ejpam-5886	351	23	.	.	PUNCT
ejpam-5886	352	1	references	reference	NOUN
ejpam-5886	352	2	[	[	X
ejpam-5886	352	3	1	1	NUM
ejpam-5886	352	4	]	]	X
ejpam-5886	352	5	sawani	sawani	NOUN
ejpam-5886	352	6	and	and	CCONJ
ejpam-5886	352	7	s	s	PROPN
ejpam-5886	352	8	khan	khan	PROPN
ejpam-5886	352	9	.	.	PUNCT
ejpam-5886	353	1	properties	property	NOUN
ejpam-5886	353	2	and	and	CCONJ
ejpam-5886	353	3	applications	application	NOUN
ejpam-5886	353	4	of	of	ADP
ejpam-5886	353	5	the	the	DET
ejpam-5886	353	6	gould	gould	PROPN
ejpam-5886	353	7	-	-	PUNCT
ejpam-5886	353	8	hopper	hopper	NOUN
ejpam-5886	353	9	-	-	PUNCT
ejpam-5886	353	10	frobenius	frobenius	NOUN
ejpam-5886	353	11	-	-	PUNCT
ejpam-5886	353	12	euler	euler	NOUN
ejpam-5886	353	13	polynomials	polynomial	NOUN
ejpam-5886	353	14	,	,	PUNCT
ejpam-5886	353	15	tbilisi	tbilisi	PROPN
ejpam-5886	353	16	math	math	PROPN
ejpam-5886	353	17	.	.	PUNCT
ejpam-5886	354	1	j	j	PROPN
ejpam-5886	354	2	,	,	PUNCT
ejpam-5886	354	3	12(1):93–104	12(1):93–104	PROPN
ejpam-5886	354	4	,	,	PUNCT
ejpam-5886	354	5	2019	2019	NUM
ejpam-5886	354	6	.	.	PUNCT
ejpam-5886	355	1	[	[	X
ejpam-5886	355	2	2	2	NUM
ejpam-5886	355	3	]	]	PUNCT
ejpam-5886	355	4	subuhi	subuhi	PROPN
ejpam-5886	355	5	khan	khan	PROPN
ejpam-5886	355	6	,	,	PUNCT
ejpam-5886	355	7	mumtaz	mumtaz	PROPN
ejpam-5886	355	8	riyasat	riyasat	PROPN
ejpam-5886	355	9	,	,	PUNCT
ejpam-5886	355	10	and	and	CCONJ
ejpam-5886	355	11	shahid	shahid	PROPN
ejpam-5886	355	12	ahmad	ahmad	PROPN
ejpam-5886	355	13	wani	wani	PROPN
ejpam-5886	355	14	.	.	PUNCT
ejpam-5886	356	1	on	on	ADP
ejpam-5886	356	2	some	some	DET
ejpam-5886	356	3	classes	class	NOUN
ejpam-5886	356	4	of	of	ADP
ejpam-5886	356	5	differential	differential	ADJ
ejpam-5886	356	6	equations	equation	NOUN
ejpam-5886	356	7	and	and	CCONJ
ejpam-5886	356	8	associated	associate	VERB
ejpam-5886	356	9	integral	integral	ADJ
ejpam-5886	356	10	equations	equation	NOUN
ejpam-5886	356	11	for	for	ADP
ejpam-5886	356	12	the	the	DET
ejpam-5886	356	13	laguerre	laguerre	NOUN
ejpam-5886	356	14	–	–	PUNCT
ejpam-5886	356	15	appell	appell	NOUN
ejpam-5886	356	16	polynomials	polynomial	NOUN
ejpam-5886	356	17	.	.	PUNCT
ejpam-5886	357	1	advances	advance	NOUN
ejpam-5886	357	2	in	in	ADP
ejpam-5886	357	3	pure	pure	ADJ
ejpam-5886	357	4	and	and	CCONJ
ejpam-5886	357	5	applied	applied	ADJ
ejpam-5886	357	6	mathematics	mathematic	NOUN
ejpam-5886	357	7	,	,	PUNCT
ejpam-5886	357	8	9(3):185–194	9(3):185–194	NOUN
ejpam-5886	357	9	,	,	PUNCT
ejpam-5886	357	10	2018	2018	NUM
ejpam-5886	357	11	.	.	PUNCT
ejpam-5886	358	1	[	[	X
ejpam-5886	358	2	3	3	X
ejpam-5886	358	3	]	]	X
ejpam-5886	358	4	w	w	PROPN
ejpam-5886	358	5	ramı́rez	ramı́rez	PROPN
ejpam-5886	358	6	and	and	CCONJ
ejpam-5886	358	7	c	c	NOUN
ejpam-5886	358	8	cesarano	cesarano	PROPN
ejpam-5886	358	9	.	.	PUNCT
ejpam-5886	359	1	some	some	DET
ejpam-5886	359	2	new	new	ADJ
ejpam-5886	359	3	classes	class	NOUN
ejpam-5886	359	4	of	of	ADP
ejpam-5886	359	5	degenerated	degenerated	ADJ
ejpam-5886	359	6	generalized	generalized	ADJ
ejpam-5886	359	7	apostolbernoulli	apostolbernoulli	NOUN
ejpam-5886	359	8	,	,	PUNCT
ejpam-5886	359	9	apostol	apostol	NOUN
ejpam-5886	359	10	-	-	PUNCT
ejpam-5886	359	11	euler	euler	NOUN
ejpam-5886	359	12	and	and	CCONJ
ejpam-5886	359	13	apostol	apostol	NOUN
ejpam-5886	359	14	-	-	PUNCT
ejpam-5886	359	15	genocchi	genocchi	PROPN
ejpam-5886	359	16	polynomials	polynomial	NOUN
ejpam-5886	359	17	.	.	PUNCT
ejpam-5886	360	1	carpathian	carpathian	ADJ
ejpam-5886	360	2	mathematical	mathematical	ADJ
ejpam-5886	360	3	publications	publication	NOUN
ejpam-5886	360	4	,	,	PUNCT
ejpam-5886	360	5	14(2):354–363	14(2):354–363	NUM
ejpam-5886	360	6	,	,	PUNCT
ejpam-5886	360	7	2022	2022	NUM
ejpam-5886	360	8	.	.	PUNCT
ejpam-5886	361	1	[	[	X
ejpam-5886	361	2	4	4	X
ejpam-5886	361	3	]	]	PUNCT
ejpam-5886	361	4	mohra	mohra	NOUN
ejpam-5886	361	5	zayed	zayed	ADJ
ejpam-5886	361	6	and	and	CCONJ
ejpam-5886	361	7	shahid	shahid	PROPN
ejpam-5886	361	8	ahmad	ahmad	PROPN
ejpam-5886	361	9	wani	wani	PROPN
ejpam-5886	361	10	.	.	PUNCT
ejpam-5886	362	1	a	a	DET
ejpam-5886	362	2	study	study	NOUN
ejpam-5886	362	3	on	on	ADP
ejpam-5886	362	4	generalized	generalized	ADJ
ejpam-5886	362	5	degenerate	degenerate	ADJ
ejpam-5886	362	6	form	form	NOUN
ejpam-5886	362	7	of	of	ADP
ejpam-5886	362	8	2d	2d	NUM
ejpam-5886	362	9	appell	appell	ADJ
ejpam-5886	362	10	polynomials	polynomial	NOUN
ejpam-5886	362	11	via	via	ADP
ejpam-5886	362	12	fractional	fractional	ADJ
ejpam-5886	362	13	operators	operator	NOUN
ejpam-5886	362	14	.	.	PUNCT
ejpam-5886	363	1	fractal	fractal	ADJ
ejpam-5886	363	2	and	and	CCONJ
ejpam-5886	363	3	fractional	fractional	ADJ
ejpam-5886	363	4	,	,	PUNCT
ejpam-5886	363	5	7(10):723	7(10):723	PROPN
ejpam-5886	363	6	,	,	PUNCT
ejpam-5886	363	7	2023	2023	NUM
ejpam-5886	363	8	.	.	PUNCT
ejpam-5886	364	1	[	[	X
ejpam-5886	364	2	5	5	NUM
ejpam-5886	364	3	]	]	PUNCT
ejpam-5886	364	4	mohra	mohra	NOUN
ejpam-5886	364	5	zayed	zaye	VERB
ejpam-5886	364	6	,	,	PUNCT
ejpam-5886	364	7	shahid	shahid	PROPN
ejpam-5886	364	8	ahmadwani	ahmadwani	VERB
ejpam-5886	364	9	,	,	PUNCT
ejpam-5886	364	10	and	and	CCONJ
ejpam-5886	364	11	yamilet	yamilet	PROPN
ejpam-5886	364	12	quintana	quintana	PROPN
ejpam-5886	364	13	.	.	PUNCT
ejpam-5886	365	1	properties	property	NOUN
ejpam-5886	365	2	of	of	ADP
ejpam-5886	365	3	multivariate	multivariate	NOUN
ejpam-5886	365	4	hermite	hermite	ADJ
ejpam-5886	365	5	polynomials	polynomial	NOUN
ejpam-5886	365	6	in	in	ADP
ejpam-5886	365	7	correlation	correlation	NOUN
ejpam-5886	365	8	with	with	ADP
ejpam-5886	365	9	frobenius	frobenius	NOUN
ejpam-5886	365	10	–	–	PUNCT
ejpam-5886	365	11	euler	euler	NOUN
ejpam-5886	365	12	polynomials	polynomial	NOUN
ejpam-5886	365	13	.	.	PUNCT
ejpam-5886	366	1	mathematics	mathematic	NOUN
ejpam-5886	366	2	,	,	PUNCT
ejpam-5886	366	3	11(16):3439	11(16):3439	NUM
ejpam-5886	366	4	,	,	PUNCT
ejpam-5886	366	5	2023	2023	NUM
ejpam-5886	366	6	.	.	PUNCT
ejpam-5886	367	1	[	[	X
ejpam-5886	367	2	6	6	NUM
ejpam-5886	367	3	]	]	PUNCT
ejpam-5886	367	4	shahid	shahid	PROPN
ejpam-5886	367	5	ahmad	ahmad	PROPN
ejpam-5886	367	6	wani	wani	PROPN
ejpam-5886	367	7	.	.	PUNCT
ejpam-5886	368	1	two	two	NUM
ejpam-5886	368	2	-	-	PUNCT
ejpam-5886	368	3	iterated	iterate	VERB
ejpam-5886	368	4	degenerate	degenerate	ADJ
ejpam-5886	368	5	appell	appell	ADJ
ejpam-5886	368	6	polynomials	polynomial	NOUN
ejpam-5886	368	7	:	:	PUNCT
ejpam-5886	368	8	properties	property	NOUN
ejpam-5886	368	9	and	and	CCONJ
ejpam-5886	368	10	applications	application	NOUN
ejpam-5886	368	11	.	.	PUNCT
ejpam-5886	369	1	arab	arab	PROPN
ejpam-5886	369	2	journal	journal	PROPN
ejpam-5886	369	3	of	of	ADP
ejpam-5886	369	4	basic	basic	ADJ
ejpam-5886	369	5	and	and	CCONJ
ejpam-5886	369	6	applied	applied	ADJ
ejpam-5886	369	7	sciences	science	NOUN
ejpam-5886	369	8	,	,	PUNCT
ejpam-5886	369	9	31(1):83–92	31(1):83–92	NUM
ejpam-5886	369	10	,	,	PUNCT
ejpam-5886	369	11	2024	2024	NUM
ejpam-5886	369	12	.	.	PUNCT
ejpam-5886	370	1	[	[	X
ejpam-5886	370	2	7	7	X
ejpam-5886	370	3	]	]	X
ejpam-5886	370	4	shahid	shahid	PROPN
ejpam-5886	370	5	ahmad	ahmad	PROPN
ejpam-5886	370	6	wani	wani	PROPN
ejpam-5886	370	7	,	,	PUNCT
ejpam-5886	370	8	kinda	kinda	ADV
ejpam-5886	370	9	abuasbeh	abuasbeh	PROPN
ejpam-5886	370	10	,	,	PUNCT
ejpam-5886	370	11	georgia	georgia	PROPN
ejpam-5886	370	12	irina	irina	PROPN
ejpam-5886	370	13	oros	oros	PROPN
ejpam-5886	370	14	,	,	PUNCT
ejpam-5886	370	15	and	and	CCONJ
ejpam-5886	370	16	salma	salma	PROPN
ejpam-5886	370	17	trabelsi	trabelsi	PROPN
ejpam-5886	370	18	.	.	PUNCT
ejpam-5886	371	1	studies	study	NOUN
ejpam-5886	371	2	on	on	ADP
ejpam-5886	371	3	special	special	ADJ
ejpam-5886	371	4	polynomials	polynomial	NOUN
ejpam-5886	371	5	involving	involve	VERB
ejpam-5886	371	6	degenerate	degenerate	ADJ
ejpam-5886	371	7	appell	appell	ADJ
ejpam-5886	371	8	polynomials	polynomial	NOUN
ejpam-5886	371	9	and	and	CCONJ
ejpam-5886	371	10	fractional	fractional	ADJ
ejpam-5886	371	11	derivative	derivative	ADJ
ejpam-5886	371	12	.	.	PUNCT
ejpam-5886	371	13	symmetry	symmetry	NOUN
ejpam-5886	371	14	,	,	PUNCT
ejpam-5886	371	15	15(4):840	15(4):840	NUM
ejpam-5886	371	16	,	,	PUNCT
ejpam-5886	371	17	2023	2023	NUM
ejpam-5886	371	18	.	.	PUNCT
ejpam-5886	372	1	[	[	X
ejpam-5886	372	2	8	8	NUM
ejpam-5886	372	3	]	]	PUNCT
ejpam-5886	372	4	sedighe	sedighe	ADJ
ejpam-5886	372	5	sadeghi	sadeghi	PROPN
ejpam-5886	372	6	roshan	roshan	PROPN
ejpam-5886	372	7	,	,	PUNCT
ejpam-5886	372	8	hossein	hossein	PROPN
ejpam-5886	372	9	jafari	jafari	PROPN
ejpam-5886	372	10	,	,	PUNCT
ejpam-5886	372	11	and	and	CCONJ
ejpam-5886	372	12	dumitru	dumitru	PROPN
ejpam-5886	372	13	baleanu	baleanu	NOUN
ejpam-5886	372	14	.	.	PUNCT
ejpam-5886	373	1	solving	solve	VERB
ejpam-5886	373	2	fdes	fde	NOUN
ejpam-5886	373	3	with	with	ADP
ejpam-5886	373	4	caputo	caputo	PROPN
ejpam-5886	373	5	-	-	PUNCT
ejpam-5886	373	6	fabrizio	fabrizio	PROPN
ejpam-5886	373	7	derivative	derivative	NOUN
ejpam-5886	373	8	by	by	ADP
ejpam-5886	373	9	operational	operational	ADJ
ejpam-5886	373	10	matrix	matrix	NOUN
ejpam-5886	373	11	based	base	VERB
ejpam-5886	373	12	on	on	ADP
ejpam-5886	373	13	genocchi	genocchi	PROPN
ejpam-5886	373	14	polynomials	polynomial	NOUN
ejpam-5886	373	15	.	.	PUNCT
ejpam-5886	374	1	mathematical	mathematical	ADJ
ejpam-5886	374	2	methods	method	NOUN
ejpam-5886	374	3	in	in	ADP
ejpam-5886	374	4	the	the	DET
ejpam-5886	374	5	applied	apply	VERB
ejpam-5886	374	6	sciences	science	NOUN
ejpam-5886	374	7	,	,	PUNCT
ejpam-5886	374	8	41(18):9134–9141	41(18):9134–9141	NUM
ejpam-5886	374	9	,	,	PUNCT
ejpam-5886	374	10	2018	2018	NUM
ejpam-5886	374	11	.	.	PUNCT
ejpam-5886	375	1	[	[	X
ejpam-5886	375	2	9	9	NUM
ejpam-5886	375	3	]	]	SYM
ejpam-5886	375	4	g	g	NOUN
ejpam-5886	375	5	dattoli	dattoli	NOUN
ejpam-5886	375	6	,	,	PUNCT
ejpam-5886	375	7	pe	pe	PROPN
ejpam-5886	375	8	ricci	ricci	PROPN
ejpam-5886	375	9	,	,	PUNCT
ejpam-5886	375	10	c	c	PROPN
ejpam-5886	375	11	cesarano	cesarano	ADJ
ejpam-5886	375	12	,	,	PUNCT
ejpam-5886	375	13	and	and	CCONJ
ejpam-5886	375	14	l	l	NOUN
ejpam-5886	375	15	vázquez	vázquez	PROPN
ejpam-5886	375	16	.	.	PROPN
ejpam-5886	375	17	special	special	ADJ
ejpam-5886	375	18	polynomials	polynomial	NOUN
ejpam-5886	375	19	and	and	CCONJ
ejpam-5886	375	20	fractional	fractional	ADJ
ejpam-5886	375	21	calculus	calculus	NOUN
ejpam-5886	375	22	.	.	PUNCT
ejpam-5886	376	1	mathematical	mathematical	ADJ
ejpam-5886	376	2	and	and	CCONJ
ejpam-5886	376	3	computer	computer	NOUN
ejpam-5886	376	4	modelling	modelling	NOUN
ejpam-5886	376	5	,	,	PUNCT
ejpam-5886	376	6	37(7	37(7	PROPN
ejpam-5886	376	7	-	-	PUNCT
ejpam-5886	376	8	8):729–733	8):729–733	NUM
ejpam-5886	376	9	,	,	PUNCT
ejpam-5886	376	10	2003	2003	NUM
ejpam-5886	376	11	.	.	PUNCT
ejpam-5886	377	1	[	[	X
ejpam-5886	377	2	10	10	NUM
ejpam-5886	377	3	]	]	X
ejpam-5886	377	4	g	g	NOUN
ejpam-5886	377	5	dattoli	dattoli	NOUN
ejpam-5886	377	6	,	,	PUNCT
ejpam-5886	377	7	s	s	VERB
ejpam-5886	377	8	lorenzutta	lorenzutta	ADJ
ejpam-5886	377	9	,	,	PUNCT
ejpam-5886	377	10	am	be	AUX
ejpam-5886	377	11	mancho	mancho	NOUN
ejpam-5886	377	12	,	,	PUNCT
ejpam-5886	377	13	and	and	CCONJ
ejpam-5886	377	14	a	a	DET
ejpam-5886	377	15	torre	torre	PROPN
ejpam-5886	377	16	.	.	PUNCT
ejpam-5886	378	1	generalized	generalize	VERB
ejpam-5886	378	2	polynomials	polynomial	NOUN
ejpam-5886	378	3	and	and	CCONJ
ejpam-5886	378	4	associated	associate	VERB
ejpam-5886	378	5	operational	operational	ADJ
ejpam-5886	378	6	identities	identity	NOUN
ejpam-5886	378	7	.	.	PUNCT
ejpam-5886	379	1	journal	journal	NOUN
ejpam-5886	379	2	of	of	ADP
ejpam-5886	379	3	computational	computational	ADJ
ejpam-5886	379	4	and	and	CCONJ
ejpam-5886	379	5	applied	applied	ADJ
ejpam-5886	379	6	mathematics	mathematic	NOUN
ejpam-5886	379	7	,	,	PUNCT
ejpam-5886	379	8	108(1	108(1	NUM
ejpam-5886	379	9	-	-	SYM
ejpam-5886	379	10	2):209–218	2):209–218	NUM
ejpam-5886	379	11	,	,	PUNCT
ejpam-5886	379	12	1999	1999	NUM
ejpam-5886	379	13	.	.	PUNCT
ejpam-5886	380	1	s.	s.	PROPN
ejpam-5886	380	2	a.	a.	PROPN
ejpam-5886	380	3	wani	wani	PROPN
ejpam-5886	380	4	et	et	PROPN
ejpam-5886	380	5	al	al	PROPN
ejpam-5886	380	6	.	.	PUNCT
ejpam-5886	380	7	/	/	SYM
ejpam-5886	380	8	eur	eur	PROPN
ejpam-5886	380	9	.	.	PUNCT
ejpam-5886	381	1	j.	j.	PROPN
ejpam-5886	381	2	pure	pure	PROPN
ejpam-5886	381	3	appl	appl	PROPN
ejpam-5886	381	4	.	.	PROPN
ejpam-5886	381	5	math	math	PROPN
ejpam-5886	381	6	,	,	PUNCT
ejpam-5886	381	7	18	18	NUM
ejpam-5886	381	8	(	(	PUNCT
ejpam-5886	381	9	2	2	NUM
ejpam-5886	381	10	)	)	PUNCT
ejpam-5886	381	11	(	(	PUNCT
ejpam-5886	381	12	2025	2025	NUM
ejpam-5886	381	13	)	)	PUNCT
ejpam-5886	381	14	,	,	PUNCT
ejpam-5886	381	15	5886	5886	NUM
ejpam-5886	381	16	16	16	NUM
ejpam-5886	381	17	of	of	ADP
ejpam-5886	381	18	16	16	NUM
ejpam-5886	382	1	[	[	X
ejpam-5886	382	2	11	11	NUM
ejpam-5886	382	3	]	]	X
ejpam-5886	382	4	giuseppe	giuseppe	PROPN
ejpam-5886	382	5	dattoli	dattoli	PROPN
ejpam-5886	382	6	,	,	PUNCT
ejpam-5886	382	7	paolo	paolo	PROPN
ejpam-5886	382	8	e	e	PROPN
ejpam-5886	382	9	ricci	ricci	PROPN
ejpam-5886	382	10	,	,	PUNCT
ejpam-5886	382	11	and	and	CCONJ
ejpam-5886	382	12	clemente	clemente	PROPN
ejpam-5886	382	13	cesarano	cesarano	PROPN
ejpam-5886	382	14	.	.	PUNCT
ejpam-5886	383	1	a	a	DET
ejpam-5886	383	2	note	note	NOUN
ejpam-5886	383	3	on	on	ADP
ejpam-5886	383	4	legendre	legendre	PROPN
ejpam-5886	383	5	polynomials	polynomial	NOUN
ejpam-5886	383	6	.	.	PUNCT
ejpam-5886	384	1	international	international	ADJ
ejpam-5886	384	2	journal	journal	PROPN
ejpam-5886	384	3	of	of	ADP
ejpam-5886	384	4	nonlinear	nonlinear	PROPN
ejpam-5886	384	5	sciences	sciences	PROPN
ejpam-5886	384	6	and	and	CCONJ
ejpam-5886	384	7	numerical	numerical	PROPN
ejpam-5886	384	8	simulation	simulation	PROPN
ejpam-5886	384	9	,	,	PUNCT
ejpam-5886	384	10	2(4):365–370	2(4):365–370	NUM
ejpam-5886	384	11	,	,	PUNCT
ejpam-5886	384	12	2001	2001	NUM
ejpam-5886	384	13	.	.	PUNCT
ejpam-5886	385	1	[	[	X
ejpam-5886	385	2	12	12	NUM
ejpam-5886	385	3	]	]	PUNCT
ejpam-5886	385	4	richard	richard	NOUN
ejpam-5886	385	5	a	a	DET
ejpam-5886	385	6	silverman	silverman	PROPN
ejpam-5886	386	1	et	et	PROPN
ejpam-5886	386	2	al	al	PROPN
ejpam-5886	386	3	.	.	PROPN
ejpam-5886	386	4	special	special	ADJ
ejpam-5886	386	5	functions	function	NOUN
ejpam-5886	386	6	and	and	CCONJ
ejpam-5886	386	7	their	their	PRON
ejpam-5886	386	8	applications	application	NOUN
ejpam-5886	386	9	.	.	PUNCT
ejpam-5886	387	1	courier	courier	NOUN
ejpam-5886	387	2	corporation	corporation	NOUN
ejpam-5886	387	3	,	,	PUNCT
ejpam-5886	387	4	1972	1972	NUM
ejpam-5886	387	5	.	.	PUNCT
ejpam-5886	388	1	[	[	X
ejpam-5886	388	2	13	13	NUM
ejpam-5886	388	3	]	]	SYM
ejpam-5886	388	4	waseem	waseem	PROPN
ejpam-5886	388	5	a	a	DET
ejpam-5886	388	6	khan	khan	PROPN
ejpam-5886	388	7	,	,	PUNCT
ejpam-5886	388	8	abdulghani	abdulghani	ADJ
ejpam-5886	388	9	muhyi	muhyi	NOUN
ejpam-5886	388	10	,	,	PUNCT
ejpam-5886	388	11	rifaqat	rifaqat	PROPN
ejpam-5886	388	12	ali	ali	PROPN
ejpam-5886	388	13	,	,	PUNCT
ejpam-5886	388	14	khaled	khaled	PROPN
ejpam-5886	388	15	ahmad	ahmad	PROPN
ejpam-5886	388	16	hassan	hassan	PROPN
ejpam-5886	388	17	alzobydi	alzobydi	PROPN
ejpam-5886	388	18	,	,	PUNCT
ejpam-5886	388	19	manoj	manoj	PROPN
ejpam-5886	388	20	singh	singh	PROPN
ejpam-5886	388	21	,	,	PUNCT
ejpam-5886	388	22	and	and	CCONJ
ejpam-5886	388	23	praveen	praveen	PROPN
ejpam-5886	388	24	agarwal	agarwal	PROPN
ejpam-5886	388	25	.	.	PUNCT
ejpam-5886	389	1	a	a	DET
ejpam-5886	389	2	new	new	ADJ
ejpam-5886	389	3	family	family	NOUN
ejpam-5886	389	4	of	of	ADP
ejpam-5886	389	5	degenerate	degenerate	ADJ
ejpam-5886	389	6	poly	poly	ADJ
ejpam-5886	389	7	-	-	PUNCT
ejpam-5886	389	8	bernoulli	bernoulli	NOUN
ejpam-5886	389	9	polynomials	polynomial	NOUN
ejpam-5886	389	10	of	of	ADP
ejpam-5886	389	11	the	the	DET
ejpam-5886	389	12	second	second	ADJ
ejpam-5886	389	13	kind	kind	NOUN
ejpam-5886	389	14	with	with	ADP
ejpam-5886	389	15	its	its	PRON
ejpam-5886	389	16	certain	certain	ADJ
ejpam-5886	389	17	related	related	ADJ
ejpam-5886	389	18	properties	property	NOUN
ejpam-5886	389	19	.	.	PUNCT
ejpam-5886	390	1	aims	aim	VERB
ejpam-5886	390	2	math	math	NOUN
ejpam-5886	390	3	,	,	PUNCT
ejpam-5886	390	4	6(11):12680–12697	6(11):12680–12697	NUM
ejpam-5886	390	5	,	,	PUNCT
ejpam-5886	390	6	2021	2021	NUM
ejpam-5886	390	7	.	.	PUNCT
ejpam-5886	391	1	[	[	X
ejpam-5886	391	2	14	14	NUM
ejpam-5886	391	3	]	]	PUNCT
ejpam-5886	391	4	waseem	waseem	PROPN
ejpam-5886	391	5	ahmad	ahmad	PROPN
ejpam-5886	391	6	khan	khan	PROPN
ejpam-5886	391	7	,	,	PUNCT
ejpam-5886	391	8	ugur	ugur	PROPN
ejpam-5886	391	9	duran	duran	PROPN
ejpam-5886	391	10	,	,	PUNCT
ejpam-5886	391	11	jihad	jihad	NOUN
ejpam-5886	391	12	younis	younis	PROPN
ejpam-5886	391	13	,	,	PUNCT
ejpam-5886	391	14	and	and	CCONJ
ejpam-5886	391	15	cheon	cheon	PROPN
ejpam-5886	391	16	seoung	seoung	PROPN
ejpam-5886	391	17	ryoo	ryoo	NOUN
ejpam-5886	391	18	.	.	PUNCT
ejpam-5886	392	1	on	on	ADP
ejpam-5886	392	2	some	some	DET
ejpam-5886	392	3	extensions	extension	NOUN
ejpam-5886	392	4	for	for	ADP
ejpam-5886	392	5	degenerate	degenerate	ADJ
ejpam-5886	392	6	frobenius	frobenius	NOUN
ejpam-5886	392	7	-	-	PUNCT
ejpam-5886	392	8	euler	euler	NOUN
ejpam-5886	392	9	-	-	PUNCT
ejpam-5886	392	10	genocchi	genocchi	PROPN
ejpam-5886	392	11	polynomials	polynomial	VERB
ejpam-5886	392	12	with	with	ADP
ejpam-5886	392	13	applications	application	NOUN
ejpam-5886	392	14	in	in	ADP
ejpam-5886	392	15	computer	computer	NOUN
ejpam-5886	392	16	modeling	modeling	NOUN
ejpam-5886	392	17	.	.	PUNCT
ejpam-5886	393	1	applied	apply	VERB
ejpam-5886	393	2	mathematics	mathematic	NOUN
ejpam-5886	393	3	in	in	ADP
ejpam-5886	393	4	science	science	NOUN
ejpam-5886	393	5	and	and	CCONJ
ejpam-5886	393	6	engineering	engineering	NOUN
ejpam-5886	393	7	,	,	PUNCT
ejpam-5886	393	8	32(1):2297072	32(1):2297072	NUM
ejpam-5886	393	9	,	,	PUNCT
ejpam-5886	393	10	2024	2024	NUM
ejpam-5886	393	11	.	.	PUNCT
ejpam-5886	394	1	[	[	X
ejpam-5886	394	2	15	15	NUM
ejpam-5886	394	3	]	]	X
ejpam-5886	394	4	waseem	waseem	PROPN
ejpam-5886	394	5	a	a	DET
ejpam-5886	394	6	khan	khan	PROPN
ejpam-5886	394	7	.	.	PUNCT
ejpam-5886	395	1	a	a	DET
ejpam-5886	395	2	note	note	NOUN
ejpam-5886	395	3	on	on	ADP
ejpam-5886	395	4	degenerate	degenerate	ADJ
ejpam-5886	395	5	hermite	hermite	ADJ
ejpam-5886	395	6	poly	poly	ADJ
ejpam-5886	395	7	-	-	PUNCT
ejpam-5886	395	8	bernoulli	bernoulli	NOUN
ejpam-5886	395	9	numbers	number	NOUN
ejpam-5886	395	10	and	and	CCONJ
ejpam-5886	395	11	polynomials	polynomial	NOUN
ejpam-5886	395	12	.	.	PUNCT
ejpam-5886	396	1	j.	j.	PROPN
ejpam-5886	396	2	class	class	PROPN
ejpam-5886	396	3	.	.	PUNCT
ejpam-5886	397	1	anal	anal	PROPN
ejpam-5886	397	2	,	,	PUNCT
ejpam-5886	397	3	8(1):65–76	8(1):65–76	NUM
ejpam-5886	397	4	,	,	PUNCT
ejpam-5886	397	5	2016	2016	NUM
ejpam-5886	397	6	.	.	PUNCT
ejpam-5886	398	1	[	[	X
ejpam-5886	398	2	16	16	NUM
ejpam-5886	398	3	]	]	PUNCT
ejpam-5886	398	4	waseem	waseem	PROPN
ejpam-5886	398	5	ahmad	ahmad	PROPN
ejpam-5886	398	6	khan	khan	PROPN
ejpam-5886	398	7	and	and	CCONJ
ejpam-5886	398	8	maryam	maryam	PROPN
ejpam-5886	398	9	salem	salem	PROPN
ejpam-5886	398	10	alatawi	alatawi	VERB
ejpam-5886	398	11	.	.	PUNCT
ejpam-5886	399	1	a	a	DET
ejpam-5886	399	2	note	note	NOUN
ejpam-5886	399	3	on	on	ADP
ejpam-5886	399	4	modified	modified	ADJ
ejpam-5886	399	5	degenerate	degenerate	ADJ
ejpam-5886	399	6	changhee	changhee	NOUN
ejpam-5886	399	7	–	–	PUNCT
ejpam-5886	399	8	genocchi	genocchi	PROPN
ejpam-5886	399	9	polynomials	polynomial	NOUN
ejpam-5886	399	10	of	of	ADP
ejpam-5886	399	11	the	the	DET
ejpam-5886	399	12	second	second	ADJ
ejpam-5886	399	13	kind	kind	NOUN
ejpam-5886	399	14	.	.	PUNCT
ejpam-5886	400	1	symmetry	symmetry	NOUN
ejpam-5886	400	2	,	,	PUNCT
ejpam-5886	400	3	15(1):136	15(1):136	NOUN
ejpam-5886	400	4	,	,	PUNCT
ejpam-5886	400	5	2023	2023	NUM
ejpam-5886	400	6	.	.	PUNCT
ejpam-5886	401	1	[	[	X
ejpam-5886	401	2	17	17	NUM
ejpam-5886	401	3	]	]	X
ejpam-5886	401	4	noor	noor	PROPN
ejpam-5886	401	5	alam	alam	PROPN
ejpam-5886	401	6	,	,	PUNCT
ejpam-5886	401	7	shahid	shahid	PROPN
ejpam-5886	401	8	ahmad	ahmad	PROPN
ejpam-5886	401	9	wani	wani	PROPN
ejpam-5886	401	10	,	,	PUNCT
ejpam-5886	401	11	waseem	waseem	PROPN
ejpam-5886	401	12	ahmad	ahmad	PROPN
ejpam-5886	401	13	khan	khan	PROPN
ejpam-5886	401	14	,	,	PUNCT
ejpam-5886	401	15	and	and	CCONJ
ejpam-5886	401	16	hasan	hasan	PROPN
ejpam-5886	401	17	nihal	nihal	PROPN
ejpam-5886	401	18	zaidi	zaidi	PROPN
ejpam-5886	401	19	.	.	PUNCT
ejpam-5886	402	1	investigating	investigate	VERB
ejpam-5886	402	2	the	the	DET
ejpam-5886	402	3	properties	property	NOUN
ejpam-5886	402	4	and	and	CCONJ
ejpam-5886	402	5	dynamic	dynamic	ADJ
ejpam-5886	402	6	applications	application	NOUN
ejpam-5886	402	7	of	of	ADP
ejpam-5886	402	8	δ	δ	PROPN
ejpam-5886	402	9	h	h	PROPN
ejpam-5886	402	10	legendre	legendre	PROPN
ejpam-5886	402	11	–	–	PUNCT
ejpam-5886	402	12	appell	appell	NOUN
ejpam-5886	402	13	polynomials	polynomial	NOUN
ejpam-5886	402	14	.	.	PUNCT
ejpam-5886	403	1	mathematics	mathematic	NOUN
ejpam-5886	403	2	,	,	PUNCT
ejpam-5886	403	3	12(13):1973	12(13):1973	NUM
ejpam-5886	403	4	,	,	PUNCT
ejpam-5886	403	5	2024	2024	NUM
ejpam-5886	403	6	.	.	PUNCT
ejpam-5886	404	1	[	[	X
ejpam-5886	404	2	18	18	NUM
ejpam-5886	404	3	]	]	X
ejpam-5886	404	4	noor	noor	PROPN
ejpam-5886	404	5	alam	alam	PROPN
ejpam-5886	404	6	,	,	PUNCT
ejpam-5886	404	7	shahid	shahid	PROPN
ejpam-5886	404	8	ahmad	ahmad	PROPN
ejpam-5886	404	9	wani	wani	PROPN
ejpam-5886	404	10	,	,	PUNCT
ejpam-5886	404	11	waseem	waseem	PROPN
ejpam-5886	404	12	ahmad	ahmad	PROPN
ejpam-5886	404	13	khan	khan	PROPN
ejpam-5886	404	14	,	,	PUNCT
ejpam-5886	404	15	fakhredine	fakhredine	PROPN
ejpam-5886	404	16	gassem	gassem	NOUN
ejpam-5886	404	17	,	,	PUNCT
ejpam-5886	404	18	and	and	CCONJ
ejpam-5886	404	19	anas	anas	PROPN
ejpam-5886	404	20	altaleb	altaleb	PROPN
ejpam-5886	404	21	.	.	PUNCT
ejpam-5886	405	1	exploring	explore	VERB
ejpam-5886	405	2	properties	property	NOUN
ejpam-5886	405	3	and	and	CCONJ
ejpam-5886	405	4	applications	application	NOUN
ejpam-5886	405	5	of	of	ADP
ejpam-5886	405	6	laguerre	laguerre	NOUN
ejpam-5886	405	7	special	special	ADJ
ejpam-5886	405	8	polynomials	polynomial	NOUN
ejpam-5886	405	9	involving	involve	VERB
ejpam-5886	405	10	the	the	DET
ejpam-5886	405	11	δ	δ	PROPN
ejpam-5886	405	12	h	h	NOUN
ejpam-5886	405	13	form	form	NOUN
ejpam-5886	405	14	.	.	PUNCT
ejpam-5886	406	1	symmetry	symmetry	NOUN
ejpam-5886	406	2	,	,	PUNCT
ejpam-5886	406	3	16(9):1154	16(9):1154	NUM
ejpam-5886	406	4	,	,	PUNCT
ejpam-5886	406	5	2024	2024	NUM
ejpam-5886	406	6	.	.	PUNCT
ejpam-5886	407	1	[	[	X
ejpam-5886	407	2	19	19	NUM
ejpam-5886	407	3	]	]	X
ejpam-5886	407	4	francesco	francesco	NOUN
ejpam-5886	407	5	a	a	DET
ejpam-5886	407	6	costabile	costabile	NOUN
ejpam-5886	407	7	and	and	CCONJ
ejpam-5886	407	8	elisabetta	elisabetta	PROPN
ejpam-5886	407	9	longo	longo	PROPN
ejpam-5886	407	10	.	.	PUNCT
ejpam-5886	408	1	δ	δ	PROPN
ejpam-5886	408	2	h	h	NOUN
ejpam-5886	408	3	-	-	PUNCT
ejpam-5886	408	4	appell	appell	ADJ
ejpam-5886	408	5	sequences	sequence	NOUN
ejpam-5886	408	6	and	and	CCONJ
ejpam-5886	408	7	related	relate	VERB
ejpam-5886	408	8	interpolation	interpolation	NOUN
ejpam-5886	408	9	problem	problem	NOUN
ejpam-5886	408	10	.	.	PUNCT
ejpam-5886	409	1	numerical	numerical	ADJ
ejpam-5886	409	2	algorithms	algorithms	PROPN
ejpam-5886	409	3	,	,	PUNCT
ejpam-5886	409	4	63:165–186	63:165–186	PROPN
ejpam-5886	409	5	,	,	PUNCT
ejpam-5886	409	6	2013	2013	NUM
ejpam-5886	409	7	.	.	PUNCT
ejpam-5886	410	1	[	[	X
ejpam-5886	410	2	20	20	NUM
ejpam-5886	410	3	]	]	PUNCT
ejpam-5886	410	4	mumtaz	mumtaz	NOUN
ejpam-5886	410	5	riyasat	riyasat	PROPN
ejpam-5886	410	6	,	,	PUNCT
ejpam-5886	410	7	amal	amal	PROPN
ejpam-5886	410	8	s	s	PART
ejpam-5886	410	9	alali	alali	ADJ
ejpam-5886	410	10	,	,	PUNCT
ejpam-5886	410	11	and	and	CCONJ
ejpam-5886	410	12	subuhi	subuhi	PROPN
ejpam-5886	410	13	khan	khan	PROPN
ejpam-5886	410	14	.	.	PUNCT
ejpam-5886	411	1	certain	certain	ADJ
ejpam-5886	411	2	properties	property	NOUN
ejpam-5886	411	3	of	of	ADP
ejpam-5886	411	4	3d	3d	NUM
ejpam-5886	411	5	degenerate	degenerate	ADJ
ejpam-5886	411	6	generalized	generalized	ADJ
ejpam-5886	411	7	fubini	fubini	ADJ
ejpam-5886	411	8	polynomials	polynomial	NOUN
ejpam-5886	411	9	and	and	CCONJ
ejpam-5886	411	10	applications	application	NOUN
ejpam-5886	411	11	.	.	PUNCT
ejpam-5886	412	1	afrika	afrika	PROPN
ejpam-5886	412	2	matematika	matematika	PROPN
ejpam-5886	412	3	,	,	PUNCT
ejpam-5886	412	4	35(2):47	35(2):47	PROPN
ejpam-5886	412	5	,	,	PUNCT
ejpam-5886	412	6	2024	2024	NUM
ejpam-5886	412	7	.	.	PUNCT
ejpam-5886	413	1	[	[	X
ejpam-5886	413	2	21	21	NUM
ejpam-5886	413	3	]	]	PUNCT
ejpam-5886	413	4	ibtehal	ibtehal	PROPN
ejpam-5886	413	5	alazman	alazman	NOUN
ejpam-5886	413	6	,	,	PUNCT
ejpam-5886	413	7	badr	badr	PROPN
ejpam-5886	413	8	saad	saad	PROPN
ejpam-5886	413	9	t	t	PROPN
ejpam-5886	413	10	alkahtani	alkahtani	PROPN
ejpam-5886	413	11	,	,	PUNCT
ejpam-5886	413	12	and	and	CCONJ
ejpam-5886	413	13	shahid	shahid	PROPN
ejpam-5886	413	14	ahmad	ahmad	PROPN
ejpam-5886	413	15	wani	wani	PROPN
ejpam-5886	413	16	.	.	PUNCT
ejpam-5886	414	1	certain	certain	ADJ
ejpam-5886	414	2	properties	property	NOUN
ejpam-5886	414	3	of	of	ADP
ejpam-5886	414	4	δ	δ	PROPN
ejpam-5886	414	5	h	h	NOUN
ejpam-5886	414	6	multi	multi	ADJ
ejpam-5886	414	7	-	-	ADJ
ejpam-5886	414	8	variate	variate	ADJ
ejpam-5886	414	9	hermite	hermite	ADJ
ejpam-5886	414	10	polynomials	polynomial	NOUN
ejpam-5886	414	11	.	.	PUNCT
ejpam-5886	415	1	symmetry	symmetry	NOUN
ejpam-5886	415	2	,	,	PUNCT
ejpam-5886	415	3	15(4):839	15(4):839	NOUN
ejpam-5886	415	4	,	,	PUNCT
ejpam-5886	415	5	2023	2023	NUM
ejpam-5886	415	6	.	.	PUNCT
ejpam-5886	416	1	[	[	X
ejpam-5886	416	2	22	22	NUM
ejpam-5886	416	3	]	]	PUNCT
ejpam-5886	416	4	shahid	shahid	PROPN
ejpam-5886	416	5	ahmad	ahmad	PROPN
ejpam-5886	416	6	wani	wani	PROPN
ejpam-5886	416	7	,	,	PUNCT
ejpam-5886	416	8	arundhati	arundhati	PROPN
ejpam-5886	416	9	warke	warke	PROPN
ejpam-5886	416	10	,	,	PUNCT
ejpam-5886	416	11	and	and	CCONJ
ejpam-5886	416	12	javid	javid	PROPN
ejpam-5886	416	13	gani	gani	PROPN
ejpam-5886	416	14	dar	dar	PROPN
ejpam-5886	416	15	.	.	PUNCT
ejpam-5886	416	16	degenerate	degenerate	ADJ
ejpam-5886	416	17	2d	2d	NUM
ejpam-5886	416	18	bivariate	bivariate	ADJ
ejpam-5886	416	19	appell	appell	ADJ
ejpam-5886	416	20	polynomials	polynomial	NOUN
ejpam-5886	416	21	:	:	PUNCT
ejpam-5886	416	22	properties	property	NOUN
ejpam-5886	416	23	and	and	CCONJ
ejpam-5886	416	24	applications	application	NOUN
ejpam-5886	416	25	.	.	PUNCT
ejpam-5886	417	1	applied	apply	VERB
ejpam-5886	417	2	mathematics	mathematic	NOUN
ejpam-5886	417	3	in	in	ADP
ejpam-5886	417	4	science	science	NOUN
ejpam-5886	417	5	and	and	CCONJ
ejpam-5886	417	6	engineering	engineering	NOUN
ejpam-5886	417	7	,	,	PUNCT
ejpam-5886	417	8	31(1):2194645	31(1):2194645	NUM
ejpam-5886	417	9	,	,	PUNCT
ejpam-5886	417	10	2023	2023	NUM
ejpam-5886	417	11	.	.	PUNCT
ejpam-5886	418	1	[	[	X
ejpam-5886	418	2	23	23	NUM
ejpam-5886	418	3	]	]	X
ejpam-5886	418	4	r	r	NOUN
ejpam-5886	418	5	alyusof	alyusof	NOUN
ejpam-5886	418	6	and	and	CCONJ
ejpam-5886	418	7	sa	sa	PROPN
ejpam-5886	418	8	wani	wani	PROPN
ejpam-5886	418	9	.	.	PUNCT
ejpam-5886	419	1	certain	certain	ADJ
ejpam-5886	419	2	properties	property	NOUN
ejpam-5886	419	3	and	and	CCONJ
ejpam-5886	419	4	applications	application	NOUN
ejpam-5886	419	5	of	of	ADP
ejpam-5886	419	6	deltah	deltah	ADJ
ejpam-5886	419	7	hybrid	hybrid	ADJ
ejpam-5886	419	8	special	special	ADJ
ejpam-5886	419	9	polynomials	polynomial	NOUN
ejpam-5886	419	10	associated	associate	VERB
ejpam-5886	419	11	with	with	ADP
ejpam-5886	419	12	appell	appell	PROPN
ejpam-5886	419	13	sequences	sequence	NOUN
ejpam-5886	419	14	,	,	PUNCT
ejpam-5886	419	15	fractal	fractal	ADJ
ejpam-5886	419	16	fract	fract	NOUN
ejpam-5886	419	17	.	.	PUNCT
ejpam-5886	420	1	,	,	PUNCT
ejpam-5886	420	2	7	7	NUM
ejpam-5886	420	3	(	(	PUNCT
ejpam-5886	420	4	2023	2023	NUM
ejpam-5886	420	5	)	)	PUNCT
ejpam-5886	420	6	,	,	PUNCT
ejpam-5886	420	7	233	233	NUM
ejpam-5886	420	8	.	.	PUNCT