id	sid	tid	token	lemma	pos
ejpam-6191	1	1	european	european	PROPN
ejpam-6191	1	2	journal	journal	PROPN
ejpam-6191	1	3	of	of	ADP
ejpam-6191	1	4	pure	pure	ADJ
ejpam-6191	1	5	and	and	CCONJ
ejpam-6191	1	6	applied	applied	ADJ
ejpam-6191	1	7	mathematics	mathematic	NOUN
ejpam-6191	1	8	2025	2025	NUM
ejpam-6191	1	9	,	,	PUNCT
ejpam-6191	1	10	vol	vol	NOUN
ejpam-6191	1	11	.	.	PROPN
ejpam-6191	1	12	18	18	NUM
ejpam-6191	1	13	,	,	PUNCT
ejpam-6191	1	14	issue	issue	NOUN
ejpam-6191	1	15	3	3	NUM
ejpam-6191	1	16	,	,	PUNCT
ejpam-6191	1	17	article	article	NOUN
ejpam-6191	1	18	number	number	NOUN
ejpam-6191	1	19	6191	6191	NUM
ejpam-6191	1	20	issn	issn	PROPN
ejpam-6191	1	21	1307	1307	NUM
ejpam-6191	1	22	-	-	SYM
ejpam-6191	1	23	5543	5543	NUM
ejpam-6191	1	24	–	–	PUNCT
ejpam-6191	1	25	ejpam.com	ejpam.com	X
ejpam-6191	1	26	published	publish	VERB
ejpam-6191	1	27	by	by	ADP
ejpam-6191	1	28	new	new	PROPN
ejpam-6191	1	29	york	york	PROPN
ejpam-6191	1	30	business	business	PROPN
ejpam-6191	1	31	global	global	PROPN
ejpam-6191	1	32	generalized	generalize	VERB
ejpam-6191	1	33	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	1	34	sum	sum	NOUN
ejpam-6191	1	35	via	via	ADP
ejpam-6191	1	36	euler	euler	PROPN
ejpam-6191	1	37	’s	’s	PART
ejpam-6191	1	38	transform	transform	NOUN
ejpam-6191	1	39	kristen	kristen	PROPN
ejpam-6191	1	40	vera	vera	NOUN
ejpam-6191	1	41	m.	m.	PROPN
ejpam-6191	1	42	manulat2	manulat2	PROPN
ejpam-6191	1	43	,	,	PUNCT
ejpam-6191	1	44	roberto	roberto	PROPN
ejpam-6191	1	45	b.	b.	PROPN
ejpam-6191	1	46	corcino1,2,∗	corcino1,2,∗	PROPN
ejpam-6191	1	47	1	1	NUM
ejpam-6191	1	48	research	research	NOUN
ejpam-6191	1	49	institute	institute	NOUN
ejpam-6191	1	50	for	for	ADP
ejpam-6191	1	51	computational	computational	ADJ
ejpam-6191	1	52	mathematics	mathematic	NOUN
ejpam-6191	1	53	and	and	CCONJ
ejpam-6191	1	54	physics	physics	NOUN
ejpam-6191	1	55	,	,	PUNCT
ejpam-6191	1	56	cebu	cebu	NOUN
ejpam-6191	1	57	normal	normal	ADJ
ejpam-6191	1	58	university	university	NOUN
ejpam-6191	1	59	,	,	PUNCT
ejpam-6191	1	60	6000	6000	NUM
ejpam-6191	1	61	cebu	cebu	NOUN
ejpam-6191	1	62	city	city	NOUN
ejpam-6191	1	63	,	,	PUNCT
ejpam-6191	1	64	philippines	philippine	NOUN
ejpam-6191	1	65	2	2	NUM
ejpam-6191	1	66	mathematics	mathematics	NOUN
ejpam-6191	1	67	department	department	NOUN
ejpam-6191	1	68	,	,	PUNCT
ejpam-6191	1	69	cebu	cebu	NOUN
ejpam-6191	1	70	normal	normal	ADJ
ejpam-6191	1	71	university	university	NOUN
ejpam-6191	1	72	,	,	PUNCT
ejpam-6191	1	73	6000	6000	NUM
ejpam-6191	1	74	cebu	cebu	NOUN
ejpam-6191	1	75	city	city	NOUN
ejpam-6191	1	76	,	,	PUNCT
ejpam-6191	1	77	philippines	philippine	NOUN
ejpam-6191	1	78	abstract	abstract	ADJ
ejpam-6191	1	79	.	.	PUNCT
ejpam-6191	2	1	in	in	ADP
ejpam-6191	2	2	this	this	DET
ejpam-6191	2	3	paper	paper	NOUN
ejpam-6191	2	4	,	,	PUNCT
ejpam-6191	2	5	we	we	PRON
ejpam-6191	2	6	present	present	VERB
ejpam-6191	2	7	and	and	CCONJ
ejpam-6191	2	8	prove	prove	VERB
ejpam-6191	2	9	a	a	DET
ejpam-6191	2	10	novel	novel	ADJ
ejpam-6191	2	11	expression	expression	NOUN
ejpam-6191	2	12	for	for	ADP
ejpam-6191	2	13	binomial	binomial	ADJ
ejpam-6191	2	14	sums	sum	NOUN
ejpam-6191	2	15	involving	involve	VERB
ejpam-6191	2	16	generalized	generalize	VERB
ejpam-6191	2	17	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	2	18	numbers	number	NOUN
ejpam-6191	2	19	.	.	PUNCT
ejpam-6191	3	1	our	our	PRON
ejpam-6191	3	2	approach	approach	NOUN
ejpam-6191	3	3	utilizes	utilize	VERB
ejpam-6191	3	4	euler	euler	PROPN
ejpam-6191	3	5	’s	’s	PART
ejpam-6191	3	6	transformation	transformation	NOUN
ejpam-6191	3	7	applied	apply	VERB
ejpam-6191	3	8	to	to	ADP
ejpam-6191	3	9	the	the	DET
ejpam-6191	3	10	ordinary	ordinary	ADJ
ejpam-6191	3	11	generating	generate	VERB
ejpam-6191	3	12	function	function	NOUN
ejpam-6191	3	13	of	of	ADP
ejpam-6191	3	14	the	the	DET
ejpam-6191	3	15	generalized	generalized	ADJ
ejpam-6191	3	16	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	3	17	numbers	number	NOUN
ejpam-6191	3	18	.	.	PUNCT
ejpam-6191	4	1	to	to	PART
ejpam-6191	4	2	demonstrate	demonstrate	VERB
ejpam-6191	4	3	the	the	DET
ejpam-6191	4	4	relevance	relevance	NOUN
ejpam-6191	4	5	of	of	ADP
ejpam-6191	4	6	this	this	DET
ejpam-6191	4	7	new	new	ADJ
ejpam-6191	4	8	expression	expression	NOUN
ejpam-6191	4	9	,	,	PUNCT
ejpam-6191	4	10	we	we	PRON
ejpam-6191	4	11	derive	derive	VERB
ejpam-6191	4	12	several	several	ADJ
ejpam-6191	4	13	identities	identity	NOUN
ejpam-6191	4	14	that	that	PRON
ejpam-6191	4	15	reveal	reveal	VERB
ejpam-6191	4	16	connections	connection	NOUN
ejpam-6191	4	17	between	between	ADP
ejpam-6191	4	18	the	the	DET
ejpam-6191	4	19	characteristic	characteristic	ADJ
ejpam-6191	4	20	equations	equation	NOUN
ejpam-6191	4	21	and	and	CCONJ
ejpam-6191	4	22	binet	binet	NOUN
ejpam-6191	4	23	forms	form	NOUN
ejpam-6191	4	24	of	of	ADP
ejpam-6191	4	25	notable	notable	ADJ
ejpam-6191	4	26	numerical	numerical	ADJ
ejpam-6191	4	27	sequences	sequence	NOUN
ejpam-6191	4	28	,	,	PUNCT
ejpam-6191	4	29	including	include	VERB
ejpam-6191	4	30	the	the	DET
ejpam-6191	4	31	fibonacci	fibonacci	NOUN
ejpam-6191	4	32	,	,	PUNCT
ejpam-6191	4	33	lucas	lucas	PROPN
ejpam-6191	4	34	,	,	PUNCT
ejpam-6191	4	35	pell	pell	INTJ
ejpam-6191	4	36	,	,	PUNCT
ejpam-6191	4	37	pell	pell	NOUN
ejpam-6191	4	38	-	-	PUNCT
ejpam-6191	4	39	lucas	lucas	NOUN
ejpam-6191	4	40	,	,	PUNCT
ejpam-6191	4	41	jacobsthal	jacobsthal	ADJ
ejpam-6191	4	42	,	,	PUNCT
ejpam-6191	4	43	jacobsthal	jacobsthal	ADJ
ejpam-6191	4	44	-	-	PUNCT
ejpam-6191	4	45	lucas	lucas	PROPN
ejpam-6191	4	46	,	,	PUNCT
ejpam-6191	4	47	mersenne	mersenne	NOUN
ejpam-6191	4	48	,	,	PUNCT
ejpam-6191	4	49	and	and	CCONJ
ejpam-6191	4	50	mersenne	mersenne	NOUN
ejpam-6191	4	51	-	-	PUNCT
ejpam-6191	4	52	lucas	lucas	PROPN
ejpam-6191	4	53	numbers	number	NOUN
ejpam-6191	4	54	.	.	PUNCT
ejpam-6191	5	1	furthermore	furthermore	ADV
ejpam-6191	5	2	,	,	PUNCT
ejpam-6191	5	3	we	we	PRON
ejpam-6191	5	4	establish	establish	VERB
ejpam-6191	5	5	the	the	DET
ejpam-6191	5	6	integer	integer	NOUN
ejpam-6191	5	7	power	power	NOUN
ejpam-6191	5	8	representation	representation	NOUN
ejpam-6191	5	9	of	of	ADP
ejpam-6191	5	10	the	the	DET
ejpam-6191	5	11	generalized	generalize	VERB
ejpam-6191	5	12	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	5	13	sums	sum	NOUN
ejpam-6191	5	14	.	.	PUNCT
ejpam-6191	6	1	as	as	ADP
ejpam-6191	6	2	an	an	DET
ejpam-6191	6	3	extension	extension	NOUN
ejpam-6191	6	4	of	of	ADP
ejpam-6191	6	5	our	our	PRON
ejpam-6191	6	6	findings	finding	NOUN
ejpam-6191	6	7	,	,	PUNCT
ejpam-6191	6	8	we	we	PRON
ejpam-6191	6	9	also	also	ADV
ejpam-6191	6	10	introduce	introduce	VERB
ejpam-6191	6	11	and	and	CCONJ
ejpam-6191	6	12	prove	prove	VERB
ejpam-6191	6	13	an	an	DET
ejpam-6191	6	14	alternative	alternative	ADJ
ejpam-6191	6	15	expression	expression	NOUN
ejpam-6191	6	16	using	use	VERB
ejpam-6191	6	17	the	the	DET
ejpam-6191	6	18	polylogarithmic	polylogarithmic	ADJ
ejpam-6191	6	19	form	form	NOUN
ejpam-6191	6	20	of	of	ADP
ejpam-6191	6	21	the	the	DET
ejpam-6191	6	22	generating	generate	VERB
ejpam-6191	6	23	function	function	NOUN
ejpam-6191	6	24	for	for	ADP
ejpam-6191	6	25	the	the	DET
ejpam-6191	6	26	generalized	generalized	ADJ
ejpam-6191	6	27	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	6	28	numbers	number	NOUN
ejpam-6191	6	29	.	.	PUNCT
ejpam-6191	7	1	2020	2020	NUM
ejpam-6191	7	2	mathematics	mathematic	NOUN
ejpam-6191	7	3	subject	subject	NOUN
ejpam-6191	7	4	classifications	classification	NOUN
ejpam-6191	7	5	:	:	PUNCT
ejpam-6191	7	6	11b68	11b68	NUM
ejpam-6191	7	7	,	,	PUNCT
ejpam-6191	7	8	11b73	11b73	NUM
ejpam-6191	7	9	,	,	PUNCT
ejpam-6191	7	10	05a15	05a15	NOUN
ejpam-6191	7	11	key	key	ADJ
ejpam-6191	7	12	words	word	NOUN
ejpam-6191	7	13	and	and	CCONJ
ejpam-6191	7	14	phrases	phrase	NOUN
ejpam-6191	7	15	:	:	PUNCT
ejpam-6191	7	16	harmonic	harmonic	ADJ
ejpam-6191	7	17	numbers	number	NOUN
ejpam-6191	7	18	,	,	PUNCT
ejpam-6191	7	19	generalized	generalize	VERB
ejpam-6191	7	20	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	7	21	numbers	number	NOUN
ejpam-6191	7	22	,	,	PUNCT
ejpam-6191	7	23	euler	euler	NOUN
ejpam-6191	7	24	’s	’s	PART
ejpam-6191	7	25	transformation	transformation	NOUN
ejpam-6191	7	26	1	1	NUM
ejpam-6191	7	27	.	.	PUNCT
ejpam-6191	7	28	introduction	introduction	NOUN
ejpam-6191	7	29	harmonic	harmonic	ADJ
ejpam-6191	7	30	numbers	number	NOUN
ejpam-6191	7	31	,	,	PUNCT
ejpam-6191	7	32	as	as	SCONJ
ejpam-6191	7	33	discussed	discuss	VERB
ejpam-6191	7	34	in	in	ADP
ejpam-6191	7	35	[	[	X
ejpam-6191	7	36	16	16	NUM
ejpam-6191	7	37	]	]	PUNCT
ejpam-6191	7	38	,	,	PUNCT
ejpam-6191	7	39	play	play	VERB
ejpam-6191	7	40	a	a	DET
ejpam-6191	7	41	fundamental	fundamental	ADJ
ejpam-6191	7	42	role	role	NOUN
ejpam-6191	7	43	in	in	ADP
ejpam-6191	7	44	combinatorial	combinatorial	ADJ
ejpam-6191	7	45	number	number	NOUN
ejpam-6191	7	46	theory	theory	NOUN
ejpam-6191	7	47	.	.	PUNCT
ejpam-6191	8	1	the	the	DET
ejpam-6191	8	2	nth	nth	NOUN
ejpam-6191	8	3	harmonic	harmonic	ADJ
ejpam-6191	8	4	number	number	NOUN
ejpam-6191	8	5	,	,	PUNCT
ejpam-6191	8	6	denoted	denote	VERB
ejpam-6191	8	7	by	by	ADP
ejpam-6191	8	8	hn	hn	PROPN
ejpam-6191	8	9	,	,	PUNCT
ejpam-6191	8	10	is	be	AUX
ejpam-6191	8	11	classically	classically	ADV
ejpam-6191	8	12	defined	define	VERB
ejpam-6191	8	13	with	with	ADP
ejpam-6191	8	14	the	the	DET
ejpam-6191	8	15	initial	initial	ADJ
ejpam-6191	8	16	condition	condition	NOUN
ejpam-6191	8	17	h0	h0	NOUN
ejpam-6191	8	18	=	=	PROPN
ejpam-6191	8	19	0	0	NUM
ejpam-6191	8	20	,	,	PUNCT
ejpam-6191	8	21	and	and	CCONJ
ejpam-6191	8	22	for	for	ADP
ejpam-6191	8	23	n	n	PROPN
ejpam-6191	8	24	>	>	X
ejpam-6191	8	25	0	0	NUM
ejpam-6191	8	26	,	,	PUNCT
ejpam-6191	8	27	it	it	PRON
ejpam-6191	8	28	can	can	AUX
ejpam-6191	8	29	be	be	AUX
ejpam-6191	8	30	expressed	express	VERB
ejpam-6191	8	31	in	in	ADP
ejpam-6191	8	32	both	both	CCONJ
ejpam-6191	8	33	recursive	recursive	ADJ
ejpam-6191	8	34	and	and	CCONJ
ejpam-6191	8	35	integral	integral	ADJ
ejpam-6191	8	36	forms	form	NOUN
ejpam-6191	8	37	as	as	SCONJ
ejpam-6191	8	38	follows	follow	VERB
ejpam-6191	8	39	:	:	PUNCT
ejpam-6191	9	1	hn	hn	PROPN
ejpam-6191	9	2	=	=	PUNCT
ejpam-6191	9	3	n∑	n∑	NOUN
ejpam-6191	9	4	k=1	k=1	NOUN
ejpam-6191	10	1	1	1	NUM
ejpam-6191	10	2	k	k	NOUN
ejpam-6191	10	3	,	,	PUNCT
ejpam-6191	10	4	hn	hn	PROPN
ejpam-6191	10	5	=	=	PUNCT
ejpam-6191	10	6	hn−1	hn−1	PROPN
ejpam-6191	10	7	+	+	CCONJ
ejpam-6191	10	8	1	1	NUM
ejpam-6191	10	9	n	n	NOUN
ejpam-6191	10	10	,	,	PUNCT
ejpam-6191	10	11	hn	hn	PROPN
ejpam-6191	10	12	=	=	SYM
ejpam-6191	10	13	∫	∫	PROPN
ejpam-6191	11	1	1	1	NUM
ejpam-6191	11	2	0	0	NUM
ejpam-6191	11	3	1−	1−	NUM
ejpam-6191	11	4	xn	xn	NUM
ejpam-6191	11	5	1−	1−	NUM
ejpam-6191	11	6	x	x	SYM
ejpam-6191	11	7	dx	dx	PROPN
ejpam-6191	11	8	,	,	PUNCT
ejpam-6191	11	9	∗corresponding	∗corresponde	VERB
ejpam-6191	11	10	author	author	NOUN
ejpam-6191	11	11	.	.	PUNCT
ejpam-6191	12	1	doi	doi	NOUN
ejpam-6191	12	2	:	:	PUNCT
ejpam-6191	12	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6191	https://doi.org/10.29020/nybg.ejpam.v18i3.6191	NOUN
ejpam-6191	12	4	email	email	NOUN
ejpam-6191	12	5	addresses	address	VERB
ejpam-6191	12	6	:	:	PUNCT
ejpam-6191	12	7	main.18000213@cnu.edu.ph	main.18000213@cnu.edu.ph	PROPN
ejpam-6191	12	8	(	(	PUNCT
ejpam-6191	12	9	k.	k.	NOUN
ejpam-6191	12	10	v.	v.	PROPN
ejpam-6191	12	11	m.	m.	PROPN
ejpam-6191	12	12	manulat	manulat	PROPN
ejpam-6191	12	13	)	)	PUNCT
ejpam-6191	12	14	,	,	PUNCT
ejpam-6191	12	15	rcorcino@yahoo.com	rcorcino@yahoo.com	PROPN
ejpam-6191	12	16	(	(	PUNCT
ejpam-6191	12	17	r.	r.	PROPN
ejpam-6191	12	18	b.	b.	PROPN
ejpam-6191	12	19	corcino	corcino	PROPN
ejpam-6191	12	20	)	)	PUNCT
ejpam-6191	12	21	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6191	13	1	1	1	NUM
ejpam-6191	13	2	copyright	copyright	NOUN
ejpam-6191	13	3	:	:	PUNCT
ejpam-6191	13	4	©	©	PROPN
ejpam-6191	13	5	2025	2025	NUM
ejpam-6191	13	6	the	the	DET
ejpam-6191	13	7	author(s	author(s	NOUN
ejpam-6191	13	8	)	)	PUNCT
ejpam-6191	13	9	.	.	PUNCT
ejpam-6191	14	1	(	(	PUNCT
ejpam-6191	14	2	cc	cc	NOUN
ejpam-6191	14	3	by	by	ADP
ejpam-6191	14	4	-	-	PUNCT
ejpam-6191	14	5	nc	nc	PROPN
ejpam-6191	14	6	4.0	4.0	NUM
ejpam-6191	14	7	)	)	PUNCT
ejpam-6191	14	8	k.	k.	PROPN
ejpam-6191	15	1	v.	v.	PROPN
ejpam-6191	15	2	m.	m.	PROPN
ejpam-6191	15	3	manulat	manulat	PROPN
ejpam-6191	15	4	,	,	PUNCT
ejpam-6191	15	5	r.	r.	PROPN
ejpam-6191	15	6	b.	b.	PROPN
ejpam-6191	15	7	corcino	corcino	PROPN
ejpam-6191	15	8	/	/	SYM
ejpam-6191	15	9	eur	eur	PROPN
ejpam-6191	15	10	.	.	PUNCT
ejpam-6191	16	1	j.	j.	PROPN
ejpam-6191	16	2	pure	pure	PROPN
ejpam-6191	16	3	appl	appl	PROPN
ejpam-6191	16	4	.	.	PROPN
ejpam-6191	16	5	math	math	PROPN
ejpam-6191	16	6	,	,	PUNCT
ejpam-6191	16	7	18	18	NUM
ejpam-6191	16	8	(	(	PUNCT
ejpam-6191	16	9	3	3	NUM
ejpam-6191	16	10	)	)	PUNCT
ejpam-6191	16	11	(	(	PUNCT
ejpam-6191	16	12	2025	2025	NUM
ejpam-6191	16	13	)	)	PUNCT
ejpam-6191	16	14	,	,	PUNCT
ejpam-6191	16	15	6191	6191	NUM
ejpam-6191	16	16	2	2	NUM
ejpam-6191	16	17	of	of	ADP
ejpam-6191	16	18	21	21	NUM
ejpam-6191	16	19	as	as	SCONJ
ejpam-6191	16	20	given	give	VERB
ejpam-6191	16	21	in	in	ADP
ejpam-6191	16	22	[	[	NOUN
ejpam-6191	16	23	14	14	NUM
ejpam-6191	16	24	]	]	PUNCT
ejpam-6191	16	25	.	.	PUNCT
ejpam-6191	17	1	these	these	DET
ejpam-6191	17	2	formulations	formulation	NOUN
ejpam-6191	17	3	not	not	PART
ejpam-6191	17	4	only	only	ADV
ejpam-6191	17	5	highlight	highlight	VERB
ejpam-6191	17	6	the	the	DET
ejpam-6191	17	7	additive	additive	ADJ
ejpam-6191	17	8	structure	structure	NOUN
ejpam-6191	17	9	of	of	ADP
ejpam-6191	17	10	harmonic	harmonic	ADJ
ejpam-6191	17	11	numbers	number	NOUN
ejpam-6191	17	12	but	but	CCONJ
ejpam-6191	17	13	also	also	ADV
ejpam-6191	17	14	provide	provide	VERB
ejpam-6191	17	15	insight	insight	NOUN
ejpam-6191	17	16	into	into	ADP
ejpam-6191	17	17	their	their	PRON
ejpam-6191	17	18	analytical	analytical	ADJ
ejpam-6191	17	19	behavior	behavior	NOUN
ejpam-6191	17	20	.	.	PUNCT
ejpam-6191	18	1	the	the	DET
ejpam-6191	18	2	generating	generate	VERB
ejpam-6191	18	3	function	function	NOUN
ejpam-6191	18	4	for	for	ADP
ejpam-6191	18	5	the	the	DET
ejpam-6191	18	6	sequence	sequence	NOUN
ejpam-6191	18	7	{	{	PUNCT
ejpam-6191	18	8	hn	hn	NOUN
ejpam-6191	18	9	}	}	PUNCT
ejpam-6191	18	10	is	be	AUX
ejpam-6191	18	11	given	give	VERB
ejpam-6191	18	12	by	by	ADP
ejpam-6191	18	13	the	the	DET
ejpam-6191	18	14	series	series	NOUN
ejpam-6191	18	15	:	:	PUNCT
ejpam-6191	18	16	∞∑	∞∑	NUM
ejpam-6191	18	17	n=0	n=0	NUM
ejpam-6191	18	18	hnz	hnz	VERB
ejpam-6191	18	19	n	n	NOUN
ejpam-6191	18	20	=	=	SYM
ejpam-6191	18	21	−	−	PROPN
ejpam-6191	19	1	ln(1−	ln(1−	PROPN
ejpam-6191	19	2	z	z	PROPN
ejpam-6191	19	3	)	)	PUNCT
ejpam-6191	19	4	1−	1−	NUM
ejpam-6191	19	5	z	z	NOUN
ejpam-6191	19	6	,	,	PUNCT
ejpam-6191	19	7	(	(	PUNCT
ejpam-6191	19	8	1.1	1.1	NUM
ejpam-6191	19	9	)	)	PUNCT
ejpam-6191	19	10	as	as	SCONJ
ejpam-6191	19	11	derived	derive	VERB
ejpam-6191	19	12	in	in	ADP
ejpam-6191	19	13	[	[	X
ejpam-6191	19	14	3	3	NUM
ejpam-6191	19	15	]	]	PUNCT
ejpam-6191	19	16	,	,	PUNCT
ejpam-6191	19	17	which	which	PRON
ejpam-6191	19	18	facilitates	facilitate	VERB
ejpam-6191	19	19	the	the	DET
ejpam-6191	19	20	manipulation	manipulation	NOUN
ejpam-6191	19	21	and	and	CCONJ
ejpam-6191	19	22	analysis	analysis	NOUN
ejpam-6191	19	23	of	of	ADP
ejpam-6191	19	24	harmonic	harmonic	ADJ
ejpam-6191	19	25	numbers	number	NOUN
ejpam-6191	19	26	in	in	ADP
ejpam-6191	19	27	a	a	DET
ejpam-6191	19	28	variety	variety	NOUN
ejpam-6191	19	29	of	of	ADP
ejpam-6191	19	30	mathematical	mathematical	ADJ
ejpam-6191	19	31	contexts	contexts	NOUN
ejpam-6191	19	32	.	.	PUNCT
ejpam-6191	20	1	over	over	ADP
ejpam-6191	20	2	recent	recent	ADJ
ejpam-6191	20	3	years	year	NOUN
ejpam-6191	20	4	,	,	PUNCT
ejpam-6191	20	5	several	several	ADJ
ejpam-6191	20	6	researchers	researcher	NOUN
ejpam-6191	20	7	have	have	AUX
ejpam-6191	20	8	extended	extend	VERB
ejpam-6191	20	9	the	the	DET
ejpam-6191	20	10	concept	concept	NOUN
ejpam-6191	20	11	of	of	ADP
ejpam-6191	20	12	harmonic	harmonic	ADJ
ejpam-6191	20	13	numbers	number	NOUN
ejpam-6191	20	14	.	.	PUNCT
ejpam-6191	21	1	one	one	NUM
ejpam-6191	21	2	such	such	ADJ
ejpam-6191	21	3	generalization	generalization	NOUN
ejpam-6191	21	4	is	be	AUX
ejpam-6191	21	5	the	the	DET
ejpam-6191	21	6	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	21	7	numbers	number	NOUN
ejpam-6191	21	8	,	,	PUNCT
ejpam-6191	21	9	introduced	introduce	VERB
ejpam-6191	21	10	in	in	ADP
ejpam-6191	21	11	[	[	X
ejpam-6191	21	12	13	13	NUM
ejpam-6191	21	13	]	]	PUNCT
ejpam-6191	21	14	.	.	PUNCT
ejpam-6191	22	1	these	these	DET
ejpam-6191	22	2	numbers	number	NOUN
ejpam-6191	22	3	are	be	AUX
ejpam-6191	22	4	defined	define	VERB
ejpam-6191	22	5	recursively	recursively	ADV
ejpam-6191	22	6	by	by	ADP
ejpam-6191	22	7	:	:	PUNCT
ejpam-6191	22	8	h(r	h(r	NOUN
ejpam-6191	22	9	)	)	PUNCT
ejpam-6191	22	10	n	n	NOUN
ejpam-6191	23	1	=	=	SYM
ejpam-6191	23	2			PROPN
ejpam-6191	23	3	∑n	∑n	PROPN
ejpam-6191	23	4	k=1h	k=1h	NOUN
ejpam-6191	23	5	(	(	PUNCT
ejpam-6191	23	6	r−1	r−1	PROPN
ejpam-6191	23	7	)	)	PUNCT
ejpam-6191	24	1	k	k	PROPN
ejpam-6191	24	2	,	,	PUNCT
ejpam-6191	24	3	for	for	ADP
ejpam-6191	24	4	n	n	CCONJ
ejpam-6191	24	5	,	,	PUNCT
ejpam-6191	24	6	r	r	NOUN
ejpam-6191	24	7	≥	≥	NUM
ejpam-6191	24	8	1	1	NUM
ejpam-6191	24	9	,	,	PUNCT
ejpam-6191	24	10	1	1	NUM
ejpam-6191	24	11	n	n	NOUN
ejpam-6191	24	12	,	,	PUNCT
ejpam-6191	24	13	for	for	ADP
ejpam-6191	24	14	r	r	NOUN
ejpam-6191	24	15	=	=	SYM
ejpam-6191	24	16	0	0	NUM
ejpam-6191	24	17	and	and	CCONJ
ejpam-6191	24	18	n	n	CCONJ
ejpam-6191	24	19	>	>	NOUN
ejpam-6191	24	20	0	0	NUM
ejpam-6191	24	21	,	,	PUNCT
ejpam-6191	24	22	0	0	NUM
ejpam-6191	24	23	,	,	PUNCT
ejpam-6191	24	24	for	for	ADP
ejpam-6191	24	25	r	r	NOUN
ejpam-6191	24	26	<	<	X
ejpam-6191	24	27	0	0	NUM
ejpam-6191	24	28	or	or	CCONJ
ejpam-6191	24	29	n	n	PRON
ejpam-6191	24	30	≤	≤	NOUN
ejpam-6191	24	31	0	0	NUM
ejpam-6191	24	32	.	.	PUNCT
ejpam-6191	25	1	the	the	DET
ejpam-6191	25	2	generating	generate	VERB
ejpam-6191	25	3	function	function	NOUN
ejpam-6191	25	4	corresponding	correspond	VERB
ejpam-6191	25	5	to	to	ADP
ejpam-6191	25	6	the	the	DET
ejpam-6191	25	7	generalized	generalize	VERB
ejpam-6191	25	8	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	25	9	numbers	number	NOUN
ejpam-6191	25	10	for	for	ADP
ejpam-6191	25	11	r	r	NOUN
ejpam-6191	25	12	≥	≥	NUM
ejpam-6191	25	13	1	1	NUM
ejpam-6191	25	14	is	be	AUX
ejpam-6191	25	15	given	give	VERB
ejpam-6191	25	16	by	by	ADP
ejpam-6191	25	17	[	[	X
ejpam-6191	25	18	8	8	NUM
ejpam-6191	25	19	,	,	PUNCT
ejpam-6191	25	20	13	13	NUM
ejpam-6191	25	21	]	]	NOUN
ejpam-6191	25	22	:	:	PUNCT
ejpam-6191	25	23	∞∑	∞∑	NUM
ejpam-6191	25	24	n=0	n=0	NUM
ejpam-6191	25	25	h(r	h(r	NOUN
ejpam-6191	25	26	)	)	PUNCT
ejpam-6191	25	27	n	n	CCONJ
ejpam-6191	25	28	zn	zn	NOUN
ejpam-6191	25	29	=	=	PUNCT
ejpam-6191	26	1	[	[	X
ejpam-6191	26	2	−	−	X
ejpam-6191	26	3	ln(1−	ln(1−	PROPN
ejpam-6191	26	4	z)]r+1	z)]r+1	PROPN
ejpam-6191	26	5	1−	1−	NUM
ejpam-6191	26	6	z	z	NOUN
ejpam-6191	26	7	.	.	PUNCT
ejpam-6191	27	1	koparal	koparal	INTJ
ejpam-6191	27	2	et	et	PROPN
ejpam-6191	27	3	al	al	PROPN
ejpam-6191	27	4	.	.	PUNCT
ejpam-6191	28	1	[	[	X
ejpam-6191	28	2	21	21	NUM
ejpam-6191	28	3	]	]	X
ejpam-6191	28	4	further	far	ADV
ejpam-6191	28	5	extended	extend	VERB
ejpam-6191	28	6	this	this	DET
ejpam-6191	28	7	framework	framework	NOUN
ejpam-6191	28	8	by	by	ADP
ejpam-6191	28	9	introducing	introduce	VERB
ejpam-6191	28	10	the	the	DET
ejpam-6191	28	11	polylogarithmic	polylogarithmic	ADJ
ejpam-6191	28	12	generating	generating	NOUN
ejpam-6191	28	13	function	function	NOUN
ejpam-6191	28	14	for	for	ADP
ejpam-6191	28	15	generalized	generalize	VERB
ejpam-6191	28	16	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	28	17	sums	sum	NOUN
ejpam-6191	28	18	.	.	PUNCT
ejpam-6191	29	1	for	for	ADP
ejpam-6191	29	2	any	any	DET
ejpam-6191	29	3	positive	positive	ADJ
ejpam-6191	29	4	integer	integer	NOUN
ejpam-6191	29	5	m	m	PROPN
ejpam-6191	29	6	∈	∈	NOUN
ejpam-6191	29	7	z+	z+	NUM
ejpam-6191	29	8	,	,	PUNCT
ejpam-6191	29	9	the	the	DET
ejpam-6191	29	10	generating	generate	VERB
ejpam-6191	29	11	function	function	NOUN
ejpam-6191	29	12	is	be	AUX
ejpam-6191	29	13	defined	define	VERB
ejpam-6191	29	14	as	as	ADP
ejpam-6191	29	15	:	:	PUNCT
ejpam-6191	29	16	∞∑	∞∑	NUM
ejpam-6191	29	17	n=1	n=1	PROPN
ejpam-6191	29	18	h(r	h(r	NOUN
ejpam-6191	29	19	)	)	PUNCT
ejpam-6191	29	20	n	n	CCONJ
ejpam-6191	29	21	,	,	PUNCT
ejpam-6191	29	22	mzn	mzn	NOUN
ejpam-6191	29	23	=	=	SYM
ejpam-6191	29	24	lim(z	lim(z	PROPN
ejpam-6191	29	25	)	)	PUNCT
ejpam-6191	29	26	(	(	PUNCT
ejpam-6191	29	27	1−	1−	NUM
ejpam-6191	29	28	z)r	z)r	NOUN
ejpam-6191	29	29	,	,	PUNCT
ejpam-6191	29	30	where	where	SCONJ
ejpam-6191	29	31	li1(z	li1(z	PROPN
ejpam-6191	29	32	)	)	PUNCT
ejpam-6191	29	33	=	=	PUNCT
ejpam-6191	30	1	∑∞	∑∞	NOUN
ejpam-6191	30	2	k=1	k=1	X
ejpam-6191	31	1	zk	zk	X
ejpam-6191	31	2	k	k	PROPN
ejpam-6191	32	1	=	=	PUNCT
ejpam-6191	33	1	−	−	PROPN
ejpam-6191	33	2	ln(1−	ln(1−	PROPN
ejpam-6191	33	3	z	z	PROPN
ejpam-6191	33	4	)	)	PUNCT
ejpam-6191	33	5	is	be	AUX
ejpam-6191	33	6	the	the	DET
ejpam-6191	33	7	classical	classical	ADJ
ejpam-6191	33	8	polylogarithm	polylogarithm	NOUN
ejpam-6191	33	9	of	of	ADP
ejpam-6191	33	10	order	order	NOUN
ejpam-6191	33	11	1	1	X
ejpam-6191	33	12	.	.	PUNCT
ejpam-6191	34	1	in	in	ADP
ejpam-6191	34	2	a	a	DET
ejpam-6191	34	3	notable	notable	ADJ
ejpam-6191	34	4	contribution	contribution	NOUN
ejpam-6191	34	5	by	by	ADP
ejpam-6191	34	6	frontczak	frontczak	NOUN
ejpam-6191	34	7	[	[	X
ejpam-6191	34	8	14	14	NUM
ejpam-6191	34	9	]	]	X
ejpam-6191	34	10	,	,	PUNCT
ejpam-6191	34	11	a	a	DET
ejpam-6191	34	12	new	new	ADJ
ejpam-6191	34	13	closed	closed	ADJ
ejpam-6191	34	14	-	-	PUNCT
ejpam-6191	34	15	form	form	NOUN
ejpam-6191	34	16	expression	expression	NOUN
ejpam-6191	34	17	for	for	ADP
ejpam-6191	34	18	binomial	binomial	ADJ
ejpam-6191	34	19	sums	sum	NOUN
ejpam-6191	34	20	involving	involve	VERB
ejpam-6191	34	21	harmonic	harmonic	ADJ
ejpam-6191	34	22	numbers	number	NOUN
ejpam-6191	34	23	was	be	AUX
ejpam-6191	34	24	derived	derive	VERB
ejpam-6191	34	25	using	use	VERB
ejpam-6191	34	26	euler	euler	NOUN
ejpam-6191	34	27	’s	’s	PART
ejpam-6191	34	28	transform	transform	NOUN
ejpam-6191	34	29	applied	apply	VERB
ejpam-6191	34	30	to	to	ADP
ejpam-6191	34	31	the	the	DET
ejpam-6191	34	32	generating	generate	VERB
ejpam-6191	34	33	function	function	NOUN
ejpam-6191	34	34	for	for	ADP
ejpam-6191	34	35	the	the	DET
ejpam-6191	34	36	harmonic	harmonic	ADJ
ejpam-6191	34	37	numbers	number	NOUN
ejpam-6191	34	38	in	in	ADP
ejpam-6191	34	39	(	(	PUNCT
ejpam-6191	34	40	1.1	1.1	NUM
ejpam-6191	34	41	)	)	PUNCT
ejpam-6191	34	42	.	.	PUNCT
ejpam-6191	35	1	this	this	DET
ejpam-6191	35	2	result	result	NOUN
ejpam-6191	35	3	is	be	AUX
ejpam-6191	35	4	expressed	express	VERB
ejpam-6191	35	5	as	as	ADP
ejpam-6191	35	6	:	:	PUNCT
ejpam-6191	35	7	∞∑	∞∑	NUM
ejpam-6191	35	8	n=1	n=1	NUM
ejpam-6191	35	9	(	(	PUNCT
ejpam-6191	35	10	n∑	n∑	INTJ
ejpam-6191	35	11	k=1	k=1	PROPN
ejpam-6191	35	12	(	(	PUNCT
ejpam-6191	35	13	n	n	X
ejpam-6191	35	14	k	k	PROPN
ejpam-6191	35	15	)	)	PUNCT
ejpam-6191	35	16	akbn−kak	akbn−kak	NOUN
ejpam-6191	35	17	)	)	PUNCT
ejpam-6191	35	18	zn	zn	NOUN
ejpam-6191	35	19	=	=	SYM
ejpam-6191	35	20	1	1	NUM
ejpam-6191	35	21	1−	1−	NUM
ejpam-6191	36	1	bz	bz	PROPN
ejpam-6191	36	2	f	f	PROPN
ejpam-6191	36	3	(	(	PUNCT
ejpam-6191	36	4	az	az	PROPN
ejpam-6191	36	5	1−	1−	NUM
ejpam-6191	36	6	bz	bz	PROPN
ejpam-6191	36	7	)	)	PUNCT
ejpam-6191	36	8	,	,	PUNCT
ejpam-6191	36	9	(	(	PUNCT
ejpam-6191	36	10	1.2	1.2	NUM
ejpam-6191	36	11	)	)	PUNCT
ejpam-6191	36	12	where	where	SCONJ
ejpam-6191	36	13	f(z	f(z	NOUN
ejpam-6191	36	14	)	)	PUNCT
ejpam-6191	36	15	=	=	PUNCT
ejpam-6191	36	16	∑∞	∑∞	NOUN
ejpam-6191	36	17	n=0	n=0	PUNCT
ejpam-6191	36	18	anz	anz	PROPN
ejpam-6191	36	19	n.	n.	NOUN
ejpam-6191	36	20	this	this	DET
ejpam-6191	36	21	formulation	formulation	NOUN
ejpam-6191	36	22	provides	provide	VERB
ejpam-6191	36	23	a	a	DET
ejpam-6191	36	24	powerful	powerful	ADJ
ejpam-6191	36	25	tool	tool	NOUN
ejpam-6191	36	26	for	for	ADP
ejpam-6191	36	27	establishing	establish	VERB
ejpam-6191	36	28	new	new	ADJ
ejpam-6191	36	29	identities	identity	NOUN
ejpam-6191	36	30	,	,	PUNCT
ejpam-6191	36	31	including	include	VERB
ejpam-6191	36	32	those	those	PRON
ejpam-6191	36	33	involving	involve	VERB
ejpam-6191	36	34	fibonacci	fibonacci	NOUN
ejpam-6191	36	35	and	and	CCONJ
ejpam-6191	36	36	lucas	lucas	PROPN
ejpam-6191	36	37	numbers	number	NOUN
ejpam-6191	36	38	.	.	PUNCT
ejpam-6191	37	1	building	build	VERB
ejpam-6191	37	2	upon	upon	SCONJ
ejpam-6191	37	3	this	this	DET
ejpam-6191	37	4	foundation	foundation	NOUN
ejpam-6191	37	5	,	,	PUNCT
ejpam-6191	37	6	the	the	DET
ejpam-6191	37	7	present	present	ADJ
ejpam-6191	37	8	paper	paper	NOUN
ejpam-6191	37	9	aims	aim	VERB
ejpam-6191	37	10	to	to	PART
ejpam-6191	37	11	derive	derive	VERB
ejpam-6191	37	12	and	and	CCONJ
ejpam-6191	37	13	prove	prove	VERB
ejpam-6191	37	14	a	a	DET
ejpam-6191	37	15	new	new	ADJ
ejpam-6191	37	16	expression	expression	NOUN
ejpam-6191	37	17	for	for	ADP
ejpam-6191	37	18	binomial	binomial	ADJ
ejpam-6191	37	19	sums	sum	NOUN
ejpam-6191	37	20	involving	involve	VERB
ejpam-6191	37	21	generalized	generalize	VERB
ejpam-6191	37	22	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	37	23	numbers	number	NOUN
ejpam-6191	37	24	,	,	PUNCT
ejpam-6191	37	25	utilizing	utilize	VERB
ejpam-6191	37	26	the	the	DET
ejpam-6191	37	27	euler	euler	ADJ
ejpam-6191	37	28	-	-	PUNCT
ejpam-6191	37	29	type	type	NOUN
ejpam-6191	37	30	transformation	transformation	NOUN
ejpam-6191	37	31	approach	approach	NOUN
ejpam-6191	37	32	outlined	outline	VERB
ejpam-6191	37	33	in	in	ADP
ejpam-6191	37	34	[	[	X
ejpam-6191	37	35	14	14	NUM
ejpam-6191	37	36	]	]	PUNCT
ejpam-6191	37	37	.	.	PUNCT
ejpam-6191	38	1	we	we	PRON
ejpam-6191	38	2	also	also	ADV
ejpam-6191	38	3	establish	establish	VERB
ejpam-6191	38	4	the	the	DET
ejpam-6191	38	5	integer	integer	NOUN
ejpam-6191	38	6	power	power	NOUN
ejpam-6191	38	7	representation	representation	NOUN
ejpam-6191	38	8	of	of	ADP
ejpam-6191	38	9	the	the	DET
ejpam-6191	38	10	derived	derived	ADJ
ejpam-6191	38	11	expression	expression	NOUN
ejpam-6191	38	12	.	.	PUNCT
ejpam-6191	39	1	to	to	PART
ejpam-6191	39	2	illustrate	illustrate	VERB
ejpam-6191	39	3	the	the	DET
ejpam-6191	39	4	applicability	applicability	NOUN
ejpam-6191	39	5	and	and	CCONJ
ejpam-6191	39	6	importance	importance	NOUN
ejpam-6191	39	7	of	of	ADP
ejpam-6191	39	8	k.	k.	PROPN
ejpam-6191	39	9	v.	v.	PROPN
ejpam-6191	39	10	m.	m.	PROPN
ejpam-6191	39	11	manulat	manulat	PROPN
ejpam-6191	39	12	,	,	PUNCT
ejpam-6191	39	13	r.	r.	PROPN
ejpam-6191	39	14	b.	b.	PROPN
ejpam-6191	39	15	corcino	corcino	PROPN
ejpam-6191	39	16	/	/	SYM
ejpam-6191	39	17	eur	eur	PROPN
ejpam-6191	39	18	.	.	PUNCT
ejpam-6191	40	1	j.	j.	PROPN
ejpam-6191	40	2	pure	pure	PROPN
ejpam-6191	40	3	appl	appl	PROPN
ejpam-6191	40	4	.	.	PROPN
ejpam-6191	40	5	math	math	PROPN
ejpam-6191	40	6	,	,	PUNCT
ejpam-6191	40	7	18	18	NUM
ejpam-6191	40	8	(	(	PUNCT
ejpam-6191	40	9	3	3	NUM
ejpam-6191	40	10	)	)	PUNCT
ejpam-6191	40	11	(	(	PUNCT
ejpam-6191	40	12	2025	2025	NUM
ejpam-6191	40	13	)	)	PUNCT
ejpam-6191	40	14	,	,	PUNCT
ejpam-6191	40	15	6191	6191	NUM
ejpam-6191	40	16	3	3	NUM
ejpam-6191	40	17	of	of	ADP
ejpam-6191	40	18	21	21	NUM
ejpam-6191	40	19	our	our	PRON
ejpam-6191	40	20	results	result	NOUN
ejpam-6191	40	21	,	,	PUNCT
ejpam-6191	40	22	we	we	PRON
ejpam-6191	40	23	deduce	deduce	VERB
ejpam-6191	40	24	a	a	DET
ejpam-6191	40	25	series	series	NOUN
ejpam-6191	40	26	of	of	ADP
ejpam-6191	40	27	identities	identity	NOUN
ejpam-6191	40	28	that	that	PRON
ejpam-6191	40	29	relate	relate	VERB
ejpam-6191	40	30	to	to	ADP
ejpam-6191	40	31	well	well	ADV
ejpam-6191	40	32	-	-	PUNCT
ejpam-6191	40	33	known	know	VERB
ejpam-6191	40	34	number	number	NOUN
ejpam-6191	40	35	sequences	sequence	NOUN
ejpam-6191	40	36	such	such	ADJ
ejpam-6191	40	37	as	as	ADP
ejpam-6191	40	38	the	the	DET
ejpam-6191	40	39	fibonacci	fibonacci	PROPN
ejpam-6191	40	40	,	,	PUNCT
ejpam-6191	40	41	lucas	lucas	PROPN
ejpam-6191	40	42	,	,	PUNCT
ejpam-6191	40	43	pell	pell	INTJ
ejpam-6191	40	44	,	,	PUNCT
ejpam-6191	40	45	pell	pell	PROPN
ejpam-6191	40	46	lucas	lucas	PROPN
ejpam-6191	40	47	,	,	PUNCT
ejpam-6191	40	48	jacobsthal	jacobsthal	ADJ
ejpam-6191	40	49	,	,	PUNCT
ejpam-6191	40	50	jacobsthal	jacobsthal	PROPN
ejpam-6191	40	51	lucas	lucas	PROPN
ejpam-6191	40	52	,	,	PUNCT
ejpam-6191	40	53	mersenne	mersenne	NOUN
ejpam-6191	40	54	,	,	PUNCT
ejpam-6191	40	55	and	and	CCONJ
ejpam-6191	40	56	mersenne	mersenne	PROPN
ejpam-6191	40	57	lucas	lucas	PROPN
ejpam-6191	40	58	numbers	number	NOUN
ejpam-6191	40	59	.	.	PUNCT
ejpam-6191	41	1	these	these	DET
ejpam-6191	41	2	identities	identity	NOUN
ejpam-6191	41	3	are	be	AUX
ejpam-6191	41	4	examined	examine	VERB
ejpam-6191	41	5	in	in	ADP
ejpam-6191	41	6	the	the	DET
ejpam-6191	41	7	context	context	NOUN
ejpam-6191	41	8	of	of	ADP
ejpam-6191	41	9	their	their	PRON
ejpam-6191	41	10	respective	respective	ADJ
ejpam-6191	41	11	characteristic	characteristic	ADJ
ejpam-6191	41	12	equations	equation	NOUN
ejpam-6191	41	13	and	and	CCONJ
ejpam-6191	41	14	binet	binet	NOUN
ejpam-6191	41	15	formulas	formula	NOUN
ejpam-6191	41	16	,	,	PUNCT
ejpam-6191	41	17	offering	offer	VERB
ejpam-6191	41	18	deeper	deep	ADJ
ejpam-6191	41	19	insights	insight	NOUN
ejpam-6191	41	20	into	into	ADP
ejpam-6191	41	21	their	their	PRON
ejpam-6191	41	22	structure	structure	NOUN
ejpam-6191	41	23	.	.	PUNCT
ejpam-6191	42	1	furthermore	furthermore	ADV
ejpam-6191	42	2	,	,	PUNCT
ejpam-6191	42	3	as	as	ADP
ejpam-6191	42	4	an	an	DET
ejpam-6191	42	5	extension	extension	NOUN
ejpam-6191	42	6	of	of	ADP
ejpam-6191	42	7	our	our	PRON
ejpam-6191	42	8	main	main	ADJ
ejpam-6191	42	9	result	result	NOUN
ejpam-6191	42	10	,	,	PUNCT
ejpam-6191	42	11	we	we	PRON
ejpam-6191	42	12	define	define	VERB
ejpam-6191	42	13	and	and	CCONJ
ejpam-6191	42	14	prove	prove	VERB
ejpam-6191	42	15	a	a	DET
ejpam-6191	42	16	new	new	ADJ
ejpam-6191	42	17	formulation	formulation	NOUN
ejpam-6191	42	18	involving	involve	VERB
ejpam-6191	42	19	the	the	DET
ejpam-6191	42	20	generalized	generalize	VERB
ejpam-6191	42	21	poly	poly	ADJ
ejpam-6191	42	22	-	-	PUNCT
ejpam-6191	42	23	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	42	24	numbers	number	NOUN
ejpam-6191	42	25	of	of	ADP
ejpam-6191	42	26	order	order	NOUN
ejpam-6191	42	27	r	r	NOUN
ejpam-6191	42	28	,	,	PUNCT
ejpam-6191	42	29	showcasing	showcase	VERB
ejpam-6191	42	30	their	their	PRON
ejpam-6191	42	31	connection	connection	NOUN
ejpam-6191	42	32	to	to	ADP
ejpam-6191	42	33	polylogarithmic	polylogarithmic	ADJ
ejpam-6191	42	34	functions	function	NOUN
ejpam-6191	42	35	and	and	CCONJ
ejpam-6191	42	36	generating	generate	VERB
ejpam-6191	42	37	function	function	NOUN
ejpam-6191	42	38	techniques	technique	NOUN
ejpam-6191	42	39	.	.	PUNCT
ejpam-6191	43	1	2	2	X
ejpam-6191	43	2	.	.	X
ejpam-6191	43	3	some	some	DET
ejpam-6191	43	4	preliminary	preliminary	ADJ
ejpam-6191	43	5	concepts	concept	NOUN
ejpam-6191	43	6	this	this	DET
ejpam-6191	43	7	section	section	NOUN
ejpam-6191	43	8	presents	present	VERB
ejpam-6191	43	9	key	key	ADJ
ejpam-6191	43	10	foundational	foundational	ADJ
ejpam-6191	43	11	concepts	concept	NOUN
ejpam-6191	43	12	that	that	PRON
ejpam-6191	43	13	will	will	AUX
ejpam-6191	43	14	be	be	AUX
ejpam-6191	43	15	referenced	reference	VERB
ejpam-6191	43	16	throughout	throughout	ADP
ejpam-6191	43	17	this	this	DET
ejpam-6191	43	18	work	work	NOUN
ejpam-6191	43	19	:	:	PUNCT
ejpam-6191	43	20	namely	namely	ADV
ejpam-6191	43	21	,	,	PUNCT
ejpam-6191	43	22	binet	binet	NOUN
ejpam-6191	43	23	formulas	formula	NOUN
ejpam-6191	43	24	,	,	PUNCT
ejpam-6191	43	25	euler	euler	NOUN
ejpam-6191	43	26	’s	’s	PART
ejpam-6191	43	27	transformation	transformation	NOUN
ejpam-6191	43	28	,	,	PUNCT
ejpam-6191	43	29	and	and	CCONJ
ejpam-6191	43	30	a	a	DET
ejpam-6191	43	31	class	class	NOUN
ejpam-6191	43	32	of	of	ADP
ejpam-6191	43	33	higher	high	ADJ
ejpam-6191	43	34	-	-	PUNCT
ejpam-6191	43	35	order	order	NOUN
ejpam-6191	43	36	differential	differential	NOUN
ejpam-6191	43	37	operators	operator	NOUN
ejpam-6191	43	38	.	.	PUNCT
ejpam-6191	44	1	each	each	PRON
ejpam-6191	44	2	of	of	ADP
ejpam-6191	44	3	these	these	DET
ejpam-6191	44	4	tools	tool	NOUN
ejpam-6191	44	5	is	be	AUX
ejpam-6191	44	6	essential	essential	ADJ
ejpam-6191	44	7	in	in	ADP
ejpam-6191	44	8	the	the	DET
ejpam-6191	44	9	study	study	NOUN
ejpam-6191	44	10	of	of	ADP
ejpam-6191	44	11	special	special	ADJ
ejpam-6191	44	12	sequences	sequence	NOUN
ejpam-6191	44	13	,	,	PUNCT
ejpam-6191	44	14	generating	generating	NOUN
ejpam-6191	44	15	functions	function	NOUN
ejpam-6191	44	16	,	,	PUNCT
ejpam-6191	44	17	and	and	CCONJ
ejpam-6191	44	18	symbolic	symbolic	ADJ
ejpam-6191	44	19	computation	computation	NOUN
ejpam-6191	44	20	.	.	PUNCT
ejpam-6191	45	1	2.1	2.1	NUM
ejpam-6191	45	2	.	.	PUNCT
ejpam-6191	46	1	binet	binet	NOUN
ejpam-6191	46	2	formulas	formula	VERB
ejpam-6191	46	3	the	the	DET
ejpam-6191	46	4	binet	binet	NOUN
ejpam-6191	46	5	formula	formula	NOUN
ejpam-6191	46	6	,	,	PUNCT
ejpam-6191	46	7	named	name	VERB
ejpam-6191	46	8	after	after	ADP
ejpam-6191	46	9	the	the	DET
ejpam-6191	46	10	french	french	ADJ
ejpam-6191	46	11	mathematician	mathematician	NOUN
ejpam-6191	46	12	jacques	jacques	PROPN
ejpam-6191	46	13	philippe	philippe	PROPN
ejpam-6191	46	14	marie	marie	PROPN
ejpam-6191	46	15	binet	binet	PROPN
ejpam-6191	46	16	,	,	PUNCT
ejpam-6191	46	17	provides	provide	VERB
ejpam-6191	46	18	a	a	DET
ejpam-6191	46	19	closed	closed	ADJ
ejpam-6191	46	20	-	-	PUNCT
ejpam-6191	46	21	form	form	NOUN
ejpam-6191	46	22	solution	solution	NOUN
ejpam-6191	46	23	to	to	ADP
ejpam-6191	46	24	certain	certain	ADJ
ejpam-6191	46	25	linear	linear	PROPN
ejpam-6191	46	26	recurrence	recurrence	NOUN
ejpam-6191	46	27	relations	relation	NOUN
ejpam-6191	46	28	.	.	PUNCT
ejpam-6191	47	1	although	although	SCONJ
ejpam-6191	47	2	commonly	commonly	ADV
ejpam-6191	47	3	attributed	attribute	VERB
ejpam-6191	47	4	to	to	ADP
ejpam-6191	47	5	binet	binet	NOUN
ejpam-6191	47	6	in	in	ADP
ejpam-6191	47	7	the	the	DET
ejpam-6191	47	8	19th	19th	ADJ
ejpam-6191	47	9	century	century	NOUN
ejpam-6191	47	10	,	,	PUNCT
ejpam-6191	47	11	the	the	DET
ejpam-6191	47	12	formula	formula	NOUN
ejpam-6191	47	13	for	for	ADP
ejpam-6191	47	14	the	the	DET
ejpam-6191	47	15	fibonacci	fibonacci	NOUN
ejpam-6191	47	16	sequence	sequence	NOUN
ejpam-6191	47	17	was	be	AUX
ejpam-6191	47	18	actually	actually	ADV
ejpam-6191	47	19	discovered	discover	VERB
ejpam-6191	47	20	earlier	early	ADV
ejpam-6191	47	21	by	by	ADP
ejpam-6191	47	22	abraham	abraham	PROPN
ejpam-6191	47	23	de	de	PROPN
ejpam-6191	47	24	moivre	moivre	NOUN
ejpam-6191	47	25	in	in	ADP
ejpam-6191	47	26	the	the	DET
ejpam-6191	47	27	18th	18th	ADJ
ejpam-6191	47	28	century	century	NOUN
ejpam-6191	47	29	.	.	PUNCT
ejpam-6191	48	1	the	the	DET
ejpam-6191	48	2	binet	binet	NOUN
ejpam-6191	48	3	method	method	NOUN
ejpam-6191	48	4	arises	arise	VERB
ejpam-6191	48	5	from	from	ADP
ejpam-6191	48	6	solving	solve	VERB
ejpam-6191	48	7	second	second	ADJ
ejpam-6191	48	8	-	-	PUNCT
ejpam-6191	48	9	order	order	NOUN
ejpam-6191	48	10	linear	linear	NOUN
ejpam-6191	48	11	recurrence	recurrence	NOUN
ejpam-6191	48	12	relations	relation	NOUN
ejpam-6191	48	13	with	with	ADP
ejpam-6191	48	14	constant	constant	ADJ
ejpam-6191	48	15	coefficients	coefficient	NOUN
ejpam-6191	48	16	using	use	VERB
ejpam-6191	48	17	the	the	DET
ejpam-6191	48	18	theory	theory	NOUN
ejpam-6191	48	19	of	of	ADP
ejpam-6191	48	20	characteristic	characteristic	ADJ
ejpam-6191	48	21	equations	equation	NOUN
ejpam-6191	48	22	.	.	PUNCT
ejpam-6191	49	1	closed	close	VERB
ejpam-6191	49	2	-	-	PUNCT
ejpam-6191	49	3	form	form	NOUN
ejpam-6191	49	4	expressions	expression	NOUN
ejpam-6191	49	5	such	such	ADJ
ejpam-6191	49	6	as	as	ADP
ejpam-6191	49	7	binet	binet	NOUN
ejpam-6191	49	8	formulas	formula	NOUN
ejpam-6191	49	9	are	be	AUX
ejpam-6191	49	10	of	of	ADP
ejpam-6191	49	11	particular	particular	ADJ
ejpam-6191	49	12	value	value	NOUN
ejpam-6191	49	13	in	in	ADP
ejpam-6191	49	14	both	both	CCONJ
ejpam-6191	49	15	theoretical	theoretical	ADJ
ejpam-6191	49	16	and	and	CCONJ
ejpam-6191	49	17	computational	computational	ADJ
ejpam-6191	49	18	mathematics	mathematic	NOUN
ejpam-6191	49	19	,	,	PUNCT
ejpam-6191	49	20	as	as	SCONJ
ejpam-6191	49	21	they	they	PRON
ejpam-6191	49	22	eliminate	eliminate	VERB
ejpam-6191	49	23	the	the	DET
ejpam-6191	49	24	need	need	NOUN
ejpam-6191	49	25	for	for	ADP
ejpam-6191	49	26	recursive	recursive	ADJ
ejpam-6191	49	27	computations	computation	NOUN
ejpam-6191	49	28	and	and	CCONJ
ejpam-6191	49	29	allow	allow	VERB
ejpam-6191	49	30	for	for	ADP
ejpam-6191	49	31	direct	direct	ADJ
ejpam-6191	49	32	evaluation	evaluation	NOUN
ejpam-6191	49	33	of	of	ADP
ejpam-6191	49	34	the	the	DET
ejpam-6191	49	35	nth	nth	ADJ
ejpam-6191	49	36	term	term	NOUN
ejpam-6191	49	37	of	of	ADP
ejpam-6191	49	38	a	a	DET
ejpam-6191	49	39	sequence	sequence	NOUN
ejpam-6191	49	40	.	.	PUNCT
ejpam-6191	50	1	over	over	ADP
ejpam-6191	50	2	time	time	NOUN
ejpam-6191	50	3	,	,	PUNCT
ejpam-6191	50	4	analogous	analogous	ADJ
ejpam-6191	50	5	formulas	formula	NOUN
ejpam-6191	50	6	have	have	AUX
ejpam-6191	50	7	been	be	AUX
ejpam-6191	50	8	derived	derive	VERB
ejpam-6191	50	9	for	for	ADP
ejpam-6191	50	10	other	other	ADJ
ejpam-6191	50	11	notable	notable	ADJ
ejpam-6191	50	12	integer	integer	NOUN
ejpam-6191	50	13	sequences	sequence	NOUN
ejpam-6191	50	14	,	,	PUNCT
ejpam-6191	50	15	including	include	VERB
ejpam-6191	50	16	the	the	DET
ejpam-6191	50	17	lucas	lucas	PROPN
ejpam-6191	50	18	numbera	numbera	PROPN
ejpam-6191	50	19	,	,	PUNCT
ejpam-6191	50	20	pel	pel	PROPN
ejpam-6191	50	21	numbers	number	NOUN
ejpam-6191	50	22	,	,	PUNCT
ejpam-6191	50	23	pell	pell	NOUN
ejpam-6191	50	24	-	-	PUNCT
ejpam-6191	50	25	lucas	lucas	NOUN
ejpam-6191	50	26	numbers	number	NOUN
ejpam-6191	50	27	,	,	PUNCT
ejpam-6191	50	28	jacobsthal	jacobsthal	ADJ
ejpam-6191	50	29	numbers	number	NOUN
ejpam-6191	50	30	,	,	PUNCT
ejpam-6191	50	31	jacobsthal	jacobsthal	ADJ
ejpam-6191	50	32	-	-	PUNCT
ejpam-6191	50	33	lucas	lucas	NOUN
ejpam-6191	50	34	numbers	number	NOUN
ejpam-6191	50	35	,	,	PUNCT
ejpam-6191	50	36	mersenne	mersenne	NOUN
ejpam-6191	50	37	numbers	number	NOUN
ejpam-6191	50	38	,	,	PUNCT
ejpam-6191	50	39	and	and	CCONJ
ejpam-6191	50	40	mersenne	mersenne	NOUN
ejpam-6191	50	41	-	-	PUNCT
ejpam-6191	50	42	lucas	lucas	PROPN
ejpam-6191	50	43	numbers	number	NOUN
ejpam-6191	50	44	.	.	PUNCT
ejpam-6191	51	1	the	the	DET
ejpam-6191	51	2	following	follow	VERB
ejpam-6191	51	3	summarizes	summarize	VERB
ejpam-6191	51	4	these	these	DET
ejpam-6191	51	5	formulations	formulation	NOUN
ejpam-6191	51	6	:	:	PUNCT
ejpam-6191	51	7	(	(	PUNCT
ejpam-6191	51	8	i	i	NOUN
ejpam-6191	51	9	)	)	PUNCT
ejpam-6191	51	10	fibonacci	fibonacci	NOUN
ejpam-6191	51	11	sequence	sequence	NOUN
ejpam-6191	51	12	:	:	PUNCT
ejpam-6191	51	13	the	the	DET
ejpam-6191	51	14	fibonacci	fibonacci	NOUN
ejpam-6191	51	15	sequence	sequence	NOUN
ejpam-6191	51	16	,	,	PUNCT
ejpam-6191	51	17	denoted	denote	VERB
ejpam-6191	51	18	by	by	ADP
ejpam-6191	51	19	(	(	PUNCT
ejpam-6191	51	20	fn	fn	NOUN
ejpam-6191	51	21	)	)	PUNCT
ejpam-6191	51	22	,	,	PUNCT
ejpam-6191	51	23	is	be	AUX
ejpam-6191	51	24	defined	define	VERB
ejpam-6191	51	25	by	by	ADP
ejpam-6191	51	26	the	the	DET
ejpam-6191	51	27	recurrence	recurrence	NOUN
ejpam-6191	51	28	relation	relation	NOUN
ejpam-6191	51	29	fn+1	fn+1	PROPN
ejpam-6191	52	1	=	=	SYM
ejpam-6191	52	2	fn−1	fn−1	PROPN
ejpam-6191	52	3	+	+	CCONJ
ejpam-6191	52	4	fn	fn	VERB
ejpam-6191	52	5	,	,	PUNCT
ejpam-6191	52	6	n	n	PRON
ejpam-6191	52	7	≥	≥	NOUN
ejpam-6191	52	8	1	1	NUM
ejpam-6191	52	9	with	with	ADP
ejpam-6191	52	10	initial	initial	ADJ
ejpam-6191	52	11	condition	condition	NOUN
ejpam-6191	52	12	f0	f0	PROPN
ejpam-6191	52	13	=	=	SYM
ejpam-6191	52	14	0	0	NUM
ejpam-6191	52	15	,	,	PUNCT
ejpam-6191	52	16	f1	f1	NOUN
ejpam-6191	52	17	=	=	NOUN
ejpam-6191	52	18	1	1	X
ejpam-6191	52	19	.	.	PUNCT
ejpam-6191	53	1	the	the	DET
ejpam-6191	53	2	binet	binet	NOUN
ejpam-6191	53	3	formula	formula	NOUN
ejpam-6191	53	4	for	for	ADP
ejpam-6191	53	5	this	this	DET
ejpam-6191	53	6	sequence	sequence	NOUN
ejpam-6191	53	7	is	be	AUX
ejpam-6191	53	8	given	give	VERB
ejpam-6191	53	9	by	by	ADP
ejpam-6191	53	10	fn	fn	NOUN
ejpam-6191	53	11	=	=	ADJ
ejpam-6191	53	12	αn	αn	NOUN
ejpam-6191	53	13	−	−	NOUN
ejpam-6191	53	14	βn	βn	NOUN
ejpam-6191	53	15	α−	α−	ADP
ejpam-6191	53	16	β	β	PROPN
ejpam-6191	53	17	,	,	PUNCT
ejpam-6191	53	18	where	where	SCONJ
ejpam-6191	53	19	α	α	NOUN
ejpam-6191	53	20	=	=	NOUN
ejpam-6191	53	21	1	1	NUM
ejpam-6191	53	22	+	+	CCONJ
ejpam-6191	53	23	√	√	NUM
ejpam-6191	53	24	5	5	NUM
ejpam-6191	53	25	2	2	NUM
ejpam-6191	53	26	,	,	PUNCT
ejpam-6191	53	27	β	β	X
ejpam-6191	53	28	=	=	SYM
ejpam-6191	53	29	1−	1−	NUM
ejpam-6191	53	30	√	√	NUM
ejpam-6191	53	31	5	5	NUM
ejpam-6191	53	32	2	2	NUM
ejpam-6191	53	33	.	.	PUNCT
ejpam-6191	54	1	(	(	PUNCT
ejpam-6191	54	2	ii	ii	NOUN
ejpam-6191	54	3	)	)	PUNCT
ejpam-6191	54	4	lucas	lucas	NOUN
ejpam-6191	54	5	numbers	number	NOUN
ejpam-6191	54	6	:	:	PUNCT
ejpam-6191	54	7	the	the	DET
ejpam-6191	54	8	lucas	lucas	PROPN
ejpam-6191	54	9	numbers	number	NOUN
ejpam-6191	54	10	,	,	PUNCT
ejpam-6191	54	11	denoted	denote	VERB
ejpam-6191	54	12	by	by	ADP
ejpam-6191	54	13	(	(	PUNCT
ejpam-6191	54	14	ln	ln	ADJ
ejpam-6191	54	15	)	)	PUNCT
ejpam-6191	54	16	,	,	PUNCT
ejpam-6191	54	17	is	be	AUX
ejpam-6191	54	18	defined	define	VERB
ejpam-6191	54	19	by	by	ADP
ejpam-6191	54	20	the	the	DET
ejpam-6191	54	21	recurrence	recurrence	NOUN
ejpam-6191	54	22	relation	relation	NOUN
ejpam-6191	54	23	ln+1	ln+1	PROPN
ejpam-6191	55	1	=	=	SYM
ejpam-6191	55	2	ln−1	ln−1	PROPN
ejpam-6191	55	3	+	+	CCONJ
ejpam-6191	56	1	ln	ln	ADJ
ejpam-6191	56	2	,	,	PUNCT
ejpam-6191	56	3	n	n	X
ejpam-6191	56	4	≥	≥	NOUN
ejpam-6191	56	5	1	1	NUM
ejpam-6191	56	6	with	with	ADP
ejpam-6191	56	7	initial	initial	ADJ
ejpam-6191	56	8	condition	condition	NOUN
ejpam-6191	56	9	l0	l0	NOUN
ejpam-6191	56	10	=	=	SYM
ejpam-6191	56	11	2	2	NUM
ejpam-6191	56	12	,	,	PUNCT
ejpam-6191	56	13	l1	l1	PROPN
ejpam-6191	56	14	=	=	PROPN
ejpam-6191	56	15	1	1	X
ejpam-6191	56	16	.	.	PUNCT
ejpam-6191	57	1	the	the	DET
ejpam-6191	57	2	binet	binet	NOUN
ejpam-6191	57	3	formula	formula	NOUN
ejpam-6191	57	4	for	for	ADP
ejpam-6191	57	5	this	this	DET
ejpam-6191	57	6	sequence	sequence	NOUN
ejpam-6191	57	7	is	be	AUX
ejpam-6191	57	8	given	give	VERB
ejpam-6191	57	9	by	by	ADP
ejpam-6191	57	10	ln	ln	ADJ
ejpam-6191	57	11	=	=	PUNCT
ejpam-6191	57	12	αn	αn	NOUN
ejpam-6191	57	13	+	+	CCONJ
ejpam-6191	57	14	βn	βn	ADJ
ejpam-6191	57	15	,	,	PUNCT
ejpam-6191	57	16	with	with	ADP
ejpam-6191	57	17	the	the	DET
ejpam-6191	57	18	same	same	ADJ
ejpam-6191	57	19	α	α	NOUN
ejpam-6191	57	20	,	,	PUNCT
ejpam-6191	57	21	β	β	X
ejpam-6191	57	22	as	as	ADP
ejpam-6191	57	23	above	above	ADV
ejpam-6191	57	24	.	.	PUNCT
ejpam-6191	58	1	k.	k.	PROPN
ejpam-6191	59	1	v.	v.	PROPN
ejpam-6191	59	2	m.	m.	PROPN
ejpam-6191	59	3	manulat	manulat	PROPN
ejpam-6191	59	4	,	,	PUNCT
ejpam-6191	59	5	r.	r.	PROPN
ejpam-6191	59	6	b.	b.	PROPN
ejpam-6191	59	7	corcino	corcino	PROPN
ejpam-6191	59	8	/	/	SYM
ejpam-6191	59	9	eur	eur	PROPN
ejpam-6191	59	10	.	.	PUNCT
ejpam-6191	60	1	j.	j.	PROPN
ejpam-6191	60	2	pure	pure	PROPN
ejpam-6191	60	3	appl	appl	PROPN
ejpam-6191	60	4	.	.	PROPN
ejpam-6191	60	5	math	math	PROPN
ejpam-6191	60	6	,	,	PUNCT
ejpam-6191	60	7	18	18	NUM
ejpam-6191	60	8	(	(	PUNCT
ejpam-6191	60	9	3	3	NUM
ejpam-6191	60	10	)	)	PUNCT
ejpam-6191	60	11	(	(	PUNCT
ejpam-6191	60	12	2025	2025	NUM
ejpam-6191	60	13	)	)	PUNCT
ejpam-6191	60	14	,	,	PUNCT
ejpam-6191	60	15	6191	6191	NUM
ejpam-6191	60	16	4	4	NUM
ejpam-6191	60	17	of	of	ADP
ejpam-6191	60	18	21	21	NUM
ejpam-6191	60	19	(	(	PUNCT
ejpam-6191	60	20	iii	iii	NOUN
ejpam-6191	60	21	)	)	PUNCT
ejpam-6191	60	22	pell	pell	NOUN
ejpam-6191	60	23	sequence	sequence	NOUN
ejpam-6191	60	24	:	:	PUNCT
ejpam-6191	60	25	the	the	DET
ejpam-6191	60	26	pell	pell	NOUN
ejpam-6191	60	27	sequence	sequence	NOUN
ejpam-6191	60	28	,	,	PUNCT
ejpam-6191	60	29	denoted	denote	VERB
ejpam-6191	60	30	by	by	ADP
ejpam-6191	60	31	(	(	PUNCT
ejpam-6191	60	32	pn	pn	NOUN
ejpam-6191	60	33	)	)	PUNCT
ejpam-6191	60	34	,	,	PUNCT
ejpam-6191	60	35	is	be	AUX
ejpam-6191	60	36	defined	define	VERB
ejpam-6191	60	37	by	by	ADP
ejpam-6191	60	38	the	the	DET
ejpam-6191	60	39	recurrence	recurrence	NOUN
ejpam-6191	60	40	relation	relation	NOUN
ejpam-6191	60	41	pn+1	pn+1	PROPN
ejpam-6191	60	42	=	=	SYM
ejpam-6191	60	43	pn−1	pn−1	PROPN
ejpam-6191	60	44	+	+	CCONJ
ejpam-6191	60	45	2pn	2pn	ADJ
ejpam-6191	60	46	,	,	PUNCT
ejpam-6191	60	47	n	n	PRON
ejpam-6191	60	48	≥	≥	NOUN
ejpam-6191	60	49	1	1	NUM
ejpam-6191	60	50	with	with	ADP
ejpam-6191	60	51	initial	initial	ADJ
ejpam-6191	60	52	condition	condition	NOUN
ejpam-6191	60	53	p0	p0	NOUN
ejpam-6191	60	54	=	=	SYM
ejpam-6191	60	55	0	0	NUM
ejpam-6191	60	56	,	,	PUNCT
ejpam-6191	60	57	p1	p1	NOUN
ejpam-6191	60	58	=	=	NOUN
ejpam-6191	60	59	1	1	X
ejpam-6191	60	60	.	.	PUNCT
ejpam-6191	61	1	the	the	DET
ejpam-6191	61	2	binet	binet	NOUN
ejpam-6191	61	3	formula	formula	NOUN
ejpam-6191	61	4	for	for	ADP
ejpam-6191	61	5	this	this	DET
ejpam-6191	61	6	sequence	sequence	NOUN
ejpam-6191	61	7	is	be	AUX
ejpam-6191	61	8	given	give	VERB
ejpam-6191	61	9	by	by	ADP
ejpam-6191	61	10	pn	pn	PROPN
ejpam-6191	61	11	=	=	PUNCT
ejpam-6191	61	12	αn	αn	NOUN
ejpam-6191	61	13	−	−	NOUN
ejpam-6191	61	14	βn	βn	NOUN
ejpam-6191	61	15	α−	α−	ADP
ejpam-6191	61	16	β	β	PROPN
ejpam-6191	61	17	,	,	PUNCT
ejpam-6191	61	18	where	where	SCONJ
ejpam-6191	61	19	α	α	NOUN
ejpam-6191	61	20	=	=	NOUN
ejpam-6191	61	21	1	1	NUM
ejpam-6191	61	22	+	+	CCONJ
ejpam-6191	61	23	√	√	NUM
ejpam-6191	61	24	2	2	NUM
ejpam-6191	61	25	,	,	PUNCT
ejpam-6191	61	26	β	β	X
ejpam-6191	61	27	=	=	SYM
ejpam-6191	61	28	1−	1−	NUM
ejpam-6191	61	29	√	√	NUM
ejpam-6191	61	30	2	2	NUM
ejpam-6191	61	31	.	.	PUNCT
ejpam-6191	61	32	(	(	PUNCT
ejpam-6191	61	33	iv	iv	X
ejpam-6191	61	34	)	)	PUNCT
ejpam-6191	61	35	pell	pell	NOUN
ejpam-6191	61	36	-	-	PUNCT
ejpam-6191	61	37	lucas	lucas	NOUN
ejpam-6191	61	38	numbers	number	NOUN
ejpam-6191	61	39	:	:	PUNCT
ejpam-6191	61	40	the	the	DET
ejpam-6191	61	41	pell	pell	NOUN
ejpam-6191	61	42	-	-	PUNCT
ejpam-6191	61	43	lucas	lucas	NOUN
ejpam-6191	61	44	sequence	sequence	NOUN
ejpam-6191	61	45	,	,	PUNCT
ejpam-6191	61	46	denoted	denote	VERB
ejpam-6191	61	47	by	by	ADP
ejpam-6191	61	48	(	(	PUNCT
ejpam-6191	61	49	qn	qn	NOUN
ejpam-6191	61	50	)	)	PUNCT
ejpam-6191	61	51	,	,	PUNCT
ejpam-6191	61	52	is	be	AUX
ejpam-6191	61	53	defined	define	VERB
ejpam-6191	61	54	by	by	ADP
ejpam-6191	61	55	the	the	DET
ejpam-6191	61	56	recurrence	recurrence	NOUN
ejpam-6191	61	57	relation	relation	NOUN
ejpam-6191	61	58	qn+1	qn+1	PROPN
ejpam-6191	61	59	=	=	SYM
ejpam-6191	61	60	qn−1	qn−1	PROPN
ejpam-6191	61	61	+	+	NOUN
ejpam-6191	61	62	2qn	2qn	ADJ
ejpam-6191	61	63	,	,	PUNCT
ejpam-6191	61	64	n	n	PRON
ejpam-6191	61	65	≥	≥	NOUN
ejpam-6191	61	66	1	1	NUM
ejpam-6191	61	67	with	with	ADP
ejpam-6191	61	68	initial	initial	ADJ
ejpam-6191	61	69	condition	condition	NOUN
ejpam-6191	61	70	q0	q0	NOUN
ejpam-6191	61	71	=	=	SYM
ejpam-6191	61	72	2	2	NUM
ejpam-6191	61	73	,	,	PUNCT
ejpam-6191	61	74	q1	q1	NOUN
ejpam-6191	61	75	=	=	SYM
ejpam-6191	61	76	2	2	X
ejpam-6191	61	77	.	.	X
ejpam-6191	61	78	the	the	DET
ejpam-6191	61	79	binet	binet	NOUN
ejpam-6191	61	80	formula	formula	NOUN
ejpam-6191	61	81	for	for	ADP
ejpam-6191	61	82	this	this	DET
ejpam-6191	61	83	sequence	sequence	NOUN
ejpam-6191	61	84	is	be	AUX
ejpam-6191	61	85	given	give	VERB
ejpam-6191	61	86	by	by	ADP
ejpam-6191	61	87	qn	qn	NOUN
ejpam-6191	61	88	=	=	ADJ
ejpam-6191	61	89	αn	αn	NOUN
ejpam-6191	61	90	+	+	CCONJ
ejpam-6191	61	91	βn	βn	ADJ
ejpam-6191	61	92	,	,	PUNCT
ejpam-6191	61	93	with	with	ADP
ejpam-6191	61	94	the	the	DET
ejpam-6191	61	95	same	same	ADJ
ejpam-6191	61	96	α	α	NOUN
ejpam-6191	61	97	,	,	PUNCT
ejpam-6191	61	98	β	β	X
ejpam-6191	61	99	as	as	ADP
ejpam-6191	61	100	in	in	ADP
ejpam-6191	61	101	pell	pell	NOUN
ejpam-6191	61	102	numbers	number	NOUN
ejpam-6191	61	103	.	.	PUNCT
ejpam-6191	62	1	(	(	PUNCT
ejpam-6191	62	2	v	v	NOUN
ejpam-6191	62	3	)	)	PUNCT
ejpam-6191	62	4	jacobsthal	jacobsthal	ADJ
ejpam-6191	62	5	sequence	sequence	NOUN
ejpam-6191	62	6	:	:	PUNCT
ejpam-6191	62	7	the	the	DET
ejpam-6191	62	8	jacobsthal	jacobsthal	ADJ
ejpam-6191	62	9	sequence	sequence	NOUN
ejpam-6191	62	10	,	,	PUNCT
ejpam-6191	62	11	denoted	denote	VERB
ejpam-6191	62	12	by	by	ADP
ejpam-6191	62	13	(	(	PUNCT
ejpam-6191	62	14	jn	jn	PROPN
ejpam-6191	62	15	)	)	PUNCT
ejpam-6191	62	16	,	,	PUNCT
ejpam-6191	62	17	is	be	AUX
ejpam-6191	62	18	defined	define	VERB
ejpam-6191	62	19	by	by	ADP
ejpam-6191	62	20	the	the	DET
ejpam-6191	62	21	recurrence	recurrence	NOUN
ejpam-6191	62	22	relation	relation	NOUN
ejpam-6191	62	23	jn+1	jn+1	PROPN
ejpam-6191	62	24	=	=	SYM
ejpam-6191	62	25	2jn−1	2jn−1	PROPN
ejpam-6191	62	26	+	+	CCONJ
ejpam-6191	62	27	jn	jn	PROPN
ejpam-6191	62	28	,	,	PUNCT
ejpam-6191	62	29	n	n	PRON
ejpam-6191	62	30	≥	≥	NOUN
ejpam-6191	62	31	1	1	NUM
ejpam-6191	62	32	with	with	ADP
ejpam-6191	62	33	initial	initial	ADJ
ejpam-6191	62	34	condition	condition	NOUN
ejpam-6191	62	35	j0	j0	PROPN
ejpam-6191	62	36	=	=	SYM
ejpam-6191	62	37	0	0	NUM
ejpam-6191	62	38	,	,	PUNCT
ejpam-6191	62	39	j1	j1	NOUN
ejpam-6191	62	40	=	=	SYM
ejpam-6191	62	41	1	1	X
ejpam-6191	62	42	.	.	PUNCT
ejpam-6191	63	1	the	the	DET
ejpam-6191	63	2	binet	binet	NOUN
ejpam-6191	63	3	formula	formula	NOUN
ejpam-6191	63	4	for	for	ADP
ejpam-6191	63	5	this	this	DET
ejpam-6191	63	6	sequence	sequence	NOUN
ejpam-6191	63	7	is	be	AUX
ejpam-6191	63	8	given	give	VERB
ejpam-6191	63	9	by	by	ADP
ejpam-6191	63	10	jn	jn	PROPN
ejpam-6191	63	11	=	=	NOUN
ejpam-6191	63	12	αn	αn	PROPN
ejpam-6191	63	13	−	−	NOUN
ejpam-6191	63	14	βn	βn	NOUN
ejpam-6191	63	15	α−	α−	ADP
ejpam-6191	63	16	β	β	PROPN
ejpam-6191	63	17	,	,	PUNCT
ejpam-6191	63	18	where	where	SCONJ
ejpam-6191	63	19	α	α	NOUN
ejpam-6191	63	20	=	=	SYM
ejpam-6191	63	21	2	2	NUM
ejpam-6191	63	22	,	,	PUNCT
ejpam-6191	63	23	β	β	X
ejpam-6191	63	24	=	=	SYM
ejpam-6191	63	25	−1	−1	NOUN
ejpam-6191	63	26	.	.	PUNCT
ejpam-6191	64	1	(	(	PUNCT
ejpam-6191	64	2	vi	vi	NOUN
ejpam-6191	64	3	)	)	PUNCT
ejpam-6191	64	4	jacobsthal	jacobsthal	ADJ
ejpam-6191	64	5	-	-	PUNCT
ejpam-6191	64	6	lucas	lucas	NOUN
ejpam-6191	64	7	sequence	sequence	NOUN
ejpam-6191	64	8	:	:	PUNCT
ejpam-6191	64	9	the	the	DET
ejpam-6191	64	10	jacobsthal	jacobsthal	ADJ
ejpam-6191	64	11	-	-	PUNCT
ejpam-6191	64	12	lucas	lucas	NOUN
ejpam-6191	64	13	sequence	sequence	NOUN
ejpam-6191	64	14	,	,	PUNCT
ejpam-6191	64	15	denoted	denote	VERB
ejpam-6191	64	16	by	by	ADP
ejpam-6191	64	17	(	(	PUNCT
ejpam-6191	64	18	tn	tn	PROPN
ejpam-6191	64	19	)	)	PUNCT
ejpam-6191	64	20	,	,	PUNCT
ejpam-6191	64	21	is	be	AUX
ejpam-6191	64	22	defined	define	VERB
ejpam-6191	64	23	by	by	ADP
ejpam-6191	64	24	the	the	DET
ejpam-6191	64	25	recurrence	recurrence	NOUN
ejpam-6191	64	26	relation	relation	NOUN
ejpam-6191	64	27	tn+1	tn+1	NOUN
ejpam-6191	64	28	=	=	SYM
ejpam-6191	64	29	2tn−1	2tn−1	PROPN
ejpam-6191	64	30	+	+	CCONJ
ejpam-6191	64	31	tn	tn	PROPN
ejpam-6191	64	32	,	,	PUNCT
ejpam-6191	64	33	n	n	PRON
ejpam-6191	64	34	≥	≥	NOUN
ejpam-6191	64	35	1	1	NUM
ejpam-6191	64	36	with	with	ADP
ejpam-6191	64	37	initial	initial	ADJ
ejpam-6191	64	38	condition	condition	NOUN
ejpam-6191	64	39	t0	t0	NOUN
ejpam-6191	64	40	=	=	SYM
ejpam-6191	64	41	2	2	NUM
ejpam-6191	64	42	,	,	PUNCT
ejpam-6191	64	43	t1	t1	NOUN
ejpam-6191	64	44	=	=	SYM
ejpam-6191	64	45	1	1	X
ejpam-6191	64	46	.	.	PUNCT
ejpam-6191	65	1	the	the	DET
ejpam-6191	65	2	binet	binet	NOUN
ejpam-6191	65	3	formula	formula	NOUN
ejpam-6191	65	4	for	for	ADP
ejpam-6191	65	5	this	this	DET
ejpam-6191	65	6	sequence	sequence	NOUN
ejpam-6191	65	7	is	be	AUX
ejpam-6191	65	8	given	give	VERB
ejpam-6191	65	9	by	by	ADP
ejpam-6191	65	10	tn	tn	NOUN
ejpam-6191	65	11	=	=	SYM
ejpam-6191	65	12	αn	αn	NOUN
ejpam-6191	65	13	+	+	CCONJ
ejpam-6191	65	14	βn	βn	ADJ
ejpam-6191	65	15	,	,	PUNCT
ejpam-6191	65	16	with	with	ADP
ejpam-6191	65	17	α	α	NOUN
ejpam-6191	65	18	=	=	SYM
ejpam-6191	65	19	2	2	NUM
ejpam-6191	65	20	,	,	PUNCT
ejpam-6191	65	21	β	β	X
ejpam-6191	65	22	=	=	SYM
ejpam-6191	65	23	−1	−1	NOUN
ejpam-6191	65	24	.	.	PUNCT
ejpam-6191	66	1	(	(	PUNCT
ejpam-6191	66	2	vii	vii	PROPN
ejpam-6191	66	3	)	)	PUNCT
ejpam-6191	66	4	mersenne	mersenne	NOUN
ejpam-6191	66	5	sequence	sequence	NOUN
ejpam-6191	66	6	:	:	PUNCT
ejpam-6191	66	7	the	the	DET
ejpam-6191	66	8	mersenne	mersenne	NOUN
ejpam-6191	66	9	sequence	sequence	NOUN
ejpam-6191	66	10	,	,	PUNCT
ejpam-6191	66	11	denoted	denote	VERB
ejpam-6191	66	12	by	by	ADP
ejpam-6191	66	13	(	(	PUNCT
ejpam-6191	66	14	mn	mn	PROPN
ejpam-6191	66	15	)	)	PUNCT
ejpam-6191	66	16	,	,	PUNCT
ejpam-6191	66	17	is	be	AUX
ejpam-6191	66	18	defined	define	VERB
ejpam-6191	66	19	by	by	ADP
ejpam-6191	66	20	the	the	DET
ejpam-6191	66	21	recurrence	recurrence	NOUN
ejpam-6191	66	22	relation	relation	NOUN
ejpam-6191	66	23	mn+1	mn+1	PROPN
ejpam-6191	66	24	=	=	PUNCT
ejpam-6191	67	1	2mn	2mn	PROPN
ejpam-6191	67	2	+	+	CCONJ
ejpam-6191	67	3	1	1	NUM
ejpam-6191	67	4	,	,	PUNCT
ejpam-6191	67	5	n	n	PRON
ejpam-6191	67	6	≥	≥	NOUN
ejpam-6191	67	7	1	1	NUM
ejpam-6191	67	8	with	with	ADP
ejpam-6191	67	9	initial	initial	ADJ
ejpam-6191	67	10	condition	condition	NOUN
ejpam-6191	67	11	m0	m0	NOUN
ejpam-6191	67	12	=	=	SYM
ejpam-6191	67	13	0,m1	0,m1	NUM
ejpam-6191	67	14	=	=	SYM
ejpam-6191	67	15	1	1	X
ejpam-6191	67	16	.	.	PUNCT
ejpam-6191	68	1	the	the	DET
ejpam-6191	68	2	binet	binet	NOUN
ejpam-6191	68	3	formula	formula	NOUN
ejpam-6191	68	4	for	for	ADP
ejpam-6191	68	5	this	this	DET
ejpam-6191	68	6	sequence	sequence	NOUN
ejpam-6191	68	7	is	be	AUX
ejpam-6191	68	8	given	give	VERB
ejpam-6191	68	9	by	by	ADP
ejpam-6191	68	10	mn	mn	PROPN
ejpam-6191	68	11	=	=	SYM
ejpam-6191	68	12	αn	αn	NOUN
ejpam-6191	68	13	−	−	NOUN
ejpam-6191	68	14	βn	βn	NOUN
ejpam-6191	68	15	α−	α−	ADP
ejpam-6191	68	16	β	β	PROPN
ejpam-6191	68	17	,	,	PUNCT
ejpam-6191	68	18	where	where	SCONJ
ejpam-6191	68	19	α	α	NOUN
ejpam-6191	68	20	=	=	SYM
ejpam-6191	68	21	−2	−2	NOUN
ejpam-6191	68	22	,	,	PUNCT
ejpam-6191	68	23	β	β	X
ejpam-6191	68	24	=	=	SYM
ejpam-6191	68	25	−1	−1	NOUN
ejpam-6191	68	26	.	.	PUNCT
ejpam-6191	69	1	(	(	PUNCT
ejpam-6191	69	2	viii	viii	NOUN
ejpam-6191	69	3	)	)	PUNCT
ejpam-6191	69	4	mersenne	mersenne	NOUN
ejpam-6191	69	5	-	-	PUNCT
ejpam-6191	69	6	lucas	lucas	PROPN
ejpam-6191	69	7	sequence	sequence	NOUN
ejpam-6191	69	8	:	:	PUNCT
ejpam-6191	69	9	the	the	DET
ejpam-6191	69	10	mersenne	mersenne	NOUN
ejpam-6191	69	11	sequence	sequence	NOUN
ejpam-6191	69	12	,	,	PUNCT
ejpam-6191	69	13	denoted	denote	VERB
ejpam-6191	69	14	by	by	ADP
ejpam-6191	69	15	(	(	PUNCT
ejpam-6191	69	16	rn	rn	NOUN
ejpam-6191	69	17	)	)	PUNCT
ejpam-6191	69	18	,	,	PUNCT
ejpam-6191	69	19	is	be	AUX
ejpam-6191	69	20	defined	define	VERB
ejpam-6191	69	21	by	by	ADP
ejpam-6191	69	22	the	the	DET
ejpam-6191	69	23	recurrence	recurrence	NOUN
ejpam-6191	69	24	relation	relation	NOUN
ejpam-6191	69	25	rn+1	rn+1	PROPN
ejpam-6191	69	26	=	=	SYM
ejpam-6191	69	27	3rn	3rn	NOUN
ejpam-6191	69	28	−	−	PROPN
ejpam-6191	69	29	2rn−1	2rn−1	NUM
ejpam-6191	69	30	,	,	PUNCT
ejpam-6191	69	31	n	n	PRON
ejpam-6191	69	32	≥	≥	NOUN
ejpam-6191	69	33	1	1	NUM
ejpam-6191	69	34	with	with	ADP
ejpam-6191	69	35	initial	initial	ADJ
ejpam-6191	69	36	condition	condition	NOUN
ejpam-6191	69	37	r0	r0	NOUN
ejpam-6191	69	38	=	=	SYM
ejpam-6191	69	39	2	2	NUM
ejpam-6191	69	40	,	,	PUNCT
ejpam-6191	69	41	r1	r1	NOUN
ejpam-6191	69	42	=	=	PUNCT
ejpam-6191	69	43	3	3	X
ejpam-6191	69	44	.	.	X
ejpam-6191	70	1	the	the	DET
ejpam-6191	70	2	binet	binet	NOUN
ejpam-6191	70	3	formula	formula	NOUN
ejpam-6191	70	4	for	for	ADP
ejpam-6191	70	5	this	this	DET
ejpam-6191	70	6	sequence	sequence	NOUN
ejpam-6191	70	7	is	be	AUX
ejpam-6191	70	8	given	give	VERB
ejpam-6191	70	9	by	by	ADP
ejpam-6191	70	10	rn	rn	PROPN
ejpam-6191	70	11	=	=	PROPN
ejpam-6191	70	12	αn	αn	NOUN
ejpam-6191	70	13	+	+	CCONJ
ejpam-6191	70	14	βn	βn	ADJ
ejpam-6191	70	15	,	,	PUNCT
ejpam-6191	70	16	where	where	SCONJ
ejpam-6191	70	17	α	α	NOUN
ejpam-6191	70	18	=	=	SYM
ejpam-6191	70	19	2	2	NUM
ejpam-6191	70	20	,	,	PUNCT
ejpam-6191	70	21	β	β	X
ejpam-6191	70	22	=	=	NOUN
ejpam-6191	70	23	1	1	X
ejpam-6191	70	24	.	.	PUNCT
ejpam-6191	71	1	each	each	PRON
ejpam-6191	71	2	of	of	ADP
ejpam-6191	71	3	these	these	DET
ejpam-6191	71	4	formulas	formula	NOUN
ejpam-6191	71	5	showcases	showcase	VERB
ejpam-6191	71	6	how	how	SCONJ
ejpam-6191	71	7	algebraic	algebraic	ADJ
ejpam-6191	71	8	techniques	technique	NOUN
ejpam-6191	71	9	and	and	CCONJ
ejpam-6191	71	10	the	the	DET
ejpam-6191	71	11	theory	theory	NOUN
ejpam-6191	71	12	of	of	ADP
ejpam-6191	71	13	recurrence	recurrence	NOUN
ejpam-6191	71	14	relations	relation	NOUN
ejpam-6191	71	15	combine	combine	VERB
ejpam-6191	71	16	to	to	PART
ejpam-6191	71	17	yield	yield	VERB
ejpam-6191	71	18	explicit	explicit	ADJ
ejpam-6191	71	19	representations	representation	NOUN
ejpam-6191	71	20	of	of	ADP
ejpam-6191	71	21	integer	integer	NOUN
ejpam-6191	71	22	sequences	sequence	NOUN
ejpam-6191	71	23	.	.	PUNCT
ejpam-6191	72	1	k.	k.	PROPN
ejpam-6191	73	1	v.	v.	ADP
ejpam-6191	73	2	m.	m.	PROPN
ejpam-6191	73	3	manulat	manulat	PROPN
ejpam-6191	73	4	,	,	PUNCT
ejpam-6191	73	5	r.	r.	PROPN
ejpam-6191	73	6	b.	b.	PROPN
ejpam-6191	73	7	corcino	corcino	PROPN
ejpam-6191	73	8	/	/	SYM
ejpam-6191	73	9	eur	eur	PROPN
ejpam-6191	73	10	.	.	PUNCT
ejpam-6191	74	1	j.	j.	PROPN
ejpam-6191	74	2	pure	pure	PROPN
ejpam-6191	74	3	appl	appl	PROPN
ejpam-6191	74	4	.	.	PROPN
ejpam-6191	74	5	math	math	PROPN
ejpam-6191	74	6	,	,	PUNCT
ejpam-6191	74	7	18	18	NUM
ejpam-6191	74	8	(	(	PUNCT
ejpam-6191	74	9	3	3	NUM
ejpam-6191	74	10	)	)	PUNCT
ejpam-6191	74	11	(	(	PUNCT
ejpam-6191	74	12	2025	2025	NUM
ejpam-6191	74	13	)	)	PUNCT
ejpam-6191	74	14	,	,	PUNCT
ejpam-6191	74	15	6191	6191	NUM
ejpam-6191	74	16	5	5	NUM
ejpam-6191	74	17	of	of	ADP
ejpam-6191	74	18	21	21	NUM
ejpam-6191	74	19	2.2	2.2	NUM
ejpam-6191	74	20	.	.	PUNCT
ejpam-6191	75	1	euler	euler	PROPN
ejpam-6191	75	2	’s	’s	PART
ejpam-6191	75	3	transformation	transformation	PROPN
ejpam-6191	75	4	euler	euler	PROPN
ejpam-6191	75	5	’s	’s	PART
ejpam-6191	75	6	transformation	transformation	NOUN
ejpam-6191	75	7	is	be	AUX
ejpam-6191	75	8	a	a	DET
ejpam-6191	75	9	classical	classical	ADJ
ejpam-6191	75	10	tool	tool	NOUN
ejpam-6191	75	11	in	in	ADP
ejpam-6191	75	12	the	the	DET
ejpam-6191	75	13	analysis	analysis	NOUN
ejpam-6191	75	14	and	and	CCONJ
ejpam-6191	75	15	manipulation	manipulation	NOUN
ejpam-6191	75	16	of	of	ADP
ejpam-6191	75	17	power	power	NOUN
ejpam-6191	75	18	series	series	NOUN
ejpam-6191	75	19	.	.	PUNCT
ejpam-6191	76	1	it	it	PRON
ejpam-6191	76	2	is	be	AUX
ejpam-6191	76	3	particularly	particularly	ADV
ejpam-6191	76	4	useful	useful	ADJ
ejpam-6191	76	5	for	for	ADP
ejpam-6191	76	6	improving	improve	VERB
ejpam-6191	76	7	the	the	DET
ejpam-6191	76	8	rate	rate	NOUN
ejpam-6191	76	9	of	of	ADP
ejpam-6191	76	10	convergence	convergence	NOUN
ejpam-6191	76	11	of	of	ADP
ejpam-6191	76	12	infinite	infinite	ADJ
ejpam-6191	76	13	series	series	NOUN
ejpam-6191	76	14	and	and	CCONJ
ejpam-6191	76	15	for	for	ADP
ejpam-6191	76	16	reformulating	reformulate	VERB
ejpam-6191	76	17	generating	generating	NOUN
ejpam-6191	76	18	functions	function	NOUN
ejpam-6191	76	19	in	in	ADP
ejpam-6191	76	20	combinatorics	combinatoric	NOUN
ejpam-6191	76	21	and	and	CCONJ
ejpam-6191	76	22	number	number	NOUN
ejpam-6191	76	23	theory	theory	NOUN
ejpam-6191	76	24	.	.	PUNCT
ejpam-6191	77	1	theorem	theorem	VERB
ejpam-6191	77	2	2.1	2.1	NUM
ejpam-6191	77	3	.	.	PUNCT
ejpam-6191	78	1	[	[	X
ejpam-6191	78	2	31	31	NUM
ejpam-6191	78	3	]	]	PUNCT
ejpam-6191	78	4	let	let	VERB
ejpam-6191	78	5	f(z	f(z	NOUN
ejpam-6191	78	6	)	)	PUNCT
ejpam-6191	78	7	=	=	PUNCT
ejpam-6191	79	1	∑∞	∑∞	NOUN
ejpam-6191	79	2	n=0	n=0	PUNCT
ejpam-6191	79	3	anz	anz	NOUN
ejpam-6191	80	1	n	n	PRON
ejpam-6191	80	2	be	be	VERB
ejpam-6191	80	3	a	a	DET
ejpam-6191	80	4	holomorphic	holomorphic	ADJ
ejpam-6191	80	5	function	function	NOUN
ejpam-6191	80	6	represented	represent	VERB
ejpam-6191	80	7	by	by	ADP
ejpam-6191	80	8	a	a	DET
ejpam-6191	80	9	power	power	NOUN
ejpam-6191	80	10	series	series	NOUN
ejpam-6191	80	11	.	.	PUNCT
ejpam-6191	81	1	then	then	ADV
ejpam-6191	81	2	euler	euler	VERB
ejpam-6191	81	3	’s	’s	PART
ejpam-6191	81	4	transformation	transformation	NOUN
ejpam-6191	81	5	yields	yield	NOUN
ejpam-6191	81	6	:	:	PUNCT
ejpam-6191	81	7	1	1	NUM
ejpam-6191	81	8	1−	1−	NUM
ejpam-6191	81	9	z	z	NOUN
ejpam-6191	81	10	f	f	NOUN
ejpam-6191	81	11	(	(	PUNCT
ejpam-6191	81	12	1	1	NUM
ejpam-6191	81	13	1−	1−	NUM
ejpam-6191	81	14	z	z	NOUN
ejpam-6191	81	15	)	)	PUNCT
ejpam-6191	82	1	=	=	PUNCT
ejpam-6191	83	1	∞∑	∞∑	NUM
ejpam-6191	83	2	n=0	n=0	NUM
ejpam-6191	83	3	zn	zn	PROPN
ejpam-6191	83	4	(	(	PUNCT
ejpam-6191	83	5	n∑	n∑	NOUN
ejpam-6191	83	6	k=0	k=0	PROPN
ejpam-6191	83	7	(	(	PUNCT
ejpam-6191	83	8	n	n	X
ejpam-6191	83	9	k	k	X
ejpam-6191	83	10	)	)	PUNCT
ejpam-6191	83	11	ak	ak	PROPN
ejpam-6191	83	12	)	)	PUNCT
ejpam-6191	83	13	.	.	PUNCT
ejpam-6191	84	1	this	this	DET
ejpam-6191	84	2	transformation	transformation	NOUN
ejpam-6191	84	3	rewrites	rewrite	VERB
ejpam-6191	84	4	the	the	DET
ejpam-6191	84	5	original	original	ADJ
ejpam-6191	84	6	series	series	NOUN
ejpam-6191	84	7	in	in	ADP
ejpam-6191	84	8	terms	term	NOUN
ejpam-6191	84	9	of	of	ADP
ejpam-6191	84	10	binomial	binomial	ADJ
ejpam-6191	84	11	-	-	PUNCT
ejpam-6191	84	12	weighted	weight	VERB
ejpam-6191	84	13	sum	sum	NOUN
ejpam-6191	84	14	of	of	ADP
ejpam-6191	84	15	its	its	PRON
ejpam-6191	84	16	coefficients	coefficient	NOUN
ejpam-6191	84	17	.	.	PUNCT
ejpam-6191	85	1	the	the	DET
ejpam-6191	85	2	new	new	ADJ
ejpam-6191	85	3	series	series	NOUN
ejpam-6191	85	4	typically	typically	ADV
ejpam-6191	85	5	converges	converge	VERB
ejpam-6191	85	6	more	more	ADV
ejpam-6191	85	7	rapidly	rapidly	ADV
ejpam-6191	85	8	and	and	CCONJ
ejpam-6191	85	9	is	be	AUX
ejpam-6191	85	10	particularly	particularly	ADV
ejpam-6191	85	11	advantageous	advantageous	ADJ
ejpam-6191	85	12	when	when	SCONJ
ejpam-6191	85	13	working	work	VERB
ejpam-6191	85	14	with	with	ADP
ejpam-6191	85	15	special	special	ADJ
ejpam-6191	85	16	functions	function	NOUN
ejpam-6191	85	17	or	or	CCONJ
ejpam-6191	85	18	combinatorial	combinatorial	ADJ
ejpam-6191	85	19	sequences	sequence	NOUN
ejpam-6191	85	20	.	.	PUNCT
ejpam-6191	86	1	remark	remark	PROPN
ejpam-6191	86	2	2.2	2.2	NUM
ejpam-6191	86	3	.	.	PUNCT
ejpam-6191	87	1	the	the	DET
ejpam-6191	87	2	formulation	formulation	NOUN
ejpam-6191	87	3	above	above	ADV
ejpam-6191	87	4	can	can	AUX
ejpam-6191	87	5	be	be	AUX
ejpam-6191	87	6	generalized	generalize	VERB
ejpam-6191	87	7	to	to	PART
ejpam-6191	87	8	incorporate	incorporate	VERB
ejpam-6191	87	9	parameterized	parameterized	ADJ
ejpam-6191	87	10	weightings	weighting	NOUN
ejpam-6191	87	11	:	:	PUNCT
ejpam-6191	88	1	∞∑	∞∑	NUM
ejpam-6191	88	2	n=0	n=0	NUM
ejpam-6191	88	3	zn	zn	PROPN
ejpam-6191	88	4	(	(	PUNCT
ejpam-6191	88	5	n∑	n∑	NOUN
ejpam-6191	88	6	k=0	k=0	PROPN
ejpam-6191	88	7	(	(	PUNCT
ejpam-6191	88	8	n	n	CCONJ
ejpam-6191	88	9	k	k	NOUN
ejpam-6191	88	10	)	)	PUNCT
ejpam-6191	88	11	αkβn−kak	αkβn−kak	NOUN
ejpam-6191	88	12	)	)	PUNCT
ejpam-6191	88	13	=	=	SYM
ejpam-6191	89	1	1	1	NUM
ejpam-6191	89	2	1−	1−	NUM
ejpam-6191	90	1	bz	bz	PROPN
ejpam-6191	91	1	f	f	PROPN
ejpam-6191	92	1	(	(	PUNCT
ejpam-6191	92	2	αz	αz	NOUN
ejpam-6191	92	3	1−	1−	NUM
ejpam-6191	92	4	βz	βz	NOUN
ejpam-6191	92	5	)	)	PUNCT
ejpam-6191	92	6	,	,	PUNCT
ejpam-6191	92	7	where	where	SCONJ
ejpam-6191	92	8	α	α	NOUN
ejpam-6191	92	9	and	and	CCONJ
ejpam-6191	92	10	β	β	X
ejpam-6191	92	11	are	be	AUX
ejpam-6191	92	12	arbitrary	arbitrary	ADJ
ejpam-6191	92	13	parameters	parameter	NOUN
ejpam-6191	92	14	.	.	PUNCT
ejpam-6191	93	1	this	this	DET
ejpam-6191	93	2	flexible	flexible	ADJ
ejpam-6191	93	3	form	form	NOUN
ejpam-6191	93	4	extends	extend	VERB
ejpam-6191	93	5	the	the	DET
ejpam-6191	93	6	utility	utility	NOUN
ejpam-6191	93	7	of	of	ADP
ejpam-6191	93	8	euler	euler	PROPN
ejpam-6191	93	9	’s	’s	PART
ejpam-6191	93	10	transformation	transformation	NOUN
ejpam-6191	93	11	to	to	ADP
ejpam-6191	93	12	a	a	DET
ejpam-6191	93	13	broader	broad	ADJ
ejpam-6191	93	14	range	range	NOUN
ejpam-6191	93	15	of	of	ADP
ejpam-6191	93	16	generating	generating	NOUN
ejpam-6191	93	17	functions	function	NOUN
ejpam-6191	93	18	and	and	CCONJ
ejpam-6191	93	19	facilitates	facilitates	AUX
ejpam-6191	93	20	weighted	weight	VERB
ejpam-6191	93	21	series	series	NOUN
ejpam-6191	93	22	transformations	transformation	NOUN
ejpam-6191	93	23	.	.	PUNCT
ejpam-6191	94	1	2.3	2.3	NUM
ejpam-6191	94	2	.	.	PUNCT
ejpam-6191	94	3	higher	high	ADJ
ejpam-6191	94	4	-	-	PUNCT
ejpam-6191	94	5	order	order	NOUN
ejpam-6191	94	6	differential	differential	NOUN
ejpam-6191	94	7	operators	operator	NOUN
ejpam-6191	94	8	we	we	PRON
ejpam-6191	94	9	now	now	ADV
ejpam-6191	94	10	consider	consider	VERB
ejpam-6191	94	11	a	a	DET
ejpam-6191	94	12	class	class	NOUN
ejpam-6191	94	13	of	of	ADP
ejpam-6191	94	14	higher	high	ADJ
ejpam-6191	94	15	-	-	PUNCT
ejpam-6191	94	16	order	order	NOUN
ejpam-6191	94	17	differential	differential	NOUN
ejpam-6191	94	18	operators	operator	NOUN
ejpam-6191	94	19	that	that	PRON
ejpam-6191	94	20	arise	arise	VERB
ejpam-6191	94	21	in	in	ADP
ejpam-6191	94	22	symbolic	symbolic	ADJ
ejpam-6191	94	23	computation	computation	NOUN
ejpam-6191	94	24	and	and	CCONJ
ejpam-6191	94	25	operator	operator	NOUN
ejpam-6191	94	26	calculus	calculus	NOUN
ejpam-6191	94	27	.	.	PUNCT
ejpam-6191	95	1	these	these	DET
ejpam-6191	95	2	operators	operator	NOUN
ejpam-6191	95	3	are	be	AUX
ejpam-6191	95	4	instrumental	instrumental	ADJ
ejpam-6191	95	5	in	in	ADP
ejpam-6191	95	6	the	the	DET
ejpam-6191	95	7	manipulation	manipulation	NOUN
ejpam-6191	95	8	of	of	ADP
ejpam-6191	95	9	generating	generating	NOUN
ejpam-6191	95	10	functions	function	NOUN
ejpam-6191	95	11	and	and	CCONJ
ejpam-6191	95	12	in	in	ADP
ejpam-6191	95	13	deriving	derive	VERB
ejpam-6191	95	14	identities	identity	NOUN
ejpam-6191	95	15	involving	involve	VERB
ejpam-6191	95	16	special	special	ADJ
ejpam-6191	95	17	sequences	sequence	NOUN
ejpam-6191	95	18	and	and	CCONJ
ejpam-6191	95	19	polynomials	polynomial	NOUN
ejpam-6191	95	20	.	.	PUNCT
ejpam-6191	96	1	let	let	VERB
ejpam-6191	96	2	m	m	PRON
ejpam-6191	96	3	be	be	AUX
ejpam-6191	96	4	a	a	DET
ejpam-6191	96	5	positive	positive	ADJ
ejpam-6191	96	6	integer	integer	NOUN
ejpam-6191	96	7	.	.	PUNCT
ejpam-6191	97	1	define	define	VERB
ejpam-6191	97	2	the	the	DET
ejpam-6191	97	3	function	function	NOUN
ejpam-6191	97	4	b(m	b(m	PROPN
ejpam-6191	97	5	,	,	PUNCT
ejpam-6191	97	6	n	n	CCONJ
ejpam-6191	97	7	,	,	PUNCT
ejpam-6191	97	8	a	a	PRON
ejpam-6191	97	9	)	)	PUNCT
ejpam-6191	97	10	by	by	ADP
ejpam-6191	97	11	applying	apply	VERB
ejpam-6191	97	12	the	the	DET
ejpam-6191	97	13	operator	operator	NOUN
ejpam-6191	97	14	(	(	PUNCT
ejpam-6191	97	15	a	a	DET
ejpam-6191	97	16	d	d	X
ejpam-6191	97	17	da	da	NOUN
ejpam-6191	97	18	)	)	PUNCT
ejpam-6191	97	19	m	m	VERB
ejpam-6191	97	20	to	to	ADP
ejpam-6191	97	21	the	the	DET
ejpam-6191	97	22	binomial	binomial	ADJ
ejpam-6191	97	23	-	-	PUNCT
ejpam-6191	97	24	type	type	NOUN
ejpam-6191	97	25	function	function	NOUN
ejpam-6191	97	26	(	(	PUNCT
ejpam-6191	97	27	a+	a+	X
ejpam-6191	97	28	1)n	1)n	NUM
ejpam-6191	97	29	:	:	PUNCT
ejpam-6191	97	30	b(m	b(m	PROPN
ejpam-6191	97	31	,	,	PUNCT
ejpam-6191	97	32	n	n	CCONJ
ejpam-6191	97	33	,	,	PUNCT
ejpam-6191	97	34	a	a	X
ejpam-6191	97	35	)	)	PUNCT
ejpam-6191	97	36	=	=	SYM
ejpam-6191	97	37	(	(	PUNCT
ejpam-6191	97	38	a	a	DET
ejpam-6191	97	39	d	d	X
ejpam-6191	97	40	da	da	NOUN
ejpam-6191	97	41	)	)	PUNCT
ejpam-6191	97	42	m	m	VERB
ejpam-6191	97	43	(	(	PUNCT
ejpam-6191	97	44	a+	a+	X
ejpam-6191	97	45	1)n	1)n	PROPN
ejpam-6191	97	46	=	=	SYM
ejpam-6191	97	47	n∑	n∑	NOUN
ejpam-6191	97	48	k=0	k=0	PROPN
ejpam-6191	97	49	(	(	PUNCT
ejpam-6191	97	50	n	n	CCONJ
ejpam-6191	97	51	k	k	PROPN
ejpam-6191	97	52	)	)	PUNCT
ejpam-6191	97	53	kmak	kmak	PROPN
ejpam-6191	97	54	.	.	PUNCT
ejpam-6191	98	1	this	this	DET
ejpam-6191	98	2	expression	expression	NOUN
ejpam-6191	98	3	reflects	reflect	VERB
ejpam-6191	98	4	a	a	DET
ejpam-6191	98	5	structured	structured	ADJ
ejpam-6191	98	6	application	application	NOUN
ejpam-6191	98	7	of	of	ADP
ejpam-6191	98	8	differentiation	differentiation	NOUN
ejpam-6191	98	9	,	,	PUNCT
ejpam-6191	98	10	producing	produce	VERB
ejpam-6191	98	11	a	a	DET
ejpam-6191	98	12	polynomial	polynomial	NOUN
ejpam-6191	98	13	whose	whose	DET
ejpam-6191	98	14	coefficients	coefficient	NOUN
ejpam-6191	98	15	involve	involve	VERB
ejpam-6191	98	16	powers	power	NOUN
ejpam-6191	98	17	of	of	ADP
ejpam-6191	98	18	the	the	DET
ejpam-6191	98	19	index	index	NOUN
ejpam-6191	98	20	k.	k.	PROPN
ejpam-6191	98	21	more	more	ADV
ejpam-6191	98	22	significantly	significantly	ADV
ejpam-6191	98	23	,	,	PUNCT
ejpam-6191	98	24	the	the	DET
ejpam-6191	98	25	same	same	ADJ
ejpam-6191	98	26	expression	expression	NOUN
ejpam-6191	98	27	can	can	AUX
ejpam-6191	98	28	be	be	AUX
ejpam-6191	98	29	recast	recast	VERB
ejpam-6191	98	30	in	in	ADP
ejpam-6191	98	31	terms	term	NOUN
ejpam-6191	98	32	of	of	ADP
ejpam-6191	98	33	stirling	stirling	NOUN
ejpam-6191	98	34	numbers	number	NOUN
ejpam-6191	98	35	of	of	ADP
ejpam-6191	98	36	the	the	DET
ejpam-6191	98	37	second	second	ADJ
ejpam-6191	98	38	kind	kind	NOUN
ejpam-6191	98	39	,	,	PUNCT
ejpam-6191	98	40	s(m	s(m	PROPN
ejpam-6191	98	41	,	,	PUNCT
ejpam-6191	98	42	k	k	NOUN
ejpam-6191	98	43	)	)	PUNCT
ejpam-6191	98	44	,	,	PUNCT
ejpam-6191	98	45	as	as	SCONJ
ejpam-6191	98	46	follows	follow	VERB
ejpam-6191	98	47	:	:	PUNCT
ejpam-6191	98	48	b(m	b(m	PROPN
ejpam-6191	98	49	,	,	PUNCT
ejpam-6191	98	50	n	n	CCONJ
ejpam-6191	98	51	,	,	PUNCT
ejpam-6191	98	52	a	a	X
ejpam-6191	98	53	)	)	PUNCT
ejpam-6191	98	54	=	=	SYM
ejpam-6191	98	55	n∑	n∑	NOUN
ejpam-6191	98	56	k=0	k=0	PROPN
ejpam-6191	98	57	(	(	PUNCT
ejpam-6191	98	58	n	n	CCONJ
ejpam-6191	98	59	k	k	PROPN
ejpam-6191	98	60	)	)	PUNCT
ejpam-6191	98	61	k!s(m	k!s(m	PROPN
ejpam-6191	98	62	,	,	PUNCT
ejpam-6191	98	63	k	k	NOUN
ejpam-6191	98	64	)	)	PUNCT
ejpam-6191	98	65	ak(1	ak(1	PROPN
ejpam-6191	98	66	+	+	PROPN
ejpam-6191	98	67	a)n−k	a)n−k	NOUN
ejpam-6191	98	68	.	.	PUNCT
ejpam-6191	99	1	this	this	DET
ejpam-6191	99	2	alternative	alternative	ADJ
ejpam-6191	99	3	formulation	formulation	NOUN
ejpam-6191	99	4	reveals	reveal	VERB
ejpam-6191	99	5	a	a	DET
ejpam-6191	99	6	deep	deep	ADJ
ejpam-6191	99	7	combinatorial	combinatorial	ADJ
ejpam-6191	99	8	structure	structure	NOUN
ejpam-6191	99	9	:	:	PUNCT
ejpam-6191	99	10	k.	k.	PROPN
ejpam-6191	100	1	v.	v.	ADP
ejpam-6191	100	2	m.	m.	PROPN
ejpam-6191	100	3	manulat	manulat	PROPN
ejpam-6191	100	4	,	,	PUNCT
ejpam-6191	100	5	r.	r.	PROPN
ejpam-6191	100	6	b.	b.	PROPN
ejpam-6191	100	7	corcino	corcino	PROPN
ejpam-6191	100	8	/	/	SYM
ejpam-6191	100	9	eur	eur	PROPN
ejpam-6191	100	10	.	.	PUNCT
ejpam-6191	101	1	j.	j.	PROPN
ejpam-6191	101	2	pure	pure	PROPN
ejpam-6191	101	3	appl	appl	PROPN
ejpam-6191	101	4	.	.	PROPN
ejpam-6191	101	5	math	math	PROPN
ejpam-6191	101	6	,	,	PUNCT
ejpam-6191	101	7	18	18	NUM
ejpam-6191	101	8	(	(	PUNCT
ejpam-6191	101	9	3	3	NUM
ejpam-6191	101	10	)	)	PUNCT
ejpam-6191	101	11	(	(	PUNCT
ejpam-6191	101	12	2025	2025	NUM
ejpam-6191	101	13	)	)	PUNCT
ejpam-6191	101	14	,	,	PUNCT
ejpam-6191	101	15	6191	6191	NUM
ejpam-6191	101	16	6	6	NUM
ejpam-6191	101	17	of	of	ADP
ejpam-6191	101	18	21	21	NUM
ejpam-6191	101	19	•	•	NUM
ejpam-6191	101	20	the	the	DET
ejpam-6191	101	21	term	term	NOUN
ejpam-6191	101	22	s(m	s(m	PROPN
ejpam-6191	101	23	,	,	PUNCT
ejpam-6191	101	24	k	k	NOUN
ejpam-6191	101	25	)	)	PUNCT
ejpam-6191	101	26	counts	count	VERB
ejpam-6191	101	27	the	the	DET
ejpam-6191	101	28	number	number	NOUN
ejpam-6191	101	29	of	of	ADP
ejpam-6191	101	30	ways	way	NOUN
ejpam-6191	101	31	to	to	PART
ejpam-6191	101	32	partition	partition	VERB
ejpam-6191	101	33	a	a	DET
ejpam-6191	101	34	set	set	NOUN
ejpam-6191	101	35	of	of	ADP
ejpam-6191	101	36	m	m	PROPN
ejpam-6191	101	37	objects	object	NOUN
ejpam-6191	101	38	into	into	ADP
ejpam-6191	101	39	k	k	PROPN
ejpam-6191	101	40	nonempty	nonempty	NOUN
ejpam-6191	101	41	subsets	subset	NOUN
ejpam-6191	101	42	.	.	PUNCT
ejpam-6191	102	1	•	•	NUM
ejpam-6191	102	2	the	the	DET
ejpam-6191	102	3	factorial	factorial	ADJ
ejpam-6191	102	4	term	term	NOUN
ejpam-6191	102	5	k	k	PROPN
ejpam-6191	102	6	!	!	PUNCT
ejpam-6191	102	7	accounts	account	VERB
ejpam-6191	102	8	for	for	ADP
ejpam-6191	102	9	permutations	permutation	NOUN
ejpam-6191	102	10	of	of	ADP
ejpam-6191	102	11	the	the	DET
ejpam-6191	102	12	subsets	subset	NOUN
ejpam-6191	102	13	.	.	PUNCT
ejpam-6191	103	1	•	•	NUM
ejpam-6191	103	2	the	the	DET
ejpam-6191	103	3	factor	factor	NOUN
ejpam-6191	103	4	(	(	PUNCT
ejpam-6191	103	5	1	1	NUM
ejpam-6191	103	6	+	+	NUM
ejpam-6191	103	7	a)n−k	a)n−k	NOUN
ejpam-6191	103	8	modulates	modulate	VERB
ejpam-6191	103	9	the	the	DET
ejpam-6191	103	10	contribution	contribution	NOUN
ejpam-6191	103	11	of	of	ADP
ejpam-6191	103	12	each	each	DET
ejpam-6191	103	13	term	term	NOUN
ejpam-6191	103	14	in	in	ADP
ejpam-6191	103	15	the	the	DET
ejpam-6191	103	16	binomial	binomial	ADJ
ejpam-6191	103	17	expansion	expansion	NOUN
ejpam-6191	103	18	.	.	PUNCT
ejpam-6191	104	1	together	together	ADV
ejpam-6191	104	2	,	,	PUNCT
ejpam-6191	104	3	these	these	DET
ejpam-6191	104	4	identities	identity	NOUN
ejpam-6191	104	5	serve	serve	VERB
ejpam-6191	104	6	as	as	ADP
ejpam-6191	104	7	powerful	powerful	ADJ
ejpam-6191	104	8	tools	tool	NOUN
ejpam-6191	104	9	in	in	ADP
ejpam-6191	104	10	enumerative	enumerative	ADJ
ejpam-6191	104	11	combinatorics	combinatoric	NOUN
ejpam-6191	104	12	,	,	PUNCT
ejpam-6191	104	13	asymptotic	asymptotic	ADJ
ejpam-6191	104	14	analysis	analysis	NOUN
ejpam-6191	104	15	,	,	PUNCT
ejpam-6191	104	16	and	and	CCONJ
ejpam-6191	104	17	the	the	DET
ejpam-6191	104	18	study	study	NOUN
ejpam-6191	104	19	of	of	ADP
ejpam-6191	104	20	generating	generating	NOUN
ejpam-6191	104	21	functions	function	NOUN
ejpam-6191	104	22	.	.	PUNCT
ejpam-6191	105	1	their	their	PRON
ejpam-6191	105	2	applications	application	NOUN
ejpam-6191	105	3	span	span	VERB
ejpam-6191	105	4	fields	field	NOUN
ejpam-6191	105	5	such	such	ADJ
ejpam-6191	105	6	as	as	ADP
ejpam-6191	105	7	discrete	discrete	ADJ
ejpam-6191	105	8	mathematics	mathematic	NOUN
ejpam-6191	105	9	,	,	PUNCT
ejpam-6191	105	10	theoretical	theoretical	ADJ
ejpam-6191	105	11	computer	computer	NOUN
ejpam-6191	105	12	science	science	NOUN
ejpam-6191	105	13	,	,	PUNCT
ejpam-6191	105	14	and	and	CCONJ
ejpam-6191	105	15	mathematical	mathematical	ADJ
ejpam-6191	105	16	physics	physics	NOUN
ejpam-6191	105	17	.	.	PUNCT
ejpam-6191	106	1	3	3	X
ejpam-6191	106	2	.	.	X
ejpam-6191	106	3	binomial	binomial	ADJ
ejpam-6191	106	4	sum	sum	NOUN
ejpam-6191	106	5	involving	involve	VERB
ejpam-6191	106	6	generalized	generalize	VERB
ejpam-6191	106	7	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	106	8	numbers	number	NOUN
ejpam-6191	106	9	in	in	ADP
ejpam-6191	106	10	this	this	DET
ejpam-6191	106	11	section	section	NOUN
ejpam-6191	106	12	,	,	PUNCT
ejpam-6191	106	13	we	we	PRON
ejpam-6191	106	14	derive	derive	VERB
ejpam-6191	106	15	a	a	DET
ejpam-6191	106	16	closed	closed	ADJ
ejpam-6191	106	17	-	-	PUNCT
ejpam-6191	106	18	form	form	NOUN
ejpam-6191	106	19	expression	expression	NOUN
ejpam-6191	106	20	for	for	ADP
ejpam-6191	106	21	a	a	DET
ejpam-6191	106	22	binomial	binomial	ADJ
ejpam-6191	106	23	sum	sum	NOUN
ejpam-6191	106	24	involving	involve	VERB
ejpam-6191	106	25	generalized	generalize	VERB
ejpam-6191	106	26	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	106	27	numbers	number	NOUN
ejpam-6191	106	28	.	.	PUNCT
ejpam-6191	107	1	these	these	DET
ejpam-6191	107	2	sums	sum	NOUN
ejpam-6191	107	3	naturally	naturally	ADV
ejpam-6191	107	4	arise	arise	VERB
ejpam-6191	107	5	in	in	ADP
ejpam-6191	107	6	various	various	ADJ
ejpam-6191	107	7	problems	problem	NOUN
ejpam-6191	107	8	involving	involve	VERB
ejpam-6191	107	9	nested	nested	ADJ
ejpam-6191	107	10	harmonic	harmonic	ADJ
ejpam-6191	107	11	structures	structure	NOUN
ejpam-6191	107	12	and	and	CCONJ
ejpam-6191	107	13	have	have	VERB
ejpam-6191	107	14	connections	connection	NOUN
ejpam-6191	107	15	to	to	ADP
ejpam-6191	107	16	combinatorics	combinatoric	NOUN
ejpam-6191	107	17	and	and	CCONJ
ejpam-6191	107	18	special	special	ADJ
ejpam-6191	107	19	functions	function	NOUN
ejpam-6191	107	20	.	.	PUNCT
ejpam-6191	108	1	by	by	ADP
ejpam-6191	108	2	utilizing	utilize	VERB
ejpam-6191	108	3	generating	generate	VERB
ejpam-6191	108	4	function	function	NOUN
ejpam-6191	108	5	techniques	technique	NOUN
ejpam-6191	108	6	and	and	CCONJ
ejpam-6191	108	7	the	the	DET
ejpam-6191	108	8	euler	euler	NOUN
ejpam-6191	108	9	transform	transform	NOUN
ejpam-6191	108	10	,	,	PUNCT
ejpam-6191	108	11	we	we	PRON
ejpam-6191	108	12	establish	establish	VERB
ejpam-6191	108	13	an	an	DET
ejpam-6191	108	14	elegant	elegant	ADJ
ejpam-6191	108	15	identity	identity	NOUN
ejpam-6191	108	16	that	that	PRON
ejpam-6191	108	17	expresses	express	VERB
ejpam-6191	108	18	the	the	DET
ejpam-6191	108	19	binomial	binomial	ADJ
ejpam-6191	108	20	-	-	PUNCT
ejpam-6191	108	21	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	108	22	convolution	convolution	NOUN
ejpam-6191	108	23	in	in	ADP
ejpam-6191	108	24	terms	term	NOUN
ejpam-6191	108	25	of	of	ADP
ejpam-6191	108	26	generalized	generalized	ADJ
ejpam-6191	108	27	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	108	28	numbers	number	NOUN
ejpam-6191	108	29	.	.	PUNCT
ejpam-6191	109	1	theorem	theorem	VERB
ejpam-6191	109	2	3.1	3.1	NUM
ejpam-6191	109	3	.	.	PUNCT
ejpam-6191	110	1	for	for	ADP
ejpam-6191	110	2	all	all	PRON
ejpam-6191	110	3	n	n	PRON
ejpam-6191	110	4	≥	≥	NUM
ejpam-6191	110	5	1	1	NUM
ejpam-6191	110	6	and	and	CCONJ
ejpam-6191	110	7	complex	complex	ADJ
ejpam-6191	110	8	numbers	number	NOUN
ejpam-6191	110	9	a	a	PRON
ejpam-6191	110	10	,	,	PUNCT
ejpam-6191	110	11	b	b	X
ejpam-6191	110	12	∈	∈	PROPN
ejpam-6191	110	13	c	c	NOUN
ejpam-6191	110	14	,	,	PUNCT
ejpam-6191	110	15	the	the	DET
ejpam-6191	110	16	following	follow	VERB
ejpam-6191	110	17	identity	identity	NOUN
ejpam-6191	110	18	holds	hold	VERB
ejpam-6191	110	19	:	:	PUNCT
ejpam-6191	110	20	sn(a	sn(a	X
ejpam-6191	110	21	,	,	PUNCT
ejpam-6191	110	22	b	b	X
ejpam-6191	110	23	)	)	PUNCT
ejpam-6191	110	24	=	=	SYM
ejpam-6191	111	1	n∑	n∑	NOUN
ejpam-6191	111	2	k=0	k=0	PROPN
ejpam-6191	111	3	(	(	PUNCT
ejpam-6191	111	4	n	n	X
ejpam-6191	111	5	k	k	NOUN
ejpam-6191	111	6	)	)	PUNCT
ejpam-6191	111	7	akbn−kh	akbn−kh	VERB
ejpam-6191	111	8	(	(	PUNCT
ejpam-6191	111	9	r	r	NOUN
ejpam-6191	111	10	)	)	PUNCT
ejpam-6191	111	11	k	k	NOUN
ejpam-6191	112	1	=	=	SYM
ejpam-6191	112	2	r+1∑	r+1∑	PROPN
ejpam-6191	112	3	k=0	k=0	PROPN
ejpam-6191	112	4	n∑	n∑	PROPN
ejpam-6191	112	5	j=0	j=0	PROPN
ejpam-6191	112	6	(	(	PUNCT
ejpam-6191	112	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	112	8	(	(	PUNCT
ejpam-6191	112	9	r	r	NOUN
ejpam-6191	112	10	+	+	NOUN
ejpam-6191	112	11	1	1	NUM
ejpam-6191	112	12	k	k	NOUN
ejpam-6191	112	13	)	)	PUNCT
ejpam-6191	112	14	(	(	PUNCT
ejpam-6191	112	15	a+	a+	PUNCT
ejpam-6191	112	16	b)jh	b)jh	PROPN
ejpam-6191	112	17	(	(	PUNCT
ejpam-6191	112	18	k−1	k−1	PROPN
ejpam-6191	112	19	)	)	PUNCT
ejpam-6191	112	20	j	j	PROPN
ejpam-6191	113	1	bn−j	bn−j	ADV
ejpam-6191	113	2	[	[	PUNCT
ejpam-6191	113	3	h	h	NOUN
ejpam-6191	113	4	(	(	PUNCT
ejpam-6191	113	5	r−k	r−k	PROPN
ejpam-6191	113	6	)	)	PUNCT
ejpam-6191	113	7	n−j	n−j	ADV
ejpam-6191	113	8	−h	−h	VERB
ejpam-6191	113	9	(	(	PUNCT
ejpam-6191	113	10	r−k	r−k	PROPN
ejpam-6191	113	11	)	)	PUNCT
ejpam-6191	113	12	n−1−j	n−1−j	NUM
ejpam-6191	113	13	]	]	PUNCT
ejpam-6191	113	14	.	.	PUNCT
ejpam-6191	114	1	(	(	PUNCT
ejpam-6191	114	2	3.1	3.1	NUM
ejpam-6191	114	3	)	)	PUNCT
ejpam-6191	114	4	proof	proof	NOUN
ejpam-6191	114	5	.	.	PUNCT
ejpam-6191	115	1	let	let	VERB
ejpam-6191	115	2	us	we	PRON
ejpam-6191	115	3	consider	consider	VERB
ejpam-6191	115	4	the	the	DET
ejpam-6191	115	5	binomial	binomial	ADJ
ejpam-6191	115	6	-	-	PUNCT
ejpam-6191	115	7	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	115	8	convolution	convolution	NOUN
ejpam-6191	115	9	sn(a	sn(a	ADP
ejpam-6191	115	10	,	,	PUNCT
ejpam-6191	115	11	b	b	X
ejpam-6191	115	12	)	)	PUNCT
ejpam-6191	116	1	=	=	SYM
ejpam-6191	116	2	n∑	n∑	NOUN
ejpam-6191	116	3	k=0	k=0	PROPN
ejpam-6191	116	4	(	(	PUNCT
ejpam-6191	116	5	n	n	X
ejpam-6191	116	6	k	k	NOUN
ejpam-6191	116	7	)	)	PUNCT
ejpam-6191	116	8	akbn−kh	akbn−kh	VERB
ejpam-6191	116	9	(	(	PUNCT
ejpam-6191	116	10	r	r	NOUN
ejpam-6191	116	11	)	)	PUNCT
ejpam-6191	116	12	k	k	NOUN
ejpam-6191	116	13	.	.	PUNCT
ejpam-6191	117	1	to	to	PART
ejpam-6191	117	2	evaluate	evaluate	VERB
ejpam-6191	117	3	this	this	DET
ejpam-6191	117	4	sum	sum	NOUN
ejpam-6191	117	5	,	,	PUNCT
ejpam-6191	117	6	we	we	PRON
ejpam-6191	117	7	begin	begin	VERB
ejpam-6191	117	8	by	by	ADP
ejpam-6191	117	9	considering	consider	VERB
ejpam-6191	117	10	the	the	DET
ejpam-6191	117	11	generating	generate	VERB
ejpam-6191	117	12	function	function	NOUN
ejpam-6191	117	13	of	of	ADP
ejpam-6191	117	14	the	the	DET
ejpam-6191	117	15	generalized	generalized	ADJ
ejpam-6191	117	16	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	117	17	numbers	number	NOUN
ejpam-6191	117	18	:	:	PUNCT
ejpam-6191	117	19	f(z	f(z	NUM
ejpam-6191	117	20	)	)	PUNCT
ejpam-6191	118	1	=	=	PUNCT
ejpam-6191	119	1	∞∑	∞∑	ADJ
ejpam-6191	119	2	n=0	n=0	NUM
ejpam-6191	119	3	h(r	h(r	NOUN
ejpam-6191	119	4	)	)	PUNCT
ejpam-6191	119	5	n	n	CCONJ
ejpam-6191	119	6	zn	zn	NOUN
ejpam-6191	119	7	=	=	PUNCT
ejpam-6191	120	1	[	[	X
ejpam-6191	120	2	−	−	X
ejpam-6191	120	3	ln(1−	ln(1−	PROPN
ejpam-6191	120	4	z)]r+1	z)]r+1	PROPN
ejpam-6191	120	5	1−	1−	NUM
ejpam-6191	120	6	z	z	NOUN
ejpam-6191	120	7	.	.	PUNCT
ejpam-6191	121	1	now	now	ADV
ejpam-6191	121	2	,	,	PUNCT
ejpam-6191	121	3	let	let	VERB
ejpam-6191	121	4	us	we	PRON
ejpam-6191	121	5	define	define	VERB
ejpam-6191	121	6	the	the	DET
ejpam-6191	121	7	generating	generate	VERB
ejpam-6191	121	8	function	function	NOUN
ejpam-6191	121	9	corresponding	correspond	VERB
ejpam-6191	121	10	to	to	ADP
ejpam-6191	121	11	the	the	DET
ejpam-6191	121	12	binomial	binomial	ADJ
ejpam-6191	121	13	sum	sum	NOUN
ejpam-6191	121	14	sn(a	sn(a	ADP
ejpam-6191	121	15	,	,	PUNCT
ejpam-6191	121	16	b	b	X
ejpam-6191	121	17	)	)	PUNCT
ejpam-6191	121	18	as	as	ADP
ejpam-6191	121	19	s(z	s(z	PROPN
ejpam-6191	121	20	)	)	PUNCT
ejpam-6191	121	21	=	=	PUNCT
ejpam-6191	122	1	∞∑	∞∑	NUM
ejpam-6191	122	2	n=0	n=0	NUM
ejpam-6191	122	3	sn(a	sn(a	NOUN
ejpam-6191	122	4	,	,	PUNCT
ejpam-6191	122	5	b)z	b)z	ADJ
ejpam-6191	122	6	n.	n.	PROPN
ejpam-6191	122	7	k.	k.	PROPN
ejpam-6191	123	1	v.	v.	ADP
ejpam-6191	123	2	m.	m.	PROPN
ejpam-6191	123	3	manulat	manulat	PROPN
ejpam-6191	123	4	,	,	PUNCT
ejpam-6191	123	5	r.	r.	PROPN
ejpam-6191	123	6	b.	b.	PROPN
ejpam-6191	123	7	corcino	corcino	PROPN
ejpam-6191	123	8	/	/	SYM
ejpam-6191	123	9	eur	eur	PROPN
ejpam-6191	123	10	.	.	PUNCT
ejpam-6191	124	1	j.	j.	PROPN
ejpam-6191	124	2	pure	pure	PROPN
ejpam-6191	124	3	appl	appl	PROPN
ejpam-6191	124	4	.	.	PROPN
ejpam-6191	124	5	math	math	PROPN
ejpam-6191	124	6	,	,	PUNCT
ejpam-6191	124	7	18	18	NUM
ejpam-6191	124	8	(	(	PUNCT
ejpam-6191	124	9	3	3	NUM
ejpam-6191	124	10	)	)	PUNCT
ejpam-6191	124	11	(	(	PUNCT
ejpam-6191	124	12	2025	2025	NUM
ejpam-6191	124	13	)	)	PUNCT
ejpam-6191	124	14	,	,	PUNCT
ejpam-6191	124	15	6191	6191	NUM
ejpam-6191	124	16	7	7	NUM
ejpam-6191	124	17	of	of	ADP
ejpam-6191	124	18	21	21	NUM
ejpam-6191	124	19	using	use	VERB
ejpam-6191	124	20	euler	euler	NOUN
ejpam-6191	124	21	’s	’s	PART
ejpam-6191	124	22	transform	transform	NOUN
ejpam-6191	124	23	for	for	ADP
ejpam-6191	124	24	binomial	binomial	ADJ
ejpam-6191	124	25	convolutions	convolution	NOUN
ejpam-6191	124	26	,	,	PUNCT
ejpam-6191	124	27	we	we	PRON
ejpam-6191	124	28	have	have	AUX
ejpam-6191	124	29	:	:	PUNCT
ejpam-6191	124	30	s(z	s(z	PROPN
ejpam-6191	124	31	)	)	PUNCT
ejpam-6191	124	32	=	=	SYM
ejpam-6191	125	1	1	1	NUM
ejpam-6191	125	2	1−	1−	NUM
ejpam-6191	125	3	bz	bz	PROPN
ejpam-6191	125	4	·	·	PUNCT
ejpam-6191	125	5	f	f	PROPN
ejpam-6191	125	6	(	(	PUNCT
ejpam-6191	125	7	az	az	PROPN
ejpam-6191	125	8	1−	1−	NUM
ejpam-6191	125	9	bz	bz	PROPN
ejpam-6191	125	10	)	)	PUNCT
ejpam-6191	125	11	=	=	PUNCT
ejpam-6191	126	1	1	1	NUM
ejpam-6191	126	2	1−	1−	NUM
ejpam-6191	126	3	bz	bz	PROPN
ejpam-6191	126	4	·	·	PUNCT
ejpam-6191	127	1	[	[	PUNCT
ejpam-6191	127	2	−	−	X
ejpam-6191	127	3	ln	ln	NOUN
ejpam-6191	127	4	(	(	PUNCT
ejpam-6191	127	5	1−	1−	NUM
ejpam-6191	127	6	az	az	PROPN
ejpam-6191	127	7	1−bz	1−bz	PROPN
ejpam-6191	127	8	)	)	PUNCT
ejpam-6191	128	1	]	]	X
ejpam-6191	128	2	r+1	r+1	PROPN
ejpam-6191	128	3	1−	1−	NUM
ejpam-6191	128	4	az	az	PROPN
ejpam-6191	128	5	1−bz	1−bz	PROPN
ejpam-6191	128	6	=	=	PUNCT
ejpam-6191	128	7	[	[	PUNCT
ejpam-6191	128	8	−	−	PROPN
ejpam-6191	128	9	ln	ln	ADJ
ejpam-6191	128	10	(	(	PUNCT
ejpam-6191	128	11	1−(a+b)z	1−(a+b)z	NUM
ejpam-6191	128	12	1−bz	1−bz	PROPN
ejpam-6191	128	13	)	)	PUNCT
ejpam-6191	128	14	]	]	X
ejpam-6191	129	1	r+1	r+1	PROPN
ejpam-6191	129	2	1−	1−	NUM
ejpam-6191	129	3	(	(	PUNCT
ejpam-6191	129	4	a+	a+	X
ejpam-6191	129	5	b)z	b)z	X
ejpam-6191	129	6	=	=	PUNCT
ejpam-6191	130	1	[	[	X
ejpam-6191	130	2	−	−	X
ejpam-6191	130	3	ln(1−	ln(1−	PROPN
ejpam-6191	130	4	(	(	PUNCT
ejpam-6191	130	5	a+	a+	PRON
ejpam-6191	130	6	b)z	b)z	NOUN
ejpam-6191	130	7	)	)	PUNCT
ejpam-6191	131	1	+	+	CCONJ
ejpam-6191	132	1	ln(1−	ln(1−	PROPN
ejpam-6191	132	2	bz)]r+1	bz)]r+1	PROPN
ejpam-6191	132	3	1−	1−	NUM
ejpam-6191	132	4	(	(	PUNCT
ejpam-6191	132	5	a+	a+	X
ejpam-6191	132	6	b)z	b)z	X
ejpam-6191	132	7	.	.	PUNCT
ejpam-6191	133	1	applying	apply	VERB
ejpam-6191	133	2	the	the	DET
ejpam-6191	133	3	binomial	binomial	ADJ
ejpam-6191	133	4	theorem	theorem	NOUN
ejpam-6191	133	5	,	,	PUNCT
ejpam-6191	133	6	s(z	s(z	PROPN
ejpam-6191	133	7	)	)	PUNCT
ejpam-6191	133	8	=	=	SYM
ejpam-6191	133	9	r+1∑	r+1∑	PROPN
ejpam-6191	133	10	k=0	k=0	PROPN
ejpam-6191	134	1	(	(	PUNCT
ejpam-6191	134	2	r	r	NOUN
ejpam-6191	134	3	+	+	PROPN
ejpam-6191	134	4	1	1	NUM
ejpam-6191	134	5	k	k	NOUN
ejpam-6191	134	6	)	)	PUNCT
ejpam-6191	135	1	[	[	X
ejpam-6191	135	2	−ln(1−	−ln(1−	X
ejpam-6191	135	3	(	(	PUNCT
ejpam-6191	135	4	a+	a+	PUNCT
ejpam-6191	135	5	b)z]k(ln(1−	b)z]k(ln(1−	PROPN
ejpam-6191	135	6	bz))r+1−k	bz))r+1−k	PROPN
ejpam-6191	135	7	1−	1−	NUM
ejpam-6191	135	8	(	(	PUNCT
ejpam-6191	135	9	a+	a+	X
ejpam-6191	135	10	b)z	b)z	X
ejpam-6191	135	11	=	=	SYM
ejpam-6191	135	12	r+1∑	r+1∑	PROPN
ejpam-6191	135	13	k=0	k=0	PROPN
ejpam-6191	135	14	(	(	PUNCT
ejpam-6191	135	15	r	r	NOUN
ejpam-6191	135	16	+	+	PROPN
ejpam-6191	135	17	1	1	NUM
ejpam-6191	135	18	k	k	NOUN
ejpam-6191	135	19	)	)	PUNCT
ejpam-6191	136	1	[	[	X
ejpam-6191	136	2	−ln(1−	−ln(1−	X
ejpam-6191	136	3	(	(	PUNCT
ejpam-6191	136	4	a+	a+	X
ejpam-6191	136	5	b)z]k	b)z]k	NOUN
ejpam-6191	136	6	1−	1−	NUM
ejpam-6191	136	7	(	(	PUNCT
ejpam-6191	136	8	a+	a+	X
ejpam-6191	136	9	b)z	b)z	X
ejpam-6191	136	10	·	·	PUNCT
ejpam-6191	136	11	(	(	PUNCT
ejpam-6191	136	12	−1)r+1−k	−1)r+1−k	NOUN
ejpam-6191	136	13	(	(	PUNCT
ejpam-6191	136	14	−ln(1−	−ln(1−	PROPN
ejpam-6191	136	15	bz))r+1−k	bz))r+1−k	NOUN
ejpam-6191	136	16	1−	1−	NUM
ejpam-6191	136	17	bz	bz	PROPN
ejpam-6191	136	18	(	(	PUNCT
ejpam-6191	136	19	1−	1−	NUM
ejpam-6191	136	20	bz	bz	PROPN
ejpam-6191	136	21	)	)	PUNCT
ejpam-6191	136	22	.	.	PUNCT
ejpam-6191	137	1	applying	apply	VERB
ejpam-6191	137	2	the	the	DET
ejpam-6191	137	3	generating	generate	VERB
ejpam-6191	137	4	function	function	NOUN
ejpam-6191	137	5	for	for	ADP
ejpam-6191	137	6	the	the	DET
ejpam-6191	137	7	generalized	generalize	VERB
ejpam-6191	137	8	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	137	9	numbers	number	NOUN
ejpam-6191	137	10	and	and	CCONJ
ejpam-6191	137	11	the	the	DET
ejpam-6191	137	12	cauchy	cauchy	ADJ
ejpam-6191	137	13	product	product	NOUN
ejpam-6191	137	14	rule	rule	NOUN
ejpam-6191	137	15	,	,	PUNCT
ejpam-6191	137	16	s(z	s(z	PROPN
ejpam-6191	137	17	)	)	PUNCT
ejpam-6191	137	18	=	=	PUNCT
ejpam-6191	138	1	(	(	PUNCT
ejpam-6191	138	2	1−	1−	NUM
ejpam-6191	138	3	bz	bz	PROPN
ejpam-6191	138	4	)	)	PUNCT
ejpam-6191	138	5	r+1∑	r+1∑	PROPN
ejpam-6191	138	6	k=0	k=0	PROPN
ejpam-6191	139	1	(	(	PUNCT
ejpam-6191	139	2	−1)r+1−k	−1)r+1−k	X
ejpam-6191	139	3	(	(	PUNCT
ejpam-6191	139	4	r	r	NOUN
ejpam-6191	139	5	+	+	NOUN
ejpam-6191	139	6	1	1	NUM
ejpam-6191	139	7	k	k	NOUN
ejpam-6191	139	8	)	)	PUNCT
ejpam-6191	139	9	(	(	PUNCT
ejpam-6191	139	10	∞∑	∞∑	NUM
ejpam-6191	139	11	n=0	n=0	NUM
ejpam-6191	139	12	(	(	PUNCT
ejpam-6191	139	13	a+	a+	X
ejpam-6191	139	14	b)nh(k−1	b)nh(k−1	PROPN
ejpam-6191	139	15	)	)	PUNCT
ejpam-6191	139	16	n	n	PROPN
ejpam-6191	139	17	zn	zn	NOUN
ejpam-6191	139	18	)	)	PUNCT
ejpam-6191	139	19	(	(	PUNCT
ejpam-6191	139	20	∞∑	∞∑	NUM
ejpam-6191	139	21	n=0	n=0	X
ejpam-6191	139	22	bnh(r−k	bnh(r−k	NOUN
ejpam-6191	139	23	)	)	PUNCT
ejpam-6191	139	24	n	n	PROPN
ejpam-6191	139	25	zn	zn	NUM
ejpam-6191	139	26	)	)	PUNCT
ejpam-6191	139	27	=	=	PUNCT
ejpam-6191	139	28	(	(	PUNCT
ejpam-6191	139	29	1−	1−	NUM
ejpam-6191	139	30	bz	bz	PROPN
ejpam-6191	139	31	)	)	PUNCT
ejpam-6191	139	32	r+1∑	r+1∑	PROPN
ejpam-6191	139	33	k=0	k=0	PROPN
ejpam-6191	139	34	(	(	PUNCT
ejpam-6191	139	35	−1)r+1−k	−1)r+1−k	X
ejpam-6191	139	36	(	(	PUNCT
ejpam-6191	139	37	r	r	NOUN
ejpam-6191	139	38	+	+	NOUN
ejpam-6191	139	39	1	1	NUM
ejpam-6191	139	40	k	k	NOUN
ejpam-6191	139	41	)	)	PUNCT
ejpam-6191	140	1	∞∑	∞∑	PRON
ejpam-6191	140	2	n=0	n=0	PUNCT
ejpam-6191	140	3			PROPN
ejpam-6191	140	4	n∑	n∑	ADJ
ejpam-6191	140	5	j=0	j=0	PROPN
ejpam-6191	140	6	(	(	PUNCT
ejpam-6191	140	7	a+	a+	PUNCT
ejpam-6191	140	8	b)jh	b)jh	PROPN
ejpam-6191	140	9	(	(	PUNCT
ejpam-6191	140	10	k−1	k−1	PROPN
ejpam-6191	140	11	)	)	PUNCT
ejpam-6191	140	12	j	j	PROPN
ejpam-6191	140	13	bn−jh	bn−jh	PROPN
ejpam-6191	140	14	(	(	PUNCT
ejpam-6191	140	15	r−k	r−k	PROPN
ejpam-6191	140	16	)	)	PUNCT
ejpam-6191	140	17	n−j	n−j	ADV
ejpam-6191	140	18			PROPN
ejpam-6191	140	19	zn	zn	X
ejpam-6191	140	20	=	=	SYM
ejpam-6191	140	21	(	(	PUNCT
ejpam-6191	140	22	1−	1−	NUM
ejpam-6191	140	23	bz	bz	NOUN
ejpam-6191	140	24	)	)	PUNCT
ejpam-6191	140	25	∞∑	∞∑	PROPN
ejpam-6191	140	26	n=0	n=0	NUM
ejpam-6191	140	27	r+1∑	r+1∑	NUM
ejpam-6191	140	28	k=0	k=0	PROPN
ejpam-6191	140	29	(	(	PUNCT
ejpam-6191	140	30	−1)r+1−k	−1)r+1−k	X
ejpam-6191	140	31	(	(	PUNCT
ejpam-6191	140	32	r	r	NOUN
ejpam-6191	140	33	+	+	NOUN
ejpam-6191	140	34	1	1	NUM
ejpam-6191	140	35	k	k	NOUN
ejpam-6191	140	36	)	)	PUNCT
ejpam-6191	140	37	n∑	n∑	NOUN
ejpam-6191	140	38	j=0	j=0	PROPN
ejpam-6191	140	39	(	(	PUNCT
ejpam-6191	140	40	a+	a+	PUNCT
ejpam-6191	140	41	b)jh	b)jh	PROPN
ejpam-6191	140	42	(	(	PUNCT
ejpam-6191	140	43	k−1	k−1	PROPN
ejpam-6191	140	44	)	)	PUNCT
ejpam-6191	141	1	j	j	PROPN
ejpam-6191	141	2	bn−jh	bn−jh	PROPN
ejpam-6191	141	3	(	(	PUNCT
ejpam-6191	141	4	r−k	r−k	PROPN
ejpam-6191	141	5	)	)	PUNCT
ejpam-6191	141	6	n−j	n−j	ADV
ejpam-6191	141	7			PROPN
ejpam-6191	141	8	zn	zn	X
ejpam-6191	141	9	=	=	PUNCT
ejpam-6191	142	1	∞∑	∞∑	PRON
ejpam-6191	142	2	n=0	n=0	NUM
ejpam-6191	142	3	r+1∑	r+1∑	NUM
ejpam-6191	142	4	k=0	k=0	PROPN
ejpam-6191	142	5	(	(	PUNCT
ejpam-6191	142	6	−1)r+1−k	−1)r+1−k	X
ejpam-6191	142	7	(	(	PUNCT
ejpam-6191	142	8	r	r	NOUN
ejpam-6191	142	9	+	+	NOUN
ejpam-6191	142	10	1	1	NUM
ejpam-6191	142	11	k	k	NOUN
ejpam-6191	142	12	)	)	PUNCT
ejpam-6191	142	13	n∑	n∑	NOUN
ejpam-6191	142	14	j=0	j=0	PROPN
ejpam-6191	142	15	(	(	PUNCT
ejpam-6191	142	16	a+	a+	PUNCT
ejpam-6191	142	17	b)jh	b)jh	PROPN
ejpam-6191	142	18	(	(	PUNCT
ejpam-6191	142	19	k−1	k−1	PROPN
ejpam-6191	142	20	)	)	PUNCT
ejpam-6191	142	21	j	j	PROPN
ejpam-6191	142	22	bn−jh	bn−jh	PROPN
ejpam-6191	142	23	(	(	PUNCT
ejpam-6191	142	24	r−k	r−k	PROPN
ejpam-6191	142	25	)	)	PUNCT
ejpam-6191	142	26	n−j	n−j	ADV
ejpam-6191	142	27			PROPN
ejpam-6191	142	28	zn	zn	PROPN
ejpam-6191	142	29	−	−	PROPN
ejpam-6191	142	30	∞∑	∞∑	PROPN
ejpam-6191	142	31	n=1	n=1	PROPN
ejpam-6191	142	32	r+1∑	r+1∑	NUM
ejpam-6191	142	33	k=0	k=0	PROPN
ejpam-6191	142	34	(	(	PUNCT
ejpam-6191	142	35	−1)r+1−k	−1)r+1−k	X
ejpam-6191	142	36	(	(	PUNCT
ejpam-6191	142	37	r	r	NOUN
ejpam-6191	142	38	+	+	NOUN
ejpam-6191	142	39	1	1	NUM
ejpam-6191	142	40	k	k	NOUN
ejpam-6191	142	41	)	)	PUNCT
ejpam-6191	142	42	n−1∑	n−1∑	PROPN
ejpam-6191	142	43	j=0	j=0	PROPN
ejpam-6191	142	44	(	(	PUNCT
ejpam-6191	142	45	a+	a+	PUNCT
ejpam-6191	142	46	b)jh	b)jh	PROPN
ejpam-6191	142	47	(	(	PUNCT
ejpam-6191	142	48	k−1	k−1	PROPN
ejpam-6191	142	49	)	)	PUNCT
ejpam-6191	143	1	j	j	PROPN
ejpam-6191	143	2	bn−jh	bn−jh	PROPN
ejpam-6191	143	3	(	(	PUNCT
ejpam-6191	143	4	r−k	r−k	PROPN
ejpam-6191	143	5	)	)	PUNCT
ejpam-6191	143	6	n−1−j	n−1−j	NUM
ejpam-6191	143	7			PROPN
ejpam-6191	143	8	zn	zn	X
ejpam-6191	143	9	.	.	PUNCT
ejpam-6191	144	1	comparing	compare	VERB
ejpam-6191	144	2	coefficients	coefficient	NOUN
ejpam-6191	144	3	of	of	ADP
ejpam-6191	144	4	zn	zn	PROPN
ejpam-6191	144	5	,	,	PUNCT
ejpam-6191	144	6	n∑	n∑	PROPN
ejpam-6191	144	7	k=0	k=0	PROPN
ejpam-6191	144	8	(	(	PUNCT
ejpam-6191	144	9	n	n	X
ejpam-6191	144	10	k	k	NOUN
ejpam-6191	144	11	)	)	PUNCT
ejpam-6191	144	12	akbn−kh	akbn−kh	VERB
ejpam-6191	144	13	(	(	PUNCT
ejpam-6191	144	14	r	r	NOUN
ejpam-6191	144	15	)	)	PUNCT
ejpam-6191	145	1	k	k	NOUN
ejpam-6191	145	2	=	=	SYM
ejpam-6191	145	3	r+1∑	r+1∑	PROPN
ejpam-6191	145	4	k=0	k=0	PROPN
ejpam-6191	145	5	(	(	PUNCT
ejpam-6191	145	6	−1)r+1−k	−1)r+1−k	X
ejpam-6191	145	7	(	(	PUNCT
ejpam-6191	145	8	r	r	NOUN
ejpam-6191	145	9	+	+	NOUN
ejpam-6191	145	10	1	1	NUM
ejpam-6191	145	11	k	k	NOUN
ejpam-6191	145	12	)	)	PUNCT
ejpam-6191	145	13	n∑	n∑	NOUN
ejpam-6191	145	14	j=0	j=0	PROPN
ejpam-6191	145	15	(	(	PUNCT
ejpam-6191	145	16	a+	a+	PUNCT
ejpam-6191	145	17	b)jh	b)jh	PROPN
ejpam-6191	145	18	(	(	PUNCT
ejpam-6191	145	19	k−1	k−1	PROPN
ejpam-6191	145	20	)	)	PUNCT
ejpam-6191	146	1	j	j	PROPN
ejpam-6191	146	2	bn−jh	bn−jh	PROPN
ejpam-6191	146	3	(	(	PUNCT
ejpam-6191	146	4	r−k	r−k	PROPN
ejpam-6191	146	5	)	)	PUNCT
ejpam-6191	146	6	n−j	n−j	PUNCT
ejpam-6191	146	7	k.	k.	PROPN
ejpam-6191	147	1	v.	v.	ADP
ejpam-6191	147	2	m.	m.	PROPN
ejpam-6191	147	3	manulat	manulat	PROPN
ejpam-6191	147	4	,	,	PUNCT
ejpam-6191	147	5	r.	r.	PROPN
ejpam-6191	147	6	b.	b.	PROPN
ejpam-6191	147	7	corcino	corcino	PROPN
ejpam-6191	147	8	/	/	SYM
ejpam-6191	147	9	eur	eur	PROPN
ejpam-6191	147	10	.	.	PUNCT
ejpam-6191	148	1	j.	j.	PROPN
ejpam-6191	148	2	pure	pure	PROPN
ejpam-6191	148	3	appl	appl	PROPN
ejpam-6191	148	4	.	.	PROPN
ejpam-6191	148	5	math	math	PROPN
ejpam-6191	148	6	,	,	PUNCT
ejpam-6191	148	7	18	18	NUM
ejpam-6191	148	8	(	(	PUNCT
ejpam-6191	148	9	3	3	NUM
ejpam-6191	148	10	)	)	PUNCT
ejpam-6191	148	11	(	(	PUNCT
ejpam-6191	148	12	2025	2025	NUM
ejpam-6191	148	13	)	)	PUNCT
ejpam-6191	148	14	,	,	PUNCT
ejpam-6191	148	15	6191	6191	NUM
ejpam-6191	148	16	8	8	NUM
ejpam-6191	148	17	of	of	ADP
ejpam-6191	148	18	21	21	NUM
ejpam-6191	148	19	−	−	PROPN
ejpam-6191	148	20	r+1∑	r+1∑	PROPN
ejpam-6191	148	21	k=0	k=0	PROPN
ejpam-6191	148	22	(	(	PUNCT
ejpam-6191	148	23	−1)r+1−k	−1)r+1−k	X
ejpam-6191	148	24	(	(	PUNCT
ejpam-6191	148	25	r	r	NOUN
ejpam-6191	148	26	+	+	NOUN
ejpam-6191	148	27	1	1	NUM
ejpam-6191	148	28	k	k	NOUN
ejpam-6191	148	29	)	)	PUNCT
ejpam-6191	148	30	n−1∑	n−1∑	PROPN
ejpam-6191	148	31	j=0	j=0	PROPN
ejpam-6191	148	32	(	(	PUNCT
ejpam-6191	148	33	a+	a+	PUNCT
ejpam-6191	148	34	b)jh	b)jh	PROPN
ejpam-6191	148	35	(	(	PUNCT
ejpam-6191	148	36	k−1	k−1	PROPN
ejpam-6191	148	37	)	)	PUNCT
ejpam-6191	149	1	j	j	PROPN
ejpam-6191	149	2	bn−jh	bn−jh	PROPN
ejpam-6191	149	3	(	(	PUNCT
ejpam-6191	149	4	r−k	r−k	PROPN
ejpam-6191	149	5	)	)	PUNCT
ejpam-6191	149	6	n−1−j	n−1−j	PUNCT
ejpam-6191	149	7	=	=	SYM
ejpam-6191	149	8	r+1∑	r+1∑	PROPN
ejpam-6191	149	9	k=0	k=0	PROPN
ejpam-6191	149	10	(	(	PUNCT
ejpam-6191	149	11	−1)r+1−k	−1)r+1−k	X
ejpam-6191	149	12	(	(	PUNCT
ejpam-6191	149	13	r	r	NOUN
ejpam-6191	149	14	+	+	NOUN
ejpam-6191	149	15	1	1	NUM
ejpam-6191	149	16	k	k	NOUN
ejpam-6191	149	17	)	)	PUNCT
ejpam-6191	149	18	n∑	n∑	NOUN
ejpam-6191	149	19	j=0	j=0	PROPN
ejpam-6191	149	20	(	(	PUNCT
ejpam-6191	149	21	a+	a+	PUNCT
ejpam-6191	149	22	b)jh	b)jh	PROPN
ejpam-6191	149	23	(	(	PUNCT
ejpam-6191	149	24	k−1	k−1	PROPN
ejpam-6191	149	25	)	)	PUNCT
ejpam-6191	150	1	j	j	PROPN
ejpam-6191	150	2	bn−jh	bn−jh	PROPN
ejpam-6191	150	3	(	(	PUNCT
ejpam-6191	150	4	r−k	r−k	PROPN
ejpam-6191	150	5	)	)	PUNCT
ejpam-6191	150	6	n−j	n−j	ADV
ejpam-6191	150	7	−	−	PROPN
ejpam-6191	150	8	r+1∑	r+1∑	PROPN
ejpam-6191	150	9	k=0	k=0	PROPN
ejpam-6191	150	10	(	(	PUNCT
ejpam-6191	150	11	−1)r+1−k	−1)r+1−k	X
ejpam-6191	150	12	(	(	PUNCT
ejpam-6191	150	13	r	r	NOUN
ejpam-6191	150	14	+	+	NOUN
ejpam-6191	150	15	1	1	NUM
ejpam-6191	150	16	k	k	NOUN
ejpam-6191	150	17	)	)	PUNCT
ejpam-6191	150	18	n∑	n∑	NOUN
ejpam-6191	150	19	j=0	j=0	PROPN
ejpam-6191	150	20	(	(	PUNCT
ejpam-6191	150	21	a+	a+	PUNCT
ejpam-6191	150	22	b)jh	b)jh	PROPN
ejpam-6191	150	23	(	(	PUNCT
ejpam-6191	150	24	k−1	k−1	PROPN
ejpam-6191	150	25	)	)	PUNCT
ejpam-6191	151	1	j	j	PROPN
ejpam-6191	151	2	bn−jh	bn−jh	PROPN
ejpam-6191	151	3	(	(	PUNCT
ejpam-6191	151	4	r−k	r−k	PROPN
ejpam-6191	151	5	)	)	PUNCT
ejpam-6191	151	6	n−1−j	n−1−j	NOUN
ejpam-6191	151	7	.	.	PUNCT
ejpam-6191	152	1	simplifying	simplify	VERB
ejpam-6191	152	2	further	far	ADV
ejpam-6191	152	3	we	we	PRON
ejpam-6191	152	4	get	get	VERB
ejpam-6191	152	5	the	the	DET
ejpam-6191	152	6	result	result	NOUN
ejpam-6191	152	7	,	,	PUNCT
ejpam-6191	152	8	sn(a	sn(a	ADV
ejpam-6191	152	9	,	,	PUNCT
ejpam-6191	152	10	b	b	X
ejpam-6191	152	11	)	)	PUNCT
ejpam-6191	153	1	=	=	SYM
ejpam-6191	153	2	n∑	n∑	NOUN
ejpam-6191	153	3	k=0	k=0	PROPN
ejpam-6191	153	4	(	(	PUNCT
ejpam-6191	153	5	n	n	X
ejpam-6191	153	6	k	k	NOUN
ejpam-6191	153	7	)	)	PUNCT
ejpam-6191	153	8	akbn−kh	akbn−kh	VERB
ejpam-6191	153	9	(	(	PUNCT
ejpam-6191	153	10	r	r	NOUN
ejpam-6191	153	11	)	)	PUNCT
ejpam-6191	153	12	k	k	NOUN
ejpam-6191	154	1	=	=	SYM
ejpam-6191	154	2	r+1∑	r+1∑	PROPN
ejpam-6191	154	3	k=0	k=0	PROPN
ejpam-6191	154	4	n∑	n∑	PROPN
ejpam-6191	154	5	j=0	j=0	PROPN
ejpam-6191	154	6	(	(	PUNCT
ejpam-6191	154	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	154	8	(	(	PUNCT
ejpam-6191	154	9	r	r	NOUN
ejpam-6191	154	10	+	+	NOUN
ejpam-6191	154	11	1	1	NUM
ejpam-6191	154	12	k	k	NOUN
ejpam-6191	154	13	)	)	PUNCT
ejpam-6191	154	14	(	(	PUNCT
ejpam-6191	154	15	a+	a+	PUNCT
ejpam-6191	154	16	b)jh	b)jh	PROPN
ejpam-6191	154	17	(	(	PUNCT
ejpam-6191	154	18	k−1	k−1	PROPN
ejpam-6191	154	19	)	)	PUNCT
ejpam-6191	154	20	j	j	PROPN
ejpam-6191	155	1	bn−j	bn−j	ADV
ejpam-6191	155	2	[	[	PUNCT
ejpam-6191	155	3	h	h	NOUN
ejpam-6191	155	4	(	(	PUNCT
ejpam-6191	155	5	r−k	r−k	PROPN
ejpam-6191	155	6	)	)	PUNCT
ejpam-6191	155	7	n−j	n−j	ADV
ejpam-6191	155	8	−h	−h	VERB
ejpam-6191	155	9	(	(	PUNCT
ejpam-6191	155	10	r−k	r−k	PROPN
ejpam-6191	155	11	)	)	PUNCT
ejpam-6191	155	12	n−1−j	n−1−j	NUM
ejpam-6191	155	13	]	]	PUNCT
ejpam-6191	155	14	.	.	PUNCT
ejpam-6191	156	1	the	the	DET
ejpam-6191	156	2	identity	identity	NOUN
ejpam-6191	156	3	established	establish	VERB
ejpam-6191	156	4	in	in	ADP
ejpam-6191	156	5	theorem	theorem	PROPN
ejpam-6191	156	6	3.1	3.1	NUM
ejpam-6191	156	7	.	.	PUNCT
ejpam-6191	156	8	serves	serve	VERB
ejpam-6191	156	9	as	as	ADP
ejpam-6191	156	10	a	a	DET
ejpam-6191	156	11	unifying	unifying	ADJ
ejpam-6191	156	12	framework	framework	NOUN
ejpam-6191	156	13	that	that	PRON
ejpam-6191	156	14	yields	yield	VERB
ejpam-6191	156	15	a	a	DET
ejpam-6191	156	16	variety	variety	NOUN
ejpam-6191	156	17	of	of	ADP
ejpam-6191	156	18	novel	novel	NOUN
ejpam-6191	156	19	and	and	CCONJ
ejpam-6191	156	20	interesting	interesting	ADJ
ejpam-6191	156	21	identities	identity	NOUN
ejpam-6191	156	22	involving	involve	VERB
ejpam-6191	156	23	several	several	ADJ
ejpam-6191	156	24	well	well	ADV
ejpam-6191	156	25	-	-	PUNCT
ejpam-6191	156	26	known	know	VERB
ejpam-6191	156	27	integer	integer	NOUN
ejpam-6191	156	28	sequences	sequence	NOUN
ejpam-6191	156	29	.	.	PUNCT
ejpam-6191	157	1	in	in	ADP
ejpam-6191	157	2	particular	particular	ADJ
ejpam-6191	157	3	,	,	PUNCT
ejpam-6191	157	4	it	it	PRON
ejpam-6191	157	5	leads	lead	VERB
ejpam-6191	157	6	to	to	ADP
ejpam-6191	157	7	new	new	ADJ
ejpam-6191	157	8	relations	relation	NOUN
ejpam-6191	157	9	for	for	ADP
ejpam-6191	157	10	fibonacci	fibonacci	NOUN
ejpam-6191	157	11	numbers	number	NOUN
ejpam-6191	157	12	,	,	PUNCT
ejpam-6191	157	13	lucas	lucas	NOUN
ejpam-6191	157	14	numbers	number	NOUN
ejpam-6191	157	15	,	,	PUNCT
ejpam-6191	157	16	pell	pell	NOUN
ejpam-6191	157	17	numbers	number	NOUN
ejpam-6191	157	18	,	,	PUNCT
ejpam-6191	157	19	pell	pell	NOUN
ejpam-6191	157	20	-	-	PUNCT
ejpam-6191	157	21	lucas	lucas	NOUN
ejpam-6191	157	22	numbers	number	NOUN
ejpam-6191	157	23	,	,	PUNCT
ejpam-6191	157	24	jacobsthal	jacobsthal	ADJ
ejpam-6191	157	25	numbers	number	NOUN
ejpam-6191	157	26	,	,	PUNCT
ejpam-6191	157	27	jacobsthal	jacobsthal	ADJ
ejpam-6191	157	28	-	-	PUNCT
ejpam-6191	157	29	lucas	lucas	NOUN
ejpam-6191	157	30	numbers	number	NOUN
ejpam-6191	157	31	,	,	PUNCT
ejpam-6191	157	32	mersenne	mersenne	NOUN
ejpam-6191	157	33	numbers	number	NOUN
ejpam-6191	157	34	,	,	PUNCT
ejpam-6191	157	35	and	and	CCONJ
ejpam-6191	157	36	mersenne	mersenne	NOUN
ejpam-6191	157	37	-	-	PUNCT
ejpam-6191	157	38	lucas	lucas	PROPN
ejpam-6191	157	39	numbers	number	NOUN
ejpam-6191	157	40	.	.	PUNCT
ejpam-6191	158	1	these	these	DET
ejpam-6191	158	2	resulting	result	VERB
ejpam-6191	158	3	identities	identity	NOUN
ejpam-6191	158	4	,	,	PUNCT
ejpam-6191	158	5	which	which	PRON
ejpam-6191	158	6	highlight	highlight	VERB
ejpam-6191	158	7	the	the	DET
ejpam-6191	158	8	structural	structural	ADJ
ejpam-6191	158	9	similarities	similarity	NOUN
ejpam-6191	158	10	and	and	CCONJ
ejpam-6191	158	11	recursive	recursive	ADJ
ejpam-6191	158	12	properties	property	NOUN
ejpam-6191	158	13	shared	share	VERB
ejpam-6191	158	14	by	by	ADP
ejpam-6191	158	15	these	these	DET
ejpam-6191	158	16	sequences	sequence	NOUN
ejpam-6191	158	17	,	,	PUNCT
ejpam-6191	158	18	are	be	AUX
ejpam-6191	158	19	systematically	systematically	ADV
ejpam-6191	158	20	presented	present	VERB
ejpam-6191	158	21	in	in	ADP
ejpam-6191	158	22	the	the	DET
ejpam-6191	158	23	subsequent	subsequent	ADJ
ejpam-6191	158	24	corollaries	corollary	NOUN
ejpam-6191	158	25	.	.	PUNCT
ejpam-6191	159	1	corollary	corollary	ADJ
ejpam-6191	159	2	3.2	3.2	NUM
ejpam-6191	159	3	.	.	PUNCT
ejpam-6191	160	1	let	let	VERB
ejpam-6191	160	2	fn	fn	NOUN
ejpam-6191	160	3	and	and	CCONJ
ejpam-6191	160	4	ln	ln	ADV
ejpam-6191	160	5	be	be	AUX
ejpam-6191	160	6	the	the	DET
ejpam-6191	160	7	fibonacci	fibonacci	NOUN
ejpam-6191	160	8	and	and	CCONJ
ejpam-6191	160	9	lucas	lucas	PROPN
ejpam-6191	160	10	numbers	number	NOUN
ejpam-6191	160	11	,	,	PUNCT
ejpam-6191	160	12	respectively	respectively	ADV
ejpam-6191	160	13	.	.	PUNCT
ejpam-6191	161	1	then	then	ADV
ejpam-6191	161	2	we	we	PRON
ejpam-6191	161	3	have	have	VERB
ejpam-6191	161	4	the	the	DET
ejpam-6191	161	5	relations	relation	NOUN
ejpam-6191	161	6	n∑	n∑	NOUN
ejpam-6191	161	7	k=0	k=0	PROPN
ejpam-6191	161	8	(	(	PUNCT
ejpam-6191	161	9	n	n	CCONJ
ejpam-6191	161	10	k	k	NOUN
ejpam-6191	161	11	)	)	PUNCT
ejpam-6191	161	12	fkh	fkh	NOUN
ejpam-6191	161	13	(	(	PUNCT
ejpam-6191	161	14	r	r	NOUN
ejpam-6191	161	15	)	)	PUNCT
ejpam-6191	161	16	k	k	NOUN
ejpam-6191	162	1	=	=	SYM
ejpam-6191	162	2	r+1∑	r+1∑	PROPN
ejpam-6191	162	3	k=0	k=0	PROPN
ejpam-6191	162	4	n∑	n∑	PROPN
ejpam-6191	162	5	j=0	j=0	PROPN
ejpam-6191	162	6	(	(	PUNCT
ejpam-6191	162	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	162	8	(	(	PUNCT
ejpam-6191	162	9	r	r	NOUN
ejpam-6191	162	10	+	+	NOUN
ejpam-6191	162	11	1	1	NUM
ejpam-6191	162	12	k	k	NOUN
ejpam-6191	162	13	)	)	PUNCT
ejpam-6191	162	14	f2jh	f2jh	PROPN
ejpam-6191	163	1	(	(	PUNCT
ejpam-6191	163	2	k−1	k−1	PROPN
ejpam-6191	163	3	)	)	PUNCT
ejpam-6191	163	4	j	j	PROPN
ejpam-6191	163	5	[	[	PUNCT
ejpam-6191	163	6	h	h	NOUN
ejpam-6191	163	7	(	(	PUNCT
ejpam-6191	163	8	r−k	r−k	PROPN
ejpam-6191	163	9	)	)	PUNCT
ejpam-6191	163	10	n−j	n−j	ADV
ejpam-6191	163	11	−h	−h	VERB
ejpam-6191	163	12	(	(	PUNCT
ejpam-6191	163	13	r−k	r−k	PROPN
ejpam-6191	163	14	)	)	PUNCT
ejpam-6191	163	15	n−1−j	n−1−j	NUM
ejpam-6191	163	16	]	]	PUNCT
ejpam-6191	163	17	.	.	PUNCT
ejpam-6191	164	1	and	and	CCONJ
ejpam-6191	164	2	,	,	PUNCT
ejpam-6191	164	3	n∑	n∑	PROPN
ejpam-6191	164	4	k=0	k=0	PROPN
ejpam-6191	164	5	(	(	PUNCT
ejpam-6191	164	6	n	n	X
ejpam-6191	164	7	k	k	NOUN
ejpam-6191	164	8	)	)	PUNCT
ejpam-6191	164	9	lkh	lkh	NOUN
ejpam-6191	164	10	(	(	PUNCT
ejpam-6191	164	11	r	r	NOUN
ejpam-6191	164	12	)	)	PUNCT
ejpam-6191	164	13	k	k	NOUN
ejpam-6191	164	14	=	=	SYM
ejpam-6191	164	15	r+1∑	r+1∑	PROPN
ejpam-6191	164	16	k=0	k=0	PROPN
ejpam-6191	164	17	n∑	n∑	PROPN
ejpam-6191	164	18	j=0	j=0	PROPN
ejpam-6191	164	19	(	(	PUNCT
ejpam-6191	164	20	−1)r+1−k	−1)r+1−k	X
ejpam-6191	164	21	(	(	PUNCT
ejpam-6191	164	22	r	r	NOUN
ejpam-6191	164	23	+	+	NOUN
ejpam-6191	164	24	1	1	NUM
ejpam-6191	164	25	k	k	NOUN
ejpam-6191	164	26	)	)	PUNCT
ejpam-6191	164	27	l2jh	l2jh	PROPN
ejpam-6191	165	1	(	(	PUNCT
ejpam-6191	165	2	k−1	k−1	PROPN
ejpam-6191	165	3	)	)	PUNCT
ejpam-6191	165	4	j	j	PROPN
ejpam-6191	165	5	[	[	PUNCT
ejpam-6191	165	6	h	h	NOUN
ejpam-6191	165	7	(	(	PUNCT
ejpam-6191	165	8	r−k	r−k	PROPN
ejpam-6191	165	9	)	)	PUNCT
ejpam-6191	165	10	n−j	n−j	ADV
ejpam-6191	165	11	−h	−h	VERB
ejpam-6191	165	12	(	(	PUNCT
ejpam-6191	165	13	r−k	r−k	PROPN
ejpam-6191	165	14	)	)	PUNCT
ejpam-6191	165	15	n−1−j	n−1−j	NUM
ejpam-6191	165	16	]	]	PUNCT
ejpam-6191	165	17	.	.	PUNCT
ejpam-6191	166	1	proof	proof	NOUN
ejpam-6191	166	2	.	.	PUNCT
ejpam-6191	167	1	setting	set	VERB
ejpam-6191	167	2	(	(	PUNCT
ejpam-6191	167	3	a	a	PRON
ejpam-6191	167	4	;	;	PUNCT
ejpam-6191	167	5	b	b	X
ejpam-6191	167	6	)	)	PUNCT
ejpam-6191	167	7	=	=	SYM
ejpam-6191	167	8	(	(	PUNCT
ejpam-6191	167	9	α	α	NOUN
ejpam-6191	167	10	;	;	PUNCT
ejpam-6191	167	11	1	1	NUM
ejpam-6191	167	12	)	)	PUNCT
ejpam-6191	167	13	and	and	CCONJ
ejpam-6191	167	14	(	(	PUNCT
ejpam-6191	167	15	a	a	PRON
ejpam-6191	167	16	;	;	PUNCT
ejpam-6191	167	17	b	b	X
ejpam-6191	167	18	)	)	PUNCT
ejpam-6191	167	19	=	=	SYM
ejpam-6191	167	20	(	(	PUNCT
ejpam-6191	167	21	β	β	NOUN
ejpam-6191	167	22	;	;	PUNCT
ejpam-6191	167	23	1	1	X
ejpam-6191	167	24	)	)	PUNCT
ejpam-6191	167	25	in	in	ADP
ejpam-6191	167	26	(	(	PUNCT
ejpam-6191	167	27	3.1	3.1	NUM
ejpam-6191	167	28	)	)	PUNCT
ejpam-6191	167	29	,	,	PUNCT
ejpam-6191	167	30	respectively	respectively	ADV
ejpam-6191	167	31	where	where	SCONJ
ejpam-6191	167	32	α	α	NOUN
ejpam-6191	167	33	=	=	SYM
ejpam-6191	167	34	1	1	NUM
ejpam-6191	167	35	+	+	NUM
ejpam-6191	167	36	√	√	NUM
ejpam-6191	167	37	5	5	NUM
ejpam-6191	167	38	2	2	NUM
ejpam-6191	167	39	and	and	CCONJ
ejpam-6191	167	40	β	β	X
ejpam-6191	167	41	=	=	SYM
ejpam-6191	167	42	1−	1−	NUM
ejpam-6191	167	43	√	√	NUM
ejpam-6191	167	44	5	5	NUM
ejpam-6191	167	45	2	2	NUM
ejpam-6191	167	46	,	,	PUNCT
ejpam-6191	167	47	and	and	CCONJ
ejpam-6191	167	48	applying	apply	VERB
ejpam-6191	167	49	the	the	DET
ejpam-6191	167	50	relations	relation	NOUN
ejpam-6191	167	51	,	,	PUNCT
ejpam-6191	167	52	α2	α2	PROPN
ejpam-6191	167	53	=	=	SYM
ejpam-6191	167	54	α+	α+	PUNCT
ejpam-6191	167	55	1	1	NUM
ejpam-6191	167	56	and	and	CCONJ
ejpam-6191	167	57	β2	β2	NOUN
ejpam-6191	167	58	=	=	SYM
ejpam-6191	167	59	β	β	PROPN
ejpam-6191	167	60	+	+	NOUN
ejpam-6191	167	61	1	1	X
ejpam-6191	167	62	.	.	PUNCT
ejpam-6191	167	63	k.	k.	PROPN
ejpam-6191	168	1	v.	v.	PROPN
ejpam-6191	168	2	m.	m.	PROPN
ejpam-6191	168	3	manulat	manulat	PROPN
ejpam-6191	168	4	,	,	PUNCT
ejpam-6191	168	5	r.	r.	PROPN
ejpam-6191	168	6	b.	b.	PROPN
ejpam-6191	168	7	corcino	corcino	PROPN
ejpam-6191	168	8	/	/	SYM
ejpam-6191	168	9	eur	eur	PROPN
ejpam-6191	168	10	.	.	PUNCT
ejpam-6191	169	1	j.	j.	PROPN
ejpam-6191	169	2	pure	pure	PROPN
ejpam-6191	169	3	appl	appl	PROPN
ejpam-6191	169	4	.	.	PROPN
ejpam-6191	169	5	math	math	PROPN
ejpam-6191	169	6	,	,	PUNCT
ejpam-6191	169	7	18	18	NUM
ejpam-6191	169	8	(	(	PUNCT
ejpam-6191	169	9	3	3	NUM
ejpam-6191	169	10	)	)	PUNCT
ejpam-6191	169	11	(	(	PUNCT
ejpam-6191	169	12	2025	2025	NUM
ejpam-6191	169	13	)	)	PUNCT
ejpam-6191	169	14	,	,	PUNCT
ejpam-6191	169	15	6191	6191	NUM
ejpam-6191	169	16	9	9	NUM
ejpam-6191	169	17	of	of	ADP
ejpam-6191	169	18	21	21	NUM
ejpam-6191	169	19	corollary	corollary	ADJ
ejpam-6191	169	20	3.3	3.3	NUM
ejpam-6191	169	21	.	.	PUNCT
ejpam-6191	170	1	let	let	VERB
ejpam-6191	170	2	fn	fn	NOUN
ejpam-6191	170	3	and	and	CCONJ
ejpam-6191	170	4	ln	ln	ADV
ejpam-6191	170	5	be	be	AUX
ejpam-6191	170	6	the	the	DET
ejpam-6191	170	7	fibonacci	fibonacci	NOUN
ejpam-6191	170	8	and	and	CCONJ
ejpam-6191	170	9	lucas	lucas	PROPN
ejpam-6191	170	10	numbers	number	NOUN
ejpam-6191	170	11	,	,	PUNCT
ejpam-6191	170	12	respectively	respectively	ADV
ejpam-6191	170	13	.	.	PUNCT
ejpam-6191	171	1	then	then	ADV
ejpam-6191	171	2	,	,	PUNCT
ejpam-6191	171	3	the	the	DET
ejpam-6191	171	4	following	follow	VERB
ejpam-6191	171	5	identities	identity	NOUN
ejpam-6191	171	6	hold	hold	VERB
ejpam-6191	171	7	n∑	n∑	NOUN
ejpam-6191	171	8	k=0	k=0	PROPN
ejpam-6191	171	9	(	(	PUNCT
ejpam-6191	171	10	n	n	X
ejpam-6191	171	11	k	k	NOUN
ejpam-6191	171	12	)	)	PUNCT
ejpam-6191	171	13	(	(	PUNCT
ejpam-6191	171	14	−1)n−kf2kh	−1)n−kf2kh	X
ejpam-6191	171	15	(	(	PUNCT
ejpam-6191	171	16	r	r	NOUN
ejpam-6191	171	17	)	)	PUNCT
ejpam-6191	171	18	k	k	NOUN
ejpam-6191	172	1	=	=	SYM
ejpam-6191	172	2	r+1∑	r+1∑	PROPN
ejpam-6191	172	3	k=0	k=0	PROPN
ejpam-6191	172	4	n∑	n∑	PROPN
ejpam-6191	172	5	j=0	j=0	PROPN
ejpam-6191	172	6	(	(	PUNCT
ejpam-6191	172	7	−1)r+1−k+n−j	−1)r+1−k+n−j	X
ejpam-6191	172	8	(	(	PUNCT
ejpam-6191	172	9	r	r	NOUN
ejpam-6191	172	10	+	+	NOUN
ejpam-6191	172	11	1	1	NUM
ejpam-6191	172	12	k	k	NOUN
ejpam-6191	172	13	)	)	PUNCT
ejpam-6191	172	14	fjh	fjh	NOUN
ejpam-6191	172	15	(	(	PUNCT
ejpam-6191	172	16	k−1	k−1	PROPN
ejpam-6191	172	17	)	)	PUNCT
ejpam-6191	172	18	j	j	PROPN
ejpam-6191	172	19	[	[	PUNCT
ejpam-6191	172	20	h	h	NOUN
ejpam-6191	172	21	(	(	PUNCT
ejpam-6191	172	22	r−k	r−k	PROPN
ejpam-6191	172	23	)	)	PUNCT
ejpam-6191	172	24	n−j	n−j	ADV
ejpam-6191	172	25	−h	−h	VERB
ejpam-6191	172	26	(	(	PUNCT
ejpam-6191	172	27	r−k	r−k	PROPN
ejpam-6191	172	28	)	)	PUNCT
ejpam-6191	172	29	n−1−j	n−1−j	X
ejpam-6191	172	30	]	]	PUNCT
ejpam-6191	172	31	and	and	CCONJ
ejpam-6191	172	32	,	,	PUNCT
ejpam-6191	172	33	n∑	n∑	PROPN
ejpam-6191	172	34	k=0	k=0	PROPN
ejpam-6191	172	35	(	(	PUNCT
ejpam-6191	172	36	n	n	X
ejpam-6191	172	37	k	k	NOUN
ejpam-6191	172	38	)	)	PUNCT
ejpam-6191	172	39	(	(	PUNCT
ejpam-6191	172	40	−1)n−kl2kh	−1)n−kl2kh	PROPN
ejpam-6191	172	41	(	(	PUNCT
ejpam-6191	172	42	r	r	NOUN
ejpam-6191	172	43	)	)	PUNCT
ejpam-6191	172	44	k	k	NOUN
ejpam-6191	172	45	=	=	SYM
ejpam-6191	172	46	r+1∑	r+1∑	PROPN
ejpam-6191	172	47	k=0	k=0	PROPN
ejpam-6191	172	48	n∑	n∑	PROPN
ejpam-6191	172	49	j=0	j=0	PROPN
ejpam-6191	172	50	(	(	PUNCT
ejpam-6191	172	51	−1)r+1−k+n−j	−1)r+1−k+n−j	X
ejpam-6191	172	52	(	(	PUNCT
ejpam-6191	172	53	r	r	NOUN
ejpam-6191	172	54	+	+	NOUN
ejpam-6191	172	55	1	1	NUM
ejpam-6191	172	56	k	k	NOUN
ejpam-6191	172	57	)	)	PUNCT
ejpam-6191	172	58	ljh	ljh	NOUN
ejpam-6191	172	59	(	(	PUNCT
ejpam-6191	172	60	k−1	k−1	PROPN
ejpam-6191	172	61	)	)	PUNCT
ejpam-6191	172	62	j	j	PROPN
ejpam-6191	172	63	[	[	PUNCT
ejpam-6191	172	64	h	h	NOUN
ejpam-6191	172	65	(	(	PUNCT
ejpam-6191	172	66	r−k	r−k	PROPN
ejpam-6191	172	67	)	)	PUNCT
ejpam-6191	172	68	n−j	n−j	ADV
ejpam-6191	172	69	−h	−h	VERB
ejpam-6191	172	70	(	(	PUNCT
ejpam-6191	172	71	r−k	r−k	PROPN
ejpam-6191	172	72	)	)	PUNCT
ejpam-6191	172	73	n−1−j	n−1−j	NUM
ejpam-6191	172	74	]	]	PUNCT
ejpam-6191	172	75	.	.	PUNCT
ejpam-6191	173	1	proof	proof	NOUN
ejpam-6191	173	2	.	.	PUNCT
ejpam-6191	174	1	setting	set	VERB
ejpam-6191	174	2	(	(	PUNCT
ejpam-6191	174	3	a	a	PRON
ejpam-6191	174	4	;	;	PUNCT
ejpam-6191	174	5	b	b	X
ejpam-6191	174	6	)	)	PUNCT
ejpam-6191	174	7	=	=	SYM
ejpam-6191	174	8	(	(	PUNCT
ejpam-6191	174	9	α2;−1	α2;−1	NOUN
ejpam-6191	174	10	)	)	PUNCT
ejpam-6191	174	11	and	and	CCONJ
ejpam-6191	174	12	(	(	PUNCT
ejpam-6191	174	13	a	a	PRON
ejpam-6191	174	14	;	;	PUNCT
ejpam-6191	174	15	b	b	X
ejpam-6191	174	16	)	)	PUNCT
ejpam-6191	174	17	=	=	SYM
ejpam-6191	174	18	(	(	PUNCT
ejpam-6191	174	19	β2;−1	β2;−1	NOUN
ejpam-6191	174	20	)	)	PUNCT
ejpam-6191	174	21	in	in	ADP
ejpam-6191	174	22	(	(	PUNCT
ejpam-6191	174	23	3.1	3.1	NUM
ejpam-6191	174	24	)	)	PUNCT
ejpam-6191	174	25	,	,	PUNCT
ejpam-6191	174	26	respectively	respectively	ADV
ejpam-6191	174	27	where	where	SCONJ
ejpam-6191	174	28	α	α	NOUN
ejpam-6191	174	29	=	=	SYM
ejpam-6191	174	30	1	1	NUM
ejpam-6191	174	31	+	+	NUM
ejpam-6191	174	32	√	√	NUM
ejpam-6191	174	33	5	5	NUM
ejpam-6191	174	34	2	2	NUM
ejpam-6191	174	35	and	and	CCONJ
ejpam-6191	174	36	β	β	X
ejpam-6191	174	37	=	=	SYM
ejpam-6191	174	38	1−	1−	NUM
ejpam-6191	174	39	√	√	NUM
ejpam-6191	174	40	5	5	NUM
ejpam-6191	174	41	2	2	NUM
ejpam-6191	174	42	,	,	PUNCT
ejpam-6191	174	43	and	and	CCONJ
ejpam-6191	174	44	applying	apply	VERB
ejpam-6191	174	45	the	the	DET
ejpam-6191	174	46	relations	relation	NOUN
ejpam-6191	174	47	,	,	PUNCT
ejpam-6191	174	48	α2	α2	PROPN
ejpam-6191	174	49	=	=	SYM
ejpam-6191	174	50	α+	α+	PUNCT
ejpam-6191	174	51	1	1	NUM
ejpam-6191	174	52	and	and	CCONJ
ejpam-6191	174	53	β2	β2	NOUN
ejpam-6191	174	54	=	=	SYM
ejpam-6191	174	55	β	β	PROPN
ejpam-6191	174	56	+	+	NOUN
ejpam-6191	174	57	1	1	X
ejpam-6191	174	58	.	.	X
ejpam-6191	174	59	corollary	corollary	ADJ
ejpam-6191	174	60	3.4	3.4	NUM
ejpam-6191	174	61	.	.	PUNCT
ejpam-6191	175	1	let	let	VERB
ejpam-6191	175	2	pn	pn	PART
ejpam-6191	175	3	be	be	AUX
ejpam-6191	175	4	the	the	DET
ejpam-6191	175	5	pell	pell	NOUN
ejpam-6191	175	6	numbers	number	NOUN
ejpam-6191	175	7	and	and	CCONJ
ejpam-6191	175	8	qn	qn	NOUN
ejpam-6191	175	9	be	be	AUX
ejpam-6191	175	10	the	the	DET
ejpam-6191	175	11	pell	pell	NOUN
ejpam-6191	175	12	-	-	PUNCT
ejpam-6191	175	13	lucas	lucas	NOUN
ejpam-6191	175	14	numbers	number	NOUN
ejpam-6191	175	15	,	,	PUNCT
ejpam-6191	175	16	respectively	respectively	ADV
ejpam-6191	175	17	.	.	PUNCT
ejpam-6191	176	1	then	then	ADV
ejpam-6191	176	2	,	,	PUNCT
ejpam-6191	176	3	the	the	DET
ejpam-6191	176	4	following	follow	VERB
ejpam-6191	176	5	holds	hold	VERB
ejpam-6191	176	6	n∑	n∑	PROPN
ejpam-6191	176	7	k=0	k=0	PROPN
ejpam-6191	176	8	(	(	PUNCT
ejpam-6191	176	9	n	n	X
ejpam-6191	176	10	k	k	PROPN
ejpam-6191	176	11	)	)	PUNCT
ejpam-6191	177	1	2kpkh	2kpkh	NUM
ejpam-6191	177	2	(	(	PUNCT
ejpam-6191	177	3	r	r	NOUN
ejpam-6191	177	4	)	)	PUNCT
ejpam-6191	177	5	k	k	NOUN
ejpam-6191	178	1	=	=	SYM
ejpam-6191	178	2	r+1∑	r+1∑	PROPN
ejpam-6191	178	3	k=0	k=0	PROPN
ejpam-6191	178	4	n∑	n∑	PROPN
ejpam-6191	178	5	j=0	j=0	PROPN
ejpam-6191	178	6	(	(	PUNCT
ejpam-6191	178	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	178	8	(	(	PUNCT
ejpam-6191	178	9	r	r	NOUN
ejpam-6191	178	10	+	+	NOUN
ejpam-6191	178	11	1	1	NUM
ejpam-6191	178	12	k	k	NOUN
ejpam-6191	178	13	)	)	PUNCT
ejpam-6191	178	14	p2jh	p2jh	PROPN
ejpam-6191	178	15	(	(	PUNCT
ejpam-6191	178	16	k−1	k−1	PROPN
ejpam-6191	178	17	)	)	PUNCT
ejpam-6191	178	18	j	j	PROPN
ejpam-6191	179	1	[	[	PUNCT
ejpam-6191	179	2	h	h	NOUN
ejpam-6191	179	3	(	(	PUNCT
ejpam-6191	179	4	r−k	r−k	PROPN
ejpam-6191	179	5	)	)	PUNCT
ejpam-6191	179	6	n−j	n−j	ADV
ejpam-6191	179	7	−h	−h	VERB
ejpam-6191	179	8	(	(	PUNCT
ejpam-6191	179	9	r−k	r−k	PROPN
ejpam-6191	179	10	)	)	PUNCT
ejpam-6191	179	11	n−1−j	n−1−j	NUM
ejpam-6191	179	12	]	]	PUNCT
ejpam-6191	179	13	.	.	PUNCT
ejpam-6191	180	1	and	and	CCONJ
ejpam-6191	180	2	,	,	PUNCT
ejpam-6191	180	3	n∑	n∑	PROPN
ejpam-6191	180	4	k=0	k=0	PROPN
ejpam-6191	180	5	(	(	PUNCT
ejpam-6191	180	6	n	n	X
ejpam-6191	180	7	k	k	PROPN
ejpam-6191	180	8	)	)	PUNCT
ejpam-6191	181	1	2kqkh	2kqkh	NUM
ejpam-6191	181	2	(	(	PUNCT
ejpam-6191	181	3	r	r	NOUN
ejpam-6191	181	4	)	)	PUNCT
ejpam-6191	181	5	k	k	NOUN
ejpam-6191	181	6	=	=	SYM
ejpam-6191	181	7	r+1∑	r+1∑	PROPN
ejpam-6191	181	8	k=0	k=0	PROPN
ejpam-6191	181	9	n∑	n∑	PROPN
ejpam-6191	181	10	j=0	j=0	PROPN
ejpam-6191	181	11	(	(	PUNCT
ejpam-6191	181	12	−1)r+1−k	−1)r+1−k	X
ejpam-6191	181	13	(	(	PUNCT
ejpam-6191	181	14	r	r	NOUN
ejpam-6191	181	15	+	+	NOUN
ejpam-6191	181	16	1	1	NUM
ejpam-6191	181	17	k	k	NOUN
ejpam-6191	181	18	)	)	PUNCT
ejpam-6191	181	19	q2jh	q2jh	PUNCT
ejpam-6191	182	1	(	(	PUNCT
ejpam-6191	182	2	k−1	k−1	PROPN
ejpam-6191	182	3	)	)	PUNCT
ejpam-6191	182	4	j	j	PROPN
ejpam-6191	182	5	[	[	PUNCT
ejpam-6191	182	6	h	h	NOUN
ejpam-6191	182	7	(	(	PUNCT
ejpam-6191	182	8	r−k	r−k	PROPN
ejpam-6191	182	9	)	)	PUNCT
ejpam-6191	182	10	n−j	n−j	ADV
ejpam-6191	182	11	−h	−h	VERB
ejpam-6191	182	12	(	(	PUNCT
ejpam-6191	182	13	r−k	r−k	PROPN
ejpam-6191	182	14	)	)	PUNCT
ejpam-6191	182	15	n−1−j	n−1−j	NUM
ejpam-6191	182	16	]	]	PUNCT
ejpam-6191	182	17	.	.	PUNCT
ejpam-6191	183	1	proof	proof	NOUN
ejpam-6191	183	2	.	.	PUNCT
ejpam-6191	184	1	setting	set	VERB
ejpam-6191	184	2	(	(	PUNCT
ejpam-6191	184	3	a	a	PRON
ejpam-6191	184	4	;	;	PUNCT
ejpam-6191	184	5	b	b	X
ejpam-6191	184	6	)	)	PUNCT
ejpam-6191	184	7	=	=	SYM
ejpam-6191	184	8	(	(	PUNCT
ejpam-6191	184	9	2α	2α	NOUN
ejpam-6191	184	10	;	;	PUNCT
ejpam-6191	184	11	1	1	NUM
ejpam-6191	184	12	)	)	PUNCT
ejpam-6191	184	13	and	and	CCONJ
ejpam-6191	184	14	(	(	PUNCT
ejpam-6191	184	15	a	a	PRON
ejpam-6191	184	16	;	;	PUNCT
ejpam-6191	184	17	b	b	X
ejpam-6191	184	18	)	)	PUNCT
ejpam-6191	184	19	=	=	SYM
ejpam-6191	184	20	(	(	PUNCT
ejpam-6191	184	21	2β	2β	NOUN
ejpam-6191	184	22	;	;	PUNCT
ejpam-6191	184	23	1	1	X
ejpam-6191	184	24	)	)	PUNCT
ejpam-6191	184	25	in	in	ADP
ejpam-6191	184	26	(	(	PUNCT
ejpam-6191	184	27	3.1	3.1	NUM
ejpam-6191	184	28	)	)	PUNCT
ejpam-6191	184	29	,	,	PUNCT
ejpam-6191	184	30	respectively	respectively	ADV
ejpam-6191	184	31	where	where	SCONJ
ejpam-6191	184	32	α	α	NOUN
ejpam-6191	184	33	=	=	NOUN
ejpam-6191	184	34	1	1	NUM
ejpam-6191	184	35	+	+	CCONJ
ejpam-6191	184	36	√	√	NUM
ejpam-6191	184	37	2	2	NUM
ejpam-6191	184	38	and	and	CCONJ
ejpam-6191	184	39	β	β	X
ejpam-6191	184	40	=	=	SYM
ejpam-6191	184	41	1−	1−	NUM
ejpam-6191	184	42	√	√	NUM
ejpam-6191	184	43	2	2	NUM
ejpam-6191	184	44	,	,	PUNCT
ejpam-6191	184	45	and	and	CCONJ
ejpam-6191	184	46	applying	apply	VERB
ejpam-6191	184	47	the	the	DET
ejpam-6191	184	48	relations	relation	NOUN
ejpam-6191	184	49	,	,	PUNCT
ejpam-6191	184	50	α2	α2	PROPN
ejpam-6191	184	51	=	=	SYM
ejpam-6191	184	52	2α+	2α+	NUM
ejpam-6191	184	53	1	1	NUM
ejpam-6191	184	54	and	and	CCONJ
ejpam-6191	184	55	β2	β2	NOUN
ejpam-6191	184	56	=	=	NOUN
ejpam-6191	184	57	2β	2β	NOUN
ejpam-6191	184	58	+	+	CCONJ
ejpam-6191	184	59	1	1	NUM
ejpam-6191	184	60	.	.	X
ejpam-6191	184	61	corollary	corollary	ADJ
ejpam-6191	184	62	3.5	3.5	NUM
ejpam-6191	184	63	.	.	PUNCT
ejpam-6191	185	1	let	let	VERB
ejpam-6191	185	2	pn	pn	PART
ejpam-6191	185	3	be	be	AUX
ejpam-6191	185	4	the	the	DET
ejpam-6191	185	5	pell	pell	NOUN
ejpam-6191	185	6	numbers	number	NOUN
ejpam-6191	185	7	and	and	CCONJ
ejpam-6191	185	8	qn	qn	NOUN
ejpam-6191	185	9	be	be	AUX
ejpam-6191	185	10	the	the	DET
ejpam-6191	185	11	pell	pell	NOUN
ejpam-6191	185	12	-	-	PUNCT
ejpam-6191	185	13	lucas	lucas	NOUN
ejpam-6191	185	14	numbers	number	NOUN
ejpam-6191	185	15	,	,	PUNCT
ejpam-6191	185	16	respectively	respectively	ADV
ejpam-6191	185	17	.	.	PUNCT
ejpam-6191	186	1	then	then	ADV
ejpam-6191	186	2	,	,	PUNCT
ejpam-6191	186	3	the	the	DET
ejpam-6191	186	4	following	follow	VERB
ejpam-6191	186	5	identity	identity	NOUN
ejpam-6191	186	6	holds	hold	VERB
ejpam-6191	186	7	n∑	n∑	NOUN
ejpam-6191	186	8	k=0	k=0	PROPN
ejpam-6191	186	9	(	(	PUNCT
ejpam-6191	186	10	n	n	X
ejpam-6191	186	11	k	k	NOUN
ejpam-6191	186	12	)	)	PUNCT
ejpam-6191	187	1	(	(	PUNCT
ejpam-6191	187	2	−1)n−kp2kh	−1)n−kp2kh	NOUN
ejpam-6191	187	3	(	(	PUNCT
ejpam-6191	187	4	r	r	NOUN
ejpam-6191	187	5	)	)	PUNCT
ejpam-6191	187	6	k	k	NOUN
ejpam-6191	187	7	=	=	SYM
ejpam-6191	187	8	r+1∑	r+1∑	PROPN
ejpam-6191	187	9	k=0	k=0	PROPN
ejpam-6191	187	10	n∑	n∑	PROPN
ejpam-6191	187	11	j=0	j=0	PROPN
ejpam-6191	187	12	(	(	PUNCT
ejpam-6191	187	13	−1)r+1−k+n−j	−1)r+1−k+n−j	X
ejpam-6191	187	14	(	(	PUNCT
ejpam-6191	187	15	r	r	NOUN
ejpam-6191	187	16	+	+	NOUN
ejpam-6191	187	17	1	1	NUM
ejpam-6191	187	18	k	k	NOUN
ejpam-6191	187	19	)	)	PUNCT
ejpam-6191	188	1	2jp2jh	2jp2jh	NUM
ejpam-6191	188	2	(	(	PUNCT
ejpam-6191	188	3	k−1	k−1	PROPN
ejpam-6191	188	4	)	)	PUNCT
ejpam-6191	188	5	j	j	PROPN
ejpam-6191	188	6	[	[	PUNCT
ejpam-6191	188	7	h	h	NOUN
ejpam-6191	188	8	(	(	PUNCT
ejpam-6191	188	9	r−k	r−k	PROPN
ejpam-6191	188	10	)	)	PUNCT
ejpam-6191	188	11	n−j	n−j	ADV
ejpam-6191	188	12	−h	−h	VERB
ejpam-6191	188	13	(	(	PUNCT
ejpam-6191	188	14	r−k	r−k	PROPN
ejpam-6191	188	15	)	)	PUNCT
ejpam-6191	188	16	n−1−j	n−1−j	NUM
ejpam-6191	188	17	]	]	PUNCT
ejpam-6191	188	18	.	.	PUNCT
ejpam-6191	189	1	k.	k.	PROPN
ejpam-6191	190	1	v.	v.	ADP
ejpam-6191	190	2	m.	m.	PROPN
ejpam-6191	190	3	manulat	manulat	PROPN
ejpam-6191	190	4	,	,	PUNCT
ejpam-6191	190	5	r.	r.	PROPN
ejpam-6191	190	6	b.	b.	PROPN
ejpam-6191	190	7	corcino	corcino	PROPN
ejpam-6191	190	8	/	/	SYM
ejpam-6191	190	9	eur	eur	PROPN
ejpam-6191	190	10	.	.	PUNCT
ejpam-6191	191	1	j.	j.	PROPN
ejpam-6191	191	2	pure	pure	PROPN
ejpam-6191	191	3	appl	appl	PROPN
ejpam-6191	191	4	.	.	PROPN
ejpam-6191	191	5	math	math	PROPN
ejpam-6191	191	6	,	,	PUNCT
ejpam-6191	191	7	18	18	NUM
ejpam-6191	191	8	(	(	PUNCT
ejpam-6191	191	9	3	3	NUM
ejpam-6191	191	10	)	)	PUNCT
ejpam-6191	191	11	(	(	PUNCT
ejpam-6191	191	12	2025	2025	NUM
ejpam-6191	191	13	)	)	PUNCT
ejpam-6191	191	14	,	,	PUNCT
ejpam-6191	191	15	6191	6191	NUM
ejpam-6191	191	16	10	10	NUM
ejpam-6191	191	17	of	of	ADP
ejpam-6191	191	18	21	21	NUM
ejpam-6191	191	19	andn∑	andn∑	PROPN
ejpam-6191	191	20	k=0	k=0	PROPN
ejpam-6191	191	21	(	(	PUNCT
ejpam-6191	191	22	n	n	X
ejpam-6191	191	23	k	k	NOUN
ejpam-6191	191	24	)	)	PUNCT
ejpam-6191	191	25	(	(	PUNCT
ejpam-6191	191	26	−1)n−kq2kh	−1)n−kq2kh	PROPN
ejpam-6191	191	27	(	(	PUNCT
ejpam-6191	191	28	r	r	NOUN
ejpam-6191	191	29	)	)	PUNCT
ejpam-6191	191	30	k	k	NOUN
ejpam-6191	192	1	=	=	SYM
ejpam-6191	192	2	r+1∑	r+1∑	PROPN
ejpam-6191	192	3	k=0	k=0	PROPN
ejpam-6191	192	4	n∑	n∑	PROPN
ejpam-6191	192	5	j=0	j=0	PROPN
ejpam-6191	192	6	(	(	PUNCT
ejpam-6191	192	7	−1)r+1−k+n−j	−1)r+1−k+n−j	X
ejpam-6191	192	8	(	(	PUNCT
ejpam-6191	192	9	r	r	NOUN
ejpam-6191	192	10	+	+	NOUN
ejpam-6191	192	11	1	1	NUM
ejpam-6191	192	12	k	k	NOUN
ejpam-6191	192	13	)	)	PUNCT
ejpam-6191	193	1	2jq2jh	2jq2jh	PROPN
ejpam-6191	193	2	(	(	PUNCT
ejpam-6191	193	3	k−1	k−1	PROPN
ejpam-6191	193	4	)	)	PUNCT
ejpam-6191	193	5	j	j	PROPN
ejpam-6191	193	6	[	[	PUNCT
ejpam-6191	193	7	h	h	NOUN
ejpam-6191	193	8	(	(	PUNCT
ejpam-6191	193	9	r−k	r−k	PROPN
ejpam-6191	193	10	)	)	PUNCT
ejpam-6191	193	11	n−j	n−j	ADV
ejpam-6191	193	12	−h	−h	VERB
ejpam-6191	193	13	(	(	PUNCT
ejpam-6191	193	14	r−k	r−k	PROPN
ejpam-6191	193	15	)	)	PUNCT
ejpam-6191	193	16	n−1−j	n−1−j	NUM
ejpam-6191	193	17	]	]	PUNCT
ejpam-6191	193	18	.	.	PUNCT
ejpam-6191	194	1	proof	proof	NOUN
ejpam-6191	194	2	.	.	PUNCT
ejpam-6191	195	1	setting	set	VERB
ejpam-6191	195	2	(	(	PUNCT
ejpam-6191	195	3	a	a	PRON
ejpam-6191	195	4	;	;	PUNCT
ejpam-6191	195	5	b	b	X
ejpam-6191	195	6	)	)	PUNCT
ejpam-6191	195	7	=	=	SYM
ejpam-6191	195	8	(	(	PUNCT
ejpam-6191	195	9	α2;−1	α2;−1	NOUN
ejpam-6191	195	10	)	)	PUNCT
ejpam-6191	195	11	and	and	CCONJ
ejpam-6191	195	12	(	(	PUNCT
ejpam-6191	195	13	a	a	PRON
ejpam-6191	195	14	;	;	PUNCT
ejpam-6191	195	15	b	b	X
ejpam-6191	195	16	)	)	PUNCT
ejpam-6191	195	17	=	=	SYM
ejpam-6191	195	18	(	(	PUNCT
ejpam-6191	195	19	β2;−1	β2;−1	NOUN
ejpam-6191	195	20	)	)	PUNCT
ejpam-6191	195	21	in	in	ADP
ejpam-6191	195	22	(	(	PUNCT
ejpam-6191	195	23	3.1	3.1	NUM
ejpam-6191	195	24	)	)	PUNCT
ejpam-6191	195	25	,	,	PUNCT
ejpam-6191	195	26	respectively	respectively	ADV
ejpam-6191	195	27	where	where	SCONJ
ejpam-6191	195	28	α	α	NOUN
ejpam-6191	195	29	=	=	NOUN
ejpam-6191	195	30	1	1	NUM
ejpam-6191	195	31	+	+	CCONJ
ejpam-6191	195	32	√	√	NUM
ejpam-6191	195	33	2	2	NUM
ejpam-6191	195	34	and	and	CCONJ
ejpam-6191	195	35	β	β	X
ejpam-6191	195	36	=	=	SYM
ejpam-6191	195	37	1−	1−	NUM
ejpam-6191	195	38	√	√	NUM
ejpam-6191	195	39	2	2	NUM
ejpam-6191	195	40	,	,	PUNCT
ejpam-6191	195	41	and	and	CCONJ
ejpam-6191	195	42	applying	apply	VERB
ejpam-6191	195	43	the	the	DET
ejpam-6191	195	44	relations	relation	NOUN
ejpam-6191	195	45	,	,	PUNCT
ejpam-6191	195	46	α2	α2	PROPN
ejpam-6191	195	47	=	=	SYM
ejpam-6191	195	48	2α+	2α+	NUM
ejpam-6191	195	49	1	1	NUM
ejpam-6191	195	50	and	and	CCONJ
ejpam-6191	195	51	β2	β2	NOUN
ejpam-6191	195	52	=	=	NOUN
ejpam-6191	195	53	2β	2β	NOUN
ejpam-6191	195	54	+	+	CCONJ
ejpam-6191	195	55	1	1	NUM
ejpam-6191	195	56	.	.	PUNCT
ejpam-6191	195	57	corollary	corollary	ADJ
ejpam-6191	195	58	3.6	3.6	NUM
ejpam-6191	195	59	.	.	PUNCT
ejpam-6191	196	1	let	let	VERB
ejpam-6191	196	2	jn	jn	PROPN
ejpam-6191	196	3	be	be	AUX
ejpam-6191	196	4	the	the	DET
ejpam-6191	196	5	jacobsthal	jacobsthal	ADJ
ejpam-6191	196	6	numbers	number	NOUN
ejpam-6191	196	7	and	and	CCONJ
ejpam-6191	196	8	tn	tn	NOUN
ejpam-6191	196	9	be	be	AUX
ejpam-6191	196	10	the	the	DET
ejpam-6191	196	11	jacobsthal	jacobsthal	ADJ
ejpam-6191	196	12	-	-	PUNCT
ejpam-6191	196	13	lucas	lucas	NOUN
ejpam-6191	196	14	numbers	number	NOUN
ejpam-6191	196	15	,	,	PUNCT
ejpam-6191	196	16	respectively	respectively	ADV
ejpam-6191	196	17	.	.	PUNCT
ejpam-6191	197	1	then	then	ADV
ejpam-6191	197	2	we	we	PRON
ejpam-6191	197	3	have	have	VERB
ejpam-6191	197	4	n∑	n∑	NOUN
ejpam-6191	197	5	k=0	k=0	PROPN
ejpam-6191	197	6	(	(	PUNCT
ejpam-6191	197	7	n	n	X
ejpam-6191	197	8	k	k	NOUN
ejpam-6191	197	9	)	)	PUNCT
ejpam-6191	197	10	2n−kjkh	2n−kjkh	NUM
ejpam-6191	197	11	(	(	PUNCT
ejpam-6191	197	12	r	r	NOUN
ejpam-6191	197	13	)	)	PUNCT
ejpam-6191	197	14	k	k	NOUN
ejpam-6191	197	15	=	=	SYM
ejpam-6191	197	16	r+1∑	r+1∑	PROPN
ejpam-6191	197	17	k=0	k=0	PROPN
ejpam-6191	197	18	n∑	n∑	PROPN
ejpam-6191	197	19	j=0	j=0	PROPN
ejpam-6191	197	20	(	(	PUNCT
ejpam-6191	197	21	−1)r+1−k	−1)r+1−k	X
ejpam-6191	197	22	(	(	PUNCT
ejpam-6191	197	23	r	r	NOUN
ejpam-6191	197	24	+	+	NOUN
ejpam-6191	197	25	1	1	NUM
ejpam-6191	197	26	k	k	NOUN
ejpam-6191	197	27	)	)	PUNCT
ejpam-6191	197	28	2n−jj2jh	2n−jj2jh	NUM
ejpam-6191	198	1	(	(	PUNCT
ejpam-6191	198	2	k−1	k−1	PROPN
ejpam-6191	198	3	)	)	PUNCT
ejpam-6191	198	4	j	j	PROPN
ejpam-6191	198	5	[	[	PUNCT
ejpam-6191	198	6	h	h	NOUN
ejpam-6191	198	7	(	(	PUNCT
ejpam-6191	198	8	r−k	r−k	PROPN
ejpam-6191	198	9	)	)	PUNCT
ejpam-6191	198	10	n−j	n−j	ADV
ejpam-6191	198	11	−h	−h	VERB
ejpam-6191	198	12	(	(	PUNCT
ejpam-6191	198	13	r−k	r−k	PROPN
ejpam-6191	198	14	)	)	PUNCT
ejpam-6191	198	15	n−1−j	n−1−j	NUM
ejpam-6191	198	16	]	]	PUNCT
ejpam-6191	198	17	.	.	PUNCT
ejpam-6191	199	1	and	and	CCONJ
ejpam-6191	199	2	,	,	PUNCT
ejpam-6191	199	3	n∑	n∑	PROPN
ejpam-6191	199	4	k=0	k=0	PROPN
ejpam-6191	199	5	(	(	PUNCT
ejpam-6191	199	6	n	n	X
ejpam-6191	199	7	k	k	NOUN
ejpam-6191	199	8	)	)	PUNCT
ejpam-6191	199	9	2n−ktkh	2n−ktkh	PROPN
ejpam-6191	199	10	(	(	PUNCT
ejpam-6191	199	11	r	r	NOUN
ejpam-6191	199	12	)	)	PUNCT
ejpam-6191	199	13	k	k	NOUN
ejpam-6191	199	14	=	=	SYM
ejpam-6191	199	15	r+1∑	r+1∑	PROPN
ejpam-6191	199	16	k=0	k=0	PROPN
ejpam-6191	199	17	n∑	n∑	PROPN
ejpam-6191	199	18	j=0	j=0	PROPN
ejpam-6191	199	19	(	(	PUNCT
ejpam-6191	199	20	−1)r+1−k	−1)r+1−k	X
ejpam-6191	199	21	(	(	PUNCT
ejpam-6191	199	22	r	r	NOUN
ejpam-6191	199	23	+	+	NOUN
ejpam-6191	199	24	1	1	NUM
ejpam-6191	199	25	k	k	NOUN
ejpam-6191	199	26	)	)	PUNCT
ejpam-6191	200	1	2n−jt2jh	2n−jt2jh	NUM
ejpam-6191	200	2	(	(	PUNCT
ejpam-6191	200	3	k−1	k−1	PROPN
ejpam-6191	200	4	)	)	PUNCT
ejpam-6191	200	5	j	j	PROPN
ejpam-6191	200	6	[	[	PUNCT
ejpam-6191	200	7	h	h	NOUN
ejpam-6191	200	8	(	(	PUNCT
ejpam-6191	200	9	r−k	r−k	PROPN
ejpam-6191	200	10	)	)	PUNCT
ejpam-6191	200	11	n−j	n−j	ADV
ejpam-6191	200	12	−h	−h	VERB
ejpam-6191	200	13	(	(	PUNCT
ejpam-6191	200	14	r−k	r−k	PROPN
ejpam-6191	200	15	)	)	PUNCT
ejpam-6191	200	16	n−1−j	n−1−j	NUM
ejpam-6191	200	17	]	]	PUNCT
ejpam-6191	200	18	.	.	PUNCT
ejpam-6191	201	1	proof	proof	NOUN
ejpam-6191	201	2	.	.	PUNCT
ejpam-6191	202	1	setting	set	VERB
ejpam-6191	202	2	(	(	PUNCT
ejpam-6191	202	3	a	a	PRON
ejpam-6191	202	4	;	;	PUNCT
ejpam-6191	202	5	b	b	X
ejpam-6191	202	6	)	)	PUNCT
ejpam-6191	202	7	=	=	SYM
ejpam-6191	202	8	(	(	PUNCT
ejpam-6191	202	9	α	α	NOUN
ejpam-6191	202	10	;	;	PUNCT
ejpam-6191	202	11	2	2	NUM
ejpam-6191	202	12	)	)	PUNCT
ejpam-6191	202	13	and	and	CCONJ
ejpam-6191	202	14	(	(	PUNCT
ejpam-6191	202	15	a	a	PRON
ejpam-6191	202	16	;	;	PUNCT
ejpam-6191	202	17	b	b	X
ejpam-6191	202	18	)	)	PUNCT
ejpam-6191	202	19	=	=	SYM
ejpam-6191	202	20	(	(	PUNCT
ejpam-6191	202	21	β	β	NOUN
ejpam-6191	202	22	;	;	PUNCT
ejpam-6191	202	23	2	2	X
ejpam-6191	202	24	)	)	PUNCT
ejpam-6191	202	25	in	in	ADP
ejpam-6191	202	26	(	(	PUNCT
ejpam-6191	202	27	3.1	3.1	NUM
ejpam-6191	202	28	)	)	PUNCT
ejpam-6191	202	29	,	,	PUNCT
ejpam-6191	202	30	respectively	respectively	ADV
ejpam-6191	202	31	where	where	SCONJ
ejpam-6191	202	32	α	α	PROPN
ejpam-6191	202	33	=	=	SYM
ejpam-6191	202	34	2	2	NUM
ejpam-6191	202	35	and	and	CCONJ
ejpam-6191	202	36	β	β	X
ejpam-6191	202	37	=	=	SYM
ejpam-6191	202	38	−1	−1	NOUN
ejpam-6191	202	39	,	,	PUNCT
ejpam-6191	202	40	and	and	CCONJ
ejpam-6191	202	41	applying	apply	VERB
ejpam-6191	202	42	the	the	DET
ejpam-6191	202	43	relations	relation	NOUN
ejpam-6191	202	44	,	,	PUNCT
ejpam-6191	202	45	α2	α2	PROPN
ejpam-6191	202	46	=	=	SYM
ejpam-6191	202	47	α+	α+	PUNCT
ejpam-6191	202	48	2	2	NUM
ejpam-6191	202	49	and	and	CCONJ
ejpam-6191	202	50	β2	β2	NOUN
ejpam-6191	202	51	=	=	SYM
ejpam-6191	202	52	β	β	PROPN
ejpam-6191	202	53	+	+	ADJ
ejpam-6191	202	54	2	2	X
ejpam-6191	202	55	.	.	X
ejpam-6191	202	56	corollary	corollary	ADJ
ejpam-6191	202	57	3.7	3.7	NUM
ejpam-6191	202	58	.	.	PUNCT
ejpam-6191	203	1	let	let	VERB
ejpam-6191	203	2	jn	jn	PROPN
ejpam-6191	203	3	be	be	AUX
ejpam-6191	203	4	the	the	DET
ejpam-6191	203	5	jacobsthal	jacobsthal	ADJ
ejpam-6191	203	6	numbers	number	NOUN
ejpam-6191	203	7	and	and	CCONJ
ejpam-6191	203	8	tn	tn	NOUN
ejpam-6191	203	9	be	be	AUX
ejpam-6191	203	10	the	the	DET
ejpam-6191	203	11	jacobsthal	jacobsthal	ADJ
ejpam-6191	203	12	-	-	PUNCT
ejpam-6191	203	13	lucas	lucas	NOUN
ejpam-6191	203	14	numbers	number	NOUN
ejpam-6191	203	15	,	,	PUNCT
ejpam-6191	203	16	respectively	respectively	ADV
ejpam-6191	203	17	.	.	PUNCT
ejpam-6191	204	1	then	then	ADV
ejpam-6191	204	2	,	,	PUNCT
ejpam-6191	204	3	the	the	DET
ejpam-6191	204	4	following	follow	VERB
ejpam-6191	204	5	identities	identity	NOUN
ejpam-6191	204	6	hold	hold	VERB
ejpam-6191	204	7	n∑	n∑	NOUN
ejpam-6191	204	8	k=0	k=0	PROPN
ejpam-6191	204	9	(	(	PUNCT
ejpam-6191	204	10	n	n	X
ejpam-6191	204	11	k	k	NOUN
ejpam-6191	204	12	)	)	PUNCT
ejpam-6191	204	13	(	(	PUNCT
ejpam-6191	204	14	−2)n−kj2kh	−2)n−kj2kh	PROPN
ejpam-6191	204	15	(	(	PUNCT
ejpam-6191	204	16	r	r	NOUN
ejpam-6191	204	17	)	)	PUNCT
ejpam-6191	204	18	k	k	NOUN
ejpam-6191	205	1	=	=	SYM
ejpam-6191	205	2	r+1∑	r+1∑	PROPN
ejpam-6191	205	3	k=0	k=0	PROPN
ejpam-6191	205	4	n∑	n∑	PROPN
ejpam-6191	205	5	j=0	j=0	PROPN
ejpam-6191	205	6	(	(	PUNCT
ejpam-6191	205	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	205	8	(	(	PUNCT
ejpam-6191	205	9	r	r	NOUN
ejpam-6191	205	10	+	+	NOUN
ejpam-6191	205	11	1	1	NUM
ejpam-6191	205	12	k	k	NOUN
ejpam-6191	205	13	)	)	PUNCT
ejpam-6191	205	14	(	(	PUNCT
ejpam-6191	205	15	−2)n−jjjh	−2)n−jjjh	PROPN
ejpam-6191	206	1	(	(	PUNCT
ejpam-6191	206	2	k−1	k−1	PROPN
ejpam-6191	206	3	)	)	PUNCT
ejpam-6191	206	4	j	j	PROPN
ejpam-6191	206	5	[	[	PUNCT
ejpam-6191	206	6	h	h	NOUN
ejpam-6191	206	7	(	(	PUNCT
ejpam-6191	206	8	r−k	r−k	PROPN
ejpam-6191	206	9	)	)	PUNCT
ejpam-6191	206	10	n−j	n−j	ADV
ejpam-6191	206	11	−h	−h	VERB
ejpam-6191	206	12	(	(	PUNCT
ejpam-6191	206	13	r−k	r−k	PROPN
ejpam-6191	206	14	)	)	PUNCT
ejpam-6191	206	15	n−1−j	n−1−j	NUM
ejpam-6191	206	16	]	]	PUNCT
ejpam-6191	206	17	.	.	PUNCT
ejpam-6191	207	1	and	and	CCONJ
ejpam-6191	207	2	,	,	PUNCT
ejpam-6191	207	3	n∑	n∑	PROPN
ejpam-6191	207	4	k=0	k=0	PROPN
ejpam-6191	207	5	(	(	PUNCT
ejpam-6191	207	6	n	n	X
ejpam-6191	207	7	k	k	NOUN
ejpam-6191	207	8	)	)	PUNCT
ejpam-6191	207	9	(	(	PUNCT
ejpam-6191	207	10	−2)n−kt2kh	−2)n−kt2kh	PROPN
ejpam-6191	207	11	(	(	PUNCT
ejpam-6191	207	12	r	r	NOUN
ejpam-6191	207	13	)	)	PUNCT
ejpam-6191	207	14	k	k	NOUN
ejpam-6191	207	15	=	=	SYM
ejpam-6191	207	16	r+1∑	r+1∑	PROPN
ejpam-6191	207	17	k=0	k=0	PROPN
ejpam-6191	207	18	n∑	n∑	PROPN
ejpam-6191	207	19	j=0	j=0	PROPN
ejpam-6191	207	20	(	(	PUNCT
ejpam-6191	207	21	−1)r+1−k	−1)r+1−k	X
ejpam-6191	207	22	(	(	PUNCT
ejpam-6191	207	23	r	r	NOUN
ejpam-6191	207	24	+	+	NOUN
ejpam-6191	207	25	1	1	NUM
ejpam-6191	207	26	k	k	NOUN
ejpam-6191	207	27	)	)	PUNCT
ejpam-6191	207	28	(	(	PUNCT
ejpam-6191	207	29	−2)n−jtjh	−2)n−jtjh	PROPN
ejpam-6191	207	30	(	(	PUNCT
ejpam-6191	207	31	k−1	k−1	PROPN
ejpam-6191	207	32	)	)	PUNCT
ejpam-6191	207	33	j	j	PROPN
ejpam-6191	207	34	[	[	PUNCT
ejpam-6191	207	35	h	h	NOUN
ejpam-6191	207	36	(	(	PUNCT
ejpam-6191	207	37	r−k	r−k	PROPN
ejpam-6191	207	38	)	)	PUNCT
ejpam-6191	207	39	n−j	n−j	ADV
ejpam-6191	207	40	−h	−h	VERB
ejpam-6191	207	41	(	(	PUNCT
ejpam-6191	207	42	r−k	r−k	PROPN
ejpam-6191	207	43	)	)	PUNCT
ejpam-6191	207	44	n−1−j	n−1−j	NUM
ejpam-6191	207	45	]	]	PUNCT
ejpam-6191	207	46	.	.	PUNCT
ejpam-6191	208	1	proof	proof	NOUN
ejpam-6191	208	2	.	.	PUNCT
ejpam-6191	209	1	setting	set	VERB
ejpam-6191	209	2	(	(	PUNCT
ejpam-6191	209	3	a	a	PRON
ejpam-6191	209	4	;	;	PUNCT
ejpam-6191	209	5	b	b	X
ejpam-6191	209	6	)	)	PUNCT
ejpam-6191	209	7	=	=	SYM
ejpam-6191	209	8	(	(	PUNCT
ejpam-6191	209	9	α2;−2	α2;−2	PROPN
ejpam-6191	209	10	)	)	PUNCT
ejpam-6191	209	11	and	and	CCONJ
ejpam-6191	209	12	(	(	PUNCT
ejpam-6191	209	13	a	a	PRON
ejpam-6191	209	14	;	;	PUNCT
ejpam-6191	209	15	b	b	X
ejpam-6191	209	16	)	)	PUNCT
ejpam-6191	209	17	=	=	SYM
ejpam-6191	209	18	(	(	PUNCT
ejpam-6191	209	19	β2;−2	β2;−2	PROPN
ejpam-6191	209	20	)	)	PUNCT
ejpam-6191	209	21	in	in	ADP
ejpam-6191	209	22	(	(	PUNCT
ejpam-6191	209	23	3.1	3.1	NUM
ejpam-6191	209	24	)	)	PUNCT
ejpam-6191	209	25	,	,	PUNCT
ejpam-6191	209	26	respectively	respectively	ADV
ejpam-6191	209	27	where	where	SCONJ
ejpam-6191	209	28	α	α	PROPN
ejpam-6191	209	29	=	=	SYM
ejpam-6191	209	30	2	2	NUM
ejpam-6191	209	31	and	and	CCONJ
ejpam-6191	209	32	β	β	X
ejpam-6191	209	33	=	=	SYM
ejpam-6191	209	34	−1	−1	NOUN
ejpam-6191	209	35	,	,	PUNCT
ejpam-6191	209	36	and	and	CCONJ
ejpam-6191	209	37	applying	apply	VERB
ejpam-6191	209	38	the	the	DET
ejpam-6191	209	39	relations	relation	NOUN
ejpam-6191	209	40	,	,	PUNCT
ejpam-6191	209	41	α2	α2	PROPN
ejpam-6191	209	42	=	=	SYM
ejpam-6191	209	43	α+	α+	PUNCT
ejpam-6191	209	44	2	2	NUM
ejpam-6191	209	45	and	and	CCONJ
ejpam-6191	209	46	β2	β2	NOUN
ejpam-6191	209	47	=	=	SYM
ejpam-6191	209	48	β	β	PROPN
ejpam-6191	209	49	+	+	ADJ
ejpam-6191	209	50	2	2	X
ejpam-6191	209	51	.	.	X
ejpam-6191	209	52	corollary	corollary	ADJ
ejpam-6191	209	53	3.8	3.8	NUM
ejpam-6191	209	54	.	.	PUNCT
ejpam-6191	210	1	let	let	VERB
ejpam-6191	210	2	mn	mn	PROPN
ejpam-6191	210	3	be	be	AUX
ejpam-6191	210	4	the	the	DET
ejpam-6191	210	5	mersenne	mersenne	NOUN
ejpam-6191	210	6	numbers	number	NOUN
ejpam-6191	210	7	and	and	CCONJ
ejpam-6191	210	8	rn	rn	AUX
ejpam-6191	210	9	be	be	AUX
ejpam-6191	210	10	the	the	DET
ejpam-6191	210	11	mersenne	mersenne	NOUN
ejpam-6191	210	12	-	-	PUNCT
ejpam-6191	210	13	lucas	lucas	PROPN
ejpam-6191	210	14	numbers	number	NOUN
ejpam-6191	210	15	,	,	PUNCT
ejpam-6191	210	16	respectively	respectively	ADV
ejpam-6191	210	17	.	.	PUNCT
ejpam-6191	211	1	then	then	ADV
ejpam-6191	211	2	we	we	PRON
ejpam-6191	211	3	have	have	VERB
ejpam-6191	211	4	k.	k.	PROPN
ejpam-6191	211	5	v.	v.	ADP
ejpam-6191	211	6	m.	m.	PROPN
ejpam-6191	211	7	manulat	manulat	PROPN
ejpam-6191	211	8	,	,	PUNCT
ejpam-6191	211	9	r.	r.	PROPN
ejpam-6191	211	10	b.	b.	PROPN
ejpam-6191	211	11	corcino	corcino	PROPN
ejpam-6191	211	12	/	/	SYM
ejpam-6191	211	13	eur	eur	PROPN
ejpam-6191	211	14	.	.	PUNCT
ejpam-6191	212	1	j.	j.	PROPN
ejpam-6191	212	2	pure	pure	PROPN
ejpam-6191	212	3	appl	appl	PROPN
ejpam-6191	212	4	.	.	PROPN
ejpam-6191	212	5	math	math	PROPN
ejpam-6191	212	6	,	,	PUNCT
ejpam-6191	212	7	18	18	NUM
ejpam-6191	212	8	(	(	PUNCT
ejpam-6191	212	9	3	3	NUM
ejpam-6191	212	10	)	)	PUNCT
ejpam-6191	212	11	(	(	PUNCT
ejpam-6191	212	12	2025	2025	NUM
ejpam-6191	212	13	)	)	PUNCT
ejpam-6191	212	14	,	,	PUNCT
ejpam-6191	212	15	6191	6191	NUM
ejpam-6191	212	16	11	11	NUM
ejpam-6191	212	17	of	of	ADP
ejpam-6191	212	18	21	21	NUM
ejpam-6191	212	19	‘	'	PUNCT
ejpam-6191	212	20	n∑	n∑	X
ejpam-6191	212	21	k=0	k=0	PROPN
ejpam-6191	212	22	(	(	PUNCT
ejpam-6191	212	23	n	n	X
ejpam-6191	212	24	k	k	PROPN
ejpam-6191	212	25	)	)	PUNCT
ejpam-6191	212	26	3k(−2)n−kmkh	3k(−2)n−kmkh	NUM
ejpam-6191	212	27	(	(	PUNCT
ejpam-6191	212	28	r	r	NOUN
ejpam-6191	212	29	)	)	PUNCT
ejpam-6191	212	30	k	k	NOUN
ejpam-6191	212	31	=	=	SYM
ejpam-6191	212	32	r+1∑	r+1∑	PROPN
ejpam-6191	212	33	k=0	k=0	PROPN
ejpam-6191	212	34	n∑	n∑	PROPN
ejpam-6191	212	35	j=0	j=0	PROPN
ejpam-6191	212	36	(	(	PUNCT
ejpam-6191	212	37	−1)r+1−k	−1)r+1−k	X
ejpam-6191	212	38	(	(	PUNCT
ejpam-6191	212	39	r	r	NOUN
ejpam-6191	212	40	+	+	NOUN
ejpam-6191	212	41	1	1	NUM
ejpam-6191	212	42	k	k	NOUN
ejpam-6191	212	43	)	)	PUNCT
ejpam-6191	212	44	(	(	PUNCT
ejpam-6191	212	45	−2)n−jm2jh	−2)n−jm2jh	PROPN
ejpam-6191	212	46	(	(	PUNCT
ejpam-6191	212	47	k−1	k−1	PROPN
ejpam-6191	212	48	)	)	PUNCT
ejpam-6191	212	49	j	j	PROPN
ejpam-6191	212	50	[	[	PUNCT
ejpam-6191	212	51	h	h	NOUN
ejpam-6191	212	52	(	(	PUNCT
ejpam-6191	212	53	r−k	r−k	PROPN
ejpam-6191	212	54	)	)	PUNCT
ejpam-6191	212	55	n−j	n−j	ADV
ejpam-6191	212	56	−h	−h	VERB
ejpam-6191	212	57	(	(	PUNCT
ejpam-6191	212	58	r−k	r−k	PROPN
ejpam-6191	212	59	)	)	PUNCT
ejpam-6191	212	60	n−1−j	n−1−j	NUM
ejpam-6191	212	61	]	]	PUNCT
ejpam-6191	212	62	.	.	PUNCT
ejpam-6191	213	1	and	and	CCONJ
ejpam-6191	213	2	,	,	PUNCT
ejpam-6191	213	3	n∑	n∑	PROPN
ejpam-6191	213	4	k=0	k=0	PROPN
ejpam-6191	213	5	(	(	PUNCT
ejpam-6191	213	6	n	n	X
ejpam-6191	213	7	k	k	PROPN
ejpam-6191	213	8	)	)	PUNCT
ejpam-6191	213	9	3k(−2)n−krkh	3k(−2)n−krkh	NUM
ejpam-6191	213	10	(	(	PUNCT
ejpam-6191	213	11	r	r	NOUN
ejpam-6191	213	12	)	)	PUNCT
ejpam-6191	213	13	k	k	NOUN
ejpam-6191	214	1	=	=	SYM
ejpam-6191	214	2	r+1∑	r+1∑	PROPN
ejpam-6191	214	3	k=0	k=0	PROPN
ejpam-6191	214	4	n∑	n∑	PROPN
ejpam-6191	214	5	j=0	j=0	PROPN
ejpam-6191	214	6	(	(	PUNCT
ejpam-6191	214	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	214	8	(	(	PUNCT
ejpam-6191	214	9	r	r	NOUN
ejpam-6191	214	10	+	+	NOUN
ejpam-6191	214	11	1	1	NUM
ejpam-6191	214	12	k	k	NOUN
ejpam-6191	214	13	)	)	PUNCT
ejpam-6191	214	14	(	(	PUNCT
ejpam-6191	214	15	−2)n−jr2jh	−2)n−jr2jh	PROPN
ejpam-6191	214	16	(	(	PUNCT
ejpam-6191	214	17	k−1	k−1	PROPN
ejpam-6191	214	18	)	)	PUNCT
ejpam-6191	214	19	j	j	PROPN
ejpam-6191	214	20	[	[	PUNCT
ejpam-6191	214	21	h	h	NOUN
ejpam-6191	214	22	(	(	PUNCT
ejpam-6191	214	23	r−k	r−k	PROPN
ejpam-6191	214	24	)	)	PUNCT
ejpam-6191	214	25	n−j	n−j	ADV
ejpam-6191	214	26	−h	−h	VERB
ejpam-6191	214	27	(	(	PUNCT
ejpam-6191	214	28	r−k	r−k	PROPN
ejpam-6191	214	29	)	)	PUNCT
ejpam-6191	214	30	n−1−j	n−1−j	NUM
ejpam-6191	214	31	]	]	PUNCT
ejpam-6191	214	32	.	.	PUNCT
ejpam-6191	215	1	proof	proof	NOUN
ejpam-6191	215	2	.	.	PUNCT
ejpam-6191	216	1	setting	set	VERB
ejpam-6191	216	2	(	(	PUNCT
ejpam-6191	216	3	a	a	PRON
ejpam-6191	216	4	;	;	PUNCT
ejpam-6191	216	5	b	b	X
ejpam-6191	216	6	)	)	PUNCT
ejpam-6191	216	7	=	=	SYM
ejpam-6191	216	8	(	(	PUNCT
ejpam-6191	216	9	3α;−2	3α;−2	PROPN
ejpam-6191	216	10	)	)	PUNCT
ejpam-6191	216	11	and	and	CCONJ
ejpam-6191	216	12	(	(	PUNCT
ejpam-6191	216	13	a	a	PRON
ejpam-6191	216	14	;	;	PUNCT
ejpam-6191	216	15	b	b	X
ejpam-6191	216	16	)	)	PUNCT
ejpam-6191	216	17	=	=	SYM
ejpam-6191	216	18	(	(	PUNCT
ejpam-6191	216	19	3β;−2	3β;−2	NUM
ejpam-6191	216	20	)	)	PUNCT
ejpam-6191	216	21	in	in	ADP
ejpam-6191	216	22	(	(	PUNCT
ejpam-6191	216	23	3.1	3.1	NUM
ejpam-6191	216	24	)	)	PUNCT
ejpam-6191	216	25	,	,	PUNCT
ejpam-6191	216	26	respectively	respectively	ADV
ejpam-6191	216	27	where	where	SCONJ
ejpam-6191	216	28	α	α	PROPN
ejpam-6191	216	29	=	=	SYM
ejpam-6191	216	30	2	2	NUM
ejpam-6191	216	31	and	and	CCONJ
ejpam-6191	216	32	β	β	X
ejpam-6191	216	33	=	=	SYM
ejpam-6191	216	34	1	1	NUM
ejpam-6191	216	35	,	,	PUNCT
ejpam-6191	216	36	and	and	CCONJ
ejpam-6191	216	37	applying	apply	VERB
ejpam-6191	216	38	the	the	DET
ejpam-6191	216	39	relations	relation	NOUN
ejpam-6191	216	40	,	,	PUNCT
ejpam-6191	216	41	α2	α2	PROPN
ejpam-6191	216	42	=	=	SYM
ejpam-6191	216	43	3α−	3α−	NUM
ejpam-6191	216	44	2	2	NUM
ejpam-6191	216	45	and	and	CCONJ
ejpam-6191	216	46	β2	β2	NOUN
ejpam-6191	216	47	=	=	PROPN
ejpam-6191	216	48	3β	3β	NUM
ejpam-6191	216	49	−	−	PROPN
ejpam-6191	216	50	2	2	X
ejpam-6191	216	51	.	.	PUNCT
ejpam-6191	216	52	corollary	corollary	ADJ
ejpam-6191	216	53	3.9	3.9	NUM
ejpam-6191	216	54	.	.	PUNCT
ejpam-6191	217	1	let	let	VERB
ejpam-6191	217	2	mn	mn	PROPN
ejpam-6191	217	3	be	be	AUX
ejpam-6191	217	4	the	the	DET
ejpam-6191	217	5	mersenne	mersenne	NOUN
ejpam-6191	217	6	numbers	number	NOUN
ejpam-6191	217	7	and	and	CCONJ
ejpam-6191	217	8	rn	rn	AUX
ejpam-6191	217	9	be	be	AUX
ejpam-6191	217	10	the	the	DET
ejpam-6191	217	11	mersenne	mersenne	NOUN
ejpam-6191	217	12	-	-	PUNCT
ejpam-6191	217	13	lucas	lucas	PROPN
ejpam-6191	217	14	numbers	number	NOUN
ejpam-6191	217	15	,	,	PUNCT
ejpam-6191	217	16	respectively	respectively	ADV
ejpam-6191	217	17	.	.	PUNCT
ejpam-6191	218	1	then	then	ADV
ejpam-6191	218	2	,	,	PUNCT
ejpam-6191	218	3	the	the	DET
ejpam-6191	218	4	following	follow	VERB
ejpam-6191	218	5	identities	identity	NOUN
ejpam-6191	218	6	hold	hold	VERB
ejpam-6191	218	7	n∑	n∑	NOUN
ejpam-6191	218	8	k=0	k=0	PROPN
ejpam-6191	218	9	(	(	PUNCT
ejpam-6191	218	10	n	n	X
ejpam-6191	218	11	k	k	PROPN
ejpam-6191	218	12	)	)	PUNCT
ejpam-6191	218	13	2n−kmkh	2n−kmkh	NUM
ejpam-6191	219	1	(	(	PUNCT
ejpam-6191	219	2	r	r	NOUN
ejpam-6191	219	3	)	)	PUNCT
ejpam-6191	219	4	k	k	NOUN
ejpam-6191	219	5	=	=	SYM
ejpam-6191	219	6	r+1∑	r+1∑	PROPN
ejpam-6191	219	7	k=0	k=0	PROPN
ejpam-6191	219	8	n∑	n∑	PROPN
ejpam-6191	219	9	j=0	j=0	PROPN
ejpam-6191	219	10	(	(	PUNCT
ejpam-6191	219	11	−1)r+1−k	−1)r+1−k	X
ejpam-6191	219	12	(	(	PUNCT
ejpam-6191	219	13	r	r	NOUN
ejpam-6191	219	14	+	+	NOUN
ejpam-6191	219	15	1	1	NUM
ejpam-6191	219	16	k	k	NOUN
ejpam-6191	219	17	)	)	PUNCT
ejpam-6191	219	18	2n−j3jmjh	2n−j3jmjh	NUM
ejpam-6191	220	1	(	(	PUNCT
ejpam-6191	220	2	k−1	k−1	PROPN
ejpam-6191	220	3	)	)	PUNCT
ejpam-6191	220	4	j	j	PROPN
ejpam-6191	220	5	[	[	PUNCT
ejpam-6191	220	6	h	h	NOUN
ejpam-6191	220	7	(	(	PUNCT
ejpam-6191	220	8	r−k	r−k	PROPN
ejpam-6191	220	9	)	)	PUNCT
ejpam-6191	220	10	n−j	n−j	ADV
ejpam-6191	220	11	−h	−h	VERB
ejpam-6191	220	12	(	(	PUNCT
ejpam-6191	220	13	r−k	r−k	PROPN
ejpam-6191	220	14	)	)	PUNCT
ejpam-6191	220	15	n−1−j	n−1−j	NUM
ejpam-6191	220	16	]	]	PUNCT
ejpam-6191	220	17	.	.	PUNCT
ejpam-6191	221	1	and	and	CCONJ
ejpam-6191	221	2	,	,	PUNCT
ejpam-6191	221	3	n∑	n∑	PROPN
ejpam-6191	221	4	k=0	k=0	PROPN
ejpam-6191	221	5	(	(	PUNCT
ejpam-6191	221	6	n	n	X
ejpam-6191	221	7	k	k	NOUN
ejpam-6191	221	8	)	)	PUNCT
ejpam-6191	221	9	2n−krkh	2n−krkh	PROPN
ejpam-6191	221	10	(	(	PUNCT
ejpam-6191	221	11	r	r	NOUN
ejpam-6191	221	12	)	)	PUNCT
ejpam-6191	221	13	k	k	NOUN
ejpam-6191	221	14	=	=	SYM
ejpam-6191	221	15	r+1∑	r+1∑	PROPN
ejpam-6191	221	16	k=0	k=0	PROPN
ejpam-6191	221	17	n∑	n∑	PROPN
ejpam-6191	221	18	j=0	j=0	PROPN
ejpam-6191	221	19	(	(	PUNCT
ejpam-6191	221	20	−1)r+1−k	−1)r+1−k	X
ejpam-6191	221	21	(	(	PUNCT
ejpam-6191	221	22	r	r	NOUN
ejpam-6191	221	23	+	+	NOUN
ejpam-6191	221	24	1	1	NUM
ejpam-6191	221	25	k	k	NOUN
ejpam-6191	221	26	)	)	PUNCT
ejpam-6191	222	1	2n−j3jrjh	2n−j3jrjh	NUM
ejpam-6191	222	2	(	(	PUNCT
ejpam-6191	222	3	k−1	k−1	PROPN
ejpam-6191	222	4	)	)	PUNCT
ejpam-6191	222	5	j	j	PROPN
ejpam-6191	223	1	[	[	PUNCT
ejpam-6191	223	2	h	h	NOUN
ejpam-6191	223	3	(	(	PUNCT
ejpam-6191	223	4	r−k	r−k	PROPN
ejpam-6191	223	5	)	)	PUNCT
ejpam-6191	223	6	n−j	n−j	ADV
ejpam-6191	223	7	−h	−h	VERB
ejpam-6191	223	8	(	(	PUNCT
ejpam-6191	223	9	r−k	r−k	PROPN
ejpam-6191	223	10	)	)	PUNCT
ejpam-6191	223	11	n−1−j	n−1−j	NUM
ejpam-6191	223	12	]	]	PUNCT
ejpam-6191	223	13	.	.	PUNCT
ejpam-6191	224	1	proof	proof	NOUN
ejpam-6191	224	2	.	.	PUNCT
ejpam-6191	225	1	setting	set	VERB
ejpam-6191	225	2	(	(	PUNCT
ejpam-6191	225	3	a	a	PRON
ejpam-6191	225	4	;	;	PUNCT
ejpam-6191	225	5	b	b	X
ejpam-6191	225	6	)	)	PUNCT
ejpam-6191	225	7	=	=	SYM
ejpam-6191	225	8	(	(	PUNCT
ejpam-6191	225	9	α2	α2	ADJ
ejpam-6191	225	10	;	;	PUNCT
ejpam-6191	225	11	2	2	X
ejpam-6191	225	12	)	)	PUNCT
ejpam-6191	225	13	and	and	CCONJ
ejpam-6191	225	14	(	(	PUNCT
ejpam-6191	225	15	a	a	PRON
ejpam-6191	225	16	;	;	PUNCT
ejpam-6191	225	17	b	b	X
ejpam-6191	225	18	)	)	PUNCT
ejpam-6191	225	19	=	=	SYM
ejpam-6191	225	20	(	(	PUNCT
ejpam-6191	225	21	β2	β2	VERB
ejpam-6191	225	22	;	;	PUNCT
ejpam-6191	225	23	2	2	X
ejpam-6191	225	24	)	)	PUNCT
ejpam-6191	225	25	in	in	ADP
ejpam-6191	225	26	(	(	PUNCT
ejpam-6191	225	27	3.1	3.1	NUM
ejpam-6191	225	28	)	)	PUNCT
ejpam-6191	225	29	,	,	PUNCT
ejpam-6191	225	30	respectively	respectively	ADV
ejpam-6191	225	31	where	where	SCONJ
ejpam-6191	225	32	α	α	PROPN
ejpam-6191	225	33	=	=	SYM
ejpam-6191	225	34	2	2	NUM
ejpam-6191	225	35	and	and	CCONJ
ejpam-6191	225	36	β	β	X
ejpam-6191	225	37	=	=	SYM
ejpam-6191	225	38	1	1	NUM
ejpam-6191	225	39	,	,	PUNCT
ejpam-6191	225	40	and	and	CCONJ
ejpam-6191	225	41	applying	apply	VERB
ejpam-6191	225	42	the	the	DET
ejpam-6191	225	43	relations	relation	NOUN
ejpam-6191	225	44	,	,	PUNCT
ejpam-6191	225	45	α2	α2	PROPN
ejpam-6191	225	46	=	=	SYM
ejpam-6191	225	47	3α−	3α−	NUM
ejpam-6191	225	48	2	2	NUM
ejpam-6191	225	49	and	and	CCONJ
ejpam-6191	225	50	β2	β2	NOUN
ejpam-6191	225	51	=	=	PROPN
ejpam-6191	225	52	3β	3β	NUM
ejpam-6191	225	53	−	−	PROPN
ejpam-6191	225	54	2	2	X
ejpam-6191	225	55	.	.	PUNCT
ejpam-6191	226	1	the	the	DET
ejpam-6191	226	2	following	follow	VERB
ejpam-6191	226	3	theorem	theorem	NOUN
ejpam-6191	226	4	contains	contain	VERB
ejpam-6191	226	5	the	the	DET
ejpam-6191	226	6	generalized	generalized	ADJ
ejpam-6191	226	7	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	226	8	sum	sum	NOUN
ejpam-6191	226	9	with	with	ADP
ejpam-6191	226	10	integer	integer	NOUN
ejpam-6191	226	11	powers	power	NOUN
ejpam-6191	226	12	.	.	PUNCT
ejpam-6191	227	1	theorem	theorem	VERB
ejpam-6191	227	2	3.10	3.10	NUM
ejpam-6191	227	3	.	.	PUNCT
ejpam-6191	228	1	for	for	ADP
ejpam-6191	228	2	all	all	DET
ejpam-6191	228	3	n	n	PRON
ejpam-6191	228	4	≥	≥	NOUN
ejpam-6191	228	5	1	1	NUM
ejpam-6191	228	6	we	we	PRON
ejpam-6191	228	7	have	have	VERB
ejpam-6191	228	8	sn(a	sn(a	ADV
ejpam-6191	228	9	,	,	PUNCT
ejpam-6191	228	10	m	m	NOUN
ejpam-6191	228	11	)	)	PUNCT
ejpam-6191	229	1	=	=	SYM
ejpam-6191	229	2	n∑	n∑	NOUN
ejpam-6191	229	3	k=0	k=0	PROPN
ejpam-6191	229	4	(	(	PUNCT
ejpam-6191	229	5	n	n	CCONJ
ejpam-6191	229	6	k	k	X
ejpam-6191	229	7	)	)	PUNCT
ejpam-6191	229	8	kmαkh	kmαkh	NOUN
ejpam-6191	229	9	(	(	PUNCT
ejpam-6191	229	10	r	r	NOUN
ejpam-6191	229	11	)	)	PUNCT
ejpam-6191	229	12	k	k	NOUN
ejpam-6191	229	13	=	=	SYM
ejpam-6191	229	14	r+1∑	r+1∑	PROPN
ejpam-6191	229	15	k=0	k=0	PROPN
ejpam-6191	229	16	n∑	n∑	PROPN
ejpam-6191	229	17	j=0	j=0	PROPN
ejpam-6191	229	18	j∑	j∑	PROPN
ejpam-6191	230	1	l=0	l=0	PROPN
ejpam-6191	230	2	(	(	PUNCT
ejpam-6191	230	3	−1)r+1−k	−1)r+1−k	X
ejpam-6191	230	4	(	(	PUNCT
ejpam-6191	230	5	r	r	NOUN
ejpam-6191	230	6	+	+	NOUN
ejpam-6191	230	7	1	1	NUM
ejpam-6191	230	8	k	k	NOUN
ejpam-6191	230	9	)	)	PUNCT
ejpam-6191	230	10	(	(	PUNCT
ejpam-6191	230	11	j	j	PROPN
ejpam-6191	230	12	l	l	NOUN
ejpam-6191	230	13	)	)	PUNCT
ejpam-6191	230	14	lmαlh	lmαlh	NOUN
ejpam-6191	230	15	(	(	PUNCT
ejpam-6191	230	16	k−1	k−1	PROPN
ejpam-6191	230	17	)	)	PUNCT
ejpam-6191	230	18	j	j	PROPN
ejpam-6191	231	1	[	[	PUNCT
ejpam-6191	231	2	h	h	NOUN
ejpam-6191	231	3	(	(	PUNCT
ejpam-6191	231	4	r−k	r−k	PROPN
ejpam-6191	231	5	)	)	PUNCT
ejpam-6191	231	6	n−j	n−j	ADV
ejpam-6191	231	7	−h	−h	VERB
ejpam-6191	231	8	(	(	PUNCT
ejpam-6191	231	9	r−k	r−k	PROPN
ejpam-6191	231	10	)	)	PUNCT
ejpam-6191	231	11	n−1−j	n−1−j	NUM
ejpam-6191	231	12	]	]	PUNCT
ejpam-6191	231	13	.	.	PUNCT
ejpam-6191	232	1	(	(	PUNCT
ejpam-6191	232	2	3.2	3.2	NUM
ejpam-6191	232	3	)	)	PUNCT
ejpam-6191	232	4	k.	k.	NOUN
ejpam-6191	233	1	v.	v.	PROPN
ejpam-6191	233	2	m.	m.	PROPN
ejpam-6191	233	3	manulat	manulat	PROPN
ejpam-6191	233	4	,	,	PUNCT
ejpam-6191	233	5	r.	r.	PROPN
ejpam-6191	233	6	b.	b.	PROPN
ejpam-6191	233	7	corcino	corcino	PROPN
ejpam-6191	233	8	/	/	SYM
ejpam-6191	233	9	eur	eur	PROPN
ejpam-6191	233	10	.	.	PUNCT
ejpam-6191	234	1	j.	j.	PROPN
ejpam-6191	234	2	pure	pure	PROPN
ejpam-6191	234	3	appl	appl	PROPN
ejpam-6191	234	4	.	.	PROPN
ejpam-6191	234	5	math	math	PROPN
ejpam-6191	234	6	,	,	PUNCT
ejpam-6191	234	7	18	18	NUM
ejpam-6191	234	8	(	(	PUNCT
ejpam-6191	234	9	3	3	NUM
ejpam-6191	234	10	)	)	PUNCT
ejpam-6191	234	11	(	(	PUNCT
ejpam-6191	234	12	2025	2025	NUM
ejpam-6191	234	13	)	)	PUNCT
ejpam-6191	234	14	,	,	PUNCT
ejpam-6191	234	15	6191	6191	NUM
ejpam-6191	234	16	12	12	NUM
ejpam-6191	234	17	of	of	ADP
ejpam-6191	234	18	21	21	NUM
ejpam-6191	234	19	proof	proof	NOUN
ejpam-6191	234	20	.	.	PUNCT
ejpam-6191	235	1	setting	set	VERB
ejpam-6191	235	2	(	(	PUNCT
ejpam-6191	235	3	a	a	PRON
ejpam-6191	235	4	;	;	PUNCT
ejpam-6191	235	5	b	b	X
ejpam-6191	235	6	)	)	PUNCT
ejpam-6191	235	7	=	=	SYM
ejpam-6191	235	8	(	(	PUNCT
ejpam-6191	235	9	a	a	NOUN
ejpam-6191	235	10	;	;	PUNCT
ejpam-6191	235	11	1	1	NUM
ejpam-6191	235	12	)	)	PUNCT
ejpam-6191	235	13	in	in	ADP
ejpam-6191	235	14	(	(	PUNCT
ejpam-6191	235	15	4.1	4.1	NUM
ejpam-6191	235	16	)	)	PUNCT
ejpam-6191	235	17	,	,	PUNCT
ejpam-6191	235	18	then	then	ADV
ejpam-6191	235	19	n∑	n∑	PROPN
ejpam-6191	235	20	k=0	k=0	PROPN
ejpam-6191	235	21	(	(	PUNCT
ejpam-6191	235	22	n	n	X
ejpam-6191	235	23	k	k	NOUN
ejpam-6191	235	24	)	)	PUNCT
ejpam-6191	235	25	akbn−kh	akbn−kh	VERB
ejpam-6191	235	26	(	(	PUNCT
ejpam-6191	235	27	r	r	NOUN
ejpam-6191	235	28	)	)	PUNCT
ejpam-6191	236	1	k	k	NOUN
ejpam-6191	236	2	=	=	SYM
ejpam-6191	236	3	r+1∑	r+1∑	PROPN
ejpam-6191	236	4	k=0	k=0	PROPN
ejpam-6191	236	5	n∑	n∑	PROPN
ejpam-6191	236	6	j=0	j=0	PROPN
ejpam-6191	236	7	(	(	PUNCT
ejpam-6191	236	8	−1)r+1−k	−1)r+1−k	X
ejpam-6191	236	9	(	(	PUNCT
ejpam-6191	236	10	r	r	NOUN
ejpam-6191	236	11	+	+	NOUN
ejpam-6191	236	12	1	1	NUM
ejpam-6191	236	13	k	k	NOUN
ejpam-6191	236	14	)	)	PUNCT
ejpam-6191	236	15	(	(	PUNCT
ejpam-6191	236	16	a+	a+	PUNCT
ejpam-6191	236	17	b)jh	b)jh	PROPN
ejpam-6191	236	18	(	(	PUNCT
ejpam-6191	236	19	k−1	k−1	PROPN
ejpam-6191	236	20	)	)	PUNCT
ejpam-6191	236	21	j	j	PROPN
ejpam-6191	237	1	bn−j	bn−j	ADV
ejpam-6191	237	2	[	[	PUNCT
ejpam-6191	237	3	h	h	NOUN
ejpam-6191	237	4	(	(	PUNCT
ejpam-6191	237	5	r−k	r−k	PROPN
ejpam-6191	237	6	)	)	PUNCT
ejpam-6191	237	7	n−j	n−j	ADV
ejpam-6191	237	8	−h	−h	VERB
ejpam-6191	237	9	(	(	PUNCT
ejpam-6191	237	10	r−k	r−k	PROPN
ejpam-6191	237	11	)	)	PUNCT
ejpam-6191	237	12	n−1−j	n−1−j	X
ejpam-6191	237	13	]	]	PUNCT
ejpam-6191	238	1	n∑	n∑	X
ejpam-6191	238	2	k=0	k=0	PROPN
ejpam-6191	238	3	(	(	PUNCT
ejpam-6191	238	4	n	n	X
ejpam-6191	238	5	k	k	X
ejpam-6191	238	6	)	)	PUNCT
ejpam-6191	238	7	akh	akh	NOUN
ejpam-6191	238	8	(	(	PUNCT
ejpam-6191	238	9	r	r	NOUN
ejpam-6191	238	10	)	)	PUNCT
ejpam-6191	238	11	k	k	NOUN
ejpam-6191	239	1	=	=	SYM
ejpam-6191	239	2	r+1∑	r+1∑	PROPN
ejpam-6191	239	3	k=0	k=0	PROPN
ejpam-6191	239	4	n∑	n∑	PROPN
ejpam-6191	239	5	j=0	j=0	PROPN
ejpam-6191	239	6	(	(	PUNCT
ejpam-6191	239	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	239	8	(	(	PUNCT
ejpam-6191	239	9	r	r	NOUN
ejpam-6191	239	10	+	+	NOUN
ejpam-6191	239	11	1	1	NUM
ejpam-6191	239	12	k	k	NOUN
ejpam-6191	239	13	)	)	PUNCT
ejpam-6191	239	14	(	(	PUNCT
ejpam-6191	239	15	a+	a+	X
ejpam-6191	239	16	1)jh	1)jh	PROPN
ejpam-6191	239	17	(	(	PUNCT
ejpam-6191	239	18	k−1	k−1	PROPN
ejpam-6191	239	19	)	)	PUNCT
ejpam-6191	239	20	j	j	PROPN
ejpam-6191	240	1	[	[	PUNCT
ejpam-6191	240	2	h	h	NOUN
ejpam-6191	240	3	(	(	PUNCT
ejpam-6191	240	4	r−k	r−k	PROPN
ejpam-6191	240	5	)	)	PUNCT
ejpam-6191	240	6	n−j	n−j	ADV
ejpam-6191	240	7	−h	−h	VERB
ejpam-6191	240	8	(	(	PUNCT
ejpam-6191	240	9	r−k	r−k	PROPN
ejpam-6191	240	10	)	)	PUNCT
ejpam-6191	240	11	n−1−j	n−1−j	NOUN
ejpam-6191	240	12	]	]	PUNCT
ejpam-6191	240	13	applying	apply	VERB
ejpam-6191	240	14	the	the	DET
ejpam-6191	240	15	differential	differential	ADJ
ejpam-6191	240	16	operator	operator	NOUN
ejpam-6191	240	17	,	,	PUNCT
ejpam-6191	240	18	(	(	PUNCT
ejpam-6191	240	19	a	a	DET
ejpam-6191	240	20	d	d	X
ejpam-6191	240	21	da	da	NOUN
ejpam-6191	240	22	)	)	PUNCT
ejpam-6191	240	23	m	m	VERB
ejpam-6191	240	24	to	to	ADP
ejpam-6191	240	25	both	both	DET
ejpam-6191	240	26	sides	side	NOUN
ejpam-6191	240	27	,	,	PUNCT
ejpam-6191	241	1	n∑	n∑	PROPN
ejpam-6191	241	2	k=0	k=0	PROPN
ejpam-6191	241	3	(	(	PUNCT
ejpam-6191	241	4	n	n	X
ejpam-6191	241	5	k	k	NOUN
ejpam-6191	241	6	)	)	PUNCT
ejpam-6191	241	7	(	(	PUNCT
ejpam-6191	241	8	a	a	DET
ejpam-6191	241	9	d	d	X
ejpam-6191	241	10	da	da	NOUN
ejpam-6191	241	11	)	)	PUNCT
ejpam-6191	241	12	m	m	PROPN
ejpam-6191	241	13	akh	akh	NOUN
ejpam-6191	241	14	(	(	PUNCT
ejpam-6191	241	15	r	r	NOUN
ejpam-6191	241	16	)	)	PUNCT
ejpam-6191	241	17	k	k	NOUN
ejpam-6191	242	1	=	=	SYM
ejpam-6191	242	2	r+1∑	r+1∑	PROPN
ejpam-6191	242	3	k=0	k=0	PROPN
ejpam-6191	242	4	n∑	n∑	PROPN
ejpam-6191	242	5	j=0	j=0	PROPN
ejpam-6191	242	6	(	(	PUNCT
ejpam-6191	243	1	−1)r+1−k	−1)r+1−k	X
ejpam-6191	243	2	(	(	PUNCT
ejpam-6191	243	3	r	r	NOUN
ejpam-6191	243	4	+	+	NOUN
ejpam-6191	243	5	1	1	NUM
ejpam-6191	243	6	k	k	NOUN
ejpam-6191	243	7	)	)	PUNCT
ejpam-6191	243	8	(	(	PUNCT
ejpam-6191	243	9	a	a	DET
ejpam-6191	243	10	d	d	X
ejpam-6191	243	11	da	da	NOUN
ejpam-6191	243	12	)	)	PUNCT
ejpam-6191	243	13	m	m	VERB
ejpam-6191	243	14	(	(	PUNCT
ejpam-6191	243	15	a+	a+	PROPN
ejpam-6191	243	16	1)jh	1)jh	PROPN
ejpam-6191	243	17	(	(	PUNCT
ejpam-6191	243	18	k−1	k−1	PROPN
ejpam-6191	243	19	)	)	PUNCT
ejpam-6191	243	20	j	j	PROPN
ejpam-6191	243	21	[	[	PUNCT
ejpam-6191	243	22	h	h	NOUN
ejpam-6191	243	23	(	(	PUNCT
ejpam-6191	243	24	r−k	r−k	PROPN
ejpam-6191	243	25	)	)	PUNCT
ejpam-6191	243	26	n−j	n−j	ADV
ejpam-6191	243	27	−h	−h	VERB
ejpam-6191	243	28	(	(	PUNCT
ejpam-6191	243	29	r−k	r−k	PROPN
ejpam-6191	243	30	)	)	PUNCT
ejpam-6191	243	31	n−1−j	n−1−j	X
ejpam-6191	243	32	]	]	PUNCT
ejpam-6191	244	1	n∑	n∑	X
ejpam-6191	244	2	k=0	k=0	PROPN
ejpam-6191	244	3	(	(	PUNCT
ejpam-6191	244	4	n	n	CCONJ
ejpam-6191	244	5	k	k	X
ejpam-6191	244	6	)	)	PUNCT
ejpam-6191	244	7	kmαkh	kmαkh	NOUN
ejpam-6191	244	8	(	(	PUNCT
ejpam-6191	244	9	r	r	NOUN
ejpam-6191	244	10	)	)	PUNCT
ejpam-6191	244	11	k	k	NOUN
ejpam-6191	245	1	=	=	SYM
ejpam-6191	245	2	r+1∑	r+1∑	PROPN
ejpam-6191	245	3	k=0	k=0	PROPN
ejpam-6191	245	4	n∑	n∑	PROPN
ejpam-6191	245	5	j=0	j=0	PROPN
ejpam-6191	245	6	(	(	PUNCT
ejpam-6191	245	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	245	8	(	(	PUNCT
ejpam-6191	245	9	r	r	NOUN
ejpam-6191	245	10	+	+	NOUN
ejpam-6191	245	11	1	1	NUM
ejpam-6191	245	12	k	k	NOUN
ejpam-6191	245	13	)	)	PUNCT
ejpam-6191	245	14	(	(	PUNCT
ejpam-6191	245	15	j∑	j∑	PROPN
ejpam-6191	245	16	l=0	l=0	PROPN
ejpam-6191	245	17	(	(	PUNCT
ejpam-6191	245	18	j	j	PROPN
ejpam-6191	245	19	l	l	NOUN
ejpam-6191	245	20	)	)	PUNCT
ejpam-6191	245	21	lmal	lmal	ADJ
ejpam-6191	245	22	)	)	PUNCT
ejpam-6191	245	23	h	h	NOUN
ejpam-6191	245	24	(	(	PUNCT
ejpam-6191	246	1	k−1	k−1	PROPN
ejpam-6191	246	2	)	)	PUNCT
ejpam-6191	246	3	j	j	PROPN
ejpam-6191	246	4	[	[	PUNCT
ejpam-6191	246	5	h	h	NOUN
ejpam-6191	246	6	(	(	PUNCT
ejpam-6191	246	7	r−k	r−k	PROPN
ejpam-6191	246	8	)	)	PUNCT
ejpam-6191	246	9	n−j	n−j	ADV
ejpam-6191	246	10	−h	−h	VERB
ejpam-6191	246	11	(	(	PUNCT
ejpam-6191	246	12	r−k	r−k	PROPN
ejpam-6191	246	13	)	)	PUNCT
ejpam-6191	246	14	n−1−j	n−1−j	X
ejpam-6191	246	15	]	]	PUNCT
ejpam-6191	246	16	,	,	PUNCT
ejpam-6191	246	17	therefore	therefore	ADV
ejpam-6191	246	18	,	,	PUNCT
ejpam-6191	246	19	we	we	PRON
ejpam-6191	246	20	obtain	obtain	VERB
ejpam-6191	246	21	the	the	DET
ejpam-6191	246	22	result	result	NOUN
ejpam-6191	246	23	sn(a	sn(a	PUNCT
ejpam-6191	246	24	,	,	PUNCT
ejpam-6191	246	25	m	m	NOUN
ejpam-6191	246	26	)	)	PUNCT
ejpam-6191	247	1	=	=	SYM
ejpam-6191	247	2	n∑	n∑	NOUN
ejpam-6191	247	3	k=0	k=0	PROPN
ejpam-6191	247	4	(	(	PUNCT
ejpam-6191	247	5	n	n	CCONJ
ejpam-6191	247	6	k	k	X
ejpam-6191	247	7	)	)	PUNCT
ejpam-6191	247	8	kmαkh	kmαkh	NOUN
ejpam-6191	247	9	(	(	PUNCT
ejpam-6191	247	10	r	r	NOUN
ejpam-6191	247	11	)	)	PUNCT
ejpam-6191	247	12	k	k	NOUN
ejpam-6191	247	13	=	=	SYM
ejpam-6191	247	14	r+1∑	r+1∑	PROPN
ejpam-6191	247	15	k=0	k=0	PROPN
ejpam-6191	247	16	n∑	n∑	PROPN
ejpam-6191	247	17	j=0	j=0	PROPN
ejpam-6191	247	18	j∑	j∑	PROPN
ejpam-6191	248	1	l=0	l=0	PROPN
ejpam-6191	248	2	(	(	PUNCT
ejpam-6191	248	3	−1)r+1−k	−1)r+1−k	X
ejpam-6191	248	4	(	(	PUNCT
ejpam-6191	248	5	r	r	NOUN
ejpam-6191	248	6	+	+	NOUN
ejpam-6191	248	7	1	1	NUM
ejpam-6191	248	8	k	k	NOUN
ejpam-6191	248	9	)	)	PUNCT
ejpam-6191	248	10	(	(	PUNCT
ejpam-6191	248	11	j	j	PROPN
ejpam-6191	248	12	l	l	NOUN
ejpam-6191	248	13	)	)	PUNCT
ejpam-6191	248	14	lmαlh	lmαlh	NOUN
ejpam-6191	248	15	(	(	PUNCT
ejpam-6191	248	16	k−1	k−1	PROPN
ejpam-6191	248	17	)	)	PUNCT
ejpam-6191	248	18	j	j	PROPN
ejpam-6191	249	1	[	[	PUNCT
ejpam-6191	249	2	h	h	NOUN
ejpam-6191	249	3	(	(	PUNCT
ejpam-6191	249	4	r−k	r−k	PROPN
ejpam-6191	249	5	)	)	PUNCT
ejpam-6191	249	6	n−j	n−j	ADV
ejpam-6191	249	7	−h	−h	VERB
ejpam-6191	249	8	(	(	PUNCT
ejpam-6191	249	9	r−k	r−k	PROPN
ejpam-6191	249	10	)	)	PUNCT
ejpam-6191	249	11	n−1−j	n−1−j	NUM
ejpam-6191	249	12	]	]	PUNCT
ejpam-6191	249	13	as	as	ADP
ejpam-6191	249	14	a	a	DET
ejpam-6191	249	15	consequence	consequence	NOUN
ejpam-6191	249	16	of	of	ADP
ejpam-6191	249	17	this	this	DET
ejpam-6191	249	18	theorem	theorem	VERB
ejpam-6191	249	19	,	,	PUNCT
ejpam-6191	249	20	additional	additional	ADJ
ejpam-6191	249	21	variants	variant	NOUN
ejpam-6191	249	22	of	of	ADP
ejpam-6191	249	23	identities	identity	NOUN
ejpam-6191	249	24	involving	involve	VERB
ejpam-6191	249	25	fibonacci	fibonacci	NOUN
ejpam-6191	249	26	and	and	CCONJ
ejpam-6191	249	27	lucas	lucas	PROPN
ejpam-6191	249	28	numbers	number	NOUN
ejpam-6191	249	29	are	be	AUX
ejpam-6191	249	30	derived	derive	VERB
ejpam-6191	249	31	.	.	PUNCT
ejpam-6191	250	1	these	these	DET
ejpam-6191	250	2	identities	identity	NOUN
ejpam-6191	250	3	are	be	AUX
ejpam-6191	250	4	systematically	systematically	ADV
ejpam-6191	250	5	presented	present	VERB
ejpam-6191	250	6	in	in	ADP
ejpam-6191	250	7	the	the	DET
ejpam-6191	250	8	corollaries	corollary	NOUN
ejpam-6191	250	9	that	that	PRON
ejpam-6191	250	10	follow	follow	VERB
ejpam-6191	250	11	,	,	PUNCT
ejpam-6191	250	12	further	far	ADV
ejpam-6191	250	13	enriching	enrich	VERB
ejpam-6191	250	14	the	the	DET
ejpam-6191	250	15	mathematical	mathematical	ADJ
ejpam-6191	250	16	relationships	relationship	NOUN
ejpam-6191	250	17	among	among	ADP
ejpam-6191	250	18	these	these	DET
ejpam-6191	250	19	classical	classical	ADJ
ejpam-6191	250	20	integer	integer	NOUN
ejpam-6191	250	21	sequences	sequence	NOUN
ejpam-6191	250	22	.	.	PUNCT
ejpam-6191	251	1	corollary	corollary	ADJ
ejpam-6191	251	2	3.11	3.11	NUM
ejpam-6191	251	3	.	.	PUNCT
ejpam-6191	252	1	for	for	ADP
ejpam-6191	252	2	fn	fn	NOUN
ejpam-6191	252	3	and	and	CCONJ
ejpam-6191	252	4	ln	ln	ADV
ejpam-6191	252	5	be	be	AUX
ejpam-6191	252	6	the	the	DET
ejpam-6191	252	7	fibonacci	fibonacci	NOUN
ejpam-6191	252	8	and	and	CCONJ
ejpam-6191	252	9	lucas	lucas	PROPN
ejpam-6191	252	10	numbers	number	NOUN
ejpam-6191	252	11	,	,	PUNCT
ejpam-6191	252	12	respectively	respectively	ADV
ejpam-6191	252	13	.	.	PUNCT
ejpam-6191	253	1	then	then	ADV
ejpam-6191	253	2	,	,	PUNCT
ejpam-6191	253	3	k.	k.	PROPN
ejpam-6191	253	4	v.	v.	PROPN
ejpam-6191	253	5	m.	m.	PROPN
ejpam-6191	253	6	manulat	manulat	PROPN
ejpam-6191	253	7	,	,	PUNCT
ejpam-6191	253	8	r.	r.	PROPN
ejpam-6191	253	9	b.	b.	PROPN
ejpam-6191	253	10	corcino	corcino	PROPN
ejpam-6191	253	11	/	/	SYM
ejpam-6191	253	12	eur	eur	PROPN
ejpam-6191	253	13	.	.	PUNCT
ejpam-6191	254	1	j.	j.	PROPN
ejpam-6191	254	2	pure	pure	PROPN
ejpam-6191	254	3	appl	appl	PROPN
ejpam-6191	254	4	.	.	PROPN
ejpam-6191	254	5	math	math	PROPN
ejpam-6191	254	6	,	,	PUNCT
ejpam-6191	254	7	18	18	NUM
ejpam-6191	254	8	(	(	PUNCT
ejpam-6191	254	9	3	3	NUM
ejpam-6191	254	10	)	)	PUNCT
ejpam-6191	254	11	(	(	PUNCT
ejpam-6191	254	12	2025	2025	NUM
ejpam-6191	254	13	)	)	PUNCT
ejpam-6191	254	14	,	,	PUNCT
ejpam-6191	254	15	6191	6191	NUM
ejpam-6191	254	16	13	13	NUM
ejpam-6191	254	17	of	of	ADP
ejpam-6191	254	18	21	21	NUM
ejpam-6191	254	19	n∑	n∑	NOUN
ejpam-6191	254	20	k=0	k=0	PROPN
ejpam-6191	254	21	(	(	PUNCT
ejpam-6191	254	22	n	n	CCONJ
ejpam-6191	254	23	k	k	NOUN
ejpam-6191	254	24	)	)	PUNCT
ejpam-6191	254	25	kfkh	kfkh	NOUN
ejpam-6191	254	26	(	(	PUNCT
ejpam-6191	254	27	r	r	NOUN
ejpam-6191	254	28	)	)	PUNCT
ejpam-6191	254	29	k	k	NOUN
ejpam-6191	255	1	=	=	SYM
ejpam-6191	255	2	r+1∑	r+1∑	PROPN
ejpam-6191	255	3	k=0	k=0	PROPN
ejpam-6191	255	4	n∑	n∑	PROPN
ejpam-6191	255	5	j=0	j=0	PROPN
ejpam-6191	255	6	(	(	PUNCT
ejpam-6191	255	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	255	8	(	(	PUNCT
ejpam-6191	255	9	r	r	NOUN
ejpam-6191	255	10	+	+	NOUN
ejpam-6191	255	11	1	1	NUM
ejpam-6191	255	12	k	k	NOUN
ejpam-6191	255	13	)	)	PUNCT
ejpam-6191	255	14	jf2j−1h	jf2j−1h	PROPN
ejpam-6191	255	15	(	(	PUNCT
ejpam-6191	256	1	k−1	k−1	PROPN
ejpam-6191	256	2	)	)	PUNCT
ejpam-6191	256	3	j	j	PROPN
ejpam-6191	256	4	[	[	PUNCT
ejpam-6191	256	5	h	h	NOUN
ejpam-6191	256	6	(	(	PUNCT
ejpam-6191	256	7	r−k	r−k	PROPN
ejpam-6191	256	8	)	)	PUNCT
ejpam-6191	256	9	n−j	n−j	ADV
ejpam-6191	256	10	−h	−h	VERB
ejpam-6191	256	11	(	(	PUNCT
ejpam-6191	256	12	r−k	r−k	PROPN
ejpam-6191	256	13	)	)	PUNCT
ejpam-6191	256	14	n−1−j	n−1−j	X
ejpam-6191	256	15	]	]	PUNCT
ejpam-6191	256	16	and	and	CCONJ
ejpam-6191	256	17	,	,	PUNCT
ejpam-6191	256	18	n∑	n∑	PROPN
ejpam-6191	256	19	k=0	k=0	PROPN
ejpam-6191	256	20	(	(	PUNCT
ejpam-6191	256	21	n	n	X
ejpam-6191	256	22	k	k	NOUN
ejpam-6191	256	23	)	)	PUNCT
ejpam-6191	256	24	klkh	klkh	NOUN
ejpam-6191	256	25	(	(	PUNCT
ejpam-6191	256	26	r	r	NOUN
ejpam-6191	256	27	)	)	PUNCT
ejpam-6191	256	28	k	k	NOUN
ejpam-6191	256	29	=	=	SYM
ejpam-6191	256	30	r+1∑	r+1∑	PROPN
ejpam-6191	256	31	k=0	k=0	PROPN
ejpam-6191	256	32	n∑	n∑	PROPN
ejpam-6191	256	33	j=0	j=0	PROPN
ejpam-6191	256	34	(	(	PUNCT
ejpam-6191	256	35	−1)r+1−k	−1)r+1−k	X
ejpam-6191	256	36	(	(	PUNCT
ejpam-6191	257	1	r	r	NOUN
ejpam-6191	257	2	+	+	NOUN
ejpam-6191	257	3	1	1	NUM
ejpam-6191	257	4	k	k	NOUN
ejpam-6191	257	5	)	)	PUNCT
ejpam-6191	257	6	jl2j−1h	jl2j−1h	PROPN
ejpam-6191	257	7	(	(	PUNCT
ejpam-6191	257	8	k−1	k−1	PROPN
ejpam-6191	257	9	)	)	PUNCT
ejpam-6191	257	10	j	j	PROPN
ejpam-6191	258	1	[	[	PUNCT
ejpam-6191	258	2	h	h	NOUN
ejpam-6191	258	3	(	(	PUNCT
ejpam-6191	258	4	r−k	r−k	PROPN
ejpam-6191	258	5	)	)	PUNCT
ejpam-6191	258	6	n−j	n−j	ADV
ejpam-6191	258	7	−h	−h	VERB
ejpam-6191	258	8	(	(	PUNCT
ejpam-6191	258	9	r−k	r−k	PROPN
ejpam-6191	258	10	)	)	PUNCT
ejpam-6191	258	11	n−1−j	n−1−j	NUM
ejpam-6191	258	12	]	]	PUNCT
ejpam-6191	258	13	.	.	PUNCT
ejpam-6191	259	1	proof	proof	NOUN
ejpam-6191	259	2	.	.	PUNCT
ejpam-6191	260	1	setting	set	VERB
ejpam-6191	260	2	(	(	PUNCT
ejpam-6191	260	3	a;m	a;m	NUM
ejpam-6191	260	4	)	)	PUNCT
ejpam-6191	260	5	=	=	SYM
ejpam-6191	260	6	(	(	PUNCT
ejpam-6191	260	7	α	α	NOUN
ejpam-6191	260	8	;	;	PUNCT
ejpam-6191	260	9	1	1	NUM
ejpam-6191	260	10	)	)	PUNCT
ejpam-6191	260	11	and	and	CCONJ
ejpam-6191	260	12	(	(	PUNCT
ejpam-6191	260	13	a;m	a;m	NUM
ejpam-6191	260	14	)	)	PUNCT
ejpam-6191	260	15	=	=	SYM
ejpam-6191	260	16	(	(	PUNCT
ejpam-6191	260	17	β	β	NOUN
ejpam-6191	260	18	;	;	PUNCT
ejpam-6191	260	19	1	1	NUM
ejpam-6191	260	20	)	)	PUNCT
ejpam-6191	260	21	,	,	PUNCT
ejpam-6191	260	22	respectively	respectively	ADV
ejpam-6191	260	23	,	,	PUNCT
ejpam-6191	260	24	y	y	PROPN
ejpam-6191	260	25	,	,	PUNCT
ejpam-6191	260	26	in	in	ADP
ejpam-6191	260	27	(	(	PUNCT
ejpam-6191	260	28	3.2	3.2	NUM
ejpam-6191	260	29	)	)	PUNCT
ejpam-6191	260	30	.	.	PUNCT
ejpam-6191	261	1	corollary	corollary	ADJ
ejpam-6191	261	2	3.12	3.12	NUM
ejpam-6191	261	3	.	.	PUNCT
ejpam-6191	262	1	for	for	ADP
ejpam-6191	262	2	n	n	PRON
ejpam-6191	262	3	≥	≥	NUM
ejpam-6191	262	4	1	1	NUM
ejpam-6191	262	5	,	,	PUNCT
ejpam-6191	262	6	the	the	DET
ejpam-6191	262	7	following	follow	VERB
ejpam-6191	262	8	relations	relation	NOUN
ejpam-6191	262	9	hold	hold	VERB
ejpam-6191	262	10	:	:	PUNCT
ejpam-6191	263	1	n∑	n∑	PROPN
ejpam-6191	263	2	k=0	k=0	PROPN
ejpam-6191	263	3	(	(	PUNCT
ejpam-6191	263	4	n	n	X
ejpam-6191	263	5	k	k	PROPN
ejpam-6191	263	6	)	)	PUNCT
ejpam-6191	263	7	k2fkh	k2fkh	PROPN
ejpam-6191	263	8	(	(	PUNCT
ejpam-6191	263	9	r	r	NOUN
ejpam-6191	263	10	)	)	PUNCT
ejpam-6191	263	11	k	k	NOUN
ejpam-6191	263	12	=	=	SYM
ejpam-6191	263	13	r+1∑	r+1∑	PROPN
ejpam-6191	263	14	k=0	k=0	PROPN
ejpam-6191	263	15	n∑	n∑	PROPN
ejpam-6191	263	16	j=0	j=0	PROPN
ejpam-6191	263	17	(	(	PUNCT
ejpam-6191	263	18	−1)r+1−k	−1)r+1−k	X
ejpam-6191	263	19	(	(	PUNCT
ejpam-6191	264	1	r	r	NOUN
ejpam-6191	264	2	+	+	NOUN
ejpam-6191	264	3	1	1	NUM
ejpam-6191	264	4	k	k	X
ejpam-6191	264	5	)	)	PUNCT
ejpam-6191	264	6	j(j	j(j	PROPN
ejpam-6191	264	7	−	−	PROPN
ejpam-6191	265	1	1)f2jh	1)f2jh	PROPN
ejpam-6191	265	2	(	(	PUNCT
ejpam-6191	265	3	k−1	k−1	PROPN
ejpam-6191	265	4	)	)	PUNCT
ejpam-6191	265	5	j	j	PROPN
ejpam-6191	265	6	[	[	PUNCT
ejpam-6191	265	7	h	h	NOUN
ejpam-6191	265	8	(	(	PUNCT
ejpam-6191	265	9	r−k	r−k	PROPN
ejpam-6191	265	10	)	)	PUNCT
ejpam-6191	265	11	n−j	n−j	ADV
ejpam-6191	265	12	−h	−h	VERB
ejpam-6191	265	13	(	(	PUNCT
ejpam-6191	265	14	r−k	r−k	PROPN
ejpam-6191	265	15	)	)	PUNCT
ejpam-6191	265	16	n−1−j	n−1−j	X
ejpam-6191	265	17	]	]	PUNCT
ejpam-6191	265	18	and	and	CCONJ
ejpam-6191	265	19	,	,	PUNCT
ejpam-6191	265	20	n∑	n∑	PROPN
ejpam-6191	265	21	k=0	k=0	PROPN
ejpam-6191	265	22	(	(	PUNCT
ejpam-6191	265	23	n	n	X
ejpam-6191	265	24	k	k	NOUN
ejpam-6191	265	25	)	)	PUNCT
ejpam-6191	265	26	k2lkh	k2lkh	PROPN
ejpam-6191	265	27	(	(	PUNCT
ejpam-6191	265	28	r	r	NOUN
ejpam-6191	265	29	)	)	PUNCT
ejpam-6191	266	1	k	k	NOUN
ejpam-6191	267	1	=	=	SYM
ejpam-6191	267	2	r+1∑	r+1∑	PROPN
ejpam-6191	267	3	k=0	k=0	PROPN
ejpam-6191	267	4	n∑	n∑	PROPN
ejpam-6191	267	5	j=0	j=0	PROPN
ejpam-6191	267	6	(	(	PUNCT
ejpam-6191	267	7	−1)r+1−k	−1)r+1−k	X
ejpam-6191	267	8	(	(	PUNCT
ejpam-6191	267	9	r	r	NOUN
ejpam-6191	267	10	+	+	NOUN
ejpam-6191	267	11	1	1	NUM
ejpam-6191	267	12	k	k	X
ejpam-6191	267	13	)	)	PUNCT
ejpam-6191	267	14	j(j	j(j	PROPN
ejpam-6191	267	15	−	−	PROPN
ejpam-6191	268	1	1)l2jh	1)l2jh	NUM
ejpam-6191	268	2	(	(	PUNCT
ejpam-6191	268	3	k−1	k−1	PROPN
ejpam-6191	268	4	)	)	PUNCT
ejpam-6191	268	5	j	j	PROPN
ejpam-6191	268	6	[	[	PUNCT
ejpam-6191	268	7	h	h	NOUN
ejpam-6191	268	8	(	(	PUNCT
ejpam-6191	268	9	r−k	r−k	PROPN
ejpam-6191	268	10	)	)	PUNCT
ejpam-6191	268	11	n−j	n−j	ADV
ejpam-6191	268	12	−h	−h	VERB
ejpam-6191	268	13	(	(	PUNCT
ejpam-6191	268	14	r−k	r−k	PROPN
ejpam-6191	268	15	)	)	PUNCT
ejpam-6191	268	16	n−1−j	n−1−j	NUM
ejpam-6191	268	17	]	]	PUNCT
ejpam-6191	268	18	.	.	PUNCT
ejpam-6191	269	1	proof	proof	NOUN
ejpam-6191	269	2	.	.	PUNCT
ejpam-6191	270	1	(	(	PUNCT
ejpam-6191	270	2	a;m	a;m	NUM
ejpam-6191	270	3	)	)	PUNCT
ejpam-6191	270	4	=	=	SYM
ejpam-6191	270	5	(	(	PUNCT
ejpam-6191	270	6	α	α	NOUN
ejpam-6191	270	7	;	;	PUNCT
ejpam-6191	270	8	2	2	NUM
ejpam-6191	270	9	)	)	PUNCT
ejpam-6191	270	10	and	and	CCONJ
ejpam-6191	270	11	(	(	PUNCT
ejpam-6191	270	12	a;m	a;m	NUM
ejpam-6191	270	13	)	)	PUNCT
ejpam-6191	270	14	=	=	SYM
ejpam-6191	270	15	(	(	PUNCT
ejpam-6191	270	16	β	β	NOUN
ejpam-6191	270	17	;	;	PUNCT
ejpam-6191	270	18	2	2	X
ejpam-6191	270	19	)	)	PUNCT
ejpam-6191	270	20	respectively	respectively	ADV
ejpam-6191	270	21	in	in	ADP
ejpam-6191	270	22	(	(	PUNCT
ejpam-6191	270	23	3.2	3.2	NUM
ejpam-6191	270	24	)	)	PUNCT
ejpam-6191	270	25	,	,	PUNCT
ejpam-6191	270	26	4	4	X
ejpam-6191	270	27	.	.	PUNCT
ejpam-6191	270	28	generalized	generalize	VERB
ejpam-6191	270	29	poly	poly	ADJ
ejpam-6191	270	30	-	-	PUNCT
ejpam-6191	270	31	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	270	32	numbers	number	NOUN
ejpam-6191	270	33	of	of	ADP
ejpam-6191	270	34	order	order	NOUN
ejpam-6191	270	35	r	r	NOUN
ejpam-6191	270	36	in	in	ADP
ejpam-6191	270	37	this	this	DET
ejpam-6191	270	38	section	section	NOUN
ejpam-6191	270	39	,	,	PUNCT
ejpam-6191	270	40	we	we	PRON
ejpam-6191	270	41	shall	shall	AUX
ejpam-6191	270	42	establish	establish	VERB
ejpam-6191	270	43	and	and	CCONJ
ejpam-6191	270	44	prove	prove	VERB
ejpam-6191	270	45	a	a	DET
ejpam-6191	270	46	new	new	ADJ
ejpam-6191	270	47	expression	expression	NOUN
ejpam-6191	270	48	for	for	ADP
ejpam-6191	270	49	the	the	DET
ejpam-6191	270	50	generalized	generalized	ADJ
ejpam-6191	270	51	polyhyperharmonic	polyhyperharmonic	ADJ
ejpam-6191	270	52	numbers	number	NOUN
ejpam-6191	270	53	of	of	ADP
ejpam-6191	270	54	order	order	NOUN
ejpam-6191	270	55	r.	r.	NOUN
ejpam-6191	270	56	the	the	DET
ejpam-6191	270	57	proof	proof	NOUN
ejpam-6191	270	58	made	make	VERB
ejpam-6191	270	59	use	use	NOUN
ejpam-6191	270	60	of	of	ADP
ejpam-6191	270	61	the	the	DET
ejpam-6191	270	62	euler	euler	NOUN
ejpam-6191	270	63	’s	’s	PART
ejpam-6191	270	64	transform	transform	NOUN
ejpam-6191	270	65	applied	apply	VERB
ejpam-6191	270	66	to	to	ADP
ejpam-6191	270	67	the	the	DET
ejpam-6191	270	68	polylogarithmic	polylogarithmic	ADJ
ejpam-6191	270	69	generating	generate	VERB
ejpam-6191	270	70	function	function	NOUN
ejpam-6191	270	71	of	of	ADP
ejpam-6191	270	72	the	the	DET
ejpam-6191	270	73	generalized	generalized	ADJ
ejpam-6191	270	74	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	270	75	number	number	NOUN
ejpam-6191	270	76	.	.	PUNCT
ejpam-6191	271	1	theorem	theorem	VERB
ejpam-6191	271	2	4.1	4.1	NUM
ejpam-6191	271	3	.	.	PUNCT
ejpam-6191	272	1	the	the	DET
ejpam-6191	272	2	following	follow	VERB
ejpam-6191	272	3	relation	relation	NOUN
ejpam-6191	272	4	holds	hold	VERB
ejpam-6191	272	5	n∑	n∑	PROPN
ejpam-6191	272	6	n=1	n=1	PROPN
ejpam-6191	272	7	(	(	PUNCT
ejpam-6191	272	8	n	n	CCONJ
ejpam-6191	272	9	k	k	NOUN
ejpam-6191	272	10	)	)	PUNCT
ejpam-6191	272	11	akbn−kh	akbn−kh	VERB
ejpam-6191	272	12	(	(	PUNCT
ejpam-6191	272	13	r	r	NOUN
ejpam-6191	272	14	)	)	PUNCT
ejpam-6191	272	15	k	k	NOUN
ejpam-6191	272	16	,	,	PUNCT
ejpam-6191	272	17	m	m	PROPN
ejpam-6191	272	18	=	=	SYM
ejpam-6191	272	19	n∑	n∑	PROPN
ejpam-6191	272	20	j=0	j=0	PROPN
ejpam-6191	272	21	j∑	j∑	PROPN
ejpam-6191	272	22	k=0	k=0	PROPN
ejpam-6191	272	23	j−k∑	j−k∑	PROPN
ejpam-6191	272	24	m=0	m=0	PROPN
ejpam-6191	272	25	∑	∑	PROPN
ejpam-6191	272	26	t+p	t+p	PUNCT
ejpam-6191	272	27	=	=	SYM
ejpam-6191	272	28	n−j	n−j	X
ejpam-6191	272	29	t≥1	t≥1	NOUN
ejpam-6191	272	30	(	(	PUNCT
ejpam-6191	272	31	m+	m+	NOUN
ejpam-6191	272	32	r	r	NOUN
ejpam-6191	272	33	−	−	NUM
ejpam-6191	272	34	1	1	NUM
ejpam-6191	272	35	m	m	NOUN
ejpam-6191	272	36	)	)	PUNCT
ejpam-6191	272	37	(	(	PUNCT
ejpam-6191	272	38	j	j	PROPN
ejpam-6191	272	39	−	−	PROPN
ejpam-6191	273	1	k	k	NOUN
ejpam-6191	274	1	−	−	PROPN
ejpam-6191	274	2	1	1	NUM
ejpam-6191	274	3	j	j	PROPN
ejpam-6191	274	4	−	−	PROPN
ejpam-6191	274	5	k	k	PROPN
ejpam-6191	274	6	−m	−m	PROPN
ejpam-6191	274	7	)	)	PUNCT
ejpam-6191	274	8	(	(	PUNCT
ejpam-6191	274	9	t+	t+	PUNCT
ejpam-6191	274	10	p−	p−	NOUN
ejpam-6191	274	11	1	1	NUM
ejpam-6191	274	12	p	p	NOUN
ejpam-6191	274	13	)	)	PUNCT
ejpam-6191	274	14	(	(	PUNCT
ejpam-6191	274	15	4.1	4.1	NUM
ejpam-6191	274	16	)	)	PUNCT
ejpam-6191	274	17	×	×	NOUN
ejpam-6191	274	18	am+tbj−m+p	am+tbj−m+p	NOUN
ejpam-6191	274	19	tr	tr	VERB
ejpam-6191	274	20	.	.	PUNCT
ejpam-6191	275	1	proof	proof	NOUN
ejpam-6191	275	2	.	.	PUNCT
ejpam-6191	276	1	given	give	VERB
ejpam-6191	276	2	the	the	DET
ejpam-6191	276	3	generating	generate	VERB
ejpam-6191	276	4	function	function	NOUN
ejpam-6191	276	5	∞∑	∞∑	NUM
ejpam-6191	276	6	n=1	n=1	PROPN
ejpam-6191	276	7	h(r	h(r	NOUN
ejpam-6191	276	8	)	)	PUNCT
ejpam-6191	276	9	n	n	CCONJ
ejpam-6191	276	10	,	,	PUNCT
ejpam-6191	276	11	mzn	mzn	NOUN
ejpam-6191	276	12	=	=	SYM
ejpam-6191	276	13	lim(z	lim(z	PROPN
ejpam-6191	276	14	)	)	PUNCT
ejpam-6191	276	15	(	(	PUNCT
ejpam-6191	276	16	1−	1−	NUM
ejpam-6191	276	17	z)r	z)r	NUM
ejpam-6191	276	18	.	.	PUNCT
ejpam-6191	277	1	k.	k.	PROPN
ejpam-6191	278	1	v.	v.	PROPN
ejpam-6191	278	2	m.	m.	PROPN
ejpam-6191	278	3	manulat	manulat	PROPN
ejpam-6191	278	4	,	,	PUNCT
ejpam-6191	278	5	r.	r.	PROPN
ejpam-6191	278	6	b.	b.	PROPN
ejpam-6191	278	7	corcino	corcino	PROPN
ejpam-6191	278	8	/	/	SYM
ejpam-6191	278	9	eur	eur	PROPN
ejpam-6191	278	10	.	.	PUNCT
ejpam-6191	279	1	j.	j.	PROPN
ejpam-6191	279	2	pure	pure	PROPN
ejpam-6191	279	3	appl	appl	PROPN
ejpam-6191	279	4	.	.	PROPN
ejpam-6191	279	5	math	math	PROPN
ejpam-6191	279	6	,	,	PUNCT
ejpam-6191	279	7	18	18	NUM
ejpam-6191	279	8	(	(	PUNCT
ejpam-6191	279	9	3	3	NUM
ejpam-6191	279	10	)	)	PUNCT
ejpam-6191	279	11	(	(	PUNCT
ejpam-6191	279	12	2025	2025	NUM
ejpam-6191	279	13	)	)	PUNCT
ejpam-6191	279	14	,	,	PUNCT
ejpam-6191	279	15	6191	6191	NUM
ejpam-6191	279	16	14	14	NUM
ejpam-6191	279	17	of	of	ADP
ejpam-6191	279	18	21	21	NUM
ejpam-6191	279	19	applying	apply	VERB
ejpam-6191	279	20	the	the	DET
ejpam-6191	279	21	euler	euler	NOUN
ejpam-6191	279	22	’s	’s	PART
ejpam-6191	279	23	transform	transform	NOUN
ejpam-6191	279	24	with	with	ADP
ejpam-6191	279	25	f(z	f(z	NOUN
ejpam-6191	279	26	)	)	PUNCT
ejpam-6191	279	27	=	=	SYM
ejpam-6191	279	28	lim(z	lim(z	PROPN
ejpam-6191	279	29	)	)	PUNCT
ejpam-6191	279	30	(	(	PUNCT
ejpam-6191	279	31	1−z)r	1−z)r	NUM
ejpam-6191	279	32	and	and	CCONJ
ejpam-6191	279	33	an	an	DET
ejpam-6191	279	34	=	=	ADJ
ejpam-6191	279	35	h	h	NOUN
ejpam-6191	279	36	(	(	PUNCT
ejpam-6191	279	37	r	r	NOUN
ejpam-6191	279	38	)	)	PUNCT
ejpam-6191	279	39	n	n	CCONJ
ejpam-6191	279	40	,	,	PUNCT
ejpam-6191	279	41	m	m	NOUN
ejpam-6191	279	42	yields	yield	NOUN
ejpam-6191	279	43	∞∑	∞∑	NUM
ejpam-6191	279	44	n=1	n=1	PROPN
ejpam-6191	279	45	zn	zn	PROPN
ejpam-6191	279	46	(	(	PUNCT
ejpam-6191	279	47	n∑	n∑	INTJ
ejpam-6191	279	48	k=1	k=1	PROPN
ejpam-6191	279	49	(	(	PUNCT
ejpam-6191	279	50	n	n	X
ejpam-6191	279	51	k	k	NOUN
ejpam-6191	279	52	)	)	PUNCT
ejpam-6191	279	53	akbn−kh	akbn−kh	VERB
ejpam-6191	279	54	(	(	PUNCT
ejpam-6191	279	55	r	r	NOUN
ejpam-6191	279	56	)	)	PUNCT
ejpam-6191	279	57	k	k	NOUN
ejpam-6191	279	58	,	,	PUNCT
ejpam-6191	279	59	m	m	NOUN
ejpam-6191	279	60	)	)	PUNCT
ejpam-6191	279	61	=	=	SYM
ejpam-6191	280	1	1	1	NUM
ejpam-6191	280	2	1−	1−	NUM
ejpam-6191	280	3	bz	bz	PROPN
ejpam-6191	280	4			PROPN
ejpam-6191	280	5	lir	lir	PROPN
ejpam-6191	280	6	(	(	PUNCT
ejpam-6191	280	7	az	az	PROPN
ejpam-6191	280	8	1−bz	1−bz	PROPN
ejpam-6191	280	9	)	)	PUNCT
ejpam-6191	280	10	(	(	PUNCT
ejpam-6191	280	11	1−	1−	NUM
ejpam-6191	280	12	az	az	PROPN
ejpam-6191	280	13	1−bz	1−bz	PROPN
ejpam-6191	280	14	)	)	PUNCT
ejpam-6191	280	15	r	r	NOUN
ejpam-6191	280	16			PROPN
ejpam-6191	280	17	=	=	SYM
ejpam-6191	280	18	1	1	NUM
ejpam-6191	280	19	1−	1−	NUM
ejpam-6191	280	20	bz	bz	PROPN
ejpam-6191	280	21	(	(	PUNCT
ejpam-6191	280	22	1	1	NUM
ejpam-6191	280	23	1−	1−	NUM
ejpam-6191	280	24	az	az	PROPN
ejpam-6191	280	25	1−bz	1−bz	PROPN
ejpam-6191	280	26	)	)	PUNCT
ejpam-6191	280	27	r	r	NOUN
ejpam-6191	280	28	lir	lir	NOUN
ejpam-6191	280	29	(	(	PUNCT
ejpam-6191	280	30	az	az	PROPN
ejpam-6191	280	31	1−	1−	NUM
ejpam-6191	280	32	bz	bz	PROPN
ejpam-6191	280	33	)	)	PUNCT
ejpam-6191	280	34	.	.	PUNCT
ejpam-6191	281	1	(	(	PUNCT
ejpam-6191	281	2	4.2	4.2	NUM
ejpam-6191	281	3	)	)	PUNCT
ejpam-6191	281	4	now	now	ADV
ejpam-6191	281	5	,	,	PUNCT
ejpam-6191	281	6	we	we	PRON
ejpam-6191	281	7	’ll	’ll	AUX
ejpam-6191	281	8	find	find	VERB
ejpam-6191	281	9	vn	vn	ADP
ejpam-6191	281	10	such	such	ADJ
ejpam-6191	281	11	that	that	SCONJ
ejpam-6191	281	12	∞∑	∞∑	NUM
ejpam-6191	281	13	n=0	n=0	NUM
ejpam-6191	281	14	vnz	vnz	NOUN
ejpam-6191	281	15	n	n	NOUN
ejpam-6191	281	16	=	=	SYM
ejpam-6191	281	17	1	1	NUM
ejpam-6191	281	18	1−	1−	NUM
ejpam-6191	281	19	bz	bz	PROPN
ejpam-6191	281	20	(	(	PUNCT
ejpam-6191	281	21	1	1	NUM
ejpam-6191	281	22	1−	1−	NUM
ejpam-6191	281	23	az	az	PROPN
ejpam-6191	281	24	1−bz	1−bz	PROPN
ejpam-6191	281	25	)	)	PUNCT
ejpam-6191	281	26	r	r	NOUN
ejpam-6191	281	27	lir	lir	NOUN
ejpam-6191	281	28	(	(	PUNCT
ejpam-6191	281	29	az	az	PROPN
ejpam-6191	281	30	1−	1−	NUM
ejpam-6191	281	31	bz	bz	PROPN
ejpam-6191	281	32	)	)	PUNCT
ejpam-6191	281	33	.	.	PUNCT
ejpam-6191	282	1	expand	expand	VERB
ejpam-6191	282	2	1	1	NUM
ejpam-6191	282	3	1−bz	1−bz	PROPN
ejpam-6191	282	4	.	.	PUNCT
ejpam-6191	283	1	the	the	DET
ejpam-6191	283	2	term	term	NOUN
ejpam-6191	283	3	1	1	NUM
ejpam-6191	283	4	1−bz	1−bz	PROPN
ejpam-6191	283	5	expands	expand	VERB
ejpam-6191	283	6	as	as	ADP
ejpam-6191	283	7	a	a	DET
ejpam-6191	283	8	geometric	geometric	ADJ
ejpam-6191	283	9	series	series	NOUN
ejpam-6191	283	10	:	:	PUNCT
ejpam-6191	283	11	1	1	NUM
ejpam-6191	283	12	1−	1−	NUM
ejpam-6191	283	13	bz	bz	PROPN
ejpam-6191	283	14	=	=	PROPN
ejpam-6191	284	1	∞∑	∞∑	DET
ejpam-6191	284	2	k=0	k=0	PROPN
ejpam-6191	284	3	bkzk	bkzk	NOUN
ejpam-6191	284	4	,	,	PUNCT
ejpam-6191	284	5	for	for	ADP
ejpam-6191	284	6	|bz|	|bz|	PROPN
ejpam-6191	284	7	<	<	X
ejpam-6191	284	8	1	1	X
ejpam-6191	284	9	.	.	X
ejpam-6191	285	1	expand	expand	VERB
ejpam-6191	285	2	(	(	PUNCT
ejpam-6191	285	3	1	1	NUM
ejpam-6191	285	4	1−	1−	NUM
ejpam-6191	285	5	az	az	PROPN
ejpam-6191	285	6	1−bz	1−bz	PROPN
ejpam-6191	285	7	)	)	PUNCT
ejpam-6191	285	8	r	r	NOUN
ejpam-6191	285	9	.	.	PUNCT
ejpam-6191	286	1	first	first	ADV
ejpam-6191	286	2	,	,	PUNCT
ejpam-6191	286	3	rewrite	rewrite	VERB
ejpam-6191	286	4	the	the	DET
ejpam-6191	286	5	denominator	denominator	NOUN
ejpam-6191	286	6	:	:	PUNCT
ejpam-6191	286	7	1−	1−	NUM
ejpam-6191	286	8	az	az	NOUN
ejpam-6191	286	9	1−	1−	NUM
ejpam-6191	287	1	bz	bz	PROPN
ejpam-6191	288	1	=	=	SYM
ejpam-6191	289	1	1−	1−	NUM
ejpam-6191	290	1	bz	bz	PROPN
ejpam-6191	291	1	−	−	PROPN
ejpam-6191	291	2	az	az	PROPN
ejpam-6191	291	3	1−	1−	NUM
ejpam-6191	291	4	bz	bz	PROPN
ejpam-6191	291	5	=	=	PRON
ejpam-6191	291	6	(	(	PUNCT
ejpam-6191	291	7	1−	1−	NUM
ejpam-6191	291	8	bz)−	bz)−	X
ejpam-6191	291	9	az	az	PROPN
ejpam-6191	291	10	1−	1−	NUM
ejpam-6191	291	11	bz	bz	PROPN
ejpam-6191	291	12	.	.	PUNCT
ejpam-6191	292	1	using	use	VERB
ejpam-6191	292	2	the	the	DET
ejpam-6191	292	3	newton	newton	PROPN
ejpam-6191	292	4	’s	’s	PART
ejpam-6191	292	5	binomial	binomial	PROPN
ejpam-6191	292	6	theorem	theorem	NOUN
ejpam-6191	292	7	for	for	ADP
ejpam-6191	292	8	(	(	PUNCT
ejpam-6191	292	9	1−	1−	NUM
ejpam-6191	292	10	x)−r	x)−r	PUNCT
ejpam-6191	293	1	when	when	SCONJ
ejpam-6191	293	2	|x|	|x|	PROPN
ejpam-6191	293	3	<	<	X
ejpam-6191	293	4	1	1	NUM
ejpam-6191	293	5	,	,	PUNCT
ejpam-6191	293	6	we	we	PRON
ejpam-6191	293	7	have	have	VERB
ejpam-6191	293	8	(	(	PUNCT
ejpam-6191	293	9	1−	1−	NUM
ejpam-6191	293	10	x)−r	x)−r	NOUN
ejpam-6191	294	1	=	=	SYM
ejpam-6191	295	1	∞∑	∞∑	NUM
ejpam-6191	295	2	m=0	m=0	PROPN
ejpam-6191	295	3	(	(	PUNCT
ejpam-6191	295	4	m+	m+	NOUN
ejpam-6191	295	5	r	r	NOUN
ejpam-6191	295	6	−	−	NUM
ejpam-6191	295	7	1	1	NUM
ejpam-6191	295	8	m	m	NOUN
ejpam-6191	295	9	)	)	PUNCT
ejpam-6191	295	10	xm	xm	PROPN
ejpam-6191	295	11	,	,	PUNCT
ejpam-6191	295	12	we	we	PRON
ejpam-6191	295	13	substitute	substitute	VERB
ejpam-6191	295	14	x	x	PUNCT
ejpam-6191	296	1	=	=	SYM
ejpam-6191	296	2	az	az	PROPN
ejpam-6191	296	3	1−bz	1−bz	PROPN
ejpam-6191	296	4	:(	:(	PUNCT
ejpam-6191	296	5	1	1	NUM
ejpam-6191	296	6	1−	1−	NUM
ejpam-6191	296	7	az	az	PROPN
ejpam-6191	296	8	1−bz	1−bz	PROPN
ejpam-6191	296	9	)	)	PUNCT
ejpam-6191	297	1	r	r	NOUN
ejpam-6191	297	2	=	=	SYM
ejpam-6191	298	1	∞∑	∞∑	NUM
ejpam-6191	298	2	m=0	m=0	PROPN
ejpam-6191	298	3	(	(	PUNCT
ejpam-6191	298	4	m+	m+	NOUN
ejpam-6191	298	5	r	r	NOUN
ejpam-6191	298	6	−	−	NUM
ejpam-6191	298	7	1	1	NUM
ejpam-6191	298	8	m	m	NOUN
ejpam-6191	298	9	)	)	PUNCT
ejpam-6191	298	10	(	(	PUNCT
ejpam-6191	298	11	az	az	PROPN
ejpam-6191	298	12	1−	1−	NUM
ejpam-6191	298	13	bz	bz	PROPN
ejpam-6191	298	14	)	)	PUNCT
ejpam-6191	298	15	m	m	VERB
ejpam-6191	298	16	.	.	PUNCT
ejpam-6191	299	1	rewriting	rewrite	VERB
ejpam-6191	299	2	,	,	PUNCT
ejpam-6191	299	3	∞∑	∞∑	PROPN
ejpam-6191	299	4	m=0	m=0	PROPN
ejpam-6191	299	5	(	(	PUNCT
ejpam-6191	299	6	m+	m+	NOUN
ejpam-6191	299	7	r	r	NOUN
ejpam-6191	299	8	−	−	NUM
ejpam-6191	299	9	1	1	NUM
ejpam-6191	299	10	m	m	NOUN
ejpam-6191	299	11	)	)	PUNCT
ejpam-6191	299	12	amzm(1−	amzm(1−	ADJ
ejpam-6191	299	13	bz)−m	bz)−m	NOUN
ejpam-6191	299	14	.	.	PUNCT
ejpam-6191	300	1	expanding	expand	VERB
ejpam-6191	300	2	(	(	PUNCT
ejpam-6191	300	3	1−	1−	NUM
ejpam-6191	300	4	bz)−m	bz)−m	NOUN
ejpam-6191	300	5	using	use	VERB
ejpam-6191	300	6	the	the	DET
ejpam-6191	300	7	binomial	binomial	ADJ
ejpam-6191	300	8	series	series	NOUN
ejpam-6191	300	9	,	,	PUNCT
ejpam-6191	300	10	k.	k.	PROPN
ejpam-6191	300	11	v.	v.	PROPN
ejpam-6191	300	12	m.	m.	PROPN
ejpam-6191	300	13	manulat	manulat	PROPN
ejpam-6191	300	14	,	,	PUNCT
ejpam-6191	300	15	r.	r.	PROPN
ejpam-6191	300	16	b.	b.	PROPN
ejpam-6191	300	17	corcino	corcino	PROPN
ejpam-6191	300	18	/	/	SYM
ejpam-6191	300	19	eur	eur	PROPN
ejpam-6191	300	20	.	.	PUNCT
ejpam-6191	301	1	j.	j.	PROPN
ejpam-6191	301	2	pure	pure	PROPN
ejpam-6191	301	3	appl	appl	PROPN
ejpam-6191	301	4	.	.	PROPN
ejpam-6191	301	5	math	math	PROPN
ejpam-6191	301	6	,	,	PUNCT
ejpam-6191	301	7	18	18	NUM
ejpam-6191	301	8	(	(	PUNCT
ejpam-6191	301	9	3	3	NUM
ejpam-6191	301	10	)	)	PUNCT
ejpam-6191	301	11	(	(	PUNCT
ejpam-6191	301	12	2025	2025	NUM
ejpam-6191	301	13	)	)	PUNCT
ejpam-6191	301	14	,	,	PUNCT
ejpam-6191	301	15	6191	6191	NUM
ejpam-6191	301	16	15	15	NUM
ejpam-6191	301	17	of	of	ADP
ejpam-6191	301	18	21	21	NUM
ejpam-6191	301	19	(	(	PUNCT
ejpam-6191	301	20	1−	1−	NUM
ejpam-6191	301	21	bz)−m	bz)−m	NOUN
ejpam-6191	301	22	=	=	NOUN
ejpam-6191	302	1	∞∑	∞∑	NUM
ejpam-6191	302	2	j=0	j=0	PROPN
ejpam-6191	302	3	(	(	PUNCT
ejpam-6191	302	4	m+	m+	NUM
ejpam-6191	302	5	j	j	NOUN
ejpam-6191	302	6	−	−	PROPN
ejpam-6191	302	7	1	1	NUM
ejpam-6191	302	8	j	j	PROPN
ejpam-6191	302	9	)	)	PUNCT
ejpam-6191	302	10	bjzj	bjzj	PROPN
ejpam-6191	302	11	,	,	PUNCT
ejpam-6191	302	12	we	we	PRON
ejpam-6191	302	13	obtain	obtain	VERB
ejpam-6191	302	14	:	:	PUNCT
ejpam-6191	302	15	∞∑	∞∑	NUM
ejpam-6191	302	16	m=0	m=0	PROPN
ejpam-6191	302	17	(	(	PUNCT
ejpam-6191	302	18	m+	m+	NOUN
ejpam-6191	302	19	r	r	NOUN
ejpam-6191	302	20	−	−	NUM
ejpam-6191	302	21	1	1	NUM
ejpam-6191	302	22	m	m	NOUN
ejpam-6191	302	23	)	)	PUNCT
ejpam-6191	302	24	amzm	amzm	PROPN
ejpam-6191	302	25	∞∑	∞∑	NUM
ejpam-6191	302	26	j=0	j=0	PROPN
ejpam-6191	302	27	(	(	PUNCT
ejpam-6191	302	28	m+	m+	NUM
ejpam-6191	302	29	j	j	NOUN
ejpam-6191	302	30	−	−	PROPN
ejpam-6191	302	31	1	1	NUM
ejpam-6191	302	32	j	j	PROPN
ejpam-6191	302	33	)	)	PUNCT
ejpam-6191	302	34	bjzj	bjzj	PROPN
ejpam-6191	302	35	.	.	PUNCT
ejpam-6191	303	1	rearranging	rearrange	VERB
ejpam-6191	303	2	the	the	DET
ejpam-6191	303	3	summations	summation	NOUN
ejpam-6191	303	4	:	:	PUNCT
ejpam-6191	303	5	∞∑	∞∑	NUM
ejpam-6191	303	6	m=0	m=0	PROPN
ejpam-6191	303	7	∞∑	∞∑	ADJ
ejpam-6191	303	8	j=0	j=0	PROPN
ejpam-6191	303	9	(	(	PUNCT
ejpam-6191	303	10	m+	m+	NOUN
ejpam-6191	303	11	r	r	NOUN
ejpam-6191	303	12	−	−	NUM
ejpam-6191	303	13	1	1	NUM
ejpam-6191	303	14	m	m	NOUN
ejpam-6191	303	15	)	)	PUNCT
ejpam-6191	303	16	(	(	PUNCT
ejpam-6191	303	17	m+	m+	NUM
ejpam-6191	303	18	j	j	NOUN
ejpam-6191	303	19	−	−	PROPN
ejpam-6191	303	20	1	1	NUM
ejpam-6191	303	21	j	j	PROPN
ejpam-6191	303	22	)	)	PUNCT
ejpam-6191	303	23	ambjzm+j	ambjzm+j	NOUN
ejpam-6191	303	24	.	.	PUNCT
ejpam-6191	304	1	expand	expand	VERB
ejpam-6191	304	2	the	the	DET
ejpam-6191	304	3	polylogarithm	polylogarithm	PROPN
ejpam-6191	304	4	function	function	PROPN
ejpam-6191	304	5	lir(x	lir(x	PROPN
ejpam-6191	304	6	)	)	PUNCT
ejpam-6191	304	7	.	.	PUNCT
ejpam-6191	305	1	the	the	DET
ejpam-6191	305	2	polylogarithm	polylogarithm	PROPN
ejpam-6191	305	3	function	function	NOUN
ejpam-6191	305	4	is	be	AUX
ejpam-6191	305	5	defined	define	VERB
ejpam-6191	305	6	as	as	ADP
ejpam-6191	305	7	:	:	PUNCT
ejpam-6191	305	8	lir(x	lir(x	PROPN
ejpam-6191	305	9	)	)	PUNCT
ejpam-6191	306	1	=	=	PUNCT
ejpam-6191	307	1	∞∑	∞∑	NUM
ejpam-6191	307	2	n=1	n=1	NUM
ejpam-6191	307	3	xn	xn	PROPN
ejpam-6191	307	4	nr	nr	PROPN
ejpam-6191	307	5	.	.	PUNCT
ejpam-6191	308	1	substituting	substitute	VERB
ejpam-6191	308	2	x	x	PUNCT
ejpam-6191	308	3	=	=	SYM
ejpam-6191	308	4	az	az	PROPN
ejpam-6191	308	5	1−bz	1−bz	PROPN
ejpam-6191	308	6	,	,	PUNCT
ejpam-6191	308	7	lir	lir	PROPN
ejpam-6191	308	8	(	(	PUNCT
ejpam-6191	308	9	az	az	PROPN
ejpam-6191	308	10	1−	1−	NUM
ejpam-6191	308	11	bz	bz	PROPN
ejpam-6191	308	12	)	)	PUNCT
ejpam-6191	309	1	=	=	PUNCT
ejpam-6191	310	1	∞∑	∞∑	NUM
ejpam-6191	310	2	n=1	n=1	PROPN
ejpam-6191	310	3	(	(	PUNCT
ejpam-6191	310	4	az)n	az)n	PROPN
ejpam-6191	310	5	(	(	PUNCT
ejpam-6191	310	6	1−	1−	NUM
ejpam-6191	310	7	bz)nnr	bz)nnr	NOUN
ejpam-6191	310	8	.	.	PUNCT
ejpam-6191	311	1	expanding	expand	VERB
ejpam-6191	311	2	(	(	PUNCT
ejpam-6191	311	3	1−	1−	NUM
ejpam-6191	311	4	bz)−n	bz)−n	PROPN
ejpam-6191	311	5	using	use	VERB
ejpam-6191	311	6	the	the	DET
ejpam-6191	311	7	binomial	binomial	ADJ
ejpam-6191	311	8	series	series	NOUN
ejpam-6191	311	9	,	,	PUNCT
ejpam-6191	311	10	(	(	PUNCT
ejpam-6191	311	11	1−	1−	NUM
ejpam-6191	311	12	bz)−n	bz)−n	NOUN
ejpam-6191	312	1	=	=	PUNCT
ejpam-6191	313	1	∞∑	∞∑	NUM
ejpam-6191	313	2	p=0	p=0	X
ejpam-6191	313	3	(	(	PUNCT
ejpam-6191	313	4	n+	n+	X
ejpam-6191	313	5	p−	p−	VERB
ejpam-6191	313	6	1	1	NUM
ejpam-6191	313	7	p	p	NOUN
ejpam-6191	313	8	)	)	PUNCT
ejpam-6191	313	9	bpzp	bpzp	NOUN
ejpam-6191	313	10	.	.	PUNCT
ejpam-6191	314	1	thus	thus	ADV
ejpam-6191	314	2	,	,	PUNCT
ejpam-6191	314	3	lir	lir	PROPN
ejpam-6191	314	4	(	(	PUNCT
ejpam-6191	314	5	az	az	PROPN
ejpam-6191	314	6	1−	1−	NUM
ejpam-6191	314	7	bz	bz	PROPN
ejpam-6191	314	8	)	)	PUNCT
ejpam-6191	315	1	=	=	PUNCT
ejpam-6191	316	1	∞∑	∞∑	NUM
ejpam-6191	316	2	n=1	n=1	PROPN
ejpam-6191	316	3	anzn	anzn	NOUN
ejpam-6191	316	4	nr	nr	NOUN
ejpam-6191	316	5	∞∑	∞∑	PROPN
ejpam-6191	316	6	p=0	p=0	PROPN
ejpam-6191	316	7	(	(	PUNCT
ejpam-6191	316	8	n+	n+	X
ejpam-6191	316	9	p−	p−	VERB
ejpam-6191	316	10	1	1	NUM
ejpam-6191	316	11	p	p	NOUN
ejpam-6191	316	12	)	)	PUNCT
ejpam-6191	316	13	bpzp	bpzp	NOUN
ejpam-6191	316	14	.	.	PUNCT
ejpam-6191	317	1	rewriting	rewrite	VERB
ejpam-6191	317	2	:	:	PUNCT
ejpam-6191	318	1	∞∑	∞∑	NUM
ejpam-6191	318	2	n=1	n=1	ADP
ejpam-6191	318	3	∞∑	∞∑	NUM
ejpam-6191	318	4	p=0	p=0	PROPN
ejpam-6191	318	5	anbp	anbp	NOUN
ejpam-6191	318	6	nr	nr	PROPN
ejpam-6191	318	7	(	(	PUNCT
ejpam-6191	318	8	n+	n+	X
ejpam-6191	318	9	p−	p−	VERB
ejpam-6191	318	10	1	1	NUM
ejpam-6191	318	11	p	p	NOUN
ejpam-6191	318	12	)	)	PUNCT
ejpam-6191	318	13	zn+p	zn+p	PROPN
ejpam-6191	318	14	.	.	PUNCT
ejpam-6191	319	1	k.	k.	PROPN
ejpam-6191	320	1	v.	v.	PROPN
ejpam-6191	320	2	m.	m.	PROPN
ejpam-6191	320	3	manulat	manulat	PROPN
ejpam-6191	320	4	,	,	PUNCT
ejpam-6191	320	5	r.	r.	PROPN
ejpam-6191	320	6	b.	b.	PROPN
ejpam-6191	320	7	corcino	corcino	PROPN
ejpam-6191	320	8	/	/	SYM
ejpam-6191	320	9	eur	eur	PROPN
ejpam-6191	320	10	.	.	PUNCT
ejpam-6191	321	1	j.	j.	PROPN
ejpam-6191	321	2	pure	pure	PROPN
ejpam-6191	321	3	appl	appl	PROPN
ejpam-6191	321	4	.	.	PROPN
ejpam-6191	321	5	math	math	PROPN
ejpam-6191	321	6	,	,	PUNCT
ejpam-6191	321	7	18	18	NUM
ejpam-6191	321	8	(	(	PUNCT
ejpam-6191	321	9	3	3	NUM
ejpam-6191	321	10	)	)	PUNCT
ejpam-6191	321	11	(	(	PUNCT
ejpam-6191	321	12	2025	2025	NUM
ejpam-6191	321	13	)	)	PUNCT
ejpam-6191	321	14	,	,	PUNCT
ejpam-6191	321	15	6191	6191	NUM
ejpam-6191	321	16	16	16	NUM
ejpam-6191	321	17	of	of	ADP
ejpam-6191	321	18	21	21	NUM
ejpam-6191	321	19	collecting	collect	VERB
ejpam-6191	321	20	terms	term	NOUN
ejpam-6191	321	21	for	for	ADP
ejpam-6191	321	22	vn	vn	NOUN
ejpam-6191	321	23	.	.	PUNCT
ejpam-6191	322	1	we	we	PRON
ejpam-6191	322	2	now	now	ADV
ejpam-6191	322	3	combine	combine	VERB
ejpam-6191	322	4	all	all	DET
ejpam-6191	322	5	expansions	expansion	NOUN
ejpam-6191	322	6	:	:	PUNCT
ejpam-6191	322	7	(	(	PUNCT
ejpam-6191	322	8	∞∑	∞∑	DET
ejpam-6191	322	9	k=0	k=0	PROPN
ejpam-6191	322	10	bkzk	bkzk	NOUN
ejpam-6191	322	11	)	)	PUNCT
ejpam-6191	323	1			PROPN
ejpam-6191	323	2	∞∑	∞∑	NUM
ejpam-6191	323	3	m=0	m=0	PROPN
ejpam-6191	323	4	∞∑	∞∑	PROPN
ejpam-6191	323	5	j=0	j=0	PROPN
ejpam-6191	323	6	(	(	PUNCT
ejpam-6191	323	7	m+	m+	NOUN
ejpam-6191	323	8	r	r	NOUN
ejpam-6191	323	9	−	−	NUM
ejpam-6191	323	10	1	1	NUM
ejpam-6191	323	11	m	m	NOUN
ejpam-6191	323	12	)	)	PUNCT
ejpam-6191	323	13	(	(	PUNCT
ejpam-6191	323	14	m+	m+	NUM
ejpam-6191	323	15	j	j	NOUN
ejpam-6191	323	16	−	−	PROPN
ejpam-6191	323	17	1	1	NUM
ejpam-6191	323	18	j	j	PROPN
ejpam-6191	323	19	)	)	PUNCT
ejpam-6191	323	20	ambjzm+j	ambjzm+j	ADJ
ejpam-6191	323	21	×	×	NOUN
ejpam-6191	323	22	(	(	PUNCT
ejpam-6191	323	23	4.3	4.3	NUM
ejpam-6191	323	24	)	)	PUNCT
ejpam-6191	323	25	×	×	NOUN
ejpam-6191	323	26			PROPN
ejpam-6191	323	27	∞∑	∞∑	NUM
ejpam-6191	323	28	n=1	n=1	ADP
ejpam-6191	324	1	∞∑	∞∑	NUM
ejpam-6191	324	2	p=0	p=0	PROPN
ejpam-6191	324	3	anbp	anbp	NOUN
ejpam-6191	324	4	nr	nr	PROPN
ejpam-6191	324	5	(	(	PUNCT
ejpam-6191	324	6	n+	n+	X
ejpam-6191	324	7	p−	p−	VERB
ejpam-6191	324	8	1	1	NUM
ejpam-6191	324	9	p	p	NOUN
ejpam-6191	324	10	)	)	PUNCT
ejpam-6191	324	11	zn+p	zn+p	PROPN
ejpam-6191	324	12			PROPN
ejpam-6191	324	13	.	.	PUNCT
ejpam-6191	325	1	(	(	PUNCT
ejpam-6191	325	2	4.4	4.4	NUM
ejpam-6191	325	3	)	)	PUNCT
ejpam-6191	325	4	we	we	PRON
ejpam-6191	325	5	denote	denote	VERB
ejpam-6191	325	6	:	:	PUNCT
ejpam-6191	325	7	a(z	a(z	NOUN
ejpam-6191	325	8	)	)	PUNCT
ejpam-6191	325	9	=	=	PUNCT
ejpam-6191	326	1	∞∑	∞∑	NUM
ejpam-6191	326	2	k=0	k=0	PROPN
ejpam-6191	326	3	bkzk	bkzk	NOUN
ejpam-6191	326	4	=	=	PUNCT
ejpam-6191	326	5	1	1	NUM
ejpam-6191	326	6	1−	1−	NUM
ejpam-6191	326	7	bz	bz	PROPN
ejpam-6191	326	8	b(z	b(z	NOUN
ejpam-6191	326	9	)	)	PUNCT
ejpam-6191	326	10	=	=	NOUN
ejpam-6191	327	1	∞∑	∞∑	NUM
ejpam-6191	327	2	m=0	m=0	PROPN
ejpam-6191	327	3	∞∑	∞∑	ADJ
ejpam-6191	327	4	j=0	j=0	PROPN
ejpam-6191	327	5	(	(	PUNCT
ejpam-6191	327	6	m+	m+	NOUN
ejpam-6191	327	7	r	r	NOUN
ejpam-6191	327	8	−	−	NUM
ejpam-6191	327	9	1	1	NUM
ejpam-6191	327	10	m	m	NOUN
ejpam-6191	327	11	)	)	PUNCT
ejpam-6191	327	12	(	(	PUNCT
ejpam-6191	327	13	m+	m+	NUM
ejpam-6191	327	14	j	j	NOUN
ejpam-6191	327	15	−	−	PROPN
ejpam-6191	327	16	1	1	NUM
ejpam-6191	327	17	j	j	PROPN
ejpam-6191	327	18	)	)	PUNCT
ejpam-6191	327	19	ambjzm+j	ambjzm+j	X
ejpam-6191	327	20	c(z	c(z	NOUN
ejpam-6191	327	21	)	)	PUNCT
ejpam-6191	327	22	=	=	PUNCT
ejpam-6191	328	1	∞∑	∞∑	PRON
ejpam-6191	328	2	t=1	t=1	ADV
ejpam-6191	328	3	∞∑	∞∑	NUM
ejpam-6191	328	4	p=0	p=0	PROPN
ejpam-6191	328	5	atbp	atbp	VERB
ejpam-6191	328	6	tr	tr	PUNCT
ejpam-6191	328	7	(	(	PUNCT
ejpam-6191	328	8	t+	t+	NOUN
ejpam-6191	328	9	p−	p−	NOUN
ejpam-6191	328	10	1	1	NUM
ejpam-6191	328	11	p	p	NOUN
ejpam-6191	328	12	)	)	PUNCT
ejpam-6191	328	13	zt+p	zt+p	PROPN
ejpam-6191	328	14	write	write	VERB
ejpam-6191	328	15	:	:	PUNCT
ejpam-6191	328	16	a(z	a(z	PROPN
ejpam-6191	328	17	)	)	PUNCT
ejpam-6191	328	18	=	=	PUNCT
ejpam-6191	329	1	∞∑	∞∑	NUM
ejpam-6191	329	2	k=0	k=0	PROPN
ejpam-6191	329	3	bkzk	bkzk	NOUN
ejpam-6191	329	4	,	,	PUNCT
ejpam-6191	329	5	b(z	b(z	NOUN
ejpam-6191	329	6	)	)	PUNCT
ejpam-6191	329	7	=	=	PUNCT
ejpam-6191	330	1	∞∑	∞∑	NUM
ejpam-6191	330	2	ℓ=0	ℓ=0	NUM
ejpam-6191	330	3	cℓz	cℓz	PROPN
ejpam-6191	330	4	ℓ	ℓ	INTJ
ejpam-6191	330	5	where	where	SCONJ
ejpam-6191	330	6	cℓ	cℓ	ADV
ejpam-6191	330	7	=	=	SYM
ejpam-6191	330	8	∑	∑	PROPN
ejpam-6191	330	9	m+j=ℓ	m+j=ℓ	X
ejpam-6191	330	10	(	(	PUNCT
ejpam-6191	330	11	m+	m+	NOUN
ejpam-6191	330	12	r	r	NOUN
ejpam-6191	330	13	−	−	NUM
ejpam-6191	330	14	1	1	NUM
ejpam-6191	330	15	m	m	NOUN
ejpam-6191	330	16	)	)	PUNCT
ejpam-6191	330	17	(	(	PUNCT
ejpam-6191	330	18	m+	m+	NUM
ejpam-6191	330	19	j	j	NOUN
ejpam-6191	330	20	−	−	PROPN
ejpam-6191	330	21	1	1	NUM
ejpam-6191	330	22	j	j	PROPN
ejpam-6191	330	23	)	)	PUNCT
ejpam-6191	330	24	ambj	ambj	PROPN
ejpam-6191	330	25	then	then	ADV
ejpam-6191	330	26	the	the	DET
ejpam-6191	330	27	cauchy	cauchy	ADJ
ejpam-6191	330	28	product	product	NOUN
ejpam-6191	330	29	becomes	become	VERB
ejpam-6191	330	30	:	:	PUNCT
ejpam-6191	330	31	a(z)b(z	a(z)b(z	NOUN
ejpam-6191	330	32	)	)	PUNCT
ejpam-6191	330	33	=	=	PUNCT
ejpam-6191	331	1	∞∑	∞∑	NUM
ejpam-6191	331	2	n=0	n=0	PROPN
ejpam-6191	331	3	zn	zn	NUM
ejpam-6191	331	4	n∑	n∑	PUNCT
ejpam-6191	331	5	k=0	k=0	PROPN
ejpam-6191	331	6	bk	bk	PROPN
ejpam-6191	331	7	·	·	PUNCT
ejpam-6191	331	8	cn−k	cn−k	VERB
ejpam-6191	332	1	so	so	SCONJ
ejpam-6191	332	2	we	we	PRON
ejpam-6191	332	3	obtain	obtain	VERB
ejpam-6191	332	4	:	:	PUNCT
ejpam-6191	332	5	a(z)b(z	a(z)b(z	NUM
ejpam-6191	332	6	)	)	PUNCT
ejpam-6191	332	7	=	=	PUNCT
ejpam-6191	333	1	∞∑	∞∑	NUM
ejpam-6191	333	2	n=0	n=0	PROPN
ejpam-6191	333	3	zn	zn	NUM
ejpam-6191	333	4	n∑	n∑	PUNCT
ejpam-6191	333	5	k=0	k=0	PROPN
ejpam-6191	333	6	bk	bk	PROPN
ejpam-6191	333	7	∑	∑	NOUN
ejpam-6191	333	8	m+j	m+j	NUM
ejpam-6191	333	9	=	=	NOUN
ejpam-6191	333	10	n−k	n−k	NOUN
ejpam-6191	333	11	(	(	PUNCT
ejpam-6191	333	12	m+	m+	NOUN
ejpam-6191	333	13	r	r	NOUN
ejpam-6191	333	14	−	−	NUM
ejpam-6191	333	15	1	1	NUM
ejpam-6191	333	16	m	m	NOUN
ejpam-6191	333	17	)	)	PUNCT
ejpam-6191	333	18	(	(	PUNCT
ejpam-6191	333	19	m+	m+	NUM
ejpam-6191	333	20	j	j	NOUN
ejpam-6191	333	21	−	−	PROPN
ejpam-6191	333	22	1	1	NUM
ejpam-6191	333	23	j	j	PROPN
ejpam-6191	333	24	)	)	PUNCT
ejpam-6191	333	25	ambj	ambj	PROPN
ejpam-6191	333	26	set	set	VERB
ejpam-6191	333	27	m+	m+	NUM
ejpam-6191	333	28	j	j	X
ejpam-6191	333	29	=	=	SYM
ejpam-6191	333	30	n−	n−	PROPN
ejpam-6191	333	31	k	k	PROPN
ejpam-6191	333	32	⇒	⇒	VERB
ejpam-6191	333	33	j	j	PROPN
ejpam-6191	334	1	=	=	PUNCT
ejpam-6191	334	2	n−	n−	PROPN
ejpam-6191	334	3	k	k	PROPN
ejpam-6191	334	4	−m	−m	NOUN
ejpam-6191	334	5	,	,	PUNCT
ejpam-6191	334	6	then	then	ADV
ejpam-6191	334	7	we	we	PRON
ejpam-6191	334	8	get	get	VERB
ejpam-6191	334	9	:	:	PUNCT
ejpam-6191	334	10	a(z)b(z	a(z)b(z	NUM
ejpam-6191	334	11	)	)	PUNCT
ejpam-6191	334	12	=	=	PUNCT
ejpam-6191	335	1	∞∑	∞∑	NUM
ejpam-6191	335	2	n=0	n=0	PROPN
ejpam-6191	335	3	zn	zn	NUM
ejpam-6191	335	4	n∑	n∑	PRON
ejpam-6191	335	5	k=0	k=0	PROPN
ejpam-6191	335	6	n−k∑	n−k∑	PROPN
ejpam-6191	335	7	m=0	m=0	PROPN
ejpam-6191	335	8	(	(	PUNCT
ejpam-6191	335	9	m+	m+	NOUN
ejpam-6191	335	10	r	r	NOUN
ejpam-6191	335	11	−	−	NUM
ejpam-6191	335	12	1	1	NUM
ejpam-6191	335	13	m	m	NOUN
ejpam-6191	335	14	)	)	PUNCT
ejpam-6191	335	15	(	(	PUNCT
ejpam-6191	335	16	m+	m+	NUM
ejpam-6191	336	1	n−	n−	NOUN
ejpam-6191	336	2	k	k	PROPN
ejpam-6191	336	3	−m−	−m−	PROPN
ejpam-6191	336	4	1	1	NUM
ejpam-6191	336	5	n−	n−	PROPN
ejpam-6191	336	6	k	k	PROPN
ejpam-6191	336	7	−m	−m	NOUN
ejpam-6191	336	8	)	)	PUNCT
ejpam-6191	337	1	ambk+n−k−m	ambk+n−k−m	PROPN
ejpam-6191	337	2	k.	k.	PROPN
ejpam-6191	337	3	v.	v.	PROPN
ejpam-6191	337	4	m.	m.	PROPN
ejpam-6191	337	5	manulat	manulat	PROPN
ejpam-6191	337	6	,	,	PUNCT
ejpam-6191	337	7	r.	r.	PROPN
ejpam-6191	337	8	b.	b.	PROPN
ejpam-6191	337	9	corcino	corcino	PROPN
ejpam-6191	337	10	/	/	SYM
ejpam-6191	337	11	eur	eur	PROPN
ejpam-6191	337	12	.	.	PUNCT
ejpam-6191	338	1	j.	j.	PROPN
ejpam-6191	338	2	pure	pure	PROPN
ejpam-6191	338	3	appl	appl	PROPN
ejpam-6191	338	4	.	.	PROPN
ejpam-6191	338	5	math	math	PROPN
ejpam-6191	338	6	,	,	PUNCT
ejpam-6191	338	7	18	18	NUM
ejpam-6191	338	8	(	(	PUNCT
ejpam-6191	338	9	3	3	NUM
ejpam-6191	338	10	)	)	PUNCT
ejpam-6191	338	11	(	(	PUNCT
ejpam-6191	338	12	2025	2025	NUM
ejpam-6191	338	13	)	)	PUNCT
ejpam-6191	338	14	,	,	PUNCT
ejpam-6191	338	15	6191	6191	NUM
ejpam-6191	338	16	17	17	NUM
ejpam-6191	338	17	of	of	ADP
ejpam-6191	338	18	21	21	NUM
ejpam-6191	338	19	=	=	SYM
ejpam-6191	338	20	∞∑	∞∑	NUM
ejpam-6191	338	21	n=0	n=0	PROPN
ejpam-6191	338	22	zn	zn	NUM
ejpam-6191	338	23	n∑	n∑	PRON
ejpam-6191	338	24	k=0	k=0	PROPN
ejpam-6191	338	25	n−k∑	n−k∑	PROPN
ejpam-6191	338	26	m=0	m=0	PROPN
ejpam-6191	338	27	(	(	PUNCT
ejpam-6191	338	28	m+	m+	NOUN
ejpam-6191	338	29	r	r	NOUN
ejpam-6191	338	30	−	−	NUM
ejpam-6191	338	31	1	1	NUM
ejpam-6191	338	32	m	m	NOUN
ejpam-6191	338	33	)	)	PUNCT
ejpam-6191	338	34	(	(	PUNCT
ejpam-6191	338	35	m+	m+	NUM
ejpam-6191	339	1	n−	n−	NOUN
ejpam-6191	339	2	k	k	PROPN
ejpam-6191	339	3	−m−	−m−	PROPN
ejpam-6191	339	4	1	1	NUM
ejpam-6191	339	5	n−	n−	PROPN
ejpam-6191	339	6	k	k	PROPN
ejpam-6191	339	7	−m	−m	NOUN
ejpam-6191	339	8	)	)	PUNCT
ejpam-6191	339	9	ambn−m	ambn−m	SCONJ
ejpam-6191	339	10	now	now	ADV
ejpam-6191	339	11	,	,	PUNCT
ejpam-6191	339	12	for	for	ADP
ejpam-6191	339	13	a(z)b(z)c(z	a(z)b(z)c(z	NUM
ejpam-6191	339	14	)	)	PUNCT
ejpam-6191	339	15	,	,	PUNCT
ejpam-6191	339	16	we	we	PRON
ejpam-6191	339	17	have	have	AUX
ejpam-6191	339	18	a(z)b(z	a(z)b(z	VERB
ejpam-6191	339	19	)	)	PUNCT
ejpam-6191	339	20	=	=	PUNCT
ejpam-6191	340	1	∞∑	∞∑	NUM
ejpam-6191	340	2	n=0	n=0	PROPN
ejpam-6191	340	3	zn	zn	NUM
ejpam-6191	340	4	n∑	n∑	PRON
ejpam-6191	340	5	k=0	k=0	PROPN
ejpam-6191	340	6	n−k∑	n−k∑	PROPN
ejpam-6191	340	7	m=0	m=0	PROPN
ejpam-6191	340	8	(	(	PUNCT
ejpam-6191	340	9	m+	m+	NOUN
ejpam-6191	340	10	r	r	NOUN
ejpam-6191	340	11	−	−	NUM
ejpam-6191	340	12	1	1	NUM
ejpam-6191	340	13	m	m	NOUN
ejpam-6191	340	14	)	)	PUNCT
ejpam-6191	340	15	(	(	PUNCT
ejpam-6191	340	16	n−	n−	NOUN
ejpam-6191	340	17	k	k	NOUN
ejpam-6191	340	18	−	−	PROPN
ejpam-6191	340	19	1	1	NUM
ejpam-6191	340	20	n−	n−	PROPN
ejpam-6191	340	21	k	k	PROPN
ejpam-6191	340	22	−m	−m	NOUN
ejpam-6191	340	23	)	)	PUNCT
ejpam-6191	340	24	ambn−m	ambn−m	SCONJ
ejpam-6191	340	25	let	let	VERB
ejpam-6191	340	26	:	:	PUNCT
ejpam-6191	340	27	an	an	DET
ejpam-6191	340	28	=	=	SYM
ejpam-6191	340	29	n∑	n∑	NOUN
ejpam-6191	340	30	k=0	k=0	PROPN
ejpam-6191	340	31	n−k∑	n−k∑	PROPN
ejpam-6191	340	32	m=0	m=0	PROPN
ejpam-6191	340	33	(	(	PUNCT
ejpam-6191	340	34	m+	m+	NOUN
ejpam-6191	340	35	r	r	NOUN
ejpam-6191	340	36	−	−	NUM
ejpam-6191	340	37	1	1	NUM
ejpam-6191	340	38	m	m	NOUN
ejpam-6191	340	39	)	)	PUNCT
ejpam-6191	340	40	(	(	PUNCT
ejpam-6191	340	41	n−	n−	NOUN
ejpam-6191	340	42	k	k	NOUN
ejpam-6191	340	43	−	−	PROPN
ejpam-6191	340	44	1	1	NUM
ejpam-6191	340	45	n−	n−	PROPN
ejpam-6191	340	46	k	k	PROPN
ejpam-6191	340	47	−m	−m	NOUN
ejpam-6191	340	48	)	)	PUNCT
ejpam-6191	340	49	ambn−m	ambn−m	ADV
ejpam-6191	340	50	then	then	ADV
ejpam-6191	340	51	:	:	PUNCT
ejpam-6191	340	52	a(z)b(z	a(z)b(z	NOUN
ejpam-6191	340	53	)	)	PUNCT
ejpam-6191	340	54	=	=	PUNCT
ejpam-6191	341	1	∞∑	∞∑	NUM
ejpam-6191	341	2	n=0	n=0	PUNCT
ejpam-6191	341	3	anz	anz	NOUN
ejpam-6191	341	4	n	n	NOUN
ejpam-6191	341	5	for	for	ADP
ejpam-6191	341	6	c(z	c(z	NOUN
ejpam-6191	341	7	)	)	PUNCT
ejpam-6191	341	8	,	,	PUNCT
ejpam-6191	341	9	let	let	VERB
ejpam-6191	341	10	bm	bm	PROPN
ejpam-6191	341	11	=	=	PRON
ejpam-6191	341	12	∑	∑	PROPN
ejpam-6191	341	13	t+p	t+p	NUM
ejpam-6191	341	14	=	=	NOUN
ejpam-6191	341	15	m	m	PROPN
ejpam-6191	341	16	t≥1	t≥1	NOUN
ejpam-6191	341	17	atbp	atbp	NOUN
ejpam-6191	341	18	tr	tr	NOUN
ejpam-6191	341	19	(	(	PUNCT
ejpam-6191	341	20	t+	t+	NOUN
ejpam-6191	341	21	p−	p−	NOUN
ejpam-6191	341	22	1	1	NUM
ejpam-6191	341	23	p	p	NOUN
ejpam-6191	341	24	)	)	PUNCT
ejpam-6191	341	25	then	then	ADV
ejpam-6191	341	26	:	:	PUNCT
ejpam-6191	341	27	c(z	c(z	X
ejpam-6191	341	28	)	)	PUNCT
ejpam-6191	341	29	=	=	PUNCT
ejpam-6191	342	1	∞∑	∞∑	NUM
ejpam-6191	342	2	m=1	m=1	VERB
ejpam-6191	342	3	bmzm	bmzm	NOUN
ejpam-6191	342	4	using	use	VERB
ejpam-6191	342	5	the	the	DET
ejpam-6191	342	6	cauchy	cauchy	ADJ
ejpam-6191	342	7	product	product	NOUN
ejpam-6191	342	8	of	of	ADP
ejpam-6191	342	9	power	power	NOUN
ejpam-6191	342	10	series	series	NOUN
ejpam-6191	342	11	:	:	PUNCT
ejpam-6191	342	12	a(z)b(z)c(z	a(z)b(z)c(z	NUM
ejpam-6191	342	13	)	)	PUNCT
ejpam-6191	342	14	=	=	PUNCT
ejpam-6191	343	1	∞∑	∞∑	PRON
ejpam-6191	343	2	n=0	n=0	NUM
ejpam-6191	343	3	cnz	cnz	NOUN
ejpam-6191	343	4	n	n	CCONJ
ejpam-6191	343	5	,	,	PUNCT
ejpam-6191	343	6	where	where	SCONJ
ejpam-6191	343	7	cn	cn	PROPN
ejpam-6191	343	8	=	=	SYM
ejpam-6191	343	9	n∑	n∑	PROPN
ejpam-6191	343	10	j=0	j=0	PROPN
ejpam-6191	343	11	ajbn−j	ajbn−j	ADV
ejpam-6191	343	12	using	use	VERB
ejpam-6191	343	13	our	our	PRON
ejpam-6191	343	14	earlier	early	ADJ
ejpam-6191	343	15	expressions	expression	NOUN
ejpam-6191	343	16	:	:	PUNCT
ejpam-6191	343	17	cn	cn	PROPN
ejpam-6191	343	18	=	=	SYM
ejpam-6191	343	19	n∑	n∑	PROPN
ejpam-6191	343	20	j=0	j=0	PROPN
ejpam-6191	343	21	ajbn−j	ajbn−j	ADV
ejpam-6191	343	22	=	=	SYM
ejpam-6191	343	23	n∑	n∑	X
ejpam-6191	343	24	j=0	j=0	PROPN
ejpam-6191	343	25	[	[	PUNCT
ejpam-6191	343	26	j∑	j∑	X
ejpam-6191	343	27	k=0	k=0	PROPN
ejpam-6191	343	28	j−k∑	j−k∑	PROPN
ejpam-6191	343	29	m=0	m=0	PROPN
ejpam-6191	343	30	(	(	PUNCT
ejpam-6191	343	31	m+	m+	NOUN
ejpam-6191	343	32	r	r	NOUN
ejpam-6191	343	33	−	−	NUM
ejpam-6191	343	34	1	1	NUM
ejpam-6191	343	35	m	m	NOUN
ejpam-6191	343	36	)	)	PUNCT
ejpam-6191	343	37	(	(	PUNCT
ejpam-6191	343	38	j	j	PROPN
ejpam-6191	343	39	−	−	PROPN
ejpam-6191	344	1	k	k	NOUN
ejpam-6191	345	1	−	−	PROPN
ejpam-6191	345	2	1	1	NUM
ejpam-6191	345	3	j	j	PROPN
ejpam-6191	345	4	−	−	PROPN
ejpam-6191	345	5	k	k	PROPN
ejpam-6191	345	6	−m	−m	PROPN
ejpam-6191	345	7	)	)	PUNCT
ejpam-6191	345	8	ambj−m	ambj−m	PROPN
ejpam-6191	345	9	]	]	PUNCT
ejpam-6191	345	10	×	×	NOUN
ejpam-6191	345	11			NOUN
ejpam-6191	345	12	∑	∑	NOUN
ejpam-6191	345	13	t+p	t+p	NOUN
ejpam-6191	345	14	=	=	SYM
ejpam-6191	345	15	n−j	n−j	NOUN
ejpam-6191	345	16	t≥1	t≥1	VERB
ejpam-6191	345	17	atbp	atbp	NOUN
ejpam-6191	345	18	tr	tr	NOUN
ejpam-6191	345	19	(	(	PUNCT
ejpam-6191	345	20	t+	t+	NOUN
ejpam-6191	345	21	p−	p−	NOUN
ejpam-6191	345	22	1	1	NUM
ejpam-6191	345	23	p	p	NOUN
ejpam-6191	345	24	)	)	PUNCT
ejpam-6191	345	25			PROPN
ejpam-6191	345	26	k.	k.	PROPN
ejpam-6191	345	27	v.	v.	ADP
ejpam-6191	345	28	m.	m.	PROPN
ejpam-6191	345	29	manulat	manulat	PROPN
ejpam-6191	345	30	,	,	PUNCT
ejpam-6191	345	31	r.	r.	PROPN
ejpam-6191	345	32	b.	b.	PROPN
ejpam-6191	345	33	corcino	corcino	PROPN
ejpam-6191	345	34	/	/	SYM
ejpam-6191	345	35	eur	eur	PROPN
ejpam-6191	345	36	.	.	PUNCT
ejpam-6191	346	1	j.	j.	PROPN
ejpam-6191	346	2	pure	pure	PROPN
ejpam-6191	346	3	appl	appl	PROPN
ejpam-6191	346	4	.	.	PROPN
ejpam-6191	346	5	math	math	PROPN
ejpam-6191	346	6	,	,	PUNCT
ejpam-6191	346	7	18	18	NUM
ejpam-6191	346	8	(	(	PUNCT
ejpam-6191	346	9	3	3	NUM
ejpam-6191	346	10	)	)	PUNCT
ejpam-6191	346	11	(	(	PUNCT
ejpam-6191	346	12	2025	2025	NUM
ejpam-6191	346	13	)	)	PUNCT
ejpam-6191	346	14	,	,	PUNCT
ejpam-6191	346	15	6191	6191	NUM
ejpam-6191	346	16	18	18	NUM
ejpam-6191	346	17	of	of	ADP
ejpam-6191	346	18	21	21	NUM
ejpam-6191	346	19	now	now	ADV
ejpam-6191	346	20	,	,	PUNCT
ejpam-6191	346	21	rearrange	rearrange	VERB
ejpam-6191	346	22	the	the	DET
ejpam-6191	346	23	entire	entire	ADJ
ejpam-6191	346	24	expression	expression	NOUN
ejpam-6191	346	25	for	for	ADP
ejpam-6191	346	26	the	the	DET
ejpam-6191	346	27	coefficient	coefficient	NOUN
ejpam-6191	346	28	cn	cn	PROPN
ejpam-6191	346	29	of	of	ADP
ejpam-6191	346	30	zn	zn	PROPN
ejpam-6191	346	31	:	:	PUNCT
ejpam-6191	346	32	a(z)b(z)c(z	a(z)b(z)c(z	NUM
ejpam-6191	346	33	)	)	PUNCT
ejpam-6191	346	34	=	=	PUNCT
ejpam-6191	347	1	∞∑	∞∑	NUM
ejpam-6191	347	2	n=0	n=0	PROPN
ejpam-6191	347	3	zn	zn	PROPN
ejpam-6191	347	4	n∑	n∑	PROPN
ejpam-6191	347	5	j=0	j=0	PROPN
ejpam-6191	347	6	j∑	j∑	PROPN
ejpam-6191	347	7	k=0	k=0	PROPN
ejpam-6191	347	8	j−k∑	j−k∑	PROPN
ejpam-6191	347	9	m=0	m=0	PROPN
ejpam-6191	347	10	∑	∑	PROPN
ejpam-6191	347	11	t+p	t+p	PUNCT
ejpam-6191	347	12	=	=	SYM
ejpam-6191	347	13	n−j	n−j	X
ejpam-6191	347	14	t≥1	t≥1	NOUN
ejpam-6191	347	15	(	(	PUNCT
ejpam-6191	347	16	m+	m+	NOUN
ejpam-6191	347	17	r	r	NOUN
ejpam-6191	347	18	−	−	NUM
ejpam-6191	347	19	1	1	NUM
ejpam-6191	347	20	m	m	NOUN
ejpam-6191	347	21	)	)	PUNCT
ejpam-6191	347	22	(	(	PUNCT
ejpam-6191	347	23	j	j	PROPN
ejpam-6191	347	24	−	−	PROPN
ejpam-6191	348	1	k	k	NOUN
ejpam-6191	349	1	−	−	PROPN
ejpam-6191	349	2	1	1	NUM
ejpam-6191	349	3	j	j	PROPN
ejpam-6191	349	4	−	−	PROPN
ejpam-6191	349	5	k	k	PROPN
ejpam-6191	349	6	−m	−m	PROPN
ejpam-6191	349	7	)	)	PUNCT
ejpam-6191	349	8	(	(	PUNCT
ejpam-6191	349	9	t+	t+	PUNCT
ejpam-6191	349	10	p−	p−	NOUN
ejpam-6191	349	11	1	1	NUM
ejpam-6191	349	12	p	p	NOUN
ejpam-6191	349	13	)	)	PUNCT
ejpam-6191	349	14	(	(	PUNCT
ejpam-6191	349	15	4.5	4.5	NUM
ejpam-6191	349	16	)	)	PUNCT
ejpam-6191	349	17	×	×	NOUN
ejpam-6191	349	18	am+tbj−m+p	am+tbj−m+p	NOUN
ejpam-6191	349	19	tr	tr	VERB
ejpam-6191	349	20	thus	thus	ADV
ejpam-6191	349	21	,	,	PUNCT
ejpam-6191	349	22	using	use	VERB
ejpam-6191	349	23	equation	equation	NOUN
ejpam-6191	349	24	(	(	PUNCT
ejpam-6191	349	25	4.5	4.5	NUM
ejpam-6191	349	26	)	)	PUNCT
ejpam-6191	349	27	,	,	PUNCT
ejpam-6191	349	28	we	we	PRON
ejpam-6191	349	29	have	have	VERB
ejpam-6191	349	30	∞∑	∞∑	NUM
ejpam-6191	349	31	n=1	n=1	PROPN
ejpam-6191	349	32	zn	zn	PROPN
ejpam-6191	349	33	(	(	PUNCT
ejpam-6191	349	34	n∑	n∑	NOUN
ejpam-6191	349	35	n=1	n=1	PROPN
ejpam-6191	349	36	(	(	PUNCT
ejpam-6191	349	37	n	n	CCONJ
ejpam-6191	349	38	k	k	NOUN
ejpam-6191	349	39	)	)	PUNCT
ejpam-6191	349	40	akbn−kh	akbn−kh	VERB
ejpam-6191	349	41	(	(	PUNCT
ejpam-6191	349	42	r	r	NOUN
ejpam-6191	349	43	)	)	PUNCT
ejpam-6191	349	44	k	k	NOUN
ejpam-6191	349	45	,	,	PUNCT
ejpam-6191	349	46	m	m	NOUN
ejpam-6191	349	47	)	)	PUNCT
ejpam-6191	350	1	=	=	PUNCT
ejpam-6191	351	1	∞∑	∞∑	PRON
ejpam-6191	351	2	n=0	n=0	NUM
ejpam-6191	351	3	vnz	vnz	NOUN
ejpam-6191	351	4	n	n	X
ejpam-6191	351	5	(	(	PUNCT
ejpam-6191	351	6	4.6	4.6	NUM
ejpam-6191	351	7	)	)	PUNCT
ejpam-6191	351	8	where	where	SCONJ
ejpam-6191	351	9	vn	vn	PROPN
ejpam-6191	351	10	=	=	PROPN
ejpam-6191	351	11	n∑	n∑	PROPN
ejpam-6191	351	12	j=0	j=0	PROPN
ejpam-6191	351	13	j∑	j∑	PROPN
ejpam-6191	351	14	k=0	k=0	PROPN
ejpam-6191	351	15	j−k∑	j−k∑	PROPN
ejpam-6191	351	16	m=0	m=0	PROPN
ejpam-6191	351	17	∑	∑	PROPN
ejpam-6191	351	18	t+p	t+p	PUNCT
ejpam-6191	351	19	=	=	SYM
ejpam-6191	351	20	n−j	n−j	X
ejpam-6191	351	21	t≥1	t≥1	NOUN
ejpam-6191	351	22	(	(	PUNCT
ejpam-6191	351	23	m+	m+	NOUN
ejpam-6191	351	24	r	r	NOUN
ejpam-6191	351	25	−	−	NUM
ejpam-6191	351	26	1	1	NUM
ejpam-6191	351	27	m	m	NOUN
ejpam-6191	351	28	)	)	PUNCT
ejpam-6191	351	29	(	(	PUNCT
ejpam-6191	351	30	j	j	PROPN
ejpam-6191	351	31	−	−	PROPN
ejpam-6191	351	32	k	k	NOUN
ejpam-6191	351	33	−	−	PROPN
ejpam-6191	351	34	1	1	NUM
ejpam-6191	351	35	j	j	PROPN
ejpam-6191	351	36	−	−	PROPN
ejpam-6191	351	37	k	k	PROPN
ejpam-6191	351	38	−m	−m	PROPN
ejpam-6191	351	39	)	)	PUNCT
ejpam-6191	351	40	(	(	PUNCT
ejpam-6191	351	41	t+	t+	PUNCT
ejpam-6191	351	42	p−	p−	NOUN
ejpam-6191	351	43	1	1	NUM
ejpam-6191	351	44	p	p	NOUN
ejpam-6191	351	45	)	)	PUNCT
ejpam-6191	351	46	×	×	NOUN
ejpam-6191	351	47	am+tbj−m+p	am+tbj−m+p	PROPN
ejpam-6191	351	48	tr	tr	VERB
ejpam-6191	351	49	.	.	PUNCT
ejpam-6191	351	50	comparing	compare	VERB
ejpam-6191	351	51	the	the	DET
ejpam-6191	351	52	coefficients	coefficient	NOUN
ejpam-6191	351	53	of	of	ADP
ejpam-6191	351	54	zn	zn	PROPN
ejpam-6191	351	55	yields	yield	NOUN
ejpam-6191	351	56	n∑	n∑	PROPN
ejpam-6191	351	57	n=1	n=1	PROPN
ejpam-6191	351	58	(	(	PUNCT
ejpam-6191	351	59	n	n	CCONJ
ejpam-6191	351	60	k	k	NOUN
ejpam-6191	351	61	)	)	PUNCT
ejpam-6191	351	62	akbn−kh	akbn−kh	VERB
ejpam-6191	351	63	(	(	PUNCT
ejpam-6191	351	64	r	r	NOUN
ejpam-6191	351	65	)	)	PUNCT
ejpam-6191	351	66	k	k	NOUN
ejpam-6191	351	67	,	,	PUNCT
ejpam-6191	351	68	m	m	PROPN
ejpam-6191	351	69	=	=	SYM
ejpam-6191	351	70	n∑	n∑	PROPN
ejpam-6191	351	71	j=0	j=0	PROPN
ejpam-6191	351	72	j∑	j∑	PROPN
ejpam-6191	351	73	k=0	k=0	PROPN
ejpam-6191	351	74	j−k∑	j−k∑	PROPN
ejpam-6191	351	75	m=0	m=0	PROPN
ejpam-6191	351	76	∑	∑	PROPN
ejpam-6191	351	77	t+p	t+p	PUNCT
ejpam-6191	351	78	=	=	SYM
ejpam-6191	351	79	n−j	n−j	X
ejpam-6191	351	80	t≥1	t≥1	NOUN
ejpam-6191	351	81	(	(	PUNCT
ejpam-6191	351	82	m+	m+	NOUN
ejpam-6191	351	83	r	r	NOUN
ejpam-6191	351	84	−	−	NUM
ejpam-6191	351	85	1	1	NUM
ejpam-6191	351	86	m	m	NOUN
ejpam-6191	351	87	)	)	PUNCT
ejpam-6191	351	88	(	(	PUNCT
ejpam-6191	351	89	j	j	PROPN
ejpam-6191	351	90	−	−	PROPN
ejpam-6191	352	1	k	k	NOUN
ejpam-6191	353	1	−	−	PROPN
ejpam-6191	353	2	1	1	NUM
ejpam-6191	353	3	j	j	PROPN
ejpam-6191	353	4	−	−	PROPN
ejpam-6191	353	5	k	k	PROPN
ejpam-6191	353	6	−m	−m	PROPN
ejpam-6191	353	7	)	)	PUNCT
ejpam-6191	353	8	(	(	PUNCT
ejpam-6191	353	9	t+	t+	PUNCT
ejpam-6191	353	10	p−	p−	NOUN
ejpam-6191	353	11	1	1	NUM
ejpam-6191	353	12	p	p	NOUN
ejpam-6191	353	13	)	)	PUNCT
ejpam-6191	353	14	×	×	NOUN
ejpam-6191	353	15	am+tbj−m+p	am+tbj−m+p	PROPN
ejpam-6191	353	16	tr	tr	VERB
ejpam-6191	353	17	.	.	PUNCT
ejpam-6191	354	1	5	5	X
ejpam-6191	354	2	.	.	X
ejpam-6191	354	3	conclusion	conclusion	NOUN
ejpam-6191	354	4	and	and	CCONJ
ejpam-6191	354	5	recommendation	recommendation	NOUN
ejpam-6191	354	6	in	in	ADP
ejpam-6191	354	7	this	this	DET
ejpam-6191	354	8	paper	paper	NOUN
ejpam-6191	354	9	,	,	PUNCT
ejpam-6191	354	10	we	we	PRON
ejpam-6191	354	11	have	have	AUX
ejpam-6191	354	12	derived	derive	VERB
ejpam-6191	354	13	a	a	DET
ejpam-6191	354	14	closed	closed	ADJ
ejpam-6191	354	15	-	-	PUNCT
ejpam-6191	354	16	form	form	NOUN
ejpam-6191	354	17	expression	expression	NOUN
ejpam-6191	354	18	for	for	ADP
ejpam-6191	354	19	a	a	DET
ejpam-6191	354	20	binomial	binomial	ADJ
ejpam-6191	354	21	sum	sum	NOUN
ejpam-6191	354	22	involving	involve	VERB
ejpam-6191	354	23	generalized	generalize	VERB
ejpam-6191	354	24	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	354	25	numbers	number	NOUN
ejpam-6191	354	26	h	h	NOUN
ejpam-6191	354	27	(	(	PUNCT
ejpam-6191	354	28	r	r	NOUN
ejpam-6191	354	29	)	)	PUNCT
ejpam-6191	354	30	k	k	NOUN
ejpam-6191	354	31	,	,	PUNCT
ejpam-6191	354	32	specifically	specifically	ADV
ejpam-6191	354	33	evaluating	evaluate	VERB
ejpam-6191	354	34	the	the	DET
ejpam-6191	354	35	convolution	convolution	NOUN
ejpam-6191	354	36	n∑	n∑	PROPN
ejpam-6191	354	37	k=0	k=0	PROPN
ejpam-6191	354	38	(	(	PUNCT
ejpam-6191	354	39	n	n	X
ejpam-6191	354	40	k	k	NOUN
ejpam-6191	354	41	)	)	PUNCT
ejpam-6191	354	42	akbn−kh	akbn−kh	VERB
ejpam-6191	354	43	(	(	PUNCT
ejpam-6191	354	44	r	r	NOUN
ejpam-6191	354	45	)	)	PUNCT
ejpam-6191	354	46	k	k	NOUN
ejpam-6191	354	47	using	use	VERB
ejpam-6191	354	48	generating	generate	VERB
ejpam-6191	354	49	function	function	NOUN
ejpam-6191	354	50	techniques	technique	NOUN
ejpam-6191	354	51	and	and	CCONJ
ejpam-6191	354	52	euler	euler	NOUN
ejpam-6191	354	53	-	-	PUNCT
ejpam-6191	354	54	type	type	NOUN
ejpam-6191	354	55	transforms	transform	VERB
ejpam-6191	354	56	.	.	PUNCT
ejpam-6191	355	1	the	the	DET
ejpam-6191	355	2	resulting	result	VERB
ejpam-6191	355	3	identity	identity	NOUN
ejpam-6191	355	4	expresses	express	VERB
ejpam-6191	355	5	the	the	DET
ejpam-6191	355	6	sum	sum	NOUN
ejpam-6191	355	7	in	in	ADP
ejpam-6191	355	8	terms	term	NOUN
ejpam-6191	355	9	of	of	ADP
ejpam-6191	355	10	a	a	DET
ejpam-6191	355	11	compact	compact	ADJ
ejpam-6191	355	12	expression	expression	NOUN
ejpam-6191	355	13	involving	involve	VERB
ejpam-6191	355	14	powers	power	NOUN
ejpam-6191	355	15	of	of	ADP
ejpam-6191	355	16	a	a	DET
ejpam-6191	355	17	+	+	NOUN
ejpam-6191	355	18	b	b	NOUN
ejpam-6191	355	19	,	,	PUNCT
ejpam-6191	355	20	b	b	NOUN
ejpam-6191	355	21	,	,	PUNCT
ejpam-6191	355	22	and	and	CCONJ
ejpam-6191	355	23	h	h	NOUN
ejpam-6191	355	24	(	(	PUNCT
ejpam-6191	355	25	r	r	NOUN
ejpam-6191	355	26	)	)	PUNCT
ejpam-6191	355	27	n	n	CCONJ
ejpam-6191	355	28	,	,	PUNCT
ejpam-6191	355	29	thereby	thereby	ADV
ejpam-6191	355	30	providing	provide	VERB
ejpam-6191	355	31	a	a	DET
ejpam-6191	355	32	deeper	deep	ADJ
ejpam-6191	355	33	analytical	analytical	ADJ
ejpam-6191	355	34	insight	insight	NOUN
ejpam-6191	355	35	into	into	ADP
ejpam-6191	355	36	the	the	DET
ejpam-6191	355	37	structure	structure	NOUN
ejpam-6191	355	38	of	of	ADP
ejpam-6191	355	39	such	such	ADJ
ejpam-6191	355	40	binomialhyperharmonic	binomialhyperharmonic	ADJ
ejpam-6191	355	41	combinations	combination	NOUN
ejpam-6191	355	42	.	.	PUNCT
ejpam-6191	356	1	k.	k.	PROPN
ejpam-6191	357	1	v.	v.	PROPN
ejpam-6191	357	2	m.	m.	PROPN
ejpam-6191	357	3	manulat	manulat	PROPN
ejpam-6191	357	4	,	,	PUNCT
ejpam-6191	357	5	r.	r.	PROPN
ejpam-6191	357	6	b.	b.	PROPN
ejpam-6191	357	7	corcino	corcino	PROPN
ejpam-6191	357	8	/	/	SYM
ejpam-6191	357	9	eur	eur	PROPN
ejpam-6191	357	10	.	.	PUNCT
ejpam-6191	358	1	j.	j.	PROPN
ejpam-6191	358	2	pure	pure	PROPN
ejpam-6191	358	3	appl	appl	PROPN
ejpam-6191	358	4	.	.	PROPN
ejpam-6191	358	5	math	math	PROPN
ejpam-6191	358	6	,	,	PUNCT
ejpam-6191	358	7	18	18	NUM
ejpam-6191	358	8	(	(	PUNCT
ejpam-6191	358	9	3	3	NUM
ejpam-6191	358	10	)	)	PUNCT
ejpam-6191	358	11	(	(	PUNCT
ejpam-6191	358	12	2025	2025	NUM
ejpam-6191	358	13	)	)	PUNCT
ejpam-6191	358	14	,	,	PUNCT
ejpam-6191	358	15	6191	6191	NUM
ejpam-6191	358	16	19	19	NUM
ejpam-6191	358	17	of	of	ADP
ejpam-6191	358	18	21	21	NUM
ejpam-6191	358	19	moreover	moreover	ADV
ejpam-6191	358	20	,	,	PUNCT
ejpam-6191	358	21	as	as	ADP
ejpam-6191	358	22	a	a	DET
ejpam-6191	358	23	significant	significant	ADJ
ejpam-6191	358	24	supplementary	supplementary	ADJ
ejpam-6191	358	25	result	result	NOUN
ejpam-6191	358	26	,	,	PUNCT
ejpam-6191	358	27	we	we	PRON
ejpam-6191	358	28	obtained	obtain	VERB
ejpam-6191	358	29	the	the	DET
ejpam-6191	358	30	polylogarithmic	polylogarithmic	ADJ
ejpam-6191	358	31	generating	generate	VERB
ejpam-6191	358	32	function	function	NOUN
ejpam-6191	358	33	of	of	ADP
ejpam-6191	358	34	the	the	DET
ejpam-6191	358	35	generalized	generalized	ADJ
ejpam-6191	358	36	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	358	37	numbers	number	NOUN
ejpam-6191	358	38	,	,	PUNCT
ejpam-6191	358	39	namely	namely	ADV
ejpam-6191	358	40	,	,	PUNCT
ejpam-6191	358	41	∞∑	∞∑	ADJ
ejpam-6191	358	42	n=1	n=1	PROPN
ejpam-6191	358	43	h(r	h(r	NOUN
ejpam-6191	358	44	)	)	PUNCT
ejpam-6191	358	45	n	n	CCONJ
ejpam-6191	358	46	,	,	PUNCT
ejpam-6191	358	47	mzn	mzn	NOUN
ejpam-6191	358	48	=	=	SYM
ejpam-6191	358	49	lim(z	lim(z	PROPN
ejpam-6191	358	50	)	)	PUNCT
ejpam-6191	358	51	(	(	PUNCT
ejpam-6191	358	52	1−	1−	NUM
ejpam-6191	358	53	z)r	z)r	NOUN
ejpam-6191	358	54	,	,	PUNCT
ejpam-6191	358	55	where	where	SCONJ
ejpam-6191	358	56	li1(z	li1(z	PROPN
ejpam-6191	358	57	)	)	PUNCT
ejpam-6191	358	58	=	=	PUNCT
ejpam-6191	359	1	−	−	PROPN
ejpam-6191	360	1	ln(1−	ln(1−	PROPN
ejpam-6191	360	2	z	z	PROPN
ejpam-6191	360	3	)	)	PUNCT
ejpam-6191	360	4	is	be	AUX
ejpam-6191	360	5	the	the	DET
ejpam-6191	360	6	polylogarithm	polylogarithm	NOUN
ejpam-6191	360	7	of	of	ADP
ejpam-6191	360	8	order	order	NOUN
ejpam-6191	360	9	1	1	X
ejpam-6191	360	10	.	.	PUNCT
ejpam-6191	361	1	this	this	DET
ejpam-6191	361	2	result	result	VERB
ejpam-6191	361	3	not	not	PART
ejpam-6191	361	4	only	only	ADV
ejpam-6191	361	5	confirms	confirm	VERB
ejpam-6191	361	6	the	the	DET
ejpam-6191	361	7	connection	connection	NOUN
ejpam-6191	361	8	between	between	ADP
ejpam-6191	361	9	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	361	10	numbers	number	NOUN
ejpam-6191	361	11	and	and	CCONJ
ejpam-6191	361	12	classical	classical	ADJ
ejpam-6191	361	13	polylogarithmic	polylogarithmic	ADJ
ejpam-6191	361	14	functions	function	NOUN
ejpam-6191	361	15	but	but	CCONJ
ejpam-6191	361	16	also	also	ADV
ejpam-6191	361	17	opens	open	VERB
ejpam-6191	361	18	avenues	avenue	NOUN
ejpam-6191	361	19	for	for	ADP
ejpam-6191	361	20	exploring	explore	VERB
ejpam-6191	361	21	their	their	PRON
ejpam-6191	361	22	analytic	analytic	ADJ
ejpam-6191	361	23	properties	property	NOUN
ejpam-6191	361	24	and	and	CCONJ
ejpam-6191	361	25	applications	application	NOUN
ejpam-6191	361	26	in	in	ADP
ejpam-6191	361	27	series	series	NOUN
ejpam-6191	361	28	transformations	transformation	NOUN
ejpam-6191	361	29	,	,	PUNCT
ejpam-6191	361	30	number	number	NOUN
ejpam-6191	361	31	theory	theory	NOUN
ejpam-6191	361	32	,	,	PUNCT
ejpam-6191	361	33	and	and	CCONJ
ejpam-6191	361	34	mathematical	mathematical	ADJ
ejpam-6191	361	35	physics	physics	NOUN
ejpam-6191	361	36	.	.	PUNCT
ejpam-6191	362	1	the	the	DET
ejpam-6191	362	2	techniques	technique	NOUN
ejpam-6191	362	3	applied	apply	VERB
ejpam-6191	362	4	throughout	throughout	ADP
ejpam-6191	362	5	the	the	DET
ejpam-6191	362	6	paper	paper	NOUN
ejpam-6191	362	7	especially	especially	ADV
ejpam-6191	362	8	those	those	PRON
ejpam-6191	362	9	involving	involve	VERB
ejpam-6191	362	10	generating	generating	NOUN
ejpam-6191	362	11	functions	function	NOUN
ejpam-6191	362	12	demonstrate	demonstrate	VERB
ejpam-6191	362	13	the	the	DET
ejpam-6191	362	14	elegance	elegance	NOUN
ejpam-6191	362	15	and	and	CCONJ
ejpam-6191	362	16	power	power	NOUN
ejpam-6191	362	17	of	of	ADP
ejpam-6191	362	18	analytic	analytic	ADJ
ejpam-6191	362	19	combinatorics	combinatoric	NOUN
ejpam-6191	362	20	in	in	ADP
ejpam-6191	362	21	deriving	derive	VERB
ejpam-6191	362	22	closed	closed	ADJ
ejpam-6191	362	23	-	-	PUNCT
ejpam-6191	362	24	form	form	NOUN
ejpam-6191	362	25	representations	representation	NOUN
ejpam-6191	362	26	of	of	ADP
ejpam-6191	362	27	complex	complex	ADJ
ejpam-6191	362	28	summations	summation	NOUN
ejpam-6191	362	29	.	.	PUNCT
ejpam-6191	363	1	in	in	ADP
ejpam-6191	363	2	light	light	NOUN
ejpam-6191	363	3	of	of	ADP
ejpam-6191	363	4	these	these	DET
ejpam-6191	363	5	findings	finding	NOUN
ejpam-6191	363	6	,	,	PUNCT
ejpam-6191	363	7	we	we	PRON
ejpam-6191	363	8	recommend	recommend	VERB
ejpam-6191	363	9	the	the	DET
ejpam-6191	363	10	following	follow	VERB
ejpam-6191	363	11	directions	direction	NOUN
ejpam-6191	363	12	for	for	ADP
ejpam-6191	363	13	future	future	ADJ
ejpam-6191	363	14	research	research	NOUN
ejpam-6191	363	15	:	:	PUNCT
ejpam-6191	363	16	(	(	PUNCT
ejpam-6191	363	17	i	i	NOUN
ejpam-6191	363	18	)	)	PUNCT
ejpam-6191	363	19	generalization	generalization	NOUN
ejpam-6191	363	20	to	to	ADP
ejpam-6191	363	21	higher	high	ADJ
ejpam-6191	363	22	-	-	PUNCT
ejpam-6191	363	23	order	order	NOUN
ejpam-6191	363	24	polylogarithms	polylogarithm	NOUN
ejpam-6191	363	25	:	:	PUNCT
ejpam-6191	363	26	extend	extend	VERB
ejpam-6191	363	27	the	the	DET
ejpam-6191	363	28	analysis	analysis	NOUN
ejpam-6191	363	29	to	to	ADP
ejpam-6191	363	30	generating	generating	NOUN
ejpam-6191	363	31	functions	function	NOUN
ejpam-6191	363	32	involving	involve	VERB
ejpam-6191	363	33	lis(z	lis(z	PROPN
ejpam-6191	363	34	)	)	PUNCT
ejpam-6191	363	35	for	for	ADP
ejpam-6191	363	36	s	s	PROPN
ejpam-6191	363	37	>	>	X
ejpam-6191	363	38	1	1	NUM
ejpam-6191	363	39	,	,	PUNCT
ejpam-6191	363	40	to	to	PART
ejpam-6191	363	41	derive	derive	VERB
ejpam-6191	363	42	identities	identity	NOUN
ejpam-6191	363	43	for	for	ADP
ejpam-6191	363	44	more	more	ADJ
ejpam-6191	363	45	generalized	generalized	ADJ
ejpam-6191	363	46	forms	form	NOUN
ejpam-6191	363	47	of	of	ADP
ejpam-6191	363	48	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	363	49	-	-	PUNCT
ejpam-6191	363	50	type	type	NOUN
ejpam-6191	363	51	sequences	sequence	NOUN
ejpam-6191	363	52	.	.	PUNCT
ejpam-6191	364	1	(	(	PUNCT
ejpam-6191	364	2	ii	ii	NOUN
ejpam-6191	364	3	)	)	PUNCT
ejpam-6191	364	4	exploration	exploration	NOUN
ejpam-6191	364	5	of	of	ADP
ejpam-6191	364	6	q	q	NOUN
ejpam-6191	364	7	-	-	PUNCT
ejpam-6191	364	8	analogues	analogue	NOUN
ejpam-6191	364	9	and	and	CCONJ
ejpam-6191	364	10	modular	modular	ADJ
ejpam-6191	364	11	connections	connection	NOUN
ejpam-6191	364	12	:	:	PUNCT
ejpam-6191	364	13	investigate	investigate	VERB
ejpam-6191	364	14	q	q	NOUN
ejpam-6191	364	15	-	-	PUNCT
ejpam-6191	364	16	analogues	analogue	NOUN
ejpam-6191	364	17	of	of	ADP
ejpam-6191	364	18	the	the	DET
ejpam-6191	364	19	generalized	generalize	VERB
ejpam-6191	364	20	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	364	21	numbers	number	NOUN
ejpam-6191	364	22	and	and	CCONJ
ejpam-6191	364	23	their	their	PRON
ejpam-6191	364	24	generating	generating	NOUN
ejpam-6191	364	25	functions	function	NOUN
ejpam-6191	364	26	,	,	PUNCT
ejpam-6191	364	27	particularly	particularly	ADV
ejpam-6191	364	28	their	their	PRON
ejpam-6191	364	29	relation	relation	NOUN
ejpam-6191	364	30	to	to	ADP
ejpam-6191	364	31	modular	modular	ADJ
ejpam-6191	364	32	forms	form	NOUN
ejpam-6191	364	33	or	or	CCONJ
ejpam-6191	364	34	q	q	NOUN
ejpam-6191	364	35	-	-	PUNCT
ejpam-6191	364	36	series	series	NOUN
ejpam-6191	364	37	.	.	PUNCT
ejpam-6191	365	1	(	(	PUNCT
ejpam-6191	365	2	iii	iii	X
ejpam-6191	365	3	)	)	PUNCT
ejpam-6191	365	4	applications	application	NOUN
ejpam-6191	365	5	in	in	ADP
ejpam-6191	365	6	analytic	analytic	ADJ
ejpam-6191	365	7	number	number	NOUN
ejpam-6191	365	8	theory	theory	NOUN
ejpam-6191	365	9	and	and	CCONJ
ejpam-6191	365	10	mathematical	mathematical	ADJ
ejpam-6191	365	11	physics	physics	NOUN
ejpam-6191	365	12	:	:	PUNCT
ejpam-6191	365	13	utilize	utilize	VERB
ejpam-6191	365	14	the	the	DET
ejpam-6191	365	15	polylogarithmic	polylogarithmic	ADJ
ejpam-6191	365	16	generating	generate	VERB
ejpam-6191	365	17	function	function	NOUN
ejpam-6191	365	18	in	in	ADP
ejpam-6191	365	19	evaluating	evaluate	VERB
ejpam-6191	365	20	special	special	ADJ
ejpam-6191	365	21	classes	class	NOUN
ejpam-6191	365	22	of	of	ADP
ejpam-6191	365	23	series	series	NOUN
ejpam-6191	365	24	,	,	PUNCT
ejpam-6191	365	25	integrals	integral	NOUN
ejpam-6191	365	26	,	,	PUNCT
ejpam-6191	365	27	or	or	CCONJ
ejpam-6191	365	28	zeta	zeta	NOUN
ejpam-6191	365	29	-	-	PUNCT
ejpam-6191	365	30	type	type	NOUN
ejpam-6191	365	31	functions	function	NOUN
ejpam-6191	365	32	that	that	PRON
ejpam-6191	365	33	appear	appear	VERB
ejpam-6191	365	34	in	in	ADP
ejpam-6191	365	35	number	number	NOUN
ejpam-6191	365	36	theory	theory	NOUN
ejpam-6191	365	37	and	and	CCONJ
ejpam-6191	365	38	quantum	quantum	ADJ
ejpam-6191	365	39	field	field	NOUN
ejpam-6191	365	40	theory	theory	NOUN
ejpam-6191	365	41	.	.	PUNCT
ejpam-6191	366	1	(	(	PUNCT
ejpam-6191	366	2	iv	iv	X
ejpam-6191	366	3	)	)	PUNCT
ejpam-6191	366	4	computational	computational	ADJ
ejpam-6191	366	5	implementations	implementation	NOUN
ejpam-6191	366	6	and	and	CCONJ
ejpam-6191	366	7	symbolic	symbolic	ADJ
ejpam-6191	366	8	analysis	analysis	NOUN
ejpam-6191	366	9	:	:	PUNCT
ejpam-6191	366	10	design	design	NOUN
ejpam-6191	366	11	symbolic	symbolic	ADJ
ejpam-6191	366	12	algorithms	algorithm	NOUN
ejpam-6191	366	13	that	that	PRON
ejpam-6191	366	14	implement	implement	VERB
ejpam-6191	366	15	the	the	DET
ejpam-6191	366	16	derived	derive	VERB
ejpam-6191	366	17	identities	identity	NOUN
ejpam-6191	366	18	and	and	CCONJ
ejpam-6191	366	19	generating	generating	NOUN
ejpam-6191	366	20	functions	function	NOUN
ejpam-6191	366	21	,	,	PUNCT
ejpam-6191	366	22	enabling	enable	VERB
ejpam-6191	366	23	automated	automate	VERB
ejpam-6191	366	24	discovery	discovery	NOUN
ejpam-6191	366	25	and	and	CCONJ
ejpam-6191	366	26	verification	verification	NOUN
ejpam-6191	366	27	of	of	ADP
ejpam-6191	366	28	related	relate	VERB
ejpam-6191	366	29	binomial	binomial	ADJ
ejpam-6191	366	30	and	and	CCONJ
ejpam-6191	366	31	harmonic	harmonic	ADJ
ejpam-6191	366	32	identities	identity	NOUN
ejpam-6191	366	33	.	.	PUNCT
ejpam-6191	367	1	through	through	ADP
ejpam-6191	367	2	these	these	DET
ejpam-6191	367	3	results	result	NOUN
ejpam-6191	367	4	,	,	PUNCT
ejpam-6191	367	5	we	we	PRON
ejpam-6191	367	6	contribute	contribute	VERB
ejpam-6191	367	7	not	not	PART
ejpam-6191	367	8	only	only	ADV
ejpam-6191	367	9	to	to	ADP
ejpam-6191	367	10	the	the	DET
ejpam-6191	367	11	combinatorial	combinatorial	ADJ
ejpam-6191	367	12	theory	theory	NOUN
ejpam-6191	367	13	of	of	ADP
ejpam-6191	367	14	special	special	ADJ
ejpam-6191	367	15	sequences	sequence	NOUN
ejpam-6191	367	16	but	but	CCONJ
ejpam-6191	367	17	also	also	ADV
ejpam-6191	367	18	to	to	ADP
ejpam-6191	367	19	the	the	DET
ejpam-6191	367	20	broader	broad	ADJ
ejpam-6191	367	21	field	field	NOUN
ejpam-6191	367	22	of	of	ADP
ejpam-6191	367	23	analytic	analytic	ADJ
ejpam-6191	367	24	combinatorics	combinatoric	NOUN
ejpam-6191	367	25	,	,	PUNCT
ejpam-6191	367	26	where	where	SCONJ
ejpam-6191	367	27	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	367	28	numbers	number	NOUN
ejpam-6191	367	29	and	and	CCONJ
ejpam-6191	367	30	polylogarithms	polylogarithm	NOUN
ejpam-6191	367	31	play	play	VERB
ejpam-6191	367	32	a	a	DET
ejpam-6191	367	33	central	central	ADJ
ejpam-6191	367	34	role	role	NOUN
ejpam-6191	367	35	.	.	PUNCT
ejpam-6191	368	1	references	reference	NOUN
ejpam-6191	368	2	[	[	X
ejpam-6191	368	3	1	1	NUM
ejpam-6191	368	4	]	]	PUNCT
ejpam-6191	368	5	m.	m.	NOUN
ejpam-6191	368	6	bahazsi	bahazsi	PROPN
ejpam-6191	368	7	and	and	CCONJ
ejpam-6191	368	8	s.	s.	PROPN
ejpam-6191	368	9	solak	solak	PROPN
ejpam-6191	368	10	,	,	PUNCT
ejpam-6191	368	11	an	an	DET
ejpam-6191	368	12	application	application	NOUN
ejpam-6191	368	13	of	of	ADP
ejpam-6191	368	14	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	368	15	numbers	number	NOUN
ejpam-6191	368	16	in	in	ADP
ejpam-6191	368	17	matrices	matrix	NOUN
ejpam-6191	368	18	,	,	PUNCT
ejpam-6191	368	19	hacet	hacet	NOUN
ejpam-6191	368	20	.	.	PUNCT
ejpam-6191	369	1	j.	j.	PROPN
ejpam-6191	369	2	math	math	PROPN
ejpam-6191	369	3	.	.	PUNCT
ejpam-6191	370	1	stat	stat	PROPN
ejpam-6191	370	2	.	.	PUNCT
ejpam-6191	371	1	42	42	NUM
ejpam-6191	371	2	(	(	PUNCT
ejpam-6191	371	3	2013	2013	NUM
ejpam-6191	371	4	)	)	PUNCT
ejpam-6191	371	5	,	,	PUNCT
ejpam-6191	371	6	387	387	NUM
ejpam-6191	371	7	-	-	SYM
ejpam-6191	371	8	393	393	NUM
ejpam-6191	371	9	.	.	PUNCT
ejpam-6191	372	1	[	[	X
ejpam-6191	372	2	2	2	NUM
ejpam-6191	372	3	]	]	X
ejpam-6191	372	4	a.t	a.t	PROPN
ejpam-6191	372	5	.	.	PROPN
ejpam-6191	372	6	benjamin	benjamin	PROPN
ejpam-6191	372	7	,	,	PUNCT
ejpam-6191	372	8	d.	d.	PROPN
ejpam-6191	372	9	gaebler	gaebler	PROPN
ejpam-6191	372	10	and	and	CCONJ
ejpam-6191	372	11	r.	r.	PROPN
ejpam-6191	372	12	gaebler	gaebler	NOUN
ejpam-6191	372	13	,	,	PUNCT
ejpam-6191	372	14	a	a	DET
ejpam-6191	372	15	combinatorial	combinatorial	ADJ
ejpam-6191	372	16	approach	approach	NOUN
ejpam-6191	372	17	to	to	ADP
ejpam-6191	372	18	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	372	19	numbers	number	NOUN
ejpam-6191	372	20	,	,	PUNCT
ejpam-6191	372	21	integers	integer	NOUN
ejpam-6191	372	22	:	:	PUNCT
ejpam-6191	372	23	electron	electron	NOUN
ejpam-6191	372	24	.	.	PUNCT
ejpam-6191	373	1	j.	j.	PROPN
ejpam-6191	373	2	combin	combin	PROPN
ejpam-6191	373	3	.	.	PUNCT
ejpam-6191	374	1	number	number	NOUN
ejpam-6191	374	2	theory	theory	NOUN
ejpam-6191	374	3	,	,	PUNCT
ejpam-6191	374	4	3	3	NUM
ejpam-6191	374	5	(	(	PUNCT
ejpam-6191	374	6	2003	2003	NUM
ejpam-6191	374	7	)	)	PUNCT
ejpam-6191	374	8	1	1	NUM
ejpam-6191	374	9	-	-	SYM
ejpam-6191	374	10	9	9	NUM
ejpam-6191	374	11	.	.	PUNCT
ejpam-6191	375	1	[	[	X
ejpam-6191	375	2	3	3	X
ejpam-6191	375	3	]	]	PUNCT
ejpam-6191	375	4	k.	k.	PROPN
ejpam-6191	375	5	n.	n.	PROPN
ejpam-6191	375	6	boyadzhiev	boyadzhiev	PROPN
ejpam-6191	375	7	,	,	PUNCT
ejpam-6191	375	8	harmonic	harmonic	ADJ
ejpam-6191	375	9	number	number	NOUN
ejpam-6191	375	10	identities	identity	NOUN
ejpam-6191	375	11	via	via	ADP
ejpam-6191	375	12	euler	euler	PROPN
ejpam-6191	375	13	’s	’s	PART
ejpam-6191	375	14	transform	transform	NOUN
ejpam-6191	375	15	.	.	PUNCT
ejpam-6191	376	1	j.	j.	PROPN
ejpam-6191	376	2	integer	integer	PROPN
ejpam-6191	376	3	sequences	sequences	PROPN
ejpam-6191	376	4	,	,	PUNCT
ejpam-6191	376	5	12(2009	12(2009	NUM
ejpam-6191	376	6	)	)	PUNCT
ejpam-6191	376	7	,	,	PUNCT
ejpam-6191	376	8	article	article	NOUN
ejpam-6191	376	9	09.6.1	09.6.1	NOUN
ejpam-6191	376	10	.	.	PUNCT
ejpam-6191	377	1	[	[	X
ejpam-6191	377	2	4	4	NUM
ejpam-6191	377	3	]	]	X
ejpam-6191	377	4	p.	p.	NOUN
ejpam-6191	377	5	catarino	catarino	PROPN
ejpam-6191	377	6	,	,	PUNCT
ejpam-6191	377	7	et	et	PROPN
ejpam-6191	377	8	al	al	PROPN
ejpam-6191	377	9	.	.	PROPN
ejpam-6191	377	10	,	,	PUNCT
ejpam-6191	377	11	on	on	ADP
ejpam-6191	377	12	the	the	DET
ejpam-6191	377	13	mersenne	mersenne	NOUN
ejpam-6191	377	14	sequence	sequence	NOUN
ejpam-6191	377	15	,	,	PUNCT
ejpam-6191	377	16	annales	annale	VERB
ejpam-6191	377	17	mathematicae	mathematicae	INTJ
ejpam-6191	377	18	et	et	PROPN
ejpam-6191	377	19	informaticae	informaticae	PROPN
ejpam-6191	377	20	.	.	PUNCT
ejpam-6191	378	1	(	(	PUNCT
ejpam-6191	378	2	2016	2016	NUM
ejpam-6191	378	3	)	)	PUNCT
ejpam-6191	378	4	,	,	PUNCT
ejpam-6191	378	5	37	37	NUM
ejpam-6191	378	6	-	-	SYM
ejpam-6191	378	7	53	53	NUM
ejpam-6191	378	8	.	.	PUNCT
ejpam-6191	379	1	k.	k.	PROPN
ejpam-6191	380	1	v.	v.	PROPN
ejpam-6191	380	2	m.	m.	PROPN
ejpam-6191	380	3	manulat	manulat	PROPN
ejpam-6191	380	4	,	,	PUNCT
ejpam-6191	380	5	r.	r.	PROPN
ejpam-6191	380	6	b.	b.	PROPN
ejpam-6191	380	7	corcino	corcino	PROPN
ejpam-6191	380	8	/	/	SYM
ejpam-6191	380	9	eur	eur	PROPN
ejpam-6191	380	10	.	.	PUNCT
ejpam-6191	381	1	j.	j.	PROPN
ejpam-6191	381	2	pure	pure	PROPN
ejpam-6191	381	3	appl	appl	PROPN
ejpam-6191	381	4	.	.	PROPN
ejpam-6191	381	5	math	math	PROPN
ejpam-6191	381	6	,	,	PUNCT
ejpam-6191	381	7	18	18	NUM
ejpam-6191	381	8	(	(	PUNCT
ejpam-6191	381	9	3	3	NUM
ejpam-6191	381	10	)	)	PUNCT
ejpam-6191	381	11	(	(	PUNCT
ejpam-6191	381	12	2025	2025	NUM
ejpam-6191	381	13	)	)	PUNCT
ejpam-6191	381	14	,	,	PUNCT
ejpam-6191	381	15	6191	6191	NUM
ejpam-6191	381	16	20	20	NUM
ejpam-6191	381	17	of	of	ADP
ejpam-6191	381	18	21	21	NUM
ejpam-6191	382	1	[	[	SYM
ejpam-6191	382	2	5	5	NUM
ejpam-6191	382	3	]	]	PUNCT
ejpam-6191	382	4	j.	j.	PROPN
ejpam-6191	382	5	l.	l.	PROPN
ejpam-6191	382	6	cereceda	cereceda	PROPN
ejpam-6191	382	7	,	,	PUNCT
ejpam-6191	382	8	an	an	DET
ejpam-6191	382	9	introduction	introduction	NOUN
ejpam-6191	382	10	to	to	ADP
ejpam-6191	382	11	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	382	12	numbers	number	NOUN
ejpam-6191	382	13	,	,	PUNCT
ejpam-6191	382	14	internat	internat	PROPN
ejpam-6191	382	15	.	.	PUNCT
ejpam-6191	383	1	j.	j.	PROPN
ejpam-6191	383	2	math	math	PROPN
ejpam-6191	383	3	.	.	PUNCT
ejpam-6191	384	1	ed	ed	NOUN
ejpam-6191	384	2	.	.	PUNCT
ejpam-6191	385	1	sci	sci	PROPN
ejpam-6191	385	2	.	.	PUNCT
ejpam-6191	385	3	tech	tech	PROPN
ejpam-6191	385	4	.	.	PUNCT
ejpam-6191	385	5	,	,	PUNCT
ejpam-6191	385	6	46(2015	46(2015	PROPN
ejpam-6191	385	7	)	)	PUNCT
ejpam-6191	385	8	,	,	PUNCT
ejpam-6191	385	9	461	461	NUM
ejpam-6191	385	10	-	-	SYM
ejpam-6191	385	11	469	469	NUM
ejpam-6191	385	12	.	.	PUNCT
ejpam-6191	386	1	[	[	X
ejpam-6191	386	2	6	6	NUM
ejpam-6191	386	3	]	]	PUNCT
ejpam-6191	386	4	j.	j.	PROPN
ejpam-6191	386	5	l.	l.	PROPN
ejpam-6191	386	6	cereceda	cereceda	PROPN
ejpam-6191	386	7	,	,	PUNCT
ejpam-6191	386	8	sums	sum	NOUN
ejpam-6191	386	9	of	of	ADP
ejpam-6191	386	10	powers	power	NOUN
ejpam-6191	386	11	of	of	ADP
ejpam-6191	386	12	integers	integer	NOUN
ejpam-6191	386	13	and	and	CCONJ
ejpam-6191	386	14	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	386	15	numbers	number	NOUN
ejpam-6191	386	16	.	.	PUNCT
ejpam-6191	387	1	notes	note	NOUN
ejpam-6191	387	2	on	on	ADP
ejpam-6191	387	3	number	number	NOUN
ejpam-6191	387	4	theory	theory	NOUN
ejpam-6191	387	5	and	and	CCONJ
ejpam-6191	387	6	discrete	discrete	ADJ
ejpam-6191	387	7	mathematics	mathematic	NOUN
ejpam-6191	387	8	.	.	PUNCT
ejpam-6191	388	1	27(2	27(2	NUM
ejpam-6191	388	2	)	)	PUNCT
ejpam-6191	388	3	(	(	PUNCT
ejpam-6191	388	4	2021	2021	NUM
ejpam-6191	388	5	)	)	PUNCT
ejpam-6191	388	6	,	,	PUNCT
ejpam-6191	388	7	101	101	NUM
ejpam-6191	388	8	-	-	SYM
ejpam-6191	388	9	110	110	NUM
ejpam-6191	388	10	.	.	PUNCT
ejpam-6191	389	1	[	[	X
ejpam-6191	389	2	7	7	X
ejpam-6191	389	3	]	]	X
ejpam-6191	389	4	j.	j.	PROPN
ejpam-6191	389	5	choi	choi	PROPN
ejpam-6191	389	6	,	,	PUNCT
ejpam-6191	389	7	certain	certain	ADJ
ejpam-6191	389	8	summation	summation	NOUN
ejpam-6191	389	9	formulas	formula	NOUN
ejpam-6191	389	10	involving	involve	VERB
ejpam-6191	389	11	harmonic	harmonic	ADJ
ejpam-6191	389	12	numbers	number	NOUN
ejpam-6191	389	13	and	and	CCONJ
ejpam-6191	389	14	generalized	generalized	ADJ
ejpam-6191	389	15	harmonic	harmonic	ADJ
ejpam-6191	389	16	nmbers	nmber	NOUN
ejpam-6191	389	17	,	,	PUNCT
ejpam-6191	389	18	appl	appl	PROPN
ejpam-6191	389	19	.	.	PROPN
ejpam-6191	389	20	math	math	PROPN
ejpam-6191	389	21	.	.	PUNCT
ejpam-6191	390	1	comput	comput	NOUN
ejpam-6191	390	2	,	,	PUNCT
ejpam-6191	390	3	218(2011),734	218(2011),734	NUM
ejpam-6191	390	4	-	-	PUNCT
ejpam-6191	390	5	740	740	NUM
ejpam-6191	390	6	.	.	PUNCT
ejpam-6191	391	1	[	[	X
ejpam-6191	391	2	8	8	NUM
ejpam-6191	391	3	]	]	X
ejpam-6191	391	4	j.	j.	PROPN
ejpam-6191	391	5	choi	choi	PROPN
ejpam-6191	391	6	,	,	PUNCT
ejpam-6191	391	7	h.m	h.m	PROPN
ejpam-6191	391	8	.	.	PROPN
ejpam-6191	391	9	srivastava	srivastava	PROPN
ejpam-6191	391	10	,	,	PUNCT
ejpam-6191	391	11	some	some	DET
ejpam-6191	391	12	summation	summation	NOUN
ejpam-6191	391	13	formulas	formula	NOUN
ejpam-6191	391	14	involving	involve	VERB
ejpam-6191	391	15	harmonic	harmonic	ADJ
ejpam-6191	391	16	numbers	number	NOUN
ejpam-6191	391	17	and	and	CCONJ
ejpam-6191	391	18	generalized	generalized	ADJ
ejpam-6191	391	19	harmonic	harmonic	ADJ
ejpam-6191	391	20	numbers	number	NOUN
ejpam-6191	391	21	.	.	PUNCT
ejpam-6191	392	1	mathematical	mathematical	ADJ
ejpam-6191	392	2	and	and	CCONJ
ejpam-6191	392	3	computer	computer	NOUN
ejpam-6191	392	4	modelling	modelling	NOUN
ejpam-6191	392	5	,	,	PUNCT
ejpam-6191	392	6	54	54	NUM
ejpam-6191	392	7	(	(	PUNCT
ejpam-6191	392	8	2011	2011	NUM
ejpam-6191	392	9	)	)	PUNCT
ejpam-6191	392	10	,	,	PUNCT
ejpam-6191	392	11	2220	2220	NUM
ejpam-6191	392	12	-	-	SYM
ejpam-6191	392	13	2234	2234	NUM
ejpam-6191	392	14	[	[	PUNCT
ejpam-6191	392	15	9	9	NUM
ejpam-6191	392	16	]	]	X
ejpam-6191	392	17	j.h	j.h	PROPN
ejpam-6191	392	18	.	.	PROPN
ejpam-6191	392	19	conway	conway	PROPN
ejpam-6191	392	20	,	,	PUNCT
ejpam-6191	392	21	r.k	r.k	PROPN
ejpam-6191	392	22	.	.	PROPN
ejpam-6191	392	23	guy	guy	PROPN
ejpam-6191	392	24	,	,	PUNCT
ejpam-6191	392	25	the	the	DET
ejpam-6191	392	26	book	book	NOUN
ejpam-6191	392	27	of	of	ADP
ejpam-6191	392	28	numbers	number	NOUN
ejpam-6191	392	29	.	.	PUNCT
ejpam-6191	393	1	copernicus	copernicus	PROPN
ejpam-6191	393	2	.	.	PUNCT
ejpam-6191	394	1	new	new	PROPN
ejpam-6191	394	2	york	york	PROPN
ejpam-6191	394	3	,	,	PUNCT
ejpam-6191	394	4	1996	1996	NUM
ejpam-6191	394	5	.	.	PUNCT
ejpam-6191	395	1	[	[	X
ejpam-6191	395	2	10	10	NUM
ejpam-6191	395	3	]	]	PUNCT
ejpam-6191	395	4	m.	m.	NOUN
ejpam-6191	395	5	c.	c.	PROPN
ejpam-6191	395	6	daglia	daglia	PROPN
ejpam-6191	395	7	,	,	PUNCT
ejpam-6191	395	8	f.	f.	PROPN
ejpam-6191	395	9	qib	qib	PROPN
ejpam-6191	395	10	,	,	PUNCT
ejpam-6191	395	11	several	several	ADJ
ejpam-6191	395	12	relations	relation	NOUN
ejpam-6191	395	13	and	and	CCONJ
ejpam-6191	395	14	identities	identity	NOUN
ejpam-6191	395	15	on	on	ADP
ejpam-6191	395	16	generalized	generalized	ADJ
ejpam-6191	395	17	derangement	derangement	NOUN
ejpam-6191	395	18	numbers	number	NOUN
ejpam-6191	395	19	,	,	PUNCT
ejpam-6191	395	20	results	result	NOUN
ejpam-6191	395	21	in	in	ADP
ejpam-6191	395	22	nonlinear	nonlinear	ADJ
ejpam-6191	395	23	analysis	analysis	NOUN
ejpam-6191	395	24	,	,	PUNCT
ejpam-6191	395	25	5	5	NUM
ejpam-6191	395	26	(	(	PUNCT
ejpam-6191	395	27	2022	2022	NUM
ejpam-6191	395	28	)	)	PUNCT
ejpam-6191	395	29	no	no	NOUN
ejpam-6191	395	30	.	.	NOUN
ejpam-6191	395	31	2	2	NUM
ejpam-6191	395	32	,	,	PUNCT
ejpam-6191	395	33	185190	185190	NUM
ejpam-6191	395	34	.	.	PUNCT
ejpam-6191	396	1	[	[	X
ejpam-6191	396	2	11	11	NUM
ejpam-6191	396	3	]	]	PUNCT
ejpam-6191	396	4	a.	a.	NOUN
ejpam-6191	396	5	dil	dil	PROPN
ejpam-6191	396	6	,	,	PUNCT
ejpam-6191	396	7	k.	k.	PROPN
ejpam-6191	396	8	boyadzhiev	boyadzhiev	PROPN
ejpam-6191	396	9	,	,	PUNCT
ejpam-6191	396	10	eulers	euler	NOUN
ejpam-6191	396	11	sums	sum	NOUN
ejpam-6191	396	12	of	of	ADP
ejpam-6191	396	13	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	396	14	numbers	number	NOUN
ejpam-6191	396	15	.	.	PUNCT
ejpam-6191	397	1	arxiv:1209.0604	arxiv:1209.0604	PROPN
ejpam-6191	397	2	,	,	PUNCT
ejpam-6191	397	3	2013	2013	NUM
ejpam-6191	397	4	.	.	PUNCT
ejpam-6191	398	1	[	[	X
ejpam-6191	398	2	12	12	NUM
ejpam-6191	398	3	]	]	PUNCT
ejpam-6191	398	4	a.	a.	NOUN
ejpam-6191	398	5	dil	dil	PROPN
ejpam-6191	398	6	,	,	PUNCT
ejpam-6191	398	7	i.	i.	NOUN
ejpam-6191	398	8	mezö	mezö	PROPN
ejpam-6191	398	9	and	and	CCONJ
ejpam-6191	398	10	m.	m.	NOUN
ejpam-6191	398	11	cenkci	cenkci	PROPN
ejpam-6191	398	12	,	,	PUNCT
ejpam-6191	398	13	evaluation	evaluation	NOUN
ejpam-6191	398	14	of	of	ADP
ejpam-6191	398	15	euler	euler	NOUN
ejpam-6191	398	16	-	-	PUNCT
ejpam-6191	398	17	like	like	ADJ
ejpam-6191	398	18	sums	sum	NOUN
ejpam-6191	398	19	via	via	ADP
ejpam-6191	398	20	hurwitz	hurwitz	PROPN
ejpam-6191	398	21	zeta	zeta	PROPN
ejpam-6191	398	22	values	value	NOUN
ejpam-6191	398	23	.	.	PUNCT
ejpam-6191	399	1	turkish	turkish	ADJ
ejpam-6191	399	2	j.	j.	PROPN
ejpam-6191	399	3	math	math	PROPN
ejpam-6191	399	4	.	.	PUNCT
ejpam-6191	399	5	,	,	PUNCT
ejpam-6191	399	6	41(6	41(6	PROPN
ejpam-6191	399	7	)	)	PUNCT
ejpam-6191	399	8	(	(	PUNCT
ejpam-6191	399	9	2017	2017	NUM
ejpam-6191	399	10	)	)	PUNCT
ejpam-6191	399	11	,	,	PUNCT
ejpam-6191	399	12	1640	1640	NUM
ejpam-6191	399	13	-	-	SYM
ejpam-6191	399	14	1655	1655	NUM
ejpam-6191	399	15	.	.	PUNCT
ejpam-6191	400	1	[	[	X
ejpam-6191	400	2	13	13	NUM
ejpam-6191	400	3	]	]	PUNCT
ejpam-6191	400	4	a.	a.	NOUN
ejpam-6191	400	5	duran	duran	PROPN
ejpam-6191	400	6	,	,	PUNCT
ejpam-6191	400	7	n.	n.	NOUN
ejpam-6191	400	8	ömür	ömür	NOUN
ejpam-6191	400	9	,	,	PUNCT
ejpam-6191	400	10	and	and	CCONJ
ejpam-6191	400	11	s.	s.	PROPN
ejpam-6191	400	12	koparal	koparal	PROPN
ejpam-6191	400	13	,	,	PUNCT
ejpam-6191	400	14	on	on	ADP
ejpam-6191	400	15	sums	sum	NOUN
ejpam-6191	400	16	with	with	ADP
ejpam-6191	400	17	generalized	generalized	ADJ
ejpam-6191	400	18	harmonic	harmonic	NOUN
ejpam-6191	400	19	,	,	PUNCT
ejpam-6191	400	20	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	400	21	and	and	CCONJ
ejpam-6191	400	22	special	special	ADJ
ejpam-6191	400	23	numbers	number	NOUN
ejpam-6191	400	24	,	,	PUNCT
ejpam-6191	400	25	miskolc	miskolc	ADJ
ejpam-6191	400	26	math	math	NOUN
ejpam-6191	400	27	.	.	PUNCT
ejpam-6191	401	1	notes	note	NOUN
ejpam-6191	401	2	,	,	PUNCT
ejpam-6191	401	3	21(2	21(2	NUM
ejpam-6191	401	4	)	)	PUNCT
ejpam-6191	401	5	(	(	PUNCT
ejpam-6191	401	6	2020	2020	NUM
ejpam-6191	401	7	)	)	PUNCT
ejpam-6191	401	8	,	,	PUNCT
ejpam-6191	401	9	791	791	NUM
ejpam-6191	401	10	-	-	SYM
ejpam-6191	401	11	803	803	NUM
ejpam-6191	401	12	;	;	PUNCT
ejpam-6191	401	13	available	available	ADJ
ejpam-6191	401	14	online	online	ADV
ejpam-6191	401	15	at	at	ADP
ejpam-6191	401	16	https://doi.org/10.18514/mmn.2020.3458	https://doi.org/10.18514/mmn.2020.3458	NOUN
ejpam-6191	401	17	.	.	PUNCT
ejpam-6191	402	1	[	[	X
ejpam-6191	402	2	14	14	NUM
ejpam-6191	402	3	]	]	X
ejpam-6191	402	4	r.	r.	PROPN
ejpam-6191	402	5	frontczak	frontczak	PROPN
ejpam-6191	402	6	,	,	PUNCT
ejpam-6191	402	7	harmonic	harmonic	ADJ
ejpam-6191	402	8	sums	sum	NOUN
ejpam-6191	402	9	via	via	ADP
ejpam-6191	402	10	euler	euler	PROPN
ejpam-6191	402	11	’s	’s	PART
ejpam-6191	402	12	transform	transform	NOUN
ejpam-6191	402	13	:	:	PUNCT
ejpam-6191	402	14	complementing	complement	VERB
ejpam-6191	402	15	the	the	DET
ejpam-6191	402	16	approach	approach	NOUN
ejpam-6191	402	17	of	of	ADP
ejpam-6191	402	18	boyadzhiev	boyadzhiev	NOUN
ejpam-6191	402	19	.	.	PUNCT
ejpam-6191	403	1	journal	journal	PROPN
ejpam-6191	403	2	of	of	ADP
ejpam-6191	403	3	integer	integer	PROPN
ejpam-6191	403	4	sequences	sequence	NOUN
ejpam-6191	403	5	,	,	PUNCT
ejpam-6191	403	6	23	23	NUM
ejpam-6191	403	7	(	(	PUNCT
ejpam-6191	403	8	2020	2020	NUM
ejpam-6191	403	9	)	)	PUNCT
ejpam-6191	403	10	,	,	PUNCT
ejpam-6191	403	11	article	article	NOUN
ejpam-6191	403	12	20.3.2	20.3.2	NUM
ejpam-6191	403	13	.	.	PUNCT
ejpam-6191	404	1	[	[	X
ejpam-6191	404	2	15	15	NUM
ejpam-6191	404	3	]	]	X
ejpam-6191	404	4	e.	e.	PROPN
ejpam-6191	404	5	gokcen	gokcen	PROPN
ejpam-6191	404	6	kocer	kocer	PROPN
ejpam-6191	404	7	and	and	CCONJ
ejpam-6191	404	8	n.	n.	PROPN
ejpam-6191	404	9	tuglu	tuglu	PROPN
ejpam-6191	404	10	,	,	PUNCT
ejpam-6191	404	11	the	the	DET
ejpam-6191	404	12	binet	binet	NOUN
ejpam-6191	404	13	formulas	formula	NOUN
ejpam-6191	404	14	for	for	ADP
ejpam-6191	404	15	the	the	DET
ejpam-6191	404	16	pell	pell	NOUN
ejpam-6191	404	17	and	and	CCONJ
ejpam-6191	404	18	pell	pell	NOUN
ejpam-6191	404	19	-	-	PUNCT
ejpam-6191	404	20	lucas	lucas	NOUN
ejpam-6191	404	21	p−	p−	NOUN
ejpam-6191	404	22	numbers	number	NOUN
ejpam-6191	404	23	,	,	PUNCT
ejpam-6191	404	24	research	research	NOUN
ejpam-6191	404	25	gate	gate	NOUN
ejpam-6191	404	26	.	.	PUNCT
ejpam-6191	404	27	85	85	NUM
ejpam-6191	404	28	(	(	PUNCT
ejpam-6191	404	29	2007	2007	NUM
ejpam-6191	404	30	)	)	PUNCT
ejpam-6191	405	1	p.	p.	NOUN
ejpam-6191	405	2	3	3	NUM
ejpam-6191	405	3	-	-	SYM
ejpam-6191	405	4	17	17	NUM
ejpam-6191	405	5	.	.	PUNCT
ejpam-6191	406	1	[	[	X
ejpam-6191	406	2	16	16	NUM
ejpam-6191	406	3	]	]	PUNCT
ejpam-6191	406	4	b.-n	b.-n	PROPN
ejpam-6191	406	5	.	.	PUNCT
ejpam-6191	407	1	guo	guo	PROPN
ejpam-6191	407	2	and	and	CCONJ
ejpam-6191	407	3	f.	f.	PROPN
ejpam-6191	407	4	qi	qi	PROPN
ejpam-6191	407	5	,	,	PUNCT
ejpam-6191	407	6	sharp	sharp	ADJ
ejpam-6191	407	7	inequalities	inequality	NOUN
ejpam-6191	407	8	for	for	ADP
ejpam-6191	407	9	the	the	DET
ejpam-6191	407	10	psi	psi	NOUN
ejpam-6191	407	11	function	function	NOUN
ejpam-6191	407	12	and	and	CCONJ
ejpam-6191	407	13	harmonic	harmonic	ADJ
ejpam-6191	407	14	numbers	number	NOUN
ejpam-6191	407	15	,	,	PUNCT
ejpam-6191	407	16	analysis	analysis	NOUN
ejpam-6191	407	17	,	,	PUNCT
ejpam-6191	407	18	34(2	34(2	NUM
ejpam-6191	407	19	)	)	PUNCT
ejpam-6191	407	20	(	(	PUNCT
ejpam-6191	407	21	2014	2014	NUM
ejpam-6191	407	22	)	)	PUNCT
ejpam-6191	407	23	,	,	PUNCT
ejpam-6191	407	24	201208	201208	NUM
ejpam-6191	407	25	;	;	PUNCT
ejpam-6191	407	26	available	available	ADJ
ejpam-6191	407	27	online	online	ADV
ejpam-6191	407	28	at	at	ADP
ejpam-6191	407	29	https://doi.org/10.1515/anly-20140001	https://doi.org/10.1515/anly-20140001	NOUN
ejpam-6191	407	30	.	.	PUNCT
ejpam-6191	408	1	[	[	X
ejpam-6191	408	2	17	17	NUM
ejpam-6191	408	3	]	]	X
ejpam-6191	408	4	d.	d.	PROPN
ejpam-6191	408	5	guo	guo	PROPN
ejpam-6191	408	6	,	,	PUNCT
ejpam-6191	408	7	w.	w.	PROPN
ejpam-6191	408	8	chu	chu	PROPN
ejpam-6191	408	9	,	,	PUNCT
ejpam-6191	408	10	summation	summation	NOUN
ejpam-6191	408	11	formulae	formulae	NOUN
ejpam-6191	408	12	involving	involve	VERB
ejpam-6191	408	13	multiple	multiple	ADJ
ejpam-6191	408	14	harmonic	harmonic	ADJ
ejpam-6191	408	15	numbers	number	NOUN
ejpam-6191	408	16	.	.	PUNCT
ejpam-6191	409	1	applicable	applicable	ADJ
ejpam-6191	409	2	analysis	analysis	NOUN
ejpam-6191	409	3	and	and	CCONJ
ejpam-6191	409	4	discrete	discrete	ADJ
ejpam-6191	409	5	mathematics	mathematic	NOUN
ejpam-6191	409	6	,	,	PUNCT
ejpam-6191	409	7	15(1	15(1	NUM
ejpam-6191	409	8	)	)	PUNCT
ejpam-6191	409	9	(	(	PUNCT
ejpam-6191	409	10	2021	2021	NUM
ejpam-6191	409	11	)	)	PUNCT
ejpam-6191	409	12	,	,	PUNCT
ejpam-6191	409	13	201	201	NUM
ejpam-6191	409	14	-	-	SYM
ejpam-6191	409	15	212	212	NUM
ejpam-6191	409	16	.	.	PUNCT
ejpam-6191	410	1	[	[	X
ejpam-6191	410	2	18	18	NUM
ejpam-6191	410	3	]	]	PUNCT
ejpam-6191	410	4	a.	a.	PROPN
ejpam-6191	410	5	f.	f.	PROPN
ejpam-6191	410	6	horadam	horadam	PROPN
ejpam-6191	410	7	,	,	PUNCT
ejpam-6191	410	8	jacobsthal	jacobsthal	ADJ
ejpam-6191	410	9	representaion	representaion	NOUN
ejpam-6191	410	10	numbers	number	NOUN
ejpam-6191	410	11	,	,	PUNCT
ejpam-6191	410	12	university	university	NOUN
ejpam-6191	410	13	of	of	ADP
ejpam-6191	410	14	new	new	PROPN
ejpam-6191	410	15	england	england	PROPN
ejpam-6191	410	16	,	,	PUNCT
ejpam-6191	410	17	armidale	armidale	PROPN
ejpam-6191	410	18	,	,	PUNCT
ejpam-6191	410	19	2351	2351	NUM
ejpam-6191	410	20	,	,	PUNCT
ejpam-6191	410	21	australia	australia	PROPN
ejpam-6191	410	22	.	.	PUNCT
ejpam-6191	411	1	[	[	X
ejpam-6191	411	2	19	19	NUM
ejpam-6191	411	3	]	]	PUNCT
ejpam-6191	411	4	k.	k.	NOUN
ejpam-6191	411	5	kamano	kamano	PROPN
ejpam-6191	411	6	,	,	PUNCT
ejpam-6191	411	7	dirichlet	dirichlet	PROPN
ejpam-6191	411	8	series	series	PROPN
ejpam-6191	411	9	associated	associate	VERB
ejpam-6191	411	10	with	with	ADP
ejpam-6191	411	11	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	411	12	numbers	number	NOUN
ejpam-6191	411	13	.	.	PUNCT
ejpam-6191	412	1	mem	mem	PROPN
ejpam-6191	412	2	.	.	PUNCT
ejpam-6191	412	3	osaka	osaka	PROPN
ejpam-6191	412	4	inst	inst	PROPN
ejpam-6191	412	5	.	.	PUNCT
ejpam-6191	412	6	tech	tech	PROPN
ejpam-6191	412	7	.	.	PUNCT
ejpam-6191	412	8	ser	ser	PROPN
ejpam-6191	412	9	.	.	PUNCT
ejpam-6191	413	1	a	a	DET
ejpam-6191	413	2	,	,	PUNCT
ejpam-6191	413	3	56(2	56(2	NUM
ejpam-6191	413	4	)	)	PUNCT
ejpam-6191	413	5	(	(	PUNCT
ejpam-6191	413	6	2011	2011	NUM
ejpam-6191	413	7	)	)	PUNCT
ejpam-6191	413	8	,	,	PUNCT
ejpam-6191	413	9	11	11	NUM
ejpam-6191	413	10	-	-	SYM
ejpam-6191	413	11	15	15	NUM
ejpam-6191	413	12	.	.	PUNCT
ejpam-6191	414	1	[	[	X
ejpam-6191	414	2	20	20	NUM
ejpam-6191	414	3	]	]	X
ejpam-6191	414	4	d.e	d.e	PROPN
ejpam-6191	414	5	.	.	PROPN
ejpam-6191	414	6	knuth	knuth	PROPN
ejpam-6191	414	7	,	,	PUNCT
ejpam-6191	414	8	the	the	DET
ejpam-6191	414	9	art	art	NOUN
ejpam-6191	414	10	of	of	ADP
ejpam-6191	414	11	computer	computer	NOUN
ejpam-6191	414	12	programming	programming	NOUN
ejpam-6191	414	13	.	.	PUNCT
ejpam-6191	415	1	vols	vol	NOUN
ejpam-6191	415	2	.	.	PUNCT
ejpam-6191	416	1	1	1	NUM
ejpam-6191	416	2	-	-	SYM
ejpam-6191	416	3	3	3	NUM
ejpam-6191	416	4	,	,	PUNCT
ejpam-6191	416	5	addisonwesley	addisonwesley	ADJ
ejpam-6191	416	6	,	,	PUNCT
ejpam-6191	416	7	reading	reading	NOUN
ejpam-6191	416	8	,	,	PUNCT
ejpam-6191	416	9	mass	mass	PROPN
ejpam-6191	416	10	.	.	PROPN
ejpam-6191	416	11	,	,	PUNCT
ejpam-6191	416	12	1968	1968	NUM
ejpam-6191	416	13	.	.	PUNCT
ejpam-6191	417	1	[	[	X
ejpam-6191	417	2	21	21	NUM
ejpam-6191	417	3	]	]	X
ejpam-6191	417	4	s.	s.	PROPN
ejpam-6191	417	5	koparal	koparal	PROPN
ejpam-6191	417	6	,	,	PUNCT
ejpam-6191	417	7	et	et	PROPN
ejpam-6191	417	8	.	.	PUNCT
ejpam-6191	418	1	al	al	PROPN
ejpam-6191	418	2	.	.	PROPN
ejpam-6191	418	3	,	,	PUNCT
ejpam-6191	418	4	on	on	ADP
ejpam-6191	418	5	generalized	generalized	ADJ
ejpam-6191	418	6	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	418	7	numbers	number	NOUN
ejpam-6191	418	8	of	of	ADP
ejpam-6191	418	9	order	order	NOUN
ejpam-6191	418	10	r	r	NOUN
ejpam-6191	418	11	,	,	PUNCT
ejpam-6191	418	12	hr	hr	NOUN
ejpam-6191	418	13	n	n	CCONJ
ejpam-6191	418	14	,	,	PUNCT
ejpam-6191	418	15	m(σ	m(σ	PROPN
ejpam-6191	418	16	)	)	PUNCT
ejpam-6191	418	17	,	,	PUNCT
ejpam-6191	418	18	notes	note	NOUN
ejpam-6191	418	19	on	on	ADP
ejpam-6191	418	20	number	number	NOUN
ejpam-6191	418	21	theory	theory	NOUN
ejpam-6191	418	22	and	and	CCONJ
ejpam-6191	418	23	discrete	discrete	ADJ
ejpam-6191	418	24	mathematics	mathematic	NOUN
ejpam-6191	418	25	,	,	PUNCT
ejpam-6191	418	26	29(4	29(4	NOUN
ejpam-6191	418	27	)	)	PUNCT
ejpam-6191	418	28	(	(	PUNCT
ejpam-6191	418	29	2023	2023	NUM
ejpam-6191	418	30	)	)	PUNCT
ejpam-6191	418	31	,	,	PUNCT
ejpam-6191	418	32	804	804	NUM
ejpam-6191	418	33	-	-	SYM
ejpam-6191	418	34	812	812	NUM
ejpam-6191	418	35	.	.	PUNCT
ejpam-6191	419	1	[	[	X
ejpam-6191	419	2	22	22	NUM
ejpam-6191	419	3	]	]	PUNCT
ejpam-6191	419	4	r.	r.	PROPN
ejpam-6191	419	5	li	li	PROPN
ejpam-6191	419	6	,	,	PUNCT
ejpam-6191	419	7	euler	euler	NOUN
ejpam-6191	419	8	sums	sum	NOUN
ejpam-6191	419	9	of	of	ADP
ejpam-6191	419	10	generalized	generalized	ADJ
ejpam-6191	419	11	hyperharmoni	hyperharmoni	NOUN
ejpam-6191	419	12	numbers	number	NOUN
ejpam-6191	419	13	.	.	PUNCT
ejpam-6191	420	1	arxiv:2103.10622	arxiv:2103.10622	PROPN
ejpam-6191	420	2	,	,	PUNCT
ejpam-6191	420	3	2020	2020	NUM
ejpam-6191	420	4	.	.	PUNCT
ejpam-6191	421	1	[	[	X
ejpam-6191	421	2	23	23	NUM
ejpam-6191	421	3	]	]	X
ejpam-6191	421	4	d.	d.	PROPN
ejpam-6191	421	5	mahajan	mahajan	PROPN
ejpam-6191	421	6	,	,	PUNCT
ejpam-6191	421	7	the	the	DET
ejpam-6191	421	8	binet	binet	NOUN
ejpam-6191	421	9	forms	form	NOUN
ejpam-6191	421	10	for	for	ADP
ejpam-6191	421	11	the	the	DET
ejpam-6191	421	12	fibonacci	fibonacci	PROPN
ejpam-6191	421	13	and	and	CCONJ
ejpam-6191	421	14	lucas	lucas	PROPN
ejpam-6191	421	15	numbers	number	NOUN
ejpam-6191	421	16	,	,	PUNCT
ejpam-6191	421	17	international	international	ADJ
ejpam-6191	421	18	journal	journal	NOUN
ejpam-6191	421	19	of	of	ADP
ejpam-6191	421	20	mathematics	mathematics	PROPN
ejpam-6191	421	21	treands	treand	NOUN
ejpam-6191	421	22	and	and	CCONJ
ejpam-6191	421	23	technology	technology	NOUN
ejpam-6191	421	24	,	,	PUNCT
ejpam-6191	421	25	10(2014	10(2014	NUM
ejpam-6191	421	26	)	)	PUNCT
ejpam-6191	421	27	,	,	PUNCT
ejpam-6191	421	28	page	page	NOUN
ejpam-6191	421	29	14	14	NUM
ejpam-6191	421	30	-	-	SYM
ejpam-6191	421	31	16	16	NUM
ejpam-6191	421	32	.	.	PUNCT
ejpam-6191	422	1	[	[	X
ejpam-6191	422	2	24	24	NUM
ejpam-6191	422	3	]	]	PUNCT
ejpam-6191	422	4	m.narayan	m.narayan	ADJ
ejpam-6191	422	5	murty	murty	NOUN
ejpam-6191	422	6	,	,	PUNCT
ejpam-6191	422	7	binayak	binayak	NOUN
ejpam-6191	422	8	padhy	padhy	NOUN
ejpam-6191	422	9	,	,	PUNCT
ejpam-6191	422	10	a	a	DET
ejpam-6191	422	11	study	study	NOUN
ejpam-6191	422	12	on	on	ADP
ejpam-6191	422	13	pell	pell	NOUN
ejpam-6191	422	14	and	and	CCONJ
ejpam-6191	422	15	pell	pell	NOUN
ejpam-6191	422	16	-	-	PUNCT
ejpam-6191	422	17	lucas	lucas	NOUN
ejpam-6191	422	18	numbers	number	NOUN
ejpam-6191	422	19	,	,	PUNCT
ejpam-6191	422	20	iosr	iosr	ADJ
ejpam-6191	422	21	journal	journal	NOUN
ejpam-6191	422	22	of	of	ADP
ejpam-6191	422	23	mathematics	mathematics	PROPN
ejpam-6191	422	24	(	(	PUNCT
ejpam-6191	422	25	iosr	iosr	PROPN
ejpam-6191	422	26	-	-	PUNCT
ejpam-6191	422	27	jm	jm	NOUN
ejpam-6191	422	28	)	)	PUNCT
ejpam-6191	422	29	,	,	PUNCT
ejpam-6191	422	30	e	e	X
ejpam-6191	422	31	-	-	PUNCT
ejpam-6191	422	32	issn	issn	NOUN
ejpam-6191	422	33	:	:	PUNCT
ejpam-6191	422	34	2278	2278	NUM
ejpam-6191	422	35	-	-	SYM
ejpam-6191	422	36	5728	5728	NUM
ejpam-6191	422	37	,	,	PUNCT
ejpam-6191	422	38	p	p	NOUN
ejpam-6191	422	39	-	-	PUNCT
ejpam-6191	422	40	issn	issn	NOUN
ejpam-6191	422	41	:	:	PUNCT
ejpam-6191	422	42	2319	2319	NUM
ejpam-6191	422	43	-	-	PUNCT
ejpam-6191	422	44	765x	765x	PROPN
ejpam-6191	422	45	.	.	PUNCT
ejpam-6191	423	1	volume	volume	NOUN
ejpam-6191	423	2	19	19	NUM
ejpam-6191	423	3	,	,	PUNCT
ejpam-6191	423	4	issue	issue	NOUN
ejpam-6191	423	5	2	2	NUM
ejpam-6191	423	6	ser	ser	NOUN
ejpam-6191	423	7	.	.	PUNCT
ejpam-6191	424	1	i	i	PRON
ejpam-6191	424	2	(	(	PUNCT
ejpam-6191	424	3	mar	mar	PROPN
ejpam-6191	424	4	.	.	PROPN
ejpam-6191	424	5	–	–	PUNCT
ejpam-6191	424	6	apr	apr	NOUN
ejpam-6191	424	7	.	.	PROPN
ejpam-6191	424	8	2023	2023	NUM
ejpam-6191	424	9	)	)	PUNCT
ejpam-6191	424	10	,	,	PUNCT
ejpam-6191	424	11	pp	pp	ADP
ejpam-6191	424	12	28	28	NUM
ejpam-6191	424	13	-	-	SYM
ejpam-6191	424	14	36	36	NUM
ejpam-6191	424	15	[	[	X
ejpam-6191	424	16	25	25	NUM
ejpam-6191	424	17	]	]	PUNCT
ejpam-6191	424	18	m.	m.	NOUN
ejpam-6191	424	19	narayan	narayan	PROPN
ejpam-6191	424	20	murty	murty	PROPN
ejpam-6191	424	21	,	,	PUNCT
ejpam-6191	424	22	”	"	PUNCT
ejpam-6191	424	23	a	a	DET
ejpam-6191	424	24	review	review	NOUN
ejpam-6191	424	25	on	on	ADP
ejpam-6191	424	26	jacobsthal	jacobsthal	ADJ
ejpam-6191	424	27	and	and	CCONJ
ejpam-6191	424	28	jacobsthal	jacobsthal	ADJ
ejpam-6191	424	29	-	-	PUNCT
ejpam-6191	424	30	lucas	lucas	NOUN
ejpam-6191	424	31	numbers	number	NOUN
ejpam-6191	424	32	”	"	PUNCT
ejpam-6191	424	33	,	,	PUNCT
ejpam-6191	424	34	iosr	iosr	PROPN
ejpam-6191	424	35	k.	k.	PROPN
ejpam-6191	424	36	v.	v.	ADP
ejpam-6191	424	37	m.	m.	PROPN
ejpam-6191	424	38	manulat	manulat	PROPN
ejpam-6191	424	39	,	,	PUNCT
ejpam-6191	424	40	r.	r.	PROPN
ejpam-6191	424	41	b.	b.	PROPN
ejpam-6191	424	42	corcino	corcino	PROPN
ejpam-6191	424	43	/	/	SYM
ejpam-6191	424	44	eur	eur	PROPN
ejpam-6191	424	45	.	.	PUNCT
ejpam-6191	425	1	j.	j.	PROPN
ejpam-6191	425	2	pure	pure	PROPN
ejpam-6191	425	3	appl	appl	PROPN
ejpam-6191	425	4	.	.	PROPN
ejpam-6191	425	5	math	math	PROPN
ejpam-6191	425	6	,	,	PUNCT
ejpam-6191	425	7	18	18	NUM
ejpam-6191	425	8	(	(	PUNCT
ejpam-6191	425	9	3	3	NUM
ejpam-6191	425	10	)	)	PUNCT
ejpam-6191	425	11	(	(	PUNCT
ejpam-6191	425	12	2025	2025	NUM
ejpam-6191	425	13	)	)	PUNCT
ejpam-6191	425	14	,	,	PUNCT
ejpam-6191	425	15	6191	6191	NUM
ejpam-6191	425	16	21	21	NUM
ejpam-6191	425	17	of	of	ADP
ejpam-6191	425	18	21	21	NUM
ejpam-6191	425	19	journal	journal	NOUN
ejpam-6191	425	20	of	of	ADP
ejpam-6191	425	21	mathematics	mathematics	PROPN
ejpam-6191	425	22	(	(	PUNCT
ejpam-6191	425	23	iosr	iosr	PROPN
ejpam-6191	425	24	-	-	PUNCT
ejpam-6191	425	25	jm	jm	NOUN
ejpam-6191	425	26	)	)	PUNCT
ejpam-6191	425	27	.	.	PUNCT
ejpam-6191	426	1	e	e	X
ejpam-6191	426	2	-	-	PUNCT
ejpam-6191	426	3	issn	issn	NOUN
ejpam-6191	426	4	:	:	PUNCT
ejpam-6191	426	5	2278	2278	NUM
ejpam-6191	426	6	-	-	SYM
ejpam-6191	426	7	5728	5728	NUM
ejpam-6191	426	8	,	,	PUNCT
ejpam-6191	426	9	p	p	NOUN
ejpam-6191	426	10	-	-	PUNCT
ejpam-6191	426	11	issn	issn	NOUN
ejpam-6191	426	12	:	:	PUNCT
ejpam-6191	426	13	2319	2319	NUM
ejpam-6191	426	14	-	-	PUNCT
ejpam-6191	426	15	765x	765x	PROPN
ejpam-6191	426	16	.	.	PUNCT
ejpam-6191	427	1	volume	volume	NOUN
ejpam-6191	427	2	18	18	NUM
ejpam-6191	427	3	,	,	PUNCT
ejpam-6191	427	4	issue	issue	VERB
ejpam-6191	427	5	3	3	NUM
ejpam-6191	427	6	ser	ser	NOUN
ejpam-6191	427	7	.	.	PUNCT
ejpam-6191	428	1	ii	ii	PROPN
ejpam-6191	428	2	(	(	PUNCT
ejpam-6191	428	3	may	may	PROPN
ejpam-6191	428	4	.	.	PUNCT
ejpam-6191	428	5	–	–	PUNCT
ejpam-6191	428	6	june	june	PROPN
ejpam-6191	428	7	.	.	PROPN
ejpam-6191	428	8	2022	2022	NUM
ejpam-6191	428	9	)	)	PUNCT
ejpam-6191	428	10	,	,	PUNCT
ejpam-6191	428	11	pp	pp	ADP
ejpam-6191	428	12	55	55	NUM
ejpam-6191	428	13	-	-	SYM
ejpam-6191	428	14	67	67	NUM
ejpam-6191	428	15	.	.	PUNCT
ejpam-6191	429	1	[	[	X
ejpam-6191	429	2	26	26	NUM
ejpam-6191	429	3	]	]	X
ejpam-6191	429	4	n.	n.	NOUN
ejpam-6191	429	5	ömür	ömür	PROPN
ejpam-6191	429	6	,	,	PUNCT
ejpam-6191	429	7	g.	g.	PROPN
ejpam-6191	429	8	bilgin	bilgin	PROPN
ejpam-6191	429	9	,	,	PUNCT
ejpam-6191	429	10	(	(	PUNCT
ejpam-6191	429	11	2018	2018	NUM
ejpam-6191	429	12	)	)	PUNCT
ejpam-6191	429	13	.	.	PUNCT
ejpam-6191	430	1	some	some	DET
ejpam-6191	430	2	applications	application	NOUN
ejpam-6191	430	3	of	of	ADP
ejpam-6191	430	4	the	the	DET
ejpam-6191	430	5	generalized	generalize	VERB
ejpam-6191	430	6	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	430	7	numbers	number	NOUN
ejpam-6191	430	8	of	of	ADP
ejpam-6191	430	9	order	order	NOUN
ejpam-6191	430	10	r	r	NOUN
ejpam-6191	430	11	,	,	PUNCT
ejpam-6191	430	12	hr	hr	NOUN
ejpam-6191	430	13	,	,	PUNCT
ejpam-6191	430	14	n.	n.	NOUN
ejpam-6191	430	15	advances	advance	NOUN
ejpam-6191	430	16	and	and	CCONJ
ejpam-6191	430	17	applications	application	NOUN
ejpam-6191	430	18	in	in	ADP
ejpam-6191	430	19	mathematical	mathematical	ADJ
ejpam-6191	430	20	sciences	science	NOUN
ejpam-6191	430	21	,	,	PUNCT
ejpam-6191	430	22	17(9	17(9	NOUN
ejpam-6191	430	23	)	)	PUNCT
ejpam-6191	430	24	,	,	PUNCT
ejpam-6191	430	25	617	617	NUM
ejpam-6191	430	26	-	-	SYM
ejpam-6191	430	27	627	627	NUM
ejpam-6191	430	28	.	.	PUNCT
ejpam-6191	431	1	[	[	X
ejpam-6191	431	2	27	27	NUM
ejpam-6191	431	3	]	]	X
ejpam-6191	431	4	n.	n.	NOUN
ejpam-6191	431	5	ömür	ömür	PROPN
ejpam-6191	431	6	,	,	PUNCT
ejpam-6191	431	7	s.	s.	PROPN
ejpam-6191	431	8	koparal	koparal	PROPN
ejpam-6191	431	9	,	,	PUNCT
ejpam-6191	431	10	(	(	PUNCT
ejpam-6191	431	11	2018	2018	NUM
ejpam-6191	431	12	)	)	PUNCT
ejpam-6191	431	13	.	.	PUNCT
ejpam-6191	432	1	on	on	ADP
ejpam-6191	432	2	the	the	DET
ejpam-6191	432	3	matrices	matrix	NOUN
ejpam-6191	432	4	with	with	ADP
ejpam-6191	432	5	the	the	DET
ejpam-6191	432	6	generalized	generalize	VERB
ejpam-6191	432	7	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	432	8	numbers	number	NOUN
ejpam-6191	432	9	of	of	ADP
ejpam-6191	432	10	order	order	NOUN
ejpam-6191	432	11	r.	r.	PROPN
ejpam-6191	432	12	asian	asian	PROPN
ejpam-6191	432	13	-	-	PUNCT
ejpam-6191	432	14	european	european	PROPN
ejpam-6191	432	15	journal	journal	NOUN
ejpam-6191	432	16	of	of	ADP
ejpam-6191	432	17	mathematics	mathematic	NOUN
ejpam-6191	432	18	,	,	PUNCT
ejpam-6191	432	19	11(3	11(3	NUM
ejpam-6191	432	20	)	)	PUNCT
ejpam-6191	432	21	,	,	PUNCT
ejpam-6191	432	22	1850045	1850045	NUM
ejpam-6191	432	23	.	.	PUNCT
ejpam-6191	433	1	[	[	X
ejpam-6191	433	2	28	28	NUM
ejpam-6191	433	3	]	]	X
ejpam-6191	433	4	n.	n.	NOUN
ejpam-6191	433	5	ömür	ömür	PROPN
ejpam-6191	433	6	,	,	PUNCT
ejpam-6191	433	7	et	et	PROPN
ejpam-6191	433	8	al	al	PROPN
ejpam-6191	433	9	.	.	PROPN
ejpam-6191	433	10	,	,	PUNCT
ejpam-6191	433	11	on	on	ADP
ejpam-6191	433	12	sums	sum	NOUN
ejpam-6191	433	13	with	with	ADP
ejpam-6191	433	14	generalized	generalized	ADJ
ejpam-6191	433	15	harmonic	harmonic	ADJ
ejpam-6191	433	16	numbers	number	NOUN
ejpam-6191	433	17	via	via	ADP
ejpam-6191	433	18	euler	euler	PROPN
ejpam-6191	433	19	’s	’s	PART
ejpam-6191	433	20	transform	transform	NOUN
ejpam-6191	433	21	.	.	PUNCT
ejpam-6191	434	1	notes	note	NOUN
ejpam-6191	434	2	on	on	ADP
ejpam-6191	434	3	number	number	NOUN
ejpam-6191	434	4	theory	theory	NOUN
ejpam-6191	434	5	and	and	CCONJ
ejpam-6191	434	6	discrete	discrete	ADJ
ejpam-6191	434	7	mathematics	mathematic	NOUN
ejpam-6191	434	8	.	.	PUNCT
ejpam-6191	435	1	volume	volume	NOUN
ejpam-6191	435	2	29(4	29(4	NOUN
ejpam-6191	435	3	)	)	PUNCT
ejpam-6191	435	4	(	(	PUNCT
ejpam-6191	435	5	2023	2023	NUM
ejpam-6191	435	6	)	)	PUNCT
ejpam-6191	435	7	,	,	PUNCT
ejpam-6191	435	8	695	695	NUM
ejpam-6191	435	9	-	-	SYM
ejpam-6191	435	10	704	704	NUM
ejpam-6191	435	11	.	.	PUNCT
ejpam-6191	436	1	[	[	X
ejpam-6191	436	2	29	29	NUM
ejpam-6191	436	3	]	]	X
ejpam-6191	436	4	e.	e.	PROPN
ejpam-6191	436	5	özkan	özkan	PROPN
ejpam-6191	436	6	,	,	PUNCT
ejpam-6191	436	7	mine	mine	PRON
ejpam-6191	436	8	uysal	uysal	ADJ
ejpam-6191	436	9	,	,	PUNCT
ejpam-6191	436	10	mersenne	mersenne	NOUN
ejpam-6191	436	11	-	-	PUNCT
ejpam-6191	436	12	lucas	lucas	PROPN
ejpam-6191	436	13	hybrid	hybrid	ADJ
ejpam-6191	436	14	number	number	NOUN
ejpam-6191	436	15	,	,	PUNCT
ejpam-6191	436	16	mathematica	mathematica	PROPN
ejpam-6191	436	17	montisnigri	montisnigri	PROPN
ejpam-6191	436	18	.	.	PUNCT
ejpam-6191	436	19	doi:10.20948	doi:10.20948	PROPN
ejpam-6191	436	20	/	/	SYM
ejpam-6191	436	21	mathmontis-2021	mathmontis-2021	NOUN
ejpam-6191	436	22	-	-	PUNCT
ejpam-6191	436	23	52	52	NUM
ejpam-6191	436	24	-	-	SYM
ejpam-6191	436	25	2	2	NUM
ejpam-6191	436	26	.	.	PUNCT
ejpam-6191	437	1	[	[	X
ejpam-6191	437	2	30	30	NUM
ejpam-6191	437	3	]	]	PUNCT
ejpam-6191	437	4	a.	a.	NOUN
ejpam-6191	437	5	sofo	sofo	NOUN
ejpam-6191	437	6	and	and	CCONJ
ejpam-6191	437	7	a.	a.	PROPN
ejpam-6191	437	8	singh	singh	PROPN
ejpam-6191	437	9	nimbran	nimbran	PROPN
ejpam-6191	437	10	,	,	PUNCT
ejpam-6191	437	11	euler	euler	NOUN
ejpam-6191	437	12	sums	sum	NOUN
ejpam-6191	437	13	and	and	CCONJ
ejpam-6191	437	14	integral	integral	ADJ
ejpam-6191	437	15	connections	connection	NOUN
ejpam-6191	437	16	,	,	PUNCT
ejpam-6191	437	17	mathematics	mathematic	NOUN
ejpam-6191	437	18	,	,	PUNCT
ejpam-6191	437	19	7(2019	7(2019	NUM
ejpam-6191	437	20	)	)	PUNCT
ejpam-6191	437	21	,	,	PUNCT
ejpam-6191	437	22	page	page	NOUN
ejpam-6191	437	23	07	07	NUM
ejpam-6191	437	24	[	[	X
ejpam-6191	437	25	31	31	NUM
ejpam-6191	437	26	]	]	PUNCT
ejpam-6191	437	27	j.	j.	PROPN
ejpam-6191	437	28	sondow	sondow	PROPN
ejpam-6191	437	29	,	,	PUNCT
ejpam-6191	437	30	analytic	analytic	ADJ
ejpam-6191	437	31	continuation	continuation	NOUN
ejpam-6191	437	32	of	of	ADP
ejpam-6191	437	33	riemann	riemann	PROPN
ejpam-6191	437	34	zeta	zeta	PROPN
ejpam-6191	437	35	function	function	PROPN
ejpam-6191	437	36	and	and	CCONJ
ejpam-6191	437	37	values	value	NOUN
ejpam-6191	437	38	at	at	ADP
ejpam-6191	437	39	negative	negative	ADJ
ejpam-6191	437	40	integers	integer	NOUN
ejpam-6191	437	41	via	via	ADP
ejpam-6191	437	42	euler	euler	PROPN
ejpam-6191	437	43	’s	’s	PART
ejpam-6191	437	44	transformation	transformation	NOUN
ejpam-6191	437	45	of	of	ADP
ejpam-6191	437	46	series	series	PROPN
ejpam-6191	437	47	,	,	PUNCT
ejpam-6191	437	48	proc	proc	PROPN
ejpam-6191	437	49	.	.	PUNCT
ejpam-6191	438	1	amer	amer	PROPN
ejpam-6191	438	2	.	.	PUNCT
ejpam-6191	438	3	math	math	PROPN
ejpam-6191	438	4	.	.	PUNCT
ejpam-6191	439	1	soc	soc	PROPN
ejpam-6191	439	2	.	.	PUNCT
ejpam-6191	439	3	,	,	PUNCT
ejpam-6191	439	4	120	120	NUM
ejpam-6191	439	5	(	(	PUNCT
ejpam-6191	439	6	1994	1994	NUM
ejpam-6191	439	7	)	)	PUNCT
ejpam-6191	439	8	,	,	PUNCT
ejpam-6191	439	9	421	421	NUM
ejpam-6191	439	10	?	?	NUM
ejpam-6191	439	11	424	424	NUM
ejpam-6191	439	12	.	.	PUNCT
ejpam-6191	440	1	[	[	X
ejpam-6191	440	2	32	32	NUM
ejpam-6191	440	3	]	]	X
ejpam-6191	440	4	uysal	uysal	PROPN
ejpam-6191	440	5	et	et	PROPN
ejpam-6191	440	6	al	al	PROPN
ejpam-6191	440	7	.	.	PROPN
ejpam-6191	440	8	,	,	PUNCT
ejpam-6191	440	9	on	on	ADP
ejpam-6191	440	10	pell	pell	NOUN
ejpam-6191	440	11	functions	function	NOUN
ejpam-6191	440	12	and	and	CCONJ
ejpam-6191	440	13	pell	pell	NOUN
ejpam-6191	440	14	numbers	number	NOUN
ejpam-6191	440	15	,	,	PUNCT
ejpam-6191	440	16	discrete	discrete	ADJ
ejpam-6191	440	17	mathematical	mathematical	ADJ
ejpam-6191	440	18	structures	structure	NOUN
ejpam-6191	440	19	,	,	PUNCT
ejpam-6191	440	20	doi	doi	NOUN
ejpam-6191	440	21	journal	journal	NOUN
ejpam-6191	440	22	:	:	PUNCT
ejpam-6191	440	23	10.37591	10.37591	NUM
ejpam-6191	440	24	/	/	SYM
ejpam-6191	440	25	rrdms	rrdms	NOUN
ejpam-6191	440	26	.	.	PUNCT
ejpam-6191	441	1	[	[	X
ejpam-6191	441	2	33	33	NUM
ejpam-6191	441	3	]	]	PUNCT
ejpam-6191	441	4	p.	p.	NOUN
ejpam-6191	441	5	vanini	vanini	NOUN
ejpam-6191	441	6	,	,	PUNCT
ejpam-6191	441	7	complex	complex	ADJ
ejpam-6191	441	8	analysis	analysis	NOUN
ejpam-6191	441	9	i	i	PRON
ejpam-6191	441	10	holomorphic	holomorphic	ADJ
ejpam-6191	441	11	functions	function	NOUN
ejpam-6191	441	12	,	,	PUNCT
ejpam-6191	441	13	theorem	theorem	NOUN
ejpam-6191	441	14	of	of	ADP
ejpam-6191	441	15	cauchy	cauchy	PROPN
ejpam-6191	441	16	,	,	PUNCT
ejpam-6191	441	17	research	research	NOUN
ejpam-6191	441	18	gate	gate	NOUN
ejpam-6191	441	19	,	,	PUNCT
ejpam-6191	441	20	1(2018	1(2018	NUM
ejpam-6191	441	21	)	)	PUNCT
ejpam-6191	441	22	4	4	NUM
ejpam-6191	441	23	-	-	SYM
ejpam-6191	441	24	5	5	NUM
ejpam-6191	441	25	.	.	PUNCT
ejpam-6191	442	1	[	[	X
ejpam-6191	442	2	34	34	NUM
ejpam-6191	442	3	]	]	X
ejpam-6191	442	4	ce	ce	PROPN
ejpam-6191	442	5	xu	xu	PROPN
ejpam-6191	442	6	,	,	PUNCT
ejpam-6191	442	7	euler	euler	NOUN
ejpam-6191	442	8	sums	sum	NOUN
ejpam-6191	442	9	of	of	ADP
ejpam-6191	442	10	generalized	generalized	ADJ
ejpam-6191	442	11	hyperharmonic	hyperharmonic	ADJ
ejpam-6191	442	12	numbers	number	NOUN
ejpam-6191	442	13	.	.	PUNCT
ejpam-6191	443	1	arxiv:2103.10622	arxiv:2103.10622	PROPN
ejpam-6191	443	2	,	,	PUNCT
ejpam-6191	443	3	2017	2017	NUM
ejpam-6191	443	4	.	.	PUNCT
ejpam-6191	444	1	[	[	X
ejpam-6191	444	2	35	35	NUM
ejpam-6191	444	3	]	]	X
ejpam-6191	444	4	d.	d.	PROPN
ejpam-6191	444	5	a.	a.	PROPN
ejpam-6191	444	6	zave	zave	PROPN
ejpam-6191	444	7	,	,	PUNCT
ejpam-6191	444	8	a	a	DET
ejpam-6191	444	9	series	series	NOUN
ejpam-6191	444	10	expansion	expansion	NOUN
ejpam-6191	444	11	involving	involve	VERB
ejpam-6191	444	12	the	the	DET
ejpam-6191	444	13	harmonic	harmonic	ADJ
ejpam-6191	444	14	numbers	number	NOUN
ejpam-6191	444	15	,	,	PUNCT
ejpam-6191	444	16	inform	inform	NOUN
ejpam-6191	444	17	.	.	PUNCT
ejpam-6191	444	18	process	process	NOUN
ejpam-6191	444	19	.	.	PUNCT
ejpam-6191	445	1	lett	lett	PROPN
ejpam-6191	445	2	.	.	PROPN
ejpam-6191	446	1	5	5	NUM
ejpam-6191	446	2	(	(	PUNCT
ejpam-6191	446	3	1976	1976	NUM
ejpam-6191	446	4	)	)	PUNCT
ejpam-6191	446	5	,	,	PUNCT
ejpam-6191	446	6	75	75	NUM
ejpam-6191	446	7	-	-	SYM
ejpam-6191	446	8	77	77	NUM
ejpam-6191	446	9	.	.	PUNCT