id	sid	tid	token	lemma	pos
ejpam-6211	1	1	european	european	PROPN
ejpam-6211	1	2	journal	journal	PROPN
ejpam-6211	1	3	of	of	ADP
ejpam-6211	1	4	pure	pure	ADJ
ejpam-6211	1	5	and	and	CCONJ
ejpam-6211	1	6	applied	applied	ADJ
ejpam-6211	1	7	mathematics	mathematic	NOUN
ejpam-6211	1	8	2025	2025	NUM
ejpam-6211	1	9	,	,	PUNCT
ejpam-6211	1	10	vol	vol	NOUN
ejpam-6211	1	11	.	.	PROPN
ejpam-6211	1	12	18	18	NUM
ejpam-6211	1	13	,	,	PUNCT
ejpam-6211	1	14	issue	issue	NOUN
ejpam-6211	1	15	3	3	NUM
ejpam-6211	1	16	,	,	PUNCT
ejpam-6211	1	17	article	article	NOUN
ejpam-6211	1	18	number	number	NOUN
ejpam-6211	1	19	6211	6211	NUM
ejpam-6211	1	20	issn	issn	VERB
ejpam-6211	1	21	1307	1307	NUM
ejpam-6211	1	22	-	-	SYM
ejpam-6211	1	23	5543	5543	NUM
ejpam-6211	1	24	–	–	PUNCT
ejpam-6211	1	25	ejpam.com	ejpam.com	X
ejpam-6211	1	26	published	publish	VERB
ejpam-6211	1	27	by	by	ADP
ejpam-6211	1	28	new	new	PROPN
ejpam-6211	1	29	york	york	PROPN
ejpam-6211	1	30	business	business	PROPN
ejpam-6211	1	31	global	global	PROPN
ejpam-6211	1	32	on	on	ADP
ejpam-6211	1	33	exploring	explore	VERB
ejpam-6211	1	34	the	the	DET
ejpam-6211	1	35	r	r	NOUN
ejpam-6211	1	36	-	-	PUNCT
ejpam-6211	1	37	stirling	stirling	NOUN
ejpam-6211	1	38	fibonacci	fibonacci	NOUN
ejpam-6211	1	39	numbers	number	NOUN
ejpam-6211	1	40	and	and	CCONJ
ejpam-6211	1	41	polynomials	polynomial	VERB
ejpam-6211	1	42	romeo	romeo	PROPN
ejpam-6211	1	43	a.	a.	PROPN
ejpam-6211	1	44	coronel2	coronel2	PROPN
ejpam-6211	1	45	,	,	PUNCT
ejpam-6211	1	46	roberto	roberto	PROPN
ejpam-6211	1	47	b.	b.	PROPN
ejpam-6211	1	48	corcino1,2,∗	corcino1,2,∗	PROPN
ejpam-6211	1	49	1	1	NUM
ejpam-6211	1	50	research	research	NOUN
ejpam-6211	1	51	institute	institute	NOUN
ejpam-6211	1	52	for	for	ADP
ejpam-6211	1	53	computational	computational	ADJ
ejpam-6211	1	54	mathematics	mathematic	NOUN
ejpam-6211	1	55	and	and	CCONJ
ejpam-6211	1	56	physics	physics	NOUN
ejpam-6211	1	57	,	,	PUNCT
ejpam-6211	1	58	cebu	cebu	NOUN
ejpam-6211	1	59	normal	normal	ADJ
ejpam-6211	1	60	university	university	NOUN
ejpam-6211	1	61	,	,	PUNCT
ejpam-6211	1	62	6000	6000	NUM
ejpam-6211	1	63	cebu	cebu	NOUN
ejpam-6211	1	64	city	city	NOUN
ejpam-6211	1	65	,	,	PUNCT
ejpam-6211	1	66	philippines	philippine	NOUN
ejpam-6211	1	67	2	2	NUM
ejpam-6211	1	68	mathematics	mathematics	NOUN
ejpam-6211	1	69	department	department	NOUN
ejpam-6211	1	70	,	,	PUNCT
ejpam-6211	1	71	cebu	cebu	NOUN
ejpam-6211	1	72	normal	normal	ADJ
ejpam-6211	1	73	university	university	NOUN
ejpam-6211	1	74	,	,	PUNCT
ejpam-6211	1	75	6000	6000	NUM
ejpam-6211	1	76	cebu	cebu	NOUN
ejpam-6211	1	77	city	city	NOUN
ejpam-6211	1	78	,	,	PUNCT
ejpam-6211	1	79	philippines	philippine	NOUN
ejpam-6211	1	80	abstract	abstract	ADJ
ejpam-6211	1	81	.	.	PUNCT
ejpam-6211	2	1	this	this	DET
ejpam-6211	2	2	paper	paper	NOUN
ejpam-6211	2	3	introduces	introduce	NOUN
ejpam-6211	2	4	and	and	CCONJ
ejpam-6211	2	5	investigates	investigate	VERB
ejpam-6211	2	6	a	a	DET
ejpam-6211	2	7	novel	novel	ADJ
ejpam-6211	2	8	class	class	NOUN
ejpam-6211	2	9	of	of	ADP
ejpam-6211	2	10	combinatorial	combinatorial	ADJ
ejpam-6211	2	11	constructs	construct	NOUN
ejpam-6211	2	12	termed	term	VERB
ejpam-6211	2	13	the	the	DET
ejpam-6211	2	14	r	r	NOUN
ejpam-6211	2	15	-	-	PUNCT
ejpam-6211	2	16	stirling	stirling	NOUN
ejpam-6211	2	17	fibonacci	fibonacci	NOUN
ejpam-6211	2	18	numbers	number	NOUN
ejpam-6211	2	19	and	and	CCONJ
ejpam-6211	2	20	polynomials	polynomial	NOUN
ejpam-6211	2	21	of	of	ADP
ejpam-6211	2	22	the	the	DET
ejpam-6211	2	23	first	first	ADJ
ejpam-6211	2	24	and	and	CCONJ
ejpam-6211	2	25	second	second	ADJ
ejpam-6211	2	26	kind	kind	NOUN
ejpam-6211	2	27	.	.	PUNCT
ejpam-6211	3	1	by	by	ADP
ejpam-6211	3	2	integrating	integrate	VERB
ejpam-6211	3	3	the	the	DET
ejpam-6211	3	4	exponential	exponential	ADJ
ejpam-6211	3	5	generating	generating	NOUN
ejpam-6211	3	6	functions	function	NOUN
ejpam-6211	3	7	of	of	ADP
ejpam-6211	3	8	classical	classical	ADJ
ejpam-6211	3	9	fibonacci	fibonacci	NOUN
ejpam-6211	3	10	numbers	number	NOUN
ejpam-6211	3	11	with	with	ADP
ejpam-6211	3	12	those	those	PRON
ejpam-6211	3	13	of	of	ADP
ejpam-6211	3	14	the	the	DET
ejpam-6211	3	15	signed	sign	VERB
ejpam-6211	3	16	r	r	NOUN
ejpam-6211	3	17	-	-	PUNCT
ejpam-6211	3	18	stirling	stirling	NOUN
ejpam-6211	3	19	numbers	number	NOUN
ejpam-6211	3	20	,	,	PUNCT
ejpam-6211	3	21	we	we	PRON
ejpam-6211	3	22	establish	establish	VERB
ejpam-6211	3	23	an	an	DET
ejpam-6211	3	24	enriched	enriched	ADJ
ejpam-6211	3	25	algebraic	algebraic	ADJ
ejpam-6211	3	26	framework	framework	NOUN
ejpam-6211	3	27	that	that	PRON
ejpam-6211	3	28	advances	advance	VERB
ejpam-6211	3	29	the	the	DET
ejpam-6211	3	30	theory	theory	NOUN
ejpam-6211	3	31	of	of	ADP
ejpam-6211	3	32	special	special	ADJ
ejpam-6211	3	33	numbers	number	NOUN
ejpam-6211	3	34	and	and	CCONJ
ejpam-6211	3	35	polynomials	polynomial	NOUN
ejpam-6211	3	36	.	.	PUNCT
ejpam-6211	4	1	the	the	DET
ejpam-6211	4	2	study	study	NOUN
ejpam-6211	4	3	yields	yield	VERB
ejpam-6211	4	4	new	new	ADJ
ejpam-6211	4	5	identities	identity	NOUN
ejpam-6211	4	6	including	include	VERB
ejpam-6211	4	7	horizontal	horizontal	ADJ
ejpam-6211	4	8	generating	generating	NOUN
ejpam-6211	4	9	functions	function	NOUN
ejpam-6211	4	10	,	,	PUNCT
ejpam-6211	4	11	explicit	explicit	ADJ
ejpam-6211	4	12	formulas	formula	NOUN
ejpam-6211	4	13	,	,	PUNCT
ejpam-6211	4	14	and	and	CCONJ
ejpam-6211	4	15	convolution	convolution	NOUN
ejpam-6211	4	16	relations	relation	NOUN
ejpam-6211	4	17	that	that	PRON
ejpam-6211	4	18	extend	extend	VERB
ejpam-6211	4	19	classical	classical	ADJ
ejpam-6211	4	20	combinatorial	combinatorial	ADJ
ejpam-6211	4	21	results	result	NOUN
ejpam-6211	4	22	.	.	PUNCT
ejpam-6211	5	1	in	in	ADP
ejpam-6211	5	2	addition	addition	NOUN
ejpam-6211	5	3	,	,	PUNCT
ejpam-6211	5	4	we	we	PRON
ejpam-6211	5	5	define	define	VERB
ejpam-6211	5	6	the	the	DET
ejpam-6211	5	7	r	r	NOUN
ejpam-6211	5	8	-	-	PUNCT
ejpam-6211	5	9	stirling	stirling	NOUN
ejpam-6211	5	10	chebyshev	chebyshev	NOUN
ejpam-6211	5	11	polynomials	polynomial	NOUN
ejpam-6211	5	12	of	of	ADP
ejpam-6211	5	13	both	both	DET
ejpam-6211	5	14	kinds	kind	NOUN
ejpam-6211	5	15	by	by	ADP
ejpam-6211	5	16	employing	employ	VERB
ejpam-6211	5	17	hyperbolic	hyperbolic	ADJ
ejpam-6211	5	18	functions	function	NOUN
ejpam-6211	5	19	and	and	CCONJ
ejpam-6211	5	20	exponential	exponential	ADJ
ejpam-6211	5	21	techniques	technique	NOUN
ejpam-6211	5	22	,	,	PUNCT
ejpam-6211	5	23	thereby	thereby	ADV
ejpam-6211	5	24	forging	forge	VERB
ejpam-6211	5	25	a	a	DET
ejpam-6211	5	26	functional	functional	ADJ
ejpam-6211	5	27	link	link	NOUN
ejpam-6211	5	28	between	between	ADP
ejpam-6211	5	29	fibonacci	fibonacci	NOUN
ejpam-6211	5	30	-	-	PUNCT
ejpam-6211	5	31	type	type	NOUN
ejpam-6211	5	32	and	and	CCONJ
ejpam-6211	5	33	stirling	stirling	NOUN
ejpam-6211	5	34	-	-	PUNCT
ejpam-6211	5	35	type	type	NOUN
ejpam-6211	5	36	sequences	sequence	NOUN
ejpam-6211	5	37	.	.	PUNCT
ejpam-6211	6	1	these	these	DET
ejpam-6211	6	2	results	result	NOUN
ejpam-6211	6	3	are	be	AUX
ejpam-6211	6	4	rigorously	rigorously	ADV
ejpam-6211	6	5	validated	validate	VERB
ejpam-6211	6	6	through	through	ADP
ejpam-6211	6	7	series	series	NOUN
ejpam-6211	6	8	expansion	expansion	NOUN
ejpam-6211	6	9	and	and	CCONJ
ejpam-6211	6	10	the	the	DET
ejpam-6211	6	11	cauchy	cauchy	ADJ
ejpam-6211	6	12	product	product	NOUN
ejpam-6211	6	13	method	method	NOUN
ejpam-6211	6	14	.	.	PUNCT
ejpam-6211	7	1	the	the	DET
ejpam-6211	7	2	theoretical	theoretical	ADJ
ejpam-6211	7	3	contributions	contribution	NOUN
ejpam-6211	7	4	of	of	ADP
ejpam-6211	7	5	this	this	DET
ejpam-6211	7	6	work	work	NOUN
ejpam-6211	7	7	highlight	highlight	VERB
ejpam-6211	7	8	the	the	DET
ejpam-6211	7	9	interplay	interplay	NOUN
ejpam-6211	7	10	between	between	ADP
ejpam-6211	7	11	combinatorics	combinatoric	NOUN
ejpam-6211	7	12	,	,	PUNCT
ejpam-6211	7	13	algebra	algebra	NOUN
ejpam-6211	7	14	,	,	PUNCT
ejpam-6211	7	15	and	and	CCONJ
ejpam-6211	7	16	analysis	analysis	NOUN
ejpam-6211	7	17	,	,	PUNCT
ejpam-6211	7	18	with	with	ADP
ejpam-6211	7	19	potential	potential	ADJ
ejpam-6211	7	20	applications	application	NOUN
ejpam-6211	7	21	in	in	ADP
ejpam-6211	7	22	number	number	NOUN
ejpam-6211	7	23	theory	theory	NOUN
ejpam-6211	7	24	,	,	PUNCT
ejpam-6211	7	25	orthogonal	orthogonal	ADJ
ejpam-6211	7	26	polynomials	polynomial	NOUN
ejpam-6211	7	27	,	,	PUNCT
ejpam-6211	7	28	and	and	CCONJ
ejpam-6211	7	29	symbolic	symbolic	ADJ
ejpam-6211	7	30	computation	computation	NOUN
ejpam-6211	7	31	.	.	PUNCT
ejpam-6211	8	1	2020	2020	NUM
ejpam-6211	8	2	mathematics	mathematic	NOUN
ejpam-6211	8	3	subject	subject	NOUN
ejpam-6211	8	4	classifications	classification	NOUN
ejpam-6211	8	5	:	:	PUNCT
ejpam-6211	8	6	11b68	11b68	NUM
ejpam-6211	8	7	,	,	PUNCT
ejpam-6211	8	8	11b73	11b73	NUM
ejpam-6211	8	9	,	,	PUNCT
ejpam-6211	8	10	05a15	05a15	NOUN
ejpam-6211	8	11	key	key	ADJ
ejpam-6211	8	12	words	word	NOUN
ejpam-6211	8	13	and	and	CCONJ
ejpam-6211	8	14	phrases	phrase	NOUN
ejpam-6211	8	15	:	:	PUNCT
ejpam-6211	8	16	r	r	X
ejpam-6211	8	17	-	-	PUNCT
ejpam-6211	8	18	stirling	stirling	NOUN
ejpam-6211	8	19	numbers	number	NOUN
ejpam-6211	8	20	of	of	ADP
ejpam-6211	8	21	the	the	DET
ejpam-6211	8	22	first	first	ADJ
ejpam-6211	8	23	kind	kind	NOUN
ejpam-6211	8	24	,	,	PUNCT
ejpam-6211	8	25	r	r	NOUN
ejpam-6211	8	26	-	-	PUNCT
ejpam-6211	8	27	stirling	stirling	NOUN
ejpam-6211	8	28	numbers	number	NOUN
ejpam-6211	8	29	of	of	ADP
ejpam-6211	8	30	the	the	DET
ejpam-6211	8	31	second	second	ADJ
ejpam-6211	8	32	kind	kind	NOUN
ejpam-6211	8	33	,	,	PUNCT
ejpam-6211	8	34	fibonacci	fibonacci	NOUN
ejpam-6211	8	35	number	number	NOUN
ejpam-6211	8	36	,	,	PUNCT
ejpam-6211	8	37	fibonacci	fibonacci	NOUN
ejpam-6211	8	38	polynomials	polynomial	NOUN
ejpam-6211	8	39	,	,	PUNCT
ejpam-6211	8	40	chebyshev	chebyshev	NOUN
ejpam-6211	8	41	polynomials	polynomial	NOUN
ejpam-6211	8	42	1	1	NUM
ejpam-6211	8	43	.	.	PUNCT
ejpam-6211	9	1	introduction	introduction	NOUN
ejpam-6211	9	2	james	james	PROPN
ejpam-6211	9	3	stirling	stirling	PROPN
ejpam-6211	9	4	,	,	PUNCT
ejpam-6211	9	5	a	a	DET
ejpam-6211	9	6	scottish	scottish	ADJ
ejpam-6211	9	7	mathematician	mathematician	NOUN
ejpam-6211	9	8	in	in	ADP
ejpam-6211	9	9	his	his	PRON
ejpam-6211	9	10	book	book	NOUN
ejpam-6211	9	11	“	"	PUNCT
ejpam-6211	9	12	methodus	methodus	NOUN
ejpam-6211	9	13	differentialis	differentialis	PROPN
ejpam-6211	9	14	”	"	PUNCT
ejpam-6211	9	15	(	(	PUNCT
ejpam-6211	9	16	stirling	stirling	NOUN
ejpam-6211	9	17	,	,	PUNCT
ejpam-6211	9	18	[	[	X
ejpam-6211	9	19	1	1	NUM
ejpam-6211	9	20	]	]	PUNCT
ejpam-6211	9	21	)	)	PUNCT
ejpam-6211	9	22	introduced	introduce	VERB
ejpam-6211	9	23	stirling	stirling	NOUN
ejpam-6211	9	24	numbers	number	NOUN
ejpam-6211	9	25	within	within	ADP
ejpam-6211	9	26	a	a	DET
ejpam-6211	9	27	purely	purely	ADV
ejpam-6211	9	28	algebraic	algebraic	ADJ
ejpam-6211	9	29	framework	framework	NOUN
ejpam-6211	9	30	.	.	PUNCT
ejpam-6211	10	1	these	these	DET
ejpam-6211	10	2	numbers	number	NOUN
ejpam-6211	10	3	are	be	AUX
ejpam-6211	10	4	considered	consider	VERB
ejpam-6211	10	5	the	the	DET
ejpam-6211	10	6	fundamental	fundamental	ADJ
ejpam-6211	10	7	concept	concept	NOUN
ejpam-6211	10	8	in	in	ADP
ejpam-6211	10	9	mathematics	mathematic	NOUN
ejpam-6211	10	10	particularly	particularly	ADV
ejpam-6211	10	11	in	in	ADP
ejpam-6211	10	12	the	the	DET
ejpam-6211	10	13	field	field	NOUN
ejpam-6211	10	14	of	of	ADP
ejpam-6211	10	15	combinatorics	combinatoric	NOUN
ejpam-6211	10	16	,	,	PUNCT
ejpam-6211	10	17	analysis	analysis	NOUN
ejpam-6211	10	18	and	and	CCONJ
ejpam-6211	10	19	algebra	algebra	NOUN
ejpam-6211	10	20	.	.	PUNCT
ejpam-6211	11	1	stirling	stirling	NOUN
ejpam-6211	11	2	numbers	number	NOUN
ejpam-6211	11	3	were	be	AUX
ejpam-6211	11	4	categorized	categorize	VERB
ejpam-6211	11	5	into	into	ADP
ejpam-6211	11	6	two	two	NUM
ejpam-6211	11	7	types	type	NOUN
ejpam-6211	11	8	;	;	PUNCT
ejpam-6211	11	9	(	(	PUNCT
ejpam-6211	11	10	1	1	X
ejpam-6211	11	11	)	)	PUNCT
ejpam-6211	11	12	first	first	ADJ
ejpam-6211	11	13	kind	kind	NOUN
ejpam-6211	11	14	stirling	stirling	NOUN
ejpam-6211	11	15	numbers	number	NOUN
ejpam-6211	11	16	which	which	PRON
ejpam-6211	11	17	enumerates	enumerate	VERB
ejpam-6211	11	18	the	the	DET
ejpam-6211	11	19	number	number	NOUN
ejpam-6211	11	20	of	of	ADP
ejpam-6211	11	21	permutations	permutation	NOUN
ejpam-6211	11	22	of	of	ADP
ejpam-6211	11	23	n	n	PRON
ejpam-6211	11	24	elements	element	NOUN
ejpam-6211	11	25	with	with	ADP
ejpam-6211	11	26	exactly	exactly	ADV
ejpam-6211	11	27	k	k	PROPN
ejpam-6211	11	28	disjoint	disjoint	PROPN
ejpam-6211	11	29	cycles	cycle	NOUN
ejpam-6211	11	30	;	;	PUNCT
ejpam-6211	11	31	(	(	PUNCT
ejpam-6211	11	32	2	2	X
ejpam-6211	11	33	)	)	PUNCT
ejpam-6211	11	34	the	the	DET
ejpam-6211	11	35	second	second	ADJ
ejpam-6211	11	36	kind	kind	NOUN
ejpam-6211	11	37	stirling	stirling	NOUN
ejpam-6211	11	38	numbers	number	NOUN
ejpam-6211	11	39	which	which	PRON
ejpam-6211	11	40	counts	count	VERB
ejpam-6211	11	41	the	the	DET
ejpam-6211	11	42	number	number	NOUN
ejpam-6211	11	43	of	of	ADP
ejpam-6211	11	44	ways	way	NOUN
ejpam-6211	11	45	to	to	PART
ejpam-6211	11	46	partition	partition	VERB
ejpam-6211	11	47	n	n	PRON
ejpam-6211	11	48	elements	element	NOUN
ejpam-6211	11	49	into	into	ADP
ejpam-6211	11	50	k	k	PROPN
ejpam-6211	11	51	non	non	ADJ
ejpam-6211	11	52	-	-	ADJ
ejpam-6211	11	53	empty	empty	ADJ
ejpam-6211	11	54	subsets	subset	NOUN
ejpam-6211	11	55	.	.	PUNCT
ejpam-6211	12	1	additionally	additionally	ADV
ejpam-6211	12	2	,	,	PUNCT
ejpam-6211	12	3	the	the	DET
ejpam-6211	12	4	stirling	stirling	NOUN
ejpam-6211	12	5	numbers	number	NOUN
ejpam-6211	12	6	are	be	AUX
ejpam-6211	12	7	the	the	DET
ejpam-6211	12	8	coefficients	coefficient	NOUN
ejpam-6211	12	9	of	of	ADP
ejpam-6211	12	10	the	the	DET
ejpam-6211	12	11	relation	relation	NOUN
ejpam-6211	12	12	∗corresponding	∗corresponde	VERB
ejpam-6211	12	13	author	author	NOUN
ejpam-6211	12	14	.	.	PUNCT
ejpam-6211	13	1	doi	doi	NOUN
ejpam-6211	13	2	:	:	PUNCT
ejpam-6211	13	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6211	https://doi.org/10.29020/nybg.ejpam.v18i3.6211	PROPN
ejpam-6211	13	4	email	email	NOUN
ejpam-6211	13	5	addresses	address	NOUN
ejpam-6211	13	6	:	:	PUNCT
ejpam-6211	13	7	coronelromeo78@gmail.com	coronelromeo78@gmail.com	X
ejpam-6211	13	8	r.a	r.a	PROPN
ejpam-6211	13	9	.	.	PROPN
ejpam-6211	13	10	coronel	coronel	PROPN
ejpam-6211	13	11	)	)	PUNCT
ejpam-6211	13	12	,	,	PUNCT
ejpam-6211	13	13	rcorcino@yahoo.com	rcorcino@yahoo.com	X
ejpam-6211	13	14	(	(	PUNCT
ejpam-6211	13	15	(	(	PUNCT
ejpam-6211	13	16	r.	r.	PROPN
ejpam-6211	13	17	b.	b.	PROPN
ejpam-6211	13	18	corcino	corcino	PROPN
ejpam-6211	13	19	)	)	PUNCT
ejpam-6211	13	20	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6211	14	1	1	1	NUM
ejpam-6211	14	2	copyright	copyright	NOUN
ejpam-6211	14	3	:	:	PUNCT
ejpam-6211	14	4	©	©	PROPN
ejpam-6211	14	5	2025	2025	NUM
ejpam-6211	14	6	the	the	DET
ejpam-6211	14	7	author(s	author(s	NOUN
ejpam-6211	14	8	)	)	PUNCT
ejpam-6211	14	9	.	.	PUNCT
ejpam-6211	15	1	(	(	PUNCT
ejpam-6211	15	2	cc	cc	NOUN
ejpam-6211	15	3	by	by	ADP
ejpam-6211	15	4	-	-	PUNCT
ejpam-6211	15	5	nc	nc	PROPN
ejpam-6211	15	6	4.0	4.0	NUM
ejpam-6211	15	7	)	)	PUNCT
ejpam-6211	15	8	2	2	NUM
ejpam-6211	15	9	of	of	ADP
ejpam-6211	15	10	23	23	NUM
ejpam-6211	15	11	xn	xn	NOUN
ejpam-6211	15	12	=	=	PUNCT
ejpam-6211	16	1	∞∑	∞∑	NUM
ejpam-6211	16	2	n=0	n=0	PUNCT
ejpam-6211	16	3	s(n	s(n	PROPN
ejpam-6211	16	4	,	,	PUNCT
ejpam-6211	16	5	k)xk	k)xk	PROPN
ejpam-6211	16	6	xn	xn	PUNCT
ejpam-6211	17	1	=	=	PUNCT
ejpam-6211	17	2	∞∑	∞∑	NUM
ejpam-6211	17	3	n=0	n=0	PUNCT
ejpam-6211	17	4	s(n	s(n	PROPN
ejpam-6211	17	5	,	,	PUNCT
ejpam-6211	17	6	k)xk	k)xk	PROPN
ejpam-6211	17	7	,	,	PUNCT
ejpam-6211	17	8	where	where	SCONJ
ejpam-6211	17	9	xn	xn	PROPN
ejpam-6211	17	10	=	=	SYM
ejpam-6211	17	11	x(x	x(x	PROPN
ejpam-6211	18	1	−	−	PROPN
ejpam-6211	18	2	1)(x	1)(x	NUM
ejpam-6211	18	3	−	−	PROPN
ejpam-6211	18	4	2	2	NUM
ejpam-6211	18	5	)	)	PUNCT
ejpam-6211	18	6	·	·	PUNCT
ejpam-6211	18	7	·	·	PUNCT
ejpam-6211	18	8	·	·	PUNCT
ejpam-6211	19	1	(	(	PUNCT
ejpam-6211	19	2	x	x	SYM
ejpam-6211	19	3	−	−	PROPN
ejpam-6211	19	4	n	n	NOUN
ejpam-6211	19	5	+	+	NOUN
ejpam-6211	19	6	1	1	NUM
ejpam-6211	19	7	)	)	PUNCT
ejpam-6211	19	8	=	=	PUNCT
ejpam-6211	19	9	∏n	∏n	ADJ
ejpam-6211	19	10	k=1(x	k=1(x	NOUN
ejpam-6211	19	11	−	−	PROPN
ejpam-6211	20	1	k	k	NOUN
ejpam-6211	21	1	+	+	PROPN
ejpam-6211	21	2	1	1	X
ejpam-6211	21	3	)	)	PUNCT
ejpam-6211	21	4	is	be	AUX
ejpam-6211	21	5	the	the	DET
ejpam-6211	21	6	falling	fall	VERB
ejpam-6211	21	7	factorial	factorial	NOUN
ejpam-6211	21	8	of	of	ADP
ejpam-6211	21	9	x	x	PUNCT
ejpam-6211	21	10	of	of	ADP
ejpam-6211	21	11	degree	degree	NOUN
ejpam-6211	21	12	n.	n.	PROPN
ejpam-6211	21	13	the	the	DET
ejpam-6211	21	14	coefficient	coefficient	NOUN
ejpam-6211	21	15	s(n	s(n	PROPN
ejpam-6211	21	16	,	,	PUNCT
ejpam-6211	21	17	k	k	NOUN
ejpam-6211	21	18	)	)	PUNCT
ejpam-6211	21	19	and	and	CCONJ
ejpam-6211	21	20	s(n	s(n	PROPN
ejpam-6211	21	21	,	,	PUNCT
ejpam-6211	21	22	k	k	NOUN
ejpam-6211	21	23	)	)	PUNCT
ejpam-6211	21	24	are	be	AUX
ejpam-6211	21	25	the	the	DET
ejpam-6211	21	26	stirling	stirling	NOUN
ejpam-6211	21	27	numbers	number	NOUN
ejpam-6211	21	28	of	of	ADP
ejpam-6211	21	29	the	the	DET
ejpam-6211	21	30	first	first	ADJ
ejpam-6211	21	31	kind	kind	NOUN
ejpam-6211	21	32	and	and	CCONJ
ejpam-6211	21	33	stirling	stirling	NOUN
ejpam-6211	21	34	numbers	number	NOUN
ejpam-6211	21	35	of	of	ADP
ejpam-6211	21	36	the	the	DET
ejpam-6211	21	37	second	second	ADJ
ejpam-6211	21	38	kind	kind	NOUN
ejpam-6211	21	39	respectively	respectively	ADV
ejpam-6211	21	40	.	.	PUNCT
ejpam-6211	22	1	over	over	ADP
ejpam-6211	22	2	the	the	DET
ejpam-6211	22	3	years	year	NOUN
ejpam-6211	22	4	,	,	PUNCT
ejpam-6211	22	5	numerous	numerous	ADJ
ejpam-6211	22	6	mathematicians	mathematician	NOUN
ejpam-6211	22	7	have	have	AUX
ejpam-6211	22	8	studied	study	VERB
ejpam-6211	22	9	these	these	DET
ejpam-6211	22	10	numbers	number	NOUN
ejpam-6211	22	11	and	and	CCONJ
ejpam-6211	22	12	explored	explore	VERB
ejpam-6211	22	13	properties	property	NOUN
ejpam-6211	22	14	,	,	PUNCT
ejpam-6211	22	15	generalizations	generalization	NOUN
ejpam-6211	22	16	and	and	CCONJ
ejpam-6211	22	17	its	its	PRON
ejpam-6211	22	18	application	application	NOUN
ejpam-6211	22	19	.	.	PUNCT
ejpam-6211	23	1	common	common	ADJ
ejpam-6211	23	2	results	result	NOUN
ejpam-6211	23	3	include	include	VERB
ejpam-6211	23	4	generating	generating	NOUN
ejpam-6211	23	5	functions	function	NOUN
ejpam-6211	23	6	,	,	PUNCT
ejpam-6211	23	7	recurrence	recurrence	NOUN
ejpam-6211	23	8	relations	relation	NOUN
ejpam-6211	23	9	,	,	PUNCT
ejpam-6211	23	10	and	and	CCONJ
ejpam-6211	23	11	explicit	explicit	ADJ
ejpam-6211	23	12	formulas	formula	NOUN
ejpam-6211	23	13	all	all	PRON
ejpam-6211	23	14	of	of	ADP
ejpam-6211	23	15	which	which	PRON
ejpam-6211	23	16	helps	help	VERB
ejpam-6211	23	17	to	to	PART
ejpam-6211	23	18	understand	understand	VERB
ejpam-6211	23	19	their	their	PRON
ejpam-6211	23	20	behavior	behavior	NOUN
ejpam-6211	23	21	.	.	PUNCT
ejpam-6211	24	1	(	(	PUNCT
ejpam-6211	24	2	see	see	VERB
ejpam-6211	24	3	[	[	X
ejpam-6211	24	4	2–7	2–7	NOUN
ejpam-6211	24	5	]	]	X
ejpam-6211	24	6	)	)	PUNCT
ejpam-6211	24	7	one	one	NUM
ejpam-6211	24	8	of	of	ADP
ejpam-6211	24	9	the	the	DET
ejpam-6211	24	10	most	most	ADV
ejpam-6211	24	11	notable	notable	ADJ
ejpam-6211	24	12	contributions	contribution	NOUN
ejpam-6211	24	13	to	to	ADP
ejpam-6211	24	14	the	the	DET
ejpam-6211	24	15	exploration	exploration	NOUN
ejpam-6211	24	16	of	of	ADP
ejpam-6211	24	17	stirling	stirling	NOUN
ejpam-6211	24	18	numbers	number	NOUN
ejpam-6211	24	19	was	be	AUX
ejpam-6211	24	20	made	make	VERB
ejpam-6211	24	21	by	by	ADP
ejpam-6211	24	22	a.z	a.z	PROPN
ejpam-6211	24	23	.	.	PROPN
ejpam-6211	24	24	broder	broder	NOUN
ejpam-6211	25	1	[	[	X
ejpam-6211	25	2	8	8	NUM
ejpam-6211	25	3	]	]	PUNCT
ejpam-6211	25	4	,	,	PUNCT
ejpam-6211	25	5	who	who	PRON
ejpam-6211	25	6	introduced	introduce	VERB
ejpam-6211	25	7	the	the	DET
ejpam-6211	25	8	rstirling	rstirle	VERB
ejpam-6211	25	9	numbers	number	NOUN
ejpam-6211	25	10	by	by	ADP
ejpam-6211	25	11	adding	add	VERB
ejpam-6211	25	12	a	a	DET
ejpam-6211	25	13	parameter	parameter	NOUN
ejpam-6211	25	14	r	r	NOUN
ejpam-6211	25	15	which	which	PRON
ejpam-6211	25	16	leads	lead	VERB
ejpam-6211	25	17	to	to	ADP
ejpam-6211	25	18	the	the	DET
ejpam-6211	25	19	rstirling	rstirle	VERB
ejpam-6211	25	20	numbers	number	NOUN
ejpam-6211	25	21	of	of	ADP
ejpam-6211	25	22	the	the	DET
ejpam-6211	25	23	first	first	ADJ
ejpam-6211	25	24	and	and	CCONJ
ejpam-6211	25	25	second	second	ADJ
ejpam-6211	25	26	kind	kind	NOUN
ejpam-6211	25	27	.	.	PUNCT
ejpam-6211	26	1	broders	broder	NOUN
ejpam-6211	26	2	’	'	PUNCT
ejpam-6211	26	3	study	study	NOUN
ejpam-6211	26	4	employs	employ	VERB
ejpam-6211	26	5	a	a	DET
ejpam-6211	26	6	combinatorial	combinatorial	ADJ
ejpam-6211	26	7	approach	approach	NOUN
ejpam-6211	26	8	in	in	ADP
ejpam-6211	26	9	deriving	derive	VERB
ejpam-6211	26	10	properties	property	NOUN
ejpam-6211	26	11	and	and	CCONJ
ejpam-6211	26	12	identities	identity	NOUN
ejpam-6211	26	13	such	such	ADJ
ejpam-6211	26	14	as	as	ADP
ejpam-6211	26	15	generating	generating	NOUN
ejpam-6211	26	16	functions	function	NOUN
ejpam-6211	26	17	(	(	PUNCT
ejpam-6211	26	18	exponential	exponential	ADJ
ejpam-6211	26	19	,	,	PUNCT
ejpam-6211	26	20	ordinary	ordinary	ADJ
ejpam-6211	26	21	,	,	PUNCT
ejpam-6211	26	22	rational	rational	ADJ
ejpam-6211	26	23	,	,	PUNCT
ejpam-6211	26	24	horizontal	horizontal	ADJ
ejpam-6211	26	25	,	,	PUNCT
ejpam-6211	26	26	vertical	vertical	ADJ
ejpam-6211	26	27	etc	etc	X
ejpam-6211	26	28	.	.	X
ejpam-6211	26	29	)	)	PUNCT
ejpam-6211	26	30	,	,	PUNCT
ejpam-6211	26	31	recurrence	recurrence	NOUN
ejpam-6211	26	32	relations	relation	NOUN
ejpam-6211	26	33	(	(	PUNCT
ejpam-6211	26	34	cross	cross	NOUN
ejpam-6211	26	35	recurrence	recurrence	NOUN
ejpam-6211	26	36	,	,	PUNCT
ejpam-6211	26	37	triangular	triangular	NOUN
ejpam-6211	26	38	,	,	PUNCT
ejpam-6211	26	39	etc	etc	X
ejpam-6211	26	40	.	.	X
ejpam-6211	26	41	)	)	PUNCT
ejpam-6211	26	42	,	,	PUNCT
ejpam-6211	26	43	explicit	explicit	ADJ
ejpam-6211	26	44	formulas	formula	NOUN
ejpam-6211	26	45	,	,	PUNCT
ejpam-6211	26	46	and	and	CCONJ
ejpam-6211	26	47	orthogonality	orthogonality	NOUN
ejpam-6211	26	48	and	and	CCONJ
ejpam-6211	26	49	inverse	inverse	NOUN
ejpam-6211	26	50	relation	relation	NOUN
ejpam-6211	26	51	and	and	CCONJ
ejpam-6211	26	52	their	their	PRON
ejpam-6211	26	53	generalizations	generalization	NOUN
ejpam-6211	26	54	.	.	PUNCT
ejpam-6211	27	1	recently	recently	ADV
ejpam-6211	27	2	,	,	PUNCT
ejpam-6211	27	3	corcino	corcino	PROPN
ejpam-6211	27	4	et	et	PROPN
ejpam-6211	27	5	al	al	PROPN
ejpam-6211	27	6	.	.	PROPN
ejpam-6211	27	7	(	(	PUNCT
ejpam-6211	27	8	2023,[9	2023,[9	NUM
ejpam-6211	27	9	]	]	PUNCT
ejpam-6211	27	10	)	)	PUNCT
ejpam-6211	27	11	modified	modify	VERB
ejpam-6211	27	12	broder	broder	PROPN
ejpam-6211	27	13	’s	’s	PART
ejpam-6211	27	14	results	result	NOUN
ejpam-6211	27	15	by	by	ADP
ejpam-6211	27	16	presenting	present	VERB
ejpam-6211	27	17	the	the	DET
ejpam-6211	27	18	r	r	NOUN
ejpam-6211	27	19	-	-	PUNCT
ejpam-6211	27	20	stirling	stirling	NOUN
ejpam-6211	27	21	numbers	number	NOUN
ejpam-6211	27	22	in	in	ADP
ejpam-6211	27	23	an	an	DET
ejpam-6211	27	24	algebraic	algebraic	ADJ
ejpam-6211	27	25	method	method	NOUN
ejpam-6211	27	26	and	and	CCONJ
ejpam-6211	27	27	obtained	obtain	VERB
ejpam-6211	27	28	results	result	NOUN
ejpam-6211	27	29	parallel	parallel	ADJ
ejpam-6211	27	30	to	to	ADP
ejpam-6211	27	31	that	that	PRON
ejpam-6211	27	32	of	of	ADP
ejpam-6211	27	33	broder	broder	PROPN
ejpam-6211	27	34	.	.	PUNCT
ejpam-6211	28	1	on	on	ADP
ejpam-6211	28	2	the	the	DET
ejpam-6211	28	3	other	other	ADJ
ejpam-6211	28	4	hand	hand	NOUN
ejpam-6211	28	5	,	,	PUNCT
ejpam-6211	28	6	fibonacci	fibonacci	NOUN
ejpam-6211	28	7	is	be	AUX
ejpam-6211	28	8	named	name	VERB
ejpam-6211	28	9	after	after	ADP
ejpam-6211	28	10	leonardo	leonardo	PROPN
ejpam-6211	28	11	pisa	pisa	PROPN
ejpam-6211	28	12	,	,	PUNCT
ejpam-6211	28	13	an	an	DET
ejpam-6211	28	14	italian	italian	ADJ
ejpam-6211	28	15	mathematician	mathematician	NOUN
ejpam-6211	28	16	who	who	PRON
ejpam-6211	28	17	introduced	introduce	VERB
ejpam-6211	28	18	the	the	DET
ejpam-6211	28	19	sequence	sequence	NOUN
ejpam-6211	28	20	in	in	ADP
ejpam-6211	28	21	his	his	PRON
ejpam-6211	28	22	book	book	NOUN
ejpam-6211	28	23	,	,	PUNCT
ejpam-6211	28	24	liber	liber	PROPN
ejpam-6211	28	25	abaci	abaci	NOUN
ejpam-6211	28	26	(	(	PUNCT
ejpam-6211	28	27	1202	1202	NUM
ejpam-6211	28	28	,	,	PUNCT
ejpam-6211	28	29	[	[	X
ejpam-6211	28	30	10	10	NUM
ejpam-6211	28	31	]	]	NUM
ejpam-6211	28	32	)	)	PUNCT
ejpam-6211	28	33	.	.	PUNCT
ejpam-6211	29	1	before	before	ADP
ejpam-6211	29	2	fibonacci	fibonacci	PROPN
ejpam-6211	29	3	,	,	PUNCT
ejpam-6211	29	4	this	this	DET
ejpam-6211	29	5	sequence	sequence	NOUN
ejpam-6211	29	6	appeared	appear	VERB
ejpam-6211	29	7	in	in	ADP
ejpam-6211	29	8	indian	indian	ADJ
ejpam-6211	29	9	mathematics	mathematic	NOUN
ejpam-6211	29	10	.	.	PUNCT
ejpam-6211	30	1	the	the	DET
ejpam-6211	30	2	exponential	exponential	ADJ
ejpam-6211	30	3	generating	generating	NOUN
ejpam-6211	30	4	function	function	NOUN
ejpam-6211	30	5	of	of	ADP
ejpam-6211	30	6	the	the	DET
ejpam-6211	30	7	fibonacci	fibonacci	NOUN
ejpam-6211	30	8	number	number	NOUN
ejpam-6211	30	9	is	be	AUX
ejpam-6211	30	10	due	due	ADJ
ejpam-6211	30	11	to	to	ADP
ejpam-6211	30	12	the	the	DET
ejpam-6211	30	13	study	study	NOUN
ejpam-6211	30	14	of	of	ADP
ejpam-6211	30	15	c.a	c.a	PROPN
ejpam-6211	30	16	church	church	PROPN
ejpam-6211	30	17	et	et	PROPN
ejpam-6211	30	18	.	.	PUNCT
ejpam-6211	31	1	al.[11	al.[11	PROPN
ejpam-6211	31	2	]	]	PUNCT
ejpam-6211	31	3	which	which	PRON
ejpam-6211	31	4	is	be	AUX
ejpam-6211	31	5	given	give	VERB
ejpam-6211	31	6	by	by	ADP
ejpam-6211	31	7	t	t	PROPN
ejpam-6211	31	8	1−	1−	NUM
ejpam-6211	31	9	t−	t−	PROPN
ejpam-6211	31	10	t2	t2	NOUN
ejpam-6211	31	11	=	=	PUNCT
ejpam-6211	32	1	∞∑	∞∑	NUM
ejpam-6211	32	2	n=0	n=0	NUM
ejpam-6211	32	3	fn	fn	NOUN
ejpam-6211	32	4	tn	tn	NOUN
ejpam-6211	32	5	n	n	PROPN
ejpam-6211	32	6	!	!	PUNCT
ejpam-6211	33	1	(	(	PUNCT
ejpam-6211	33	2	1.1	1.1	NUM
ejpam-6211	33	3	)	)	PUNCT
ejpam-6211	33	4	this	this	PRON
ejpam-6211	33	5	can	can	AUX
ejpam-6211	33	6	further	far	ADV
ejpam-6211	33	7	be	be	AUX
ejpam-6211	33	8	expressed	express	VERB
ejpam-6211	33	9	as	as	SCONJ
ejpam-6211	33	10	follows	follow	VERB
ejpam-6211	33	11	eαt	eαt	ADV
ejpam-6211	33	12	−	−	PROPN
ejpam-6211	33	13	eβt	eβt	NOUN
ejpam-6211	33	14	α−	α−	ADP
ejpam-6211	33	15	β	β	NOUN
ejpam-6211	33	16	=	=	PUNCT
ejpam-6211	33	17	∞∑	∞∑	PROPN
ejpam-6211	33	18	n=0	n=0	NUM
ejpam-6211	33	19	fn	fn	NOUN
ejpam-6211	33	20	tn	tn	NOUN
ejpam-6211	33	21	n	n	PROPN
ejpam-6211	33	22	!	!	PUNCT
ejpam-6211	33	23	where	where	SCONJ
ejpam-6211	33	24	f1	f1	NOUN
ejpam-6211	33	25	=	=	SYM
ejpam-6211	33	26	f2	f2	PROPN
ejpam-6211	33	27	=	=	SYM
ejpam-6211	33	28	1	1	NUM
ejpam-6211	33	29	,	,	PUNCT
ejpam-6211	33	30	fn+1	fn+1	X
ejpam-6211	33	31	=	=	SYM
ejpam-6211	33	32	fn	fn	NOUN
ejpam-6211	34	1	+	+	CCONJ
ejpam-6211	34	2	fn−1	fn−1	ADJ
ejpam-6211	34	3	and	and	CCONJ
ejpam-6211	34	4	α	α	NOUN
ejpam-6211	34	5	=	=	SYM
ejpam-6211	34	6	(	(	PUNCT
ejpam-6211	34	7	1	1	NUM
ejpam-6211	34	8	+	+	NUM
ejpam-6211	34	9	√	√	NUM
ejpam-6211	34	10	5	5	NUM
ejpam-6211	34	11	)	)	PUNCT
ejpam-6211	34	12	2	2	NUM
ejpam-6211	34	13	,	,	PUNCT
ejpam-6211	34	14	β	β	X
ejpam-6211	34	15	=	=	SYM
ejpam-6211	34	16	(	(	PUNCT
ejpam-6211	34	17	1−	1−	NUM
ejpam-6211	34	18	√	√	NUM
ejpam-6211	34	19	5	5	NUM
ejpam-6211	34	20	)	)	PUNCT
ejpam-6211	34	21	2	2	NUM
ejpam-6211	34	22	.	.	PUNCT
ejpam-6211	35	1	in	in	ADP
ejpam-6211	35	2	mathematics	mathematic	NOUN
ejpam-6211	35	3	,	,	PUNCT
ejpam-6211	35	4	special	special	ADJ
ejpam-6211	35	5	functions	function	NOUN
ejpam-6211	35	6	and	and	CCONJ
ejpam-6211	35	7	number	number	NOUN
ejpam-6211	35	8	sequences	sequence	NOUN
ejpam-6211	35	9	can	can	AUX
ejpam-6211	35	10	be	be	AUX
ejpam-6211	35	11	generalized	generalize	VERB
ejpam-6211	35	12	through	through	ADP
ejpam-6211	35	13	various	various	ADJ
ejpam-6211	35	14	methods	method	NOUN
ejpam-6211	35	15	.	.	PUNCT
ejpam-6211	36	1	one	one	NUM
ejpam-6211	36	2	common	common	ADJ
ejpam-6211	36	3	approach	approach	NOUN
ejpam-6211	36	4	is	be	AUX
ejpam-6211	36	5	to	to	PART
ejpam-6211	36	6	integrate	integrate	VERB
ejpam-6211	36	7	them	they	PRON
ejpam-6211	36	8	with	with	ADP
ejpam-6211	36	9	concepts	concept	NOUN
ejpam-6211	36	10	from	from	ADP
ejpam-6211	36	11	other	other	ADJ
ejpam-6211	36	12	well	well	ADV
ejpam-6211	36	13	-	-	PUNCT
ejpam-6211	36	14	known	know	VERB
ejpam-6211	36	15	functions	function	NOUN
ejpam-6211	36	16	.	.	PUNCT
ejpam-6211	37	1	another	another	DET
ejpam-6211	37	2	method	method	NOUN
ejpam-6211	37	3	involves	involve	VERB
ejpam-6211	37	4	introducing	introduce	VERB
ejpam-6211	37	5	parameters	parameter	NOUN
ejpam-6211	37	6	by	by	ADP
ejpam-6211	37	7	modifying	modify	VERB
ejpam-6211	37	8	the	the	DET
ejpam-6211	37	9	defining	define	VERB
ejpam-6211	37	10	generating	generate	VERB
ejpam-6211	37	11	function	function	NOUN
ejpam-6211	37	12	either	either	CCONJ
ejpam-6211	37	13	through	through	ADP
ejpam-6211	37	14	addition	addition	NOUN
ejpam-6211	37	15	or	or	CCONJ
ejpam-6211	37	16	multiplication	multiplication	NOUN
ejpam-6211	37	17	with	with	ADP
ejpam-6211	37	18	certain	certain	ADJ
ejpam-6211	37	19	expressions	expression	NOUN
ejpam-6211	37	20	.	.	PUNCT
ejpam-6211	38	1	for	for	ADP
ejpam-6211	38	2	example	example	NOUN
ejpam-6211	38	3	,	,	PUNCT
ejpam-6211	38	4	the	the	DET
ejpam-6211	38	5	exponential	exponential	ADJ
ejpam-6211	38	6	generating	generating	NOUN
ejpam-6211	38	7	function	function	NOUN
ejpam-6211	38	8	of	of	ADP
ejpam-6211	38	9	the	the	DET
ejpam-6211	38	10	fibonacci	fibonacci	NOUN
ejpam-6211	38	11	numbers	number	NOUN
ejpam-6211	38	12	can	can	AUX
ejpam-6211	38	13	be	be	AUX
ejpam-6211	38	14	extended	extend	VERB
ejpam-6211	38	15	or	or	CCONJ
ejpam-6211	38	16	combined	combine	VERB
ejpam-6211	38	17	with	with	ADP
ejpam-6211	38	18	other	other	ADJ
ejpam-6211	38	19	numerical	numerical	ADJ
ejpam-6211	38	20	sequences	sequence	NOUN
ejpam-6211	38	21	.	.	PUNCT
ejpam-6211	39	1	similar	similar	ADJ
ejpam-6211	39	2	techniques	technique	NOUN
ejpam-6211	39	3	are	be	AUX
ejpam-6211	39	4	3	3	NUM
ejpam-6211	39	5	of	of	ADP
ejpam-6211	39	6	23	23	NUM
ejpam-6211	39	7	employed	employ	VERB
ejpam-6211	39	8	in	in	ADP
ejpam-6211	39	9	works	work	NOUN
ejpam-6211	39	10	such	such	ADJ
ejpam-6211	39	11	as	as	ADP
ejpam-6211	39	12	[	[	X
ejpam-6211	39	13	12–14	12–14	NUM
ejpam-6211	39	14	]	]	X
ejpam-6211	39	15	,	,	PUNCT
ejpam-6211	39	16	where	where	SCONJ
ejpam-6211	39	17	the	the	DET
ejpam-6211	39	18	generating	generating	NOUN
ejpam-6211	39	19	functions	function	NOUN
ejpam-6211	39	20	of	of	ADP
ejpam-6211	39	21	degenerate	degenerate	ADJ
ejpam-6211	39	22	hermite	hermite	PROPN
ejpam-6211	39	23	,	,	PUNCT
ejpam-6211	39	24	bernoulli	bernoulli	PROPN
ejpam-6211	39	25	,	,	PUNCT
ejpam-6211	39	26	euler	euler	NOUN
ejpam-6211	39	27	,	,	PUNCT
ejpam-6211	39	28	and	and	CCONJ
ejpam-6211	39	29	genocchi	genocchi	PROPN
ejpam-6211	39	30	polynomials	polynomial	NOUN
ejpam-6211	39	31	(	(	PUNCT
ejpam-6211	39	32	1	1	NUM
ejpam-6211	39	33	+	+	NUM
ejpam-6211	39	34	αt	αt	NOUN
ejpam-6211	39	35	)	)	PUNCT
ejpam-6211	39	36	x	x	X
ejpam-6211	39	37	α	α	PROPN
ejpam-6211	39	38	(	(	PUNCT
ejpam-6211	39	39	1	1	NUM
ejpam-6211	39	40	+	+	NUM
ejpam-6211	39	41	αt2	αt2	NOUN
ejpam-6211	39	42	)	)	PUNCT
ejpam-6211	39	43	y	y	PROPN
ejpam-6211	39	44	α	α	NOUN
ejpam-6211	39	45	=	=	PUNCT
ejpam-6211	40	1	∞∑	∞∑	PRON
ejpam-6211	40	2	n=0	n=0	NUM
ejpam-6211	40	3	hn(x	hn(x	X
ejpam-6211	40	4	,	,	PUNCT
ejpam-6211	40	5	y	y	PROPN
ejpam-6211	40	6	,	,	PUNCT
ejpam-6211	40	7	α	α	PROPN
ejpam-6211	40	8	)	)	PUNCT
ejpam-6211	40	9	tn	tn	PROPN
ejpam-6211	40	10	n	n	PROPN
ejpam-6211	41	1	!	!	PROPN
ejpam-6211	41	2	t	t	PROPN
ejpam-6211	41	3	(	(	PUNCT
ejpam-6211	41	4	1	1	NUM
ejpam-6211	41	5	+	+	NUM
ejpam-6211	41	6	αt	αt	NOUN
ejpam-6211	41	7	)	)	PUNCT
ejpam-6211	41	8	1	1	NUM
ejpam-6211	41	9	α	α	NOUN
ejpam-6211	41	10	−	−	NOUN
ejpam-6211	41	11	1	1	NUM
ejpam-6211	41	12	(	(	PUNCT
ejpam-6211	41	13	1	1	NUM
ejpam-6211	41	14	+	+	NUM
ejpam-6211	41	15	αt2	αt2	NOUN
ejpam-6211	41	16	)	)	PUNCT
ejpam-6211	41	17	x	x	SYM
ejpam-6211	41	18	α	α	NOUN
ejpam-6211	41	19	=	=	PUNCT
ejpam-6211	42	1	∞∑	∞∑	PRON
ejpam-6211	42	2	n=0	n=0	NUM
ejpam-6211	42	3	bn(x	bn(x	NUM
ejpam-6211	42	4	,	,	PUNCT
ejpam-6211	42	5	α	α	X
ejpam-6211	42	6	)	)	PUNCT
ejpam-6211	42	7	tn	tn	PROPN
ejpam-6211	42	8	n	n	PROPN
ejpam-6211	42	9	!	!	PROPN
ejpam-6211	42	10	2	2	NUM
ejpam-6211	42	11	(	(	PUNCT
ejpam-6211	42	12	1	1	NUM
ejpam-6211	42	13	+	+	NUM
ejpam-6211	42	14	αt	αt	NOUN
ejpam-6211	42	15	)	)	PUNCT
ejpam-6211	42	16	1	1	NUM
ejpam-6211	42	17	α	α	NOUN
ejpam-6211	42	18	+	+	NOUN
ejpam-6211	42	19	1	1	NUM
ejpam-6211	42	20	(	(	PUNCT
ejpam-6211	42	21	1	1	NUM
ejpam-6211	42	22	+	+	NUM
ejpam-6211	42	23	αt2	αt2	NOUN
ejpam-6211	42	24	)	)	PUNCT
ejpam-6211	42	25	x	x	SYM
ejpam-6211	42	26	α	α	NOUN
ejpam-6211	42	27	=	=	PUNCT
ejpam-6211	42	28	∞∑	∞∑	NOUN
ejpam-6211	42	29	n=0	n=0	NUM
ejpam-6211	42	30	en(x	en(x	ADP
ejpam-6211	42	31	,	,	PUNCT
ejpam-6211	42	32	α	α	NOUN
ejpam-6211	42	33	)	)	PUNCT
ejpam-6211	42	34	tn	tn	PROPN
ejpam-6211	42	35	n	n	NOUN
ejpam-6211	42	36	!	!	PROPN
ejpam-6211	43	1	2	2	NUM
ejpam-6211	43	2	t	t	NOUN
ejpam-6211	43	3	(	(	PUNCT
ejpam-6211	43	4	1	1	NUM
ejpam-6211	43	5	+	+	NUM
ejpam-6211	43	6	αt	αt	NOUN
ejpam-6211	43	7	)	)	PUNCT
ejpam-6211	43	8	1	1	NUM
ejpam-6211	43	9	α	α	NOUN
ejpam-6211	43	10	+	+	NOUN
ejpam-6211	43	11	1	1	NUM
ejpam-6211	43	12	(	(	PUNCT
ejpam-6211	43	13	1	1	NUM
ejpam-6211	43	14	+	+	NUM
ejpam-6211	43	15	αt2	αt2	NOUN
ejpam-6211	43	16	)	)	PUNCT
ejpam-6211	43	17	x	x	SYM
ejpam-6211	44	1	α	α	NOUN
ejpam-6211	44	2	=	=	PUNCT
ejpam-6211	44	3	∞∑	∞∑	NUM
ejpam-6211	44	4	n=0	n=0	NUM
ejpam-6211	44	5	gn(x	gn(x	X
ejpam-6211	44	6	,	,	PUNCT
ejpam-6211	44	7	α	α	NOUN
ejpam-6211	44	8	)	)	PUNCT
ejpam-6211	44	9	tn	tn	PROPN
ejpam-6211	44	10	n	n	PROPN
ejpam-6211	44	11	!	!	PROPN
ejpam-6211	44	12	are	be	AUX
ejpam-6211	44	13	modified	modify	VERB
ejpam-6211	44	14	by	by	ADP
ejpam-6211	44	15	introducing	introduce	VERB
ejpam-6211	44	16	new	new	ADJ
ejpam-6211	44	17	parameter	parameter	NOUN
ejpam-6211	44	18	λ	λ	PROPN
ejpam-6211	44	19	as	as	SCONJ
ejpam-6211	44	20	follows	follow	VERB
ejpam-6211	44	21	:(	:(	PUNCT
ejpam-6211	45	1	2µtν	2µtν	NUM
ejpam-6211	45	2	λ(1	λ(1	PROPN
ejpam-6211	46	1	+	+	CCONJ
ejpam-6211	46	2	at	at	ADP
ejpam-6211	46	3	)	)	PUNCT
ejpam-6211	46	4	1	1	NUM
ejpam-6211	46	5	a	a	DET
ejpam-6211	46	6	+	+	NOUN
ejpam-6211	46	7	1	1	NUM
ejpam-6211	46	8	)	)	PUNCT
ejpam-6211	46	9	α	α	NOUN
ejpam-6211	46	10	(	(	PUNCT
ejpam-6211	46	11	1	1	NUM
ejpam-6211	46	12	+	+	CCONJ
ejpam-6211	46	13	at	at	ADP
ejpam-6211	46	14	)	)	PUNCT
ejpam-6211	46	15	x	x	X
ejpam-6211	46	16	a	a	PRON
ejpam-6211	46	17	(	(	PUNCT
ejpam-6211	46	18	1	1	NUM
ejpam-6211	46	19	+	+	NUM
ejpam-6211	46	20	at2	at2	PROPN
ejpam-6211	46	21	)	)	PUNCT
ejpam-6211	46	22	y	y	PROPN
ejpam-6211	47	1	a	a	X
ejpam-6211	47	2	=	=	PRON
ejpam-6211	47	3	∞∑	∞∑	PROPN
ejpam-6211	47	4	n=0	n=0	NUM
ejpam-6211	47	5	hp(α	hp(α	NOUN
ejpam-6211	47	6	)	)	PUNCT
ejpam-6211	47	7	n	n	CCONJ
ejpam-6211	47	8	(	(	PUNCT
ejpam-6211	47	9	x	x	X
ejpam-6211	47	10	,	,	PUNCT
ejpam-6211	47	11	y	y	PROPN
ejpam-6211	47	12	;	;	PUNCT
ejpam-6211	47	13	a;λ;µ	a;λ;µ	NOUN
ejpam-6211	47	14	;	;	PUNCT
ejpam-6211	47	15	ν	ν	X
ejpam-6211	47	16	)	)	PUNCT
ejpam-6211	47	17	tn	tn	PROPN
ejpam-6211	47	18	n	n	PROPN
ejpam-6211	47	19	!	!	PUNCT
ejpam-6211	48	1	(	(	PUNCT
ejpam-6211	48	2	tm	tm	PROPN
ejpam-6211	48	3	λ(1	λ(1	PROPN
ejpam-6211	48	4	+	+	CCONJ
ejpam-6211	48	5	at	at	ADP
ejpam-6211	48	6	)	)	PUNCT
ejpam-6211	48	7	1	1	NUM
ejpam-6211	48	8	a	a	DET
ejpam-6211	48	9	−	−	X
ejpam-6211	48	10	∑m−1	∑m−1	NOUN
ejpam-6211	49	1	l=0	l=0	PROPN
ejpam-6211	49	2	(	(	PUNCT
ejpam-6211	49	3	t	t	PROPN
ejpam-6211	49	4	log	log	VERB
ejpam-6211	49	5	b)l	b)l	NOUN
ejpam-6211	49	6	l	l	NOUN
ejpam-6211	49	7	!	!	PUNCT
ejpam-6211	49	8	)	)	PUNCT
ejpam-6211	50	1	α	α	NOUN
ejpam-6211	50	2	(	(	PUNCT
ejpam-6211	50	3	1	1	NUM
ejpam-6211	50	4	+	+	NUM
ejpam-6211	50	5	at2	at2	PROPN
ejpam-6211	50	6	)	)	PUNCT
ejpam-6211	50	7	x	x	X
ejpam-6211	51	1	a	a	DET
ejpam-6211	51	2	=	=	VERB
ejpam-6211	51	3	∞∑	∞∑	ADJ
ejpam-6211	51	4	n=0	n=0	ADJ
ejpam-6211	51	5	b[m−1,α	b[m−1,α	NOUN
ejpam-6211	51	6	]	]	X
ejpam-6211	51	7	n	n	CCONJ
ejpam-6211	51	8	(	(	PUNCT
ejpam-6211	51	9	x	x	X
ejpam-6211	51	10	,	,	PUNCT
ejpam-6211	51	11	a	a	DET
ejpam-6211	51	12	,	,	PUNCT
ejpam-6211	51	13	b;λ	b;λ	NOUN
ejpam-6211	51	14	)	)	PUNCT
ejpam-6211	51	15	tn	tn	PROPN
ejpam-6211	51	16	n	n	PROPN
ejpam-6211	51	17	!	!	PUNCT
ejpam-6211	51	18	(	(	PUNCT
ejpam-6211	51	19	2	2	NUM
ejpam-6211	51	20	m	m	VERB
ejpam-6211	51	21	λ(1	λ(1	NOUN
ejpam-6211	51	22	+	+	CCONJ
ejpam-6211	51	23	at	at	ADP
ejpam-6211	51	24	)	)	PUNCT
ejpam-6211	51	25	1	1	NUM
ejpam-6211	51	26	a	a	DET
ejpam-6211	51	27	+	+	NOUN
ejpam-6211	51	28	∑m−1	∑m−1	ADJ
ejpam-6211	51	29	l=0	l=0	PROPN
ejpam-6211	51	30	(	(	PUNCT
ejpam-6211	51	31	t	t	PROPN
ejpam-6211	51	32	log	log	VERB
ejpam-6211	51	33	b)l	b)l	NOUN
ejpam-6211	51	34	l	l	NOUN
ejpam-6211	51	35	!	!	PUNCT
ejpam-6211	51	36	)	)	PUNCT
ejpam-6211	52	1	α	α	NOUN
ejpam-6211	52	2	(	(	PUNCT
ejpam-6211	52	3	1	1	NUM
ejpam-6211	52	4	+	+	NUM
ejpam-6211	52	5	at2	at2	PROPN
ejpam-6211	52	6	)	)	PUNCT
ejpam-6211	52	7	x	x	X
ejpam-6211	53	1	a	a	DET
ejpam-6211	53	2	=	=	PRON
ejpam-6211	53	3	∞∑	∞∑	PROPN
ejpam-6211	53	4	n=0	n=0	NUM
ejpam-6211	53	5	en(x	en(x	ADP
ejpam-6211	53	6	,	,	PUNCT
ejpam-6211	53	7	a	a	DET
ejpam-6211	53	8	,	,	PUNCT
ejpam-6211	53	9	b;λ	b;λ	NOUN
ejpam-6211	53	10	)	)	PUNCT
ejpam-6211	53	11	tn	tn	PROPN
ejpam-6211	53	12	n	n	PROPN
ejpam-6211	53	13	!	!	PUNCT
ejpam-6211	54	1	(	(	PUNCT
ejpam-6211	54	2	(	(	PUNCT
ejpam-6211	54	3	2t)m	2t)m	NUM
ejpam-6211	54	4	λ(1	λ(1	PROPN
ejpam-6211	55	1	+	+	CCONJ
ejpam-6211	55	2	at	at	ADP
ejpam-6211	55	3	)	)	PUNCT
ejpam-6211	55	4	1	1	NUM
ejpam-6211	55	5	a	a	DET
ejpam-6211	55	6	+	+	NOUN
ejpam-6211	55	7	∑m−1	∑m−1	ADJ
ejpam-6211	55	8	l=0	l=0	PROPN
ejpam-6211	55	9	(	(	PUNCT
ejpam-6211	55	10	t	t	PROPN
ejpam-6211	55	11	log	log	VERB
ejpam-6211	55	12	b)l	b)l	NOUN
ejpam-6211	55	13	l	l	NOUN
ejpam-6211	55	14	!	!	PUNCT
ejpam-6211	55	15	)	)	PUNCT
ejpam-6211	56	1	α	α	NOUN
ejpam-6211	56	2	(	(	PUNCT
ejpam-6211	56	3	1	1	NUM
ejpam-6211	56	4	+	+	NUM
ejpam-6211	56	5	at2	at2	PROPN
ejpam-6211	56	6	)	)	PUNCT
ejpam-6211	56	7	x	x	X
ejpam-6211	56	8	a	a	PRON
ejpam-6211	56	9	=	=	PUNCT
ejpam-6211	56	10	∞∑	∞∑	NUM
ejpam-6211	56	11	n=0	n=0	NOUN
ejpam-6211	56	12	gn(x	gn(x	X
ejpam-6211	56	13	,	,	PUNCT
ejpam-6211	56	14	a	a	DET
ejpam-6211	56	15	,	,	PUNCT
ejpam-6211	56	16	b;λ	b;λ	NOUN
ejpam-6211	56	17	)	)	PUNCT
ejpam-6211	56	18	tn	tn	PROPN
ejpam-6211	56	19	n	n	PROPN
ejpam-6211	56	20	!	!	PUNCT
ejpam-6211	56	21	.	.	PUNCT
ejpam-6211	57	1	in	in	ADP
ejpam-6211	57	2	this	this	DET
ejpam-6211	57	3	paper	paper	NOUN
ejpam-6211	57	4	we	we	PRON
ejpam-6211	57	5	will	will	AUX
ejpam-6211	57	6	introduce	introduce	VERB
ejpam-6211	57	7	another	another	DET
ejpam-6211	57	8	novel	novel	ADJ
ejpam-6211	57	9	variant	variant	NOUN
ejpam-6211	57	10	of	of	ADP
ejpam-6211	57	11	r	r	NOUN
ejpam-6211	57	12	-	-	PUNCT
ejpam-6211	57	13	stirling	stirling	NOUN
ejpam-6211	57	14	numbers	number	NOUN
ejpam-6211	57	15	which	which	PRON
ejpam-6211	57	16	combines	combine	VERB
ejpam-6211	57	17	the	the	DET
ejpam-6211	57	18	exponential	exponential	ADJ
ejpam-6211	57	19	generating	generating	NOUN
ejpam-6211	57	20	function	function	NOUN
ejpam-6211	57	21	of	of	ADP
ejpam-6211	57	22	the	the	DET
ejpam-6211	57	23	fibonacci	fibonacci	NOUN
ejpam-6211	57	24	number	number	NOUN
ejpam-6211	57	25	to	to	ADP
ejpam-6211	57	26	the	the	DET
ejpam-6211	57	27	exponential	exponential	ADJ
ejpam-6211	57	28	generating	generating	NOUN
ejpam-6211	57	29	function	function	NOUN
ejpam-6211	57	30	of	of	ADP
ejpam-6211	57	31	the	the	DET
ejpam-6211	57	32	signed	sign	VERB
ejpam-6211	57	33	r	r	NOUN
ejpam-6211	57	34	-	-	PUNCT
ejpam-6211	57	35	stirling	stirling	NOUN
ejpam-6211	57	36	numbers	number	NOUN
ejpam-6211	57	37	of	of	ADP
ejpam-6211	57	38	the	the	DET
ejpam-6211	57	39	first	first	ADJ
ejpam-6211	57	40	kind	kind	NOUN
ejpam-6211	57	41	and	and	CCONJ
ejpam-6211	57	42	r	r	NOUN
ejpam-6211	57	43	-	-	PUNCT
ejpam-6211	57	44	stirling	stirling	NOUN
ejpam-6211	57	45	number	number	NOUN
ejpam-6211	57	46	of	of	ADP
ejpam-6211	57	47	the	the	DET
ejpam-6211	57	48	second	second	ADJ
ejpam-6211	57	49	kind	kind	NOUN
ejpam-6211	57	50	.	.	PUNCT
ejpam-6211	58	1	these	these	DET
ejpam-6211	58	2	numbers	number	NOUN
ejpam-6211	58	3	are	be	AUX
ejpam-6211	58	4	called	call	VERB
ejpam-6211	58	5	the	the	DET
ejpam-6211	58	6	r	r	NOUN
ejpam-6211	58	7	-	-	PUNCT
ejpam-6211	58	8	stirling	stirling	NOUN
ejpam-6211	58	9	fibonacci	fibonacci	NOUN
ejpam-6211	58	10	numbers	number	NOUN
ejpam-6211	58	11	.	.	PUNCT
ejpam-6211	59	1	various	various	ADJ
ejpam-6211	59	2	formulas	formula	NOUN
ejpam-6211	59	3	were	be	AUX
ejpam-6211	59	4	derived	derive	VERB
ejpam-6211	59	5	including	include	VERB
ejpam-6211	59	6	horizontal	horizontal	ADJ
ejpam-6211	59	7	generating	generating	NOUN
ejpam-6211	59	8	function	function	NOUN
ejpam-6211	59	9	,	,	PUNCT
ejpam-6211	59	10	explicit	explicit	ADJ
ejpam-6211	59	11	formulas	formula	NOUN
ejpam-6211	59	12	and	and	CCONJ
ejpam-6211	59	13	convolution	convolution	NOUN
ejpam-6211	59	14	.	.	PUNCT
ejpam-6211	60	1	2	2	X
ejpam-6211	60	2	.	.	X
ejpam-6211	60	3	fibonacci	fibonacci	NOUN
ejpam-6211	60	4	polynomials	polynomials	PROPN
ejpam-6211	60	5	fibonacci	fibonacci	PROPN
ejpam-6211	60	6	polynomials	polynomials	PROPN
ejpam-6211	60	7	f	f	PROPN
ejpam-6211	60	8	(	(	PUNCT
ejpam-6211	60	9	1	1	NUM
ejpam-6211	60	10	)	)	PUNCT
ejpam-6211	60	11	n	n	CCONJ
ejpam-6211	60	12	(	(	PUNCT
ejpam-6211	60	13	x	x	X
ejpam-6211	60	14	)	)	PUNCT
ejpam-6211	60	15	are	be	AUX
ejpam-6211	60	16	a	a	DET
ejpam-6211	60	17	family	family	NOUN
ejpam-6211	60	18	of	of	ADP
ejpam-6211	60	19	polynomials	polynomial	NOUN
ejpam-6211	60	20	that	that	PRON
ejpam-6211	60	21	extend	extend	VERB
ejpam-6211	60	22	the	the	DET
ejpam-6211	60	23	fibonacci	fibonacci	NOUN
ejpam-6211	60	24	sequence	sequence	NOUN
ejpam-6211	60	25	into	into	ADP
ejpam-6211	60	26	the	the	DET
ejpam-6211	60	27	realm	realm	NOUN
ejpam-6211	60	28	of	of	ADP
ejpam-6211	60	29	algebra	algebra	NOUN
ejpam-6211	60	30	.	.	PUNCT
ejpam-6211	61	1	instead	instead	ADV
ejpam-6211	61	2	of	of	ADP
ejpam-6211	61	3	producing	produce	VERB
ejpam-6211	61	4	just	just	ADV
ejpam-6211	61	5	numbers	number	NOUN
ejpam-6211	61	6	,	,	PUNCT
ejpam-6211	61	7	they	they	PRON
ejpam-6211	61	8	produce	produce	VERB
ejpam-6211	61	9	polynomials	polynomial	NOUN
ejpam-6211	61	10	in	in	ADP
ejpam-6211	61	11	x	x	NOUN
ejpam-6211	61	12	,	,	PUNCT
ejpam-6211	61	13	while	while	SCONJ
ejpam-6211	61	14	still	still	ADV
ejpam-6211	61	15	obeying	obey	VERB
ejpam-6211	61	16	a	a	DET
ejpam-6211	61	17	fibonacci	fibonacci	NOUN
ejpam-6211	61	18	-	-	PUNCT
ejpam-6211	61	19	style	style	NOUN
ejpam-6211	61	20	recurrence	recurrence	NOUN
ejpam-6211	61	21	.	.	PUNCT
ejpam-6211	62	1	that	that	PRON
ejpam-6211	62	2	is	be	AUX
ejpam-6211	62	3	,	,	PUNCT
ejpam-6211	62	4	f	f	PROPN
ejpam-6211	62	5	(	(	PUNCT
ejpam-6211	62	6	1	1	NUM
ejpam-6211	62	7	)	)	PUNCT
ejpam-6211	62	8	0	0	NUM
ejpam-6211	63	1	(	(	PUNCT
ejpam-6211	63	2	x	x	X
ejpam-6211	63	3	)	)	PUNCT
ejpam-6211	63	4	=	=	SYM
ejpam-6211	63	5	0	0	NUM
ejpam-6211	63	6	,	,	PUNCT
ejpam-6211	63	7	f	f	X
ejpam-6211	63	8	(	(	PUNCT
ejpam-6211	63	9	1	1	NUM
ejpam-6211	63	10	)	)	PUNCT
ejpam-6211	63	11	1	1	NUM
ejpam-6211	63	12	(	(	PUNCT
ejpam-6211	63	13	x	x	NOUN
ejpam-6211	63	14	)	)	PUNCT
ejpam-6211	63	15	=	=	SYM
ejpam-6211	63	16	1	1	NUM
ejpam-6211	63	17	,	,	PUNCT
ejpam-6211	63	18	f	f	X
ejpam-6211	63	19	(	(	PUNCT
ejpam-6211	63	20	1	1	NUM
ejpam-6211	63	21	)	)	PUNCT
ejpam-6211	63	22	n	n	PROPN
ejpam-6211	63	23	(	(	PUNCT
ejpam-6211	63	24	x	x	X
ejpam-6211	63	25	)	)	PUNCT
ejpam-6211	63	26	=	=	SYM
ejpam-6211	63	27	xf	xf	PROPN
ejpam-6211	63	28	(	(	PUNCT
ejpam-6211	63	29	1	1	NUM
ejpam-6211	63	30	)	)	PUNCT
ejpam-6211	63	31	n−1(x	n−1(x	PROPN
ejpam-6211	63	32	)	)	PUNCT
ejpam-6211	64	1	+	+	NUM
ejpam-6211	64	2	f	f	X
ejpam-6211	64	3	(	(	PUNCT
ejpam-6211	64	4	1	1	NUM
ejpam-6211	64	5	)	)	PUNCT
ejpam-6211	64	6	n−2(x	n−2(x	PROPN
ejpam-6211	64	7	)	)	PUNCT
ejpam-6211	64	8	,	,	PUNCT
ejpam-6211	64	9	for	for	ADP
ejpam-6211	64	10	n	n	X
ejpam-6211	64	11	(	(	PUNCT
ejpam-6211	64	12	2.1	2.1	NUM
ejpam-6211	64	13	)	)	PUNCT
ejpam-6211	64	14	4	4	NUM
ejpam-6211	64	15	of	of	ADP
ejpam-6211	64	16	23	23	NUM
ejpam-6211	64	17	this	this	DET
ejpam-6211	64	18	recurrence	recurrence	NOUN
ejpam-6211	64	19	is	be	AUX
ejpam-6211	64	20	linear	linear	ADJ
ejpam-6211	64	21	and	and	CCONJ
ejpam-6211	64	22	non	non	ADJ
ejpam-6211	64	23	-	-	ADJ
ejpam-6211	64	24	homogeneous	homogeneous	ADJ
ejpam-6211	64	25	,	,	PUNCT
ejpam-6211	64	26	and	and	CCONJ
ejpam-6211	64	27	it	it	PRON
ejpam-6211	64	28	reflects	reflect	VERB
ejpam-6211	64	29	a	a	DET
ejpam-6211	64	30	combination	combination	NOUN
ejpam-6211	64	31	of	of	ADP
ejpam-6211	64	32	growth	growth	NOUN
ejpam-6211	64	33	(	(	PUNCT
ejpam-6211	64	34	via	via	ADP
ejpam-6211	64	35	multiplication	multiplication	NOUN
ejpam-6211	64	36	by	by	ADP
ejpam-6211	64	37	x	x	NOUN
ejpam-6211	64	38	)	)	PUNCT
ejpam-6211	64	39	and	and	CCONJ
ejpam-6211	64	40	memory	memory	NOUN
ejpam-6211	64	41	(	(	PUNCT
ejpam-6211	64	42	via	via	ADP
ejpam-6211	64	43	the	the	DET
ejpam-6211	64	44	f	f	PROPN
ejpam-6211	64	45	(	(	PUNCT
ejpam-6211	64	46	1	1	NUM
ejpam-6211	64	47	)	)	PUNCT
ejpam-6211	64	48	n−2(x	n−2(x	PROPN
ejpam-6211	64	49	)	)	PUNCT
ejpam-6211	64	50	term	term	NOUN
ejpam-6211	64	51	)	)	PUNCT
ejpam-6211	64	52	.	.	PUNCT
ejpam-6211	65	1	in	in	ADP
ejpam-6211	65	2	the	the	DET
ejpam-6211	65	3	same	same	ADJ
ejpam-6211	65	4	way	way	NOUN
ejpam-6211	65	5	the	the	DET
ejpam-6211	65	6	fibonacci	fibonacci	NOUN
ejpam-6211	65	7	numbers	number	NOUN
ejpam-6211	65	8	appear	appear	VERB
ejpam-6211	65	9	in	in	ADP
ejpam-6211	65	10	natural	natural	ADJ
ejpam-6211	65	11	growth	growth	NOUN
ejpam-6211	65	12	,	,	PUNCT
ejpam-6211	65	13	recursion	recursion	NOUN
ejpam-6211	65	14	,	,	PUNCT
ejpam-6211	65	15	and	and	CCONJ
ejpam-6211	65	16	combinatorics	combinatoric	NOUN
ejpam-6211	65	17	,	,	PUNCT
ejpam-6211	65	18	fibonacci	fibonacci	NOUN
ejpam-6211	65	19	polynomials	polynomial	NOUN
ejpam-6211	65	20	provide	provide	VERB
ejpam-6211	65	21	a	a	DET
ejpam-6211	65	22	parameterized	parameterized	ADJ
ejpam-6211	65	23	version	version	NOUN
ejpam-6211	65	24	of	of	ADP
ejpam-6211	65	25	fibonacci	fibonacci	NOUN
ejpam-6211	65	26	numbers	number	NOUN
ejpam-6211	65	27	.	.	PUNCT
ejpam-6211	66	1	when	when	SCONJ
ejpam-6211	66	2	we	we	PRON
ejpam-6211	66	3	evaluate	evaluate	VERB
ejpam-6211	66	4	f	f	PROPN
ejpam-6211	66	5	(	(	PUNCT
ejpam-6211	66	6	1	1	NUM
ejpam-6211	66	7	)	)	PUNCT
ejpam-6211	66	8	n	n	PROPN
ejpam-6211	66	9	(	(	PUNCT
ejpam-6211	66	10	x	x	X
ejpam-6211	66	11	)	)	PUNCT
ejpam-6211	66	12	at	at	ADP
ejpam-6211	66	13	specific	specific	ADJ
ejpam-6211	66	14	values	value	NOUN
ejpam-6211	66	15	of	of	ADP
ejpam-6211	66	16	x	x	PRON
ejpam-6211	66	17	,	,	PUNCT
ejpam-6211	66	18	we	we	PRON
ejpam-6211	66	19	recover	recover	VERB
ejpam-6211	66	20	numerical	numerical	ADJ
ejpam-6211	66	21	sequences	sequence	NOUN
ejpam-6211	66	22	with	with	ADP
ejpam-6211	66	23	meaningful	meaningful	ADJ
ejpam-6211	66	24	interpretations	interpretation	NOUN
ejpam-6211	66	25	.	.	PUNCT
ejpam-6211	67	1	for	for	ADP
ejpam-6211	67	2	instance	instance	NOUN
ejpam-6211	67	3	:	:	PUNCT
ejpam-6211	67	4	•	•	NUM
ejpam-6211	67	5	f	f	X
ejpam-6211	67	6	(	(	PUNCT
ejpam-6211	67	7	1	1	NUM
ejpam-6211	67	8	)	)	PUNCT
ejpam-6211	67	9	n	n	CCONJ
ejpam-6211	67	10	(	(	PUNCT
ejpam-6211	67	11	1	1	X
ejpam-6211	67	12	)	)	PUNCT
ejpam-6211	67	13	gives	give	VERB
ejpam-6211	67	14	the	the	DET
ejpam-6211	67	15	standard	standard	ADJ
ejpam-6211	67	16	fibonacci	fibonacci	NOUN
ejpam-6211	67	17	numbers	number	NOUN
ejpam-6211	67	18	.	.	PUNCT
ejpam-6211	68	1	•	•	NUM
ejpam-6211	68	2	f	f	PROPN
ejpam-6211	68	3	(	(	PUNCT
ejpam-6211	68	4	1	1	NUM
ejpam-6211	68	5	)	)	PUNCT
ejpam-6211	68	6	n	n	CCONJ
ejpam-6211	68	7	(	(	PUNCT
ejpam-6211	68	8	2	2	X
ejpam-6211	68	9	)	)	PUNCT
ejpam-6211	68	10	gives	give	VERB
ejpam-6211	68	11	the	the	DET
ejpam-6211	68	12	pell	pell	NOUN
ejpam-6211	68	13	numbers	number	NOUN
ejpam-6211	68	14	.	.	PUNCT
ejpam-6211	69	1	•	•	ADV
ejpam-6211	69	2	they	they	PRON
ejpam-6211	69	3	also	also	ADV
ejpam-6211	69	4	show	show	VERB
ejpam-6211	69	5	up	up	ADP
ejpam-6211	69	6	in	in	ADP
ejpam-6211	69	7	algebraic	algebraic	ADJ
ejpam-6211	69	8	identities	identity	NOUN
ejpam-6211	69	9	and	and	CCONJ
ejpam-6211	69	10	generating	generate	VERB
ejpam-6211	69	11	function	function	NOUN
ejpam-6211	69	12	theory	theory	NOUN
ejpam-6211	69	13	.	.	PUNCT
ejpam-6211	70	1	first	first	ADJ
ejpam-6211	70	2	few	few	ADJ
ejpam-6211	70	3	terms	term	NOUN
ejpam-6211	70	4	are	be	AUX
ejpam-6211	70	5	given	give	VERB
ejpam-6211	70	6	as	as	SCONJ
ejpam-6211	70	7	follows	follow	VERB
ejpam-6211	70	8	f	f	PROPN
ejpam-6211	70	9	(	(	PUNCT
ejpam-6211	70	10	1	1	NUM
ejpam-6211	70	11	)	)	PUNCT
ejpam-6211	70	12	0	0	NUM
ejpam-6211	71	1	(	(	PUNCT
ejpam-6211	71	2	x	x	X
ejpam-6211	71	3	)	)	PUNCT
ejpam-6211	71	4	=	=	SYM
ejpam-6211	72	1	0	0	NUM
ejpam-6211	72	2	f	f	X
ejpam-6211	72	3	(	(	PUNCT
ejpam-6211	72	4	1	1	NUM
ejpam-6211	72	5	)	)	PUNCT
ejpam-6211	72	6	1	1	NUM
ejpam-6211	72	7	(	(	PUNCT
ejpam-6211	72	8	x	x	NOUN
ejpam-6211	72	9	)	)	PUNCT
ejpam-6211	72	10	=	=	SYM
ejpam-6211	72	11	1	1	NUM
ejpam-6211	72	12	f	f	X
ejpam-6211	72	13	(	(	PUNCT
ejpam-6211	72	14	1	1	NUM
ejpam-6211	72	15	)	)	SYM
ejpam-6211	72	16	2	2	NUM
ejpam-6211	72	17	(	(	PUNCT
ejpam-6211	72	18	x	x	NOUN
ejpam-6211	72	19	)	)	PUNCT
ejpam-6211	72	20	=	=	PUNCT
ejpam-6211	72	21	x	x	SYM
ejpam-6211	72	22	f	f	X
ejpam-6211	72	23	(	(	PUNCT
ejpam-6211	72	24	1	1	NUM
ejpam-6211	72	25	)	)	PUNCT
ejpam-6211	72	26	3	3	NUM
ejpam-6211	72	27	(	(	PUNCT
ejpam-6211	72	28	x	x	NOUN
ejpam-6211	72	29	)	)	PUNCT
ejpam-6211	72	30	=	=	SYM
ejpam-6211	73	1	x2	x2	PROPN
ejpam-6211	74	1	+	+	CCONJ
ejpam-6211	74	2	1	1	NUM
ejpam-6211	74	3	f	f	X
ejpam-6211	74	4	(	(	PUNCT
ejpam-6211	74	5	1	1	NUM
ejpam-6211	74	6	)	)	PUNCT
ejpam-6211	74	7	4	4	NUM
ejpam-6211	74	8	(	(	PUNCT
ejpam-6211	74	9	x	x	NOUN
ejpam-6211	74	10	)	)	PUNCT
ejpam-6211	74	11	=	=	SYM
ejpam-6211	75	1	x3	x3	PROPN
ejpam-6211	76	1	+	+	CCONJ
ejpam-6211	76	2	2x	2x	NUM
ejpam-6211	76	3	f	f	X
ejpam-6211	76	4	(	(	PUNCT
ejpam-6211	76	5	1	1	NUM
ejpam-6211	76	6	)	)	PUNCT
ejpam-6211	76	7	5	5	NUM
ejpam-6211	76	8	(	(	PUNCT
ejpam-6211	76	9	x	x	NOUN
ejpam-6211	76	10	)	)	PUNCT
ejpam-6211	76	11	=	=	SYM
ejpam-6211	76	12	x4	x4	PROPN
ejpam-6211	76	13	+	+	NUM
ejpam-6211	76	14	3x2	3x2	NUM
ejpam-6211	76	15	+	+	SYM
ejpam-6211	76	16	1	1	NUM
ejpam-6211	76	17	f	f	NOUN
ejpam-6211	76	18	(	(	PUNCT
ejpam-6211	76	19	1	1	NUM
ejpam-6211	76	20	)	)	PUNCT
ejpam-6211	76	21	6	6	NUM
ejpam-6211	76	22	(	(	PUNCT
ejpam-6211	76	23	x	x	NOUN
ejpam-6211	76	24	)	)	PUNCT
ejpam-6211	76	25	=	=	SYM
ejpam-6211	76	26	x5	x5	NOUN
ejpam-6211	76	27	+	+	NUM
ejpam-6211	76	28	4x3	4x3	NUM
ejpam-6211	76	29	+	+	CCONJ
ejpam-6211	76	30	3x	3x	NUM
ejpam-6211	76	31	these	these	PRON
ejpam-6211	76	32	are	be	AUX
ejpam-6211	76	33	monic	monic	ADJ
ejpam-6211	76	34	polynomials	polynomial	NOUN
ejpam-6211	76	35	(	(	PUNCT
ejpam-6211	76	36	leading	lead	VERB
ejpam-6211	76	37	coefficient	coefficient	NOUN
ejpam-6211	76	38	is	be	AUX
ejpam-6211	76	39	1	1	NUM
ejpam-6211	76	40	)	)	PUNCT
ejpam-6211	76	41	and	and	CCONJ
ejpam-6211	76	42	have	have	AUX
ejpam-6211	76	43	alternating	alternate	VERB
ejpam-6211	76	44	degrees	degree	NOUN
ejpam-6211	76	45	,	,	PUNCT
ejpam-6211	76	46	increasing	increase	VERB
ejpam-6211	76	47	by	by	ADP
ejpam-6211	76	48	1	1	NUM
ejpam-6211	76	49	with	with	ADP
ejpam-6211	76	50	each	each	DET
ejpam-6211	76	51	n.	n.	NOUN
ejpam-6211	76	52	the	the	DET
ejpam-6211	76	53	closed	close	VERB
ejpam-6211	76	54	form	form	NOUN
ejpam-6211	76	55	uses	use	VERB
ejpam-6211	76	56	the	the	DET
ejpam-6211	76	57	roots	root	NOUN
ejpam-6211	76	58	of	of	ADP
ejpam-6211	76	59	the	the	DET
ejpam-6211	76	60	characteristic	characteristic	ADJ
ejpam-6211	76	61	equation	equation	NOUN
ejpam-6211	76	62	associated	associate	VERB
ejpam-6211	76	63	with	with	ADP
ejpam-6211	76	64	the	the	DET
ejpam-6211	76	65	recurrence	recurrence	NOUN
ejpam-6211	76	66	:	:	PUNCT
ejpam-6211	76	67	characteristic	characteristic	ADJ
ejpam-6211	76	68	equation	equation	NOUN
ejpam-6211	76	69	:	:	PUNCT
ejpam-6211	77	1	r2	r2	PROPN
ejpam-6211	77	2	−	−	PROPN
ejpam-6211	77	3	xr	xr	PROPN
ejpam-6211	77	4	−	−	PROPN
ejpam-6211	77	5	1	1	NUM
ejpam-6211	77	6	=	=	SYM
ejpam-6211	77	7	0	0	NUM
ejpam-6211	77	8	⇒	⇒	NOUN
ejpam-6211	77	9	r	r	NOUN
ejpam-6211	77	10	=	=	PUNCT
ejpam-6211	77	11	x±	x±	PROPN
ejpam-6211	77	12	√	√	NUM
ejpam-6211	77	13	x2	x2	PROPN
ejpam-6211	78	1	+	+	CCONJ
ejpam-6211	78	2	4	4	NUM
ejpam-6211	78	3	2	2	NUM
ejpam-6211	78	4	let	let	VERB
ejpam-6211	78	5	:	:	PUNCT
ejpam-6211	78	6	α(x	α(x	NUM
ejpam-6211	78	7	)	)	PUNCT
ejpam-6211	78	8	=	=	PUNCT
ejpam-6211	79	1	x+	x+	PUNCT
ejpam-6211	80	1	√	√	PUNCT
ejpam-6211	80	2	x2	x2	NOUN
ejpam-6211	81	1	+	+	CCONJ
ejpam-6211	81	2	4	4	NUM
ejpam-6211	81	3	2	2	NUM
ejpam-6211	81	4	,	,	PUNCT
ejpam-6211	81	5	β(x	β(x	NOUN
ejpam-6211	81	6	)	)	PUNCT
ejpam-6211	81	7	=	=	SYM
ejpam-6211	82	1	x−	x−	PROPN
ejpam-6211	82	2	√	√	NUM
ejpam-6211	82	3	x2	x2	PROPN
ejpam-6211	83	1	+	+	CCONJ
ejpam-6211	83	2	4	4	NUM
ejpam-6211	83	3	2	2	NUM
ejpam-6211	83	4	then	then	ADV
ejpam-6211	83	5	:	:	PUNCT
ejpam-6211	83	6	f	f	PROPN
ejpam-6211	83	7	(	(	PUNCT
ejpam-6211	83	8	1	1	NUM
ejpam-6211	83	9	)	)	PUNCT
ejpam-6211	83	10	n	n	PROPN
ejpam-6211	83	11	(	(	PUNCT
ejpam-6211	83	12	x	x	X
ejpam-6211	83	13	)	)	PUNCT
ejpam-6211	83	14	=	=	SYM
ejpam-6211	83	15	αn(x)−	αn(x)−	PROPN
ejpam-6211	83	16	βn(x	βn(x	PUNCT
ejpam-6211	83	17	)	)	PUNCT
ejpam-6211	83	18	α(x)−	α(x)−	PROPN
ejpam-6211	83	19	β(x	β(x	PROPN
ejpam-6211	83	20	)	)	PUNCT
ejpam-6211	83	21	this	this	PRON
ejpam-6211	83	22	generalizes	generalize	VERB
ejpam-6211	83	23	the	the	DET
ejpam-6211	83	24	binet	binet	NOUN
ejpam-6211	83	25	formula	formula	NOUN
ejpam-6211	83	26	for	for	ADP
ejpam-6211	83	27	fibonacci	fibonacci	NOUN
ejpam-6211	83	28	numbers	number	NOUN
ejpam-6211	83	29	and	and	CCONJ
ejpam-6211	83	30	offers	offer	VERB
ejpam-6211	83	31	a	a	DET
ejpam-6211	83	32	fast	fast	ADJ
ejpam-6211	83	33	way	way	NOUN
ejpam-6211	83	34	to	to	PART
ejpam-6211	83	35	compute	compute	VERB
ejpam-6211	83	36	fibonacci	fibonacci	NOUN
ejpam-6211	83	37	polynomials	polynomial	NOUN
ejpam-6211	83	38	for	for	ADP
ejpam-6211	83	39	large	large	ADJ
ejpam-6211	83	40	n.	n.	NOUN
ejpam-6211	83	41	generating	generating	NOUN
ejpam-6211	83	42	function	function	NOUN
ejpam-6211	83	43	5	5	NUM
ejpam-6211	83	44	of	of	ADP
ejpam-6211	83	45	23	23	NUM
ejpam-6211	83	46	the	the	DET
ejpam-6211	83	47	generating	generate	VERB
ejpam-6211	83	48	function	function	NOUN
ejpam-6211	83	49	encodes	encode	NOUN
ejpam-6211	83	50	all	all	DET
ejpam-6211	83	51	fibonacci	fibonacci	NOUN
ejpam-6211	83	52	polynomials	polynomial	NOUN
ejpam-6211	83	53	into	into	ADP
ejpam-6211	83	54	a	a	DET
ejpam-6211	83	55	single	single	ADJ
ejpam-6211	83	56	expression	expression	NOUN
ejpam-6211	83	57	:	:	PUNCT
ejpam-6211	83	58	g(x	g(x	NOUN
ejpam-6211	83	59	,	,	PUNCT
ejpam-6211	83	60	t	t	PROPN
ejpam-6211	83	61	)	)	PUNCT
ejpam-6211	83	62	=	=	PUNCT
ejpam-6211	84	1	∞∑	∞∑	NUM
ejpam-6211	84	2	n=0	n=0	NUM
ejpam-6211	84	3	f	f	X
ejpam-6211	84	4	(	(	PUNCT
ejpam-6211	84	5	1	1	NUM
ejpam-6211	84	6	)	)	PUNCT
ejpam-6211	84	7	n	n	PROPN
ejpam-6211	84	8	(	(	PUNCT
ejpam-6211	84	9	x)tn	x)tn	PROPN
ejpam-6211	84	10	=	=	PROPN
ejpam-6211	84	11	t	t	PROPN
ejpam-6211	84	12	1−	1−	NUM
ejpam-6211	84	13	xt−	xt−	PUNCT
ejpam-6211	84	14	t2	t2	NOUN
ejpam-6211	84	15	(	(	PUNCT
ejpam-6211	84	16	2.2	2.2	NUM
ejpam-6211	84	17	)	)	PUNCT
ejpam-6211	84	18	this	this	DET
ejpam-6211	84	19	rational	rational	ADJ
ejpam-6211	84	20	function	function	NOUN
ejpam-6211	84	21	captures	capture	VERB
ejpam-6211	84	22	all	all	PRON
ejpam-6211	84	23	of	of	ADP
ejpam-6211	84	24	the	the	DET
ejpam-6211	84	25	recurrence	recurrence	NOUN
ejpam-6211	84	26	information	information	NOUN
ejpam-6211	84	27	and	and	CCONJ
ejpam-6211	84	28	is	be	AUX
ejpam-6211	84	29	extremely	extremely	ADV
ejpam-6211	84	30	useful	useful	ADJ
ejpam-6211	84	31	in	in	ADP
ejpam-6211	84	32	solving	solve	VERB
ejpam-6211	84	33	recurrence	recurrence	NOUN
ejpam-6211	84	34	relations	relation	NOUN
ejpam-6211	84	35	,	,	PUNCT
ejpam-6211	84	36	analyzing	analyze	VERB
ejpam-6211	84	37	growth	growth	NOUN
ejpam-6211	84	38	behavior	behavior	NOUN
ejpam-6211	84	39	and	and	CCONJ
ejpam-6211	84	40	deriving	derive	VERB
ejpam-6211	84	41	identities	identity	NOUN
ejpam-6211	84	42	.	.	PUNCT
ejpam-6211	85	1	properties	property	NOUN
ejpam-6211	85	2	and	and	CCONJ
ejpam-6211	85	3	identities	identity	NOUN
ejpam-6211	85	4	some	some	DET
ejpam-6211	85	5	elegant	elegant	ADJ
ejpam-6211	85	6	identities	identity	NOUN
ejpam-6211	85	7	include	include	VERB
ejpam-6211	85	8	:	:	PUNCT
ejpam-6211	85	9	(	(	PUNCT
ejpam-6211	85	10	i	i	NOUN
ejpam-6211	85	11	)	)	PUNCT
ejpam-6211	85	12	derivative	derivative	ADJ
ejpam-6211	85	13	identity	identity	NOUN
ejpam-6211	85	14	:	:	PUNCT
ejpam-6211	86	1	d	d	X
ejpam-6211	86	2	dx	dx	PROPN
ejpam-6211	86	3	f	f	PROPN
ejpam-6211	86	4	(	(	PUNCT
ejpam-6211	86	5	1	1	NUM
ejpam-6211	86	6	)	)	PUNCT
ejpam-6211	86	7	n	n	PROPN
ejpam-6211	86	8	(	(	PUNCT
ejpam-6211	86	9	x	x	X
ejpam-6211	86	10	)	)	PUNCT
ejpam-6211	86	11	=	=	SYM
ejpam-6211	86	12	n−1∑	n−1∑	NOUN
ejpam-6211	86	13	k=1	k=1	PUNCT
ejpam-6211	86	14	f	f	X
ejpam-6211	86	15	(	(	PUNCT
ejpam-6211	86	16	1	1	NUM
ejpam-6211	86	17	)	)	PUNCT
ejpam-6211	86	18	k	k	NOUN
ejpam-6211	86	19	(	(	PUNCT
ejpam-6211	86	20	x)f	x)f	X
ejpam-6211	86	21	(	(	PUNCT
ejpam-6211	86	22	1	1	X
ejpam-6211	86	23	)	)	PUNCT
ejpam-6211	86	24	n−k(x	n−k(x	NOUN
ejpam-6211	86	25	)	)	PUNCT
ejpam-6211	86	26	(	(	PUNCT
ejpam-6211	86	27	ii	ii	NOUN
ejpam-6211	86	28	)	)	PUNCT
ejpam-6211	86	29	addition	addition	NOUN
ejpam-6211	86	30	formula	formula	NOUN
ejpam-6211	86	31	:	:	PUNCT
ejpam-6211	86	32	f	f	PROPN
ejpam-6211	86	33	(	(	PUNCT
ejpam-6211	86	34	1	1	NUM
ejpam-6211	86	35	)	)	PUNCT
ejpam-6211	86	36	m+n(x	m+n(x	X
ejpam-6211	86	37	)	)	PUNCT
ejpam-6211	86	38	=	=	SYM
ejpam-6211	86	39	f	f	PROPN
ejpam-6211	86	40	(	(	PUNCT
ejpam-6211	86	41	1	1	X
ejpam-6211	86	42	)	)	PUNCT
ejpam-6211	86	43	m	m	VERB
ejpam-6211	86	44	(	(	PUNCT
ejpam-6211	86	45	x)f	x)f	X
ejpam-6211	86	46	(	(	PUNCT
ejpam-6211	86	47	1	1	X
ejpam-6211	86	48	)	)	PUNCT
ejpam-6211	86	49	n+1(x	n+1(x	NOUN
ejpam-6211	86	50	)	)	PUNCT
ejpam-6211	87	1	+	+	CCONJ
ejpam-6211	87	2	f	f	X
ejpam-6211	87	3	(	(	PUNCT
ejpam-6211	87	4	1	1	X
ejpam-6211	87	5	)	)	PUNCT
ejpam-6211	87	6	m−1(x)f	m−1(x)f	NOUN
ejpam-6211	87	7	(	(	PUNCT
ejpam-6211	87	8	1	1	NUM
ejpam-6211	87	9	)	)	PUNCT
ejpam-6211	87	10	n	n	PROPN
ejpam-6211	87	11	(	(	PUNCT
ejpam-6211	87	12	x	x	X
ejpam-6211	87	13	)	)	PUNCT
ejpam-6211	87	14	these	these	PRON
ejpam-6211	87	15	resemble	resemble	VERB
ejpam-6211	87	16	classical	classical	ADJ
ejpam-6211	87	17	fibonacci	fibonacci	NOUN
ejpam-6211	87	18	identities	identity	NOUN
ejpam-6211	87	19	and	and	CCONJ
ejpam-6211	87	20	are	be	AUX
ejpam-6211	87	21	useful	useful	ADJ
ejpam-6211	87	22	in	in	ADP
ejpam-6211	87	23	proving	prove	VERB
ejpam-6211	87	24	combinatorial	combinatorial	ADJ
ejpam-6211	87	25	or	or	CCONJ
ejpam-6211	87	26	algebraic	algebraic	ADJ
ejpam-6211	87	27	results	result	NOUN
ejpam-6211	87	28	.	.	PUNCT
ejpam-6211	88	1	in	in	ADP
ejpam-6211	88	2	this	this	DET
ejpam-6211	88	3	section	section	NOUN
ejpam-6211	88	4	,	,	PUNCT
ejpam-6211	88	5	we	we	PRON
ejpam-6211	88	6	define	define	VERB
ejpam-6211	88	7	another	another	DET
ejpam-6211	88	8	form	form	NOUN
ejpam-6211	88	9	of	of	ADP
ejpam-6211	88	10	fibonacci	fibonacci	NOUN
ejpam-6211	88	11	polynomials	polynomial	NOUN
ejpam-6211	88	12	and	and	CCONJ
ejpam-6211	88	13	call	call	VERB
ejpam-6211	88	14	this	this	PRON
ejpam-6211	88	15	as	as	ADP
ejpam-6211	88	16	the	the	DET
ejpam-6211	88	17	second	second	ADJ
ejpam-6211	88	18	form	form	NOUN
ejpam-6211	88	19	of	of	ADP
ejpam-6211	88	20	fibonacci	fibonacci	NOUN
ejpam-6211	88	21	polynomials	polynomial	NOUN
ejpam-6211	88	22	.	.	PUNCT
ejpam-6211	89	1	throughout	throughout	ADP
ejpam-6211	89	2	this	this	DET
ejpam-6211	89	3	paper	paper	NOUN
ejpam-6211	89	4	,	,	PUNCT
ejpam-6211	89	5	we	we	PRON
ejpam-6211	89	6	call	call	VERB
ejpam-6211	89	7	the	the	DET
ejpam-6211	89	8	fibonacci	fibonacci	NOUN
ejpam-6211	89	9	polynomials	polynomial	NOUN
ejpam-6211	89	10	defined	define	VERB
ejpam-6211	89	11	in	in	ADP
ejpam-6211	89	12	(	(	PUNCT
ejpam-6211	89	13	2.2	2.2	NUM
ejpam-6211	89	14	)	)	PUNCT
ejpam-6211	89	15	as	as	ADP
ejpam-6211	89	16	the	the	DET
ejpam-6211	89	17	first	first	ADJ
ejpam-6211	89	18	form	form	NOUN
ejpam-6211	89	19	of	of	ADP
ejpam-6211	89	20	fibonacci	fibonacci	NOUN
ejpam-6211	89	21	polynomials	polynomial	NOUN
ejpam-6211	89	22	.	.	PUNCT
ejpam-6211	90	1	definition	definition	NOUN
ejpam-6211	90	2	2.1	2.1	NUM
ejpam-6211	90	3	.	.	PUNCT
ejpam-6211	91	1	the	the	DET
ejpam-6211	91	2	second	second	ADJ
ejpam-6211	91	3	form	form	NOUN
ejpam-6211	91	4	of	of	ADP
ejpam-6211	91	5	fibonacci	fibonacci	NOUN
ejpam-6211	91	6	polynomials	polynomial	NOUN
ejpam-6211	91	7	,	,	PUNCT
ejpam-6211	91	8	denoted	denote	VERB
ejpam-6211	91	9	by	by	ADP
ejpam-6211	91	10	f	f	PROPN
ejpam-6211	91	11	(	(	PUNCT
ejpam-6211	91	12	2	2	NUM
ejpam-6211	91	13	)	)	PUNCT
ejpam-6211	91	14	n	n	NOUN
ejpam-6211	91	15	(	(	PUNCT
ejpam-6211	91	16	x	x	NOUN
ejpam-6211	91	17	)	)	PUNCT
ejpam-6211	91	18	,	,	PUNCT
ejpam-6211	91	19	is	be	AUX
ejpam-6211	91	20	defined	define	VERB
ejpam-6211	91	21	by	by	ADP
ejpam-6211	91	22	t	t	PROPN
ejpam-6211	91	23	1−	1−	NUM
ejpam-6211	91	24	t−	t−	PROPN
ejpam-6211	91	25	t2	t2	PROPN
ejpam-6211	91	26	ext	ext	NOUN
ejpam-6211	91	27	=	=	PUNCT
ejpam-6211	92	1	∞∑	∞∑	NUM
ejpam-6211	92	2	n=0	n=0	NUM
ejpam-6211	92	3	f	f	NOUN
ejpam-6211	92	4	(	(	PUNCT
ejpam-6211	92	5	2	2	NUM
ejpam-6211	92	6	)	)	PUNCT
ejpam-6211	92	7	n	n	NOUN
ejpam-6211	92	8	(	(	PUNCT
ejpam-6211	92	9	x	x	X
ejpam-6211	92	10	)	)	PUNCT
ejpam-6211	92	11	tn	tn	PROPN
ejpam-6211	92	12	n	n	NUM
ejpam-6211	92	13	!	!	PUNCT
ejpam-6211	92	14	.	.	PUNCT
ejpam-6211	93	1	(	(	PUNCT
ejpam-6211	93	2	2.3	2.3	NUM
ejpam-6211	93	3	)	)	PUNCT
ejpam-6211	93	4	note	note	VERB
ejpam-6211	93	5	that	that	SCONJ
ejpam-6211	93	6	,	,	PUNCT
ejpam-6211	93	7	when	when	SCONJ
ejpam-6211	93	8	x	x	X
ejpam-6211	93	9	=	=	SYM
ejpam-6211	93	10	0	0	NUM
ejpam-6211	93	11	,	,	PUNCT
ejpam-6211	93	12	the	the	DET
ejpam-6211	93	13	generating	generate	VERB
ejpam-6211	93	14	function	function	NOUN
ejpam-6211	93	15	in	in	ADP
ejpam-6211	93	16	(	(	PUNCT
ejpam-6211	93	17	2.3	2.3	NUM
ejpam-6211	93	18	)	)	PUNCT
ejpam-6211	93	19	reduces	reduce	VERB
ejpam-6211	93	20	to	to	ADP
ejpam-6211	93	21	the	the	DET
ejpam-6211	93	22	generating	generate	VERB
ejpam-6211	93	23	function	function	NOUN
ejpam-6211	93	24	of	of	ADP
ejpam-6211	93	25	fibonacci	fibonacci	NOUN
ejpam-6211	93	26	numbers	number	NOUN
ejpam-6211	93	27	in	in	ADP
ejpam-6211	93	28	(	(	PUNCT
ejpam-6211	93	29	1.1	1.1	NUM
ejpam-6211	93	30	)	)	PUNCT
ejpam-6211	93	31	.	.	PUNCT
ejpam-6211	94	1	this	this	PRON
ejpam-6211	94	2	implies	imply	VERB
ejpam-6211	94	3	that	that	SCONJ
ejpam-6211	94	4	f	f	PROPN
ejpam-6211	94	5	(	(	PUNCT
ejpam-6211	94	6	2	2	NUM
ejpam-6211	94	7	)	)	PUNCT
ejpam-6211	94	8	n	n	CCONJ
ejpam-6211	94	9	(	(	PUNCT
ejpam-6211	94	10	0	0	NUM
ejpam-6211	94	11	)	)	PUNCT
ejpam-6211	95	1	=	=	VERB
ejpam-6211	95	2	fn	fn	ADV
ejpam-6211	95	3	now	now	ADV
ejpam-6211	95	4	,	,	PUNCT
ejpam-6211	95	5	using	use	VERB
ejpam-6211	95	6	cauchy	cauchy	PROPN
ejpam-6211	95	7	’s	’s	PART
ejpam-6211	95	8	rule	rule	NOUN
ejpam-6211	95	9	for	for	ADP
ejpam-6211	95	10	the	the	DET
ejpam-6211	95	11	product	product	NOUN
ejpam-6211	95	12	of	of	ADP
ejpam-6211	95	13	two	two	NUM
ejpam-6211	95	14	power	power	NOUN
ejpam-6211	95	15	series	series	NOUN
ejpam-6211	95	16	,	,	PUNCT
ejpam-6211	95	17	we	we	PRON
ejpam-6211	95	18	have	have	VERB
ejpam-6211	95	19	∞∑	∞∑	NUM
ejpam-6211	95	20	n=0	n=0	NUM
ejpam-6211	95	21	f	f	X
ejpam-6211	95	22	(	(	PUNCT
ejpam-6211	95	23	2	2	NUM
ejpam-6211	95	24	)	)	PUNCT
ejpam-6211	95	25	n	n	NOUN
ejpam-6211	95	26	(	(	PUNCT
ejpam-6211	95	27	x	x	X
ejpam-6211	95	28	)	)	PUNCT
ejpam-6211	95	29	tn	tn	PROPN
ejpam-6211	95	30	n	n	NOUN
ejpam-6211	95	31	!	!	PUNCT
ejpam-6211	96	1	=	=	PUNCT
ejpam-6211	97	1	(	(	PUNCT
ejpam-6211	97	2	∞∑	∞∑	NUM
ejpam-6211	97	3	n=0	n=0	NUM
ejpam-6211	97	4	fn	fn	NOUN
ejpam-6211	97	5	tn	tn	NOUN
ejpam-6211	97	6	n	n	PROPN
ejpam-6211	97	7	!	!	PUNCT
ejpam-6211	97	8	)	)	PUNCT
ejpam-6211	98	1	(	(	PUNCT
ejpam-6211	98	2	∞∑	∞∑	NUM
ejpam-6211	98	3	n=0	n=0	NUM
ejpam-6211	98	4	xn	xn	PROPN
ejpam-6211	98	5	tn	tn	PROPN
ejpam-6211	98	6	n	n	PROPN
ejpam-6211	98	7	!	!	PUNCT
ejpam-6211	98	8	)	)	PUNCT
ejpam-6211	99	1	=	=	PUNCT
ejpam-6211	100	1	∞∑	∞∑	NUM
ejpam-6211	100	2	n=0	n=0	PUNCT
ejpam-6211	100	3			PROPN
ejpam-6211	100	4	n∑	n∑	ADJ
ejpam-6211	100	5	j=0	j=0	PROPN
ejpam-6211	100	6	(	(	PUNCT
ejpam-6211	100	7	n	n	CCONJ
ejpam-6211	100	8	j	j	PROPN
ejpam-6211	100	9	)	)	PUNCT
ejpam-6211	100	10	fn−jx	fn−jx	PROPN
ejpam-6211	100	11	j	j	PROPN
ejpam-6211	100	12			PROPN
ejpam-6211	100	13	tn	tn	PROPN
ejpam-6211	100	14	n	n	X
ejpam-6211	100	15	!	!	PUNCT
ejpam-6211	100	16	.	.	PUNCT
ejpam-6211	101	1	comparing	compare	VERB
ejpam-6211	101	2	the	the	DET
ejpam-6211	101	3	coefficients	coefficient	NOUN
ejpam-6211	101	4	of	of	ADP
ejpam-6211	101	5	tn	tn	NOUN
ejpam-6211	101	6	n	n	ADP
ejpam-6211	101	7	!	!	PROPN
ejpam-6211	101	8	completes	complete	VERB
ejpam-6211	101	9	the	the	DET
ejpam-6211	101	10	proof	proof	NOUN
ejpam-6211	101	11	of	of	ADP
ejpam-6211	101	12	the	the	DET
ejpam-6211	101	13	following	follow	VERB
ejpam-6211	101	14	theorem	theorem	NOUN
ejpam-6211	101	15	.	.	PROPN
ejpam-6211	101	16	6	6	NUM
ejpam-6211	101	17	of	of	ADP
ejpam-6211	101	18	23	23	NUM
ejpam-6211	101	19	theorem	theorem	VERB
ejpam-6211	101	20	2.2	2.2	NUM
ejpam-6211	101	21	.	.	PUNCT
ejpam-6211	102	1	the	the	DET
ejpam-6211	102	2	second	second	ADJ
ejpam-6211	102	3	form	form	NOUN
ejpam-6211	102	4	of	of	ADP
ejpam-6211	102	5	fibonacci	fibonacci	NOUN
ejpam-6211	102	6	polynomials	polynomial	NOUN
ejpam-6211	102	7	is	be	AUX
ejpam-6211	102	8	given	give	VERB
ejpam-6211	102	9	by	by	ADP
ejpam-6211	102	10	f	f	PROPN
ejpam-6211	102	11	(	(	PUNCT
ejpam-6211	102	12	2	2	NUM
ejpam-6211	102	13	)	)	PUNCT
ejpam-6211	102	14	n	n	NOUN
ejpam-6211	102	15	(	(	PUNCT
ejpam-6211	102	16	x	x	X
ejpam-6211	102	17	)	)	PUNCT
ejpam-6211	102	18	=	=	SYM
ejpam-6211	102	19	n∑	n∑	NOUN
ejpam-6211	102	20	j=0	j=0	PROPN
ejpam-6211	102	21	(	(	PUNCT
ejpam-6211	102	22	n	n	CCONJ
ejpam-6211	102	23	j	j	PROPN
ejpam-6211	102	24	)	)	PUNCT
ejpam-6211	102	25	fn−jx	fn−jx	PROPN
ejpam-6211	102	26	j	j	PROPN
ejpam-6211	102	27	.	.	PUNCT
ejpam-6211	103	1	(	(	PUNCT
ejpam-6211	103	2	2.4	2.4	NUM
ejpam-6211	103	3	)	)	PUNCT
ejpam-6211	103	4	the	the	DET
ejpam-6211	103	5	following	follow	VERB
ejpam-6211	103	6	theorem	theorem	NOUN
ejpam-6211	103	7	gives	give	VERB
ejpam-6211	103	8	the	the	DET
ejpam-6211	103	9	addition	addition	NOUN
ejpam-6211	103	10	formula	formula	NOUN
ejpam-6211	103	11	for	for	ADP
ejpam-6211	103	12	the	the	DET
ejpam-6211	103	13	second	second	ADJ
ejpam-6211	103	14	form	form	NOUN
ejpam-6211	103	15	of	of	ADP
ejpam-6211	103	16	fibonacci	fibonacci	NOUN
ejpam-6211	103	17	polynomials	polynomial	NOUN
ejpam-6211	103	18	.	.	PUNCT
ejpam-6211	104	1	theorem	theorem	VERB
ejpam-6211	104	2	2.3	2.3	NUM
ejpam-6211	104	3	.	.	PUNCT
ejpam-6211	105	1	the	the	DET
ejpam-6211	105	2	second	second	ADJ
ejpam-6211	105	3	form	form	NOUN
ejpam-6211	105	4	of	of	ADP
ejpam-6211	105	5	fibonacci	fibonacci	NOUN
ejpam-6211	105	6	polynomials	polynomial	NOUN
ejpam-6211	105	7	is	be	AUX
ejpam-6211	105	8	given	give	VERB
ejpam-6211	105	9	by	by	ADP
ejpam-6211	105	10	f	f	PROPN
ejpam-6211	105	11	(	(	PUNCT
ejpam-6211	105	12	2	2	NUM
ejpam-6211	105	13	)	)	PUNCT
ejpam-6211	105	14	n	n	CCONJ
ejpam-6211	105	15	(	(	PUNCT
ejpam-6211	105	16	x+	x+	PROPN
ejpam-6211	105	17	y	y	NOUN
ejpam-6211	105	18	)	)	PUNCT
ejpam-6211	106	1	=	=	SYM
ejpam-6211	106	2	n∑	n∑	X
ejpam-6211	106	3	j=0	j=0	PROPN
ejpam-6211	106	4	(	(	PUNCT
ejpam-6211	106	5	n	n	X
ejpam-6211	106	6	j	j	PROPN
ejpam-6211	106	7	)	)	PUNCT
ejpam-6211	106	8	f	f	PROPN
ejpam-6211	106	9	(	(	PUNCT
ejpam-6211	106	10	2	2	X
ejpam-6211	106	11	)	)	PUNCT
ejpam-6211	106	12	n−j(x)y	n−j(x)y	PROPN
ejpam-6211	106	13	j	j	PROPN
ejpam-6211	106	14	.	.	PUNCT
ejpam-6211	107	1	(	(	PUNCT
ejpam-6211	107	2	2.5	2.5	NUM
ejpam-6211	107	3	)	)	PUNCT
ejpam-6211	107	4	proof	proof	NOUN
ejpam-6211	107	5	.	.	PUNCT
ejpam-6211	108	1	∞∑	∞∑	PRON
ejpam-6211	108	2	n=0	n=0	NUM
ejpam-6211	108	3	f	f	X
ejpam-6211	108	4	(	(	PUNCT
ejpam-6211	108	5	2	2	NUM
ejpam-6211	108	6	)	)	PUNCT
ejpam-6211	108	7	n	n	CCONJ
ejpam-6211	108	8	(	(	PUNCT
ejpam-6211	108	9	x+	x+	PROPN
ejpam-6211	108	10	y	y	PROPN
ejpam-6211	108	11	)	)	PUNCT
ejpam-6211	108	12	tn	tn	PROPN
ejpam-6211	108	13	n	n	PROPN
ejpam-6211	108	14	!	!	PUNCT
ejpam-6211	108	15	=	=	PUNCT
ejpam-6211	109	1	(	(	PUNCT
ejpam-6211	109	2	∞∑	∞∑	NUM
ejpam-6211	109	3	n=0	n=0	NUM
ejpam-6211	109	4	f	f	NOUN
ejpam-6211	109	5	(	(	PUNCT
ejpam-6211	109	6	2	2	NUM
ejpam-6211	109	7	)	)	PUNCT
ejpam-6211	109	8	n	n	NOUN
ejpam-6211	109	9	(	(	PUNCT
ejpam-6211	109	10	x	x	X
ejpam-6211	109	11	)	)	PUNCT
ejpam-6211	109	12	tn	tn	PROPN
ejpam-6211	109	13	n	n	PROPN
ejpam-6211	109	14	!	!	PUNCT
ejpam-6211	109	15	)	)	PUNCT
ejpam-6211	110	1	(	(	PUNCT
ejpam-6211	110	2	∞∑	∞∑	PROPN
ejpam-6211	110	3	n=0	n=0	NUM
ejpam-6211	110	4	yn	yn	PROPN
ejpam-6211	110	5	tn	tn	PROPN
ejpam-6211	110	6	n	n	PROPN
ejpam-6211	110	7	!	!	PUNCT
ejpam-6211	110	8	)	)	PUNCT
ejpam-6211	111	1	=	=	PUNCT
ejpam-6211	112	1	∞∑	∞∑	NUM
ejpam-6211	112	2	n=0	n=0	PUNCT
ejpam-6211	112	3			PROPN
ejpam-6211	112	4	n∑	n∑	ADJ
ejpam-6211	112	5	j=0	j=0	PROPN
ejpam-6211	112	6	(	(	PUNCT
ejpam-6211	112	7	n	n	X
ejpam-6211	112	8	j	j	PROPN
ejpam-6211	112	9	)	)	PUNCT
ejpam-6211	112	10	f	f	PROPN
ejpam-6211	112	11	(	(	PUNCT
ejpam-6211	112	12	2	2	X
ejpam-6211	112	13	)	)	PUNCT
ejpam-6211	112	14	n−j(x)y	n−j(x)y	PROPN
ejpam-6211	112	15	j	j	PROPN
ejpam-6211	113	1			PROPN
ejpam-6211	113	2	tn	tn	PROPN
ejpam-6211	113	3	n	n	X
ejpam-6211	113	4	!	!	PUNCT
ejpam-6211	113	5	.	.	PUNCT
ejpam-6211	114	1	comparing	compare	VERB
ejpam-6211	114	2	the	the	DET
ejpam-6211	114	3	coefficients	coefficient	NOUN
ejpam-6211	114	4	of	of	ADP
ejpam-6211	114	5	tn	tn	NOUN
ejpam-6211	114	6	n	n	ADP
ejpam-6211	114	7	!	!	PROPN
ejpam-6211	114	8	completes	complete	VERB
ejpam-6211	114	9	the	the	DET
ejpam-6211	114	10	proof	proof	NOUN
ejpam-6211	114	11	of	of	ADP
ejpam-6211	114	12	the	the	DET
ejpam-6211	114	13	following	follow	VERB
ejpam-6211	114	14	theorem	theorem	VERB
ejpam-6211	114	15	.	.	PUNCT
ejpam-6211	114	16	applying	apply	VERB
ejpam-6211	114	17	derivative	derivative	NOUN
ejpam-6211	114	18	to	to	ADP
ejpam-6211	114	19	both	both	DET
ejpam-6211	114	20	sides	side	NOUN
ejpam-6211	114	21	of	of	ADP
ejpam-6211	114	22	(	(	PUNCT
ejpam-6211	114	23	2.3	2.3	NUM
ejpam-6211	114	24	)	)	PUNCT
ejpam-6211	114	25	with	with	ADP
ejpam-6211	114	26	respect	respect	NOUN
ejpam-6211	114	27	to	to	ADP
ejpam-6211	114	28	x	x	SYM
ejpam-6211	114	29	yields	yield	NOUN
ejpam-6211	114	30	∞∑	∞∑	NOUN
ejpam-6211	114	31	n=0	n=0	NUM
ejpam-6211	114	32	d	d	NOUN
ejpam-6211	114	33	dx	dx	PROPN
ejpam-6211	114	34	f	f	PROPN
ejpam-6211	114	35	(	(	PUNCT
ejpam-6211	114	36	2	2	NUM
ejpam-6211	114	37	)	)	PUNCT
ejpam-6211	114	38	n	n	NOUN
ejpam-6211	114	39	(	(	PUNCT
ejpam-6211	114	40	x	x	X
ejpam-6211	114	41	)	)	PUNCT
ejpam-6211	114	42	tn	tn	PROPN
ejpam-6211	114	43	n	n	NOUN
ejpam-6211	114	44	!	!	PUNCT
ejpam-6211	115	1	=	=	PUNCT
ejpam-6211	115	2	t	t	PROPN
ejpam-6211	115	3	∞∑	∞∑	PROPN
ejpam-6211	115	4	n=0	n=0	PROPN
ejpam-6211	115	5	f	f	X
ejpam-6211	115	6	(	(	PUNCT
ejpam-6211	115	7	2	2	NUM
ejpam-6211	115	8	)	)	PUNCT
ejpam-6211	115	9	n	n	NOUN
ejpam-6211	115	10	(	(	PUNCT
ejpam-6211	115	11	x	x	X
ejpam-6211	115	12	)	)	PUNCT
ejpam-6211	115	13	tn	tn	PROPN
ejpam-6211	115	14	n	n	NOUN
ejpam-6211	115	15	!	!	PUNCT
ejpam-6211	116	1	=	=	NOUN
ejpam-6211	117	1	∞∑	∞∑	PRON
ejpam-6211	117	2	n=0	n=0	NUM
ejpam-6211	117	3	f	f	X
ejpam-6211	117	4	(	(	PUNCT
ejpam-6211	117	5	2	2	NUM
ejpam-6211	117	6	)	)	PUNCT
ejpam-6211	117	7	n	n	NOUN
ejpam-6211	117	8	(	(	PUNCT
ejpam-6211	117	9	x	x	X
ejpam-6211	117	10	)	)	PUNCT
ejpam-6211	117	11	tn+1	tn+1	NOUN
ejpam-6211	117	12	n	n	X
ejpam-6211	117	13	!	!	PUNCT
ejpam-6211	118	1	=	=	NOUN
ejpam-6211	119	1	∞∑	∞∑	NUM
ejpam-6211	119	2	n=1	n=1	NUM
ejpam-6211	119	3	nf	nf	X
ejpam-6211	119	4	(	(	PUNCT
ejpam-6211	119	5	2	2	NUM
ejpam-6211	119	6	)	)	PUNCT
ejpam-6211	119	7	n−1(x	n−1(x	PROPN
ejpam-6211	119	8	)	)	PUNCT
ejpam-6211	119	9	tn	tn	PROPN
ejpam-6211	119	10	n	n	PROPN
ejpam-6211	119	11	!	!	PUNCT
ejpam-6211	119	12	comparing	compare	VERB
ejpam-6211	119	13	the	the	DET
ejpam-6211	119	14	coefficients	coefficient	NOUN
ejpam-6211	119	15	of	of	ADP
ejpam-6211	119	16	tn	tn	NOUN
ejpam-6211	119	17	n	n	ADP
ejpam-6211	119	18	!	!	PROPN
ejpam-6211	119	19	completes	complete	VERB
ejpam-6211	119	20	the	the	DET
ejpam-6211	119	21	proof	proof	NOUN
ejpam-6211	119	22	of	of	ADP
ejpam-6211	119	23	the	the	DET
ejpam-6211	119	24	following	follow	VERB
ejpam-6211	119	25	theorem	theorem	PROPN
ejpam-6211	119	26	.	.	PUNCT
ejpam-6211	119	27	theorem	theorem	PROPN
ejpam-6211	119	28	2.4	2.4	NUM
ejpam-6211	119	29	.	.	PUNCT
ejpam-6211	120	1	the	the	DET
ejpam-6211	120	2	second	second	ADJ
ejpam-6211	120	3	form	form	NOUN
ejpam-6211	120	4	of	of	ADP
ejpam-6211	120	5	fibonacci	fibonacci	NOUN
ejpam-6211	120	6	polynomials	polynomial	NOUN
ejpam-6211	120	7	satisfies	satisfy	VERB
ejpam-6211	120	8	the	the	DET
ejpam-6211	120	9	following	follow	VERB
ejpam-6211	120	10	differential	differential	ADJ
ejpam-6211	120	11	equation	equation	NOUN
ejpam-6211	120	12	d	d	PROPN
ejpam-6211	120	13	dx	dx	PROPN
ejpam-6211	120	14	f	f	PROPN
ejpam-6211	120	15	(	(	PUNCT
ejpam-6211	120	16	2	2	NUM
ejpam-6211	120	17	)	)	PUNCT
ejpam-6211	120	18	n	n	NOUN
ejpam-6211	120	19	(	(	PUNCT
ejpam-6211	120	20	x	x	X
ejpam-6211	120	21	)	)	PUNCT
ejpam-6211	120	22	=	=	SYM
ejpam-6211	120	23	nf	nf	INTJ
ejpam-6211	120	24	(	(	PUNCT
ejpam-6211	120	25	2	2	NUM
ejpam-6211	120	26	)	)	PUNCT
ejpam-6211	120	27	n−1(x	n−1(x	PROPN
ejpam-6211	120	28	)	)	PUNCT
ejpam-6211	120	29	.	.	PUNCT
ejpam-6211	121	1	(	(	PUNCT
ejpam-6211	121	2	2.6	2.6	NUM
ejpam-6211	121	3	)	)	PUNCT
ejpam-6211	121	4	this	this	PRON
ejpam-6211	121	5	means	mean	VERB
ejpam-6211	121	6	that	that	SCONJ
ejpam-6211	121	7	the	the	DET
ejpam-6211	121	8	second	second	ADJ
ejpam-6211	121	9	form	form	NOUN
ejpam-6211	121	10	of	of	ADP
ejpam-6211	121	11	fibonacci	fibonacci	NOUN
ejpam-6211	121	12	polynomials	polynomial	NOUN
ejpam-6211	121	13	can	can	AUX
ejpam-6211	121	14	be	be	AUX
ejpam-6211	121	15	classified	classify	VERB
ejpam-6211	121	16	as	as	ADP
ejpam-6211	121	17	appell	appell	NOUN
ejpam-6211	121	18	polynomials	polynomial	NOUN
ejpam-6211	121	19	.	.	PUNCT
ejpam-6211	122	1	7	7	NUM
ejpam-6211	122	2	of	of	ADP
ejpam-6211	122	3	23	23	NUM
ejpam-6211	122	4	definition	definition	NOUN
ejpam-6211	122	5	2.5	2.5	NUM
ejpam-6211	122	6	.	.	PUNCT
ejpam-6211	123	1	the	the	DET
ejpam-6211	123	2	third	third	ADJ
ejpam-6211	123	3	form	form	NOUN
ejpam-6211	123	4	of	of	ADP
ejpam-6211	123	5	fibonacci	fibonacci	NOUN
ejpam-6211	123	6	polynomials	polynomial	NOUN
ejpam-6211	123	7	,	,	PUNCT
ejpam-6211	123	8	denoted	denote	VERB
ejpam-6211	123	9	by	by	ADP
ejpam-6211	123	10	f	f	PROPN
ejpam-6211	123	11	(	(	PUNCT
ejpam-6211	123	12	3	3	NUM
ejpam-6211	123	13	)	)	PUNCT
ejpam-6211	123	14	n	n	PROPN
ejpam-6211	123	15	(	(	PUNCT
ejpam-6211	123	16	x	x	NOUN
ejpam-6211	123	17	)	)	PUNCT
ejpam-6211	123	18	,	,	PUNCT
ejpam-6211	123	19	is	be	AUX
ejpam-6211	123	20	defined	define	VERB
ejpam-6211	123	21	by	by	ADP
ejpam-6211	123	22	t	t	PROPN
ejpam-6211	123	23	1−	1−	NUM
ejpam-6211	123	24	t−	t−	PROPN
ejpam-6211	123	25	t2	t2	NOUN
ejpam-6211	123	26	(	(	PUNCT
ejpam-6211	123	27	1	1	NUM
ejpam-6211	123	28	+	+	CCONJ
ejpam-6211	123	29	t)x	t)x	PUNCT
ejpam-6211	123	30	=	=	PUNCT
ejpam-6211	123	31	∞∑	∞∑	NUM
ejpam-6211	123	32	n=0	n=0	NUM
ejpam-6211	123	33	f	f	NOUN
ejpam-6211	123	34	(	(	PUNCT
ejpam-6211	123	35	3	3	NUM
ejpam-6211	123	36	)	)	PUNCT
ejpam-6211	123	37	n	n	NOUN
ejpam-6211	123	38	(	(	PUNCT
ejpam-6211	123	39	x	x	X
ejpam-6211	123	40	)	)	PUNCT
ejpam-6211	123	41	tn	tn	PROPN
ejpam-6211	123	42	n	n	NUM
ejpam-6211	123	43	!	!	PUNCT
ejpam-6211	123	44	.	.	PUNCT
ejpam-6211	124	1	(	(	PUNCT
ejpam-6211	124	2	2.7	2.7	NUM
ejpam-6211	124	3	)	)	PUNCT
ejpam-6211	124	4	note	note	VERB
ejpam-6211	124	5	that	that	SCONJ
ejpam-6211	124	6	,	,	PUNCT
ejpam-6211	124	7	when	when	SCONJ
ejpam-6211	124	8	x	x	X
ejpam-6211	124	9	=	=	SYM
ejpam-6211	124	10	0	0	NUM
ejpam-6211	124	11	,	,	PUNCT
ejpam-6211	124	12	the	the	DET
ejpam-6211	124	13	generating	generate	VERB
ejpam-6211	124	14	function	function	NOUN
ejpam-6211	124	15	in	in	ADP
ejpam-6211	124	16	(	(	PUNCT
ejpam-6211	124	17	2.3	2.3	NUM
ejpam-6211	124	18	)	)	PUNCT
ejpam-6211	124	19	reduces	reduce	VERB
ejpam-6211	124	20	to	to	ADP
ejpam-6211	124	21	the	the	DET
ejpam-6211	124	22	generating	generate	VERB
ejpam-6211	124	23	function	function	NOUN
ejpam-6211	124	24	of	of	ADP
ejpam-6211	124	25	fibonacci	fibonacci	NOUN
ejpam-6211	124	26	numbers	number	NOUN
ejpam-6211	124	27	in	in	ADP
ejpam-6211	124	28	(	(	PUNCT
ejpam-6211	124	29	1.1	1.1	NUM
ejpam-6211	124	30	)	)	PUNCT
ejpam-6211	124	31	.	.	PUNCT
ejpam-6211	125	1	this	this	PRON
ejpam-6211	125	2	implies	imply	VERB
ejpam-6211	125	3	that	that	SCONJ
ejpam-6211	125	4	f	f	PROPN
ejpam-6211	125	5	(	(	PUNCT
ejpam-6211	125	6	3	3	NUM
ejpam-6211	125	7	)	)	PUNCT
ejpam-6211	125	8	n	n	CCONJ
ejpam-6211	125	9	(	(	PUNCT
ejpam-6211	125	10	0	0	NUM
ejpam-6211	125	11	)	)	PUNCT
ejpam-6211	126	1	=	=	VERB
ejpam-6211	126	2	fn	fn	ADV
ejpam-6211	126	3	now	now	ADV
ejpam-6211	126	4	,	,	PUNCT
ejpam-6211	126	5	using	use	VERB
ejpam-6211	126	6	cauchy	cauchy	PROPN
ejpam-6211	126	7	’s	’s	PART
ejpam-6211	126	8	rule	rule	NOUN
ejpam-6211	126	9	for	for	ADP
ejpam-6211	126	10	the	the	DET
ejpam-6211	126	11	product	product	NOUN
ejpam-6211	126	12	of	of	ADP
ejpam-6211	126	13	two	two	NUM
ejpam-6211	126	14	power	power	NOUN
ejpam-6211	126	15	series	series	NOUN
ejpam-6211	126	16	,	,	PUNCT
ejpam-6211	126	17	we	we	PRON
ejpam-6211	126	18	have	have	VERB
ejpam-6211	126	19	∞∑	∞∑	NUM
ejpam-6211	126	20	n=0	n=0	NUM
ejpam-6211	126	21	f	f	NOUN
ejpam-6211	126	22	(	(	PUNCT
ejpam-6211	126	23	3	3	NUM
ejpam-6211	126	24	)	)	PUNCT
ejpam-6211	126	25	n	n	NOUN
ejpam-6211	126	26	(	(	PUNCT
ejpam-6211	126	27	x	x	X
ejpam-6211	126	28	)	)	PUNCT
ejpam-6211	126	29	tn	tn	PROPN
ejpam-6211	126	30	n	n	NOUN
ejpam-6211	126	31	!	!	PUNCT
ejpam-6211	127	1	=	=	PUNCT
ejpam-6211	128	1	(	(	PUNCT
ejpam-6211	128	2	∞∑	∞∑	NUM
ejpam-6211	128	3	n=0	n=0	NUM
ejpam-6211	128	4	fn	fn	NOUN
ejpam-6211	128	5	tn	tn	NOUN
ejpam-6211	128	6	n	n	PROPN
ejpam-6211	128	7	!	!	PUNCT
ejpam-6211	128	8	)	)	PUNCT
ejpam-6211	129	1	(	(	PUNCT
ejpam-6211	129	2	∞∑	∞∑	NUM
ejpam-6211	129	3	n=0	n=0	NUM
ejpam-6211	129	4	(	(	PUNCT
ejpam-6211	129	5	x)n	x)n	PUNCT
ejpam-6211	129	6	n	n	CCONJ
ejpam-6211	129	7	!	!	PUNCT
ejpam-6211	129	8	tn	tn	X
ejpam-6211	129	9	)	)	PUNCT
ejpam-6211	130	1	=	=	PUNCT
ejpam-6211	131	1	∞∑	∞∑	NUM
ejpam-6211	131	2	n=0	n=0	PUNCT
ejpam-6211	131	3			PROPN
ejpam-6211	131	4	n∑	n∑	ADJ
ejpam-6211	131	5	j=0	j=0	PROPN
ejpam-6211	131	6	(	(	PUNCT
ejpam-6211	131	7	n	n	X
ejpam-6211	131	8	j	j	NOUN
ejpam-6211	131	9	)	)	PUNCT
ejpam-6211	131	10	(	(	PUNCT
ejpam-6211	131	11	x)jfn−j	x)jfn−j	CCONJ
ejpam-6211	131	12			PROPN
ejpam-6211	131	13	tn	tn	PROPN
ejpam-6211	131	14	n	n	X
ejpam-6211	131	15	!	!	PUNCT
ejpam-6211	131	16	.	.	PUNCT
ejpam-6211	132	1	comparing	compare	VERB
ejpam-6211	132	2	the	the	DET
ejpam-6211	132	3	coefficients	coefficient	NOUN
ejpam-6211	132	4	of	of	ADP
ejpam-6211	132	5	tn	tn	NOUN
ejpam-6211	132	6	n	n	ADP
ejpam-6211	132	7	!	!	PROPN
ejpam-6211	132	8	completes	complete	VERB
ejpam-6211	132	9	the	the	DET
ejpam-6211	132	10	proof	proof	NOUN
ejpam-6211	132	11	of	of	ADP
ejpam-6211	132	12	the	the	DET
ejpam-6211	132	13	following	follow	VERB
ejpam-6211	132	14	theorem	theorem	PROPN
ejpam-6211	132	15	.	.	PUNCT
ejpam-6211	132	16	theorem	theorem	VERB
ejpam-6211	132	17	2.6	2.6	NUM
ejpam-6211	132	18	.	.	PUNCT
ejpam-6211	133	1	the	the	DET
ejpam-6211	133	2	third	third	ADJ
ejpam-6211	133	3	form	form	NOUN
ejpam-6211	133	4	of	of	ADP
ejpam-6211	133	5	fibonacci	fibonacci	NOUN
ejpam-6211	133	6	polynomials	polynomial	NOUN
ejpam-6211	133	7	is	be	AUX
ejpam-6211	133	8	given	give	VERB
ejpam-6211	133	9	by	by	ADP
ejpam-6211	133	10	f	f	PROPN
ejpam-6211	133	11	(	(	PUNCT
ejpam-6211	133	12	3	3	NUM
ejpam-6211	133	13	)	)	PUNCT
ejpam-6211	133	14	n	n	PROPN
ejpam-6211	133	15	(	(	PUNCT
ejpam-6211	133	16	x	x	X
ejpam-6211	133	17	)	)	PUNCT
ejpam-6211	133	18	=	=	SYM
ejpam-6211	133	19	n∑	n∑	NOUN
ejpam-6211	133	20	j=0	j=0	PROPN
ejpam-6211	133	21	(	(	PUNCT
ejpam-6211	133	22	n	n	X
ejpam-6211	133	23	j	j	NOUN
ejpam-6211	133	24	)	)	PUNCT
ejpam-6211	133	25	(	(	PUNCT
ejpam-6211	133	26	x)jfn−j	x)jfn−j	PROPN
ejpam-6211	133	27	.	.	PUNCT
ejpam-6211	134	1	(	(	PUNCT
ejpam-6211	134	2	2.8	2.8	NUM
ejpam-6211	134	3	)	)	PUNCT
ejpam-6211	134	4	the	the	DET
ejpam-6211	134	5	following	follow	VERB
ejpam-6211	134	6	theorem	theorem	NOUN
ejpam-6211	134	7	gives	give	VERB
ejpam-6211	134	8	the	the	DET
ejpam-6211	134	9	addition	addition	NOUN
ejpam-6211	134	10	formula	formula	NOUN
ejpam-6211	134	11	for	for	ADP
ejpam-6211	134	12	the	the	DET
ejpam-6211	134	13	third	third	ADJ
ejpam-6211	134	14	form	form	NOUN
ejpam-6211	134	15	of	of	ADP
ejpam-6211	134	16	fibonacci	fibonacci	NOUN
ejpam-6211	134	17	polynomials	polynomial	NOUN
ejpam-6211	134	18	.	.	PUNCT
ejpam-6211	135	1	theorem	theorem	VERB
ejpam-6211	135	2	2.7	2.7	NUM
ejpam-6211	135	3	.	.	PUNCT
ejpam-6211	136	1	the	the	DET
ejpam-6211	136	2	third	third	ADJ
ejpam-6211	136	3	form	form	NOUN
ejpam-6211	136	4	of	of	ADP
ejpam-6211	136	5	fibonacci	fibonacci	NOUN
ejpam-6211	136	6	polynomials	polynomial	NOUN
ejpam-6211	136	7	is	be	AUX
ejpam-6211	136	8	given	give	VERB
ejpam-6211	136	9	by	by	ADP
ejpam-6211	136	10	f	f	PROPN
ejpam-6211	136	11	(	(	PUNCT
ejpam-6211	136	12	2	2	NUM
ejpam-6211	136	13	)	)	PUNCT
ejpam-6211	136	14	n	n	CCONJ
ejpam-6211	136	15	(	(	PUNCT
ejpam-6211	136	16	x+	x+	PROPN
ejpam-6211	136	17	y	y	NOUN
ejpam-6211	136	18	)	)	PUNCT
ejpam-6211	136	19	=	=	SYM
ejpam-6211	136	20	n∑	n∑	X
ejpam-6211	136	21	j=0	j=0	PROPN
ejpam-6211	136	22	(	(	PUNCT
ejpam-6211	136	23	n	n	X
ejpam-6211	136	24	j	j	NOUN
ejpam-6211	136	25	)	)	PUNCT
ejpam-6211	137	1	(	(	PUNCT
ejpam-6211	137	2	y)jf	y)jf	NOUN
ejpam-6211	137	3	(	(	PUNCT
ejpam-6211	137	4	2	2	NUM
ejpam-6211	137	5	)	)	PUNCT
ejpam-6211	137	6	n−j(x	n−j(x	NOUN
ejpam-6211	137	7	)	)	PUNCT
ejpam-6211	137	8	.	.	PUNCT
ejpam-6211	138	1	(	(	PUNCT
ejpam-6211	138	2	2.9	2.9	NUM
ejpam-6211	138	3	)	)	PUNCT
ejpam-6211	138	4	proof	proof	NOUN
ejpam-6211	138	5	.	.	PUNCT
ejpam-6211	139	1	∞∑	∞∑	PRON
ejpam-6211	139	2	n=0	n=0	NUM
ejpam-6211	139	3	f	f	NOUN
ejpam-6211	139	4	(	(	PUNCT
ejpam-6211	139	5	3	3	NUM
ejpam-6211	139	6	)	)	PUNCT
ejpam-6211	139	7	n	n	CCONJ
ejpam-6211	139	8	(	(	PUNCT
ejpam-6211	139	9	x+	x+	PROPN
ejpam-6211	139	10	y	y	PROPN
ejpam-6211	139	11	)	)	PUNCT
ejpam-6211	139	12	tn	tn	PROPN
ejpam-6211	139	13	n	n	PROPN
ejpam-6211	139	14	!	!	PUNCT
ejpam-6211	140	1	=	=	PUNCT
ejpam-6211	140	2	(	(	PUNCT
ejpam-6211	140	3	∞∑	∞∑	NUM
ejpam-6211	140	4	n=0	n=0	NUM
ejpam-6211	140	5	f	f	NOUN
ejpam-6211	140	6	(	(	PUNCT
ejpam-6211	140	7	3	3	NUM
ejpam-6211	140	8	)	)	PUNCT
ejpam-6211	140	9	n	n	NOUN
ejpam-6211	140	10	(	(	PUNCT
ejpam-6211	140	11	x	x	X
ejpam-6211	140	12	)	)	PUNCT
ejpam-6211	140	13	tn	tn	PROPN
ejpam-6211	140	14	n	n	PROPN
ejpam-6211	140	15	!	!	PUNCT
ejpam-6211	140	16	)	)	PUNCT
ejpam-6211	141	1	(	(	PUNCT
ejpam-6211	141	2	∞∑	∞∑	NUM
ejpam-6211	141	3	n=0	n=0	NUM
ejpam-6211	141	4	(	(	PUNCT
ejpam-6211	141	5	y)n	y)n	NUM
ejpam-6211	141	6	tn	tn	PROPN
ejpam-6211	141	7	n	n	X
ejpam-6211	141	8	!	!	PUNCT
ejpam-6211	141	9	)	)	PUNCT
ejpam-6211	142	1	=	=	PUNCT
ejpam-6211	143	1	∞∑	∞∑	NUM
ejpam-6211	143	2	n=0	n=0	PUNCT
ejpam-6211	143	3			PROPN
ejpam-6211	143	4	n∑	n∑	ADJ
ejpam-6211	143	5	j=0	j=0	PROPN
ejpam-6211	143	6	(	(	PUNCT
ejpam-6211	143	7	n	n	X
ejpam-6211	143	8	j	j	NOUN
ejpam-6211	143	9	)	)	PUNCT
ejpam-6211	144	1	(	(	PUNCT
ejpam-6211	144	2	y)jf	y)jf	NOUN
ejpam-6211	144	3	(	(	PUNCT
ejpam-6211	144	4	2	2	NUM
ejpam-6211	144	5	)	)	PUNCT
ejpam-6211	144	6	n−j(x	n−j(x	NOUN
ejpam-6211	144	7	)	)	PUNCT
ejpam-6211	144	8			PROPN
ejpam-6211	144	9	tn	tn	PROPN
ejpam-6211	144	10	n	n	NOUN
ejpam-6211	144	11	!	!	PUNCT
ejpam-6211	144	12	.	.	PUNCT
ejpam-6211	145	1	comparing	compare	VERB
ejpam-6211	145	2	the	the	DET
ejpam-6211	145	3	coefficients	coefficient	NOUN
ejpam-6211	145	4	of	of	ADP
ejpam-6211	145	5	tn	tn	NOUN
ejpam-6211	145	6	n	n	ADP
ejpam-6211	145	7	!	!	PROPN
ejpam-6211	145	8	completes	complete	VERB
ejpam-6211	145	9	the	the	DET
ejpam-6211	145	10	proof	proof	NOUN
ejpam-6211	145	11	of	of	ADP
ejpam-6211	145	12	the	the	DET
ejpam-6211	145	13	following	follow	VERB
ejpam-6211	145	14	theorem	theorem	PROPN
ejpam-6211	145	15	.	.	PUNCT
ejpam-6211	146	1	note	note	VERB
ejpam-6211	146	2	that	that	SCONJ
ejpam-6211	147	1	d	d	PROPN
ejpam-6211	147	2	dx	dx	PROPN
ejpam-6211	147	3	(	(	PUNCT
ejpam-6211	147	4	1	1	NUM
ejpam-6211	147	5	+	+	CCONJ
ejpam-6211	147	6	t)x	t)x	PUNCT
ejpam-6211	147	7	=	=	SYM
ejpam-6211	147	8	(	(	PUNCT
ejpam-6211	147	9	1	1	NUM
ejpam-6211	147	10	+	+	CCONJ
ejpam-6211	147	11	t)xln(1	t)xln(1	SYM
ejpam-6211	147	12	+	+	NUM
ejpam-6211	147	13	t	t	PROPN
ejpam-6211	147	14	)	)	PUNCT
ejpam-6211	147	15	.	.	PUNCT
ejpam-6211	148	1	8	8	NUM
ejpam-6211	148	2	of	of	ADP
ejpam-6211	148	3	23	23	NUM
ejpam-6211	148	4	hence	hence	ADV
ejpam-6211	148	5	,	,	PUNCT
ejpam-6211	148	6	applying	apply	VERB
ejpam-6211	148	7	derivative	derivative	NOUN
ejpam-6211	148	8	to	to	ADP
ejpam-6211	148	9	both	both	DET
ejpam-6211	148	10	sides	side	NOUN
ejpam-6211	148	11	of	of	ADP
ejpam-6211	148	12	(	(	PUNCT
ejpam-6211	148	13	2.7	2.7	NUM
ejpam-6211	148	14	)	)	PUNCT
ejpam-6211	148	15	with	with	ADP
ejpam-6211	148	16	respect	respect	NOUN
ejpam-6211	148	17	to	to	ADP
ejpam-6211	148	18	x	x	SYM
ejpam-6211	148	19	yields	yield	NOUN
ejpam-6211	148	20	∞∑	∞∑	NOUN
ejpam-6211	148	21	n=0	n=0	NUM
ejpam-6211	148	22	d	d	NOUN
ejpam-6211	148	23	dx	dx	PROPN
ejpam-6211	148	24	f	f	PROPN
ejpam-6211	148	25	(	(	PUNCT
ejpam-6211	148	26	3	3	NUM
ejpam-6211	148	27	)	)	PUNCT
ejpam-6211	148	28	n	n	NOUN
ejpam-6211	148	29	(	(	PUNCT
ejpam-6211	148	30	x	x	X
ejpam-6211	148	31	)	)	PUNCT
ejpam-6211	148	32	tn	tn	PROPN
ejpam-6211	148	33	n	n	NOUN
ejpam-6211	148	34	!	!	PUNCT
ejpam-6211	149	1	=	=	PUNCT
ejpam-6211	150	1	ln(1	ln(1	PROPN
ejpam-6211	150	2	+	+	NUM
ejpam-6211	150	3	t	t	NOUN
ejpam-6211	150	4	)	)	PUNCT
ejpam-6211	150	5	∞∑	∞∑	PROPN
ejpam-6211	150	6	n=0	n=0	NUM
ejpam-6211	150	7	f	f	NOUN
ejpam-6211	150	8	(	(	PUNCT
ejpam-6211	150	9	3	3	NUM
ejpam-6211	150	10	)	)	PUNCT
ejpam-6211	150	11	n	n	NOUN
ejpam-6211	150	12	(	(	PUNCT
ejpam-6211	150	13	x	x	X
ejpam-6211	150	14	)	)	PUNCT
ejpam-6211	150	15	tn	tn	PROPN
ejpam-6211	150	16	n	n	NOUN
ejpam-6211	150	17	!	!	PUNCT
ejpam-6211	151	1	=	=	PUNCT
ejpam-6211	152	1	∞∑	∞∑	NUM
ejpam-6211	152	2	n=0	n=0	NUM
ejpam-6211	152	3	(	(	PUNCT
ejpam-6211	152	4	n	n	X
ejpam-6211	152	5	!	!	PUNCT
ejpam-6211	152	6	n∑	n∑	INTJ
ejpam-6211	153	1	k=1	k=1	X
ejpam-6211	153	2	(	(	PUNCT
ejpam-6211	153	3	−1)k+1	−1)k+1	VERB
ejpam-6211	153	4	k	k	X
ejpam-6211	153	5	·	·	PUNCT
ejpam-6211	153	6	f	f	X
ejpam-6211	153	7	(	(	PUNCT
ejpam-6211	153	8	3	3	X
ejpam-6211	153	9	)	)	PUNCT
ejpam-6211	153	10	n−k(x	n−k(x	NOUN
ejpam-6211	153	11	)	)	PUNCT
ejpam-6211	153	12	(	(	PUNCT
ejpam-6211	153	13	n−	n−	NOUN
ejpam-6211	153	14	k	k	NOUN
ejpam-6211	153	15	)	)	PUNCT
ejpam-6211	153	16	!	!	PUNCT
ejpam-6211	153	17	)	)	PUNCT
ejpam-6211	154	1	tn	tn	PROPN
ejpam-6211	154	2	n	n	PROPN
ejpam-6211	154	3	!	!	PUNCT
ejpam-6211	154	4	.	.	PUNCT
ejpam-6211	155	1	comparing	compare	VERB
ejpam-6211	155	2	the	the	DET
ejpam-6211	155	3	coefficients	coefficient	NOUN
ejpam-6211	155	4	of	of	ADP
ejpam-6211	155	5	tn	tn	NOUN
ejpam-6211	155	6	n	n	ADP
ejpam-6211	155	7	!	!	PROPN
ejpam-6211	155	8	completes	complete	VERB
ejpam-6211	155	9	the	the	DET
ejpam-6211	155	10	proof	proof	NOUN
ejpam-6211	155	11	of	of	ADP
ejpam-6211	155	12	the	the	DET
ejpam-6211	155	13	following	follow	VERB
ejpam-6211	155	14	theorem	theorem	PROPN
ejpam-6211	155	15	.	.	PUNCT
ejpam-6211	155	16	theorem	theorem	VERB
ejpam-6211	155	17	2.8	2.8	NUM
ejpam-6211	155	18	.	.	PUNCT
ejpam-6211	156	1	the	the	DET
ejpam-6211	156	2	second	second	ADJ
ejpam-6211	156	3	form	form	NOUN
ejpam-6211	156	4	of	of	ADP
ejpam-6211	156	5	fibonacci	fibonacci	NOUN
ejpam-6211	156	6	polynomials	polynomial	NOUN
ejpam-6211	156	7	satisfies	satisfy	VERB
ejpam-6211	156	8	the	the	DET
ejpam-6211	156	9	following	follow	VERB
ejpam-6211	156	10	differential	differential	ADJ
ejpam-6211	156	11	equation	equation	NOUN
ejpam-6211	156	12	d	d	PROPN
ejpam-6211	156	13	dx	dx	PROPN
ejpam-6211	156	14	f	f	PROPN
ejpam-6211	156	15	(	(	PUNCT
ejpam-6211	156	16	3	3	NUM
ejpam-6211	156	17	)	)	PUNCT
ejpam-6211	156	18	n	n	PROPN
ejpam-6211	156	19	(	(	PUNCT
ejpam-6211	156	20	x	x	X
ejpam-6211	156	21	)	)	PUNCT
ejpam-6211	156	22	=	=	SYM
ejpam-6211	156	23	n	n	X
ejpam-6211	156	24	!	!	PUNCT
ejpam-6211	157	1	n∑	n∑	INTJ
ejpam-6211	158	1	k=1	k=1	X
ejpam-6211	158	2	(	(	PUNCT
ejpam-6211	158	3	−1)k+1	−1)k+1	VERB
ejpam-6211	158	4	k	k	X
ejpam-6211	158	5	·	·	PUNCT
ejpam-6211	158	6	f	f	X
ejpam-6211	158	7	(	(	PUNCT
ejpam-6211	158	8	3	3	X
ejpam-6211	158	9	)	)	PUNCT
ejpam-6211	158	10	n−k(x	n−k(x	NOUN
ejpam-6211	158	11	)	)	PUNCT
ejpam-6211	158	12	(	(	PUNCT
ejpam-6211	158	13	n−	n−	NOUN
ejpam-6211	158	14	k	k	NOUN
ejpam-6211	158	15	)	)	PUNCT
ejpam-6211	158	16	!	!	PUNCT
ejpam-6211	158	17	.	.	PUNCT
ejpam-6211	159	1	(	(	PUNCT
ejpam-6211	159	2	2.10	2.10	NUM
ejpam-6211	159	3	)	)	PUNCT
ejpam-6211	159	4	3	3	NUM
ejpam-6211	159	5	.	.	X
ejpam-6211	159	6	formulas	formula	NOUN
ejpam-6211	159	7	for	for	ADP
ejpam-6211	159	8	r	r	NOUN
ejpam-6211	159	9	-	-	PUNCT
ejpam-6211	159	10	stirling	stirling	NOUN
ejpam-6211	159	11	fibonacci	fibonacci	NOUN
ejpam-6211	159	12	number	number	NOUN
ejpam-6211	159	13	in	in	ADP
ejpam-6211	159	14	this	this	DET
ejpam-6211	159	15	section	section	NOUN
ejpam-6211	159	16	,	,	PUNCT
ejpam-6211	159	17	we	we	PRON
ejpam-6211	159	18	introduce	introduce	VERB
ejpam-6211	159	19	and	and	CCONJ
ejpam-6211	159	20	explore	explore	VERB
ejpam-6211	159	21	new	new	ADJ
ejpam-6211	159	22	classes	class	NOUN
ejpam-6211	159	23	of	of	ADP
ejpam-6211	159	24	combinatorial	combinatorial	ADJ
ejpam-6211	159	25	numbers	number	NOUN
ejpam-6211	159	26	which	which	PRON
ejpam-6211	159	27	we	we	PRON
ejpam-6211	159	28	call	call	VERB
ejpam-6211	159	29	the	the	DET
ejpam-6211	159	30	r	r	NOUN
ejpam-6211	159	31	-	-	PUNCT
ejpam-6211	159	32	stirling	stirling	NOUN
ejpam-6211	159	33	fibonacci	fibonacci	NOUN
ejpam-6211	159	34	numbers	number	NOUN
ejpam-6211	159	35	of	of	ADP
ejpam-6211	159	36	the	the	DET
ejpam-6211	159	37	first	first	ADJ
ejpam-6211	159	38	and	and	CCONJ
ejpam-6211	159	39	second	second	ADJ
ejpam-6211	159	40	kinds	kind	NOUN
ejpam-6211	159	41	.	.	PUNCT
ejpam-6211	160	1	these	these	DET
ejpam-6211	160	2	numbers	number	NOUN
ejpam-6211	160	3	are	be	AUX
ejpam-6211	160	4	defined	define	VERB
ejpam-6211	160	5	through	through	ADP
ejpam-6211	160	6	exponential	exponential	ADJ
ejpam-6211	160	7	generating	generating	NOUN
ejpam-6211	160	8	functions	function	NOUN
ejpam-6211	160	9	involving	involve	VERB
ejpam-6211	160	10	the	the	DET
ejpam-6211	160	11	classical	classical	ADJ
ejpam-6211	160	12	fibonacci	fibonacci	NOUN
ejpam-6211	160	13	numbers	number	NOUN
ejpam-6211	160	14	and	and	CCONJ
ejpam-6211	160	15	generalized	generalize	VERB
ejpam-6211	160	16	r	r	NOUN
ejpam-6211	160	17	-	-	PUNCT
ejpam-6211	160	18	stirling	stirling	NOUN
ejpam-6211	160	19	numbers	number	NOUN
ejpam-6211	160	20	.	.	PUNCT
ejpam-6211	161	1	the	the	DET
ejpam-6211	161	2	definitions	definition	NOUN
ejpam-6211	161	3	extend	extend	VERB
ejpam-6211	161	4	the	the	DET
ejpam-6211	161	5	framework	framework	NOUN
ejpam-6211	161	6	of	of	ADP
ejpam-6211	161	7	both	both	DET
ejpam-6211	161	8	fibonacci	fibonacci	NOUN
ejpam-6211	161	9	and	and	CCONJ
ejpam-6211	161	10	r	r	NOUN
ejpam-6211	161	11	-	-	PUNCT
ejpam-6211	161	12	stirling	stirling	NOUN
ejpam-6211	161	13	numbers	number	NOUN
ejpam-6211	161	14	by	by	ADP
ejpam-6211	161	15	integrating	integrate	VERB
ejpam-6211	161	16	their	their	PRON
ejpam-6211	161	17	combinatorial	combinatorial	ADJ
ejpam-6211	161	18	structures	structure	NOUN
ejpam-6211	161	19	.	.	PUNCT
ejpam-6211	162	1	we	we	PRON
ejpam-6211	162	2	derive	derive	VERB
ejpam-6211	162	3	convolution	convolution	NOUN
ejpam-6211	162	4	formulas	formula	NOUN
ejpam-6211	162	5	that	that	PRON
ejpam-6211	162	6	relate	relate	VERB
ejpam-6211	162	7	these	these	DET
ejpam-6211	162	8	numbers	number	NOUN
ejpam-6211	162	9	to	to	ADP
ejpam-6211	162	10	the	the	DET
ejpam-6211	162	11	classical	classical	ADJ
ejpam-6211	162	12	fibonacci	fibonacci	NOUN
ejpam-6211	162	13	sequence	sequence	NOUN
ejpam-6211	162	14	and	and	CCONJ
ejpam-6211	162	15	investigate	investigate	VERB
ejpam-6211	162	16	their	their	PRON
ejpam-6211	162	17	generating	generating	NOUN
ejpam-6211	162	18	functions	function	NOUN
ejpam-6211	162	19	.	.	PUNCT
ejpam-6211	163	1	in	in	ADP
ejpam-6211	163	2	particular	particular	ADJ
ejpam-6211	163	3	,	,	PUNCT
ejpam-6211	163	4	we	we	PRON
ejpam-6211	163	5	establish	establish	VERB
ejpam-6211	163	6	horizontal	horizontal	ADJ
ejpam-6211	163	7	generating	generating	NOUN
ejpam-6211	163	8	functions	function	NOUN
ejpam-6211	163	9	and	and	CCONJ
ejpam-6211	163	10	a	a	DET
ejpam-6211	163	11	schlömilch	schlömilch	ADJ
ejpam-6211	163	12	-	-	ADJ
ejpam-6211	163	13	type	type	NOUN
ejpam-6211	163	14	formula	formula	NOUN
ejpam-6211	163	15	that	that	PRON
ejpam-6211	163	16	highlights	highlight	VERB
ejpam-6211	163	17	their	their	PRON
ejpam-6211	163	18	intricate	intricate	ADJ
ejpam-6211	163	19	combinatorial	combinatorial	ADJ
ejpam-6211	163	20	nature	nature	NOUN
ejpam-6211	163	21	.	.	PUNCT
ejpam-6211	164	1	definition	definition	NOUN
ejpam-6211	164	2	3.1	3.1	NUM
ejpam-6211	164	3	.	.	PUNCT
ejpam-6211	165	1	the	the	DET
ejpam-6211	165	2	r	r	NOUN
ejpam-6211	165	3	-	-	PUNCT
ejpam-6211	165	4	stirling	stirling	NOUN
ejpam-6211	165	5	fibonacci	fibonacci	NOUN
ejpam-6211	165	6	number	number	NOUN
ejpam-6211	165	7	of	of	ADP
ejpam-6211	165	8	the	the	DET
ejpam-6211	165	9	first	first	ADJ
ejpam-6211	165	10	kind	kind	NOUN
ejpam-6211	165	11	is	be	AUX
ejpam-6211	165	12	defined	define	VERB
ejpam-6211	165	13	by	by	ADP
ejpam-6211	165	14	means	mean	NOUN
ejpam-6211	165	15	of	of	ADP
ejpam-6211	165	16	the	the	DET
ejpam-6211	165	17	following	follow	VERB
ejpam-6211	165	18	exponential	exponential	ADJ
ejpam-6211	165	19	generating	generating	NOUN
ejpam-6211	165	20	functions	function	NOUN
ejpam-6211	165	21	:	:	PUNCT
ejpam-6211	165	22	∞∑	∞∑	NUM
ejpam-6211	165	23	n=0	n=0	NUM
ejpam-6211	165	24	sf	sf	VERB
ejpam-6211	165	25	1	1	NUM
ejpam-6211	165	26	n	n	CCONJ
ejpam-6211	165	27	(	(	PUNCT
ejpam-6211	165	28	k	k	NOUN
ejpam-6211	165	29	;	;	PUNCT
ejpam-6211	165	30	r	r	X
ejpam-6211	165	31	)	)	PUNCT
ejpam-6211	165	32	tn	tn	NOUN
ejpam-6211	165	33	n	n	NOUN
ejpam-6211	165	34	!	!	PUNCT
ejpam-6211	166	1	=	=	PUNCT
ejpam-6211	166	2	(	(	PUNCT
ejpam-6211	166	3	1	1	NUM
ejpam-6211	166	4	1	1	NUM
ejpam-6211	166	5	+	+	NUM
ejpam-6211	166	6	t	t	NOUN
ejpam-6211	166	7	)	)	PUNCT
ejpam-6211	166	8	r	r	NOUN
ejpam-6211	166	9	(	(	PUNCT
ejpam-6211	166	10	eαt	eαt	ADV
ejpam-6211	166	11	−	−	PROPN
ejpam-6211	166	12	eβt)(lnk	eβt)(lnk	ADJ
ejpam-6211	166	13	(	(	PUNCT
ejpam-6211	166	14	1	1	NUM
ejpam-6211	166	15	+	+	NUM
ejpam-6211	166	16	t	t	PROPN
ejpam-6211	166	17	)	)	PUNCT
ejpam-6211	166	18	)	)	PUNCT
ejpam-6211	167	1	k	k	X
ejpam-6211	167	2	!	!	PUNCT
ejpam-6211	168	1	(	(	PUNCT
ejpam-6211	168	2	α−	α−	ADP
ejpam-6211	168	3	β	β	NOUN
ejpam-6211	168	4	)	)	PUNCT
ejpam-6211	168	5	(	(	PUNCT
ejpam-6211	168	6	3.1	3.1	NUM
ejpam-6211	168	7	)	)	PUNCT
ejpam-6211	168	8	where	where	SCONJ
ejpam-6211	168	9	α	α	NOUN
ejpam-6211	168	10	=	=	X
ejpam-6211	168	11	(	(	PUNCT
ejpam-6211	168	12	1	1	NUM
ejpam-6211	168	13	+	+	NUM
ejpam-6211	168	14	√	√	NUM
ejpam-6211	168	15	5	5	NUM
ejpam-6211	168	16	)	)	PUNCT
ejpam-6211	168	17	2	2	NUM
ejpam-6211	168	18	,	,	PUNCT
ejpam-6211	168	19	β	β	X
ejpam-6211	168	20	=	=	SYM
ejpam-6211	168	21	(	(	PUNCT
ejpam-6211	168	22	1−	1−	NUM
ejpam-6211	168	23	√	√	NUM
ejpam-6211	168	24	5	5	NUM
ejpam-6211	168	25	)	)	PUNCT
ejpam-6211	168	26	2	2	NUM
ejpam-6211	168	27	and	and	CCONJ
ejpam-6211	168	28	α−	α−	ADP
ejpam-6211	168	29	β	β	X
ejpam-6211	168	30	=	=	NOUN
ejpam-6211	168	31	√	√	NUM
ejpam-6211	168	32	5	5	NUM
ejpam-6211	168	33	.	.	PUNCT
ejpam-6211	168	34	definition	definition	NOUN
ejpam-6211	168	35	3.2	3.2	NUM
ejpam-6211	168	36	.	.	PUNCT
ejpam-6211	169	1	the	the	DET
ejpam-6211	169	2	r	r	NOUN
ejpam-6211	169	3	-	-	PUNCT
ejpam-6211	169	4	stirling	stirling	NOUN
ejpam-6211	169	5	fibonacci	fibonacci	NOUN
ejpam-6211	169	6	number	number	NOUN
ejpam-6211	169	7	of	of	ADP
ejpam-6211	169	8	the	the	DET
ejpam-6211	169	9	second	second	ADJ
ejpam-6211	169	10	kind	kind	NOUN
ejpam-6211	169	11	is	be	AUX
ejpam-6211	169	12	defined	define	VERB
ejpam-6211	169	13	by	by	ADP
ejpam-6211	169	14	means	mean	NOUN
ejpam-6211	169	15	of	of	ADP
ejpam-6211	169	16	the	the	DET
ejpam-6211	169	17	following	follow	VERB
ejpam-6211	169	18	exponential	exponential	ADJ
ejpam-6211	169	19	generating	generating	NOUN
ejpam-6211	169	20	functions	function	NOUN
ejpam-6211	169	21	:	:	PUNCT
ejpam-6211	169	22	∞∑	∞∑	NUM
ejpam-6211	169	23	n=0	n=0	NUM
ejpam-6211	169	24	sf	sf	VERB
ejpam-6211	169	25	2	2	NUM
ejpam-6211	169	26	n	n	NOUN
ejpam-6211	169	27	(	(	PUNCT
ejpam-6211	169	28	k	k	NOUN
ejpam-6211	169	29	;	;	PUNCT
ejpam-6211	169	30	r	r	X
ejpam-6211	169	31	)	)	PUNCT
ejpam-6211	169	32	tn	tn	NOUN
ejpam-6211	169	33	n	n	NOUN
ejpam-6211	169	34	!	!	PUNCT
ejpam-6211	170	1	=	=	PUNCT
ejpam-6211	171	1	(	(	PUNCT
ejpam-6211	171	2	eαt	eαt	ADV
ejpam-6211	171	3	−	−	PRON
ejpam-6211	171	4	eβt)(ert(et	eβt)(ert(et	NOUN
ejpam-6211	171	5	−	−	PROPN
ejpam-6211	171	6	1)k	1)k	NUM
ejpam-6211	171	7	)	)	PUNCT
ejpam-6211	172	1	k	k	X
ejpam-6211	172	2	!	!	PUNCT
ejpam-6211	173	1	(	(	PUNCT
ejpam-6211	173	2	α−	α−	ADP
ejpam-6211	173	3	β	β	NOUN
ejpam-6211	173	4	)	)	PUNCT
ejpam-6211	173	5	(	(	PUNCT
ejpam-6211	173	6	3.2	3.2	NUM
ejpam-6211	173	7	)	)	PUNCT
ejpam-6211	173	8	where	where	SCONJ
ejpam-6211	173	9	α	α	NOUN
ejpam-6211	173	10	=	=	X
ejpam-6211	173	11	(	(	PUNCT
ejpam-6211	173	12	1	1	NUM
ejpam-6211	173	13	+	+	NUM
ejpam-6211	173	14	√	√	NUM
ejpam-6211	173	15	5	5	NUM
ejpam-6211	173	16	)	)	PUNCT
ejpam-6211	173	17	2	2	NUM
ejpam-6211	173	18	,	,	PUNCT
ejpam-6211	173	19	β	β	X
ejpam-6211	173	20	=	=	SYM
ejpam-6211	173	21	(	(	PUNCT
ejpam-6211	173	22	1−	1−	NUM
ejpam-6211	173	23	√	√	NUM
ejpam-6211	173	24	5	5	NUM
ejpam-6211	173	25	)	)	PUNCT
ejpam-6211	173	26	2	2	NUM
ejpam-6211	173	27	and	and	CCONJ
ejpam-6211	173	28	α−	α−	ADP
ejpam-6211	173	29	β	β	X
ejpam-6211	173	30	=	=	NOUN
ejpam-6211	173	31	√	√	NUM
ejpam-6211	173	32	5	5	NUM
ejpam-6211	173	33	.	.	NOUN
ejpam-6211	173	34	9	9	NUM
ejpam-6211	173	35	of	of	ADP
ejpam-6211	173	36	23	23	NUM
ejpam-6211	173	37	theorem	theorem	VERB
ejpam-6211	173	38	3.3	3.3	NUM
ejpam-6211	173	39	.	.	PUNCT
ejpam-6211	174	1	the	the	DET
ejpam-6211	174	2	r	r	NOUN
ejpam-6211	174	3	-	-	PUNCT
ejpam-6211	174	4	stirling	stirling	NOUN
ejpam-6211	174	5	fibonacci	fibonacci	NOUN
ejpam-6211	174	6	number	number	NOUN
ejpam-6211	174	7	of	of	ADP
ejpam-6211	174	8	the	the	DET
ejpam-6211	174	9	first	first	ADJ
ejpam-6211	174	10	and	and	CCONJ
ejpam-6211	174	11	second	second	ADJ
ejpam-6211	174	12	kind	kind	NOUN
ejpam-6211	174	13	satisfy	satisfy	VERB
ejpam-6211	174	14	the	the	DET
ejpam-6211	174	15	convolution	convolution	NOUN
ejpam-6211	174	16	formula	formula	NOUN
ejpam-6211	174	17	respectively	respectively	ADV
ejpam-6211	174	18	;	;	PUNCT
ejpam-6211	174	19	sf	sf	PROPN
ejpam-6211	174	20	1	1	NUM
ejpam-6211	174	21	n(k	n(k	PROPN
ejpam-6211	174	22	;	;	PUNCT
ejpam-6211	174	23	r	r	X
ejpam-6211	174	24	)	)	PUNCT
ejpam-6211	174	25	=	=	PUNCT
ejpam-6211	174	26	n∑	n∑	NOUN
ejpam-6211	174	27	m	m	PROPN
ejpam-6211	174	28	=	=	PROPN
ejpam-6211	174	29	k	k	X
ejpam-6211	175	1	̂[m+	̂[m+	NOUN
ejpam-6211	176	1	r	r	NOUN
ejpam-6211	176	2	k	k	NOUN
ejpam-6211	177	1	+	+	CCONJ
ejpam-6211	177	2	r	r	NOUN
ejpam-6211	177	3	]	]	X
ejpam-6211	177	4	r	r	NOUN
ejpam-6211	177	5	(	(	PUNCT
ejpam-6211	177	6	n	n	NOUN
ejpam-6211	177	7	m	m	NOUN
ejpam-6211	177	8	)	)	PUNCT
ejpam-6211	177	9	fn−m	fn−m	NOUN
ejpam-6211	177	10	(	(	PUNCT
ejpam-6211	177	11	3.3	3.3	NUM
ejpam-6211	177	12	)	)	PUNCT
ejpam-6211	177	13	sf	sf	NOUN
ejpam-6211	177	14	2	2	NUM
ejpam-6211	177	15	n(k	n(k	PROPN
ejpam-6211	177	16	;	;	PUNCT
ejpam-6211	177	17	r	r	X
ejpam-6211	177	18	)	)	PUNCT
ejpam-6211	177	19	=	=	PUNCT
ejpam-6211	178	1	n∑	n∑	NOUN
ejpam-6211	178	2	m	m	PROPN
ejpam-6211	178	3	=	=	VERB
ejpam-6211	178	4	k	k	X
ejpam-6211	178	5	{	{	PUNCT
ejpam-6211	178	6	m+	m+	NOUN
ejpam-6211	179	1	r	r	NOUN
ejpam-6211	179	2	k	k	PROPN
ejpam-6211	179	3	+	+	CCONJ
ejpam-6211	179	4	r	r	NOUN
ejpam-6211	179	5	}	}	PUNCT
ejpam-6211	179	6	r	r	NOUN
ejpam-6211	179	7	(	(	PUNCT
ejpam-6211	179	8	n	n	NOUN
ejpam-6211	179	9	m	m	NOUN
ejpam-6211	179	10	)	)	PUNCT
ejpam-6211	179	11	fn−m	fn−m	NOUN
ejpam-6211	179	12	(	(	PUNCT
ejpam-6211	179	13	3.4	3.4	NUM
ejpam-6211	179	14	)	)	PUNCT
ejpam-6211	179	15	where	where	SCONJ
ejpam-6211	179	16	n	n	NUM
ejpam-6211	179	17	≥	≥	X
ejpam-6211	179	18	k	k	NOUN
ejpam-6211	179	19	,	,	PUNCT
ejpam-6211	179	20	otherwise	otherwise	ADV
ejpam-6211	179	21	sf	sf	PROPN
ejpam-6211	179	22	1	1	NUM
ejpam-6211	179	23	n(k	n(k	PROPN
ejpam-6211	179	24	;	;	PUNCT
ejpam-6211	179	25	r	r	X
ejpam-6211	179	26	)	)	PUNCT
ejpam-6211	179	27	=	=	SYM
ejpam-6211	180	1	sf	sf	PROPN
ejpam-6211	180	2	2	2	NUM
ejpam-6211	180	3	n(k	n(k	PROPN
ejpam-6211	180	4	;	;	PUNCT
ejpam-6211	180	5	r	r	X
ejpam-6211	180	6	)	)	PUNCT
ejpam-6211	180	7	=	=	SYM
ejpam-6211	180	8	0	0	X
ejpam-6211	180	9	.	.	PUNCT
ejpam-6211	181	1	proof	proof	NOUN
ejpam-6211	181	2	the	the	DET
ejpam-6211	181	3	exponential	exponential	ADJ
ejpam-6211	181	4	generating	generating	NOUN
ejpam-6211	181	5	function	function	NOUN
ejpam-6211	181	6	(	(	PUNCT
ejpam-6211	181	7	3.1	3.1	NUM
ejpam-6211	181	8	)	)	PUNCT
ejpam-6211	181	9	can	can	AUX
ejpam-6211	181	10	be	be	AUX
ejpam-6211	181	11	written	write	VERB
ejpam-6211	181	12	as	as	ADP
ejpam-6211	181	13	:	:	PUNCT
ejpam-6211	181	14	∞∑	∞∑	NUM
ejpam-6211	181	15	n=0	n=0	NUM
ejpam-6211	181	16	sf	sf	VERB
ejpam-6211	181	17	1	1	NUM
ejpam-6211	181	18	n	n	CCONJ
ejpam-6211	181	19	(	(	PUNCT
ejpam-6211	181	20	k	k	NOUN
ejpam-6211	181	21	;	;	PUNCT
ejpam-6211	181	22	r	r	X
ejpam-6211	181	23	)	)	PUNCT
ejpam-6211	181	24	tn	tn	NOUN
ejpam-6211	181	25	n	n	NOUN
ejpam-6211	181	26	!	!	PUNCT
ejpam-6211	182	1	=	=	PUNCT
ejpam-6211	182	2	(	(	PUNCT
ejpam-6211	182	3	∞∑	∞∑	NUM
ejpam-6211	182	4	n=0	n=0	NUM
ejpam-6211	183	1	̂[n+	̂[n+	ADP
ejpam-6211	183	2	r	r	NOUN
ejpam-6211	183	3	k	k	NOUN
ejpam-6211	184	1	+	+	CCONJ
ejpam-6211	184	2	r	r	NOUN
ejpam-6211	184	3	]	]	PUNCT
ejpam-6211	184	4	r	r	NOUN
ejpam-6211	184	5	tn	tn	PROPN
ejpam-6211	184	6	n	n	CCONJ
ejpam-6211	184	7	!	!	PUNCT
ejpam-6211	184	8	)	)	PUNCT
ejpam-6211	185	1	(	(	PUNCT
ejpam-6211	185	2	∞∑	∞∑	NUM
ejpam-6211	185	3	n=0	n=0	NUM
ejpam-6211	185	4	fn	fn	NOUN
ejpam-6211	185	5	tn	tn	NOUN
ejpam-6211	185	6	n	n	PROPN
ejpam-6211	185	7	!	!	PUNCT
ejpam-6211	185	8	)	)	PUNCT
ejpam-6211	185	9	by	by	ADP
ejpam-6211	185	10	cauchy	cauchy	NOUN
ejpam-6211	185	11	product	product	NOUN
ejpam-6211	185	12	we	we	PRON
ejpam-6211	185	13	have	have	VERB
ejpam-6211	185	14	,	,	PUNCT
ejpam-6211	185	15	=	=	SYM
ejpam-6211	185	16	∞∑	∞∑	NUM
ejpam-6211	185	17	n=0	n=0	NUM
ejpam-6211	185	18	{	{	PUNCT
ejpam-6211	185	19	n∑	n∑	NOUN
ejpam-6211	185	20	m=0	m=0	PROPN
ejpam-6211	185	21	̂[m+	̂[m+	NOUN
ejpam-6211	186	1	r	r	NOUN
ejpam-6211	186	2	k	k	NOUN
ejpam-6211	187	1	+	+	CCONJ
ejpam-6211	187	2	r	r	NOUN
ejpam-6211	187	3	]	]	X
ejpam-6211	187	4	r	r	NOUN
ejpam-6211	187	5	(	(	PUNCT
ejpam-6211	187	6	n	n	NOUN
ejpam-6211	187	7	m	m	NOUN
ejpam-6211	187	8	)	)	PUNCT
ejpam-6211	187	9	fn−m	fn−m	PROPN
ejpam-6211	187	10	}	}	PUNCT
ejpam-6211	187	11	tn	tn	PROPN
ejpam-6211	187	12	n	n	CCONJ
ejpam-6211	187	13	!	!	NOUN
ejpam-6211	187	14	comparing	compare	VERB
ejpam-6211	187	15	coefficients	coefficient	NOUN
ejpam-6211	187	16	of	of	ADP
ejpam-6211	187	17	tn	tn	NOUN
ejpam-6211	187	18	n	n	CCONJ
ejpam-6211	187	19	!	!	PUNCT
ejpam-6211	188	1	we	we	PRON
ejpam-6211	188	2	have	have	VERB
ejpam-6211	188	3	,	,	PUNCT
ejpam-6211	188	4	sf	sf	PROPN
ejpam-6211	188	5	1	1	NUM
ejpam-6211	188	6	n(k	n(k	ADJ
ejpam-6211	188	7	;	;	PUNCT
ejpam-6211	188	8	r	r	X
ejpam-6211	188	9	)	)	PUNCT
ejpam-6211	188	10	=	=	SYM
ejpam-6211	188	11	n∑	n∑	PROPN
ejpam-6211	188	12	m=0	m=0	PROPN
ejpam-6211	188	13	̂[m+	̂[m+	NOUN
ejpam-6211	189	1	r	r	NOUN
ejpam-6211	189	2	k	k	NOUN
ejpam-6211	190	1	+	+	CCONJ
ejpam-6211	190	2	r	r	NOUN
ejpam-6211	190	3	]	]	X
ejpam-6211	190	4	r	r	NOUN
ejpam-6211	190	5	(	(	PUNCT
ejpam-6211	190	6	n	n	NOUN
ejpam-6211	190	7	m	m	NOUN
ejpam-6211	190	8	)	)	PUNCT
ejpam-6211	190	9	fn−m	fn−m	NOUN
ejpam-6211	190	10	=	=	PUNCT
ejpam-6211	191	1	n∑	n∑	NOUN
ejpam-6211	191	2	m	m	PROPN
ejpam-6211	191	3	=	=	PROPN
ejpam-6211	191	4	k	k	X
ejpam-6211	191	5	̂[m+	̂[m+	NOUN
ejpam-6211	192	1	r	r	NOUN
ejpam-6211	192	2	k	k	NOUN
ejpam-6211	193	1	+	+	CCONJ
ejpam-6211	193	2	r	r	NOUN
ejpam-6211	193	3	]	]	X
ejpam-6211	193	4	r	r	NOUN
ejpam-6211	193	5	(	(	PUNCT
ejpam-6211	193	6	n	n	NOUN
ejpam-6211	193	7	m	m	PROPN
ejpam-6211	193	8	)	)	PUNCT
ejpam-6211	193	9	fn−m	fn−m	PROPN
ejpam-6211	193	10	similary	similary	NOUN
ejpam-6211	193	11	,	,	PUNCT
ejpam-6211	193	12	(	(	PUNCT
ejpam-6211	193	13	3.2	3.2	NUM
ejpam-6211	193	14	)	)	PUNCT
ejpam-6211	193	15	can	can	AUX
ejpam-6211	193	16	be	be	AUX
ejpam-6211	193	17	written	write	VERB
ejpam-6211	193	18	as	as	ADP
ejpam-6211	193	19	:	:	PUNCT
ejpam-6211	193	20	∞∑	∞∑	NUM
ejpam-6211	193	21	n=0	n=0	NUM
ejpam-6211	193	22	sf	sf	VERB
ejpam-6211	193	23	2	2	NUM
ejpam-6211	193	24	n	n	NOUN
ejpam-6211	193	25	(	(	PUNCT
ejpam-6211	193	26	k	k	NOUN
ejpam-6211	193	27	;	;	PUNCT
ejpam-6211	193	28	r	r	X
ejpam-6211	193	29	)	)	PUNCT
ejpam-6211	193	30	tn	tn	NOUN
ejpam-6211	193	31	n	n	NOUN
ejpam-6211	193	32	!	!	PUNCT
ejpam-6211	194	1	=	=	PUNCT
ejpam-6211	194	2	(	(	PUNCT
ejpam-6211	194	3	∞∑	∞∑	NUM
ejpam-6211	194	4	n=0	n=0	NUM
ejpam-6211	194	5	{	{	PUNCT
ejpam-6211	194	6	n+	n+	ADP
ejpam-6211	194	7	r	r	NOUN
ejpam-6211	194	8	k	k	NOUN
ejpam-6211	195	1	+	+	CCONJ
ejpam-6211	195	2	r	r	NOUN
ejpam-6211	195	3	}	}	PUNCT
ejpam-6211	195	4	r	r	NOUN
ejpam-6211	195	5	tn	tn	NOUN
ejpam-6211	195	6	n	n	CCONJ
ejpam-6211	195	7	!	!	PUNCT
ejpam-6211	195	8	)	)	PUNCT
ejpam-6211	196	1	(	(	PUNCT
ejpam-6211	196	2	∞∑	∞∑	NUM
ejpam-6211	196	3	n=0	n=0	NUM
ejpam-6211	196	4	fn	fn	NOUN
ejpam-6211	196	5	tn	tn	NOUN
ejpam-6211	196	6	n	n	PROPN
ejpam-6211	196	7	!	!	PUNCT
ejpam-6211	196	8	)	)	PUNCT
ejpam-6211	196	9	by	by	ADP
ejpam-6211	196	10	cauchy	cauchy	NOUN
ejpam-6211	196	11	product	product	NOUN
ejpam-6211	196	12	we	we	PRON
ejpam-6211	196	13	have	have	VERB
ejpam-6211	196	14	,	,	PUNCT
ejpam-6211	196	15	=	=	SYM
ejpam-6211	196	16	∞∑	∞∑	NUM
ejpam-6211	196	17	n=0	n=0	NUM
ejpam-6211	196	18	{	{	PUNCT
ejpam-6211	196	19	n∑	n∑	PROPN
ejpam-6211	196	20	m=0	m=0	PROPN
ejpam-6211	196	21	{	{	PUNCT
ejpam-6211	197	1	n+	n+	ADP
ejpam-6211	197	2	r	r	NOUN
ejpam-6211	197	3	k	k	NOUN
ejpam-6211	198	1	+	+	CCONJ
ejpam-6211	198	2	r	r	NOUN
ejpam-6211	198	3	}	}	PUNCT
ejpam-6211	198	4	r	r	NOUN
ejpam-6211	198	5	(	(	PUNCT
ejpam-6211	198	6	n	n	NOUN
ejpam-6211	198	7	m	m	NOUN
ejpam-6211	198	8	)	)	PUNCT
ejpam-6211	198	9	fn−m	fn−m	PROPN
ejpam-6211	198	10	}	}	PUNCT
ejpam-6211	198	11	tn	tn	PROPN
ejpam-6211	198	12	n	n	CCONJ
ejpam-6211	198	13	!	!	NOUN
ejpam-6211	198	14	comparing	compare	VERB
ejpam-6211	198	15	coefficients	coefficient	NOUN
ejpam-6211	198	16	of	of	ADP
ejpam-6211	198	17	tn	tn	NOUN
ejpam-6211	198	18	n	n	CCONJ
ejpam-6211	198	19	!	!	PUNCT
ejpam-6211	199	1	we	we	PRON
ejpam-6211	199	2	have	have	VERB
ejpam-6211	199	3	,	,	PUNCT
ejpam-6211	199	4	sf	sf	PROPN
ejpam-6211	199	5	2	2	NUM
ejpam-6211	199	6	n(k	n(k	PROPN
ejpam-6211	199	7	;	;	PUNCT
ejpam-6211	199	8	r	r	X
ejpam-6211	199	9	)	)	PUNCT
ejpam-6211	199	10	=	=	SYM
ejpam-6211	200	1	n∑	n∑	PROPN
ejpam-6211	200	2	m=0	m=0	PROPN
ejpam-6211	200	3	{	{	PUNCT
ejpam-6211	201	1	n+	n+	ADP
ejpam-6211	201	2	r	r	NOUN
ejpam-6211	201	3	k	k	NOUN
ejpam-6211	202	1	+	+	CCONJ
ejpam-6211	202	2	r	r	NOUN
ejpam-6211	202	3	}	}	PUNCT
ejpam-6211	202	4	r	r	NOUN
ejpam-6211	202	5	(	(	PUNCT
ejpam-6211	202	6	n	n	NOUN
ejpam-6211	202	7	m	m	NOUN
ejpam-6211	202	8	)	)	PUNCT
ejpam-6211	202	9	fn−m	fn−m	NOUN
ejpam-6211	202	10	=	=	PUNCT
ejpam-6211	203	1	n∑	n∑	NOUN
ejpam-6211	203	2	m	m	PROPN
ejpam-6211	204	1	=	=	VERB
ejpam-6211	204	2	k	k	X
ejpam-6211	204	3	{	{	PUNCT
ejpam-6211	204	4	n+	n+	ADP
ejpam-6211	204	5	r	r	NOUN
ejpam-6211	204	6	k	k	NOUN
ejpam-6211	205	1	+	+	CCONJ
ejpam-6211	205	2	r	r	NOUN
ejpam-6211	205	3	}	}	PUNCT
ejpam-6211	205	4	r	r	NOUN
ejpam-6211	205	5	(	(	PUNCT
ejpam-6211	205	6	n	n	NOUN
ejpam-6211	205	7	m	m	PROPN
ejpam-6211	205	8	)	)	PUNCT
ejpam-6211	205	9	fn−m	fn−m	NOUN
ejpam-6211	205	10	10	10	NUM
ejpam-6211	205	11	of	of	ADP
ejpam-6211	205	12	23	23	NUM
ejpam-6211	205	13	theorem	theorem	VERB
ejpam-6211	205	14	3.4	3.4	NUM
ejpam-6211	205	15	.	.	PUNCT
ejpam-6211	206	1	the	the	DET
ejpam-6211	206	2	horizontal	horizontal	ADJ
ejpam-6211	206	3	generating	generate	VERB
ejpam-6211	206	4	function	function	NOUN
ejpam-6211	206	5	of	of	ADP
ejpam-6211	206	6	the	the	DET
ejpam-6211	206	7	r	r	NOUN
ejpam-6211	206	8	-	-	PUNCT
ejpam-6211	206	9	stirling	stirling	NOUN
ejpam-6211	206	10	fibonacci	fibonacci	NOUN
ejpam-6211	206	11	number	number	NOUN
ejpam-6211	206	12	of	of	ADP
ejpam-6211	206	13	the	the	DET
ejpam-6211	206	14	first	first	ADJ
ejpam-6211	206	15	and	and	CCONJ
ejpam-6211	206	16	second	second	ADJ
ejpam-6211	206	17	kind	kind	NOUN
ejpam-6211	206	18	is	be	AUX
ejpam-6211	206	19	given	give	VERB
ejpam-6211	206	20	by	by	ADP
ejpam-6211	206	21	respectively	respectively	ADV
ejpam-6211	206	22	;	;	PUNCT
ejpam-6211	206	23	n∑	n∑	PROPN
ejpam-6211	206	24	k=0	k=0	PROPN
ejpam-6211	206	25	sf	sf	PROPN
ejpam-6211	206	26	1	1	NUM
ejpam-6211	206	27	n	n	CCONJ
ejpam-6211	206	28	(	(	PUNCT
ejpam-6211	206	29	k	k	NOUN
ejpam-6211	206	30	;	;	PUNCT
ejpam-6211	206	31	r	r	X
ejpam-6211	206	32	)	)	PUNCT
ejpam-6211	206	33	zk	zk	PROPN
ejpam-6211	206	34	=	=	SYM
ejpam-6211	206	35	f	f	PROPN
ejpam-6211	206	36	(	(	PUNCT
ejpam-6211	206	37	3	3	NUM
ejpam-6211	206	38	)	)	PUNCT
ejpam-6211	206	39	n	n	NOUN
ejpam-6211	206	40	(	(	PUNCT
ejpam-6211	206	41	z	z	NOUN
ejpam-6211	206	42	−	−	NOUN
ejpam-6211	206	43	r	r	NOUN
ejpam-6211	206	44	)	)	PUNCT
ejpam-6211	206	45	(	(	PUNCT
ejpam-6211	206	46	3.5	3.5	NUM
ejpam-6211	206	47	)	)	PUNCT
ejpam-6211	206	48	n∑	n∑	PART
ejpam-6211	207	1	k=0	k=0	PROPN
ejpam-6211	207	2	sf	sf	NOUN
ejpam-6211	207	3	2	2	NUM
ejpam-6211	207	4	n	n	NOUN
ejpam-6211	207	5	(	(	PUNCT
ejpam-6211	207	6	k	k	NOUN
ejpam-6211	207	7	;	;	PUNCT
ejpam-6211	207	8	r	r	X
ejpam-6211	207	9	)	)	PUNCT
ejpam-6211	207	10	zk	zk	PROPN
ejpam-6211	207	11	=	=	SYM
ejpam-6211	207	12	f	f	PROPN
ejpam-6211	207	13	(	(	PUNCT
ejpam-6211	207	14	2	2	NUM
ejpam-6211	207	15	)	)	PUNCT
ejpam-6211	207	16	n	n	NOUN
ejpam-6211	207	17	(	(	PUNCT
ejpam-6211	207	18	z	z	NOUN
ejpam-6211	207	19	+	+	CCONJ
ejpam-6211	207	20	r	r	NOUN
ejpam-6211	207	21	)	)	PUNCT
ejpam-6211	207	22	(	(	PUNCT
ejpam-6211	207	23	3.6	3.6	NUM
ejpam-6211	207	24	)	)	PUNCT
ejpam-6211	207	25	proof	proof	NOUN
ejpam-6211	207	26	the	the	DET
ejpam-6211	207	27	exponential	exponential	ADJ
ejpam-6211	207	28	generating	generating	NOUN
ejpam-6211	207	29	function	function	NOUN
ejpam-6211	207	30	in	in	ADP
ejpam-6211	207	31	(	(	PUNCT
ejpam-6211	207	32	3.1	3.1	NUM
ejpam-6211	207	33	)	)	PUNCT
ejpam-6211	207	34	can	can	AUX
ejpam-6211	207	35	be	be	AUX
ejpam-6211	207	36	written	write	VERB
ejpam-6211	207	37	as	as	ADP
ejpam-6211	207	38	∞∑	∞∑	NUM
ejpam-6211	207	39	k=0	k=0	PROPN
ejpam-6211	207	40	{	{	PUNCT
ejpam-6211	207	41	∞∑	∞∑	PROPN
ejpam-6211	207	42	n	n	CCONJ
ejpam-6211	207	43	=	=	SYM
ejpam-6211	207	44	k	k	NOUN
ejpam-6211	207	45	sf	sf	NOUN
ejpam-6211	207	46	1	1	NUM
ejpam-6211	207	47	n	n	CCONJ
ejpam-6211	207	48	(	(	PUNCT
ejpam-6211	207	49	k	k	NOUN
ejpam-6211	207	50	;	;	PUNCT
ejpam-6211	207	51	r	r	X
ejpam-6211	207	52	)	)	PUNCT
ejpam-6211	207	53	tn	tn	NOUN
ejpam-6211	207	54	n	n	NOUN
ejpam-6211	207	55	!	!	PUNCT
ejpam-6211	207	56	}	}	PUNCT
ejpam-6211	207	57	zk	zk	PROPN
ejpam-6211	208	1	=	=	PUNCT
ejpam-6211	208	2			PUNCT
ejpam-6211	208	3	(	(	PUNCT
ejpam-6211	208	4	1	1	NUM
ejpam-6211	208	5	1	1	NUM
ejpam-6211	208	6	+	+	NUM
ejpam-6211	208	7	t	t	NOUN
ejpam-6211	208	8	)	)	PUNCT
ejpam-6211	208	9	r	r	NOUN
ejpam-6211	208	10	(	(	PUNCT
ejpam-6211	208	11	eαt	eαt	ADV
ejpam-6211	208	12	−	−	PROPN
ejpam-6211	208	13	eβt	eβt	NOUN
ejpam-6211	208	14	α−	α−	ADP
ejpam-6211	208	15	β	β	PROPN
ejpam-6211	208	16	)	)	PUNCT
ejpam-6211	208	17	∑	∑	PROPN
ejpam-6211	208	18	k≥0	k≥0	PROPN
ejpam-6211	208	19	lnk	lnk	NOUN
ejpam-6211	208	20	(	(	PUNCT
ejpam-6211	208	21	1	1	NUM
ejpam-6211	208	22	+	+	NUM
ejpam-6211	208	23	t	t	PROPN
ejpam-6211	208	24	)	)	PUNCT
ejpam-6211	208	25	k	k	NOUN
ejpam-6211	208	26	!	!	PUNCT
ejpam-6211	209	1			NOUN
ejpam-6211	209	2	zk	zk	X
ejpam-6211	210	1	=	=	PUNCT
ejpam-6211	210	2	∑	∑	PROPN
ejpam-6211	210	3	n≥0	n≥0	PROPN
ejpam-6211	210	4	(	(	PUNCT
ejpam-6211	210	5	z	z	NOUN
ejpam-6211	210	6	−	−	NOUN
ejpam-6211	210	7	r)n	r)n	NOUN
ejpam-6211	210	8	tn	tn	NOUN
ejpam-6211	210	9	n	n	X
ejpam-6211	210	10	!	!	PUNCT
ejpam-6211	211	1			PROPN
ejpam-6211	211	2	(	(	PUNCT
ejpam-6211	211	3	∞∑	∞∑	PROPN
ejpam-6211	211	4	n=0	n=0	NUM
ejpam-6211	211	5	fn	fn	NOUN
ejpam-6211	211	6	tn	tn	NOUN
ejpam-6211	211	7	n	n	PROPN
ejpam-6211	211	8	!	!	PUNCT
ejpam-6211	211	9	)	)	PUNCT
ejpam-6211	212	1	by	by	ADP
ejpam-6211	212	2	cauchy	cauchy	NOUN
ejpam-6211	212	3	product	product	NOUN
ejpam-6211	212	4	the	the	DET
ejpam-6211	212	5	rhs	rhs	PROPN
ejpam-6211	212	6	becomes	become	VERB
ejpam-6211	212	7	,	,	PUNCT
ejpam-6211	212	8	∞∑	∞∑	PROPN
ejpam-6211	212	9	n=0	n=0	PROPN
ejpam-6211	212	10	{	{	PUNCT
ejpam-6211	212	11	n∑	n∑	NOUN
ejpam-6211	212	12	k=0	k=0	PROPN
ejpam-6211	212	13	sf	sf	PROPN
ejpam-6211	212	14	1	1	NUM
ejpam-6211	212	15	n	n	CCONJ
ejpam-6211	212	16	(	(	PUNCT
ejpam-6211	212	17	k	k	NOUN
ejpam-6211	212	18	;	;	PUNCT
ejpam-6211	212	19	r	r	X
ejpam-6211	212	20	)	)	PUNCT
ejpam-6211	212	21	zk	zk	PROPN
ejpam-6211	212	22	}	}	PUNCT
ejpam-6211	212	23	tn	tn	PROPN
ejpam-6211	212	24	n	n	CCONJ
ejpam-6211	212	25	!	!	PUNCT
ejpam-6211	212	26	=	=	NOUN
ejpam-6211	213	1	∞∑	∞∑	PRON
ejpam-6211	213	2	n=0	n=0	NUM
ejpam-6211	213	3	{	{	PUNCT
ejpam-6211	213	4	n∑	n∑	PROPN
ejpam-6211	213	5	m=0	m=0	PROPN
ejpam-6211	213	6	(	(	PUNCT
ejpam-6211	213	7	z	z	NOUN
ejpam-6211	213	8	−	−	NOUN
ejpam-6211	213	9	r)m	r)m	NOUN
ejpam-6211	213	10	(	(	PUNCT
ejpam-6211	213	11	n	n	X
ejpam-6211	213	12	m	m	NOUN
ejpam-6211	213	13	)	)	PUNCT
ejpam-6211	213	14	fn−m	fn−m	PROPN
ejpam-6211	213	15	}	}	PUNCT
ejpam-6211	213	16	tn	tn	PROPN
ejpam-6211	213	17	n	n	CCONJ
ejpam-6211	213	18	!	!	NOUN
ejpam-6211	213	19	comparing	compare	VERB
ejpam-6211	213	20	coefficients	coefficient	NOUN
ejpam-6211	213	21	of	of	ADP
ejpam-6211	213	22	tn	tn	NOUN
ejpam-6211	213	23	n	n	CCONJ
ejpam-6211	213	24	!	!	PUNCT
ejpam-6211	214	1	we	we	PRON
ejpam-6211	214	2	have	have	VERB
ejpam-6211	214	3	,	,	PUNCT
ejpam-6211	214	4	n∑	n∑	PROPN
ejpam-6211	214	5	k=0	k=0	PROPN
ejpam-6211	214	6	sf	sf	PROPN
ejpam-6211	214	7	1	1	NUM
ejpam-6211	214	8	n	n	CCONJ
ejpam-6211	214	9	(	(	PUNCT
ejpam-6211	214	10	k	k	NOUN
ejpam-6211	214	11	;	;	PUNCT
ejpam-6211	214	12	r	r	X
ejpam-6211	214	13	)	)	PUNCT
ejpam-6211	214	14	zk	zk	PROPN
ejpam-6211	215	1	=	=	SYM
ejpam-6211	215	2	n∑	n∑	PROPN
ejpam-6211	215	3	m=0	m=0	PROPN
ejpam-6211	215	4	(	(	PUNCT
ejpam-6211	215	5	z	z	NOUN
ejpam-6211	215	6	−	−	NOUN
ejpam-6211	215	7	r)m	r)m	NOUN
ejpam-6211	215	8	(	(	PUNCT
ejpam-6211	215	9	n	n	X
ejpam-6211	215	10	m	m	NOUN
ejpam-6211	215	11	)	)	PUNCT
ejpam-6211	215	12	fn−m	fn−m	NOUN
ejpam-6211	215	13	=	=	PUNCT
ejpam-6211	215	14	f	f	PROPN
ejpam-6211	215	15	(	(	PUNCT
ejpam-6211	215	16	3	3	NUM
ejpam-6211	215	17	)	)	PUNCT
ejpam-6211	215	18	n	n	NOUN
ejpam-6211	215	19	(	(	PUNCT
ejpam-6211	215	20	z	z	NOUN
ejpam-6211	215	21	−	−	NOUN
ejpam-6211	216	1	r	r	NOUN
ejpam-6211	216	2	)	)	PUNCT
ejpam-6211	216	3	similarly	similarly	ADV
ejpam-6211	216	4	,	,	PUNCT
ejpam-6211	216	5	the	the	DET
ejpam-6211	216	6	exponential	exponential	ADJ
ejpam-6211	216	7	generating	generating	NOUN
ejpam-6211	216	8	function	function	NOUN
ejpam-6211	216	9	in	in	ADP
ejpam-6211	216	10	(	(	PUNCT
ejpam-6211	216	11	3.2	3.2	NUM
ejpam-6211	216	12	)	)	PUNCT
ejpam-6211	216	13	can	can	AUX
ejpam-6211	216	14	be	be	AUX
ejpam-6211	216	15	written	write	VERB
ejpam-6211	216	16	as	as	ADP
ejpam-6211	216	17	∞∑	∞∑	NUM
ejpam-6211	216	18	k=0	k=0	PROPN
ejpam-6211	216	19	{	{	PUNCT
ejpam-6211	216	20	∞∑	∞∑	PROPN
ejpam-6211	216	21	n	n	CCONJ
ejpam-6211	216	22	=	=	SYM
ejpam-6211	216	23	k	k	NOUN
ejpam-6211	216	24	sf	sf	NOUN
ejpam-6211	216	25	2	2	NUM
ejpam-6211	216	26	n	n	NOUN
ejpam-6211	216	27	(	(	PUNCT
ejpam-6211	216	28	k	k	NOUN
ejpam-6211	216	29	;	;	PUNCT
ejpam-6211	216	30	r	r	X
ejpam-6211	216	31	)	)	PUNCT
ejpam-6211	216	32	tn	tn	NOUN
ejpam-6211	216	33	n	n	NOUN
ejpam-6211	216	34	!	!	PUNCT
ejpam-6211	216	35	}	}	PUNCT
ejpam-6211	217	1	zk	zk	PROPN
ejpam-6211	218	1	=	=	PUNCT
ejpam-6211	219	1	(	(	PUNCT
ejpam-6211	219	2	eαt	eαt	ADV
ejpam-6211	219	3	−	−	PROPN
ejpam-6211	219	4	eβt	eβt	NOUN
ejpam-6211	219	5	α−	α−	ADP
ejpam-6211	219	6	β	β	NOUN
ejpam-6211	219	7	)	)	PUNCT
ejpam-6211	219	8	(	(	PUNCT
ejpam-6211	219	9	∞∑	∞∑	NUM
ejpam-6211	219	10	k=0	k=0	PROPN
ejpam-6211	219	11	ert(et	ert(et	NOUN
ejpam-6211	219	12	−	−	PROPN
ejpam-6211	219	13	1)k	1)k	NUM
ejpam-6211	219	14	k	k	NOUN
ejpam-6211	219	15	!	!	PUNCT
ejpam-6211	219	16	zk	zk	PROPN
ejpam-6211	219	17	)	)	PUNCT
ejpam-6211	220	1	=	=	PRON
ejpam-6211	220	2	(	(	PUNCT
ejpam-6211	220	3	eαt	eαt	ADV
ejpam-6211	220	4	−	−	PROPN
ejpam-6211	220	5	eβt	eβt	NOUN
ejpam-6211	220	6	α−	α−	ADP
ejpam-6211	220	7	β	β	PROPN
ejpam-6211	220	8	)	)	PUNCT
ejpam-6211	220	9	(	(	PUNCT
ejpam-6211	220	10	ert	ert	VERB
ejpam-6211	220	11	∞∑	∞∑	PRON
ejpam-6211	220	12	k=0	k=0	PROPN
ejpam-6211	220	13	zk	zk	PROPN
ejpam-6211	220	14	k	k	PROPN
ejpam-6211	220	15	!	!	PUNCT
ejpam-6211	221	1	(	(	PUNCT
ejpam-6211	221	2	et	et	X
ejpam-6211	221	3	−	−	PROPN
ejpam-6211	221	4	1)k	1)k	NUM
ejpam-6211	221	5	)	)	PUNCT
ejpam-6211	222	1	=	=	SYM
ejpam-6211	222	2	(	(	PUNCT
ejpam-6211	222	3	eαt	eαt	ADV
ejpam-6211	222	4	−	−	PROPN
ejpam-6211	222	5	eβt	eβt	NOUN
ejpam-6211	222	6	α−	α−	ADP
ejpam-6211	222	7	β	β	PROPN
ejpam-6211	222	8	)	)	PUNCT
ejpam-6211	222	9	ert(1−	ert(1−	PROPN
ejpam-6211	222	10	et	et	NOUN
ejpam-6211	222	11	−	−	PROPN
ejpam-6211	222	12	1)z	1)z	NUM
ejpam-6211	222	13	=	=	PUNCT
ejpam-6211	222	14	eαt	eαt	ADP
ejpam-6211	222	15	−	−	PROPN
ejpam-6211	222	16	eβt	eβt	NOUN
ejpam-6211	222	17	α−	α−	ADP
ejpam-6211	222	18	β	β	PROPN
ejpam-6211	222	19	ert+zt	ert+zt	X
ejpam-6211	222	20	=	=	SYM
ejpam-6211	222	21	(	(	PUNCT
ejpam-6211	222	22	∞∑	∞∑	PROPN
ejpam-6211	222	23	n=0	n=0	NUM
ejpam-6211	222	24	(	(	PUNCT
ejpam-6211	222	25	z	z	NOUN
ejpam-6211	222	26	+	+	NOUN
ejpam-6211	222	27	r)n	r)n	X
ejpam-6211	222	28	tn	tn	NOUN
ejpam-6211	222	29	n	n	CCONJ
ejpam-6211	222	30	!	!	PUNCT
ejpam-6211	222	31	)	)	PUNCT
ejpam-6211	223	1	(	(	PUNCT
ejpam-6211	223	2	∞∑	∞∑	NUM
ejpam-6211	223	3	n=0	n=0	NUM
ejpam-6211	223	4	fn	fn	NOUN
ejpam-6211	223	5	tn	tn	NOUN
ejpam-6211	223	6	n	n	PROPN
ejpam-6211	223	7	!	!	PUNCT
ejpam-6211	223	8	)	)	PUNCT
ejpam-6211	224	1	11	11	NUM
ejpam-6211	224	2	of	of	ADP
ejpam-6211	224	3	23	23	NUM
ejpam-6211	224	4	by	by	ADP
ejpam-6211	224	5	cauchy	cauchy	NOUN
ejpam-6211	224	6	product	product	NOUN
ejpam-6211	224	7	the	the	DET
ejpam-6211	224	8	rhs	rhs	PROPN
ejpam-6211	224	9	becomes	become	VERB
ejpam-6211	224	10	,	,	PUNCT
ejpam-6211	224	11	=	=	SYM
ejpam-6211	224	12	∞∑	∞∑	NUM
ejpam-6211	224	13	n=0	n=0	NUM
ejpam-6211	224	14	{	{	PUNCT
ejpam-6211	224	15	n∑	n∑	PROPN
ejpam-6211	224	16	m=0	m=0	PROPN
ejpam-6211	224	17	(	(	PUNCT
ejpam-6211	224	18	z	z	NOUN
ejpam-6211	224	19	+	+	NOUN
ejpam-6211	224	20	r)m	r)m	NOUN
ejpam-6211	224	21	(	(	PUNCT
ejpam-6211	224	22	n	n	X
ejpam-6211	224	23	m	m	NOUN
ejpam-6211	224	24	)	)	PUNCT
ejpam-6211	224	25	fn−m	fn−m	PROPN
ejpam-6211	224	26	}	}	PUNCT
ejpam-6211	224	27	tn	tn	PROPN
ejpam-6211	224	28	n	n	CCONJ
ejpam-6211	224	29	!	!	NOUN
ejpam-6211	224	30	comparing	compare	VERB
ejpam-6211	224	31	coefficients	coefficient	NOUN
ejpam-6211	224	32	of	of	ADP
ejpam-6211	224	33	tn	tn	NOUN
ejpam-6211	224	34	n	n	CCONJ
ejpam-6211	224	35	!	!	PUNCT
ejpam-6211	225	1	we	we	PRON
ejpam-6211	225	2	have	have	VERB
ejpam-6211	225	3	,	,	PUNCT
ejpam-6211	225	4	n∑	n∑	PROPN
ejpam-6211	225	5	k=0	k=0	PROPN
ejpam-6211	225	6	sf	sf	PROPN
ejpam-6211	225	7	2	2	NUM
ejpam-6211	225	8	n	n	NOUN
ejpam-6211	225	9	(	(	PUNCT
ejpam-6211	225	10	k	k	NOUN
ejpam-6211	225	11	;	;	PUNCT
ejpam-6211	225	12	r	r	X
ejpam-6211	225	13	)	)	PUNCT
ejpam-6211	225	14	zk	zk	PROPN
ejpam-6211	226	1	=	=	SYM
ejpam-6211	226	2	n∑	n∑	PROPN
ejpam-6211	226	3	m=0	m=0	PROPN
ejpam-6211	226	4	(	(	PUNCT
ejpam-6211	226	5	z	z	NOUN
ejpam-6211	226	6	+	+	NOUN
ejpam-6211	226	7	r)m	r)m	NOUN
ejpam-6211	226	8	(	(	PUNCT
ejpam-6211	226	9	n	n	X
ejpam-6211	226	10	m	m	NOUN
ejpam-6211	226	11	)	)	PUNCT
ejpam-6211	226	12	fn−m	fn−m	NOUN
ejpam-6211	227	1	=	=	PUNCT
ejpam-6211	227	2	f	f	PROPN
ejpam-6211	227	3	(	(	PUNCT
ejpam-6211	227	4	2	2	NUM
ejpam-6211	227	5	)	)	PUNCT
ejpam-6211	227	6	n	n	NOUN
ejpam-6211	227	7	(	(	PUNCT
ejpam-6211	227	8	z	z	NOUN
ejpam-6211	227	9	+	+	CCONJ
ejpam-6211	227	10	r	r	NOUN
ejpam-6211	227	11	)	)	PUNCT
ejpam-6211	227	12	theorem	theorem	NOUN
ejpam-6211	227	13	3.5	3.5	NUM
ejpam-6211	227	14	.	.	PUNCT
ejpam-6211	228	1	the	the	DET
ejpam-6211	228	2	schlömilch	schlömilch	ADJ
ejpam-6211	228	3	-	-	PUNCT
ejpam-6211	228	4	type	type	NOUN
ejpam-6211	228	5	formula	formula	NOUN
ejpam-6211	228	6	of	of	ADP
ejpam-6211	228	7	the	the	DET
ejpam-6211	228	8	r	r	NOUN
ejpam-6211	228	9	-	-	PUNCT
ejpam-6211	228	10	stirling	stirling	NOUN
ejpam-6211	228	11	fibonacci	fibonacci	NOUN
ejpam-6211	228	12	number	number	NOUN
ejpam-6211	228	13	is	be	AUX
ejpam-6211	228	14	given	give	VERB
ejpam-6211	228	15	by	by	ADP
ejpam-6211	228	16	sf	sf	PROPN
ejpam-6211	228	17	1	1	NUM
ejpam-6211	228	18	n	n	NOUN
ejpam-6211	228	19	(	(	PUNCT
ejpam-6211	228	20	k	k	NOUN
ejpam-6211	228	21	;	;	PUNCT
ejpam-6211	228	22	r	r	X
ejpam-6211	228	23	)	)	PUNCT
ejpam-6211	228	24	=	=	PUNCT
ejpam-6211	228	25	n∑	n∑	NOUN
ejpam-6211	228	26	m	m	PROPN
ejpam-6211	228	27	=	=	AUX
ejpam-6211	228	28	k	k	X
ejpam-6211	228	29	m∑	m∑	ADV
ejpam-6211	228	30	j=0	j=0	PROPN
ejpam-6211	228	31	m−k∑	m−k∑	X
ejpam-6211	229	1	a=0	a=0	PRON
ejpam-6211	229	2	a∑	a∑	PROPN
ejpam-6211	229	3	b=0	b=0	PROPN
ejpam-6211	229	4	(	(	PUNCT
ejpam-6211	229	5	−1)n−j+b+a	−1)n−j+b+a	PROPN
ejpam-6211	229	6	(	(	PUNCT
ejpam-6211	229	7	a	a	DET
ejpam-6211	229	8	b	b	NOUN
ejpam-6211	229	9	)	)	PUNCT
ejpam-6211	229	10	(	(	PUNCT
ejpam-6211	229	11	n	n	CCONJ
ejpam-6211	229	12	j	j	NOUN
ejpam-6211	229	13	)	)	PUNCT
ejpam-6211	229	14	(	(	PUNCT
ejpam-6211	229	15	j	j	PROPN
ejpam-6211	229	16	m	m	NOUN
ejpam-6211	229	17	)	)	PUNCT
ejpam-6211	229	18	fj−m	fj−m	NOUN
ejpam-6211	229	19	(	(	PUNCT
ejpam-6211	229	20	m−	m−	PROPN
ejpam-6211	229	21	1	1	NUM
ejpam-6211	229	22	+	+	CCONJ
ejpam-6211	229	23	a	a	DET
ejpam-6211	229	24	m−	m−	PROPN
ejpam-6211	229	25	k	k	PROPN
ejpam-6211	229	26	+	+	CCONJ
ejpam-6211	229	27	a	a	X
ejpam-6211	229	28	)	)	PUNCT
ejpam-6211	229	29	(	(	PUNCT
ejpam-6211	229	30	3.7	3.7	NUM
ejpam-6211	229	31	)	)	PUNCT
ejpam-6211	229	32	×	×	NOUN
ejpam-6211	229	33	(	(	PUNCT
ejpam-6211	229	34	2m−	2m−	NUM
ejpam-6211	229	35	k	k	NOUN
ejpam-6211	229	36	m−	m−	PROPN
ejpam-6211	230	1	k	k	PROPN
ejpam-6211	230	2	−	−	PROPN
ejpam-6211	230	3	a	a	X
ejpam-6211	230	4	)	)	PUNCT
ejpam-6211	230	5	(	(	PUNCT
ejpam-6211	230	6	a−	a−	PROPN
ejpam-6211	230	7	b)m−k+a	b)m−k+a	PROPN
ejpam-6211	230	8	a	a	PRON
ejpam-6211	230	9	!	!	PUNCT
ejpam-6211	231	1	rn−j	rn−j	NOUN
ejpam-6211	231	2	proof	proof	NOUN
ejpam-6211	231	3	to	to	PART
ejpam-6211	231	4	derive	derive	VERB
ejpam-6211	231	5	the	the	DET
ejpam-6211	231	6	schlömilch	schlömilch	ADJ
ejpam-6211	231	7	-	-	PUNCT
ejpam-6211	231	8	type	type	NOUN
ejpam-6211	231	9	formula	formula	NOUN
ejpam-6211	231	10	we	we	PRON
ejpam-6211	231	11	have	have	VERB
ejpam-6211	231	12	to	to	PART
ejpam-6211	231	13	decompose	decompose	VERB
ejpam-6211	231	14	the	the	DET
ejpam-6211	231	15	exponential	exponential	ADJ
ejpam-6211	231	16	generating	generating	NOUN
ejpam-6211	231	17	function	function	NOUN
ejpam-6211	231	18	in	in	ADP
ejpam-6211	231	19	(	(	PUNCT
ejpam-6211	231	20	3.1	3.1	NUM
ejpam-6211	231	21	)	)	PUNCT
ejpam-6211	231	22	into	into	ADP
ejpam-6211	231	23	product	product	NOUN
ejpam-6211	231	24	of	of	ADP
ejpam-6211	231	25	three	three	NUM
ejpam-6211	231	26	functions	function	NOUN
ejpam-6211	231	27	as	as	SCONJ
ejpam-6211	231	28	follows	follow	VERB
ejpam-6211	231	29	:	:	PUNCT
ejpam-6211	231	30	the	the	DET
ejpam-6211	231	31	first	first	ADJ
ejpam-6211	231	32	function	function	NOUN
ejpam-6211	231	33	can	can	AUX
ejpam-6211	231	34	be	be	AUX
ejpam-6211	231	35	expressed	express	VERB
ejpam-6211	231	36	as	as	ADP
ejpam-6211	231	37	:	:	PUNCT
ejpam-6211	231	38	(	(	PUNCT
ejpam-6211	231	39	1	1	NUM
ejpam-6211	231	40	1	1	NUM
ejpam-6211	231	41	+	+	NUM
ejpam-6211	231	42	t	t	NOUN
ejpam-6211	231	43	)	)	PUNCT
ejpam-6211	231	44	r	r	NOUN
ejpam-6211	231	45	=	=	SYM
ejpam-6211	231	46	(	(	PUNCT
ejpam-6211	231	47	1	1	NUM
ejpam-6211	231	48	+	+	NUM
ejpam-6211	231	49	t)−r	t)−r	NOUN
ejpam-6211	232	1	=	=	PUNCT
ejpam-6211	232	2	∑	∑	PUNCT
ejpam-6211	232	3	n≥0	n≥0	PROPN
ejpam-6211	232	4	(	(	PUNCT
ejpam-6211	232	5	−r	−r	ADJ
ejpam-6211	232	6	n	n	CCONJ
ejpam-6211	232	7	)	)	PUNCT
ejpam-6211	232	8	tn	tn	PROPN
ejpam-6211	232	9	=	=	SYM
ejpam-6211	232	10	(	(	PUNCT
ejpam-6211	232	11	−r	−r	PROPN
ejpam-6211	232	12	0	0	NUM
ejpam-6211	232	13	)	)	PUNCT
ejpam-6211	232	14	(	(	PUNCT
ejpam-6211	232	15	−t)0	−t)0	NOUN
ejpam-6211	233	1	+	+	CCONJ
ejpam-6211	233	2	∑	∑	PROPN
ejpam-6211	233	3	n≥0	n≥0	PROPN
ejpam-6211	233	4	(	(	PUNCT
ejpam-6211	233	5	−r	−r	ADJ
ejpam-6211	233	6	n	n	X
ejpam-6211	233	7	)	)	PUNCT
ejpam-6211	233	8	tn	tn	NOUN
ejpam-6211	233	9	where	where	SCONJ
ejpam-6211	233	10	(	(	PUNCT
ejpam-6211	233	11	−r	−r	PROPN
ejpam-6211	233	12	n	n	CCONJ
ejpam-6211	233	13	)	)	PUNCT
ejpam-6211	233	14	is	be	AUX
ejpam-6211	233	15	the	the	DET
ejpam-6211	233	16	newton	newton	PROPN
ejpam-6211	233	17	’s	’s	PART
ejpam-6211	233	18	generalized	generalize	VERB
ejpam-6211	233	19	binomial	binomial	ADJ
ejpam-6211	233	20	coefficients	coefficient	NOUN
ejpam-6211	233	21	.	.	PUNCT
ejpam-6211	234	1	by	by	ADP
ejpam-6211	234	2	newton	newton	PROPN
ejpam-6211	234	3	’s	’s	PART
ejpam-6211	234	4	binomial	binomial	PROPN
ejpam-6211	234	5	theorem	theorem	NOUN
ejpam-6211	234	6	this	this	PRON
ejpam-6211	234	7	can	can	AUX
ejpam-6211	234	8	be	be	AUX
ejpam-6211	234	9	further	far	ADV
ejpam-6211	234	10	computed	compute	VERB
ejpam-6211	234	11	as	as	ADP
ejpam-6211	234	12	(	(	PUNCT
ejpam-6211	234	13	1	1	NUM
ejpam-6211	234	14	1	1	NUM
ejpam-6211	234	15	+	+	NUM
ejpam-6211	234	16	t	t	NOUN
ejpam-6211	234	17	)	)	PUNCT
ejpam-6211	234	18	r	r	NOUN
ejpam-6211	234	19	=	=	SYM
ejpam-6211	234	20	1	1	NUM
ejpam-6211	234	21	+	+	CCONJ
ejpam-6211	234	22	∑	∑	PROPN
ejpam-6211	234	23	n>0	n>0	PROPN
ejpam-6211	234	24	(	(	PUNCT
ejpam-6211	234	25	−r	−r	PROPN
ejpam-6211	234	26	)	)	PUNCT
ejpam-6211	234	27	(	(	PUNCT
ejpam-6211	234	28	−r	−r	ADJ
ejpam-6211	234	29	−	−	NOUN
ejpam-6211	234	30	1	1	NUM
ejpam-6211	234	31	)	)	PUNCT
ejpam-6211	234	32	·	·	PUNCT
ejpam-6211	234	33	·	·	PUNCT
ejpam-6211	234	34	·	·	PUNCT
ejpam-6211	234	35	(	(	PUNCT
ejpam-6211	234	36	−r	−r	ADJ
ejpam-6211	234	37	−	−	PROPN
ejpam-6211	234	38	n+	n+	NOUN
ejpam-6211	234	39	1	1	NUM
ejpam-6211	234	40	)	)	PUNCT
ejpam-6211	234	41	n	n	CCONJ
ejpam-6211	234	42	!	!	PUNCT
ejpam-6211	234	43	tn	tn	NOUN
ejpam-6211	235	1	=	=	SYM
ejpam-6211	235	2	1	1	NUM
ejpam-6211	236	1	+	+	CCONJ
ejpam-6211	236	2	∑	∑	PROPN
ejpam-6211	236	3	n>0	n>0	PROPN
ejpam-6211	236	4	(	(	PUNCT
ejpam-6211	236	5	−1)n	−1)n	X
ejpam-6211	236	6	(	(	PUNCT
ejpam-6211	236	7	r	r	NOUN
ejpam-6211	236	8	)	)	PUNCT
ejpam-6211	236	9	(	(	PUNCT
ejpam-6211	236	10	r	r	NOUN
ejpam-6211	236	11	+	+	NOUN
ejpam-6211	236	12	1	1	NUM
ejpam-6211	236	13	)	)	PUNCT
ejpam-6211	236	14	·	·	PUNCT
ejpam-6211	236	15	·	·	PUNCT
ejpam-6211	236	16	·	·	PUNCT
ejpam-6211	237	1	(	(	PUNCT
ejpam-6211	237	2	r	r	NOUN
ejpam-6211	237	3	+	+	NUM
ejpam-6211	237	4	n−	n−	NOUN
ejpam-6211	237	5	1	1	NUM
ejpam-6211	237	6	)	)	PUNCT
ejpam-6211	237	7	n	n	CCONJ
ejpam-6211	237	8	!	!	PUNCT
ejpam-6211	237	9	tn	tn	NOUN
ejpam-6211	238	1	=	=	SYM
ejpam-6211	238	2	1	1	NUM
ejpam-6211	239	1	+	+	CCONJ
ejpam-6211	239	2	∑	∑	PROPN
ejpam-6211	239	3	n>0	n>0	PROPN
ejpam-6211	239	4	(	(	PUNCT
ejpam-6211	239	5	−1)n	−1)n	X
ejpam-6211	239	6	(	(	PUNCT
ejpam-6211	239	7	(	(	PUNCT
ejpam-6211	239	8	r	r	NOUN
ejpam-6211	239	9	)	)	PUNCT
ejpam-6211	239	10	(	(	PUNCT
ejpam-6211	239	11	r	r	NOUN
ejpam-6211	239	12	+	+	NOUN
ejpam-6211	239	13	1	1	NUM
ejpam-6211	239	14	)	)	PUNCT
ejpam-6211	239	15	·	·	PUNCT
ejpam-6211	239	16	·	·	PUNCT
ejpam-6211	239	17	·	·	PUNCT
ejpam-6211	240	1	(	(	PUNCT
ejpam-6211	240	2	r	r	NOUN
ejpam-6211	240	3	+	+	NUM
ejpam-6211	240	4	n−	n−	NOUN
ejpam-6211	240	5	1	1	NUM
ejpam-6211	240	6	)	)	PUNCT
ejpam-6211	240	7	)	)	PUNCT
ejpam-6211	240	8	tn	tn	PROPN
ejpam-6211	240	9	n	n	CCONJ
ejpam-6211	240	10	!	!	PUNCT
ejpam-6211	241	1	by	by	ADP
ejpam-6211	241	2	the	the	DET
ejpam-6211	241	3	definition	definition	NOUN
ejpam-6211	241	4	of	of	ADP
ejpam-6211	241	5	the	the	DET
ejpam-6211	241	6	rising	rise	VERB
ejpam-6211	241	7	factorial	factorial	NOUN
ejpam-6211	241	8	rn	rn	PROPN
ejpam-6211	241	9	=	=	PROPN
ejpam-6211	241	10	r(r	r(r	PROPN
ejpam-6211	241	11	+	+	CCONJ
ejpam-6211	241	12	1)(r	1)(r	NUM
ejpam-6211	241	13	+	+	CCONJ
ejpam-6211	241	14	1	1	NUM
ejpam-6211	241	15	)	)	PUNCT
ejpam-6211	241	16	·	·	PUNCT
ejpam-6211	241	17	·	·	PUNCT
ejpam-6211	241	18	·	·	PUNCT
ejpam-6211	241	19	(	(	PUNCT
ejpam-6211	241	20	r	r	NOUN
ejpam-6211	241	21	+	+	NUM
ejpam-6211	241	22	n−	n−	NOUN
ejpam-6211	241	23	1	1	NUM
ejpam-6211	241	24	)	)	PUNCT
ejpam-6211	241	25	we	we	PRON
ejpam-6211	241	26	have	have	VERB
ejpam-6211	241	27	,	,	PUNCT
ejpam-6211	241	28	(	(	PUNCT
ejpam-6211	241	29	1	1	NUM
ejpam-6211	241	30	1	1	NUM
ejpam-6211	241	31	+	+	NUM
ejpam-6211	241	32	t	t	NOUN
ejpam-6211	241	33	)	)	PUNCT
ejpam-6211	241	34	r	r	NOUN
ejpam-6211	241	35	=	=	SYM
ejpam-6211	241	36	1	1	NUM
ejpam-6211	241	37	+	+	CCONJ
ejpam-6211	241	38	∑	∑	PROPN
ejpam-6211	241	39	n>0	n>0	PROPN
ejpam-6211	241	40	(	(	PUNCT
ejpam-6211	241	41	−1)n	−1)n	PROPN
ejpam-6211	241	42	rn	rn	PROPN
ejpam-6211	241	43	tn	tn	PROPN
ejpam-6211	241	44	n	n	PROPN
ejpam-6211	241	45	!	!	PROPN
ejpam-6211	241	46	12	12	NUM
ejpam-6211	241	47	of	of	ADP
ejpam-6211	241	48	23	23	NUM
ejpam-6211	241	49	=	=	SYM
ejpam-6211	241	50	∑	∑	PUNCT
ejpam-6211	241	51	n≥0	n≥0	PROPN
ejpam-6211	241	52	(	(	PUNCT
ejpam-6211	241	53	−1)n	−1)n	PROPN
ejpam-6211	241	54	rn	rn	PROPN
ejpam-6211	241	55	tn	tn	PROPN
ejpam-6211	241	56	n	n	PROPN
ejpam-6211	241	57	!	!	PROPN
ejpam-6211	241	58	.	.	PUNCT
ejpam-6211	242	1	the	the	DET
ejpam-6211	242	2	second	second	ADJ
ejpam-6211	242	3	function	function	NOUN
ejpam-6211	242	4	is	be	AUX
ejpam-6211	242	5	the	the	DET
ejpam-6211	242	6	fibonacci	fibonacci	NOUN
ejpam-6211	242	7	polynomial	polynomial	ADJ
ejpam-6211	242	8	which	which	PRON
ejpam-6211	242	9	can	can	AUX
ejpam-6211	242	10	be	be	AUX
ejpam-6211	242	11	expressed	express	VERB
ejpam-6211	242	12	as	as	ADP
ejpam-6211	242	13	its	its	PRON
ejpam-6211	242	14	exponential	exponential	ADJ
ejpam-6211	242	15	generating	generating	NOUN
ejpam-6211	242	16	function	function	NOUN
ejpam-6211	242	17	given	give	VERB
ejpam-6211	242	18	by	by	ADP
ejpam-6211	242	19	;	;	PUNCT
ejpam-6211	242	20	eαt	eαt	ADV
ejpam-6211	242	21	−	−	PROPN
ejpam-6211	242	22	eβt	eβt	NOUN
ejpam-6211	242	23	α−	α−	ADP
ejpam-6211	242	24	β	β	NOUN
ejpam-6211	242	25	=	=	PUNCT
ejpam-6211	243	1	∞∑	∞∑	PROPN
ejpam-6211	243	2	n=0	n=0	NUM
ejpam-6211	243	3	fn	fn	NOUN
ejpam-6211	243	4	tn	tn	NOUN
ejpam-6211	243	5	n	n	X
ejpam-6211	243	6	!	!	PUNCT
ejpam-6211	243	7	.	.	PUNCT
ejpam-6211	244	1	lastly	lastly	ADV
ejpam-6211	244	2	,	,	PUNCT
ejpam-6211	244	3	the	the	DET
ejpam-6211	244	4	third	third	ADJ
ejpam-6211	244	5	function	function	NOUN
ejpam-6211	244	6	can	can	AUX
ejpam-6211	244	7	be	be	AUX
ejpam-6211	244	8	written	write	VERB
ejpam-6211	244	9	as	as	ADP
ejpam-6211	244	10	[	[	X
ejpam-6211	244	11	ln	ln	ADJ
ejpam-6211	244	12	(	(	PUNCT
ejpam-6211	244	13	1	1	NUM
ejpam-6211	244	14	+	+	CCONJ
ejpam-6211	244	15	t)]k	t)]k	NOUN
ejpam-6211	244	16	k	k	NOUN
ejpam-6211	244	17	!	!	PUNCT
ejpam-6211	244	18	=	=	PUNCT
ejpam-6211	245	1	∑	∑	PUNCT
ejpam-6211	245	2	n≥k	n≥k	PROPN
ejpam-6211	245	3	s(n	s(n	PROPN
ejpam-6211	245	4	,	,	PUNCT
ejpam-6211	245	5	k	k	NOUN
ejpam-6211	245	6	)	)	PUNCT
ejpam-6211	245	7	tn	tn	PROPN
ejpam-6211	245	8	n	n	PROPN
ejpam-6211	245	9	!	!	PUNCT
ejpam-6211	245	10	.	.	PUNCT
ejpam-6211	246	1	hence	hence	ADV
ejpam-6211	246	2	,	,	PUNCT
ejpam-6211	246	3	using	use	VERB
ejpam-6211	246	4	cauchy	cauchy	PROPN
ejpam-6211	246	5	’s	’s	PART
ejpam-6211	246	6	rule	rule	NOUN
ejpam-6211	246	7	for	for	ADP
ejpam-6211	246	8	the	the	DET
ejpam-6211	246	9	product	product	NOUN
ejpam-6211	246	10	we	we	PRON
ejpam-6211	246	11	have	have	VERB
ejpam-6211	246	12	,	,	PUNCT
ejpam-6211	246	13	∞∑	∞∑	PROPN
ejpam-6211	246	14	k=0	k=0	PROPN
ejpam-6211	246	15	sf	sf	ADP
ejpam-6211	246	16	1	1	NUM
ejpam-6211	246	17	n	n	NOUN
ejpam-6211	246	18	(	(	PUNCT
ejpam-6211	246	19	k	k	NOUN
ejpam-6211	246	20	;	;	PUNCT
ejpam-6211	246	21	r	r	X
ejpam-6211	246	22	)	)	PUNCT
ejpam-6211	246	23	tn	tn	NOUN
ejpam-6211	246	24	n	n	NOUN
ejpam-6211	246	25	!	!	PUNCT
ejpam-6211	246	26	=	=	PUNCT
ejpam-6211	247	1	∑	∑	NOUN
ejpam-6211	247	2	n≥0	n≥0	PROPN
ejpam-6211	247	3	(	(	PUNCT
ejpam-6211	247	4	−1)n	−1)n	PROPN
ejpam-6211	247	5	rn	rn	PROPN
ejpam-6211	247	6	tn	tn	PROPN
ejpam-6211	247	7	n	n	PROPN
ejpam-6211	247	8	!	!	PUNCT
ejpam-6211	248	1			PROPN
ejpam-6211	248	2	(	(	PUNCT
ejpam-6211	248	3	∞∑	∞∑	PROPN
ejpam-6211	248	4	n=0	n=0	NUM
ejpam-6211	248	5	fn	fn	NOUN
ejpam-6211	248	6	tn	tn	NOUN
ejpam-6211	248	7	n	n	PROPN
ejpam-6211	248	8	!	!	PUNCT
ejpam-6211	248	9	)	)	PUNCT
ejpam-6211	249	1	∑	∑	NOUN
ejpam-6211	249	2	n≥k	n≥k	PROPN
ejpam-6211	249	3	s(n	s(n	PROPN
ejpam-6211	249	4	,	,	PUNCT
ejpam-6211	249	5	k	k	NOUN
ejpam-6211	249	6	)	)	PUNCT
ejpam-6211	249	7	tn	tn	PROPN
ejpam-6211	250	1	n	n	NOUN
ejpam-6211	250	2	!	!	PUNCT
ejpam-6211	251	1			PROPN
ejpam-6211	252	1	=	=	PUNCT
ejpam-6211	252	2	∑	∑	NOUN
ejpam-6211	252	3	n≥0	n≥0	PROPN
ejpam-6211	252	4	(	(	PUNCT
ejpam-6211	252	5	−1)n	−1)n	PROPN
ejpam-6211	252	6	rn	rn	PROPN
ejpam-6211	252	7	tn	tn	PROPN
ejpam-6211	252	8	n	n	PROPN
ejpam-6211	252	9	!	!	PUNCT
ejpam-6211	253	1			PROPN
ejpam-6211	253	2	(	(	PUNCT
ejpam-6211	253	3	∞∑	∞∑	PROPN
ejpam-6211	253	4	n=0	n=0	PROPN
ejpam-6211	253	5	{	{	PUNCT
ejpam-6211	253	6	n∑	n∑	NOUN
ejpam-6211	253	7	m	m	PROPN
ejpam-6211	253	8	=	=	PROPN
ejpam-6211	253	9	k	k	PROPN
ejpam-6211	253	10	s(m	s(m	PROPN
ejpam-6211	253	11	,	,	PUNCT
ejpam-6211	253	12	k	k	NOUN
ejpam-6211	253	13	)	)	PUNCT
ejpam-6211	253	14	(	(	PUNCT
ejpam-6211	253	15	n	n	X
ejpam-6211	253	16	m	m	NOUN
ejpam-6211	253	17	)	)	PUNCT
ejpam-6211	253	18	fn−m	fn−m	PROPN
ejpam-6211	253	19	}	}	PUNCT
ejpam-6211	253	20	tn	tn	PROPN
ejpam-6211	253	21	n	n	CCONJ
ejpam-6211	253	22	!	!	PUNCT
ejpam-6211	253	23	)	)	PUNCT
ejpam-6211	254	1	=	=	PUNCT
ejpam-6211	255	1	∞∑	∞∑	NUM
ejpam-6211	255	2	n=0	n=0	NUM
ejpam-6211	255	3			PUNCT
ejpam-6211	255	4	n∑	n∑	PROPN
ejpam-6211	255	5	m=0	m=0	PROPN
ejpam-6211	255	6	n∑	n∑	PROPN
ejpam-6211	255	7	j	j	PROPN
ejpam-6211	256	1	=	=	NOUN
ejpam-6211	256	2	m	m	PROPN
ejpam-6211	256	3	(	(	PUNCT
ejpam-6211	256	4	−1)n−j	−1)n−j	X
ejpam-6211	256	5	rn−j	rn−j	X
ejpam-6211	256	6	(	(	PUNCT
ejpam-6211	256	7	n	n	CCONJ
ejpam-6211	256	8	j	j	PROPN
ejpam-6211	256	9	)	)	PUNCT
ejpam-6211	256	10	s(m	s(m	PROPN
ejpam-6211	256	11	,	,	PUNCT
ejpam-6211	256	12	k	k	NOUN
ejpam-6211	256	13	)	)	PUNCT
ejpam-6211	256	14	(	(	PUNCT
ejpam-6211	256	15	j	j	PROPN
ejpam-6211	256	16	m	m	VERB
ejpam-6211	256	17	)	)	PUNCT
ejpam-6211	256	18	fj−m	fj−m	NOUN
ejpam-6211	256	19			PROPN
ejpam-6211	256	20	tn	tn	PROPN
ejpam-6211	256	21	n	n	X
ejpam-6211	256	22	!	!	X
ejpam-6211	256	23	comparing	compare	VERB
ejpam-6211	256	24	coefficients	coefficient	NOUN
ejpam-6211	256	25	of	of	ADP
ejpam-6211	256	26	tn	tn	NOUN
ejpam-6211	256	27	n	n	CCONJ
ejpam-6211	256	28	!	!	PUNCT
ejpam-6211	257	1	we	we	PRON
ejpam-6211	257	2	have	have	VERB
ejpam-6211	257	3	,	,	PUNCT
ejpam-6211	257	4	sf	sf	PROPN
ejpam-6211	257	5	1	1	NUM
ejpam-6211	257	6	n	n	CCONJ
ejpam-6211	257	7	(	(	PUNCT
ejpam-6211	257	8	k	k	NOUN
ejpam-6211	257	9	;	;	PUNCT
ejpam-6211	257	10	r	r	X
ejpam-6211	257	11	)	)	PUNCT
ejpam-6211	257	12	=	=	SYM
ejpam-6211	258	1	n∑	n∑	PROPN
ejpam-6211	258	2	m=0	m=0	PROPN
ejpam-6211	258	3	n∑	n∑	PROPN
ejpam-6211	258	4	j	j	PROPN
ejpam-6211	259	1	=	=	NOUN
ejpam-6211	259	2	m	m	PROPN
ejpam-6211	259	3	(	(	PUNCT
ejpam-6211	259	4	−1)n−j	−1)n−j	PUNCT
ejpam-6211	259	5	s(m	s(m	PROPN
ejpam-6211	259	6	,	,	PUNCT
ejpam-6211	259	7	k	k	NOUN
ejpam-6211	259	8	)	)	PUNCT
ejpam-6211	259	9	(	(	PUNCT
ejpam-6211	259	10	n	n	X
ejpam-6211	259	11	j	j	NOUN
ejpam-6211	259	12	)	)	PUNCT
ejpam-6211	259	13	(	(	PUNCT
ejpam-6211	259	14	j	j	PROPN
ejpam-6211	259	15	m	m	VERB
ejpam-6211	259	16	)	)	PUNCT
ejpam-6211	259	17	fj−mrn−j	fj−mrn−j	ADV
ejpam-6211	259	18	the	the	DET
ejpam-6211	259	19	schlömilch	schlömilch	ADJ
ejpam-6211	259	20	-	-	ADJ
ejpam-6211	259	21	type	type	NOUN
ejpam-6211	259	22	formula	formula	NOUN
ejpam-6211	259	23	for	for	ADP
ejpam-6211	259	24	the	the	DET
ejpam-6211	259	25	signed	sign	VERB
ejpam-6211	259	26	r	r	NOUN
ejpam-6211	259	27	-	-	PUNCT
ejpam-6211	259	28	stirling	stirling	NOUN
ejpam-6211	259	29	number	number	NOUN
ejpam-6211	259	30	of	of	ADP
ejpam-6211	259	31	the	the	DET
ejpam-6211	259	32	first	first	ADJ
ejpam-6211	259	33	kind	kind	NOUN
ejpam-6211	259	34	is	be	AUX
ejpam-6211	259	35	given	give	VERB
ejpam-6211	259	36	by[15	by[15	PROPN
ejpam-6211	259	37	]	]	X
ejpam-6211	259	38	,	,	PUNCT
ejpam-6211	259	39	s(n	s(n	PROPN
ejpam-6211	259	40	,	,	PUNCT
ejpam-6211	259	41	k	k	NOUN
ejpam-6211	259	42	)	)	PUNCT
ejpam-6211	259	43	=	=	PUNCT
ejpam-6211	260	1	n−k∑	n−k∑	NOUN
ejpam-6211	260	2	r=0	r=0	VERB
ejpam-6211	260	3	r∑	r∑	X
ejpam-6211	260	4	j=0	j=0	PROPN
ejpam-6211	260	5	(	(	PUNCT
ejpam-6211	260	6	−1)j+r	−1)j+r	X
ejpam-6211	260	7	(	(	PUNCT
ejpam-6211	260	8	r	r	NOUN
ejpam-6211	260	9	j	j	PROPN
ejpam-6211	260	10	)	)	PUNCT
ejpam-6211	260	11	(	(	PUNCT
ejpam-6211	260	12	n−	n−	NOUN
ejpam-6211	260	13	1	1	NUM
ejpam-6211	260	14	+	+	CCONJ
ejpam-6211	260	15	r	r	NOUN
ejpam-6211	260	16	n−	n−	NOUN
ejpam-6211	260	17	k	k	NOUN
ejpam-6211	260	18	+	+	CCONJ
ejpam-6211	260	19	r	r	NOUN
ejpam-6211	260	20	)	)	PUNCT
ejpam-6211	260	21	(	(	PUNCT
ejpam-6211	261	1	2n−	2n−	NUM
ejpam-6211	261	2	k	k	NOUN
ejpam-6211	261	3	n−	n−	NOUN
ejpam-6211	261	4	k	k	NOUN
ejpam-6211	261	5	−	−	NOUN
ejpam-6211	261	6	r	r	NOUN
ejpam-6211	261	7	)	)	PUNCT
ejpam-6211	261	8	(	(	PUNCT
ejpam-6211	261	9	r	r	NOUN
ejpam-6211	261	10	−	−	PROPN
ejpam-6211	261	11	j)n−k+r	j)n−k+r	PROPN
ejpam-6211	261	12	r	r	NOUN
ejpam-6211	261	13	!	!	PUNCT
ejpam-6211	262	1	so	so	ADV
ejpam-6211	262	2	the	the	DET
ejpam-6211	262	3	schlömilch	schlömilch	ADJ
ejpam-6211	262	4	-	-	ADJ
ejpam-6211	262	5	type	type	NOUN
ejpam-6211	262	6	formula	formula	NOUN
ejpam-6211	262	7	for	for	ADP
ejpam-6211	262	8	sf	sf	PROPN
ejpam-6211	262	9	1	1	NUM
ejpam-6211	262	10	n(k	n(k	PROPN
ejpam-6211	262	11	;	;	PUNCT
ejpam-6211	262	12	r	r	X
ejpam-6211	262	13	)	)	PUNCT
ejpam-6211	262	14	is	be	AUX
ejpam-6211	262	15	sf	sf	PROPN
ejpam-6211	262	16	1	1	NUM
ejpam-6211	262	17	n(k	n(k	PROPN
ejpam-6211	262	18	;	;	PUNCT
ejpam-6211	262	19	r	r	X
ejpam-6211	262	20	)	)	PUNCT
ejpam-6211	262	21	=	=	PUNCT
ejpam-6211	263	1	n∑	n∑	NOUN
ejpam-6211	263	2	m	m	PROPN
ejpam-6211	263	3	=	=	AUX
ejpam-6211	263	4	k	k	X
ejpam-6211	263	5	m∑	m∑	ADV
ejpam-6211	263	6	j=0	j=0	PROPN
ejpam-6211	263	7	m−k∑	m−k∑	X
ejpam-6211	264	1	a=0	a=0	PRON
ejpam-6211	264	2	a∑	a∑	PROPN
ejpam-6211	264	3	b=0	b=0	PROPN
ejpam-6211	264	4	(	(	PUNCT
ejpam-6211	264	5	−1)n−j+b+a	−1)n−j+b+a	PROPN
ejpam-6211	264	6	(	(	PUNCT
ejpam-6211	264	7	a	a	DET
ejpam-6211	264	8	b	b	NOUN
ejpam-6211	264	9	)	)	PUNCT
ejpam-6211	264	10	(	(	PUNCT
ejpam-6211	264	11	n	n	CCONJ
ejpam-6211	264	12	j	j	NOUN
ejpam-6211	264	13	)	)	PUNCT
ejpam-6211	264	14	(	(	PUNCT
ejpam-6211	264	15	j	j	PROPN
ejpam-6211	264	16	m	m	NOUN
ejpam-6211	264	17	)	)	PUNCT
ejpam-6211	264	18	fj−m	fj−m	NOUN
ejpam-6211	264	19	(	(	PUNCT
ejpam-6211	264	20	m−	m−	PROPN
ejpam-6211	264	21	1	1	NUM
ejpam-6211	264	22	+	+	CCONJ
ejpam-6211	264	23	a	a	DET
ejpam-6211	264	24	m−	m−	PROPN
ejpam-6211	264	25	k	k	PROPN
ejpam-6211	264	26	+	+	CCONJ
ejpam-6211	264	27	a	a	X
ejpam-6211	264	28	)	)	PUNCT
ejpam-6211	264	29	×	×	NOUN
ejpam-6211	264	30	(	(	PUNCT
ejpam-6211	264	31	2m−	2m−	NUM
ejpam-6211	264	32	k	k	NOUN
ejpam-6211	264	33	m−	m−	PROPN
ejpam-6211	265	1	k	k	PROPN
ejpam-6211	265	2	−	−	PROPN
ejpam-6211	265	3	a	a	X
ejpam-6211	265	4	)	)	PUNCT
ejpam-6211	265	5	(	(	PUNCT
ejpam-6211	265	6	a−	a−	PROPN
ejpam-6211	265	7	b)m−k+a	b)m−k+a	PROPN
ejpam-6211	265	8	a	a	X
ejpam-6211	265	9	!	!	PUNCT
ejpam-6211	266	1	rn−j	rn−j	NOUN
ejpam-6211	266	2	13	13	NUM
ejpam-6211	266	3	of	of	ADP
ejpam-6211	266	4	23	23	NUM
ejpam-6211	266	5	theorem	theorem	NOUN
ejpam-6211	266	6	3.6	3.6	NUM
ejpam-6211	266	7	.	.	PUNCT
ejpam-6211	267	1	the	the	DET
ejpam-6211	267	2	explicit	explicit	ADJ
ejpam-6211	267	3	formula	formula	NOUN
ejpam-6211	267	4	of	of	ADP
ejpam-6211	267	5	the	the	DET
ejpam-6211	267	6	r	r	NOUN
ejpam-6211	267	7	-	-	PUNCT
ejpam-6211	267	8	stirling	stirling	NOUN
ejpam-6211	267	9	fibonacci	fibonacci	NOUN
ejpam-6211	267	10	number	number	NOUN
ejpam-6211	267	11	of	of	ADP
ejpam-6211	267	12	the	the	DET
ejpam-6211	267	13	second	second	ADJ
ejpam-6211	267	14	kind	kind	NOUN
ejpam-6211	267	15	is	be	AUX
ejpam-6211	267	16	given	give	VERB
ejpam-6211	267	17	by	by	ADP
ejpam-6211	267	18	sf	sf	PROPN
ejpam-6211	267	19	2	2	NUM
ejpam-6211	267	20	n	n	NOUN
ejpam-6211	267	21	(	(	PUNCT
ejpam-6211	267	22	k	k	NOUN
ejpam-6211	267	23	;	;	PUNCT
ejpam-6211	267	24	r	r	X
ejpam-6211	267	25	)	)	PUNCT
ejpam-6211	267	26	=	=	SYM
ejpam-6211	267	27	n∑	n∑	PROPN
ejpam-6211	267	28	m=0	m=0	PROPN
ejpam-6211	267	29	{	{	PUNCT
ejpam-6211	268	1	n+	n+	ADP
ejpam-6211	268	2	r	r	NOUN
ejpam-6211	268	3	k	k	NOUN
ejpam-6211	269	1	+	+	CCONJ
ejpam-6211	269	2	r	r	NOUN
ejpam-6211	269	3	}	}	PUNCT
ejpam-6211	269	4	r	r	NOUN
ejpam-6211	269	5	(	(	PUNCT
ejpam-6211	269	6	n	n	NOUN
ejpam-6211	269	7	m	m	NOUN
ejpam-6211	269	8	)	)	PUNCT
ejpam-6211	269	9	fn−m	fn−m	NOUN
ejpam-6211	269	10	(	(	PUNCT
ejpam-6211	269	11	3.8	3.8	NUM
ejpam-6211	269	12	)	)	PUNCT
ejpam-6211	269	13	proof	proof	NOUN
ejpam-6211	269	14	the	the	DET
ejpam-6211	269	15	exponential	exponential	ADJ
ejpam-6211	269	16	generating	generating	NOUN
ejpam-6211	269	17	function	function	NOUN
ejpam-6211	269	18	(	(	PUNCT
ejpam-6211	269	19	3.2	3.2	NUM
ejpam-6211	269	20	)	)	PUNCT
ejpam-6211	269	21	k	k	NOUN
ejpam-6211	269	22	!	!	PUNCT
ejpam-6211	270	1	∞∑	∞∑	ADJ
ejpam-6211	270	2	n=0	n=0	PROPN
ejpam-6211	270	3	sf	sf	NOUN
ejpam-6211	270	4	2	2	NUM
ejpam-6211	270	5	n	n	NOUN
ejpam-6211	270	6	(	(	PUNCT
ejpam-6211	270	7	k	k	NOUN
ejpam-6211	270	8	;	;	PUNCT
ejpam-6211	270	9	r	r	X
ejpam-6211	270	10	,	,	PUNCT
ejpam-6211	270	11	x	x	NOUN
ejpam-6211	270	12	)	)	PUNCT
ejpam-6211	270	13	tn	tn	PROPN
ejpam-6211	270	14	n	n	NOUN
ejpam-6211	270	15	!	!	PUNCT
ejpam-6211	271	1	=	=	PUNCT
ejpam-6211	271	2	(	(	PUNCT
ejpam-6211	271	3	eαt	eαt	ADV
ejpam-6211	271	4	−	−	PRON
ejpam-6211	271	5	eβt)(ert(et	eβt)(ert(et	NOUN
ejpam-6211	271	6	−	−	PROPN
ejpam-6211	271	7	1)k	1)k	NUM
ejpam-6211	271	8	)	)	PUNCT
ejpam-6211	271	9	(	(	PUNCT
ejpam-6211	271	10	α−	α−	ADP
ejpam-6211	271	11	β	β	NOUN
ejpam-6211	271	12	)	)	PUNCT
ejpam-6211	271	13	=	=	SYM
ejpam-6211	272	1	(	(	PUNCT
ejpam-6211	272	2	∞∑	∞∑	NUM
ejpam-6211	272	3	n=0	n=0	NUM
ejpam-6211	272	4	{	{	PUNCT
ejpam-6211	272	5	(	(	PUNCT
ejpam-6211	272	6	−1)i	−1)i	X
ejpam-6211	272	7	(	(	PUNCT
ejpam-6211	272	8	k	k	X
ejpam-6211	272	9	i	i	PROPN
ejpam-6211	272	10	)	)	PUNCT
ejpam-6211	272	11	(	(	PUNCT
ejpam-6211	272	12	(	(	PUNCT
ejpam-6211	272	13	k	k	X
ejpam-6211	272	14	−	−	PROPN
ejpam-6211	272	15	i	i	PROPN
ejpam-6211	272	16	)	)	PUNCT
ejpam-6211	272	17	+	+	PUNCT
ejpam-6211	272	18	r)n	r)n	X
ejpam-6211	272	19	}	}	PUNCT
ejpam-6211	272	20	tn	tn	PROPN
ejpam-6211	272	21	n	n	NOUN
ejpam-6211	272	22	!	!	PUNCT
ejpam-6211	272	23	)	)	PUNCT
ejpam-6211	273	1	(	(	PUNCT
ejpam-6211	273	2	∞∑	∞∑	NUM
ejpam-6211	273	3	n=0	n=0	NUM
ejpam-6211	273	4	fn	fn	NOUN
ejpam-6211	273	5	tn	tn	NOUN
ejpam-6211	273	6	n	n	NOUN
ejpam-6211	273	7	!	!	PUNCT
ejpam-6211	273	8	)	)	PUNCT
ejpam-6211	274	1	=	=	PUNCT
ejpam-6211	274	2	(	(	PUNCT
ejpam-6211	274	3	∞∑	∞∑	PROPN
ejpam-6211	274	4	n=0	n=0	PROPN
ejpam-6211	274	5	k	k	X
ejpam-6211	274	6	!	!	PUNCT
ejpam-6211	274	7	{	{	PUNCT
ejpam-6211	275	1	n+	n+	ADP
ejpam-6211	275	2	r	r	NOUN
ejpam-6211	275	3	k	k	NOUN
ejpam-6211	276	1	+	+	CCONJ
ejpam-6211	276	2	r	r	NOUN
ejpam-6211	276	3	}	}	PUNCT
ejpam-6211	276	4	r	r	NOUN
ejpam-6211	276	5	tn	tn	NOUN
ejpam-6211	276	6	n	n	CCONJ
ejpam-6211	276	7	!	!	PUNCT
ejpam-6211	276	8	)	)	PUNCT
ejpam-6211	277	1	(	(	PUNCT
ejpam-6211	277	2	∞∑	∞∑	NUM
ejpam-6211	277	3	n=0	n=0	NUM
ejpam-6211	277	4	fn	fn	NOUN
ejpam-6211	277	5	tn	tn	NOUN
ejpam-6211	277	6	n	n	PROPN
ejpam-6211	277	7	!	!	PUNCT
ejpam-6211	277	8	)	)	PUNCT
ejpam-6211	277	9	by	by	ADP
ejpam-6211	277	10	cauchy	cauchy	NOUN
ejpam-6211	277	11	product	product	NOUN
ejpam-6211	277	12	we	we	PRON
ejpam-6211	277	13	have	have	VERB
ejpam-6211	277	14	,	,	PUNCT
ejpam-6211	277	15	sf	sf	PROPN
ejpam-6211	277	16	2	2	NUM
ejpam-6211	277	17	n	n	NOUN
ejpam-6211	277	18	(	(	PUNCT
ejpam-6211	277	19	k	k	NOUN
ejpam-6211	277	20	;	;	PUNCT
ejpam-6211	277	21	r	r	X
ejpam-6211	277	22	,	,	PUNCT
ejpam-6211	277	23	x	x	NOUN
ejpam-6211	277	24	)	)	PUNCT
ejpam-6211	277	25	=	=	SYM
ejpam-6211	277	26	n∑	n∑	PROPN
ejpam-6211	277	27	m=0	m=0	PROPN
ejpam-6211	277	28	{	{	PUNCT
ejpam-6211	278	1	m+	m+	NOUN
ejpam-6211	278	2	r	r	NOUN
ejpam-6211	278	3	k	k	PROPN
ejpam-6211	279	1	+	+	CCONJ
ejpam-6211	279	2	r	r	NOUN
ejpam-6211	279	3	}	}	PUNCT
ejpam-6211	279	4	r	r	NOUN
ejpam-6211	279	5	(	(	PUNCT
ejpam-6211	279	6	n	n	NOUN
ejpam-6211	279	7	k	k	NOUN
ejpam-6211	279	8	)	)	PUNCT
ejpam-6211	279	9	fn−m	fn−m	NOUN
ejpam-6211	279	10	4	4	NUM
ejpam-6211	279	11	.	.	PUNCT
ejpam-6211	280	1	formulas	formula	NOUN
ejpam-6211	280	2	r	r	NOUN
ejpam-6211	280	3	-	-	PUNCT
ejpam-6211	280	4	stirling	stirling	NOUN
ejpam-6211	280	5	fibonacci	fibonacci	NOUN
ejpam-6211	280	6	polynomials	polynomial	NOUN
ejpam-6211	280	7	in	in	ADP
ejpam-6211	280	8	this	this	DET
ejpam-6211	280	9	section	section	NOUN
ejpam-6211	280	10	,	,	PUNCT
ejpam-6211	280	11	we	we	PRON
ejpam-6211	280	12	introduce	introduce	VERB
ejpam-6211	280	13	and	and	CCONJ
ejpam-6211	280	14	derive	derive	VERB
ejpam-6211	280	15	various	various	ADJ
ejpam-6211	280	16	identities	identity	NOUN
ejpam-6211	280	17	and	and	CCONJ
ejpam-6211	280	18	properties	property	NOUN
ejpam-6211	280	19	associated	associate	VERB
ejpam-6211	280	20	with	with	ADP
ejpam-6211	280	21	the	the	DET
ejpam-6211	280	22	r	r	NOUN
ejpam-6211	280	23	-	-	PUNCT
ejpam-6211	280	24	stirling	stirling	NOUN
ejpam-6211	280	25	fibonacci	fibonacci	NOUN
ejpam-6211	280	26	polynomials	polynomial	NOUN
ejpam-6211	280	27	of	of	ADP
ejpam-6211	280	28	the	the	DET
ejpam-6211	280	29	first	first	ADJ
ejpam-6211	280	30	and	and	CCONJ
ejpam-6211	280	31	second	second	ADJ
ejpam-6211	280	32	kinds	kind	NOUN
ejpam-6211	280	33	.	.	PUNCT
ejpam-6211	281	1	these	these	DET
ejpam-6211	281	2	polynomials	polynomial	NOUN
ejpam-6211	281	3	are	be	AUX
ejpam-6211	281	4	generalizations	generalization	NOUN
ejpam-6211	281	5	that	that	PRON
ejpam-6211	281	6	intertwine	intertwine	VERB
ejpam-6211	281	7	the	the	DET
ejpam-6211	281	8	concepts	concept	NOUN
ejpam-6211	281	9	of	of	ADP
ejpam-6211	281	10	r	r	NOUN
ejpam-6211	281	11	-	-	PUNCT
ejpam-6211	281	12	stirling	stirling	NOUN
ejpam-6211	281	13	numbers	number	NOUN
ejpam-6211	281	14	and	and	CCONJ
ejpam-6211	281	15	classical	classical	ADJ
ejpam-6211	281	16	fibonacci	fibonacci	NOUN
ejpam-6211	281	17	polynomials	polynomial	NOUN
ejpam-6211	281	18	.	.	PUNCT
ejpam-6211	282	1	their	their	PRON
ejpam-6211	282	2	structure	structure	NOUN
ejpam-6211	282	3	is	be	AUX
ejpam-6211	282	4	made	make	VERB
ejpam-6211	282	5	explicit	explicit	ADJ
ejpam-6211	282	6	through	through	ADP
ejpam-6211	282	7	exponential	exponential	ADJ
ejpam-6211	282	8	generating	generating	NOUN
ejpam-6211	282	9	functions	function	NOUN
ejpam-6211	282	10	and	and	CCONJ
ejpam-6211	282	11	convolution	convolution	NOUN
ejpam-6211	282	12	-	-	PUNCT
ejpam-6211	282	13	type	type	NOUN
ejpam-6211	282	14	formulas	formula	NOUN
ejpam-6211	282	15	that	that	PRON
ejpam-6211	282	16	extend	extend	VERB
ejpam-6211	282	17	known	know	VERB
ejpam-6211	282	18	results	result	NOUN
ejpam-6211	282	19	in	in	ADP
ejpam-6211	282	20	combinatorics	combinatoric	NOUN
ejpam-6211	282	21	and	and	CCONJ
ejpam-6211	282	22	special	special	ADJ
ejpam-6211	282	23	functions	function	NOUN
ejpam-6211	282	24	.	.	PUNCT
ejpam-6211	283	1	we	we	PRON
ejpam-6211	283	2	also	also	ADV
ejpam-6211	283	3	establish	establish	VERB
ejpam-6211	283	4	their	their	PRON
ejpam-6211	283	5	horizontal	horizontal	ADJ
ejpam-6211	283	6	generating	generating	NOUN
ejpam-6211	283	7	functions	function	NOUN
ejpam-6211	283	8	,	,	PUNCT
ejpam-6211	283	9	a	a	DET
ejpam-6211	283	10	schlömilchtype	schlömilchtype	NOUN
ejpam-6211	283	11	expansion	expansion	NOUN
ejpam-6211	283	12	,	,	PUNCT
ejpam-6211	283	13	and	and	CCONJ
ejpam-6211	283	14	an	an	DET
ejpam-6211	283	15	explicit	explicit	ADJ
ejpam-6211	283	16	representation	representation	NOUN
ejpam-6211	283	17	,	,	PUNCT
ejpam-6211	283	18	all	all	PRON
ejpam-6211	283	19	of	of	ADP
ejpam-6211	283	20	which	which	PRON
ejpam-6211	283	21	showcase	showcase	VERB
ejpam-6211	283	22	the	the	DET
ejpam-6211	283	23	richness	richness	NOUN
ejpam-6211	283	24	and	and	CCONJ
ejpam-6211	283	25	applicability	applicability	NOUN
ejpam-6211	283	26	of	of	ADP
ejpam-6211	283	27	these	these	DET
ejpam-6211	283	28	polynomials	polynomial	NOUN
ejpam-6211	283	29	in	in	ADP
ejpam-6211	283	30	combinatorial	combinatorial	ADJ
ejpam-6211	283	31	analysis	analysis	NOUN
ejpam-6211	283	32	.	.	PUNCT
ejpam-6211	284	1	definition	definition	NOUN
ejpam-6211	284	2	4.1	4.1	NUM
ejpam-6211	284	3	.	.	PUNCT
ejpam-6211	285	1	the	the	DET
ejpam-6211	285	2	r	r	NOUN
ejpam-6211	285	3	-	-	PUNCT
ejpam-6211	285	4	stirling	stirling	NOUN
ejpam-6211	285	5	fibonacci	fibonacci	NOUN
ejpam-6211	285	6	polynomials	polynomial	NOUN
ejpam-6211	285	7	of	of	ADP
ejpam-6211	285	8	the	the	DET
ejpam-6211	285	9	first	first	ADJ
ejpam-6211	285	10	and	and	CCONJ
ejpam-6211	285	11	second	second	ADJ
ejpam-6211	285	12	kinds	kind	NOUN
ejpam-6211	285	13	are	be	AUX
ejpam-6211	285	14	respectively	respectively	ADV
ejpam-6211	285	15	defined	define	VERB
ejpam-6211	285	16	by	by	ADP
ejpam-6211	285	17	means	mean	NOUN
ejpam-6211	285	18	of	of	ADP
ejpam-6211	285	19	the	the	DET
ejpam-6211	285	20	following	follow	VERB
ejpam-6211	285	21	exponential	exponential	ADJ
ejpam-6211	285	22	generating	generating	NOUN
ejpam-6211	285	23	functions	function	NOUN
ejpam-6211	285	24	:	:	PUNCT
ejpam-6211	285	25	ϕ(t	ϕ(t	NUM
ejpam-6211	285	26	,	,	PUNCT
ejpam-6211	285	27	x	x	X
ejpam-6211	285	28	)	)	PUNCT
ejpam-6211	285	29	=	=	SYM
ejpam-6211	286	1	∞∑	∞∑	NUM
ejpam-6211	286	2	n=0	n=0	NUM
ejpam-6211	286	3	sf	sf	NOUN
ejpam-6211	286	4	1	1	NUM
ejpam-6211	286	5	n	n	CCONJ
ejpam-6211	286	6	(	(	PUNCT
ejpam-6211	286	7	k	k	NOUN
ejpam-6211	286	8	;	;	PUNCT
ejpam-6211	286	9	r	r	X
ejpam-6211	286	10	,	,	PUNCT
ejpam-6211	286	11	x	x	NOUN
ejpam-6211	286	12	)	)	PUNCT
ejpam-6211	286	13	tn	tn	PROPN
ejpam-6211	286	14	n	n	NOUN
ejpam-6211	286	15	!	!	PUNCT
ejpam-6211	286	16	=	=	PUNCT
ejpam-6211	287	1	(	(	PUNCT
ejpam-6211	287	2	1	1	NUM
ejpam-6211	287	3	1	1	NUM
ejpam-6211	287	4	+	+	NUM
ejpam-6211	287	5	t	t	NOUN
ejpam-6211	287	6	)	)	PUNCT
ejpam-6211	287	7	r	r	NOUN
ejpam-6211	287	8	(	(	PUNCT
ejpam-6211	287	9	2e	2e	NUM
ejpam-6211	287	10	xt	xt	NUM
ejpam-6211	287	11	2	2	NUM
ejpam-6211	287	12	sinh	sinh	NOUN
ejpam-6211	287	13	(	(	PUNCT
ejpam-6211	287	14	√	√	PROPN
ejpam-6211	287	15	x2	x2	PROPN
ejpam-6211	287	16	+	+	PROPN
ejpam-6211	287	17	4	4	NUM
ejpam-6211	287	18	2	2	NUM
ejpam-6211	287	19	t	t	NOUN
ejpam-6211	287	20	)	)	PUNCT
ejpam-6211	287	21	)	)	PUNCT
ejpam-6211	288	1	(	(	PUNCT
ejpam-6211	288	2	lnk	lnk	NOUN
ejpam-6211	288	3	(	(	PUNCT
ejpam-6211	288	4	1	1	NUM
ejpam-6211	288	5	+	+	NUM
ejpam-6211	288	6	t	t	PROPN
ejpam-6211	288	7	)	)	PUNCT
ejpam-6211	288	8	)	)	PUNCT
ejpam-6211	289	1	k	k	X
ejpam-6211	289	2	!	!	PUNCT
ejpam-6211	290	1	(	(	PUNCT
ejpam-6211	290	2	√	√	NUM
ejpam-6211	290	3	x2	x2	NOUN
ejpam-6211	291	1	+	+	CCONJ
ejpam-6211	291	2	4	4	NUM
ejpam-6211	291	3	)	)	PUNCT
ejpam-6211	291	4	(	(	PUNCT
ejpam-6211	291	5	4.1	4.1	NUM
ejpam-6211	291	6	)	)	PUNCT
ejpam-6211	292	1	γ(t	γ(t	NOUN
ejpam-6211	292	2	,	,	PUNCT
ejpam-6211	292	3	x	x	X
ejpam-6211	292	4	)	)	PUNCT
ejpam-6211	292	5	=	=	SYM
ejpam-6211	293	1	∞∑	∞∑	NUM
ejpam-6211	293	2	n=0	n=0	PUNCT
ejpam-6211	293	3	sf	sf	NOUN
ejpam-6211	293	4	2	2	NUM
ejpam-6211	293	5	n	n	NOUN
ejpam-6211	293	6	(	(	PUNCT
ejpam-6211	293	7	k	k	NOUN
ejpam-6211	293	8	;	;	PUNCT
ejpam-6211	293	9	r	r	X
ejpam-6211	293	10	,	,	PUNCT
ejpam-6211	293	11	x	x	NOUN
ejpam-6211	293	12	)	)	PUNCT
ejpam-6211	293	13	tn	tn	PROPN
ejpam-6211	293	14	n	n	NOUN
ejpam-6211	293	15	!	!	PUNCT
ejpam-6211	294	1	=	=	PUNCT
ejpam-6211	294	2	(	(	PUNCT
ejpam-6211	294	3	2e	2e	NUM
ejpam-6211	294	4	xt	xt	NUM
ejpam-6211	294	5	2	2	NUM
ejpam-6211	294	6	sinh	sinh	NOUN
ejpam-6211	294	7	(	(	PUNCT
ejpam-6211	294	8	√	√	PROPN
ejpam-6211	294	9	x2	x2	PROPN
ejpam-6211	294	10	+	+	PROPN
ejpam-6211	294	11	4	4	NUM
ejpam-6211	294	12	2	2	NUM
ejpam-6211	294	13	t	t	NOUN
ejpam-6211	294	14	)	)	PUNCT
ejpam-6211	294	15	)	)	PUNCT
ejpam-6211	294	16	ert(et	ert(et	X
ejpam-6211	294	17	−	−	PROPN
ejpam-6211	294	18	1)k	1)k	NUM
ejpam-6211	294	19	k	k	NOUN
ejpam-6211	294	20	!	!	PUNCT
ejpam-6211	295	1	(	(	PUNCT
ejpam-6211	295	2	√	√	NUM
ejpam-6211	295	3	x2	x2	NOUN
ejpam-6211	296	1	+	+	CCONJ
ejpam-6211	296	2	4	4	NUM
ejpam-6211	296	3	)	)	PUNCT
ejpam-6211	296	4	(	(	PUNCT
ejpam-6211	296	5	4.2	4.2	NUM
ejpam-6211	296	6	)	)	PUNCT
ejpam-6211	296	7	14	14	NUM
ejpam-6211	296	8	of	of	ADP
ejpam-6211	296	9	23	23	NUM
ejpam-6211	296	10	theorem	theorem	VERB
ejpam-6211	296	11	4.2	4.2	NUM
ejpam-6211	296	12	.	.	PUNCT
ejpam-6211	297	1	the	the	DET
ejpam-6211	297	2	r	r	NOUN
ejpam-6211	297	3	-	-	PUNCT
ejpam-6211	297	4	stirling	stirling	NOUN
ejpam-6211	297	5	fibonacci	fibonacci	NOUN
ejpam-6211	297	6	polynomial	polynomial	NOUN
ejpam-6211	297	7	of	of	ADP
ejpam-6211	297	8	the	the	DET
ejpam-6211	297	9	first	first	ADJ
ejpam-6211	297	10	and	and	CCONJ
ejpam-6211	297	11	second	second	ADJ
ejpam-6211	297	12	kind	kind	NOUN
ejpam-6211	297	13	satisfy	satisfy	VERB
ejpam-6211	297	14	the	the	DET
ejpam-6211	297	15	convolution	convolution	NOUN
ejpam-6211	297	16	formula	formula	NOUN
ejpam-6211	297	17	respectively	respectively	ADV
ejpam-6211	297	18	;	;	PUNCT
ejpam-6211	297	19	sf	sf	PROPN
ejpam-6211	297	20	1	1	NUM
ejpam-6211	297	21	n(k	n(k	PROPN
ejpam-6211	297	22	;	;	PUNCT
ejpam-6211	297	23	r	r	NOUN
ejpam-6211	297	24	,	,	PUNCT
ejpam-6211	297	25	x	x	NOUN
ejpam-6211	297	26	)	)	PUNCT
ejpam-6211	297	27	=	=	PUNCT
ejpam-6211	297	28	n∑	n∑	NOUN
ejpam-6211	297	29	m	m	PROPN
ejpam-6211	297	30	=	=	PROPN
ejpam-6211	297	31	k	k	X
ejpam-6211	298	1	̂[m+	̂[m+	NOUN
ejpam-6211	299	1	r	r	NOUN
ejpam-6211	299	2	k	k	NOUN
ejpam-6211	300	1	+	+	CCONJ
ejpam-6211	300	2	r	r	NOUN
ejpam-6211	300	3	]	]	X
ejpam-6211	300	4	r	r	NOUN
ejpam-6211	300	5	(	(	PUNCT
ejpam-6211	300	6	n	n	NOUN
ejpam-6211	300	7	m	m	VERB
ejpam-6211	300	8	)	)	PUNCT
ejpam-6211	300	9	fn−m(x	fn−m(x	X
ejpam-6211	300	10	)	)	PUNCT
ejpam-6211	300	11	(	(	PUNCT
ejpam-6211	300	12	4.3	4.3	NUM
ejpam-6211	300	13	)	)	PUNCT
ejpam-6211	300	14	sf	sf	NOUN
ejpam-6211	300	15	2	2	NUM
ejpam-6211	300	16	n(k	n(k	PROPN
ejpam-6211	300	17	;	;	PUNCT
ejpam-6211	300	18	r	r	NOUN
ejpam-6211	300	19	,	,	PUNCT
ejpam-6211	300	20	x	x	NOUN
ejpam-6211	300	21	)	)	PUNCT
ejpam-6211	300	22	=	=	PUNCT
ejpam-6211	301	1	n∑	n∑	NOUN
ejpam-6211	301	2	m	m	PROPN
ejpam-6211	302	1	=	=	VERB
ejpam-6211	302	2	k	k	X
ejpam-6211	302	3	{	{	PUNCT
ejpam-6211	302	4	m+	m+	NOUN
ejpam-6211	302	5	r	r	NOUN
ejpam-6211	302	6	k	k	PROPN
ejpam-6211	302	7	+	+	CCONJ
ejpam-6211	302	8	r	r	NOUN
ejpam-6211	302	9	}	}	PUNCT
ejpam-6211	302	10	r	r	NOUN
ejpam-6211	302	11	(	(	PUNCT
ejpam-6211	302	12	n	n	NOUN
ejpam-6211	302	13	m	m	VERB
ejpam-6211	302	14	)	)	PUNCT
ejpam-6211	302	15	fn−m(x	fn−m(x	X
ejpam-6211	302	16	)	)	PUNCT
ejpam-6211	302	17	(	(	PUNCT
ejpam-6211	302	18	4.4	4.4	NUM
ejpam-6211	302	19	)	)	PUNCT
ejpam-6211	302	20	where	where	SCONJ
ejpam-6211	302	21	n	n	PRON
ejpam-6211	302	22	≥	≥	X
ejpam-6211	302	23	k	k	NOUN
ejpam-6211	302	24	,	,	PUNCT
ejpam-6211	302	25	otherwise	otherwise	ADV
ejpam-6211	302	26	sf	sf	PROPN
ejpam-6211	302	27	1	1	NUM
ejpam-6211	302	28	n(k	n(k	PROPN
ejpam-6211	302	29	;	;	PUNCT
ejpam-6211	302	30	r	r	NOUN
ejpam-6211	302	31	,	,	PUNCT
ejpam-6211	302	32	x	x	NOUN
ejpam-6211	302	33	)	)	PUNCT
ejpam-6211	303	1	=	=	SYM
ejpam-6211	303	2	sf	sf	PROPN
ejpam-6211	303	3	2	2	NUM
ejpam-6211	303	4	n(k	n(k	PROPN
ejpam-6211	303	5	;	;	PUNCT
ejpam-6211	303	6	r	r	NOUN
ejpam-6211	303	7	,	,	PUNCT
ejpam-6211	303	8	x	x	NOUN
ejpam-6211	303	9	)	)	PUNCT
ejpam-6211	303	10	=	=	SYM
ejpam-6211	303	11	0	0	X
ejpam-6211	303	12	.	.	PUNCT
ejpam-6211	304	1	proof	proof	NOUN
ejpam-6211	304	2	the	the	DET
ejpam-6211	304	3	proof	proof	NOUN
ejpam-6211	304	4	follows	follow	VERB
ejpam-6211	304	5	similarly	similarly	ADV
ejpam-6211	304	6	to	to	PART
ejpam-6211	304	7	theorem	theorem	VERB
ejpam-6211	304	8	2.3	2.3	NUM
ejpam-6211	304	9	.	.	PUNCT
ejpam-6211	304	10	theorem	theorem	VERB
ejpam-6211	304	11	4.3	4.3	NUM
ejpam-6211	304	12	.	.	PUNCT
ejpam-6211	305	1	the	the	DET
ejpam-6211	305	2	horizontal	horizontal	ADJ
ejpam-6211	305	3	generating	generate	VERB
ejpam-6211	305	4	function	function	NOUN
ejpam-6211	305	5	of	of	ADP
ejpam-6211	305	6	the	the	DET
ejpam-6211	305	7	r	r	NOUN
ejpam-6211	305	8	-	-	PUNCT
ejpam-6211	305	9	stirling	stirling	NOUN
ejpam-6211	305	10	fibonacci	fibonacci	NOUN
ejpam-6211	305	11	polynomials	polynomial	NOUN
ejpam-6211	305	12	of	of	ADP
ejpam-6211	305	13	the	the	DET
ejpam-6211	305	14	first	first	ADJ
ejpam-6211	305	15	and	and	CCONJ
ejpam-6211	305	16	second	second	ADJ
ejpam-6211	305	17	kind	kind	NOUN
ejpam-6211	305	18	is	be	AUX
ejpam-6211	305	19	given	give	VERB
ejpam-6211	305	20	by	by	ADP
ejpam-6211	305	21	respectively	respectively	ADV
ejpam-6211	305	22	;	;	PUNCT
ejpam-6211	305	23	∞∑	∞∑	NUM
ejpam-6211	305	24	n=0	n=0	NUM
ejpam-6211	305	25	sf	sf	VERB
ejpam-6211	305	26	1	1	NUM
ejpam-6211	305	27	n	n	CCONJ
ejpam-6211	305	28	(	(	PUNCT
ejpam-6211	305	29	k	k	NOUN
ejpam-6211	305	30	;	;	PUNCT
ejpam-6211	305	31	r	r	X
ejpam-6211	305	32	,	,	PUNCT
ejpam-6211	305	33	x	x	NOUN
ejpam-6211	305	34	)	)	PUNCT
ejpam-6211	305	35	zk	zk	PROPN
ejpam-6211	306	1	=	=	SYM
ejpam-6211	306	2	f	f	PROPN
ejpam-6211	306	3	(	(	PUNCT
ejpam-6211	306	4	3	3	NUM
ejpam-6211	306	5	)	)	PUNCT
ejpam-6211	306	6	n	n	CCONJ
ejpam-6211	306	7	(	(	PUNCT
ejpam-6211	306	8	x+	x+	X
ejpam-6211	306	9	z	z	NOUN
ejpam-6211	306	10	−	−	NOUN
ejpam-6211	306	11	r	r	NOUN
ejpam-6211	306	12	)	)	PUNCT
ejpam-6211	306	13	(	(	PUNCT
ejpam-6211	306	14	4.5	4.5	NUM
ejpam-6211	306	15	)	)	PUNCT
ejpam-6211	306	16	∞∑	∞∑	PROPN
ejpam-6211	306	17	n=0	n=0	PUNCT
ejpam-6211	306	18	sf	sf	NOUN
ejpam-6211	306	19	2	2	NUM
ejpam-6211	306	20	n	n	NOUN
ejpam-6211	306	21	(	(	PUNCT
ejpam-6211	306	22	k	k	NOUN
ejpam-6211	306	23	;	;	PUNCT
ejpam-6211	306	24	r	r	X
ejpam-6211	306	25	,	,	PUNCT
ejpam-6211	306	26	x	x	NOUN
ejpam-6211	306	27	)	)	PUNCT
ejpam-6211	306	28	zk	zk	PROPN
ejpam-6211	307	1	=	=	SYM
ejpam-6211	307	2	f	f	PROPN
ejpam-6211	307	3	(	(	PUNCT
ejpam-6211	307	4	2	2	NUM
ejpam-6211	307	5	)	)	PUNCT
ejpam-6211	307	6	n	n	CCONJ
ejpam-6211	307	7	(	(	PUNCT
ejpam-6211	307	8	x+	x+	X
ejpam-6211	307	9	z	z	NOUN
ejpam-6211	307	10	+	+	CCONJ
ejpam-6211	307	11	r	r	X
ejpam-6211	307	12	)	)	PUNCT
ejpam-6211	307	13	(	(	PUNCT
ejpam-6211	307	14	4.6	4.6	NUM
ejpam-6211	307	15	)	)	PUNCT
ejpam-6211	307	16	proof	proof	NOUN
ejpam-6211	307	17	.	.	PUNCT
ejpam-6211	308	1	the	the	DET
ejpam-6211	308	2	proof	proof	NOUN
ejpam-6211	308	3	follows	follow	VERB
ejpam-6211	308	4	similarly	similarly	ADV
ejpam-6211	308	5	to	to	PART
ejpam-6211	308	6	theorem	theorem	VERB
ejpam-6211	308	7	2.4	2.4	NUM
ejpam-6211	308	8	.	.	PUNCT
ejpam-6211	309	1	theorem	theorem	VERB
ejpam-6211	309	2	4.4	4.4	NUM
ejpam-6211	309	3	.	.	PUNCT
ejpam-6211	310	1	the	the	DET
ejpam-6211	310	2	schlömilch	schlömilch	ADJ
ejpam-6211	310	3	-	-	ADJ
ejpam-6211	310	4	type	type	NOUN
ejpam-6211	310	5	formula	formula	NOUN
ejpam-6211	310	6	of	of	ADP
ejpam-6211	310	7	the	the	DET
ejpam-6211	310	8	r	r	NOUN
ejpam-6211	310	9	-	-	PUNCT
ejpam-6211	310	10	stirling	stirling	NOUN
ejpam-6211	310	11	fibonacci	fibonacci	NOUN
ejpam-6211	310	12	polynomial	polynomial	NOUN
ejpam-6211	310	13	is	be	AUX
ejpam-6211	310	14	given	give	VERB
ejpam-6211	310	15	by	by	ADP
ejpam-6211	310	16	sf	sf	PROPN
ejpam-6211	310	17	1	1	NUM
ejpam-6211	310	18	n(k	n(k	PROPN
ejpam-6211	310	19	;	;	PUNCT
ejpam-6211	310	20	r	r	NOUN
ejpam-6211	310	21	,	,	PUNCT
ejpam-6211	310	22	x	x	NOUN
ejpam-6211	310	23	)	)	PUNCT
ejpam-6211	310	24	=	=	PUNCT
ejpam-6211	310	25	n∑	n∑	NOUN
ejpam-6211	310	26	m	m	PROPN
ejpam-6211	310	27	=	=	AUX
ejpam-6211	310	28	k	k	X
ejpam-6211	310	29	m∑	m∑	ADV
ejpam-6211	310	30	j=0	j=0	PROPN
ejpam-6211	310	31	m−k∑	m−k∑	X
ejpam-6211	311	1	a=0	a=0	PRON
ejpam-6211	311	2	a∑	a∑	PROPN
ejpam-6211	311	3	b=0	b=0	PROPN
ejpam-6211	311	4	(	(	PUNCT
ejpam-6211	311	5	−1)n−j+b+a	−1)n−j+b+a	PROPN
ejpam-6211	311	6	(	(	PUNCT
ejpam-6211	311	7	a	a	DET
ejpam-6211	311	8	b	b	NOUN
ejpam-6211	311	9	)	)	PUNCT
ejpam-6211	311	10	(	(	PUNCT
ejpam-6211	311	11	n	n	CCONJ
ejpam-6211	311	12	j	j	NOUN
ejpam-6211	311	13	)	)	PUNCT
ejpam-6211	311	14	(	(	PUNCT
ejpam-6211	311	15	j	j	PROPN
ejpam-6211	311	16	m	m	VERB
ejpam-6211	311	17	)	)	PUNCT
ejpam-6211	311	18	fj−m(x	fj−m(x	PROPN
ejpam-6211	311	19	)	)	PUNCT
ejpam-6211	311	20	(	(	PUNCT
ejpam-6211	311	21	4.7	4.7	NUM
ejpam-6211	311	22	)	)	PUNCT
ejpam-6211	311	23	×	×	NOUN
ejpam-6211	311	24	(	(	PUNCT
ejpam-6211	311	25	m−	m−	PROPN
ejpam-6211	311	26	1	1	NUM
ejpam-6211	311	27	+	+	CCONJ
ejpam-6211	311	28	a	a	DET
ejpam-6211	311	29	m−	m−	PROPN
ejpam-6211	311	30	k	k	PROPN
ejpam-6211	311	31	+	+	CCONJ
ejpam-6211	311	32	a	a	X
ejpam-6211	311	33	)	)	PUNCT
ejpam-6211	311	34	(	(	PUNCT
ejpam-6211	312	1	2m−	2m−	NUM
ejpam-6211	312	2	k	k	NOUN
ejpam-6211	312	3	m−	m−	PROPN
ejpam-6211	313	1	k	k	PROPN
ejpam-6211	313	2	−	−	PROPN
ejpam-6211	313	3	a	a	X
ejpam-6211	313	4	)	)	PUNCT
ejpam-6211	313	5	(	(	PUNCT
ejpam-6211	313	6	a−	a−	PROPN
ejpam-6211	313	7	b)m−k+a	b)m−k+a	PROPN
ejpam-6211	313	8	a	a	PRON
ejpam-6211	313	9	!	!	PUNCT
ejpam-6211	314	1	rn−j	rn−j	NOUN
ejpam-6211	314	2	proof	proof	NOUN
ejpam-6211	314	3	the	the	DET
ejpam-6211	314	4	proof	proof	NOUN
ejpam-6211	314	5	follows	follow	VERB
ejpam-6211	314	6	similarly	similarly	ADV
ejpam-6211	314	7	to	to	PART
ejpam-6211	314	8	theorem	theorem	VERB
ejpam-6211	314	9	2.5	2.5	NUM
ejpam-6211	314	10	.	.	PUNCT
ejpam-6211	315	1	theorem	theorem	VERB
ejpam-6211	315	2	4.5	4.5	NUM
ejpam-6211	315	3	.	.	PUNCT
ejpam-6211	316	1	the	the	DET
ejpam-6211	316	2	explicit	explicit	ADJ
ejpam-6211	316	3	formula	formula	NOUN
ejpam-6211	316	4	of	of	ADP
ejpam-6211	316	5	the	the	DET
ejpam-6211	316	6	r	r	NOUN
ejpam-6211	316	7	-	-	PUNCT
ejpam-6211	316	8	stirling	stirling	NOUN
ejpam-6211	316	9	fibonacci	fibonacci	NOUN
ejpam-6211	316	10	polynomial	polynomial	NOUN
ejpam-6211	316	11	of	of	ADP
ejpam-6211	316	12	the	the	DET
ejpam-6211	316	13	second	second	ADJ
ejpam-6211	316	14	kind	kind	NOUN
ejpam-6211	316	15	is	be	AUX
ejpam-6211	316	16	given	give	VERB
ejpam-6211	316	17	by	by	ADP
ejpam-6211	316	18	sf	sf	PROPN
ejpam-6211	316	19	2	2	NUM
ejpam-6211	316	20	n	n	NOUN
ejpam-6211	316	21	(	(	PUNCT
ejpam-6211	316	22	k	k	NOUN
ejpam-6211	316	23	;	;	PUNCT
ejpam-6211	316	24	r	r	X
ejpam-6211	316	25	,	,	PUNCT
ejpam-6211	316	26	x	x	NOUN
ejpam-6211	316	27	)	)	PUNCT
ejpam-6211	316	28	=	=	SYM
ejpam-6211	316	29	n∑	n∑	PROPN
ejpam-6211	316	30	m=0	m=0	PROPN
ejpam-6211	316	31	{	{	PUNCT
ejpam-6211	316	32	n+	n+	ADP
ejpam-6211	316	33	r	r	NOUN
ejpam-6211	316	34	k	k	NOUN
ejpam-6211	317	1	+	+	CCONJ
ejpam-6211	317	2	r	r	NOUN
ejpam-6211	317	3	}	}	PUNCT
ejpam-6211	317	4	r	r	NOUN
ejpam-6211	317	5	(	(	PUNCT
ejpam-6211	317	6	n	n	NOUN
ejpam-6211	317	7	m	m	VERB
ejpam-6211	317	8	)	)	PUNCT
ejpam-6211	317	9	fn−m(x	fn−m(x	X
ejpam-6211	317	10	)	)	PUNCT
ejpam-6211	317	11	(	(	PUNCT
ejpam-6211	317	12	4.8	4.8	NUM
ejpam-6211	317	13	)	)	PUNCT
ejpam-6211	317	14	proof	proof	NOUN
ejpam-6211	317	15	the	the	DET
ejpam-6211	317	16	proof	proof	NOUN
ejpam-6211	317	17	follows	follow	VERB
ejpam-6211	317	18	similarly	similarly	ADV
ejpam-6211	317	19	to	to	PART
ejpam-6211	317	20	theorem	theorem	VERB
ejpam-6211	317	21	2.6	2.6	NUM
ejpam-6211	317	22	.	.	NOUN
ejpam-6211	317	23	15	15	NUM
ejpam-6211	317	24	of	of	ADP
ejpam-6211	317	25	23	23	NUM
ejpam-6211	317	26	5	5	NUM
ejpam-6211	317	27	.	.	PUNCT
ejpam-6211	317	28	r	r	X
ejpam-6211	317	29	-	-	PUNCT
ejpam-6211	317	30	stirling	stirling	NOUN
ejpam-6211	317	31	fibonacci	fibonacci	NOUN
ejpam-6211	317	32	polynomials	polynomial	VERB
ejpam-6211	317	33	identities	identity	NOUN
ejpam-6211	317	34	via	via	ADP
ejpam-6211	317	35	exponential	exponential	ADJ
ejpam-6211	317	36	generating	generating	NOUN
ejpam-6211	317	37	function	function	NOUN
ejpam-6211	317	38	in	in	ADP
ejpam-6211	317	39	this	this	DET
ejpam-6211	317	40	section	section	NOUN
ejpam-6211	317	41	,	,	PUNCT
ejpam-6211	317	42	we	we	PRON
ejpam-6211	317	43	derive	derive	VERB
ejpam-6211	317	44	new	new	ADJ
ejpam-6211	317	45	identities	identity	NOUN
ejpam-6211	317	46	for	for	ADP
ejpam-6211	317	47	the	the	DET
ejpam-6211	317	48	r	r	NOUN
ejpam-6211	317	49	-	-	PUNCT
ejpam-6211	317	50	stirling	stirling	NOUN
ejpam-6211	317	51	fibonacci	fibonacci	NOUN
ejpam-6211	317	52	polynomials	polynomial	NOUN
ejpam-6211	317	53	by	by	ADP
ejpam-6211	317	54	employing	employ	VERB
ejpam-6211	317	55	exponential	exponential	ADJ
ejpam-6211	317	56	generating	generating	NOUN
ejpam-6211	317	57	functions	function	NOUN
ejpam-6211	317	58	.	.	PUNCT
ejpam-6211	318	1	these	these	DET
ejpam-6211	318	2	identities	identity	NOUN
ejpam-6211	318	3	establish	establish	VERB
ejpam-6211	318	4	explicit	explicit	ADJ
ejpam-6211	318	5	connections	connection	NOUN
ejpam-6211	318	6	between	between	ADP
ejpam-6211	318	7	modified	modify	VERB
ejpam-6211	318	8	stirling	stirling	NOUN
ejpam-6211	318	9	-	-	PUNCT
ejpam-6211	318	10	type	type	NOUN
ejpam-6211	318	11	polynomials	polynomial	NOUN
ejpam-6211	318	12	and	and	CCONJ
ejpam-6211	318	13	fibonacci	fibonacci	NOUN
ejpam-6211	318	14	-	-	PUNCT
ejpam-6211	318	15	type	type	NOUN
ejpam-6211	318	16	structures	structure	NOUN
ejpam-6211	318	17	enriched	enrich	VERB
ejpam-6211	318	18	with	with	ADP
ejpam-6211	318	19	chebyshev	chebyshev	NOUN
ejpam-6211	318	20	polynomial	polynomial	ADJ
ejpam-6211	318	21	components	component	NOUN
ejpam-6211	318	22	.	.	PUNCT
ejpam-6211	319	1	we	we	PRON
ejpam-6211	319	2	begin	begin	VERB
ejpam-6211	319	3	by	by	ADP
ejpam-6211	319	4	introducing	introduce	VERB
ejpam-6211	319	5	two	two	NUM
ejpam-6211	319	6	forms	form	NOUN
ejpam-6211	319	7	of	of	ADP
ejpam-6211	319	8	r	r	NOUN
ejpam-6211	319	9	-	-	PUNCT
ejpam-6211	319	10	stirling	stirling	NOUN
ejpam-6211	319	11	chebyshev	chebyshev	NOUN
ejpam-6211	319	12	polynomials	polynomial	NOUN
ejpam-6211	319	13	of	of	ADP
ejpam-6211	319	14	the	the	DET
ejpam-6211	319	15	first	first	ADJ
ejpam-6211	319	16	and	and	CCONJ
ejpam-6211	319	17	second	second	ADJ
ejpam-6211	319	18	kind	kind	NOUN
ejpam-6211	319	19	,	,	PUNCT
ejpam-6211	319	20	which	which	PRON
ejpam-6211	319	21	serve	serve	VERB
ejpam-6211	319	22	as	as	ADP
ejpam-6211	319	23	essential	essential	ADJ
ejpam-6211	319	24	generating	generating	NOUN
ejpam-6211	319	25	tools	tool	NOUN
ejpam-6211	319	26	in	in	ADP
ejpam-6211	319	27	the	the	DET
ejpam-6211	319	28	subsequent	subsequent	ADJ
ejpam-6211	319	29	derivations	derivation	NOUN
ejpam-6211	319	30	.	.	PUNCT
ejpam-6211	320	1	these	these	DET
ejpam-6211	320	2	definitions	definition	NOUN
ejpam-6211	320	3	form	form	VERB
ejpam-6211	320	4	the	the	DET
ejpam-6211	320	5	foundation	foundation	NOUN
ejpam-6211	320	6	for	for	ADP
ejpam-6211	320	7	constructing	construct	VERB
ejpam-6211	320	8	closed	closed	ADJ
ejpam-6211	320	9	-	-	PUNCT
ejpam-6211	320	10	form	form	NOUN
ejpam-6211	320	11	expressions	expression	NOUN
ejpam-6211	320	12	and	and	CCONJ
ejpam-6211	320	13	recurrence	recurrence	NOUN
ejpam-6211	320	14	-	-	PUNCT
ejpam-6211	320	15	type	type	NOUN
ejpam-6211	320	16	relations	relation	NOUN
ejpam-6211	320	17	that	that	PRON
ejpam-6211	320	18	generalize	generalize	VERB
ejpam-6211	320	19	classical	classical	ADJ
ejpam-6211	320	20	results	result	NOUN
ejpam-6211	320	21	through	through	ADP
ejpam-6211	320	22	the	the	DET
ejpam-6211	320	23	lens	lens	NOUN
ejpam-6211	320	24	of	of	ADP
ejpam-6211	320	25	hyperbolic	hyperbolic	ADJ
ejpam-6211	320	26	and	and	CCONJ
ejpam-6211	320	27	exponential	exponential	ADJ
ejpam-6211	320	28	function	function	NOUN
ejpam-6211	320	29	techniques	technique	NOUN
ejpam-6211	320	30	.	.	PUNCT
ejpam-6211	321	1	the	the	DET
ejpam-6211	321	2	identities	identity	NOUN
ejpam-6211	321	3	are	be	AUX
ejpam-6211	321	4	validated	validate	VERB
ejpam-6211	321	5	using	use	VERB
ejpam-6211	321	6	series	series	NOUN
ejpam-6211	321	7	expansions	expansion	NOUN
ejpam-6211	321	8	and	and	CCONJ
ejpam-6211	321	9	cauchy	cauchy	ADJ
ejpam-6211	321	10	product	product	NOUN
ejpam-6211	321	11	methods	method	NOUN
ejpam-6211	321	12	,	,	PUNCT
ejpam-6211	321	13	culminating	culminate	VERB
ejpam-6211	321	14	in	in	ADP
ejpam-6211	321	15	elegant	elegant	ADJ
ejpam-6211	321	16	representations	representation	NOUN
ejpam-6211	321	17	that	that	PRON
ejpam-6211	321	18	link	link	VERB
ejpam-6211	321	19	special	special	ADJ
ejpam-6211	321	20	polynomial	polynomial	ADJ
ejpam-6211	321	21	sequences	sequence	NOUN
ejpam-6211	321	22	through	through	ADP
ejpam-6211	321	23	combinatorial	combinatorial	ADJ
ejpam-6211	321	24	and	and	CCONJ
ejpam-6211	321	25	analytic	analytic	ADJ
ejpam-6211	321	26	perspectives	perspective	NOUN
ejpam-6211	321	27	.	.	PUNCT
ejpam-6211	322	1	definition	definition	NOUN
ejpam-6211	322	2	5.1	5.1	NUM
ejpam-6211	322	3	.	.	PUNCT
ejpam-6211	323	1	the	the	DET
ejpam-6211	323	2	first	first	ADJ
ejpam-6211	323	3	form	form	NOUN
ejpam-6211	323	4	of	of	ADP
ejpam-6211	323	5	r	r	NOUN
ejpam-6211	323	6	-	-	PUNCT
ejpam-6211	323	7	stirling	stirling	NOUN
ejpam-6211	323	8	chebyshev	chebyshev	NOUN
ejpam-6211	323	9	polynomial	polynomial	NOUN
ejpam-6211	323	10	of	of	ADP
ejpam-6211	323	11	the	the	DET
ejpam-6211	323	12	first	first	ADJ
ejpam-6211	323	13	and	and	CCONJ
ejpam-6211	323	14	second	second	ADJ
ejpam-6211	323	15	kind	kind	NOUN
ejpam-6211	323	16	is	be	AUX
ejpam-6211	323	17	defined	define	VERB
ejpam-6211	323	18	by	by	ADP
ejpam-6211	323	19	;	;	PUNCT
ejpam-6211	323	20	τ1(t	τ1(t	PROPN
ejpam-6211	323	21	,	,	PUNCT
ejpam-6211	323	22	x	x	X
ejpam-6211	323	23	)	)	PUNCT
ejpam-6211	323	24	=	=	SYM
ejpam-6211	324	1	∞∑	∞∑	NUM
ejpam-6211	324	2	n=0	n=0	NUM
ejpam-6211	324	3	st	st	NOUN
ejpam-6211	324	4	1	1	NUM
ejpam-6211	324	5	n	n	PROPN
ejpam-6211	324	6	(	(	PUNCT
ejpam-6211	324	7	k	k	NOUN
ejpam-6211	324	8	;	;	PUNCT
ejpam-6211	324	9	r	r	X
ejpam-6211	324	10	,	,	PUNCT
ejpam-6211	324	11	x	x	NOUN
ejpam-6211	324	12	)	)	PUNCT
ejpam-6211	324	13	tn	tn	PROPN
ejpam-6211	324	14	n	n	PROPN
ejpam-6211	324	15	!	!	PUNCT
ejpam-6211	325	1	(	(	PUNCT
ejpam-6211	325	2	5.1	5.1	NUM
ejpam-6211	325	3	)	)	PUNCT
ejpam-6211	325	4	=	=	PUNCT
ejpam-6211	326	1	(	(	PUNCT
ejpam-6211	326	2	1	1	NUM
ejpam-6211	326	3	1	1	NUM
ejpam-6211	326	4	+	+	NUM
ejpam-6211	326	5	t	t	NOUN
ejpam-6211	326	6	)	)	PUNCT
ejpam-6211	326	7	r	r	NOUN
ejpam-6211	326	8	(	(	PUNCT
ejpam-6211	326	9	2ext	2ext	NUM
ejpam-6211	326	10	cosh	cosh	NOUN
ejpam-6211	326	11	(	(	PUNCT
ejpam-6211	326	12	√	√	PROPN
ejpam-6211	326	13	x2	x2	INTJ
ejpam-6211	326	14	−	−	PROPN
ejpam-6211	326	15	1	1	NUM
ejpam-6211	326	16	t	t	NOUN
ejpam-6211	326	17	)	)	PUNCT
ejpam-6211	326	18	)	)	PUNCT
ejpam-6211	327	1	(	(	PUNCT
ejpam-6211	327	2	lnk	lnk	NOUN
ejpam-6211	327	3	(	(	PUNCT
ejpam-6211	327	4	1	1	NUM
ejpam-6211	327	5	+	+	NUM
ejpam-6211	327	6	t	t	PROPN
ejpam-6211	327	7	)	)	PUNCT
ejpam-6211	327	8	)	)	PUNCT
ejpam-6211	328	1	k	k	X
ejpam-6211	328	2	!	!	PUNCT
ejpam-6211	329	1	τ2(t	τ2(t	NUM
ejpam-6211	329	2	,	,	PUNCT
ejpam-6211	329	3	x	x	X
ejpam-6211	329	4	)	)	PUNCT
ejpam-6211	330	1	=	=	SYM
ejpam-6211	331	1	∞∑	∞∑	NUM
ejpam-6211	331	2	n=0	n=0	NUM
ejpam-6211	331	3	st	st	NOUN
ejpam-6211	331	4	2	2	NUM
ejpam-6211	331	5	n	n	NOUN
ejpam-6211	331	6	(	(	PUNCT
ejpam-6211	331	7	k	k	NOUN
ejpam-6211	331	8	;	;	PUNCT
ejpam-6211	331	9	r	r	X
ejpam-6211	331	10	,	,	PUNCT
ejpam-6211	331	11	x	x	NOUN
ejpam-6211	331	12	)	)	PUNCT
ejpam-6211	331	13	tn	tn	PROPN
ejpam-6211	331	14	n	n	PROPN
ejpam-6211	331	15	!	!	PUNCT
ejpam-6211	332	1	(	(	PUNCT
ejpam-6211	332	2	5.2	5.2	NUM
ejpam-6211	332	3	)	)	PUNCT
ejpam-6211	332	4	=	=	NOUN
ejpam-6211	333	1	(	(	PUNCT
ejpam-6211	333	2	2ext	2ext	NUM
ejpam-6211	333	3	cosh	cosh	NOUN
ejpam-6211	333	4	(	(	PUNCT
ejpam-6211	333	5	√	√	PROPN
ejpam-6211	333	6	x2	x2	INTJ
ejpam-6211	333	7	−	−	PROPN
ejpam-6211	333	8	1	1	NUM
ejpam-6211	333	9	t	t	NOUN
ejpam-6211	333	10	)	)	PUNCT
ejpam-6211	333	11	)	)	PUNCT
ejpam-6211	333	12	(	(	PUNCT
ejpam-6211	333	13	ert(et	ert(et	NOUN
ejpam-6211	333	14	−	−	NOUN
ejpam-6211	333	15	1	1	NUM
ejpam-6211	333	16	)	)	PUNCT
ejpam-6211	333	17	)	)	PUNCT
ejpam-6211	334	1	k	k	PROPN
ejpam-6211	335	1	k	k	X
ejpam-6211	335	2	!	!	PUNCT
ejpam-6211	335	3	definition	definition	NOUN
ejpam-6211	335	4	5.2	5.2	NUM
ejpam-6211	335	5	.	.	PUNCT
ejpam-6211	336	1	the	the	DET
ejpam-6211	336	2	second	second	ADJ
ejpam-6211	336	3	form	form	NOUN
ejpam-6211	336	4	of	of	ADP
ejpam-6211	336	5	r	r	NOUN
ejpam-6211	336	6	-	-	PUNCT
ejpam-6211	336	7	stirling	stirling	NOUN
ejpam-6211	336	8	chebyshev	chebyshev	NOUN
ejpam-6211	336	9	polynomial	polynomial	NOUN
ejpam-6211	336	10	of	of	ADP
ejpam-6211	336	11	the	the	DET
ejpam-6211	336	12	first	first	ADJ
ejpam-6211	336	13	and	and	CCONJ
ejpam-6211	336	14	second	second	ADJ
ejpam-6211	336	15	kind	kind	NOUN
ejpam-6211	336	16	is	be	AUX
ejpam-6211	336	17	defined	define	VERB
ejpam-6211	336	18	by	by	ADP
ejpam-6211	336	19	;	;	PUNCT
ejpam-6211	336	20	ω1(t	ω1(t	NUM
ejpam-6211	336	21	,	,	PUNCT
ejpam-6211	336	22	x	x	X
ejpam-6211	336	23	)	)	PUNCT
ejpam-6211	336	24	=	=	SYM
ejpam-6211	337	1	∞∑	∞∑	PRON
ejpam-6211	337	2	n=0	n=0	NUM
ejpam-6211	337	3	su1	su1	NOUN
ejpam-6211	337	4	n	n	CCONJ
ejpam-6211	337	5	(	(	PUNCT
ejpam-6211	337	6	k	k	X
ejpam-6211	337	7	;	;	PUNCT
ejpam-6211	337	8	r	r	X
ejpam-6211	337	9	,	,	PUNCT
ejpam-6211	337	10	x	x	NOUN
ejpam-6211	337	11	)	)	PUNCT
ejpam-6211	337	12	tn	tn	PROPN
ejpam-6211	337	13	n	n	PROPN
ejpam-6211	337	14	!	!	PUNCT
ejpam-6211	338	1	(	(	PUNCT
ejpam-6211	338	2	5.3	5.3	NUM
ejpam-6211	338	3	)	)	PUNCT
ejpam-6211	338	4	=	=	NOUN
ejpam-6211	339	1	(	(	PUNCT
ejpam-6211	339	2	1	1	NUM
ejpam-6211	339	3	1	1	NUM
ejpam-6211	339	4	+	+	NUM
ejpam-6211	339	5	t	t	NOUN
ejpam-6211	339	6	)	)	PUNCT
ejpam-6211	339	7	r	r	NOUN
ejpam-6211	339	8	ext	ext	NOUN
ejpam-6211	339	9	(	(	PUNCT
ejpam-6211	339	10	x	x	PART
ejpam-6211	339	11	sinh	sinh	PROPN
ejpam-6211	339	12	(	(	PUNCT
ejpam-6211	339	13	√	√	PROPN
ejpam-6211	339	14	x2	x2	INTJ
ejpam-6211	339	15	−	−	PROPN
ejpam-6211	339	16	1	1	NUM
ejpam-6211	339	17	t	t	NOUN
ejpam-6211	339	18	)	)	PUNCT
ejpam-6211	340	1	+	+	CCONJ
ejpam-6211	340	2	√	√	ADJ
ejpam-6211	340	3	x2	x2	NUM
ejpam-6211	340	4	−	−	PROPN
ejpam-6211	340	5	1	1	NUM
ejpam-6211	340	6	cosh	cosh	NOUN
ejpam-6211	340	7	(	(	PUNCT
ejpam-6211	340	8	√	√	PROPN
ejpam-6211	340	9	x2	x2	NUM
ejpam-6211	340	10	−	−	PROPN
ejpam-6211	340	11	1	1	NUM
ejpam-6211	340	12	t	t	NOUN
ejpam-6211	340	13	)	)	PUNCT
ejpam-6211	340	14	)	)	PUNCT
ejpam-6211	341	1	(	(	PUNCT
ejpam-6211	341	2	lnk	lnk	NOUN
ejpam-6211	341	3	(	(	PUNCT
ejpam-6211	341	4	1	1	NUM
ejpam-6211	341	5	+	+	NUM
ejpam-6211	341	6	t	t	PROPN
ejpam-6211	341	7	)	)	PUNCT
ejpam-6211	341	8	)	)	PUNCT
ejpam-6211	342	1	k	k	X
ejpam-6211	342	2	!	!	PUNCT
ejpam-6211	342	3	ω2(t	ω2(t	PROPN
ejpam-6211	342	4	,	,	PUNCT
ejpam-6211	342	5	x	x	NOUN
ejpam-6211	342	6	)	)	PUNCT
ejpam-6211	342	7	=	=	SYM
ejpam-6211	343	1	∞∑	∞∑	NUM
ejpam-6211	343	2	n=0	n=0	NUM
ejpam-6211	343	3	su2	su2	PROPN
ejpam-6211	344	1	n	n	CCONJ
ejpam-6211	344	2	(	(	PUNCT
ejpam-6211	344	3	k	k	X
ejpam-6211	344	4	;	;	PUNCT
ejpam-6211	344	5	r	r	X
ejpam-6211	344	6	,	,	PUNCT
ejpam-6211	344	7	x	x	NOUN
ejpam-6211	344	8	)	)	PUNCT
ejpam-6211	344	9	tn	tn	PROPN
ejpam-6211	344	10	n	n	PROPN
ejpam-6211	344	11	!	!	PUNCT
ejpam-6211	345	1	(	(	PUNCT
ejpam-6211	345	2	5.4	5.4	NUM
ejpam-6211	345	3	)	)	PUNCT
ejpam-6211	345	4	=	=	NOUN
ejpam-6211	345	5	ext	ext	NOUN
ejpam-6211	345	6	(	(	PUNCT
ejpam-6211	345	7	x	x	PART
ejpam-6211	345	8	sinh	sinh	PROPN
ejpam-6211	345	9	(	(	PUNCT
ejpam-6211	345	10	√	√	PROPN
ejpam-6211	345	11	x2	x2	INTJ
ejpam-6211	345	12	−	−	PROPN
ejpam-6211	345	13	1	1	NUM
ejpam-6211	345	14	t	t	NOUN
ejpam-6211	345	15	)	)	PUNCT
ejpam-6211	346	1	+	+	CCONJ
ejpam-6211	346	2	√	√	ADJ
ejpam-6211	346	3	x2	x2	NUM
ejpam-6211	346	4	−	−	PROPN
ejpam-6211	346	5	1	1	NUM
ejpam-6211	346	6	cosh	cosh	NOUN
ejpam-6211	346	7	(	(	PUNCT
ejpam-6211	346	8	√	√	PROPN
ejpam-6211	346	9	x2	x2	NUM
ejpam-6211	346	10	−	−	PROPN
ejpam-6211	346	11	1	1	NUM
ejpam-6211	346	12	t	t	NOUN
ejpam-6211	346	13	)	)	PUNCT
ejpam-6211	346	14	)	)	PUNCT
ejpam-6211	347	1	(	(	PUNCT
ejpam-6211	347	2	ert(et	ert(et	NOUN
ejpam-6211	347	3	−	−	PROPN
ejpam-6211	347	4	1)k	1)k	NUM
ejpam-6211	347	5	)	)	PUNCT
ejpam-6211	348	1	k	k	X
ejpam-6211	348	2	!	!	PROPN
ejpam-6211	349	1	16	16	NUM
ejpam-6211	349	2	of	of	ADP
ejpam-6211	349	3	23	23	NUM
ejpam-6211	349	4	theorem	theorem	VERB
ejpam-6211	349	5	5.3	5.3	NUM
ejpam-6211	349	6	.	.	PUNCT
ejpam-6211	350	1	for	for	ADP
ejpam-6211	350	2	n	n	PRON
ejpam-6211	350	3	≥	≥	X
ejpam-6211	350	4	0	0	NUM
ejpam-6211	350	5	the	the	DET
ejpam-6211	350	6	formula	formula	NOUN
ejpam-6211	350	7	holds	hold	VERB
ejpam-6211	350	8	,	,	PUNCT
ejpam-6211	350	9	n−1∑	n−1∑	PROPN
ejpam-6211	350	10	k=0	k=0	PROPN
ejpam-6211	350	11	st	st	PROPN
ejpam-6211	350	12	1	1	NUM
ejpam-6211	350	13	k	k	X
ejpam-6211	350	14	(	(	PUNCT
ejpam-6211	350	15	m	m	PROPN
ejpam-6211	350	16	;	;	PUNCT
ejpam-6211	350	17	r	r	X
ejpam-6211	350	18	,	,	PUNCT
ejpam-6211	350	19	x	x	NOUN
ejpam-6211	350	20	)	)	PUNCT
ejpam-6211	350	21	(	(	PUNCT
ejpam-6211	350	22	n	n	X
ejpam-6211	350	23	k	k	NOUN
ejpam-6211	350	24	)	)	PUNCT
ejpam-6211	350	25	(	(	PUNCT
ejpam-6211	350	26	(	(	PUNCT
ejpam-6211	350	27	√	√	INTJ
ejpam-6211	350	28	x2	x2	NOUN
ejpam-6211	351	1	+	+	CCONJ
ejpam-6211	351	2	4	4	X
ejpam-6211	351	3	)	)	PUNCT
ejpam-6211	351	4	n−k−1	n−k−1	PROPN
ejpam-6211	351	5	(	(	PUNCT
ejpam-6211	351	6	1−	1−	NUM
ejpam-6211	351	7	(	(	PUNCT
ejpam-6211	351	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	351	9	)	)	PUNCT
ejpam-6211	351	10	)	)	PUNCT
ejpam-6211	351	11	(	(	PUNCT
ejpam-6211	351	12	5.5	5.5	NUM
ejpam-6211	351	13	)	)	PUNCT
ejpam-6211	351	14	=	=	SYM
ejpam-6211	352	1	n∑	n∑	NOUN
ejpam-6211	352	2	k=1	k=1	X
ejpam-6211	353	1	(	(	PUNCT
ejpam-6211	353	2	2	2	NUM
ejpam-6211	353	3	+	+	SYM
ejpam-6211	353	4	2	2	NUM
ejpam-6211	353	5	t	t	NOUN
ejpam-6211	353	6	2	2	NUM
ejpam-6211	353	7	+	+	NUM
ejpam-6211	353	8	t	t	NOUN
ejpam-6211	353	9	)	)	PUNCT
ejpam-6211	353	10	r	r	NOUN
ejpam-6211	353	11	lnm	lnm	NOUN
ejpam-6211	353	12	(	(	PUNCT
ejpam-6211	353	13	−t	−t	PROPN
ejpam-6211	353	14	2	2	X
ejpam-6211	353	15	)	)	PUNCT
ejpam-6211	353	16	sf	sf	NOUN
ejpam-6211	353	17	1	1	NUM
ejpam-6211	353	18	k	k	X
ejpam-6211	353	19	(	(	PUNCT
ejpam-6211	353	20	m	m	PROPN
ejpam-6211	353	21	;	;	PUNCT
ejpam-6211	353	22	r	r	X
ejpam-6211	353	23	,	,	PUNCT
ejpam-6211	353	24	x	x	NOUN
ejpam-6211	353	25	)	)	PUNCT
ejpam-6211	353	26	(	(	PUNCT
ejpam-6211	353	27	n	n	X
ejpam-6211	353	28	k	k	X
ejpam-6211	353	29	)	)	PUNCT
ejpam-6211	353	30	2k−1	2k−1	NUM
ejpam-6211	353	31	(	(	PUNCT
ejpam-6211	353	32	(	(	PUNCT
ejpam-6211	353	33	√	√	INTJ
ejpam-6211	354	1	x2	x2	NUM
ejpam-6211	355	1	−	−	PROPN
ejpam-6211	355	2	1	1	X
ejpam-6211	355	3	)	)	PUNCT
ejpam-6211	355	4	n−k	n−k	NOUN
ejpam-6211	355	5	(	(	PUNCT
ejpam-6211	355	6	1	1	NUM
ejpam-6211	355	7	+	+	CCONJ
ejpam-6211	355	8	(	(	PUNCT
ejpam-6211	355	9	−1)n−k	−1)n−k	NOUN
ejpam-6211	355	10	)	)	PUNCT
ejpam-6211	355	11	)	)	PUNCT
ejpam-6211	355	12	(	(	PUNCT
ejpam-6211	355	13	5.6	5.6	NUM
ejpam-6211	355	14	)	)	PUNCT
ejpam-6211	355	15	where	where	SCONJ
ejpam-6211	355	16	t	t	PROPN
ejpam-6211	355	17	<	<	X
ejpam-6211	355	18	0	0	PUNCT
ejpam-6211	355	19	and	and	CCONJ
ejpam-6211	355	20	t	t	PROPN
ejpam-6211	355	21	̸=	̸=	PROPN
ejpam-6211	355	22	−2	−2	NOUN
ejpam-6211	355	23	.	.	PUNCT
ejpam-6211	356	1	proof	proof	NOUN
ejpam-6211	356	2	to	to	PART
ejpam-6211	356	3	prove	prove	VERB
ejpam-6211	356	4	(	(	PUNCT
ejpam-6211	356	5	5.5	5.5	NUM
ejpam-6211	356	6	)	)	PUNCT
ejpam-6211	356	7	we	we	PRON
ejpam-6211	356	8	use	use	VERB
ejpam-6211	356	9	the	the	DET
ejpam-6211	356	10	exponential	exponential	ADJ
ejpam-6211	356	11	generating	generating	NOUN
ejpam-6211	356	12	functions	function	NOUN
ejpam-6211	356	13	in	in	ADP
ejpam-6211	356	14	(	(	PUNCT
ejpam-6211	356	15	4.1	4.1	NUM
ejpam-6211	356	16	)	)	PUNCT
ejpam-6211	356	17	and	and	CCONJ
ejpam-6211	356	18	(	(	PUNCT
ejpam-6211	356	19	5.1	5.1	NUM
ejpam-6211	356	20	)	)	PUNCT
ejpam-6211	356	21	,	,	PUNCT
ejpam-6211	356	22	which	which	PRON
ejpam-6211	356	23	gives	give	VERB
ejpam-6211	356	24	a	a	DET
ejpam-6211	356	25	functional	functional	ADJ
ejpam-6211	356	26	equation	equation	NOUN
ejpam-6211	356	27	,	,	PUNCT
ejpam-6211	356	28	2τ1	2τ1	NUM
ejpam-6211	356	29	(	(	PUNCT
ejpam-6211	356	30	t	t	PROPN
ejpam-6211	356	31	2	2	NUM
ejpam-6211	356	32	,	,	PUNCT
ejpam-6211	356	33	x	x	SYM
ejpam-6211	356	34	)	)	PUNCT
ejpam-6211	356	35	sinh	sinh	NOUN
ejpam-6211	356	36	(	(	PUNCT
ejpam-6211	356	37	√	√	PROPN
ejpam-6211	356	38	x2	x2	PROPN
ejpam-6211	357	1	+	+	CCONJ
ejpam-6211	357	2	4	4	NUM
ejpam-6211	357	3	2	2	NUM
ejpam-6211	357	4	t	t	NOUN
ejpam-6211	357	5	)	)	PUNCT
ejpam-6211	358	1	=	=	SYM
ejpam-6211	358	2	2r	2r	NUM
ejpam-6211	358	3	(	(	PUNCT
ejpam-6211	358	4	1	1	NUM
ejpam-6211	358	5	+	+	NUM
ejpam-6211	358	6	t)r	t)r	ADJ
ejpam-6211	358	7	lnk	lnk	NOUN
ejpam-6211	358	8	(	(	PUNCT
ejpam-6211	358	9	2+t	2+t	NUM
ejpam-6211	358	10	2	2	NUM
ejpam-6211	358	11	)	)	PUNCT
ejpam-6211	358	12	√	√	PROPN
ejpam-6211	359	1	x2	x2	NOUN
ejpam-6211	360	1	+	+	CCONJ
ejpam-6211	360	2	4	4	NUM
ejpam-6211	360	3	cosh	cosh	NOUN
ejpam-6211	360	4	(	(	PUNCT
ejpam-6211	360	5	√	√	NOUN
ejpam-6211	360	6	x2−1	x2−1	PROPN
ejpam-6211	360	7	2	2	NUM
ejpam-6211	360	8	t	t	PROPN
ejpam-6211	360	9	)	)	PUNCT
ejpam-6211	360	10	ϕ	ϕ	PROPN
ejpam-6211	360	11	(	(	PUNCT
ejpam-6211	360	12	t	t	PROPN
ejpam-6211	360	13	,	,	PUNCT
ejpam-6211	360	14	x	x	NOUN
ejpam-6211	360	15	)	)	PUNCT
ejpam-6211	360	16	(	(	PUNCT
ejpam-6211	360	17	2	2	NUM
ejpam-6211	360	18	+	+	NUM
ejpam-6211	360	19	t)r	t)r	ADJ
ejpam-6211	360	20	lnk	lnk	NOUN
ejpam-6211	360	21	(	(	PUNCT
ejpam-6211	360	22	1	1	NUM
ejpam-6211	360	23	+	+	NUM
ejpam-6211	360	24	t	t	PROPN
ejpam-6211	360	25	)	)	PUNCT
ejpam-6211	360	26	.	.	PUNCT
ejpam-6211	361	1	(	(	PUNCT
ejpam-6211	361	2	5.7	5.7	NUM
ejpam-6211	361	3	)	)	PUNCT
ejpam-6211	361	4	after	after	ADP
ejpam-6211	361	5	solving	solve	VERB
ejpam-6211	361	6	the	the	DET
ejpam-6211	361	7	equation	equation	NOUN
ejpam-6211	361	8	by	by	ADP
ejpam-6211	361	9	expressing	express	VERB
ejpam-6211	361	10	the	the	DET
ejpam-6211	361	11	hyperbolic	hyperbolic	ADJ
ejpam-6211	361	12	functions	function	NOUN
ejpam-6211	361	13	in	in	ADP
ejpam-6211	361	14	terms	term	NOUN
ejpam-6211	361	15	of	of	ADP
ejpam-6211	361	16	exponential	exponential	ADJ
ejpam-6211	361	17	functions	function	NOUN
ejpam-6211	361	18	and	and	CCONJ
ejpam-6211	361	19	cauchy	cauchy	NOUN
ejpam-6211	361	20	product	product	NOUN
ejpam-6211	361	21	yields	yield	VERB
ejpam-6211	361	22	lhs	lhs	PROPN
ejpam-6211	361	23	and	and	CCONJ
ejpam-6211	361	24	rhs	rhs	PROPN
ejpam-6211	361	25	expressions	expression	NOUN
ejpam-6211	361	26	,	,	PUNCT
ejpam-6211	361	27	for	for	ADP
ejpam-6211	361	28	lhs	lhs	PROPN
ejpam-6211	361	29	,	,	PUNCT
ejpam-6211	361	30	lhs	lhs	PROPN
ejpam-6211	361	31	=	=	PROPN
ejpam-6211	362	1	∞∑	∞∑	NUM
ejpam-6211	362	2	n=0	n=0	PROPN
ejpam-6211	362	3	{	{	PUNCT
ejpam-6211	362	4	n−1∑	n−1∑	PROPN
ejpam-6211	362	5	k=0	k=0	PROPN
ejpam-6211	362	6	st	st	PROPN
ejpam-6211	362	7	1	1	NUM
ejpam-6211	362	8	k	k	X
ejpam-6211	362	9	(	(	PUNCT
ejpam-6211	362	10	m	m	PROPN
ejpam-6211	362	11	;	;	PUNCT
ejpam-6211	362	12	r	r	X
ejpam-6211	362	13	,	,	PUNCT
ejpam-6211	362	14	x	x	NOUN
ejpam-6211	362	15	)	)	PUNCT
ejpam-6211	362	16	(	(	PUNCT
ejpam-6211	362	17	n	n	X
ejpam-6211	362	18	k	k	X
ejpam-6211	362	19	)	)	PUNCT
ejpam-6211	362	20	1	1	NUM
ejpam-6211	362	21	2n	2n	NUM
ejpam-6211	362	22	(	(	PUNCT
ejpam-6211	362	23	(	(	PUNCT
ejpam-6211	362	24	√	√	INTJ
ejpam-6211	362	25	x2	x2	NOUN
ejpam-6211	363	1	+	+	CCONJ
ejpam-6211	363	2	4	4	X
ejpam-6211	363	3	)	)	PUNCT
ejpam-6211	363	4	n−k	n−k	NOUN
ejpam-6211	363	5	(	(	PUNCT
ejpam-6211	363	6	1−	1−	NUM
ejpam-6211	363	7	(	(	PUNCT
ejpam-6211	363	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	363	9	)	)	PUNCT
ejpam-6211	363	10	)	)	PUNCT
ejpam-6211	363	11	}	}	PUNCT
ejpam-6211	363	12	tn	tn	PROPN
ejpam-6211	363	13	n	n	CCONJ
ejpam-6211	363	14	!	!	PUNCT
ejpam-6211	364	1	similarly	similarly	ADV
ejpam-6211	364	2	,	,	PUNCT
ejpam-6211	364	3	the	the	DET
ejpam-6211	364	4	rhs	rhs	PROPN
ejpam-6211	364	5	,	,	PUNCT
ejpam-6211	364	6	rhs	rhs	PROPN
ejpam-6211	364	7	=	=	PUNCT
ejpam-6211	365	1	(	(	PUNCT
ejpam-6211	365	2	2	2	NUM
ejpam-6211	365	3	+	+	SYM
ejpam-6211	365	4	2	2	NUM
ejpam-6211	365	5	t	t	NOUN
ejpam-6211	365	6	2	2	NUM
ejpam-6211	365	7	+	+	NUM
ejpam-6211	365	8	t	t	NOUN
ejpam-6211	365	9	)	)	PUNCT
ejpam-6211	365	10	r	r	NOUN
ejpam-6211	365	11	(	(	PUNCT
ejpam-6211	365	12	lnm	lnm	PROPN
ejpam-6211	365	13	(	(	PUNCT
ejpam-6211	365	14	−t	−t	PROPN
ejpam-6211	365	15	2	2	NUM
ejpam-6211	365	16	)	)	PUNCT
ejpam-6211	365	17	)	)	PUNCT
ejpam-6211	366	1	√	√	PUNCT
ejpam-6211	367	1	x2	x2	INTJ
ejpam-6211	368	1	+	+	CCONJ
ejpam-6211	368	2	4	4	NUM
ejpam-6211	368	3	2	2	NUM
ejpam-6211	368	4	∞∑	∞∑	NUM
ejpam-6211	368	5	n=0	n=0	NUM
ejpam-6211	368	6	n∑	n∑	NOUN
ejpam-6211	368	7	k=1	k=1	PUNCT
ejpam-6211	369	1	sf	sf	NOUN
ejpam-6211	369	2	1	1	NUM
ejpam-6211	369	3	k	k	X
ejpam-6211	369	4	(	(	PUNCT
ejpam-6211	369	5	m	m	PROPN
ejpam-6211	369	6	;	;	PUNCT
ejpam-6211	369	7	r	r	X
ejpam-6211	369	8	,	,	PUNCT
ejpam-6211	369	9	x	x	NOUN
ejpam-6211	369	10	)	)	PUNCT
ejpam-6211	369	11	(	(	PUNCT
ejpam-6211	369	12	n	n	X
ejpam-6211	369	13	k	k	X
ejpam-6211	369	14	)	)	PUNCT
ejpam-6211	369	15	×	×	PROPN
ejpam-6211	369	16	(√	(√	VERB
ejpam-6211	369	17	x2	x2	ADJ
ejpam-6211	369	18	−	−	NUM
ejpam-6211	369	19	1	1	NUM
ejpam-6211	369	20	2	2	NUM
ejpam-6211	369	21	)	)	PUNCT
ejpam-6211	369	22	n−k	n−k	NOUN
ejpam-6211	369	23	(	(	PUNCT
ejpam-6211	369	24	1	1	NUM
ejpam-6211	369	25	+	+	CCONJ
ejpam-6211	369	26	(	(	PUNCT
ejpam-6211	369	27	−1)n−k	−1)n−k	NOUN
ejpam-6211	369	28	)	)	PUNCT
ejpam-6211	370	1			PROPN
ejpam-6211	370	2	tn	tn	PROPN
ejpam-6211	370	3	n	n	NOUN
ejpam-6211	370	4	!	!	PUNCT
ejpam-6211	371	1	now	now	ADV
ejpam-6211	371	2	,	,	PUNCT
ejpam-6211	371	3	by	by	ADP
ejpam-6211	371	4	(	(	PUNCT
ejpam-6211	371	5	5.7	5.7	NUM
ejpam-6211	371	6	)	)	PUNCT
ejpam-6211	371	7	we	we	PRON
ejpam-6211	371	8	have	have	VERB
ejpam-6211	371	9	∞∑	∞∑	NUM
ejpam-6211	371	10	n=0	n=0	PROPN
ejpam-6211	371	11	{	{	PUNCT
ejpam-6211	371	12	n−1∑	n−1∑	PROPN
ejpam-6211	371	13	k=0	k=0	PROPN
ejpam-6211	371	14	st	st	PROPN
ejpam-6211	371	15	1	1	NUM
ejpam-6211	371	16	k	k	X
ejpam-6211	371	17	(	(	PUNCT
ejpam-6211	371	18	m	m	PROPN
ejpam-6211	371	19	;	;	PUNCT
ejpam-6211	371	20	r	r	X
ejpam-6211	371	21	,	,	PUNCT
ejpam-6211	371	22	x	x	NOUN
ejpam-6211	371	23	)	)	PUNCT
ejpam-6211	371	24	(	(	PUNCT
ejpam-6211	371	25	n	n	X
ejpam-6211	371	26	k	k	X
ejpam-6211	371	27	)	)	PUNCT
ejpam-6211	371	28	1	1	NUM
ejpam-6211	371	29	2n	2n	NUM
ejpam-6211	371	30	(	(	PUNCT
ejpam-6211	371	31	(	(	PUNCT
ejpam-6211	371	32	√	√	INTJ
ejpam-6211	371	33	x2	x2	NOUN
ejpam-6211	372	1	+	+	CCONJ
ejpam-6211	372	2	4	4	X
ejpam-6211	372	3	)	)	PUNCT
ejpam-6211	372	4	n−k	n−k	NOUN
ejpam-6211	372	5	(	(	PUNCT
ejpam-6211	372	6	1−	1−	NUM
ejpam-6211	372	7	(	(	PUNCT
ejpam-6211	372	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	372	9	)	)	PUNCT
ejpam-6211	372	10	)	)	PUNCT
ejpam-6211	372	11	}	}	PUNCT
ejpam-6211	372	12	tn	tn	PROPN
ejpam-6211	372	13	n	n	NOUN
ejpam-6211	372	14	!	!	PUNCT
ejpam-6211	372	15	=	=	PUNCT
ejpam-6211	373	1	(	(	PUNCT
ejpam-6211	373	2	2	2	NUM
ejpam-6211	373	3	+	+	SYM
ejpam-6211	373	4	2	2	NUM
ejpam-6211	373	5	t	t	NOUN
ejpam-6211	373	6	2	2	NUM
ejpam-6211	373	7	+	+	NUM
ejpam-6211	373	8	t	t	NOUN
ejpam-6211	373	9	)	)	PUNCT
ejpam-6211	373	10	r	r	NOUN
ejpam-6211	373	11	(	(	PUNCT
ejpam-6211	373	12	lnm	lnm	PROPN
ejpam-6211	373	13	(	(	PUNCT
ejpam-6211	373	14	−t	−t	PROPN
ejpam-6211	373	15	2	2	NUM
ejpam-6211	373	16	)	)	PUNCT
ejpam-6211	373	17	)	)	PUNCT
ejpam-6211	374	1	√	√	PUNCT
ejpam-6211	375	1	x2	x2	INTJ
ejpam-6211	376	1	+	+	CCONJ
ejpam-6211	376	2	4	4	NUM
ejpam-6211	376	3	2	2	NUM
ejpam-6211	376	4	∞∑	∞∑	NUM
ejpam-6211	376	5	n=0	n=0	NUM
ejpam-6211	376	6	n∑	n∑	NOUN
ejpam-6211	376	7	k=1	k=1	PUNCT
ejpam-6211	377	1	sf	sf	NOUN
ejpam-6211	377	2	1	1	NUM
ejpam-6211	377	3	k	k	X
ejpam-6211	377	4	(	(	PUNCT
ejpam-6211	377	5	m	m	PROPN
ejpam-6211	377	6	;	;	PUNCT
ejpam-6211	377	7	r	r	X
ejpam-6211	377	8	,	,	PUNCT
ejpam-6211	377	9	x	x	NOUN
ejpam-6211	377	10	)	)	PUNCT
ejpam-6211	377	11	(	(	PUNCT
ejpam-6211	377	12	n	n	X
ejpam-6211	377	13	k	k	NOUN
ejpam-6211	377	14	)	)	PUNCT
ejpam-6211	377	15	17	17	NUM
ejpam-6211	377	16	of	of	ADP
ejpam-6211	377	17	23	23	NUM
ejpam-6211	377	18	×	×	NOUN
ejpam-6211	377	19	(√	(√	ADJ
ejpam-6211	377	20	x2	x2	ADJ
ejpam-6211	377	21	−	−	NUM
ejpam-6211	377	22	1	1	NUM
ejpam-6211	377	23	2	2	NUM
ejpam-6211	377	24	)	)	PUNCT
ejpam-6211	377	25	n−k	n−k	NOUN
ejpam-6211	377	26	(	(	PUNCT
ejpam-6211	377	27	1	1	NUM
ejpam-6211	377	28	+	+	CCONJ
ejpam-6211	377	29	(	(	PUNCT
ejpam-6211	377	30	−1)n−k	−1)n−k	NOUN
ejpam-6211	377	31	)	)	PUNCT
ejpam-6211	378	1			PROPN
ejpam-6211	378	2	tn	tn	PROPN
ejpam-6211	378	3	n	n	CCONJ
ejpam-6211	378	4	!	!	X
ejpam-6211	378	5	comparing	compare	VERB
ejpam-6211	378	6	coefficients	coefficient	NOUN
ejpam-6211	378	7	of	of	ADP
ejpam-6211	378	8	tn	tn	NOUN
ejpam-6211	378	9	n	n	CCONJ
ejpam-6211	378	10	!	!	PUNCT
ejpam-6211	379	1	and	and	CCONJ
ejpam-6211	379	2	expressing	express	VERB
ejpam-6211	379	3	2n−k+1	2n−k+1	NUM
ejpam-6211	379	4	as	as	ADP
ejpam-6211	379	5	2n2k−1	2n2k−1	NUM
ejpam-6211	379	6	yields	yield	NOUN
ejpam-6211	379	7	,	,	PUNCT
ejpam-6211	379	8	n−1∑	n−1∑	PROPN
ejpam-6211	379	9	k=0	k=0	PROPN
ejpam-6211	379	10	st	st	PROPN
ejpam-6211	379	11	1	1	NUM
ejpam-6211	379	12	k	k	X
ejpam-6211	379	13	(	(	PUNCT
ejpam-6211	379	14	m	m	PROPN
ejpam-6211	379	15	;	;	PUNCT
ejpam-6211	379	16	r	r	X
ejpam-6211	379	17	,	,	PUNCT
ejpam-6211	379	18	x	x	NOUN
ejpam-6211	379	19	)	)	PUNCT
ejpam-6211	379	20	(	(	PUNCT
ejpam-6211	379	21	n	n	X
ejpam-6211	379	22	k	k	X
ejpam-6211	379	23	)	)	PUNCT
ejpam-6211	379	24	1	1	NUM
ejpam-6211	379	25	2n	2n	NUM
ejpam-6211	379	26	(	(	PUNCT
ejpam-6211	379	27	(	(	PUNCT
ejpam-6211	379	28	√	√	INTJ
ejpam-6211	379	29	x2	x2	NOUN
ejpam-6211	380	1	+	+	CCONJ
ejpam-6211	380	2	4	4	X
ejpam-6211	380	3	)	)	PUNCT
ejpam-6211	380	4	n−k	n−k	NOUN
ejpam-6211	380	5	(	(	PUNCT
ejpam-6211	380	6	1−	1−	NUM
ejpam-6211	380	7	(	(	PUNCT
ejpam-6211	380	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	380	9	)	)	PUNCT
ejpam-6211	380	10	)	)	PUNCT
ejpam-6211	381	1	=	=	PUNCT
ejpam-6211	381	2	(	(	PUNCT
ejpam-6211	381	3	2	2	NUM
ejpam-6211	381	4	+	+	SYM
ejpam-6211	381	5	2	2	NUM
ejpam-6211	381	6	t	t	NOUN
ejpam-6211	381	7	2	2	NUM
ejpam-6211	381	8	+	+	NUM
ejpam-6211	381	9	t	t	NOUN
ejpam-6211	381	10	)	)	PUNCT
ejpam-6211	381	11	r	r	NOUN
ejpam-6211	381	12	(	(	PUNCT
ejpam-6211	381	13	lnm	lnm	PROPN
ejpam-6211	381	14	(	(	PUNCT
ejpam-6211	381	15	−t	−t	PROPN
ejpam-6211	381	16	2	2	NUM
ejpam-6211	381	17	)	)	PUNCT
ejpam-6211	381	18	)	)	PUNCT
ejpam-6211	382	1	n∑	n∑	INTJ
ejpam-6211	382	2	k=1	k=1	PUNCT
ejpam-6211	383	1	sf	sf	NOUN
ejpam-6211	383	2	1	1	NUM
ejpam-6211	383	3	k	k	X
ejpam-6211	383	4	(	(	PUNCT
ejpam-6211	383	5	m	m	PROPN
ejpam-6211	383	6	;	;	PUNCT
ejpam-6211	383	7	r	r	X
ejpam-6211	383	8	,	,	PUNCT
ejpam-6211	383	9	x	x	NOUN
ejpam-6211	383	10	)	)	PUNCT
ejpam-6211	383	11	(	(	PUNCT
ejpam-6211	383	12	n	n	X
ejpam-6211	383	13	k	k	NOUN
ejpam-6211	383	14	)	)	PUNCT
ejpam-6211	383	15	√	√	PROPN
ejpam-6211	384	1	x2	x2	NOUN
ejpam-6211	385	1	+	+	CCONJ
ejpam-6211	385	2	4	4	NUM
ejpam-6211	385	3	2n	2n	NUM
ejpam-6211	385	4	2k−1	2k−1	NUM
ejpam-6211	385	5	×	×	NOUN
ejpam-6211	385	6	(	(	PUNCT
ejpam-6211	385	7	(	(	PUNCT
ejpam-6211	385	8	√	√	INTJ
ejpam-6211	385	9	x2	x2	NUM
ejpam-6211	386	1	−	−	PROPN
ejpam-6211	386	2	1	1	X
ejpam-6211	386	3	)	)	PUNCT
ejpam-6211	386	4	n−k	n−k	NOUN
ejpam-6211	386	5	(	(	PUNCT
ejpam-6211	386	6	1	1	NUM
ejpam-6211	386	7	+	+	CCONJ
ejpam-6211	386	8	(	(	PUNCT
ejpam-6211	386	9	−1)n−k	−1)n−k	NOUN
ejpam-6211	386	10	)	)	PUNCT
ejpam-6211	386	11	)	)	PUNCT
ejpam-6211	387	1	multiplying	multiply	VERB
ejpam-6211	387	2	2n√	2n√	NUM
ejpam-6211	387	3	x2	x2	PUNCT
ejpam-6211	388	1	+	+	NOUN
ejpam-6211	388	2	4	4	NUM
ejpam-6211	388	3	both	both	DET
ejpam-6211	388	4	sides	side	NOUN
ejpam-6211	388	5	yields	yield	NOUN
ejpam-6211	388	6	(	(	PUNCT
ejpam-6211	388	7	5.5	5.5	NUM
ejpam-6211	388	8	)	)	PUNCT
ejpam-6211	388	9	remark	remark	NOUN
ejpam-6211	388	10	5.4	5.4	NUM
ejpam-6211	388	11	.	.	PUNCT
ejpam-6211	389	1	the	the	DET
ejpam-6211	389	2	term	term	NOUN
ejpam-6211	389	3	(	(	PUNCT
ejpam-6211	389	4	2(1	2(1	NUM
ejpam-6211	389	5	+	+	NUM
ejpam-6211	389	6	t	t	NOUN
ejpam-6211	389	7	)	)	PUNCT
ejpam-6211	389	8	2	2	NUM
ejpam-6211	390	1	+	+	NUM
ejpam-6211	390	2	t	t	NOUN
ejpam-6211	390	3	)	)	PUNCT
ejpam-6211	390	4	r	r	NOUN
ejpam-6211	390	5	(	(	PUNCT
ejpam-6211	390	6	lnm	lnm	PROPN
ejpam-6211	390	7	(	(	PUNCT
ejpam-6211	390	8	−t	−t	PROPN
ejpam-6211	390	9	2	2	NUM
ejpam-6211	390	10	)	)	PUNCT
ejpam-6211	390	11	)	)	PUNCT
ejpam-6211	390	12	is	be	AUX
ejpam-6211	390	13	well	well	ADV
ejpam-6211	390	14	defined	define	VERB
ejpam-6211	390	15	when	when	SCONJ
ejpam-6211	390	16	t	t	PROPN
ejpam-6211	390	17	<	<	X
ejpam-6211	390	18	0	0	PUNCT
ejpam-6211	390	19	and	and	CCONJ
ejpam-6211	390	20	t	t	PROPN
ejpam-6211	390	21	̸=	̸=	PROPN
ejpam-6211	390	22	−2	−2	NOUN
ejpam-6211	390	23	.	.	PUNCT
ejpam-6211	391	1	theorem	theorem	VERB
ejpam-6211	391	2	5.5	5.5	NUM
ejpam-6211	391	3	.	.	PUNCT
ejpam-6211	392	1	for	for	SCONJ
ejpam-6211	392	2	n	n	PRON
ejpam-6211	392	3	≥	≥	X
ejpam-6211	392	4	0	0	NUM
ejpam-6211	392	5	the	the	DET
ejpam-6211	392	6	formula	formula	NOUN
ejpam-6211	392	7	holds	hold	VERB
ejpam-6211	392	8	,	,	PUNCT
ejpam-6211	392	9	n−1∑	n−1∑	PROPN
ejpam-6211	392	10	k=0	k=0	PROPN
ejpam-6211	392	11	(	(	PUNCT
ejpam-6211	392	12	n	n	X
ejpam-6211	392	13	k	k	NOUN
ejpam-6211	392	14	)	)	PUNCT
ejpam-6211	392	15	(	(	PUNCT
ejpam-6211	392	16	√	√	NUM
ejpam-6211	392	17	x2	x2	PROPN
ejpam-6211	393	1	+	+	CCONJ
ejpam-6211	393	2	4	4	X
ejpam-6211	393	3	)	)	PUNCT
ejpam-6211	393	4	n−k−1	n−k−1	PROPN
ejpam-6211	393	5	(	(	PUNCT
ejpam-6211	393	6	1−	1−	NUM
ejpam-6211	393	7	(	(	PUNCT
ejpam-6211	393	8	−1)n−k)st	−1)n−k)st	NOUN
ejpam-6211	393	9	2	2	NUM
ejpam-6211	393	10	k	k	X
ejpam-6211	393	11	(	(	PUNCT
ejpam-6211	393	12	m	m	PROPN
ejpam-6211	393	13	;	;	PUNCT
ejpam-6211	394	1	r	r	X
ejpam-6211	394	2	,	,	PUNCT
ejpam-6211	394	3	x	x	NOUN
ejpam-6211	394	4	)	)	PUNCT
ejpam-6211	394	5	=	=	SYM
ejpam-6211	395	1	n∑	n∑	NOUN
ejpam-6211	395	2	k=1	k=1	PUNCT
ejpam-6211	396	1	sf	sf	NOUN
ejpam-6211	396	2	2	2	NUM
ejpam-6211	396	3	k	k	X
ejpam-6211	396	4	(	(	PUNCT
ejpam-6211	396	5	m	m	PROPN
ejpam-6211	396	6	;	;	PUNCT
ejpam-6211	396	7	r	r	X
ejpam-6211	396	8	,	,	PUNCT
ejpam-6211	396	9	x	x	NOUN
ejpam-6211	396	10	)	)	PUNCT
ejpam-6211	396	11	e	e	NOUN
ejpam-6211	396	12	−rt	−rt	NOUN
ejpam-6211	396	13	2	2	NUM
ejpam-6211	396	14	(	(	PUNCT
ejpam-6211	396	15	e	e	NOUN
ejpam-6211	396	16	t	t	PROPN
ejpam-6211	396	17	2	2	NUM
ejpam-6211	396	18	−	−	PROPN
ejpam-6211	396	19	1	1	NUM
ejpam-6211	396	20	et	et	NOUN
ejpam-6211	396	21	−	−	NOUN
ejpam-6211	396	22	1	1	NUM
ejpam-6211	396	23	)	)	PUNCT
ejpam-6211	396	24	m	m	PROPN
ejpam-6211	396	25	(	(	PUNCT
ejpam-6211	396	26	n	n	PROPN
ejpam-6211	396	27	k	k	PROPN
ejpam-6211	396	28	)	)	PUNCT
ejpam-6211	396	29	2k−1	2k−1	NUM
ejpam-6211	396	30	×	×	NOUN
ejpam-6211	396	31	(	(	PUNCT
ejpam-6211	396	32	(	(	PUNCT
ejpam-6211	396	33	√	√	INTJ
ejpam-6211	396	34	x2	x2	NUM
ejpam-6211	396	35	−	−	PROPN
ejpam-6211	396	36	1	1	X
ejpam-6211	396	37	)	)	PUNCT
ejpam-6211	396	38	n−k	n−k	NOUN
ejpam-6211	396	39	(	(	PUNCT
ejpam-6211	396	40	1	1	NUM
ejpam-6211	396	41	+	+	CCONJ
ejpam-6211	396	42	(	(	PUNCT
ejpam-6211	396	43	−1)n−k	−1)n−k	NOUN
ejpam-6211	396	44	)	)	PUNCT
ejpam-6211	396	45	)	)	PUNCT
ejpam-6211	396	46	(	(	PUNCT
ejpam-6211	396	47	5.8	5.8	NUM
ejpam-6211	396	48	)	)	PUNCT
ejpam-6211	396	49	where	where	SCONJ
ejpam-6211	396	50	t	t	PROPN
ejpam-6211	396	51	≥	≥	NOUN
ejpam-6211	396	52	0	0	NUM
ejpam-6211	396	53	.	.	PUNCT
ejpam-6211	397	1	proof	proof	NOUN
ejpam-6211	397	2	to	to	PART
ejpam-6211	397	3	prove	prove	VERB
ejpam-6211	397	4	(	(	PUNCT
ejpam-6211	397	5	5.8	5.8	NUM
ejpam-6211	397	6	)	)	PUNCT
ejpam-6211	397	7	we	we	PRON
ejpam-6211	397	8	use	use	VERB
ejpam-6211	397	9	the	the	DET
ejpam-6211	397	10	exponential	exponential	ADJ
ejpam-6211	397	11	generating	generating	NOUN
ejpam-6211	397	12	functions	function	NOUN
ejpam-6211	397	13	in	in	ADP
ejpam-6211	397	14	(	(	PUNCT
ejpam-6211	397	15	4.2	4.2	NUM
ejpam-6211	397	16	)	)	PUNCT
ejpam-6211	397	17	and	and	CCONJ
ejpam-6211	397	18	(	(	PUNCT
ejpam-6211	397	19	5.2),which	5.2),which	PRON
ejpam-6211	397	20	gives	give	VERB
ejpam-6211	397	21	a	a	DET
ejpam-6211	397	22	functional	functional	ADJ
ejpam-6211	397	23	equation	equation	NOUN
ejpam-6211	397	24	,	,	PUNCT
ejpam-6211	397	25	2τ2	2τ2	NUM
ejpam-6211	398	1	(	(	PUNCT
ejpam-6211	398	2	t	t	PROPN
ejpam-6211	398	3	2	2	NUM
ejpam-6211	398	4	,	,	PUNCT
ejpam-6211	398	5	x	x	SYM
ejpam-6211	398	6	)	)	PUNCT
ejpam-6211	398	7	sinh	sinh	NOUN
ejpam-6211	398	8	(	(	PUNCT
ejpam-6211	398	9	√	√	PROPN
ejpam-6211	398	10	x2	x2	PROPN
ejpam-6211	399	1	+	+	CCONJ
ejpam-6211	399	2	4	4	NUM
ejpam-6211	399	3	2	2	NUM
ejpam-6211	399	4	t	t	NOUN
ejpam-6211	399	5	)	)	PUNCT
ejpam-6211	400	1	=	=	PUNCT
ejpam-6211	401	1	e	e	X
ejpam-6211	401	2	−rt	−rt	X
ejpam-6211	401	3	2	2	NUM
ejpam-6211	401	4	(	(	PUNCT
ejpam-6211	401	5	e	e	NOUN
ejpam-6211	401	6	t	t	PROPN
ejpam-6211	401	7	2	2	NUM
ejpam-6211	401	8	−	−	PROPN
ejpam-6211	401	9	1	1	NUM
ejpam-6211	401	10	et	et	NOUN
ejpam-6211	401	11	−	−	NOUN
ejpam-6211	401	12	1	1	NUM
ejpam-6211	401	13	)	)	PUNCT
ejpam-6211	401	14	k	k	PROPN
ejpam-6211	401	15	γ	γ	X
ejpam-6211	401	16	(	(	PUNCT
ejpam-6211	401	17	t	t	PROPN
ejpam-6211	401	18	,	,	PUNCT
ejpam-6211	401	19	x	x	NOUN
ejpam-6211	401	20	)	)	PUNCT
ejpam-6211	401	21	√	√	PUNCT
ejpam-6211	401	22	x2	x2	INTJ
ejpam-6211	402	1	+	+	CCONJ
ejpam-6211	402	2	4	4	NUM
ejpam-6211	402	3	cosh	cosh	NOUN
ejpam-6211	402	4	(	(	PUNCT
ejpam-6211	402	5	√	√	PROPN
ejpam-6211	402	6	x2	x2	INTJ
ejpam-6211	402	7	−	−	NOUN
ejpam-6211	402	8	1	1	NUM
ejpam-6211	402	9	2	2	NUM
ejpam-6211	402	10	t	t	NOUN
ejpam-6211	402	11	)	)	PUNCT
ejpam-6211	402	12	(	(	PUNCT
ejpam-6211	402	13	5.9	5.9	NUM
ejpam-6211	402	14	)	)	PUNCT
ejpam-6211	402	15	18	18	NUM
ejpam-6211	402	16	of	of	ADP
ejpam-6211	402	17	23	23	NUM
ejpam-6211	402	18	after	after	ADP
ejpam-6211	402	19	solving	solve	VERB
ejpam-6211	402	20	the	the	DET
ejpam-6211	402	21	equation	equation	NOUN
ejpam-6211	402	22	by	by	ADP
ejpam-6211	402	23	expressing	express	VERB
ejpam-6211	402	24	the	the	DET
ejpam-6211	402	25	hyperbolic	hyperbolic	ADJ
ejpam-6211	402	26	functions	function	NOUN
ejpam-6211	402	27	in	in	ADP
ejpam-6211	402	28	terms	term	NOUN
ejpam-6211	402	29	of	of	ADP
ejpam-6211	402	30	exponential	exponential	ADJ
ejpam-6211	402	31	functions	function	NOUN
ejpam-6211	402	32	and	and	CCONJ
ejpam-6211	402	33	cauchy	cauchy	NOUN
ejpam-6211	402	34	product	product	NOUN
ejpam-6211	402	35	yields	yield	VERB
ejpam-6211	402	36	lhs	lhs	PROPN
ejpam-6211	402	37	and	and	CCONJ
ejpam-6211	402	38	rhs	rhs	PROPN
ejpam-6211	402	39	expressions	expression	NOUN
ejpam-6211	402	40	,	,	PUNCT
ejpam-6211	402	41	for	for	ADP
ejpam-6211	402	42	lhs	lhs	PROPN
ejpam-6211	402	43	,	,	PUNCT
ejpam-6211	402	44	lhs	lhs	PROPN
ejpam-6211	402	45	=	=	PROPN
ejpam-6211	403	1	∞∑	∞∑	NUM
ejpam-6211	403	2	n=0	n=0	PROPN
ejpam-6211	403	3	{	{	PUNCT
ejpam-6211	403	4	n−1∑	n−1∑	PROPN
ejpam-6211	403	5	k=0	k=0	PROPN
ejpam-6211	403	6	st	st	PROPN
ejpam-6211	403	7	2	2	NUM
ejpam-6211	403	8	k	k	X
ejpam-6211	403	9	(	(	PUNCT
ejpam-6211	403	10	m	m	PROPN
ejpam-6211	403	11	;	;	PUNCT
ejpam-6211	403	12	r	r	X
ejpam-6211	403	13	,	,	PUNCT
ejpam-6211	403	14	x	x	NOUN
ejpam-6211	403	15	)	)	PUNCT
ejpam-6211	403	16	(	(	PUNCT
ejpam-6211	403	17	n	n	X
ejpam-6211	403	18	k	k	X
ejpam-6211	403	19	)	)	PUNCT
ejpam-6211	403	20	1	1	NUM
ejpam-6211	403	21	2n	2n	NUM
ejpam-6211	403	22	(	(	PUNCT
ejpam-6211	403	23	(	(	PUNCT
ejpam-6211	403	24	√	√	INTJ
ejpam-6211	403	25	x2	x2	NOUN
ejpam-6211	404	1	+	+	CCONJ
ejpam-6211	404	2	4	4	X
ejpam-6211	404	3	)	)	PUNCT
ejpam-6211	404	4	n−k	n−k	NOUN
ejpam-6211	404	5	(	(	PUNCT
ejpam-6211	404	6	1−	1−	NUM
ejpam-6211	404	7	(	(	PUNCT
ejpam-6211	404	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	404	9	)	)	PUNCT
ejpam-6211	404	10	)	)	PUNCT
ejpam-6211	404	11	}	}	PUNCT
ejpam-6211	404	12	tn	tn	PROPN
ejpam-6211	404	13	n	n	CCONJ
ejpam-6211	404	14	!	!	PUNCT
ejpam-6211	405	1	similarly	similarly	ADV
ejpam-6211	405	2	the	the	DET
ejpam-6211	405	3	rhs	rhs	PROPN
ejpam-6211	405	4	,	,	PUNCT
ejpam-6211	405	5	rhs	rhs	X
ejpam-6211	405	6	=	=	PUNCT
ejpam-6211	406	1	e	e	X
ejpam-6211	406	2	−rt	−rt	X
ejpam-6211	406	3	2	2	NUM
ejpam-6211	406	4	(	(	PUNCT
ejpam-6211	406	5	e	e	NOUN
ejpam-6211	406	6	t	t	PROPN
ejpam-6211	406	7	2	2	NUM
ejpam-6211	406	8	−	−	PROPN
ejpam-6211	406	9	1	1	NUM
ejpam-6211	406	10	et	et	NOUN
ejpam-6211	406	11	−	−	NOUN
ejpam-6211	406	12	1	1	NUM
ejpam-6211	406	13	)	)	PUNCT
ejpam-6211	406	14	m	m	VERB
ejpam-6211	406	15	√	√	NUM
ejpam-6211	406	16	x2	x2	INTJ
ejpam-6211	407	1	+	+	CCONJ
ejpam-6211	407	2	4	4	NUM
ejpam-6211	407	3	2	2	NUM
ejpam-6211	407	4	∞∑	∞∑	NUM
ejpam-6211	407	5	n=0	n=0	NUM
ejpam-6211	407	6	n∑	n∑	NOUN
ejpam-6211	407	7	k=1	k=1	PUNCT
ejpam-6211	408	1	sf	sf	NOUN
ejpam-6211	408	2	2	2	NUM
ejpam-6211	408	3	k	k	X
ejpam-6211	408	4	(	(	PUNCT
ejpam-6211	408	5	m	m	PROPN
ejpam-6211	408	6	;	;	PUNCT
ejpam-6211	408	7	r	r	X
ejpam-6211	408	8	,	,	PUNCT
ejpam-6211	408	9	x	x	NOUN
ejpam-6211	408	10	)	)	PUNCT
ejpam-6211	408	11	(	(	PUNCT
ejpam-6211	408	12	n	n	X
ejpam-6211	408	13	k	k	X
ejpam-6211	408	14	)	)	PUNCT
ejpam-6211	408	15	×	×	NOUN
ejpam-6211	408	16	(	(	PUNCT
ejpam-6211	408	17	√	√	PROPN
ejpam-6211	409	1	x2	x2	NUM
ejpam-6211	410	1	−	−	NUM
ejpam-6211	410	2	1	1	NUM
ejpam-6211	410	3	2	2	NUM
ejpam-6211	410	4	)	)	PUNCT
ejpam-6211	410	5	n−k	n−k	NOUN
ejpam-6211	410	6	(	(	PUNCT
ejpam-6211	410	7	1	1	NUM
ejpam-6211	410	8	+	+	CCONJ
ejpam-6211	410	9	(	(	PUNCT
ejpam-6211	410	10	−1)n−k	−1)n−k	PROPN
ejpam-6211	410	11	)	)	PUNCT
ejpam-6211	410	12	tn	tn	PROPN
ejpam-6211	410	13	n	n	PROPN
ejpam-6211	410	14	!	!	PUNCT
ejpam-6211	411	1	now	now	ADV
ejpam-6211	411	2	,	,	PUNCT
ejpam-6211	411	3	by	by	ADP
ejpam-6211	411	4	(	(	PUNCT
ejpam-6211	411	5	5.9	5.9	NUM
ejpam-6211	411	6	)	)	PUNCT
ejpam-6211	411	7	we	we	PRON
ejpam-6211	411	8	have	have	VERB
ejpam-6211	411	9	∞∑	∞∑	NUM
ejpam-6211	411	10	n=0	n=0	PROPN
ejpam-6211	411	11	{	{	PUNCT
ejpam-6211	411	12	n−1∑	n−1∑	PROPN
ejpam-6211	411	13	k=0	k=0	PROPN
ejpam-6211	411	14	st	st	PROPN
ejpam-6211	411	15	2	2	NUM
ejpam-6211	411	16	k	k	X
ejpam-6211	411	17	(	(	PUNCT
ejpam-6211	411	18	m	m	PROPN
ejpam-6211	411	19	;	;	PUNCT
ejpam-6211	411	20	r	r	X
ejpam-6211	411	21	,	,	PUNCT
ejpam-6211	411	22	x	x	NOUN
ejpam-6211	411	23	)	)	PUNCT
ejpam-6211	411	24	(	(	PUNCT
ejpam-6211	411	25	n	n	X
ejpam-6211	411	26	k	k	X
ejpam-6211	411	27	)	)	PUNCT
ejpam-6211	411	28	1	1	NUM
ejpam-6211	411	29	2n	2n	NUM
ejpam-6211	411	30	(	(	PUNCT
ejpam-6211	411	31	(	(	PUNCT
ejpam-6211	411	32	√	√	INTJ
ejpam-6211	411	33	x2	x2	NOUN
ejpam-6211	412	1	+	+	CCONJ
ejpam-6211	412	2	4	4	X
ejpam-6211	412	3	)	)	PUNCT
ejpam-6211	412	4	n−k	n−k	NOUN
ejpam-6211	412	5	(	(	PUNCT
ejpam-6211	412	6	1−	1−	NUM
ejpam-6211	412	7	(	(	PUNCT
ejpam-6211	412	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	412	9	)	)	PUNCT
ejpam-6211	412	10	)	)	PUNCT
ejpam-6211	412	11	}	}	PUNCT
ejpam-6211	412	12	tn	tn	PROPN
ejpam-6211	412	13	n	n	NOUN
ejpam-6211	412	14	!	!	PUNCT
ejpam-6211	412	15	=	=	PUNCT
ejpam-6211	413	1	e	e	NOUN
ejpam-6211	413	2	−rt	−rt	X
ejpam-6211	413	3	2	2	NUM
ejpam-6211	413	4	(	(	PUNCT
ejpam-6211	413	5	e	e	NOUN
ejpam-6211	413	6	t	t	PROPN
ejpam-6211	413	7	2	2	NUM
ejpam-6211	413	8	−	−	PROPN
ejpam-6211	413	9	1	1	NUM
ejpam-6211	413	10	et	et	NOUN
ejpam-6211	413	11	−	−	NOUN
ejpam-6211	413	12	1	1	NUM
ejpam-6211	413	13	)	)	PUNCT
ejpam-6211	413	14	m	m	VERB
ejpam-6211	413	15	√	√	NUM
ejpam-6211	413	16	x2	x2	INTJ
ejpam-6211	414	1	+	+	CCONJ
ejpam-6211	414	2	4	4	NUM
ejpam-6211	414	3	2	2	NUM
ejpam-6211	414	4	∞∑	∞∑	NUM
ejpam-6211	414	5	n=0	n=0	PROPN
ejpam-6211	414	6	{	{	PUNCT
ejpam-6211	414	7	n∑	n∑	NOUN
ejpam-6211	414	8	k=1	k=1	PUNCT
ejpam-6211	415	1	sf	sf	NOUN
ejpam-6211	415	2	2	2	NUM
ejpam-6211	415	3	k	k	X
ejpam-6211	415	4	(	(	PUNCT
ejpam-6211	415	5	m	m	PROPN
ejpam-6211	415	6	;	;	PUNCT
ejpam-6211	415	7	r	r	X
ejpam-6211	415	8	,	,	PUNCT
ejpam-6211	415	9	x	x	NOUN
ejpam-6211	415	10	)	)	PUNCT
ejpam-6211	415	11	(	(	PUNCT
ejpam-6211	415	12	n	n	X
ejpam-6211	415	13	k	k	NOUN
ejpam-6211	415	14	)	)	PUNCT
ejpam-6211	415	15	(√	(√	VERB
ejpam-6211	416	1	x2	x2	ADJ
ejpam-6211	416	2	−	−	NUM
ejpam-6211	416	3	1	1	NUM
ejpam-6211	416	4	2	2	NUM
ejpam-6211	416	5	)	)	PUNCT
ejpam-6211	416	6	n−k	n−k	NOUN
ejpam-6211	416	7	(	(	PUNCT
ejpam-6211	416	8	1	1	NUM
ejpam-6211	416	9	+	+	CCONJ
ejpam-6211	416	10	(	(	PUNCT
ejpam-6211	416	11	−1)n−k	−1)n−k	NOUN
ejpam-6211	416	12	)	)	PUNCT
ejpam-6211	416	13			X
ejpam-6211	416	14	tn	tn	PROPN
ejpam-6211	416	15	n	n	CCONJ
ejpam-6211	416	16	!	!	X
ejpam-6211	416	17	comparing	compare	VERB
ejpam-6211	416	18	coefficients	coefficient	NOUN
ejpam-6211	416	19	of	of	ADP
ejpam-6211	416	20	tn	tn	NOUN
ejpam-6211	416	21	n	n	CCONJ
ejpam-6211	416	22	!	!	PUNCT
ejpam-6211	417	1	and	and	CCONJ
ejpam-6211	417	2	expressing	express	VERB
ejpam-6211	417	3	2n−k+1	2n−k+1	NUM
ejpam-6211	417	4	as	as	ADP
ejpam-6211	417	5	2n2k−1	2n2k−1	NUM
ejpam-6211	417	6	yields	yield	NOUN
ejpam-6211	417	7	,	,	PUNCT
ejpam-6211	417	8	n−1∑	n−1∑	PROPN
ejpam-6211	417	9	k=0	k=0	PROPN
ejpam-6211	417	10	st	st	PROPN
ejpam-6211	417	11	2	2	NUM
ejpam-6211	417	12	k	k	X
ejpam-6211	417	13	(	(	PUNCT
ejpam-6211	417	14	m	m	PROPN
ejpam-6211	417	15	;	;	PUNCT
ejpam-6211	417	16	r	r	X
ejpam-6211	417	17	,	,	PUNCT
ejpam-6211	417	18	x	x	NOUN
ejpam-6211	417	19	)	)	PUNCT
ejpam-6211	417	20	(	(	PUNCT
ejpam-6211	417	21	n	n	X
ejpam-6211	417	22	k	k	X
ejpam-6211	417	23	)	)	PUNCT
ejpam-6211	417	24	1	1	NUM
ejpam-6211	417	25	2n	2n	NUM
ejpam-6211	417	26	(	(	PUNCT
ejpam-6211	417	27	(	(	PUNCT
ejpam-6211	417	28	√	√	INTJ
ejpam-6211	417	29	x2	x2	NOUN
ejpam-6211	418	1	+	+	CCONJ
ejpam-6211	418	2	4	4	X
ejpam-6211	418	3	)	)	PUNCT
ejpam-6211	418	4	n−k	n−k	NOUN
ejpam-6211	418	5	(	(	PUNCT
ejpam-6211	418	6	1−	1−	NUM
ejpam-6211	418	7	(	(	PUNCT
ejpam-6211	418	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	418	9	)	)	PUNCT
ejpam-6211	418	10	)	)	PUNCT
ejpam-6211	419	1	=	=	PUNCT
ejpam-6211	419	2	e	e	X
ejpam-6211	419	3	−rt	−rt	X
ejpam-6211	419	4	2	2	NUM
ejpam-6211	419	5	(	(	PUNCT
ejpam-6211	419	6	e	e	NOUN
ejpam-6211	419	7	t	t	PROPN
ejpam-6211	419	8	2	2	NUM
ejpam-6211	419	9	−	−	PROPN
ejpam-6211	419	10	1	1	NUM
ejpam-6211	419	11	et	et	NOUN
ejpam-6211	419	12	−	−	NOUN
ejpam-6211	419	13	1	1	NUM
ejpam-6211	419	14	)	)	PUNCT
ejpam-6211	419	15	m	m	VERB
ejpam-6211	419	16	√	√	NUM
ejpam-6211	419	17	x2	x2	INTJ
ejpam-6211	419	18	+	+	CCONJ
ejpam-6211	419	19	4	4	NUM
ejpam-6211	419	20	2	2	NUM
ejpam-6211	419	21	n∑	n∑	NOUN
ejpam-6211	419	22	k=1	k=1	PUNCT
ejpam-6211	420	1	sf	sf	NOUN
ejpam-6211	420	2	2	2	NUM
ejpam-6211	420	3	k	k	X
ejpam-6211	420	4	(	(	PUNCT
ejpam-6211	420	5	m	m	PROPN
ejpam-6211	420	6	;	;	PUNCT
ejpam-6211	420	7	r	r	X
ejpam-6211	420	8	,	,	PUNCT
ejpam-6211	420	9	x	x	NOUN
ejpam-6211	420	10	)	)	PUNCT
ejpam-6211	420	11	(	(	PUNCT
ejpam-6211	420	12	n	n	X
ejpam-6211	420	13	k	k	X
ejpam-6211	420	14	)	)	PUNCT
ejpam-6211	420	15	×	×	PROPN
ejpam-6211	420	16	(√	(√	VERB
ejpam-6211	420	17	x2	x2	ADJ
ejpam-6211	420	18	−	−	NUM
ejpam-6211	420	19	1	1	NUM
ejpam-6211	420	20	2	2	NUM
ejpam-6211	420	21	)	)	PUNCT
ejpam-6211	420	22	n−k	n−k	NOUN
ejpam-6211	420	23	(	(	PUNCT
ejpam-6211	420	24	1	1	NUM
ejpam-6211	420	25	+	+	CCONJ
ejpam-6211	420	26	(	(	PUNCT
ejpam-6211	420	27	−1)n−k	−1)n−k	NOUN
ejpam-6211	420	28	)	)	PUNCT
ejpam-6211	421	1			PROPN
ejpam-6211	421	2	multiplying	multiply	VERB
ejpam-6211	421	3	2n√	2n√	NUM
ejpam-6211	421	4	x2	x2	PRON
ejpam-6211	422	1	+	+	NOUN
ejpam-6211	422	2	4	4	NUM
ejpam-6211	422	3	both	both	DET
ejpam-6211	422	4	sides	side	NOUN
ejpam-6211	422	5	yields	yield	NOUN
ejpam-6211	422	6	(	(	PUNCT
ejpam-6211	422	7	5.8	5.8	NUM
ejpam-6211	422	8	)	)	PUNCT
ejpam-6211	422	9	theorem	theorem	VERB
ejpam-6211	422	10	5.6	5.6	NUM
ejpam-6211	422	11	.	.	PUNCT
ejpam-6211	423	1	for	for	ADP
ejpam-6211	423	2	n	n	PRON
ejpam-6211	423	3	≥	≥	X
ejpam-6211	423	4	0	0	NUM
ejpam-6211	423	5	the	the	DET
ejpam-6211	423	6	formula	formula	NOUN
ejpam-6211	423	7	holds	hold	VERB
ejpam-6211	423	8	,	,	PUNCT
ejpam-6211	423	9	n−1∑	n−1∑	NUM
ejpam-6211	423	10	k=0	k=0	PROPN
ejpam-6211	423	11	su1	su1	X
ejpam-6211	424	1	k	k	X
ejpam-6211	424	2	(	(	PUNCT
ejpam-6211	424	3	m	m	PROPN
ejpam-6211	424	4	;	;	PUNCT
ejpam-6211	424	5	r	r	X
ejpam-6211	424	6	,	,	PUNCT
ejpam-6211	424	7	x	x	NOUN
ejpam-6211	424	8	)	)	PUNCT
ejpam-6211	424	9	(	(	PUNCT
ejpam-6211	424	10	n	n	X
ejpam-6211	424	11	k	k	NOUN
ejpam-6211	424	12	)	)	PUNCT
ejpam-6211	424	13	(	(	PUNCT
ejpam-6211	424	14	(	(	PUNCT
ejpam-6211	424	15	√	√	INTJ
ejpam-6211	424	16	x2	x2	NOUN
ejpam-6211	425	1	+	+	CCONJ
ejpam-6211	425	2	4	4	X
ejpam-6211	425	3	)	)	PUNCT
ejpam-6211	425	4	n−k−1	n−k−1	PROPN
ejpam-6211	425	5	(	(	PUNCT
ejpam-6211	425	6	1−	1−	NUM
ejpam-6211	425	7	(	(	PUNCT
ejpam-6211	425	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	425	9	)	)	PUNCT
ejpam-6211	425	10	)	)	PUNCT
ejpam-6211	426	1	19	19	NUM
ejpam-6211	426	2	of	of	ADP
ejpam-6211	426	3	23	23	NUM
ejpam-6211	426	4	=	=	SYM
ejpam-6211	426	5	n∑	n∑	NOUN
ejpam-6211	426	6	k=1	k=1	PUNCT
ejpam-6211	427	1	sf	sf	NOUN
ejpam-6211	427	2	1	1	NUM
ejpam-6211	427	3	k	k	X
ejpam-6211	427	4	(	(	PUNCT
ejpam-6211	427	5	m	m	PROPN
ejpam-6211	427	6	;	;	PUNCT
ejpam-6211	427	7	r	r	X
ejpam-6211	427	8	,	,	PUNCT
ejpam-6211	427	9	x	x	NOUN
ejpam-6211	427	10	)	)	PUNCT
ejpam-6211	427	11	(	(	PUNCT
ejpam-6211	427	12	n	n	X
ejpam-6211	427	13	k	k	NOUN
ejpam-6211	427	14	)	)	PUNCT
ejpam-6211	427	15	(	(	PUNCT
ejpam-6211	427	16	2	2	NUM
ejpam-6211	427	17	+	+	SYM
ejpam-6211	427	18	2	2	NUM
ejpam-6211	427	19	t	t	NOUN
ejpam-6211	427	20	2	2	NUM
ejpam-6211	427	21	+	+	NUM
ejpam-6211	427	22	t	t	NOUN
ejpam-6211	427	23	)	)	PUNCT
ejpam-6211	427	24	r	r	NOUN
ejpam-6211	427	25	lnm	lnm	NOUN
ejpam-6211	427	26	(	(	PUNCT
ejpam-6211	427	27	−t	−t	PROPN
ejpam-6211	427	28	2	2	NUM
ejpam-6211	427	29	)	)	PUNCT
ejpam-6211	427	30	2k−1	2k−1	NUM
ejpam-6211	427	31	×	×	NOUN
ejpam-6211	427	32	(	(	PUNCT
ejpam-6211	427	33	(	(	PUNCT
ejpam-6211	427	34	√	√	INTJ
ejpam-6211	427	35	x2	x2	NUM
ejpam-6211	427	36	−	−	NOUN
ejpam-6211	427	37	1	1	X
ejpam-6211	427	38	)	)	PUNCT
ejpam-6211	427	39	n−k	n−k	NOUN
ejpam-6211	427	40	(	(	PUNCT
ejpam-6211	427	41	α(x)−	α(x)−	PROPN
ejpam-6211	427	42	β(x)(−1)n−k	β(x)(−1)n−k	PUNCT
ejpam-6211	427	43	)	)	PUNCT
ejpam-6211	427	44	)	)	PUNCT
ejpam-6211	427	45	(	(	PUNCT
ejpam-6211	427	46	5.10	5.10	NUM
ejpam-6211	427	47	)	)	PUNCT
ejpam-6211	427	48	where	where	SCONJ
ejpam-6211	427	49	α(x	α(x	NOUN
ejpam-6211	427	50	)	)	PUNCT
ejpam-6211	427	51	=	=	PUNCT
ejpam-6211	427	52	x+	x+	PUNCT
ejpam-6211	427	53	√	√	PUNCT
ejpam-6211	427	54	x2	x2	INTJ
ejpam-6211	427	55	−	−	PROPN
ejpam-6211	427	56	1	1	NUM
ejpam-6211	427	57	,	,	PUNCT
ejpam-6211	427	58	β(x	β(x	NOUN
ejpam-6211	427	59	)	)	PUNCT
ejpam-6211	427	60	=	=	SYM
ejpam-6211	427	61	x−	x−	PROPN
ejpam-6211	427	62	√	√	NUM
ejpam-6211	428	1	x2	x2	NUM
ejpam-6211	429	1	−	−	PROPN
ejpam-6211	429	2	1,t	1,t	X
ejpam-6211	429	3	<	<	X
ejpam-6211	429	4	0	0	NUM
ejpam-6211	429	5	and	and	CCONJ
ejpam-6211	429	6	t	t	PROPN
ejpam-6211	429	7	̸=	̸=	PROPN
ejpam-6211	429	8	−2	−2	NOUN
ejpam-6211	429	9	.	.	PUNCT
ejpam-6211	430	1	proof	proof	NOUN
ejpam-6211	430	2	to	to	PART
ejpam-6211	430	3	prove	prove	VERB
ejpam-6211	430	4	(	(	PUNCT
ejpam-6211	430	5	5.10	5.10	NUM
ejpam-6211	430	6	)	)	PUNCT
ejpam-6211	430	7	we	we	PRON
ejpam-6211	430	8	use	use	VERB
ejpam-6211	430	9	the	the	DET
ejpam-6211	430	10	exponential	exponential	ADJ
ejpam-6211	430	11	generating	generating	NOUN
ejpam-6211	430	12	functions	function	NOUN
ejpam-6211	430	13	in	in	ADP
ejpam-6211	430	14	(	(	PUNCT
ejpam-6211	430	15	4.1	4.1	NUM
ejpam-6211	430	16	)	)	PUNCT
ejpam-6211	430	17	and	and	CCONJ
ejpam-6211	430	18	(	(	PUNCT
ejpam-6211	430	19	5.3),which	5.3),which	PRON
ejpam-6211	430	20	gives	give	VERB
ejpam-6211	430	21	a	a	DET
ejpam-6211	430	22	functional	functional	ADJ
ejpam-6211	430	23	equation	equation	NOUN
ejpam-6211	430	24	,	,	PUNCT
ejpam-6211	430	25	2	2	NUM
ejpam-6211	430	26	√	√	NUM
ejpam-6211	430	27	x2	x2	NUM
ejpam-6211	430	28	−	−	PROPN
ejpam-6211	430	29	1	1	NUM
ejpam-6211	430	30	sinh	sinh	NOUN
ejpam-6211	430	31	(	(	PUNCT
ejpam-6211	430	32	√	√	PROPN
ejpam-6211	430	33	x2	x2	PROPN
ejpam-6211	431	1	+	+	CCONJ
ejpam-6211	431	2	4	4	NUM
ejpam-6211	431	3	2	2	NUM
ejpam-6211	431	4	t	t	NOUN
ejpam-6211	431	5	)	)	PUNCT
ejpam-6211	431	6	ω1	ω1	PROPN
ejpam-6211	431	7	(	(	PUNCT
ejpam-6211	431	8	t	t	PROPN
ejpam-6211	431	9	2	2	NUM
ejpam-6211	431	10	,	,	PUNCT
ejpam-6211	431	11	x	x	X
ejpam-6211	431	12	)	)	PUNCT
ejpam-6211	432	1	=	=	SYM
ejpam-6211	432	2	2r(1	2r(1	X
ejpam-6211	433	1	+	+	NUM
ejpam-6211	433	2	t)r	t)r	X
ejpam-6211	433	3	lnk	lnk	NOUN
ejpam-6211	433	4	(	(	PUNCT
ejpam-6211	433	5	2+t	2+t	NUM
ejpam-6211	433	6	2	2	NUM
ejpam-6211	433	7	)	)	PUNCT
ejpam-6211	433	8	(	(	PUNCT
ejpam-6211	433	9	2	2	NUM
ejpam-6211	433	10	+	+	CCONJ
ejpam-6211	433	11	t)r	t)r	ADJ
ejpam-6211	433	12	lnk(1	lnk(1	NOUN
ejpam-6211	433	13	+	+	CCONJ
ejpam-6211	433	14	t	t	NOUN
ejpam-6211	433	15	)	)	PUNCT
ejpam-6211	433	16	√	√	PUNCT
ejpam-6211	433	17	x2	x2	PROPN
ejpam-6211	434	1	+	+	CCONJ
ejpam-6211	434	2	4	4	NUM
ejpam-6211	434	3	(	(	PUNCT
ejpam-6211	434	4	x	x	NOUN
ejpam-6211	434	5	sinh	sinh	NOUN
ejpam-6211	434	6	(	(	PUNCT
ejpam-6211	434	7	√	√	PROPN
ejpam-6211	434	8	x2	x2	INTJ
ejpam-6211	434	9	−	−	NOUN
ejpam-6211	434	10	1	1	NUM
ejpam-6211	434	11	2	2	NUM
ejpam-6211	434	12	t	t	NOUN
ejpam-6211	434	13	)	)	PUNCT
ejpam-6211	435	1	+	+	CCONJ
ejpam-6211	435	2	√	√	ADJ
ejpam-6211	435	3	x2	x2	NUM
ejpam-6211	435	4	−	−	PROPN
ejpam-6211	435	5	1	1	NUM
ejpam-6211	435	6	cosh	cosh	NOUN
ejpam-6211	435	7	(	(	PUNCT
ejpam-6211	435	8	√	√	PROPN
ejpam-6211	435	9	x2	x2	INTJ
ejpam-6211	435	10	−	−	NOUN
ejpam-6211	435	11	1	1	NUM
ejpam-6211	435	12	2	2	NUM
ejpam-6211	435	13	t	t	NOUN
ejpam-6211	435	14	)	)	PUNCT
ejpam-6211	435	15	)	)	PUNCT
ejpam-6211	436	1	ϕ(t	ϕ(t	NUM
ejpam-6211	436	2	,	,	PUNCT
ejpam-6211	436	3	x	x	X
ejpam-6211	436	4	)	)	PUNCT
ejpam-6211	436	5	(	(	PUNCT
ejpam-6211	436	6	5.11	5.11	NUM
ejpam-6211	436	7	)	)	PUNCT
ejpam-6211	436	8	after	after	ADP
ejpam-6211	436	9	solving	solve	VERB
ejpam-6211	436	10	the	the	DET
ejpam-6211	436	11	equation	equation	NOUN
ejpam-6211	436	12	by	by	ADP
ejpam-6211	436	13	expressing	express	VERB
ejpam-6211	436	14	the	the	DET
ejpam-6211	436	15	hyperbolic	hyperbolic	ADJ
ejpam-6211	436	16	functions	function	NOUN
ejpam-6211	436	17	in	in	ADP
ejpam-6211	436	18	terms	term	NOUN
ejpam-6211	436	19	of	of	ADP
ejpam-6211	436	20	exponential	exponential	ADJ
ejpam-6211	436	21	functions	function	NOUN
ejpam-6211	436	22	and	and	CCONJ
ejpam-6211	436	23	cauchy	cauchy	NOUN
ejpam-6211	436	24	product	product	NOUN
ejpam-6211	436	25	yields	yield	VERB
ejpam-6211	436	26	lhs	lhs	PROPN
ejpam-6211	436	27	and	and	CCONJ
ejpam-6211	436	28	rhs	rhs	PROPN
ejpam-6211	436	29	expressions	expression	NOUN
ejpam-6211	436	30	,	,	PUNCT
ejpam-6211	436	31	for	for	ADP
ejpam-6211	436	32	lhs	lhs	PROPN
ejpam-6211	436	33	,	,	PUNCT
ejpam-6211	436	34	lhs	lhs	PROPN
ejpam-6211	436	35	=	=	PROPN
ejpam-6211	437	1	∞∑	∞∑	NUM
ejpam-6211	437	2	n=0	n=0	PROPN
ejpam-6211	437	3	{	{	PUNCT
ejpam-6211	437	4	n−1∑	n−1∑	PROPN
ejpam-6211	437	5	k=0	k=0	PROPN
ejpam-6211	437	6	su1	su1	X
ejpam-6211	437	7	k	k	X
ejpam-6211	437	8	(	(	PUNCT
ejpam-6211	437	9	m	m	PROPN
ejpam-6211	437	10	;	;	PUNCT
ejpam-6211	437	11	r	r	X
ejpam-6211	437	12	,	,	PUNCT
ejpam-6211	437	13	x	x	NOUN
ejpam-6211	437	14	)	)	PUNCT
ejpam-6211	437	15	1	1	NUM
ejpam-6211	437	16	2n	2n	NUM
ejpam-6211	437	17	(	(	PUNCT
ejpam-6211	437	18	n	n	X
ejpam-6211	437	19	k	k	NOUN
ejpam-6211	437	20	)	)	PUNCT
ejpam-6211	437	21	(	(	PUNCT
ejpam-6211	437	22	(	(	PUNCT
ejpam-6211	437	23	√	√	INTJ
ejpam-6211	437	24	x2	x2	NOUN
ejpam-6211	438	1	+	+	CCONJ
ejpam-6211	438	2	4	4	X
ejpam-6211	438	3	)	)	PUNCT
ejpam-6211	438	4	n−k	n−k	NOUN
ejpam-6211	438	5	(	(	PUNCT
ejpam-6211	438	6	1−	1−	NUM
ejpam-6211	438	7	(	(	PUNCT
ejpam-6211	438	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	438	9	)	)	PUNCT
ejpam-6211	438	10	)	)	PUNCT
ejpam-6211	438	11	}	}	PUNCT
ejpam-6211	438	12	tn	tn	PROPN
ejpam-6211	438	13	n	n	CCONJ
ejpam-6211	438	14	!	!	PUNCT
ejpam-6211	439	1	similarly	similarly	ADV
ejpam-6211	439	2	the	the	DET
ejpam-6211	439	3	rhs	rhs	PROPN
ejpam-6211	439	4	,	,	PUNCT
ejpam-6211	439	5	rhs	rhs	X
ejpam-6211	439	6	=	=	PUNCT
ejpam-6211	440	1	∞∑	∞∑	ADJ
ejpam-6211	440	2	n=0	n=0	NUM
ejpam-6211	440	3	n∑	n∑	NOUN
ejpam-6211	440	4	k=0	k=0	PROPN
ejpam-6211	440	5	sf	sf	PROPN
ejpam-6211	440	6	1	1	NUM
ejpam-6211	440	7	k	k	X
ejpam-6211	440	8	(	(	PUNCT
ejpam-6211	440	9	m	m	PROPN
ejpam-6211	440	10	;	;	PUNCT
ejpam-6211	440	11	r	r	X
ejpam-6211	440	12	,	,	PUNCT
ejpam-6211	440	13	x	x	NOUN
ejpam-6211	440	14	)	)	PUNCT
ejpam-6211	440	15	n	n	CCONJ
ejpam-6211	440	16	!	!	PUNCT
ejpam-6211	440	17	k!(n−	k!(n−	PROPN
ejpam-6211	440	18	k	k	X
ejpam-6211	440	19	)	)	PUNCT
ejpam-6211	440	20	!	!	PUNCT
ejpam-6211	441	1	(	(	PUNCT
ejpam-6211	441	2	2	2	NUM
ejpam-6211	441	3	+	+	SYM
ejpam-6211	441	4	2	2	NUM
ejpam-6211	441	5	t	t	NOUN
ejpam-6211	441	6	2	2	NUM
ejpam-6211	441	7	+	+	NUM
ejpam-6211	441	8	t	t	NOUN
ejpam-6211	441	9	)	)	PUNCT
ejpam-6211	441	10	r	r	NOUN
ejpam-6211	441	11	(	(	PUNCT
ejpam-6211	441	12	lnm	lnm	PROPN
ejpam-6211	441	13	(	(	PUNCT
ejpam-6211	441	14	−t	−t	PROPN
ejpam-6211	441	15	2	2	NUM
ejpam-6211	441	16	)	)	PUNCT
ejpam-6211	441	17	)	)	PUNCT
ejpam-6211	442	1	×	×	NOUN
ejpam-6211	443	1	√	√	INTJ
ejpam-6211	444	1	x2	x2	PROPN
ejpam-6211	445	1	+	+	CCONJ
ejpam-6211	445	2	4	4	NUM
ejpam-6211	445	3	2	2	NUM
ejpam-6211	445	4	α(x	α(x	NOUN
ejpam-6211	445	5	)	)	PUNCT
ejpam-6211	445	6	(	(	PUNCT
ejpam-6211	445	7	√	√	NUM
ejpam-6211	446	1	x2	x2	INTJ
ejpam-6211	446	2	−	−	NUM
ejpam-6211	446	3	1	1	NUM
ejpam-6211	446	4	2	2	NUM
ejpam-6211	446	5	)	)	PUNCT
ejpam-6211	446	6	n−k	n−k	NOUN
ejpam-6211	446	7	−	−	PROPN
ejpam-6211	446	8	β(x	β(x	NOUN
ejpam-6211	446	9	)	)	PUNCT
ejpam-6211	446	10	(	(	PUNCT
ejpam-6211	446	11	√	√	NUM
ejpam-6211	447	1	x2	x2	INTJ
ejpam-6211	447	2	−	−	NUM
ejpam-6211	447	3	1	1	NUM
ejpam-6211	447	4	2	2	NUM
ejpam-6211	447	5	)	)	PUNCT
ejpam-6211	447	6	n−k	n−k	NOUN
ejpam-6211	447	7			PROPN
ejpam-6211	447	8	tn	tn	PROPN
ejpam-6211	447	9	n	n	X
ejpam-6211	447	10	!	!	PUNCT
ejpam-6211	447	11	.	.	PUNCT
ejpam-6211	448	1	now	now	ADV
ejpam-6211	448	2	,	,	PUNCT
ejpam-6211	448	3	by	by	ADP
ejpam-6211	448	4	(	(	PUNCT
ejpam-6211	448	5	5.11	5.11	NUM
ejpam-6211	448	6	)	)	PUNCT
ejpam-6211	448	7	we	we	PRON
ejpam-6211	448	8	have	have	VERB
ejpam-6211	448	9	∞∑	∞∑	NUM
ejpam-6211	448	10	n=0	n=0	PROPN
ejpam-6211	448	11	{	{	PUNCT
ejpam-6211	448	12	n−1∑	n−1∑	PROPN
ejpam-6211	448	13	k=0	k=0	PROPN
ejpam-6211	448	14	su1	su1	X
ejpam-6211	448	15	k	k	X
ejpam-6211	448	16	(	(	PUNCT
ejpam-6211	448	17	m	m	PROPN
ejpam-6211	448	18	;	;	PUNCT
ejpam-6211	449	1	r	r	X
ejpam-6211	449	2	,	,	PUNCT
ejpam-6211	449	3	x	x	NOUN
ejpam-6211	449	4	)	)	PUNCT
ejpam-6211	449	5	1	1	NUM
ejpam-6211	449	6	2n	2n	NUM
ejpam-6211	449	7	(	(	PUNCT
ejpam-6211	449	8	n	n	X
ejpam-6211	449	9	k	k	NOUN
ejpam-6211	449	10	)	)	PUNCT
ejpam-6211	449	11	(	(	PUNCT
ejpam-6211	449	12	(	(	PUNCT
ejpam-6211	449	13	√	√	INTJ
ejpam-6211	449	14	x2	x2	NOUN
ejpam-6211	450	1	+	+	CCONJ
ejpam-6211	450	2	4	4	X
ejpam-6211	450	3	)	)	PUNCT
ejpam-6211	450	4	n−k	n−k	NOUN
ejpam-6211	450	5	(	(	PUNCT
ejpam-6211	450	6	1−	1−	NUM
ejpam-6211	450	7	(	(	PUNCT
ejpam-6211	450	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	450	9	)	)	PUNCT
ejpam-6211	450	10	)	)	PUNCT
ejpam-6211	450	11	}	}	PUNCT
ejpam-6211	450	12	tn	tn	PROPN
ejpam-6211	450	13	n	n	NOUN
ejpam-6211	450	14	!	!	PUNCT
ejpam-6211	450	15	=	=	NOUN
ejpam-6211	451	1	∞∑	∞∑	PRON
ejpam-6211	451	2	n=0	n=0	NUM
ejpam-6211	451	3	n∑	n∑	NOUN
ejpam-6211	451	4	k=1	k=1	PUNCT
ejpam-6211	452	1	sf	sf	NOUN
ejpam-6211	452	2	1	1	NUM
ejpam-6211	452	3	k	k	X
ejpam-6211	452	4	(	(	PUNCT
ejpam-6211	452	5	m	m	PROPN
ejpam-6211	452	6	;	;	PUNCT
ejpam-6211	452	7	r	r	X
ejpam-6211	452	8	,	,	PUNCT
ejpam-6211	452	9	x	x	NOUN
ejpam-6211	452	10	)	)	PUNCT
ejpam-6211	452	11	(	(	PUNCT
ejpam-6211	452	12	n	n	X
ejpam-6211	452	13	k	k	NOUN
ejpam-6211	452	14	)	)	PUNCT
ejpam-6211	452	15	(	(	PUNCT
ejpam-6211	452	16	2	2	NUM
ejpam-6211	452	17	+	+	SYM
ejpam-6211	452	18	2	2	NUM
ejpam-6211	452	19	t	t	NOUN
ejpam-6211	452	20	2	2	NUM
ejpam-6211	452	21	+	+	NUM
ejpam-6211	452	22	t	t	NOUN
ejpam-6211	452	23	)	)	PUNCT
ejpam-6211	452	24	r	r	NOUN
ejpam-6211	452	25	(	(	PUNCT
ejpam-6211	452	26	lnm	lnm	PROPN
ejpam-6211	452	27	(	(	PUNCT
ejpam-6211	452	28	−t	−t	PROPN
ejpam-6211	452	29	2	2	NUM
ejpam-6211	452	30	)	)	PUNCT
ejpam-6211	452	31	)	)	PUNCT
ejpam-6211	453	1	×	×	NOUN
ejpam-6211	454	1	√	√	INTJ
ejpam-6211	455	1	x2	x2	PROPN
ejpam-6211	456	1	+	+	CCONJ
ejpam-6211	456	2	4	4	NUM
ejpam-6211	456	3	2n−k+1	2n−k+1	NUM
ejpam-6211	456	4	(	(	PUNCT
ejpam-6211	456	5	(	(	PUNCT
ejpam-6211	456	6	√	√	INTJ
ejpam-6211	456	7	x2	x2	NUM
ejpam-6211	457	1	−	−	NOUN
ejpam-6211	457	2	1	1	X
ejpam-6211	457	3	)	)	PUNCT
ejpam-6211	457	4	n−k	n−k	NOUN
ejpam-6211	457	5	(	(	PUNCT
ejpam-6211	457	6	α(x)−	α(x)−	PROPN
ejpam-6211	457	7	β(x)(−1)n−k	β(x)(−1)n−k	PUNCT
ejpam-6211	457	8	)	)	PUNCT
ejpam-6211	457	9	)	)	PUNCT
ejpam-6211	457	10	tn	tn	PROPN
ejpam-6211	457	11	n	n	CCONJ
ejpam-6211	457	12	!	!	PROPN
ejpam-6211	457	13	20	20	NUM
ejpam-6211	457	14	of	of	ADP
ejpam-6211	457	15	23	23	NUM
ejpam-6211	457	16	comparing	compare	VERB
ejpam-6211	457	17	coefficients	coefficient	NOUN
ejpam-6211	457	18	of	of	ADP
ejpam-6211	457	19	tn	tn	NOUN
ejpam-6211	457	20	n	n	CCONJ
ejpam-6211	457	21	!	!	PUNCT
ejpam-6211	458	1	and	and	CCONJ
ejpam-6211	458	2	expressing	express	VERB
ejpam-6211	458	3	2n−k+1	2n−k+1	NUM
ejpam-6211	458	4	as	as	ADP
ejpam-6211	458	5	2n2k−1	2n2k−1	NUM
ejpam-6211	458	6	yields	yield	NOUN
ejpam-6211	458	7	,	,	PUNCT
ejpam-6211	458	8	n−1∑	n−1∑	NUM
ejpam-6211	458	9	k=0	k=0	PROPN
ejpam-6211	458	10	su1	su1	X
ejpam-6211	459	1	k	k	X
ejpam-6211	459	2	(	(	PUNCT
ejpam-6211	459	3	m	m	PROPN
ejpam-6211	459	4	;	;	PUNCT
ejpam-6211	459	5	r	r	X
ejpam-6211	459	6	,	,	PUNCT
ejpam-6211	459	7	x	x	NOUN
ejpam-6211	459	8	)	)	PUNCT
ejpam-6211	459	9	1	1	NUM
ejpam-6211	459	10	2n	2n	NUM
ejpam-6211	459	11	(	(	PUNCT
ejpam-6211	459	12	n	n	X
ejpam-6211	459	13	k	k	NOUN
ejpam-6211	459	14	)	)	PUNCT
ejpam-6211	459	15	(	(	PUNCT
ejpam-6211	459	16	(	(	PUNCT
ejpam-6211	459	17	√	√	INTJ
ejpam-6211	459	18	x2	x2	NOUN
ejpam-6211	460	1	+	+	CCONJ
ejpam-6211	460	2	4	4	X
ejpam-6211	460	3	)	)	PUNCT
ejpam-6211	460	4	n−k	n−k	NOUN
ejpam-6211	460	5	(	(	PUNCT
ejpam-6211	460	6	1−	1−	NUM
ejpam-6211	460	7	(	(	PUNCT
ejpam-6211	460	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	460	9	)	)	PUNCT
ejpam-6211	460	10	)	)	PUNCT
ejpam-6211	461	1	=	=	PUNCT
ejpam-6211	462	1	n∑	n∑	INTJ
ejpam-6211	462	2	k=1	k=1	PUNCT
ejpam-6211	463	1	sf	sf	NOUN
ejpam-6211	463	2	1	1	NUM
ejpam-6211	463	3	k	k	X
ejpam-6211	463	4	(	(	PUNCT
ejpam-6211	463	5	m	m	PROPN
ejpam-6211	463	6	;	;	PUNCT
ejpam-6211	463	7	r	r	X
ejpam-6211	463	8	,	,	PUNCT
ejpam-6211	463	9	x	x	NOUN
ejpam-6211	463	10	)	)	PUNCT
ejpam-6211	463	11	(	(	PUNCT
ejpam-6211	463	12	n	n	X
ejpam-6211	463	13	k	k	NOUN
ejpam-6211	463	14	)	)	PUNCT
ejpam-6211	463	15	(	(	PUNCT
ejpam-6211	463	16	2	2	NUM
ejpam-6211	463	17	+	+	SYM
ejpam-6211	463	18	2	2	NUM
ejpam-6211	463	19	t	t	NOUN
ejpam-6211	463	20	2	2	NUM
ejpam-6211	463	21	+	+	NUM
ejpam-6211	463	22	t	t	NOUN
ejpam-6211	463	23	)	)	PUNCT
ejpam-6211	463	24	r	r	NOUN
ejpam-6211	463	25	(	(	PUNCT
ejpam-6211	463	26	lnm	lnm	PROPN
ejpam-6211	463	27	(	(	PUNCT
ejpam-6211	463	28	−t	−t	PROPN
ejpam-6211	463	29	2	2	NUM
ejpam-6211	463	30	)	)	PUNCT
ejpam-6211	463	31	)	)	PUNCT
ejpam-6211	464	1	×	×	NOUN
ejpam-6211	465	1	√	√	INTJ
ejpam-6211	466	1	x2	x2	PROPN
ejpam-6211	467	1	+	+	CCONJ
ejpam-6211	467	2	4	4	NUM
ejpam-6211	467	3	2n	2n	NUM
ejpam-6211	467	4	2k−1	2k−1	NUM
ejpam-6211	467	5	(	(	PUNCT
ejpam-6211	467	6	(	(	PUNCT
ejpam-6211	467	7	√	√	INTJ
ejpam-6211	467	8	x2	x2	NUM
ejpam-6211	467	9	−	−	NOUN
ejpam-6211	467	10	1	1	X
ejpam-6211	467	11	)	)	PUNCT
ejpam-6211	467	12	n−k	n−k	NOUN
ejpam-6211	467	13	(	(	PUNCT
ejpam-6211	467	14	α(x)−	α(x)−	PROPN
ejpam-6211	467	15	β(x)(−1)n−k	β(x)(−1)n−k	PUNCT
ejpam-6211	467	16	)	)	PUNCT
ejpam-6211	467	17	)	)	PUNCT
ejpam-6211	468	1	multiplying	multiply	VERB
ejpam-6211	468	2	2n√	2n√	NUM
ejpam-6211	468	3	x2	x2	PUNCT
ejpam-6211	469	1	+	+	NOUN
ejpam-6211	469	2	4	4	NUM
ejpam-6211	469	3	both	both	DET
ejpam-6211	469	4	sides	side	NOUN
ejpam-6211	469	5	yields	yield	NOUN
ejpam-6211	469	6	(	(	PUNCT
ejpam-6211	469	7	5.10	5.10	NUM
ejpam-6211	469	8	)	)	PUNCT
ejpam-6211	469	9	remark	remark	NOUN
ejpam-6211	469	10	5.7	5.7	NUM
ejpam-6211	469	11	.	.	PUNCT
ejpam-6211	470	1	the	the	DET
ejpam-6211	470	2	term	term	NOUN
ejpam-6211	470	3	(	(	PUNCT
ejpam-6211	470	4	2(1	2(1	NUM
ejpam-6211	470	5	+	+	NUM
ejpam-6211	470	6	t	t	NOUN
ejpam-6211	470	7	)	)	PUNCT
ejpam-6211	470	8	2	2	NUM
ejpam-6211	471	1	+	+	NUM
ejpam-6211	471	2	t	t	NOUN
ejpam-6211	471	3	)	)	PUNCT
ejpam-6211	471	4	r	r	NOUN
ejpam-6211	471	5	(	(	PUNCT
ejpam-6211	471	6	lnm	lnm	PROPN
ejpam-6211	471	7	(	(	PUNCT
ejpam-6211	471	8	−t	−t	PROPN
ejpam-6211	471	9	2	2	NUM
ejpam-6211	471	10	)	)	PUNCT
ejpam-6211	471	11	)	)	PUNCT
ejpam-6211	471	12	is	be	AUX
ejpam-6211	471	13	well	well	ADV
ejpam-6211	471	14	defined	define	VERB
ejpam-6211	471	15	when	when	SCONJ
ejpam-6211	471	16	t	t	PROPN
ejpam-6211	471	17	<	<	X
ejpam-6211	471	18	0	0	PUNCT
ejpam-6211	471	19	and	and	CCONJ
ejpam-6211	471	20	t	t	PROPN
ejpam-6211	471	21	̸=	̸=	PROPN
ejpam-6211	471	22	−2	−2	NOUN
ejpam-6211	471	23	.	.	PUNCT
ejpam-6211	472	1	theorem	theorem	VERB
ejpam-6211	472	2	5.8	5.8	NUM
ejpam-6211	472	3	.	.	PUNCT
ejpam-6211	473	1	for	for	ADP
ejpam-6211	473	2	n	n	PRON
ejpam-6211	473	3	≥	≥	X
ejpam-6211	473	4	0	0	NUM
ejpam-6211	473	5	the	the	DET
ejpam-6211	473	6	formula	formula	NOUN
ejpam-6211	473	7	holds	hold	VERB
ejpam-6211	473	8	,	,	PUNCT
ejpam-6211	473	9	n−1∑	n−1∑	NUM
ejpam-6211	473	10	k=0	k=0	PROPN
ejpam-6211	473	11	su1	su1	X
ejpam-6211	474	1	k	k	X
ejpam-6211	474	2	(	(	PUNCT
ejpam-6211	474	3	m	m	PROPN
ejpam-6211	474	4	;	;	PUNCT
ejpam-6211	474	5	r	r	X
ejpam-6211	474	6	,	,	PUNCT
ejpam-6211	474	7	x	x	NOUN
ejpam-6211	474	8	)	)	PUNCT
ejpam-6211	474	9	(	(	PUNCT
ejpam-6211	474	10	n	n	X
ejpam-6211	474	11	k	k	NOUN
ejpam-6211	474	12	)	)	PUNCT
ejpam-6211	474	13	(	(	PUNCT
ejpam-6211	474	14	(	(	PUNCT
ejpam-6211	474	15	√	√	INTJ
ejpam-6211	474	16	x2	x2	NOUN
ejpam-6211	475	1	+	+	CCONJ
ejpam-6211	475	2	4	4	X
ejpam-6211	475	3	)	)	PUNCT
ejpam-6211	475	4	n−k−1	n−k−1	PROPN
ejpam-6211	475	5	(	(	PUNCT
ejpam-6211	475	6	1−	1−	NUM
ejpam-6211	475	7	(	(	PUNCT
ejpam-6211	475	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	475	9	)	)	PUNCT
ejpam-6211	475	10	)	)	PUNCT
ejpam-6211	476	1	=	=	PUNCT
ejpam-6211	477	1	n∑	n∑	INTJ
ejpam-6211	477	2	k=1	k=1	PUNCT
ejpam-6211	478	1	sf	sf	NOUN
ejpam-6211	478	2	1	1	NUM
ejpam-6211	478	3	k	k	X
ejpam-6211	478	4	(	(	PUNCT
ejpam-6211	478	5	m	m	PROPN
ejpam-6211	478	6	;	;	PUNCT
ejpam-6211	478	7	r	r	X
ejpam-6211	478	8	,	,	PUNCT
ejpam-6211	478	9	x	x	NOUN
ejpam-6211	478	10	)	)	PUNCT
ejpam-6211	478	11	(	(	PUNCT
ejpam-6211	478	12	n	n	X
ejpam-6211	478	13	k	k	NOUN
ejpam-6211	478	14	)	)	PUNCT
ejpam-6211	478	15	(	(	PUNCT
ejpam-6211	478	16	e	e	X
ejpam-6211	478	17	−rt	−rt	X
ejpam-6211	478	18	2	2	NUM
ejpam-6211	478	19	(	(	PUNCT
ejpam-6211	478	20	e	e	NOUN
ejpam-6211	478	21	t	t	PROPN
ejpam-6211	478	22	2	2	NUM
ejpam-6211	478	23	−	−	PROPN
ejpam-6211	478	24	1	1	NUM
ejpam-6211	478	25	et	et	NOUN
ejpam-6211	478	26	−	−	NOUN
ejpam-6211	478	27	1	1	NUM
ejpam-6211	478	28	)	)	PUNCT
ejpam-6211	478	29	m	m	NOUN
ejpam-6211	478	30	)	)	PUNCT
ejpam-6211	478	31	2k−1	2k−1	NUM
ejpam-6211	478	32	(	(	PUNCT
ejpam-6211	478	33	(	(	PUNCT
ejpam-6211	478	34	√	√	INTJ
ejpam-6211	478	35	x2	x2	NUM
ejpam-6211	478	36	−	−	NOUN
ejpam-6211	478	37	1	1	X
ejpam-6211	478	38	)	)	PUNCT
ejpam-6211	478	39	n−k	n−k	NOUN
ejpam-6211	478	40	(	(	PUNCT
ejpam-6211	478	41	α(x)−	α(x)−	PROPN
ejpam-6211	478	42	β(x)(−1)n−k	β(x)(−1)n−k	PUNCT
ejpam-6211	478	43	)	)	PUNCT
ejpam-6211	478	44	)	)	PUNCT
ejpam-6211	478	45	(	(	PUNCT
ejpam-6211	478	46	5.12	5.12	NUM
ejpam-6211	478	47	)	)	PUNCT
ejpam-6211	478	48	where	where	SCONJ
ejpam-6211	478	49	α(x	α(x	NOUN
ejpam-6211	478	50	)	)	PUNCT
ejpam-6211	478	51	=	=	PUNCT
ejpam-6211	478	52	x+	x+	PUNCT
ejpam-6211	478	53	√	√	PUNCT
ejpam-6211	478	54	x2	x2	INTJ
ejpam-6211	478	55	−	−	PROPN
ejpam-6211	478	56	1	1	NUM
ejpam-6211	478	57	,	,	PUNCT
ejpam-6211	478	58	β(x	β(x	NOUN
ejpam-6211	478	59	)	)	PUNCT
ejpam-6211	478	60	=	=	SYM
ejpam-6211	478	61	x−	x−	PROPN
ejpam-6211	478	62	√	√	PROPN
ejpam-6211	479	1	x2	x2	INTJ
ejpam-6211	480	1	−	−	NOUN
ejpam-6211	481	1	1	1	X
ejpam-6211	481	2	.	.	PUNCT
ejpam-6211	481	3	proof	proof	NOUN
ejpam-6211	481	4	to	to	PART
ejpam-6211	481	5	prove	prove	VERB
ejpam-6211	481	6	(	(	PUNCT
ejpam-6211	481	7	5.12)we	5.12)we	NOUN
ejpam-6211	481	8	use	use	VERB
ejpam-6211	481	9	the	the	DET
ejpam-6211	481	10	exponential	exponential	ADJ
ejpam-6211	481	11	generating	generating	NOUN
ejpam-6211	481	12	functions	function	NOUN
ejpam-6211	481	13	in	in	ADP
ejpam-6211	481	14	eq	eq	ADP
ejpam-6211	481	15	.	.	PUNCT
ejpam-6211	482	1	(	(	PUNCT
ejpam-6211	482	2	4.2	4.2	NUM
ejpam-6211	482	3	)	)	PUNCT
ejpam-6211	482	4	and	and	CCONJ
ejpam-6211	482	5	(	(	PUNCT
ejpam-6211	482	6	5.4),which	5.4),which	PRON
ejpam-6211	482	7	gives	give	VERB
ejpam-6211	482	8	a	a	DET
ejpam-6211	482	9	functional	functional	ADJ
ejpam-6211	482	10	equation	equation	NOUN
ejpam-6211	482	11	,	,	PUNCT
ejpam-6211	482	12	2	2	NUM
ejpam-6211	482	13	√	√	NUM
ejpam-6211	482	14	x2	x2	NUM
ejpam-6211	482	15	−	−	PROPN
ejpam-6211	482	16	1	1	NUM
ejpam-6211	482	17	sinh	sinh	NOUN
ejpam-6211	482	18	(	(	PUNCT
ejpam-6211	482	19	√	√	PROPN
ejpam-6211	482	20	x2	x2	PROPN
ejpam-6211	483	1	+	+	CCONJ
ejpam-6211	483	2	4	4	NUM
ejpam-6211	483	3	2	2	NUM
ejpam-6211	483	4	t	t	NOUN
ejpam-6211	483	5	)	)	PUNCT
ejpam-6211	483	6	ω2	ω2	CCONJ
ejpam-6211	483	7	(	(	PUNCT
ejpam-6211	483	8	t	t	PROPN
ejpam-6211	483	9	2	2	NUM
ejpam-6211	483	10	,	,	PUNCT
ejpam-6211	483	11	x	x	X
ejpam-6211	483	12	)	)	PUNCT
ejpam-6211	484	1	=	=	PUNCT
ejpam-6211	484	2	e	e	X
ejpam-6211	484	3	−rt	−rt	X
ejpam-6211	484	4	2	2	NUM
ejpam-6211	484	5	(	(	PUNCT
ejpam-6211	484	6	e	e	NOUN
ejpam-6211	484	7	t	t	PROPN
ejpam-6211	484	8	2	2	NUM
ejpam-6211	484	9	−	−	PROPN
ejpam-6211	484	10	1	1	NUM
ejpam-6211	484	11	et	et	NOUN
ejpam-6211	484	12	−	−	NOUN
ejpam-6211	484	13	1	1	NUM
ejpam-6211	484	14	)	)	PUNCT
ejpam-6211	484	15	k√	k√	NOUN
ejpam-6211	484	16	x2	x2	PROPN
ejpam-6211	485	1	+	+	CCONJ
ejpam-6211	485	2	4	4	NUM
ejpam-6211	485	3	(	(	PUNCT
ejpam-6211	485	4	x	x	NOUN
ejpam-6211	485	5	sinh	sinh	NOUN
ejpam-6211	485	6	(	(	PUNCT
ejpam-6211	485	7	√	√	PROPN
ejpam-6211	485	8	x2	x2	INTJ
ejpam-6211	485	9	−	−	NOUN
ejpam-6211	485	10	1	1	NUM
ejpam-6211	485	11	2	2	NUM
ejpam-6211	485	12	t	t	NOUN
ejpam-6211	485	13	)	)	PUNCT
ejpam-6211	486	1	+	+	CCONJ
ejpam-6211	486	2	√	√	ADJ
ejpam-6211	486	3	x2	x2	NUM
ejpam-6211	486	4	−	−	PROPN
ejpam-6211	486	5	1	1	NUM
ejpam-6211	486	6	cosh	cosh	NOUN
ejpam-6211	486	7	(	(	PUNCT
ejpam-6211	486	8	√	√	PROPN
ejpam-6211	486	9	x2	x2	INTJ
ejpam-6211	486	10	−	−	NOUN
ejpam-6211	486	11	1	1	NUM
ejpam-6211	486	12	2	2	NUM
ejpam-6211	486	13	t	t	NOUN
ejpam-6211	486	14	)	)	PUNCT
ejpam-6211	486	15	)	)	PUNCT
ejpam-6211	487	1	γ(t	γ(t	NOUN
ejpam-6211	487	2	,	,	PUNCT
ejpam-6211	487	3	x	x	X
ejpam-6211	487	4	)	)	PUNCT
ejpam-6211	487	5	(	(	PUNCT
ejpam-6211	487	6	5.13	5.13	NUM
ejpam-6211	487	7	)	)	PUNCT
ejpam-6211	487	8	after	after	ADP
ejpam-6211	487	9	solving	solve	VERB
ejpam-6211	487	10	the	the	DET
ejpam-6211	487	11	equation	equation	NOUN
ejpam-6211	487	12	by	by	ADP
ejpam-6211	487	13	expressing	express	VERB
ejpam-6211	487	14	the	the	DET
ejpam-6211	487	15	hyperbolic	hyperbolic	ADJ
ejpam-6211	487	16	functions	function	NOUN
ejpam-6211	487	17	in	in	ADP
ejpam-6211	487	18	terms	term	NOUN
ejpam-6211	487	19	of	of	ADP
ejpam-6211	487	20	exponential	exponential	ADJ
ejpam-6211	487	21	functions	function	NOUN
ejpam-6211	487	22	and	and	CCONJ
ejpam-6211	487	23	cauchy	cauchy	NOUN
ejpam-6211	487	24	product	product	NOUN
ejpam-6211	487	25	yields	yield	VERB
ejpam-6211	487	26	lhs	lhs	PROPN
ejpam-6211	487	27	and	and	CCONJ
ejpam-6211	487	28	rhs	rhs	PROPN
ejpam-6211	487	29	expressions	expression	NOUN
ejpam-6211	487	30	,	,	PUNCT
ejpam-6211	487	31	21	21	NUM
ejpam-6211	487	32	of	of	ADP
ejpam-6211	487	33	23	23	NUM
ejpam-6211	487	34	for	for	ADP
ejpam-6211	487	35	lhs	lhs	PROPN
ejpam-6211	487	36	,	,	PUNCT
ejpam-6211	487	37	lhs	lhs	PROPN
ejpam-6211	487	38	=	=	PROPN
ejpam-6211	488	1	∞∑	∞∑	NUM
ejpam-6211	488	2	n=0	n=0	PROPN
ejpam-6211	488	3	{	{	PUNCT
ejpam-6211	488	4	n−1∑	n−1∑	PROPN
ejpam-6211	488	5	k=0	k=0	PROPN
ejpam-6211	488	6	su2	su2	PROPN
ejpam-6211	489	1	k	k	PROPN
ejpam-6211	489	2	(	(	PUNCT
ejpam-6211	489	3	m	m	PROPN
ejpam-6211	489	4	;	;	PUNCT
ejpam-6211	489	5	r	r	X
ejpam-6211	489	6	,	,	PUNCT
ejpam-6211	489	7	x	x	NOUN
ejpam-6211	489	8	)	)	PUNCT
ejpam-6211	489	9	1	1	NUM
ejpam-6211	489	10	2n	2n	NUM
ejpam-6211	489	11	(	(	PUNCT
ejpam-6211	489	12	n	n	X
ejpam-6211	489	13	k	k	NOUN
ejpam-6211	489	14	)	)	PUNCT
ejpam-6211	489	15	(	(	PUNCT
ejpam-6211	489	16	(	(	PUNCT
ejpam-6211	489	17	√	√	INTJ
ejpam-6211	489	18	x2	x2	NOUN
ejpam-6211	490	1	+	+	CCONJ
ejpam-6211	490	2	4	4	X
ejpam-6211	490	3	)	)	PUNCT
ejpam-6211	490	4	n−k	n−k	NOUN
ejpam-6211	490	5	(	(	PUNCT
ejpam-6211	490	6	1−	1−	NUM
ejpam-6211	490	7	(	(	PUNCT
ejpam-6211	490	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	490	9	)	)	PUNCT
ejpam-6211	490	10	)	)	PUNCT
ejpam-6211	490	11	}	}	PUNCT
ejpam-6211	490	12	tn	tn	PROPN
ejpam-6211	490	13	n	n	CCONJ
ejpam-6211	490	14	!	!	PUNCT
ejpam-6211	491	1	similarly	similarly	ADV
ejpam-6211	491	2	the	the	DET
ejpam-6211	491	3	rhs	rhs	PROPN
ejpam-6211	491	4	,	,	PUNCT
ejpam-6211	491	5	rhs	rhs	X
ejpam-6211	491	6	=	=	PUNCT
ejpam-6211	492	1	∞∑	∞∑	ADJ
ejpam-6211	492	2	n=0	n=0	NUM
ejpam-6211	492	3	n∑	n∑	NOUN
ejpam-6211	492	4	k=0	k=0	PROPN
ejpam-6211	492	5	sf	sf	PROPN
ejpam-6211	492	6	2	2	NUM
ejpam-6211	492	7	k	k	X
ejpam-6211	492	8	(	(	PUNCT
ejpam-6211	492	9	m	m	PROPN
ejpam-6211	492	10	;	;	PUNCT
ejpam-6211	492	11	r	r	X
ejpam-6211	492	12	,	,	PUNCT
ejpam-6211	492	13	x	x	NOUN
ejpam-6211	492	14	)	)	PUNCT
ejpam-6211	492	15	(	(	PUNCT
ejpam-6211	492	16	n	n	X
ejpam-6211	492	17	k	k	NOUN
ejpam-6211	492	18	)	)	PUNCT
ejpam-6211	492	19	(	(	PUNCT
ejpam-6211	492	20	e	e	X
ejpam-6211	492	21	−rt	−rt	X
ejpam-6211	492	22	2	2	NUM
ejpam-6211	492	23	(	(	PUNCT
ejpam-6211	492	24	e	e	NOUN
ejpam-6211	492	25	t	t	PROPN
ejpam-6211	492	26	2	2	NUM
ejpam-6211	492	27	−	−	PROPN
ejpam-6211	492	28	1	1	NUM
ejpam-6211	492	29	et	et	NOUN
ejpam-6211	492	30	−	−	NOUN
ejpam-6211	492	31	1	1	NUM
ejpam-6211	492	32	)	)	PUNCT
ejpam-6211	492	33	m	m	NOUN
ejpam-6211	492	34	)	)	PUNCT
ejpam-6211	492	35	×	×	NOUN
ejpam-6211	492	36	√	√	PUNCT
ejpam-6211	493	1	x2	x2	PROPN
ejpam-6211	494	1	+	+	CCONJ
ejpam-6211	494	2	4	4	NUM
ejpam-6211	494	3	2	2	NUM
ejpam-6211	494	4	α(x	α(x	NOUN
ejpam-6211	494	5	)	)	PUNCT
ejpam-6211	494	6	(	(	PUNCT
ejpam-6211	494	7	√	√	NUM
ejpam-6211	495	1	x2	x2	INTJ
ejpam-6211	495	2	−	−	NUM
ejpam-6211	495	3	1	1	NUM
ejpam-6211	495	4	2	2	NUM
ejpam-6211	495	5	)	)	PUNCT
ejpam-6211	495	6	n−k	n−k	NOUN
ejpam-6211	495	7	−	−	PROPN
ejpam-6211	495	8	β(x	β(x	NOUN
ejpam-6211	495	9	)	)	PUNCT
ejpam-6211	495	10	(	(	PUNCT
ejpam-6211	495	11	√	√	NUM
ejpam-6211	496	1	x2	x2	INTJ
ejpam-6211	496	2	−	−	NUM
ejpam-6211	496	3	1	1	NUM
ejpam-6211	496	4	2	2	NUM
ejpam-6211	496	5	)	)	PUNCT
ejpam-6211	496	6	n−k	n−k	NOUN
ejpam-6211	496	7			PROPN
ejpam-6211	496	8	tn	tn	PROPN
ejpam-6211	496	9	n	n	NOUN
ejpam-6211	496	10	!	!	PUNCT
ejpam-6211	497	1	now	now	ADV
ejpam-6211	497	2	,	,	PUNCT
ejpam-6211	497	3	by	by	ADP
ejpam-6211	497	4	(	(	PUNCT
ejpam-6211	497	5	5.13	5.13	NUM
ejpam-6211	497	6	)	)	PUNCT
ejpam-6211	497	7	we	we	PRON
ejpam-6211	497	8	have	have	VERB
ejpam-6211	497	9	∞∑	∞∑	NUM
ejpam-6211	497	10	n=0	n=0	PROPN
ejpam-6211	497	11	{	{	PUNCT
ejpam-6211	497	12	n−1∑	n−1∑	PROPN
ejpam-6211	497	13	k=0	k=0	PROPN
ejpam-6211	497	14	su2	su2	PROPN
ejpam-6211	497	15	k	k	PROPN
ejpam-6211	497	16	(	(	PUNCT
ejpam-6211	497	17	m	m	PROPN
ejpam-6211	497	18	;	;	PUNCT
ejpam-6211	497	19	r	r	X
ejpam-6211	497	20	,	,	PUNCT
ejpam-6211	497	21	x	x	NOUN
ejpam-6211	497	22	)	)	PUNCT
ejpam-6211	497	23	1	1	NUM
ejpam-6211	497	24	2n	2n	NUM
ejpam-6211	497	25	(	(	PUNCT
ejpam-6211	497	26	n	n	X
ejpam-6211	497	27	k	k	NOUN
ejpam-6211	497	28	)	)	PUNCT
ejpam-6211	497	29	(	(	PUNCT
ejpam-6211	497	30	(	(	PUNCT
ejpam-6211	497	31	√	√	INTJ
ejpam-6211	497	32	x2	x2	NOUN
ejpam-6211	498	1	+	+	CCONJ
ejpam-6211	498	2	4	4	X
ejpam-6211	498	3	)	)	PUNCT
ejpam-6211	498	4	n−k	n−k	NOUN
ejpam-6211	498	5	(	(	PUNCT
ejpam-6211	498	6	1−	1−	NUM
ejpam-6211	498	7	(	(	PUNCT
ejpam-6211	498	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	498	9	)	)	PUNCT
ejpam-6211	498	10	)	)	PUNCT
ejpam-6211	498	11	}	}	PUNCT
ejpam-6211	498	12	tn	tn	PROPN
ejpam-6211	498	13	n	n	NOUN
ejpam-6211	498	14	!	!	PUNCT
ejpam-6211	498	15	=	=	NOUN
ejpam-6211	499	1	∞∑	∞∑	PRON
ejpam-6211	499	2	n=0	n=0	NUM
ejpam-6211	499	3	n∑	n∑	NOUN
ejpam-6211	499	4	k=1	k=1	PUNCT
ejpam-6211	500	1	sf	sf	NOUN
ejpam-6211	500	2	2	2	NUM
ejpam-6211	500	3	k	k	X
ejpam-6211	500	4	(	(	PUNCT
ejpam-6211	500	5	m	m	PROPN
ejpam-6211	500	6	;	;	PUNCT
ejpam-6211	500	7	r	r	X
ejpam-6211	500	8	,	,	PUNCT
ejpam-6211	500	9	x	x	NOUN
ejpam-6211	500	10	)	)	PUNCT
ejpam-6211	500	11	(	(	PUNCT
ejpam-6211	500	12	n	n	X
ejpam-6211	500	13	k	k	NOUN
ejpam-6211	500	14	)	)	PUNCT
ejpam-6211	500	15	(	(	PUNCT
ejpam-6211	500	16	e	e	X
ejpam-6211	500	17	−rt	−rt	X
ejpam-6211	500	18	2	2	NUM
ejpam-6211	500	19	(	(	PUNCT
ejpam-6211	500	20	e	e	NOUN
ejpam-6211	500	21	t	t	PROPN
ejpam-6211	500	22	2	2	NUM
ejpam-6211	500	23	−	−	PROPN
ejpam-6211	500	24	1	1	NUM
ejpam-6211	500	25	et	et	NOUN
ejpam-6211	500	26	−	−	NOUN
ejpam-6211	500	27	1	1	NUM
ejpam-6211	500	28	)	)	PUNCT
ejpam-6211	500	29	m	m	NOUN
ejpam-6211	500	30	)	)	PUNCT
ejpam-6211	500	31	×	×	NOUN
ejpam-6211	500	32	√	√	PUNCT
ejpam-6211	500	33	x2	x2	PROPN
ejpam-6211	501	1	+	+	CCONJ
ejpam-6211	501	2	4	4	NUM
ejpam-6211	501	3	2n−k+1	2n−k+1	NUM
ejpam-6211	501	4	(	(	PUNCT
ejpam-6211	501	5	(	(	PUNCT
ejpam-6211	501	6	√	√	INTJ
ejpam-6211	501	7	x2	x2	NUM
ejpam-6211	502	1	−	−	NOUN
ejpam-6211	502	2	1	1	X
ejpam-6211	502	3	)	)	PUNCT
ejpam-6211	502	4	n−k	n−k	NOUN
ejpam-6211	502	5	(	(	PUNCT
ejpam-6211	502	6	α(x)−	α(x)−	PROPN
ejpam-6211	502	7	β(x)(−1)n−k	β(x)(−1)n−k	PUNCT
ejpam-6211	502	8	)	)	PUNCT
ejpam-6211	502	9	)	)	PUNCT
ejpam-6211	502	10	tn	tn	PROPN
ejpam-6211	503	1	n	n	CCONJ
ejpam-6211	503	2	!	!	X
ejpam-6211	504	1	comparing	compare	VERB
ejpam-6211	504	2	coefficients	coefficient	NOUN
ejpam-6211	504	3	of	of	ADP
ejpam-6211	504	4	tn	tn	NOUN
ejpam-6211	504	5	n	n	CCONJ
ejpam-6211	504	6	!	!	PUNCT
ejpam-6211	505	1	and	and	CCONJ
ejpam-6211	505	2	expressing	express	VERB
ejpam-6211	505	3	2n−k+1	2n−k+1	NUM
ejpam-6211	505	4	as	as	ADP
ejpam-6211	505	5	2n2k−1	2n2k−1	NUM
ejpam-6211	505	6	yields	yield	NOUN
ejpam-6211	505	7	,	,	PUNCT
ejpam-6211	505	8	n−1∑	n−1∑	NUM
ejpam-6211	505	9	k=0	k=0	PROPN
ejpam-6211	505	10	su1	su1	X
ejpam-6211	506	1	k	k	X
ejpam-6211	506	2	(	(	PUNCT
ejpam-6211	506	3	m	m	PROPN
ejpam-6211	506	4	;	;	PUNCT
ejpam-6211	506	5	r	r	X
ejpam-6211	506	6	,	,	PUNCT
ejpam-6211	506	7	x	x	NOUN
ejpam-6211	506	8	)	)	PUNCT
ejpam-6211	506	9	1	1	NUM
ejpam-6211	506	10	2n	2n	NUM
ejpam-6211	506	11	(	(	PUNCT
ejpam-6211	506	12	n	n	X
ejpam-6211	506	13	k	k	NOUN
ejpam-6211	506	14	)	)	PUNCT
ejpam-6211	506	15	(	(	PUNCT
ejpam-6211	506	16	(	(	PUNCT
ejpam-6211	506	17	√	√	INTJ
ejpam-6211	506	18	x2	x2	NOUN
ejpam-6211	507	1	+	+	CCONJ
ejpam-6211	507	2	4	4	X
ejpam-6211	507	3	)	)	PUNCT
ejpam-6211	507	4	n−k	n−k	NOUN
ejpam-6211	507	5	(	(	PUNCT
ejpam-6211	507	6	1−	1−	NUM
ejpam-6211	507	7	(	(	PUNCT
ejpam-6211	507	8	−1)n−k	−1)n−k	PROPN
ejpam-6211	507	9	)	)	PUNCT
ejpam-6211	507	10	)	)	PUNCT
ejpam-6211	508	1	=	=	PUNCT
ejpam-6211	509	1	n∑	n∑	INTJ
ejpam-6211	509	2	k=1	k=1	PUNCT
ejpam-6211	510	1	sf	sf	NOUN
ejpam-6211	510	2	1	1	NUM
ejpam-6211	510	3	k	k	X
ejpam-6211	510	4	(	(	PUNCT
ejpam-6211	510	5	m	m	PROPN
ejpam-6211	510	6	;	;	PUNCT
ejpam-6211	510	7	r	r	X
ejpam-6211	510	8	,	,	PUNCT
ejpam-6211	510	9	x	x	NOUN
ejpam-6211	510	10	)	)	PUNCT
ejpam-6211	510	11	(	(	PUNCT
ejpam-6211	510	12	n	n	X
ejpam-6211	510	13	k	k	NOUN
ejpam-6211	510	14	)	)	PUNCT
ejpam-6211	510	15	(	(	PUNCT
ejpam-6211	510	16	e	e	X
ejpam-6211	510	17	−rt	−rt	X
ejpam-6211	510	18	2	2	NUM
ejpam-6211	510	19	(	(	PUNCT
ejpam-6211	510	20	e	e	NOUN
ejpam-6211	510	21	t	t	PROPN
ejpam-6211	510	22	2	2	NUM
ejpam-6211	510	23	−	−	PROPN
ejpam-6211	510	24	1	1	NUM
ejpam-6211	510	25	et	et	NOUN
ejpam-6211	510	26	−	−	NOUN
ejpam-6211	510	27	1	1	NUM
ejpam-6211	510	28	)	)	PUNCT
ejpam-6211	510	29	m	m	NOUN
ejpam-6211	510	30	)	)	PUNCT
ejpam-6211	510	31	×	×	NOUN
ejpam-6211	511	1	√	√	PUNCT
ejpam-6211	511	2	x2	x2	PROPN
ejpam-6211	512	1	+	+	CCONJ
ejpam-6211	512	2	4	4	NUM
ejpam-6211	512	3	2n	2n	NUM
ejpam-6211	512	4	2k−1	2k−1	NUM
ejpam-6211	512	5	(	(	PUNCT
ejpam-6211	512	6	(	(	PUNCT
ejpam-6211	512	7	√	√	INTJ
ejpam-6211	512	8	x2	x2	NUM
ejpam-6211	512	9	−	−	NOUN
ejpam-6211	512	10	1	1	X
ejpam-6211	512	11	)	)	PUNCT
ejpam-6211	512	12	n−k	n−k	NOUN
ejpam-6211	512	13	(	(	PUNCT
ejpam-6211	512	14	α(x)−	α(x)−	PROPN
ejpam-6211	512	15	β(x)(−1)n−k	β(x)(−1)n−k	PUNCT
ejpam-6211	512	16	)	)	PUNCT
ejpam-6211	512	17	)	)	PUNCT
ejpam-6211	513	1	multiplying	multiply	VERB
ejpam-6211	513	2	2n√	2n√	NUM
ejpam-6211	513	3	x2	x2	PUNCT
ejpam-6211	514	1	+	+	NOUN
ejpam-6211	514	2	4	4	NUM
ejpam-6211	514	3	both	both	DET
ejpam-6211	514	4	sides	side	NOUN
ejpam-6211	514	5	yields	yield	NOUN
ejpam-6211	514	6	(	(	PUNCT
ejpam-6211	514	7	5.12	5.12	NUM
ejpam-6211	514	8	)	)	PUNCT
ejpam-6211	514	9	6	6	NUM
ejpam-6211	514	10	.	.	PUNCT
ejpam-6211	515	1	conclusion	conclusion	NOUN
ejpam-6211	515	2	and	and	CCONJ
ejpam-6211	515	3	recommendations	recommendation	NOUN
ejpam-6211	515	4	this	this	DET
ejpam-6211	515	5	study	study	NOUN
ejpam-6211	515	6	introduced	introduce	VERB
ejpam-6211	515	7	and	and	CCONJ
ejpam-6211	515	8	developed	develop	VERB
ejpam-6211	515	9	a	a	DET
ejpam-6211	515	10	novel	novel	ADJ
ejpam-6211	515	11	class	class	NOUN
ejpam-6211	515	12	of	of	ADP
ejpam-6211	515	13	combinatorial	combinatorial	ADJ
ejpam-6211	515	14	structures	structure	NOUN
ejpam-6211	515	15	known	know	VERB
ejpam-6211	515	16	as	as	ADP
ejpam-6211	515	17	the	the	DET
ejpam-6211	515	18	r	r	NOUN
ejpam-6211	515	19	-	-	PUNCT
ejpam-6211	515	20	stirling	stirling	NOUN
ejpam-6211	515	21	fibonacci	fibonacci	NOUN
ejpam-6211	515	22	numbers	number	NOUN
ejpam-6211	515	23	and	and	CCONJ
ejpam-6211	515	24	polynomials	polynomial	NOUN
ejpam-6211	515	25	of	of	ADP
ejpam-6211	515	26	the	the	DET
ejpam-6211	515	27	first	first	ADJ
ejpam-6211	515	28	and	and	CCONJ
ejpam-6211	515	29	second	second	ADJ
ejpam-6211	515	30	kind	kind	NOUN
ejpam-6211	515	31	.	.	PUNCT
ejpam-6211	516	1	by	by	ADP
ejpam-6211	516	2	integrating	integrate	VERB
ejpam-6211	516	3	the	the	DET
ejpam-6211	516	4	exponential	exponential	ADJ
ejpam-6211	516	5	generating	generating	NOUN
ejpam-6211	516	6	functions	function	NOUN
ejpam-6211	516	7	of	of	ADP
ejpam-6211	516	8	fibonacci	fibonacci	NOUN
ejpam-6211	516	9	numbers	number	NOUN
ejpam-6211	516	10	with	with	ADP
ejpam-6211	516	11	those	those	PRON
ejpam-6211	516	12	of	of	ADP
ejpam-6211	516	13	the	the	DET
ejpam-6211	516	14	signed	sign	VERB
ejpam-6211	516	15	r	r	NOUN
ejpam-6211	516	16	-	-	PUNCT
ejpam-6211	516	17	stirling	stirling	NOUN
ejpam-6211	516	18	numbers	number	NOUN
ejpam-6211	516	19	,	,	PUNCT
ejpam-6211	516	20	we	we	PRON
ejpam-6211	516	21	established	establish	VERB
ejpam-6211	516	22	an	an	DET
ejpam-6211	516	23	enriched	enrich	VERB
ejpam-6211	516	24	framework	framework	NOUN
ejpam-6211	516	25	that	that	PRON
ejpam-6211	516	26	deepens	deepen	VERB
ejpam-6211	516	27	the	the	DET
ejpam-6211	516	28	mathematical	mathematical	ADJ
ejpam-6211	516	29	theory	theory	NOUN
ejpam-6211	516	30	of	of	ADP
ejpam-6211	516	31	special	special	ADJ
ejpam-6211	516	32	numbers	number	NOUN
ejpam-6211	516	33	and	and	CCONJ
ejpam-6211	516	34	polynomials	polynomial	NOUN
ejpam-6211	516	35	.	.	PUNCT
ejpam-6211	517	1	22	22	NUM
ejpam-6211	517	2	of	of	ADP
ejpam-6211	517	3	23	23	NUM
ejpam-6211	517	4	through	through	ADP
ejpam-6211	517	5	this	this	DET
ejpam-6211	517	6	construction	construction	NOUN
ejpam-6211	517	7	,	,	PUNCT
ejpam-6211	517	8	we	we	PRON
ejpam-6211	517	9	derived	derive	VERB
ejpam-6211	517	10	new	new	ADJ
ejpam-6211	517	11	and	and	CCONJ
ejpam-6211	517	12	elegant	elegant	ADJ
ejpam-6211	517	13	identities	identity	NOUN
ejpam-6211	517	14	,	,	PUNCT
ejpam-6211	517	15	including	include	VERB
ejpam-6211	517	16	horizontal	horizontal	ADJ
ejpam-6211	517	17	generating	generating	NOUN
ejpam-6211	517	18	functions	function	NOUN
ejpam-6211	517	19	,	,	PUNCT
ejpam-6211	517	20	explicit	explicit	ADJ
ejpam-6211	517	21	expressions	expression	NOUN
ejpam-6211	517	22	,	,	PUNCT
ejpam-6211	517	23	and	and	CCONJ
ejpam-6211	517	24	convolution	convolution	NOUN
ejpam-6211	517	25	formulas	formula	NOUN
ejpam-6211	517	26	that	that	PRON
ejpam-6211	517	27	generalize	generalize	VERB
ejpam-6211	517	28	classical	classical	ADJ
ejpam-6211	517	29	results	result	NOUN
ejpam-6211	517	30	in	in	ADP
ejpam-6211	517	31	combinatorics	combinatoric	NOUN
ejpam-6211	517	32	.	.	PUNCT
ejpam-6211	518	1	furthermore	furthermore	ADV
ejpam-6211	518	2	,	,	PUNCT
ejpam-6211	518	3	we	we	PRON
ejpam-6211	518	4	introduced	introduce	VERB
ejpam-6211	518	5	the	the	DET
ejpam-6211	518	6	r	r	NOUN
ejpam-6211	518	7	-	-	PUNCT
ejpam-6211	518	8	stirling	stirling	NOUN
ejpam-6211	518	9	chebyshev	chebyshev	NOUN
ejpam-6211	518	10	polynomials	polynomial	NOUN
ejpam-6211	518	11	of	of	ADP
ejpam-6211	518	12	the	the	DET
ejpam-6211	518	13	first	first	ADJ
ejpam-6211	518	14	and	and	CCONJ
ejpam-6211	518	15	second	second	ADJ
ejpam-6211	518	16	kind	kind	NOUN
ejpam-6211	518	17	,	,	PUNCT
ejpam-6211	518	18	using	use	VERB
ejpam-6211	518	19	hyperbolic	hyperbolic	ADJ
ejpam-6211	518	20	functions	function	NOUN
ejpam-6211	518	21	and	and	CCONJ
ejpam-6211	518	22	exponential	exponential	ADJ
ejpam-6211	518	23	techniques	technique	NOUN
ejpam-6211	518	24	to	to	PART
ejpam-6211	518	25	create	create	VERB
ejpam-6211	518	26	a	a	DET
ejpam-6211	518	27	functional	functional	ADJ
ejpam-6211	518	28	bridge	bridge	NOUN
ejpam-6211	518	29	between	between	ADP
ejpam-6211	518	30	fibonacci	fibonacci	NOUN
ejpam-6211	518	31	-	-	PUNCT
ejpam-6211	518	32	type	type	NOUN
ejpam-6211	518	33	and	and	CCONJ
ejpam-6211	518	34	stirling	stirling	NOUN
ejpam-6211	518	35	-	-	PUNCT
ejpam-6211	518	36	type	type	NOUN
ejpam-6211	518	37	sequences	sequence	NOUN
ejpam-6211	518	38	.	.	PUNCT
ejpam-6211	519	1	these	these	DET
ejpam-6211	519	2	derivations	derivation	NOUN
ejpam-6211	519	3	were	be	AUX
ejpam-6211	519	4	rigorously	rigorously	ADV
ejpam-6211	519	5	validated	validate	VERB
ejpam-6211	519	6	through	through	ADP
ejpam-6211	519	7	series	series	NOUN
ejpam-6211	519	8	expansions	expansion	NOUN
ejpam-6211	519	9	and	and	CCONJ
ejpam-6211	519	10	the	the	DET
ejpam-6211	519	11	cauchy	cauchy	ADJ
ejpam-6211	519	12	product	product	NOUN
ejpam-6211	519	13	rule	rule	NOUN
ejpam-6211	519	14	,	,	PUNCT
ejpam-6211	519	15	confirming	confirm	VERB
ejpam-6211	519	16	their	their	PRON
ejpam-6211	519	17	consistency	consistency	NOUN
ejpam-6211	519	18	and	and	CCONJ
ejpam-6211	519	19	opening	open	VERB
ejpam-6211	519	20	new	new	ADJ
ejpam-6211	519	21	pathways	pathway	NOUN
ejpam-6211	519	22	in	in	ADP
ejpam-6211	519	23	discrete	discrete	ADJ
ejpam-6211	519	24	mathematics	mathematic	NOUN
ejpam-6211	519	25	.	.	PUNCT
ejpam-6211	520	1	the	the	DET
ejpam-6211	520	2	results	result	NOUN
ejpam-6211	520	3	of	of	ADP
ejpam-6211	520	4	this	this	DET
ejpam-6211	520	5	research	research	NOUN
ejpam-6211	520	6	offer	offer	VERB
ejpam-6211	520	7	significant	significant	ADJ
ejpam-6211	520	8	insights	insight	NOUN
ejpam-6211	520	9	into	into	ADP
ejpam-6211	520	10	the	the	DET
ejpam-6211	520	11	interplay	interplay	NOUN
ejpam-6211	520	12	between	between	ADP
ejpam-6211	520	13	combinatorics	combinatoric	NOUN
ejpam-6211	520	14	,	,	PUNCT
ejpam-6211	520	15	algebra	algebra	NOUN
ejpam-6211	520	16	,	,	PUNCT
ejpam-6211	520	17	and	and	CCONJ
ejpam-6211	520	18	analysis	analysis	NOUN
ejpam-6211	520	19	.	.	PUNCT
ejpam-6211	521	1	the	the	DET
ejpam-6211	521	2	structures	structure	NOUN
ejpam-6211	521	3	revealed	reveal	VERB
ejpam-6211	521	4	by	by	ADP
ejpam-6211	521	5	the	the	DET
ejpam-6211	521	6	presence	presence	NOUN
ejpam-6211	521	7	of	of	ADP
ejpam-6211	521	8	alternating	alternate	VERB
ejpam-6211	521	9	signs	sign	NOUN
ejpam-6211	521	10	,	,	PUNCT
ejpam-6211	521	11	binomial	binomial	ADJ
ejpam-6211	521	12	coefficients	coefficient	NOUN
ejpam-6211	521	13	,	,	PUNCT
ejpam-6211	521	14	and	and	CCONJ
ejpam-6211	521	15	hyperbolic	hyperbolic	ADJ
ejpam-6211	521	16	identities	identity	NOUN
ejpam-6211	521	17	not	not	PART
ejpam-6211	521	18	only	only	ADV
ejpam-6211	521	19	broaden	broaden	VERB
ejpam-6211	521	20	our	our	PRON
ejpam-6211	521	21	understanding	understanding	NOUN
ejpam-6211	521	22	of	of	ADP
ejpam-6211	521	23	fibonacci	fibonacci	NOUN
ejpam-6211	521	24	-	-	PUNCT
ejpam-6211	521	25	related	relate	VERB
ejpam-6211	521	26	sequences	sequence	NOUN
ejpam-6211	521	27	but	but	CCONJ
ejpam-6211	521	28	also	also	ADV
ejpam-6211	521	29	suggest	suggest	VERB
ejpam-6211	521	30	deeper	deep	ADJ
ejpam-6211	521	31	connections	connection	NOUN
ejpam-6211	521	32	to	to	ADP
ejpam-6211	521	33	number	number	NOUN
ejpam-6211	521	34	theory	theory	NOUN
ejpam-6211	521	35	,	,	PUNCT
ejpam-6211	521	36	orthogonal	orthogonal	ADJ
ejpam-6211	521	37	polynomials	polynomial	NOUN
ejpam-6211	521	38	,	,	PUNCT
ejpam-6211	521	39	and	and	CCONJ
ejpam-6211	521	40	symbolic	symbolic	ADJ
ejpam-6211	521	41	computation	computation	NOUN
ejpam-6211	521	42	.	.	PUNCT
ejpam-6211	522	1	given	give	VERB
ejpam-6211	522	2	the	the	DET
ejpam-6211	522	3	promising	promising	ADJ
ejpam-6211	522	4	findings	finding	NOUN
ejpam-6211	522	5	of	of	ADP
ejpam-6211	522	6	this	this	DET
ejpam-6211	522	7	work	work	NOUN
ejpam-6211	522	8	,	,	PUNCT
ejpam-6211	522	9	we	we	PRON
ejpam-6211	522	10	propose	propose	VERB
ejpam-6211	522	11	the	the	DET
ejpam-6211	522	12	following	follow	VERB
ejpam-6211	522	13	directions	direction	NOUN
ejpam-6211	522	14	for	for	ADP
ejpam-6211	522	15	further	further	ADJ
ejpam-6211	522	16	research	research	NOUN
ejpam-6211	522	17	and	and	CCONJ
ejpam-6211	522	18	exploration	exploration	NOUN
ejpam-6211	522	19	:	:	PUNCT
ejpam-6211	522	20	(	(	PUNCT
ejpam-6211	522	21	i	i	NOUN
ejpam-6211	522	22	)	)	PUNCT
ejpam-6211	522	23	generalizations	generalization	NOUN
ejpam-6211	522	24	via	via	ADP
ejpam-6211	522	25	q	q	NOUN
ejpam-6211	522	26	-	-	PUNCT
ejpam-6211	522	27	analogues	analogue	NOUN
ejpam-6211	522	28	and	and	CCONJ
ejpam-6211	522	29	(	(	PUNCT
ejpam-6211	522	30	p	p	X
ejpam-6211	522	31	,	,	PUNCT
ejpam-6211	522	32	q)-extensions	q)-extension	NOUN
ejpam-6211	522	33	.	.	PUNCT
ejpam-6211	523	1	we	we	PRON
ejpam-6211	523	2	recommend	recommend	VERB
ejpam-6211	523	3	investigating	investigate	VERB
ejpam-6211	523	4	q	q	NOUN
ejpam-6211	523	5	-	-	PUNCT
ejpam-6211	523	6	analogues	analogue	NOUN
ejpam-6211	523	7	or	or	CCONJ
ejpam-6211	523	8	(	(	PUNCT
ejpam-6211	523	9	p	p	X
ejpam-6211	523	10	,	,	PUNCT
ejpam-6211	523	11	q)-extensions	q)-extension	NOUN
ejpam-6211	523	12	of	of	ADP
ejpam-6211	523	13	the	the	DET
ejpam-6211	523	14	r	r	NOUN
ejpam-6211	523	15	-	-	PUNCT
ejpam-6211	523	16	stirling	stirling	NOUN
ejpam-6211	523	17	fibonacci	fibonacci	NOUN
ejpam-6211	523	18	numbers	number	NOUN
ejpam-6211	523	19	and	and	CCONJ
ejpam-6211	523	20	polynomials	polynomial	NOUN
ejpam-6211	523	21	.	.	PUNCT
ejpam-6211	524	1	these	these	DET
ejpam-6211	524	2	generalizations	generalization	NOUN
ejpam-6211	524	3	can	can	AUX
ejpam-6211	524	4	yield	yield	VERB
ejpam-6211	524	5	new	new	ADJ
ejpam-6211	524	6	recurrence	recurrence	NOUN
ejpam-6211	524	7	relations	relation	NOUN
ejpam-6211	524	8	,	,	PUNCT
ejpam-6211	524	9	identities	identity	NOUN
ejpam-6211	524	10	,	,	PUNCT
ejpam-6211	524	11	and	and	CCONJ
ejpam-6211	524	12	applications	application	NOUN
ejpam-6211	524	13	,	,	PUNCT
ejpam-6211	524	14	particularly	particularly	ADV
ejpam-6211	524	15	in	in	ADP
ejpam-6211	524	16	the	the	DET
ejpam-6211	524	17	context	context	NOUN
ejpam-6211	524	18	of	of	ADP
ejpam-6211	524	19	quantum	quantum	NOUN
ejpam-6211	524	20	algebra	algebra	NOUN
ejpam-6211	524	21	and	and	CCONJ
ejpam-6211	524	22	combinatorial	combinatorial	ADJ
ejpam-6211	524	23	enumeration	enumeration	NOUN
ejpam-6211	524	24	.	.	PUNCT
ejpam-6211	525	1	(	(	PUNCT
ejpam-6211	525	2	ii	ii	NOUN
ejpam-6211	525	3	)	)	PUNCT
ejpam-6211	525	4	combinatorial	combinatorial	ADJ
ejpam-6211	525	5	models	model	NOUN
ejpam-6211	525	6	and	and	CCONJ
ejpam-6211	525	7	interpretations	interpretation	NOUN
ejpam-6211	525	8	.	.	PUNCT
ejpam-6211	526	1	it	it	PRON
ejpam-6211	526	2	is	be	AUX
ejpam-6211	526	3	essential	essential	ADJ
ejpam-6211	526	4	to	to	PART
ejpam-6211	526	5	explore	explore	VERB
ejpam-6211	526	6	combinatorial	combinatorial	ADJ
ejpam-6211	526	7	models	model	NOUN
ejpam-6211	526	8	such	such	ADJ
ejpam-6211	526	9	as	as	ADP
ejpam-6211	526	10	labeled	label	VERB
ejpam-6211	526	11	graphs	graph	NOUN
ejpam-6211	526	12	,	,	PUNCT
ejpam-6211	526	13	partition	partition	NOUN
ejpam-6211	526	14	structures	structure	NOUN
ejpam-6211	526	15	,	,	PUNCT
ejpam-6211	526	16	or	or	CCONJ
ejpam-6211	526	17	lattice	lattice	NOUN
ejpam-6211	526	18	paths	path	NOUN
ejpam-6211	526	19	that	that	PRON
ejpam-6211	526	20	offer	offer	VERB
ejpam-6211	526	21	tangible	tangible	ADJ
ejpam-6211	526	22	interpretations	interpretation	NOUN
ejpam-6211	526	23	of	of	ADP
ejpam-6211	526	24	the	the	DET
ejpam-6211	526	25	r	r	NOUN
ejpam-6211	526	26	-	-	PUNCT
ejpam-6211	526	27	stirling	stirling	NOUN
ejpam-6211	526	28	fibonacci	fibonacci	NOUN
ejpam-6211	526	29	sequences	sequence	NOUN
ejpam-6211	526	30	and	and	CCONJ
ejpam-6211	526	31	polynomials	polynomial	NOUN
ejpam-6211	526	32	.	.	PUNCT
ejpam-6211	527	1	such	such	ADJ
ejpam-6211	527	2	models	model	NOUN
ejpam-6211	527	3	can	can	AUX
ejpam-6211	527	4	enrich	enrich	VERB
ejpam-6211	527	5	their	their	PRON
ejpam-6211	527	6	applicability	applicability	NOUN
ejpam-6211	527	7	and	and	CCONJ
ejpam-6211	527	8	enhance	enhance	VERB
ejpam-6211	527	9	intuition	intuition	NOUN
ejpam-6211	527	10	for	for	ADP
ejpam-6211	527	11	their	their	PRON
ejpam-6211	527	12	properties	property	NOUN
ejpam-6211	527	13	.	.	PUNCT
ejpam-6211	528	1	(	(	PUNCT
ejpam-6211	528	2	iii	iii	X
ejpam-6211	528	3	)	)	PUNCT
ejpam-6211	528	4	applications	application	NOUN
ejpam-6211	528	5	to	to	AUX
ejpam-6211	528	6	number	number	NOUN
ejpam-6211	528	7	theory	theory	NOUN
ejpam-6211	528	8	and	and	CCONJ
ejpam-6211	528	9	discrete	discrete	ADJ
ejpam-6211	528	10	structures	structure	NOUN
ejpam-6211	528	11	.	.	PUNCT
ejpam-6211	529	1	the	the	DET
ejpam-6211	529	2	derived	derive	VERB
ejpam-6211	529	3	sequences	sequence	NOUN
ejpam-6211	529	4	have	have	VERB
ejpam-6211	529	5	potential	potential	ADJ
ejpam-6211	529	6	applications	application	NOUN
ejpam-6211	529	7	in	in	ADP
ejpam-6211	529	8	number	number	NOUN
ejpam-6211	529	9	theory	theory	NOUN
ejpam-6211	529	10	,	,	PUNCT
ejpam-6211	529	11	tiling	tile	VERB
ejpam-6211	529	12	problems	problem	NOUN
ejpam-6211	529	13	,	,	PUNCT
ejpam-6211	529	14	recurrence	recurrence	NOUN
ejpam-6211	529	15	relations	relation	NOUN
ejpam-6211	529	16	,	,	PUNCT
ejpam-6211	529	17	and	and	CCONJ
ejpam-6211	529	18	discrete	discrete	ADJ
ejpam-6211	529	19	dynamical	dynamical	ADJ
ejpam-6211	529	20	systems	system	NOUN
ejpam-6211	529	21	.	.	PUNCT
ejpam-6211	530	1	we	we	PRON
ejpam-6211	530	2	recommend	recommend	VERB
ejpam-6211	530	3	investigating	investigate	VERB
ejpam-6211	530	4	their	their	PRON
ejpam-6211	530	5	role	role	NOUN
ejpam-6211	530	6	in	in	ADP
ejpam-6211	530	7	coding	code	VERB
ejpam-6211	530	8	theory	theory	NOUN
ejpam-6211	530	9	,	,	PUNCT
ejpam-6211	530	10	integer	integer	NOUN
ejpam-6211	530	11	partitions	partition	NOUN
ejpam-6211	530	12	,	,	PUNCT
ejpam-6211	530	13	and	and	CCONJ
ejpam-6211	530	14	modular	modular	ADJ
ejpam-6211	530	15	arithmetic	arithmetic	ADJ
ejpam-6211	530	16	.	.	PUNCT
ejpam-6211	531	1	(	(	PUNCT
ejpam-6211	531	2	iv	iv	X
ejpam-6211	531	3	)	)	PUNCT
ejpam-6211	531	4	extension	extension	NOUN
ejpam-6211	531	5	to	to	ADP
ejpam-6211	531	6	other	other	ADJ
ejpam-6211	531	7	polynomial	polynomial	ADJ
ejpam-6211	531	8	families	family	NOUN
ejpam-6211	531	9	.	.	PUNCT
ejpam-6211	532	1	the	the	DET
ejpam-6211	532	2	exponential	exponential	ADJ
ejpam-6211	532	3	generating	generate	VERB
ejpam-6211	532	4	function	function	NOUN
ejpam-6211	532	5	framework	framework	NOUN
ejpam-6211	532	6	used	use	VERB
ejpam-6211	532	7	here	here	ADV
ejpam-6211	532	8	may	may	AUX
ejpam-6211	532	9	be	be	AUX
ejpam-6211	532	10	applied	apply	VERB
ejpam-6211	532	11	to	to	ADP
ejpam-6211	532	12	other	other	ADJ
ejpam-6211	532	13	orthogonal	orthogonal	ADJ
ejpam-6211	532	14	polynomial	polynomial	ADJ
ejpam-6211	532	15	families	family	NOUN
ejpam-6211	532	16	such	such	ADJ
ejpam-6211	532	17	as	as	ADP
ejpam-6211	532	18	legendre	legendre	PROPN
ejpam-6211	532	19	,	,	PUNCT
ejpam-6211	532	20	hermite	hermite	PROPN
ejpam-6211	532	21	,	,	PUNCT
ejpam-6211	532	22	laguerre	laguerre	NOUN
ejpam-6211	532	23	,	,	PUNCT
ejpam-6211	532	24	and	and	CCONJ
ejpam-6211	532	25	jacobi	jacobi	PROPN
ejpam-6211	532	26	polynomials	polynomial	VERB
ejpam-6211	532	27	by	by	ADP
ejpam-6211	532	28	introducing	introduce	VERB
ejpam-6211	532	29	r	r	NOUN
ejpam-6211	532	30	-	-	PUNCT
ejpam-6211	532	31	stirling	stirling	NOUN
ejpam-6211	532	32	analogues	analogue	NOUN
ejpam-6211	532	33	.	.	PUNCT
ejpam-6211	533	1	this	this	PRON
ejpam-6211	533	2	could	could	AUX
ejpam-6211	533	3	uncover	uncover	VERB
ejpam-6211	533	4	new	new	ADJ
ejpam-6211	533	5	algebraic	algebraic	ADJ
ejpam-6211	533	6	and	and	CCONJ
ejpam-6211	533	7	combinatorial	combinatorial	ADJ
ejpam-6211	533	8	relationships	relationship	NOUN
ejpam-6211	533	9	and	and	CCONJ
ejpam-6211	533	10	extend	extend	VERB
ejpam-6211	533	11	the	the	DET
ejpam-6211	533	12	utility	utility	NOUN
ejpam-6211	533	13	of	of	ADP
ejpam-6211	533	14	the	the	DET
ejpam-6211	533	15	approach	approach	NOUN
ejpam-6211	533	16	.	.	PUNCT
ejpam-6211	534	1	(	(	PUNCT
ejpam-6211	534	2	v	v	NOUN
ejpam-6211	534	3	)	)	PUNCT
ejpam-6211	534	4	symbolic	symbolic	ADJ
ejpam-6211	534	5	and	and	CCONJ
ejpam-6211	534	6	computational	computational	ADJ
ejpam-6211	534	7	implementations	implementation	NOUN
ejpam-6211	534	8	.	.	PUNCT
ejpam-6211	535	1	developing	develop	VERB
ejpam-6211	535	2	efficient	efficient	ADJ
ejpam-6211	535	3	symbolic	symbolic	ADJ
ejpam-6211	535	4	algorithms	algorithms	NOUN
ejpam-6211	535	5	for	for	ADP
ejpam-6211	535	6	computing	compute	VERB
ejpam-6211	535	7	r	r	NOUN
ejpam-6211	535	8	-	-	PUNCT
ejpam-6211	535	9	stirling	stirling	NOUN
ejpam-6211	535	10	fibonacci	fibonacci	NOUN
ejpam-6211	535	11	numbers	number	NOUN
ejpam-6211	535	12	and	and	CCONJ
ejpam-6211	535	13	polynomials	polynomial	NOUN
ejpam-6211	535	14	will	will	AUX
ejpam-6211	535	15	support	support	VERB
ejpam-6211	535	16	further	further	ADJ
ejpam-6211	535	17	research	research	NOUN
ejpam-6211	535	18	and	and	CCONJ
ejpam-6211	535	19	educational	educational	ADJ
ejpam-6211	535	20	use	use	NOUN
ejpam-6211	535	21	.	.	PUNCT
ejpam-6211	536	1	tools	tool	NOUN
ejpam-6211	536	2	such	such	ADJ
ejpam-6211	536	3	as	as	ADP
ejpam-6211	536	4	mathematica	mathematica	PROPN
ejpam-6211	536	5	,	,	PUNCT
ejpam-6211	536	6	maple	maple	NOUN
ejpam-6211	536	7	,	,	PUNCT
ejpam-6211	536	8	and	and	CCONJ
ejpam-6211	536	9	sagemath	sagemath	NOUN
ejpam-6211	536	10	may	may	AUX
ejpam-6211	536	11	be	be	AUX
ejpam-6211	536	12	utilized	utilize	VERB
ejpam-6211	536	13	to	to	PART
ejpam-6211	536	14	visualize	visualize	VERB
ejpam-6211	536	15	,	,	PUNCT
ejpam-6211	536	16	manipulate	manipulate	VERB
ejpam-6211	536	17	,	,	PUNCT
ejpam-6211	536	18	and	and	CCONJ
ejpam-6211	536	19	analyze	analyze	VERB
ejpam-6211	536	20	these	these	DET
ejpam-6211	536	21	sequences	sequence	NOUN
ejpam-6211	536	22	at	at	ADP
ejpam-6211	536	23	higher	high	ADJ
ejpam-6211	536	24	orders	order	NOUN
ejpam-6211	536	25	.	.	PUNCT
ejpam-6211	537	1	(	(	PUNCT
ejpam-6211	537	2	vi	vi	NOUN
ejpam-6211	537	3	)	)	PUNCT
ejpam-6211	537	4	multivariate	multivariate	NOUN
ejpam-6211	537	5	and	and	CCONJ
ejpam-6211	537	6	matrix	matrix	NOUN
ejpam-6211	537	7	extensions	extension	NOUN
ejpam-6211	537	8	.	.	PUNCT
ejpam-6211	538	1	future	future	ADJ
ejpam-6211	538	2	work	work	NOUN
ejpam-6211	538	3	may	may	AUX
ejpam-6211	538	4	explore	explore	VERB
ejpam-6211	538	5	multivariate	multivariate	NOUN
ejpam-6211	538	6	versions	version	NOUN
ejpam-6211	538	7	or	or	CCONJ
ejpam-6211	538	8	matrix	matrix	NOUN
ejpam-6211	538	9	representations	representation	NOUN
ejpam-6211	538	10	of	of	ADP
ejpam-6211	538	11	these	these	DET
ejpam-6211	538	12	identities	identity	NOUN
ejpam-6211	538	13	,	,	PUNCT
ejpam-6211	538	14	which	which	PRON
ejpam-6211	538	15	could	could	AUX
ejpam-6211	538	16	reveal	reveal	VERB
ejpam-6211	538	17	richer	rich	ADJ
ejpam-6211	538	18	symmetries	symmetry	NOUN
ejpam-6211	538	19	and	and	CCONJ
ejpam-6211	538	20	deeper	deep	ADJ
ejpam-6211	538	21	structural	structural	ADJ
ejpam-6211	538	22	insights	insight	NOUN
ejpam-6211	538	23	-	-	PUNCT
ejpam-6211	538	24	especially	especially	ADV
ejpam-6211	538	25	relevant	relevant	ADJ
ejpam-6211	538	26	in	in	ADP
ejpam-6211	538	27	linear	linear	PROPN
ejpam-6211	538	28	algebra	algebra	NOUN
ejpam-6211	538	29	and	and	CCONJ
ejpam-6211	538	30	applied	apply	VERB
ejpam-6211	538	31	fields	field	NOUN
ejpam-6211	538	32	.	.	PUNCT
ejpam-6211	539	1	23	23	NUM
ejpam-6211	539	2	of	of	ADP
ejpam-6211	539	3	23	23	NUM
ejpam-6211	539	4	by	by	ADP
ejpam-6211	539	5	pursuing	pursue	VERB
ejpam-6211	539	6	these	these	DET
ejpam-6211	539	7	directions	direction	NOUN
ejpam-6211	539	8	,	,	PUNCT
ejpam-6211	539	9	researchers	researcher	NOUN
ejpam-6211	539	10	can	can	AUX
ejpam-6211	539	11	build	build	VERB
ejpam-6211	539	12	upon	upon	SCONJ
ejpam-6211	539	13	the	the	DET
ejpam-6211	539	14	foundational	foundational	ADJ
ejpam-6211	539	15	results	result	NOUN
ejpam-6211	539	16	of	of	ADP
ejpam-6211	539	17	this	this	DET
ejpam-6211	539	18	study	study	NOUN
ejpam-6211	539	19	and	and	CCONJ
ejpam-6211	539	20	continue	continue	VERB
ejpam-6211	539	21	to	to	PART
ejpam-6211	539	22	expand	expand	VERB
ejpam-6211	539	23	the	the	DET
ejpam-6211	539	24	theoretical	theoretical	ADJ
ejpam-6211	539	25	and	and	CCONJ
ejpam-6211	539	26	practical	practical	ADJ
ejpam-6211	539	27	landscape	landscape	NOUN
ejpam-6211	539	28	of	of	ADP
ejpam-6211	539	29	r	r	NOUN
ejpam-6211	539	30	-	-	PUNCT
ejpam-6211	539	31	stirling	stirling	NOUN
ejpam-6211	539	32	fibonacci	fibonacci	NOUN
ejpam-6211	539	33	numbers	number	NOUN
ejpam-6211	539	34	and	and	CCONJ
ejpam-6211	539	35	polynomials	polynomial	NOUN
ejpam-6211	539	36	within	within	ADP
ejpam-6211	539	37	mathematics	mathematic	NOUN
ejpam-6211	539	38	and	and	CCONJ
ejpam-6211	539	39	related	related	ADJ
ejpam-6211	539	40	disciplines	discipline	NOUN
ejpam-6211	539	41	.	.	PUNCT
ejpam-6211	540	1	references	reference	NOUN
ejpam-6211	540	2	[	[	X
ejpam-6211	540	3	1	1	X
ejpam-6211	540	4	]	]	PUNCT
ejpam-6211	540	5	j.	j.	PROPN
ejpam-6211	540	6	stirling	stirling	PROPN
ejpam-6211	540	7	.	.	PUNCT
ejpam-6211	541	1	methodus	methodus	PROPN
ejpam-6211	541	2	differentialis	differentialis	PROPN
ejpam-6211	541	3	,	,	PUNCT
ejpam-6211	541	4	sire	sire	NOUN
ejpam-6211	541	5	tractatus	tractatus	PROPN
ejpam-6211	541	6	de	de	ADP
ejpam-6211	541	7	summatione	summatione	NOUN
ejpam-6211	541	8	et	et	NOUN
ejpam-6211	541	9	interpolatione	interpolatione	ADJ
ejpam-6211	541	10	serierum	serierum	PROPN
ejpam-6211	541	11	infinitorum	infinitorum	PROPN
ejpam-6211	541	12	.	.	PUNCT
ejpam-6211	541	13	london	london	PROPN
ejpam-6211	541	14	,	,	PUNCT
ejpam-6211	541	15	1730	1730	NUM
ejpam-6211	541	16	.	.	PUNCT
ejpam-6211	542	1	[	[	X
ejpam-6211	542	2	2	2	NUM
ejpam-6211	542	3	]	]	PUNCT
ejpam-6211	542	4	r.	r.	PROPN
ejpam-6211	542	5	b.	b.	PROPN
ejpam-6211	542	6	corcino	corcino	PROPN
ejpam-6211	542	7	and	and	CCONJ
ejpam-6211	542	8	c.	c.	PROPN
ejpam-6211	542	9	b.	b.	PROPN
ejpam-6211	542	10	corcino	corcino	PROPN
ejpam-6211	542	11	.	.	PUNCT
ejpam-6211	543	1	the	the	DET
ejpam-6211	543	2	hankel	hankel	NOUN
ejpam-6211	543	3	transform	transform	NOUN
ejpam-6211	543	4	of	of	ADP
ejpam-6211	543	5	generalized	generalized	ADJ
ejpam-6211	543	6	bell	bell	NOUN
ejpam-6211	543	7	numbers	number	NOUN
ejpam-6211	543	8	and	and	CCONJ
ejpam-6211	543	9	its	its	PRON
ejpam-6211	543	10	q	q	NOUN
ejpam-6211	543	11	-	-	PUNCT
ejpam-6211	543	12	analogue	analogue	NOUN
ejpam-6211	543	13	.	.	PUNCT
ejpam-6211	544	1	utilitas	utilitas	PROPN
ejpam-6211	544	2	mathematica	mathematica	PROPN
ejpam-6211	544	3	,	,	PUNCT
ejpam-6211	544	4	89:297–309	89:297–309	PROPN
ejpam-6211	544	5	,	,	PUNCT
ejpam-6211	544	6	2012	2012	NUM
ejpam-6211	544	7	.	.	PUNCT
ejpam-6211	545	1	[	[	X
ejpam-6211	545	2	3	3	X
ejpam-6211	545	3	]	]	X
ejpam-6211	545	4	r.	r.	PROPN
ejpam-6211	545	5	b.	b.	PROPN
ejpam-6211	545	6	corcino	corcino	PROPN
ejpam-6211	545	7	and	and	CCONJ
ejpam-6211	545	8	c.	c.	PROPN
ejpam-6211	545	9	b.	b.	PROPN
ejpam-6211	545	10	montero	montero	PROPN
ejpam-6211	545	11	.	.	PUNCT
ejpam-6211	546	1	a	a	DET
ejpam-6211	546	2	q	q	NOUN
ejpam-6211	546	3	-	-	PUNCT
ejpam-6211	546	4	analogue	analogue	NOUN
ejpam-6211	546	5	of	of	ADP
ejpam-6211	546	6	rucinski	rucinski	ADJ
ejpam-6211	546	7	-	-	PUNCT
ejpam-6211	546	8	voigt	voigt	NOUN
ejpam-6211	546	9	numbers	number	NOUN
ejpam-6211	546	10	.	.	PUNCT
ejpam-6211	547	1	isrn	isrn	NOUN
ejpam-6211	547	2	discrete	discrete	VERB
ejpam-6211	547	3	mathematics	mathematic	NOUN
ejpam-6211	547	4	,	,	PUNCT
ejpam-6211	547	5	page	page	NOUN
ejpam-6211	547	6	818	818	NUM
ejpam-6211	547	7	,	,	PUNCT
ejpam-6211	547	8	2012	2012	NUM
ejpam-6211	547	9	.	.	PUNCT
ejpam-6211	548	1	[	[	X
ejpam-6211	548	2	4	4	NUM
ejpam-6211	548	3	]	]	PUNCT
ejpam-6211	548	4	r.	r.	PROPN
ejpam-6211	548	5	b.	b.	PROPN
ejpam-6211	548	6	corcino	corcino	PROPN
ejpam-6211	548	7	and	and	CCONJ
ejpam-6211	548	8	c.	c.	PROPN
ejpam-6211	548	9	barrientos	barrientos	PROPN
ejpam-6211	548	10	.	.	PUNCT
ejpam-6211	549	1	some	some	DET
ejpam-6211	549	2	theorems	theorem	NOUN
ejpam-6211	549	3	on	on	ADP
ejpam-6211	549	4	the	the	DET
ejpam-6211	549	5	q	q	NOUN
ejpam-6211	549	6	-	-	PUNCT
ejpam-6211	549	7	analogue	analogue	NOUN
ejpam-6211	549	8	of	of	ADP
ejpam-6211	549	9	the	the	DET
ejpam-6211	549	10	generalized	generalized	ADJ
ejpam-6211	549	11	stirling	stirling	NOUN
ejpam-6211	549	12	numbers	number	NOUN
ejpam-6211	549	13	.	.	PUNCT
ejpam-6211	550	1	bulletin	bulletin	NOUN
ejpam-6211	550	2	of	of	ADP
ejpam-6211	550	3	the	the	DET
ejpam-6211	550	4	malaysian	malaysian	PROPN
ejpam-6211	550	5	mathematical	mathematical	PROPN
ejpam-6211	550	6	sciences	sciences	PROPN
ejpam-6211	550	7	society	society	NOUN
ejpam-6211	550	8	,	,	PUNCT
ejpam-6211	550	9	34(3):487	34(3):487	NUM
ejpam-6211	550	10	–	–	PUNCT
ejpam-6211	550	11	501	501	NUM
ejpam-6211	550	12	,	,	PUNCT
ejpam-6211	550	13	2011	2011	NUM
ejpam-6211	550	14	.	.	PUNCT
ejpam-6211	551	1	[	[	X
ejpam-6211	551	2	5	5	NUM
ejpam-6211	551	3	]	]	PUNCT
ejpam-6211	551	4	r.	r.	PROPN
ejpam-6211	551	5	b.	b.	PROPN
ejpam-6211	551	6	corcino	corcino	PROPN
ejpam-6211	551	7	,	,	PUNCT
ejpam-6211	551	8	l.	l.	PROPN
ejpam-6211	551	9	c.	c.	PROPN
ejpam-6211	551	10	hsu	hsu	PROPN
ejpam-6211	551	11	,	,	PUNCT
ejpam-6211	551	12	and	and	CCONJ
ejpam-6211	551	13	e.	e.	PROPN
ejpam-6211	551	14	l.	l.	PROPN
ejpam-6211	551	15	tan	tan	PROPN
ejpam-6211	551	16	.	.	PUNCT
ejpam-6211	552	1	a	a	DET
ejpam-6211	552	2	q	q	NOUN
ejpam-6211	552	3	-	-	PUNCT
ejpam-6211	552	4	analogue	analogue	NOUN
ejpam-6211	552	5	of	of	ADP
ejpam-6211	552	6	generalized	generalized	ADJ
ejpam-6211	552	7	stirling	stirling	NOUN
ejpam-6211	552	8	numbers	number	NOUN
ejpam-6211	552	9	.	.	PUNCT
ejpam-6211	553	1	the	the	DET
ejpam-6211	553	2	fibonacci	fibonacci	NOUN
ejpam-6211	553	3	quarterly	quarterly	PROPN
ejpam-6211	553	4	,	,	PUNCT
ejpam-6211	553	5	44(2):154–165	44(2):154–165	PROPN
ejpam-6211	553	6	,	,	PUNCT
ejpam-6211	553	7	2006	2006	NUM
ejpam-6211	553	8	.	.	PUNCT
ejpam-6211	554	1	[	[	X
ejpam-6211	554	2	6	6	NUM
ejpam-6211	554	3	]	]	PUNCT
ejpam-6211	554	4	r.	r.	PROPN
ejpam-6211	554	5	b.	b.	PROPN
ejpam-6211	554	6	corcino	corcino	PROPN
ejpam-6211	554	7	and	and	CCONJ
ejpam-6211	554	8	c.	c.	PROPN
ejpam-6211	554	9	b.	b.	PROPN
ejpam-6211	554	10	corcino	corcino	PROPN
ejpam-6211	554	11	.	.	PUNCT
ejpam-6211	555	1	an	an	DET
ejpam-6211	555	2	asymptotic	asymptotic	ADJ
ejpam-6211	555	3	formula	formula	NOUN
ejpam-6211	555	4	for	for	ADP
ejpam-6211	555	5	the	the	DET
ejpam-6211	555	6	r	r	NOUN
ejpam-6211	555	7	-	-	PUNCT
ejpam-6211	555	8	bell	bell	NOUN
ejpam-6211	555	9	numbers	number	NOUN
ejpam-6211	555	10	.	.	PUNCT
ejpam-6211	556	1	matimyas	matimyas	PROPN
ejpam-6211	556	2	matematika	matematika	PROPN
ejpam-6211	556	3	,	,	PUNCT
ejpam-6211	556	4	24(1):9–18	24(1):9–18	NUM
ejpam-6211	556	5	,	,	PUNCT
ejpam-6211	556	6	2001	2001	NUM
ejpam-6211	556	7	.	.	PUNCT
ejpam-6211	557	1	[	[	X
ejpam-6211	557	2	7	7	X
ejpam-6211	557	3	]	]	X
ejpam-6211	557	4	i.	i.	PROPN
ejpam-6211	557	5	mező	mező	PROPN
ejpam-6211	557	6	and	and	CCONJ
ejpam-6211	557	7	r.	r.	PROPN
ejpam-6211	557	8	b.	b.	PROPN
ejpam-6211	557	9	corcino	corcino	PROPN
ejpam-6211	557	10	.	.	PUNCT
ejpam-6211	558	1	the	the	DET
ejpam-6211	558	2	estimation	estimation	NOUN
ejpam-6211	558	3	of	of	ADP
ejpam-6211	558	4	the	the	DET
ejpam-6211	558	5	zeros	zero	NOUN
ejpam-6211	558	6	of	of	ADP
ejpam-6211	558	7	the	the	DET
ejpam-6211	558	8	bell	bell	NOUN
ejpam-6211	558	9	and	and	CCONJ
ejpam-6211	558	10	r	r	NOUN
ejpam-6211	558	11	-	-	PUNCT
ejpam-6211	558	12	bell	bell	NOUN
ejpam-6211	558	13	polynomials	polynomial	NOUN
ejpam-6211	558	14	.	.	PUNCT
ejpam-6211	559	1	applied	apply	VERB
ejpam-6211	559	2	mathematics	mathematic	NOUN
ejpam-6211	559	3	and	and	CCONJ
ejpam-6211	559	4	computation	computation	NOUN
ejpam-6211	559	5	,	,	PUNCT
ejpam-6211	559	6	250:727–732	250:727–732	NUM
ejpam-6211	559	7	,	,	PUNCT
ejpam-6211	559	8	2015	2015	NUM
ejpam-6211	559	9	.	.	PUNCT
ejpam-6211	560	1	[	[	X
ejpam-6211	560	2	8	8	NUM
ejpam-6211	560	3	]	]	PUNCT
ejpam-6211	560	4	a.	a.	NOUN
ejpam-6211	560	5	z.	z.	PROPN
ejpam-6211	560	6	broder	broder	PROPN
ejpam-6211	560	7	.	.	PUNCT
ejpam-6211	561	1	the	the	DET
ejpam-6211	561	2	r	r	NOUN
ejpam-6211	561	3	-	-	PUNCT
ejpam-6211	561	4	stirling	stirling	NOUN
ejpam-6211	561	5	numbers	number	NOUN
ejpam-6211	561	6	.	.	PUNCT
ejpam-6211	562	1	discrete	discrete	ADJ
ejpam-6211	562	2	mathematics	mathematic	NOUN
ejpam-6211	562	3	,	,	PUNCT
ejpam-6211	562	4	49(3):241–259	49(3):241–259	PROPN
ejpam-6211	562	5	,	,	PUNCT
ejpam-6211	562	6	1984	1984	NUM
ejpam-6211	562	7	.	.	PUNCT
ejpam-6211	563	1	[	[	X
ejpam-6211	563	2	9	9	NUM
ejpam-6211	563	3	]	]	X
ejpam-6211	563	4	r.	r.	PROPN
ejpam-6211	563	5	corcino	corcino	PROPN
ejpam-6211	563	6	,	,	PUNCT
ejpam-6211	563	7	v.	v.	ADP
ejpam-6211	563	8	dechosa	dechosa	PROPN
ejpam-6211	563	9	,	,	PUNCT
ejpam-6211	563	10	r.	r.	PROPN
ejpam-6211	563	11	coronel	coronel	PROPN
ejpam-6211	563	12	,	,	PUNCT
ejpam-6211	563	13	and	and	CCONJ
ejpam-6211	563	14	v.	v.	ADP
ejpam-6211	563	15	m.	m.	NOUN
ejpam-6211	563	16	deveraturda	deveraturda	PROPN
ejpam-6211	563	17	.	.	PUNCT
ejpam-6211	564	1	the	the	DET
ejpam-6211	564	2	sm	sm	PROPN
ejpam-6211	564	3	r	r	PROPN
ejpam-6211	564	4	-	-	PUNCT
ejpam-6211	564	5	stirling	stirling	NOUN
ejpam-6211	564	6	numbers	number	NOUN
ejpam-6211	564	7	:	:	PUNCT
ejpam-6211	564	8	an	an	DET
ejpam-6211	564	9	algebraic	algebraic	ADJ
ejpam-6211	564	10	approach	approach	NOUN
ejpam-6211	564	11	.	.	PUNCT
ejpam-6211	565	1	cnu	cnu	PROPN
ejpam-6211	565	2	-	-	PUNCT
ejpam-6211	565	3	journal	journal	PROPN
ejpam-6211	565	4	of	of	ADP
ejpam-6211	565	5	higher	high	ADJ
ejpam-6211	565	6	education	education	NOUN
ejpam-6211	565	7	,	,	PUNCT
ejpam-6211	565	8	17:1–14	17:1–14	NUM
ejpam-6211	565	9	,	,	PUNCT
ejpam-6211	565	10	2023	2023	NUM
ejpam-6211	565	11	.	.	PUNCT
ejpam-6211	566	1	[	[	X
ejpam-6211	566	2	10	10	NUM
ejpam-6211	566	3	]	]	X
ejpam-6211	566	4	l.	l.	PROPN
ejpam-6211	566	5	sigler	sigler	PROPN
ejpam-6211	566	6	.	.	PUNCT
ejpam-6211	567	1	fibonacci	fibonacci	PROPN
ejpam-6211	567	2	’s	’s	PART
ejpam-6211	567	3	liber	liber	PROPN
ejpam-6211	567	4	abaci	abaci	NOUN
ejpam-6211	567	5	:	:	PUNCT
ejpam-6211	567	6	a	a	DET
ejpam-6211	567	7	translation	translation	NOUN
ejpam-6211	567	8	into	into	ADP
ejpam-6211	567	9	modern	modern	ADJ
ejpam-6211	567	10	english	english	PROPN
ejpam-6211	567	11	of	of	ADP
ejpam-6211	567	12	leonardo	leonardo	PROPN
ejpam-6211	567	13	pisano	pisano	PROPN
ejpam-6211	567	14	’s	’s	PART
ejpam-6211	567	15	book	book	NOUN
ejpam-6211	567	16	of	of	ADP
ejpam-6211	567	17	calculation	calculation	NOUN
ejpam-6211	567	18	.	.	PUNCT
ejpam-6211	568	1	springer	springer	NOUN
ejpam-6211	568	2	,	,	PUNCT
ejpam-6211	568	3	new	new	PROPN
ejpam-6211	568	4	york	york	PROPN
ejpam-6211	568	5	,	,	PUNCT
ejpam-6211	568	6	2002	2002	NUM
ejpam-6211	568	7	.	.	PUNCT
ejpam-6211	569	1	[	[	X
ejpam-6211	569	2	11	11	NUM
ejpam-6211	569	3	]	]	X
ejpam-6211	569	4	c.	c.	PROPN
ejpam-6211	569	5	a.	a.	NOUN
ejpam-6211	569	6	church	church	PROPN
ejpam-6211	569	7	and	and	CCONJ
ejpam-6211	569	8	m.	m.	NOUN
ejpam-6211	569	9	bicknell	bicknell	NOUN
ejpam-6211	569	10	.	.	PUNCT
ejpam-6211	570	1	exponential	exponential	ADJ
ejpam-6211	570	2	generating	generating	NOUN
ejpam-6211	570	3	functions	function	NOUN
ejpam-6211	570	4	for	for	ADP
ejpam-6211	570	5	fibonacci	fibonacci	NOUN
ejpam-6211	570	6	identities	identity	NOUN
ejpam-6211	570	7	.	.	PUNCT
ejpam-6211	571	1	the	the	DET
ejpam-6211	571	2	fibonacci	fibonacci	NOUN
ejpam-6211	571	3	quarterly	quarterly	PROPN
ejpam-6211	571	4	,	,	PUNCT
ejpam-6211	571	5	11(3):275–281	11(3):275–281	NUM
ejpam-6211	571	6	,	,	PUNCT
ejpam-6211	571	7	1973	1973	NUM
ejpam-6211	571	8	.	.	PUNCT
ejpam-6211	572	1	[	[	X
ejpam-6211	572	2	12	12	NUM
ejpam-6211	572	3	]	]	X
ejpam-6211	572	4	c.	c.	PROPN
ejpam-6211	572	5	cesarano	cesarano	PROPN
ejpam-6211	572	6	,	,	PUNCT
ejpam-6211	572	7	w.	w.	PROPN
ejpam-6211	572	8	ramirez	ramirez	PROPN
ejpam-6211	572	9	,	,	PUNCT
ejpam-6211	572	10	and	and	CCONJ
ejpam-6211	572	11	s.	s.	PROPN
ejpam-6211	572	12	khan	khan	PROPN
ejpam-6211	572	13	.	.	PUNCT
ejpam-6211	573	1	a	a	DET
ejpam-6211	573	2	new	new	ADJ
ejpam-6211	573	3	class	class	NOUN
ejpam-6211	573	4	of	of	ADP
ejpam-6211	573	5	degenerate	degenerate	ADJ
ejpam-6211	573	6	apostol	apostol	NOUN
ejpam-6211	573	7	-	-	PUNCT
ejpam-6211	573	8	type	type	NOUN
ejpam-6211	573	9	hermite	hermite	ADJ
ejpam-6211	573	10	polynomials	polynomial	NOUN
ejpam-6211	573	11	and	and	CCONJ
ejpam-6211	573	12	applications	application	NOUN
ejpam-6211	573	13	.	.	PUNCT
ejpam-6211	574	1	dolomites	dolomite	NOUN
ejpam-6211	574	2	research	research	NOUN
ejpam-6211	574	3	notes	note	NOUN
ejpam-6211	574	4	on	on	ADP
ejpam-6211	574	5	approximation	approximation	NOUN
ejpam-6211	574	6	,	,	PUNCT
ejpam-6211	574	7	15(1):1–10	15(1):1–10	NUM
ejpam-6211	574	8	,	,	PUNCT
ejpam-6211	574	9	2022	2022	NUM
ejpam-6211	574	10	.	.	PUNCT
ejpam-6211	575	1	[	[	X
ejpam-6211	575	2	13	13	NUM
ejpam-6211	575	3	]	]	PUNCT
ejpam-6211	575	4	w.	w.	PROPN
ejpam-6211	575	5	ramirez	ramirez	PROPN
ejpam-6211	575	6	and	and	CCONJ
ejpam-6211	575	7	c.	c.	PROPN
ejpam-6211	575	8	cesarano	cesarano	PROPN
ejpam-6211	575	9	.	.	PUNCT
ejpam-6211	576	1	some	some	DET
ejpam-6211	576	2	new	new	ADJ
ejpam-6211	576	3	classes	class	NOUN
ejpam-6211	576	4	of	of	ADP
ejpam-6211	576	5	degenerated	degenerated	ADJ
ejpam-6211	576	6	generalized	generalized	ADJ
ejpam-6211	576	7	apostolbernoulli	apostolbernoulli	NOUN
ejpam-6211	576	8	,	,	PUNCT
ejpam-6211	576	9	apostol	apostol	NOUN
ejpam-6211	576	10	-	-	PUNCT
ejpam-6211	576	11	euler	euler	NOUN
ejpam-6211	576	12	and	and	CCONJ
ejpam-6211	576	13	apostol	apostol	NOUN
ejpam-6211	576	14	-	-	PUNCT
ejpam-6211	576	15	genocchi	genocchi	PROPN
ejpam-6211	576	16	polynomials	polynomial	NOUN
ejpam-6211	576	17	.	.	PUNCT
ejpam-6211	577	1	carpathian	carpathian	ADJ
ejpam-6211	577	2	mathematical	mathematical	ADJ
ejpam-6211	577	3	publications	publication	NOUN
ejpam-6211	577	4	,	,	PUNCT
ejpam-6211	577	5	14(2):354–363	14(2):354–363	NUM
ejpam-6211	577	6	,	,	PUNCT
ejpam-6211	577	7	2022	2022	NUM
ejpam-6211	577	8	.	.	PUNCT
ejpam-6211	578	1	[	[	X
ejpam-6211	578	2	14	14	NUM
ejpam-6211	578	3	]	]	X
ejpam-6211	578	4	w.	w.	PROPN
ejpam-6211	578	5	ramirez	ramirez	PROPN
ejpam-6211	578	6	,	,	PUNCT
ejpam-6211	578	7	c.	c.	PROPN
ejpam-6211	578	8	cesarano	cesarano	PROPN
ejpam-6211	578	9	,	,	PUNCT
ejpam-6211	578	10	and	and	CCONJ
ejpam-6211	578	11	s.	s.	PROPN
ejpam-6211	578	12	diaz	diaz	PROPN
ejpam-6211	578	13	.	.	PUNCT
ejpam-6211	579	1	new	new	ADJ
ejpam-6211	579	2	results	result	NOUN
ejpam-6211	579	3	for	for	ADP
ejpam-6211	579	4	degenerated	degenerated	ADJ
ejpam-6211	579	5	generalized	generalized	ADJ
ejpam-6211	579	6	apostol	apostol	NOUN
ejpam-6211	579	7	-	-	PUNCT
ejpam-6211	579	8	bernoulli	bernoulli	NOUN
ejpam-6211	579	9	,	,	PUNCT
ejpam-6211	579	10	apostol	apostol	NOUN
ejpam-6211	579	11	-	-	PUNCT
ejpam-6211	579	12	euler	euler	NOUN
ejpam-6211	579	13	and	and	CCONJ
ejpam-6211	579	14	apostol	apostol	NOUN
ejpam-6211	579	15	-	-	PUNCT
ejpam-6211	579	16	genocchi	genocchi	PROPN
ejpam-6211	579	17	polynomials	polynomial	NOUN
ejpam-6211	579	18	.	.	PUNCT
ejpam-6211	580	1	wseas	wseas	VERB
ejpam-6211	580	2	transactions	transaction	NOUN
ejpam-6211	580	3	on	on	ADP
ejpam-6211	580	4	mathematics	mathematic	NOUN
ejpam-6211	580	5	,	,	PUNCT
ejpam-6211	580	6	21:604–608	21:604–608	NUM
ejpam-6211	580	7	,	,	PUNCT
ejpam-6211	580	8	2022	2022	NUM
ejpam-6211	580	9	.	.	PUNCT
ejpam-6211	581	1	[	[	X
ejpam-6211	581	2	15	15	NUM
ejpam-6211	581	3	]	]	X
ejpam-6211	581	4	r.	r.	PROPN
ejpam-6211	581	5	b.	b.	PROPN
ejpam-6211	581	6	corcino	corcino	PROPN
ejpam-6211	581	7	,	,	PUNCT
ejpam-6211	581	8	m.	m.	PROPN
ejpam-6211	581	9	b.	b.	PROPN
ejpam-6211	581	10	montero	montero	PROPN
ejpam-6211	581	11	,	,	PUNCT
ejpam-6211	581	12	and	and	CCONJ
ejpam-6211	581	13	s.	s.	PROPN
ejpam-6211	581	14	l.	l.	PROPN
ejpam-6211	581	15	ballenas	ballenas	PROPN
ejpam-6211	581	16	.	.	PUNCT
ejpam-6211	582	1	schlömilch	schlömilch	ADJ
ejpam-6211	582	2	-	-	NOUN
ejpam-6211	582	3	type	type	NOUN
ejpam-6211	582	4	formula	formula	NOUN
ejpam-6211	582	5	for	for	ADP
ejpam-6211	582	6	r	r	NOUN
ejpam-6211	582	7	-	-	PUNCT
ejpam-6211	582	8	whitney	whitney	NOUN
ejpam-6211	582	9	numbers	number	NOUN
ejpam-6211	582	10	of	of	ADP
ejpam-6211	582	11	the	the	DET
ejpam-6211	582	12	first	first	ADJ
ejpam-6211	582	13	kind	kind	NOUN
ejpam-6211	582	14	.	.	PUNCT
ejpam-6211	583	1	european	european	PROPN
ejpam-6211	583	2	journal	journal	PROPN
ejpam-6211	583	3	of	of	ADP
ejpam-6211	583	4	pure	pure	ADJ
ejpam-6211	583	5	and	and	CCONJ
ejpam-6211	583	6	applied	applied	ADJ
ejpam-6211	583	7	mathematics	mathematic	NOUN
ejpam-6211	583	8	,	,	PUNCT
ejpam-6211	583	9	2024	2024	NUM
ejpam-6211	583	10	.	.	PUNCT