id	sid	tid	token	lemma	pos
cana-5491	1	1	communications	communication	NOUN
cana-5491	1	2	on	on	ADP
cana-5491	1	3	applied	apply	VERB
cana-5491	1	4	nonlinear	nonlinear	ADJ
cana-5491	1	5	analysis	analysis	NOUN
cana-5491	1	6	issn	issn	NOUN
cana-5491	1	7	:	:	PUNCT
cana-5491	1	8	1074	1074	NUM
cana-5491	1	9	-	-	PUNCT
cana-5491	1	10	133x	133x	NUM
cana-5491	1	11	vol	vol	VERB
cana-5491	1	12	32	32	NUM
cana-5491	1	13	no	no	NOUN
cana-5491	1	14	.	.	PUNCT
cana-5491	2	1	10s	10	NOUN
cana-5491	2	2	(	(	PUNCT
cana-5491	2	3	2025	2025	NUM
cana-5491	2	4	)	)	PUNCT
cana-5491	2	5	2434	2434	NUM
cana-5491	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	3	1	caputo	caputo	PROPN
cana-5491	3	2	derivative	derivative	ADJ
cana-5491	3	3	formulas	formula	NOUN
cana-5491	3	4	of	of	ADP
cana-5491	3	5	hurwitz	hurwitz	PROPN
cana-5491	3	6	-	-	PUNCT
cana-5491	3	7	lerch	lerch	PROPN
cana-5491	3	8	zeta	zeta	PROPN
cana-5491	3	9	function	function	PROPN
cana-5491	3	10	and	and	CCONJ
cana-5491	3	11	applications	application	NOUN
cana-5491	3	12	sandeep	sandeep	PROPN
cana-5491	3	13	kumar1	kumar1	PROPN
cana-5491	3	14	,	,	PUNCT
cana-5491	3	15	owais	owais	PROPN
cana-5491	3	16	khan2	khan2	PROPN
cana-5491	3	17	*	*	PROPN
cana-5491	3	18	,	,	PUNCT
cana-5491	4	1	n.	n.	PROPN
cana-5491	4	2	u.	u.	PROPN
cana-5491	4	3	khan3	khan3	PROPN
cana-5491	4	4	and	and	CCONJ
cana-5491	4	5	n.	n.	PROPN
cana-5491	4	6	ahmad4	ahmad4	PROPN
cana-5491	4	7	1,2,4department	1,2,4department	NUM
cana-5491	4	8	of	of	ADP
cana-5491	4	9	mathematics	mathematic	NOUN
cana-5491	4	10	and	and	CCONJ
cana-5491	4	11	statistics	statistic	NOUN
cana-5491	4	12	,	,	PUNCT
cana-5491	4	13	integral	integral	ADJ
cana-5491	4	14	university	university	NOUN
cana-5491	4	15	,	,	PUNCT
cana-5491	4	16	lucknow-226026,india	lucknow-226026,india	PROPN
cana-5491	4	17	emails	email	NOUN
cana-5491	4	18	:	:	PUNCT
cana-5491	4	19	1sandeep8603@gmail.com	1sandeep8603@gmail.com	NUM
cana-5491	4	20	,	,	PUNCT
cana-5491	4	21	2owkhan05@gmail.com	2owkhan05@gmail.com	NUM
cana-5491	4	22	,	,	PUNCT
cana-5491	4	23	4najmuddinahmad33@gmail.com	4najmuddinahmad33@gmail.com	X
cana-5491	5	1	3department	3department	NUM
cana-5491	5	2	of	of	ADP
cana-5491	5	3	applied	apply	VERB
cana-5491	5	4	mathematics	mathematic	NOUN
cana-5491	5	5	,	,	PUNCT
cana-5491	5	6	aligarh	aligarh	PROPN
cana-5491	5	7	muslim	muslim	PROPN
cana-5491	5	8	university	university	PROPN
cana-5491	5	9	,	,	PUNCT
cana-5491	5	10	aligarh-202002	aligarh-202002	NOUN
cana-5491	5	11	,	,	PUNCT
cana-5491	5	12	india	india	PROPN
cana-5491	5	13	email:3nukhanmath@gmail.com	email:3nukhanmath@gmail.com	PROPN
cana-5491	5	14	article	article	NOUN
cana-5491	5	15	history	history	NOUN
cana-5491	5	16	:	:	PUNCT
cana-5491	5	17	received	receive	VERB
cana-5491	5	18	:	:	PUNCT
cana-5491	5	19	12	12	NUM
cana-5491	5	20	-	-	SYM
cana-5491	5	21	01	01	NUM
cana-5491	5	22	-	-	PUNCT
cana-5491	5	23	2025	2025	NUM
cana-5491	5	24	revised	revise	VERB
cana-5491	5	25	:	:	PUNCT
cana-5491	5	26	15	15	NUM
cana-5491	5	27	-	-	NUM
cana-5491	5	28	02	02	NUM
cana-5491	5	29	-	-	PUNCT
cana-5491	5	30	2025	2025	NUM
cana-5491	5	31	accepted	accept	VERB
cana-5491	5	32	:	:	PUNCT
cana-5491	5	33	01	01	NUM
cana-5491	5	34	-	-	SYM
cana-5491	5	35	03	03	NUM
cana-5491	5	36	-	-	PUNCT
cana-5491	5	37	2025	2025	NUM
cana-5491	5	38	abstract	abstract	NOUN
cana-5491	5	39	:	:	PUNCT
cana-5491	5	40	in	in	ADP
cana-5491	5	41	this	this	DET
cana-5491	5	42	paper	paper	NOUN
cana-5491	5	43	,	,	PUNCT
cana-5491	5	44	we	we	PRON
cana-5491	5	45	find	find	VERB
cana-5491	5	46	the	the	DET
cana-5491	5	47	fractional	fractional	ADJ
cana-5491	5	48	derivative	derivative	ADJ
cana-5491	5	49	formulas	formula	NOUN
cana-5491	5	50	of	of	ADP
cana-5491	5	51	hurwitz	hurwitz	PROPN
cana-5491	5	52	-	-	PUNCT
cana-5491	5	53	lerch	lerch	PROPN
cana-5491	5	54	zeta	zeta	PROPN
cana-5491	5	55	function	function	PROPN
cana-5491	5	56	.	.	PUNCT
cana-5491	6	1	further	far	ADV
cana-5491	6	2	,	,	PUNCT
cana-5491	6	3	we	we	PRON
cana-5491	6	4	compute	compute	VERB
cana-5491	6	5	the	the	DET
cana-5491	6	6	solution	solution	NOUN
cana-5491	6	7	of	of	ADP
cana-5491	6	8	fractional	fractional	ADJ
cana-5491	6	9	differential	differential	ADJ
cana-5491	6	10	equations	equation	NOUN
cana-5491	6	11	involving	involve	VERB
cana-5491	6	12	hurwitz	hurwitz	PROPN
cana-5491	6	13	-	-	PUNCT
cana-5491	6	14	lerch	lerch	PROPN
cana-5491	6	15	zeta	zeta	PROPN
cana-5491	6	16	function	function	PROPN
cana-5491	6	17	.	.	PUNCT
cana-5491	7	1	keywords	keyword	NOUN
cana-5491	7	2	:	:	PUNCT
cana-5491	7	3	hurwitz	hurwitz	PROPN
cana-5491	7	4	-	-	PUNCT
cana-5491	7	5	lerch	lerch	PROPN
cana-5491	7	6	zeta	zeta	PROPN
cana-5491	7	7	function	function	PROPN
cana-5491	7	8	,	,	PUNCT
cana-5491	7	9	hypergeometric	hypergeometric	ADJ
cana-5491	7	10	function	function	NOUN
cana-5491	7	11	and	and	CCONJ
cana-5491	7	12	fractional	fractional	ADJ
cana-5491	7	13	derivatives	derivative	NOUN
cana-5491	7	14	.	.	PUNCT
cana-5491	8	1	1	1	X
cana-5491	8	2	.	.	X
cana-5491	8	3	introduction	introduction	NOUN
cana-5491	8	4	fractional	fractional	ADJ
cana-5491	8	5	calculus	calculus	NOUN
cana-5491	8	6	serves	serve	VERB
cana-5491	8	7	as	as	ADP
cana-5491	8	8	an	an	DET
cana-5491	8	9	excellent	excellent	ADJ
cana-5491	8	10	tool	tool	NOUN
cana-5491	8	11	for	for	ADP
cana-5491	8	12	studying	study	VERB
cana-5491	8	13	fractional	fractional	ADJ
cana-5491	8	14	order	order	NOUN
cana-5491	8	15	integrals	integral	NOUN
cana-5491	8	16	and	and	CCONJ
cana-5491	8	17	derivatives	derivative	NOUN
cana-5491	8	18	.	.	PUNCT
cana-5491	9	1	there	there	PRON
cana-5491	9	2	are	be	VERB
cana-5491	9	3	a	a	DET
cana-5491	9	4	lot	lot	NOUN
cana-5491	9	5	of	of	ADP
cana-5491	9	6	disciplines	discipline	NOUN
cana-5491	9	7	in	in	ADP
cana-5491	9	8	science	science	NOUN
cana-5491	9	9	and	and	CCONJ
cana-5491	9	10	engineering	engineering	NOUN
cana-5491	9	11	that	that	PRON
cana-5491	9	12	benefit	benefit	VERB
cana-5491	9	13	from	from	ADP
cana-5491	9	14	fractional	fractional	ADJ
cana-5491	9	15	calculus	calculus	NOUN
cana-5491	9	16	.	.	PUNCT
cana-5491	10	1	in	in	ADP
cana-5491	10	2	a	a	DET
cana-5491	10	3	variety	variety	NOUN
cana-5491	10	4	of	of	ADP
cana-5491	10	5	fields	field	NOUN
cana-5491	10	6	,	,	PUNCT
cana-5491	10	7	fractional	fractional	ADJ
cana-5491	10	8	differential	differential	ADJ
cana-5491	10	9	equations	equation	NOUN
cana-5491	10	10	and	and	CCONJ
cana-5491	10	11	their	their	PRON
cana-5491	10	12	applications	application	NOUN
cana-5491	10	13	have	have	AUX
cana-5491	10	14	played	play	VERB
cana-5491	10	15	a	a	DET
cana-5491	10	16	significant	significant	ADJ
cana-5491	10	17	role	role	NOUN
cana-5491	10	18	.	.	PUNCT
cana-5491	11	1	these	these	PRON
cana-5491	11	2	include	include	VERB
cana-5491	11	3	applied	apply	VERB
cana-5491	11	4	science	science	NOUN
cana-5491	11	5	,	,	PUNCT
cana-5491	11	6	physics	physics	NOUN
cana-5491	11	7	,	,	PUNCT
cana-5491	11	8	biology	biology	NOUN
cana-5491	11	9	,	,	PUNCT
cana-5491	11	10	chemistry	chemistry	NOUN
cana-5491	11	11	and	and	CCONJ
cana-5491	11	12	engineering	engineering	NOUN
cana-5491	11	13	science	science	NOUN
cana-5491	11	14	.	.	PUNCT
cana-5491	12	1	as	as	ADP
cana-5491	12	2	a	a	DET
cana-5491	12	3	system	system	NOUN
cana-5491	12	4	of	of	ADP
cana-5491	12	5	differential	differential	ADJ
cana-5491	12	6	,	,	PUNCT
cana-5491	12	7	kinetic	kinetic	ADJ
cana-5491	12	8	equations	equation	NOUN
cana-5491	12	9	provide	provide	VERB
cana-5491	12	10	a	a	DET
cana-5491	12	11	description	description	NOUN
cana-5491	12	12	of	of	ADP
cana-5491	12	13	the	the	DET
cana-5491	12	14	rate	rate	NOUN
cana-5491	12	15	at	at	ADP
cana-5491	12	16	which	which	PRON
cana-5491	12	17	changes	change	NOUN
cana-5491	12	18	in	in	ADP
cana-5491	12	19	the	the	DET
cana-5491	12	20	chemical	chemical	NOUN
cana-5491	12	21	composition	composition	NOUN
cana-5491	12	22	of	of	ADP
cana-5491	12	23	a	a	DET
cana-5491	12	24	star	star	NOUN
cana-5491	12	25	occur	occur	VERB
cana-5491	12	26	.	.	PUNCT
cana-5491	13	1	fractional	fractional	ADJ
cana-5491	13	2	differential	differential	ADJ
cana-5491	13	3	equations	equation	NOUN
cana-5491	13	4	have	have	AUX
cana-5491	13	5	been	be	AUX
cana-5491	13	6	widely	widely	ADV
cana-5491	13	7	and	and	CCONJ
cana-5491	13	8	successfully	successfully	ADV
cana-5491	13	9	used	use	VERB
cana-5491	13	10	to	to	PART
cana-5491	13	11	describe	describe	VERB
cana-5491	13	12	and	and	CCONJ
cana-5491	13	13	solve	solve	VERB
cana-5491	13	14	many	many	ADJ
cana-5491	13	15	problems	problem	NOUN
cana-5491	13	16	in	in	ADP
cana-5491	13	17	physics	physics	NOUN
cana-5491	13	18	and	and	CCONJ
cana-5491	13	19	astrophysics	astrophysic	NOUN
cana-5491	13	20	over	over	ADP
cana-5491	13	21	the	the	DET
cana-5491	13	22	past	past	ADJ
cana-5491	13	23	several	several	ADJ
cana-5491	13	24	decades	decade	NOUN
cana-5491	13	25	.	.	PUNCT
cana-5491	14	1	in	in	ADP
cana-5491	14	2	mathematics	mathematics	PROPN
cana-5491	14	3	and	and	CCONJ
cana-5491	14	4	mathematical	mathematical	ADJ
cana-5491	14	5	physics	physics	NOUN
cana-5491	14	6	,	,	PUNCT
cana-5491	14	7	the	the	DET
cana-5491	14	8	special	special	ADJ
cana-5491	14	9	functions	function	NOUN
cana-5491	14	10	are	be	AUX
cana-5491	14	11	useful	useful	ADJ
cana-5491	14	12	for	for	ADP
cana-5491	14	13	the	the	DET
cana-5491	14	14	solution	solution	NOUN
cana-5491	14	15	of	of	ADP
cana-5491	14	16	fractional	fractional	ADJ
cana-5491	14	17	integral	integral	ADJ
cana-5491	14	18	and	and	CCONJ
cana-5491	14	19	differential	differential	ADJ
cana-5491	14	20	equation	equation	NOUN
cana-5491	14	21	problems	problem	NOUN
cana-5491	14	22	.	.	PUNCT
cana-5491	15	1	in	in	ADP
cana-5491	15	2	order	order	NOUN
cana-5491	15	3	to	to	PART
cana-5491	15	4	incorporate	incorporate	VERB
cana-5491	15	5	fractional	fractional	ADJ
cana-5491	15	6	derivatives	derivative	NOUN
cana-5491	15	7	into	into	ADP
cana-5491	15	8	differential	differential	ADJ
cana-5491	15	9	equations	equation	NOUN
cana-5491	15	10	,	,	PUNCT
cana-5491	15	11	fractional	fractional	ADJ
cana-5491	15	12	differential	differential	ADJ
cana-5491	15	13	equations	equation	NOUN
cana-5491	15	14	were	be	AUX
cana-5491	15	15	developed	develop	VERB
cana-5491	15	16	.	.	PUNCT
cana-5491	16	1	when	when	SCONJ
cana-5491	16	2	f(w	f(w	PROPN
cana-5491	16	3	)	)	PUNCT
cana-5491	16	4	is	be	AUX
cana-5491	16	5	a	a	DET
cana-5491	16	6	function	function	NOUN
cana-5491	16	7	of	of	ADP
cana-5491	16	8	order	order	NOUN
cana-5491	16	9	alpha	alpha	NOUN
cana-5491	16	10	.	.	PUNCT
cana-5491	17	1	its	its	PRON
cana-5491	17	2	fractional	fractional	ADJ
cana-5491	17	3	derivatives	derivative	NOUN
cana-5491	17	4	are	be	AUX
cana-5491	17	5	represented	represent	VERB
cana-5491	17	6	mathematically	mathematically	ADV
cana-5491	17	7	as	as	ADP
cana-5491	17	8	dαf(w	dαf(w	PROPN
cana-5491	17	9	)	)	PUNCT
cana-5491	17	10	,	,	PUNCT
cana-5491	17	11	where	where	SCONJ
cana-5491	17	12	α	α	NOUN
cana-5491	17	13	is	be	AUX
cana-5491	17	14	a	a	DET
cana-5491	17	15	non	non	X
cana-5491	17	16	integer	integer	NOUN
cana-5491	17	17	.	.	PUNCT
cana-5491	18	1	many	many	ADJ
cana-5491	18	2	fractional	fractional	ADJ
cana-5491	18	3	derivatives	derivative	NOUN
cana-5491	18	4	exist	exist	VERB
cana-5491	18	5	,	,	PUNCT
cana-5491	18	6	including	include	VERB
cana-5491	18	7	riemann	riemann	PROPN
cana-5491	18	8	-	-	PUNCT
cana-5491	18	9	lionville	lionville	PROPN
cana-5491	18	10	,	,	PUNCT
cana-5491	18	11	caputo	caputo	PROPN
cana-5491	18	12	and	and	CCONJ
cana-5491	18	13	grunwald	grunwald	NOUN
cana-5491	18	14	-	-	PUNCT
cana-5491	18	15	letnikov	letnikov	NOUN
cana-5491	18	16	derivatives	derivative	NOUN
cana-5491	18	17	.	.	PUNCT
cana-5491	19	1	they	they	PRON
cana-5491	19	2	each	each	PRON
cana-5491	19	3	have	have	VERB
cana-5491	19	4	their	their	PRON
cana-5491	19	5	own	own	ADJ
cana-5491	19	6	advantage	advantage	NOUN
cana-5491	19	7	and	and	CCONJ
cana-5491	19	8	uses	use	VERB
cana-5491	19	9	.	.	PUNCT
cana-5491	20	1	the	the	DET
cana-5491	20	2	fractional	fractional	ADJ
cana-5491	20	3	derivative	derivative	NOUN
cana-5491	20	4	of	of	ADP
cana-5491	20	5	f(w	f(w	PROPN
cana-5491	20	6	)	)	PUNCT
cana-5491	20	7	in	in	ADP
cana-5491	20	8	the	the	DET
cana-5491	20	9	caputo	caputo	PROPN
cana-5491	20	10	sense	sense	NOUN
cana-5491	20	11	is	be	AUX
cana-5491	20	12	defined	define	VERB
cana-5491	20	13	as	as	ADP
cana-5491	20	14	:	:	PUNCT
cana-5491	20	15	dαf(w	dαf(w	PROPN
cana-5491	20	16	)	)	PUNCT
cana-5491	21	1	=	=	SYM
cana-5491	21	2	il−αdlf(w	il−αdlf(w	NOUN
cana-5491	21	3	)	)	PUNCT
cana-5491	21	4	(	(	PUNCT
cana-5491	21	5	1	1	X
cana-5491	21	6	)	)	PUNCT
cana-5491	21	7	dαf(ω	dαf(ω	PROPN
cana-5491	21	8	)	)	PUNCT
cana-5491	21	9	=	=	NOUN
cana-5491	21	10	1	1	NUM
cana-5491	21	11	γ(l−α	γ(l−α	NUM
cana-5491	21	12	)	)	PUNCT
cana-5491	21	13	∫	∫	PROPN
cana-5491	22	1	(	(	PUNCT
cana-5491	22	2	ω	ω	NUM
cana-5491	22	3	−	−	PROPN
cana-5491	22	4	u)l−α−1f	u)l−α−1f	PROPN
cana-5491	22	5	lw	lw	NOUN
cana-5491	22	6	0	0	NUM
cana-5491	23	1	(	(	PUNCT
cana-5491	23	2	u)du	u)du	PROPN
cana-5491	23	3	.	.	PROPN
cana-5491	23	4	.	.	PUNCT
cana-5491	24	1	(	(	PUNCT
cana-5491	24	2	2	2	X
cana-5491	24	3	)	)	PUNCT
cana-5491	24	4	for	for	ADP
cana-5491	24	5	l	l	NOUN
cana-5491	24	6	−	−	PROPN
cana-5491	24	7	1	1	NUM
cana-5491	24	8	˂	˂	NOUN
cana-5491	24	9	α	α	NOUN
cana-5491	24	10	≤	≤	ADJ
cana-5491	24	11	l	l	NOUN
cana-5491	24	12	,	,	PUNCT
cana-5491	24	13	l	l	PROPN
cana-5491	24	14	∈	∈	PROPN
cana-5491	24	15	n	n	CCONJ
cana-5491	24	16	,	,	PUNCT
cana-5491	24	17	ω	ω	NUM
cana-5491	24	18	˃	˃	NOUN
cana-5491	24	19	0	0	NUM
cana-5491	24	20	.	.	PUNCT
cana-5491	25	1	the	the	DET
cana-5491	25	2	caputo	caputo	PROPN
cana-5491	25	3	derivative	derivative	NOUN
cana-5491	25	4	,	,	PUNCT
cana-5491	25	5	we	we	PRON
cana-5491	25	6	have	have	VERB
cana-5491	25	7	dαc	dαc	NOUN
cana-5491	25	8	=	=	SYM
cana-5491	25	9	0	0	PROPN
cana-5491	25	10	,	,	PUNCT
cana-5491	25	11	c	c	PROPN
cana-5491	25	12	is	be	AUX
cana-5491	25	13	constant	constant	ADJ
cana-5491	25	14	.	.	PUNCT
cana-5491	26	1	mailto:1sandeep8603@gmail.com	mailto:1sandeep8603@gmail.com	PROPN
cana-5491	26	2	mailto:2owkhan05@gmail.com	mailto:2owkhan05@gmail.com	X
cana-5491	26	3	mailto:4najmuddinahmad33@gmail.com	mailto:4najmuddinahmad33@gmail.com	X
cana-5491	27	1	mailto:3nukhanmath@gmail.com	mailto:3nukhanmath@gmail.com	PROPN
cana-5491	27	2	communications	communication	NOUN
cana-5491	27	3	on	on	ADP
cana-5491	27	4	applied	apply	VERB
cana-5491	27	5	nonlinear	nonlinear	ADJ
cana-5491	27	6	analysis	analysis	NOUN
cana-5491	27	7	issn	issn	NOUN
cana-5491	27	8	:	:	PUNCT
cana-5491	27	9	1074	1074	NUM
cana-5491	27	10	-	-	PUNCT
cana-5491	27	11	133x	133x	NUM
cana-5491	27	12	vol	vol	VERB
cana-5491	27	13	32	32	NUM
cana-5491	27	14	no	no	NOUN
cana-5491	27	15	.	.	PUNCT
cana-5491	28	1	10s	10	NOUN
cana-5491	28	2	(	(	PUNCT
cana-5491	28	3	2025	2025	NUM
cana-5491	28	4	)	)	PUNCT
cana-5491	28	5	2435	2435	NUM
cana-5491	28	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	28	7	dαrr	dαrr	NOUN
cana-5491	28	8	=	=	PUNCT
cana-5491	28	9	{	{	PUNCT
cana-5491	28	10	0	0	NUM
cana-5491	28	11	,	,	PUNCT
cana-5491	28	12	r	r	NOUN
cana-5491	28	13	≤	≤	PUNCT
cana-5491	29	1	α	α	NOUN
cana-5491	29	2	−	−	PROPN
cana-5491	29	3	1	1	NUM
cana-5491	29	4	γ(r+1	γ(r+1	NUM
cana-5491	29	5	)	)	PUNCT
cana-5491	29	6	γ(r−α+1	γ(r−α+1	NOUN
cana-5491	29	7	)	)	PUNCT
cana-5491	29	8	tr−α	tr−α	NOUN
cana-5491	29	9	,	,	PUNCT
cana-5491	29	10	r	r	NOUN
cana-5491	29	11	˃	˃	PROPN
cana-5491	29	12	(	(	PUNCT
cana-5491	29	13	α	α	NOUN
cana-5491	29	14	−	−	PROPN
cana-5491	29	15	1	1	NUM
cana-5491	29	16	)	)	PUNCT
cana-5491	29	17	.	.	PUNCT
cana-5491	30	1	(	(	PUNCT
cana-5491	30	2	3	3	X
cana-5491	30	3	)	)	PUNCT
cana-5491	30	4	mathematical	mathematical	ADJ
cana-5491	30	5	special	special	ADJ
cana-5491	30	6	functions	function	NOUN
cana-5491	30	7	or	or	CCONJ
cana-5491	30	8	sfs	sfs	ADJ
cana-5491	30	9	date	date	NOUN
cana-5491	30	10	back	back	ADV
cana-5491	30	11	to	to	ADP
cana-5491	30	12	the	the	DET
cana-5491	30	13	nineteenth	nineteenth	ADJ
cana-5491	30	14	century	century	NOUN
cana-5491	30	15	,	,	PUNCT
cana-5491	30	16	when	when	SCONJ
cana-5491	30	17	they	they	PRON
cana-5491	30	18	were	be	AUX
cana-5491	30	19	developed	develop	VERB
cana-5491	30	20	as	as	ADP
cana-5491	30	21	a	a	DET
cana-5491	30	22	unified	unified	ADJ
cana-5491	30	23	and	and	CCONJ
cana-5491	30	24	complete	complete	ADJ
cana-5491	30	25	theory	theory	NOUN
cana-5491	30	26	.	.	PUNCT
cana-5491	31	1	researchers	researcher	NOUN
cana-5491	31	2	and	and	CCONJ
cana-5491	31	3	engineers	engineer	NOUN
cana-5491	31	4	working	work	VERB
cana-5491	31	5	with	with	ADP
cana-5491	31	6	differential	differential	ADJ
cana-5491	31	7	equations	equation	NOUN
cana-5491	31	8	are	be	AUX
cana-5491	31	9	well	well	ADV
cana-5491	31	10	aware	aware	ADJ
cana-5491	31	11	of	of	ADP
cana-5491	31	12	the	the	DET
cana-5491	31	13	value	value	NOUN
cana-5491	31	14	of	of	ADP
cana-5491	31	15	sfs	sfs	NOUN
cana-5491	31	16	as	as	ADP
cana-5491	31	17	a	a	DET
cana-5491	31	18	tool	tool	NOUN
cana-5491	31	19	for	for	ADP
cana-5491	31	20	mathematical	mathematical	ADJ
cana-5491	31	21	analysis	analysis	NOUN
cana-5491	31	22	.	.	PUNCT
cana-5491	32	1	several	several	ADJ
cana-5491	32	2	of	of	ADP
cana-5491	32	3	these	these	DET
cana-5491	32	4	named	name	VERB
cana-5491	32	5	functions	function	NOUN
cana-5491	32	6	are	be	AUX
cana-5491	32	7	formulated	formulate	VERB
cana-5491	32	8	as	as	ADP
cana-5491	32	9	mathematical	mathematical	ADJ
cana-5491	32	10	models	model	NOUN
cana-5491	32	11	by	by	ADP
cana-5491	32	12	solving	solve	VERB
cana-5491	32	13	differential	differential	ADJ
cana-5491	32	14	equations	equation	NOUN
cana-5491	32	15	and	and	CCONJ
cana-5491	32	16	systems	system	NOUN
cana-5491	32	17	of	of	ADP
cana-5491	32	18	integer	integer	NOUN
cana-5491	32	19	order	order	NOUN
cana-5491	32	20	.	.	PUNCT
cana-5491	33	1	as	as	ADP
cana-5491	33	2	a	a	DET
cana-5491	33	3	result	result	NOUN
cana-5491	33	4	of	of	ADP
cana-5491	33	5	the	the	DET
cana-5491	33	6	growing	grow	VERB
cana-5491	33	7	by	by	ADP
cana-5491	33	8	interest	interest	NOUN
cana-5491	33	9	and	and	CCONJ
cana-5491	33	10	widespread	widespread	ADJ
cana-5491	33	11	application	application	NOUN
cana-5491	33	12	of	of	ADP
cana-5491	33	13	differential	differential	ADJ
cana-5491	33	14	equations	equation	NOUN
cana-5491	33	15	and	and	CCONJ
cana-5491	33	16	fractional	fractional	ADJ
cana-5491	33	17	order	order	NOUN
cana-5491	33	18	system	system	NOUN
cana-5491	33	19	,	,	PUNCT
cana-5491	33	20	many	many	ADJ
cana-5491	33	21	physical	physical	ADJ
cana-5491	33	22	,	,	PUNCT
cana-5491	33	23	engineering	engineering	NOUN
cana-5491	33	24	,	,	PUNCT
cana-5491	33	25	automation	automation	NOUN
cana-5491	33	26	,	,	PUNCT
cana-5491	33	27	biological	biological	ADJ
cana-5491	33	28	,	,	PUNCT
cana-5491	33	29	chemical	chemical	NOUN
cana-5491	33	30	,	,	PUNCT
cana-5491	33	31	earth	earth	NOUN
cana-5491	33	32	science	science	NOUN
cana-5491	33	33	,	,	PUNCT
cana-5491	33	34	economic	economic	ADJ
cana-5491	33	35	phenomena	phenomenon	NOUN
cana-5491	33	36	have	have	AUX
cana-5491	33	37	been	be	AUX
cana-5491	33	38	better	well	ADV
cana-5491	33	39	represented	represent	VERB
cana-5491	33	40	in	in	ADP
cana-5491	33	41	the	the	DET
cana-5491	33	42	function	function	NOUN
cana-5491	33	43	,	,	PUNCT
cana-5491	33	44	euler	euler	NOUN
cana-5491	33	45	beta	beta	NOUN
cana-5491	33	46	function	function	NOUN
cana-5491	33	47	,	,	PUNCT
cana-5491	33	48	and	and	CCONJ
cana-5491	33	49	many	many	ADJ
cana-5491	33	50	more	more	ADJ
cana-5491	33	51	function	function	NOUN
cana-5491	33	52	have	have	AUX
cana-5491	33	53	all	all	PRON
cana-5491	33	54	recently	recently	ADV
cana-5491	33	55	seen	see	VERB
cana-5491	33	56	extensions	extension	NOUN
cana-5491	33	57	created	create	VERB
cana-5491	33	58	by	by	ADP
cana-5491	33	59	numerous	numerous	ADJ
cana-5491	33	60	writers	writer	NOUN
cana-5491	33	61	.	.	PUNCT
cana-5491	34	1	we	we	PRON
cana-5491	34	2	are	be	AUX
cana-5491	34	3	familiar	familiar	ADJ
cana-5491	34	4	with	with	ADP
cana-5491	34	5	the	the	DET
cana-5491	34	6	hurwitz	hurwitz	PROPN
cana-5491	34	7	-	-	PUNCT
cana-5491	34	8	lerch	lerch	PROPN
cana-5491	34	9	zeta	zeta	PROPN
cana-5491	34	10	function	function	PROPN
cana-5491	34	11	φ(w	φ(w	PROPN
cana-5491	34	12	,	,	PUNCT
cana-5491	34	13	k	k	NOUN
cana-5491	34	14	,	,	PUNCT
cana-5491	34	15	r	r	NOUN
cana-5491	34	16	)	)	PUNCT
cana-5491	34	17	defined	define	VERB
cana-5491	34	18	as	as	ADP
cana-5491	34	19	:	:	PUNCT
cana-5491	34	20	𝜑(𝑤	𝜑(𝑤	PROPN
cana-5491	34	21	,	,	PUNCT
cana-5491	34	22	𝑘	𝑘	NOUN
cana-5491	34	23	,	,	PUNCT
cana-5491	34	24	𝑡	𝑡	NOUN
cana-5491	34	25	)	)	PUNCT
cana-5491	34	26	=	=	SYM
cana-5491	34	27	∑	∑	PUNCT
cana-5491	34	28	𝑤𝑚	𝑤𝑚	NOUN
cana-5491	34	29	(	(	PUNCT
cana-5491	34	30	𝑚+𝑟)𝑘	𝑚+𝑟)𝑘	X
cana-5491	34	31	,	,	PUNCT
cana-5491	34	32	∞	∞	NUM
cana-5491	34	33	𝑚=0	𝑚=0	PUNCT
cana-5491	34	34	(	(	PUNCT
cana-5491	34	35	4	4	NUM
cana-5491	34	36	)	)	PUNCT
cana-5491	34	37	(	(	PUNCT
cana-5491	34	38	𝑟	𝑟	X
cana-5491	34	39	∈	∈	PROPN
cana-5491	34	40	𝑍+	𝑍+	NOUN
cana-5491	34	41	:	:	PUNCT
cana-5491	34	42	𝑘	𝑘	PROPN
cana-5491	34	43	∈	∈	PROPN
cana-5491	34	44	𝐶	𝐶	PROPN
cana-5491	34	45	𝑤ℎ𝑒𝑛	𝑤ℎ𝑒𝑛	VERB
cana-5491	34	46	|𝑤|	|𝑤|	PROPN
cana-5491	34	47	<	<	X
cana-5491	34	48	1	1	NUM
cana-5491	34	49	:	:	PUNCT
cana-5491	34	50	𝑅(𝑘	𝑅(𝑘	NUM
cana-5491	34	51	)	)	PUNCT
cana-5491	34	52	>	>	SYM
cana-5491	34	53	1	1	NUM
cana-5491	34	54	𝑤ℎ𝑒𝑛	𝑤ℎ𝑒𝑛	VERB
cana-5491	34	55	|𝑤|	|𝑤|	PROPN
cana-5491	34	56	=	=	SYM
cana-5491	34	57	1	1	NUM
cana-5491	34	58	)	)	PUNCT
cana-5491	34	59	.	.	PUNCT
cana-5491	35	1	various	various	ADJ
cana-5491	35	2	generalizations	generalization	NOUN
cana-5491	35	3	of	of	ADP
cana-5491	35	4	the	the	DET
cana-5491	35	5	hurwitz	hurwitz	PROPN
cana-5491	35	6	-	-	PUNCT
cana-5491	35	7	lerch	lerch	PROPN
cana-5491	35	8	zeta	zeta	PROPN
cana-5491	35	9	functions	function	NOUN
cana-5491	35	10	have	have	AUX
cana-5491	35	11	been	be	AUX
cana-5491	35	12	given	give	VERB
cana-5491	35	13	by	by	ADP
cana-5491	35	14	the	the	DET
cana-5491	35	15	researchers	researcher	NOUN
cana-5491	35	16	.	.	PUNCT
cana-5491	36	1	for	for	ADP
cana-5491	36	2	example	example	NOUN
cana-5491	36	3	,	,	PUNCT
cana-5491	36	4	goyal	goyal	PROPN
cana-5491	36	5	et.al	et.al	PROPN
cana-5491	36	6	.	.	PUNCT
cana-5491	36	7	has	have	AUX
cana-5491	36	8	introduced	introduce	VERB
cana-5491	36	9	an	an	DET
cana-5491	36	10	extension	extension	NOUN
cana-5491	36	11	of	of	ADP
cana-5491	36	12	hurwitz	hurwitz	PROPN
cana-5491	36	13	-	-	PUNCT
cana-5491	36	14	lerch	lerch	PROPN
cana-5491	36	15	zeta	zeta	PROPN
cana-5491	36	16	function	function	NOUN
cana-5491	36	17	defined	define	VERB
cana-5491	36	18	as	as	ADP
cana-5491	36	19	:	:	PUNCT
cana-5491	36	20	𝜑𝜃1	𝜑𝜃1	PROPN
cana-5491	36	21	∗	∗	NOUN
cana-5491	36	22	(	(	PUNCT
cana-5491	36	23	𝑤	𝑤	ADP
cana-5491	36	24	,	,	PUNCT
cana-5491	36	25	𝑘	𝑘	PROPN
cana-5491	36	26	,	,	PUNCT
cana-5491	36	27	𝑡	𝑡	NOUN
cana-5491	36	28	)	)	PUNCT
cana-5491	36	29	=	=	SYM
cana-5491	36	30	∑	∑	PUNCT
cana-5491	36	31	(	(	PUNCT
cana-5491	37	1	𝜃1)𝑚	𝜃1)𝑚	NOUN
cana-5491	37	2	𝑚	𝑚	PROPN
cana-5491	37	3	!	!	PUNCT
cana-5491	37	4	𝑤𝑚	𝑤𝑚	NOUN
cana-5491	37	5	(	(	PUNCT
cana-5491	37	6	𝑚	𝑚	PROPN
cana-5491	37	7	+	+	X
cana-5491	37	8	𝑟)𝑘	𝑟)𝑘	NUM
cana-5491	37	9	,	,	PUNCT
cana-5491	37	10	(	(	PUNCT
cana-5491	37	11	5	5	X
cana-5491	37	12	)	)	PUNCT
cana-5491	37	13	∞	∞	NUM
cana-5491	37	14	𝑚=0	𝑚=0	PUNCT
cana-5491	37	15	(	(	PUNCT
cana-5491	37	16	𝜃1	𝜃1	VERB
cana-5491	37	17	∈	∈	PROPN
cana-5491	37	18	𝐶	𝐶	PROPN
cana-5491	37	19	;	;	PUNCT
cana-5491	37	20	𝑟	𝑟	X
cana-5491	37	21	∈	∈	PROPN
cana-5491	37	22	𝑍+	𝑍+	NOUN
cana-5491	37	23	;	;	PUNCT
cana-5491	37	24	𝑘	𝑘	PROPN
cana-5491	37	25	∈	∈	PROPN
cana-5491	37	26	𝐶	𝐶	PROPN
cana-5491	37	27	𝑖𝑓	𝑖𝑓	ADP
cana-5491	37	28	|𝑤|	|𝑤|	PROPN
cana-5491	37	29	<	<	X
cana-5491	37	30	1	1	NUM
cana-5491	37	31	;	;	PUNCT
cana-5491	37	32	𝑅(𝑘	𝑅(𝑘	ADP
cana-5491	37	33	−	−	PROPN
cana-5491	37	34	𝛿	𝛿	X
cana-5491	37	35	)	)	PUNCT
cana-5491	37	36	>	>	X
cana-5491	37	37	1	1	NUM
cana-5491	37	38	𝑖𝑓	𝑖𝑓	ADP
cana-5491	37	39	|𝑤|	|𝑤|	PROPN
cana-5491	37	40	=	=	NOUN
cana-5491	37	41	1	1	NUM
cana-5491	37	42	)	)	PUNCT
cana-5491	37	43	.	.	PUNCT
cana-5491	38	1	lin	lin	PROPN
cana-5491	38	2	et	et	PROPN
cana-5491	38	3	.	.	PUNCT
cana-5491	39	1	al	al	PROPN
cana-5491	39	2	.	.	PUNCT
cana-5491	40	1	[	[	X
cana-5491	40	2	13	13	NUM
cana-5491	40	3	]	]	PUNCT
cana-5491	40	4	also	also	ADV
cana-5491	40	5	defined	define	VERB
cana-5491	40	6	the	the	DET
cana-5491	40	7	hurwitz	hurwitz	PROPN
cana-5491	40	8	-	-	PUNCT
cana-5491	40	9	lerch	lerch	PROPN
cana-5491	40	10	zeta	zeta	PROPN
cana-5491	40	11	function	function	PROPN
cana-5491	40	12	as	as	ADP
cana-5491	40	13	:	:	PUNCT
cana-5491	40	14	𝜑𝜃1,𝜃2	𝜑𝜃1,𝜃2	ADJ
cana-5491	40	15	,	,	PUNCT
cana-5491	40	16	𝛽,𝛿	𝛽,𝛿	PRON
cana-5491	40	17	(	(	PUNCT
cana-5491	40	18	𝑤	𝑤	ADP
cana-5491	40	19	,	,	PUNCT
cana-5491	40	20	𝑘	𝑘	PROPN
cana-5491	40	21	,	,	PUNCT
cana-5491	40	22	𝑝	𝑝	NOUN
cana-5491	40	23	)	)	PUNCT
cana-5491	40	24	=	=	SYM
cana-5491	40	25	∑	∑	PROPN
cana-5491	40	26	(	(	PUNCT
cana-5491	40	27	𝜃1)𝛽𝑚	𝜃1)𝛽𝑚	PROPN
cana-5491	40	28	(	(	PUNCT
cana-5491	40	29	𝜃2)𝛿𝑚	𝜃2)𝛿𝑚	NOUN
cana-5491	40	30	𝑤𝑚	𝑤𝑚	NOUN
cana-5491	40	31	(	(	PUNCT
cana-5491	40	32	𝑚	𝑚	PROPN
cana-5491	40	33	+	+	X
cana-5491	40	34	𝑝)𝑘	𝑝)𝑘	NOUN
cana-5491	40	35	,	,	PUNCT
cana-5491	40	36	(	(	PUNCT
cana-5491	40	37	6	6	NUM
cana-5491	40	38	)	)	PUNCT
cana-5491	40	39	∞	∞	NUM
cana-5491	40	40	𝑚=0	𝑚=0	PUNCT
cana-5491	40	41	(	(	PUNCT
cana-5491	40	42	𝜃1	𝜃1	VERB
cana-5491	40	43	∈	∈	PROPN
cana-5491	40	44	𝐶	𝐶	PROPN
cana-5491	40	45	;	;	PUNCT
cana-5491	40	46	𝑝	𝑝	NUM
cana-5491	40	47	,	,	PUNCT
cana-5491	40	48	𝜃2	𝜃2	PROPN
cana-5491	40	49	∈	∈	PROPN
cana-5491	40	50	𝑍+	𝑍+	PRON
cana-5491	40	51	;	;	PUNCT
cana-5491	40	52	𝛽	𝛽	NOUN
cana-5491	40	53	,	,	PUNCT
cana-5491	40	54	𝑤	𝑤	ADP
cana-5491	40	55	∈	∈	PROPN
cana-5491	40	56	𝑅	𝑅	PROPN
cana-5491	40	57	+	+	PROPN
cana-5491	40	58	;	;	PUNCT
cana-5491	40	59	𝛽	𝛽	NOUN
cana-5491	40	60	<	<	X
cana-5491	40	61	𝑤	𝑤	PART
cana-5491	40	62	𝑖𝑓𝑠	𝑖𝑓𝑠	PROPN
cana-5491	40	63	,	,	PUNCT
cana-5491	40	64	𝑤	𝑤	ADP
cana-5491	40	65	∈	∈	PROPN
cana-5491	40	66	𝐶	𝐶	PROPN
cana-5491	40	67	;	;	PUNCT
cana-5491	40	68	𝛽	𝛽	NOUN
cana-5491	40	69	=	=	SYM
cana-5491	40	70	𝛿	𝛿	PROPN
cana-5491	40	71	,	,	PUNCT
cana-5491	40	72	𝑠	𝑠	PROPN
cana-5491	40	73	𝜖	𝜖	PROPN
cana-5491	40	74	𝐶	𝐶	PROPN
cana-5491	40	75	𝑖𝑓	𝑖𝑓	ADP
cana-5491	40	76	|𝑤|	|𝑤|	PROPN
cana-5491	40	77	<	<	X
cana-5491	40	78	1	1	NUM
cana-5491	40	79	;	;	PUNCT
cana-5491	40	80	𝑅(𝑠	𝑅(𝑠	NUM
cana-5491	40	81	−	−	NOUN
cana-5491	40	82	𝜃1	𝜃1	NOUN
cana-5491	40	83	+	+	CCONJ
cana-5491	40	84	𝜃2	𝜃2	NOUN
cana-5491	40	85	)	)	PUNCT
cana-5491	40	86	>	>	X
cana-5491	40	87	1	1	NUM
cana-5491	40	88	,	,	PUNCT
cana-5491	40	89	|𝑤|	|𝑤|	PROPN
cana-5491	40	90	=	=	SYM
cana-5491	40	91	1	1	X
cana-5491	40	92	)	)	PUNCT
cana-5491	40	93	garg	garg	NOUN
cana-5491	40	94	et	et	PROPN
cana-5491	40	95	.	.	PUNCT
cana-5491	41	1	al	al	PROPN
cana-5491	41	2	.	.	PUNCT
cana-5491	42	1	[	[	X
cana-5491	42	2	3	3	X
cana-5491	42	3	]	]	PUNCT
cana-5491	42	4	also	also	ADV
cana-5491	42	5	introduced	introduce	VERB
cana-5491	42	6	hurwitz	hurwitz	PROPN
cana-5491	42	7	-	-	PUNCT
cana-5491	42	8	lerch	lerch	PROPN
cana-5491	42	9	zeta	zeta	PROPN
cana-5491	42	10	function	function	PROPN
cana-5491	42	11	as	as	ADP
cana-5491	42	12	:	:	PUNCT
cana-5491	42	13	𝜑𝜃1,𝜃2,𝜃3	𝜑𝜃1,𝜃2,𝜃3	PROPN
cana-5491	42	14	𝛽,𝛾,𝛿	𝛽,𝛾,𝛿	PROPN
cana-5491	42	15	(	(	PUNCT
cana-5491	42	16	𝑤	𝑤	PROPN
cana-5491	42	17	,	,	PUNCT
cana-5491	42	18	𝑠	𝑠	PROPN
cana-5491	42	19	,	,	PUNCT
cana-5491	42	20	𝑝	𝑝	NOUN
cana-5491	42	21	)	)	PUNCT
cana-5491	42	22	=	=	SYM
cana-5491	42	23	∑	∑	PROPN
cana-5491	42	24	(	(	PUNCT
cana-5491	42	25	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	42	26	(	(	PUNCT
cana-5491	42	27	𝜃2)𝑟𝛿	𝜃2)𝑟𝛿	PROPN
cana-5491	42	28	(	(	PUNCT
cana-5491	42	29	𝜃2)𝑟𝛾(𝑤𝛼)𝑟	𝜃2)𝑟𝛾(𝑤𝛼)𝑟	NUM
cana-5491	42	30	(	(	PUNCT
cana-5491	42	31	𝑟	𝑟	NOUN
cana-5491	42	32	+	+	CCONJ
cana-5491	42	33	𝑝)𝑠	𝑝)𝑠	ADV
cana-5491	42	34	,	,	PUNCT
cana-5491	42	35	(	(	PUNCT
cana-5491	42	36	7	7	X
cana-5491	42	37	)	)	PUNCT
cana-5491	42	38	∞	∞	NUM
cana-5491	42	39	𝑚=0	𝑚=0	PUNCT
cana-5491	42	40	(	(	PUNCT
cana-5491	42	41	𝜃1	𝜃1	VERB
cana-5491	42	42	,	,	PUNCT
cana-5491	42	43	𝜃2	𝜃2	PROPN
cana-5491	42	44	∈	∈	PROPN
cana-5491	42	45	𝐶	𝐶	PROPN
cana-5491	42	46	;	;	PUNCT
cana-5491	42	47	𝜃3	𝜃3	ADJ
cana-5491	42	48	,	,	PUNCT
cana-5491	42	49	𝑝	𝑝	PROPN
cana-5491	42	50	∈	∈	PROPN
cana-5491	42	51	𝑍+	𝑍+	NOUN
cana-5491	42	52	;	;	PUNCT
cana-5491	42	53	𝑠	𝑠	PROPN
cana-5491	42	54	∈	∈	PROPN
cana-5491	42	55	𝐶	𝐶	PROPN
cana-5491	42	56	;	;	PUNCT
cana-5491	42	57	𝑖𝑓	𝑖𝑓	ADP
cana-5491	42	58	|𝑤|	|𝑤|	PROPN
cana-5491	42	59	<	<	X
cana-5491	42	60	1	1	NUM
cana-5491	42	61	;	;	PUNCT
cana-5491	42	62	𝑅(𝑠	𝑅(𝑠	SYM
cana-5491	42	63	+	+	CCONJ
cana-5491	42	64	𝜃3	𝜃3	ADJ
cana-5491	42	65	−	−	NOUN
cana-5491	42	66	𝜃1	𝜃1	NOUN
cana-5491	42	67	−	−	PROPN
cana-5491	42	68	𝜃2	𝜃2	PROPN
cana-5491	42	69	)	)	PUNCT
cana-5491	42	70	>	>	X
cana-5491	42	71	1	1	NUM
cana-5491	42	72	,	,	PUNCT
cana-5491	42	73	|𝑤|	|𝑤|	PROPN
cana-5491	42	74	=	=	SYM
cana-5491	42	75	1	1	NUM
cana-5491	42	76	)	)	PUNCT
cana-5491	42	77	in	in	ADP
cana-5491	42	78	order	order	NOUN
cana-5491	42	79	to	to	PART
cana-5491	42	80	address	address	VERB
cana-5491	42	81	the	the	DET
cana-5491	42	82	short	short	ADJ
cana-5491	42	83	comings	coming	NOUN
cana-5491	42	84	associated	associate	VERB
cana-5491	42	85	with	with	ADP
cana-5491	42	86	fractional	fractional	ADJ
cana-5491	42	87	derivatives	derivative	NOUN
cana-5491	42	88	in	in	ADP
cana-5491	42	89	mathematics	mathematic	NOUN
cana-5491	42	90	and	and	CCONJ
cana-5491	42	91	physics	physics	PROPN
cana-5491	42	92	,	,	PUNCT
cana-5491	42	93	the	the	DET
cana-5491	42	94	author	author	NOUN
cana-5491	42	95	developed	develop	VERB
cana-5491	42	96	formulas	formula	NOUN
cana-5491	42	97	for	for	ADP
cana-5491	42	98	the	the	DET
cana-5491	42	99	caputo	caputo	PROPN
cana-5491	42	100	derivative	derivative	NOUN
cana-5491	42	101	of	of	ADP
cana-5491	42	102	the	the	DET
cana-5491	42	103	hurwitz	hurwitz	PROPN
cana-5491	42	104	-	-	PUNCT
cana-5491	42	105	lerch	lerch	PROPN
cana-5491	42	106	zeta	zeta	PROPN
cana-5491	42	107	function	function	PROPN
cana-5491	42	108	.	.	PUNCT
cana-5491	43	1	as	as	ADP
cana-5491	43	2	a	a	DET
cana-5491	43	3	result	result	NOUN
cana-5491	43	4	of	of	ADP
cana-5491	43	5	these	these	DET
cana-5491	43	6	derivatives	derivative	NOUN
cana-5491	43	7	formulas	formula	NOUN
cana-5491	43	8	,	,	PUNCT
cana-5491	43	9	we	we	PRON
cana-5491	43	10	are	be	AUX
cana-5491	43	11	able	able	ADJ
cana-5491	43	12	to	to	PART
cana-5491	43	13	compute	compute	VERB
cana-5491	43	14	solutions	solution	NOUN
cana-5491	43	15	to	to	ADP
cana-5491	43	16	fractional	fractional	ADJ
cana-5491	43	17	differential	differential	ADJ
cana-5491	43	18	equations	equation	NOUN
cana-5491	43	19	.	.	PUNCT
cana-5491	44	1	communications	communication	NOUN
cana-5491	44	2	on	on	ADP
cana-5491	44	3	applied	apply	VERB
cana-5491	44	4	nonlinear	nonlinear	ADJ
cana-5491	44	5	analysis	analysis	NOUN
cana-5491	44	6	issn	issn	NOUN
cana-5491	44	7	:	:	PUNCT
cana-5491	44	8	1074	1074	NUM
cana-5491	44	9	-	-	PUNCT
cana-5491	44	10	133x	133x	NUM
cana-5491	44	11	vol	vol	VERB
cana-5491	44	12	32	32	NUM
cana-5491	44	13	no	no	NOUN
cana-5491	44	14	.	.	PUNCT
cana-5491	45	1	10s	10	NOUN
cana-5491	45	2	(	(	PUNCT
cana-5491	45	3	2025	2025	NUM
cana-5491	45	4	)	)	PUNCT
cana-5491	45	5	2436	2436	NUM
cana-5491	45	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	45	7	2	2	X
cana-5491	45	8	.	.	PUNCT
cana-5491	45	9	analysis	analysis	NOUN
cana-5491	45	10	of	of	ADP
cana-5491	45	11	method	method	NOUN
cana-5491	45	12	in	in	ADP
cana-5491	45	13	mathematics	mathematic	NOUN
cana-5491	45	14	and	and	CCONJ
cana-5491	45	15	mathematics	mathematic	NOUN
cana-5491	45	16	physics	physic	NOUN
cana-5491	45	17	,	,	PUNCT
cana-5491	45	18	the	the	DET
cana-5491	45	19	special	special	ADJ
cana-5491	45	20	functions	function	NOUN
cana-5491	45	21	are	be	AUX
cana-5491	45	22	useful	useful	ADJ
cana-5491	45	23	for	for	ADP
cana-5491	45	24	the	the	DET
cana-5491	45	25	solution	solution	NOUN
cana-5491	45	26	of	of	ADP
cana-5491	45	27	fractional	fractional	ADJ
cana-5491	45	28	integral	integral	ADJ
cana-5491	45	29	and	and	CCONJ
cana-5491	45	30	differential	differential	ADJ
cana-5491	45	31	equation	equation	NOUN
cana-5491	45	32	problems	problem	NOUN
cana-5491	45	33	.	.	PUNCT
cana-5491	46	1	in	in	ADP
cana-5491	46	2	order	order	NOUN
cana-5491	46	3	to	to	PART
cana-5491	46	4	incorporate	incorporate	VERB
cana-5491	46	5	fractional	fractional	ADJ
cana-5491	46	6	derivatives	derivative	NOUN
cana-5491	46	7	into	into	ADP
cana-5491	46	8	differential	differential	ADJ
cana-5491	46	9	equations	equation	NOUN
cana-5491	46	10	,	,	PUNCT
cana-5491	46	11	fractional	fractional	ADJ
cana-5491	46	12	differential	differential	ADJ
cana-5491	46	13	equations	equation	NOUN
cana-5491	46	14	were	be	AUX
cana-5491	46	15	developed	develop	VERB
cana-5491	46	16	[	[	X
cana-5491	46	17	2	2	NUM
cana-5491	46	18	,	,	PUNCT
cana-5491	46	19	12	12	NUM
cana-5491	46	20	,	,	PUNCT
cana-5491	46	21	15	15	NUM
cana-5491	46	22	]	]	PUNCT
cana-5491	46	23	.	.	PUNCT
cana-5491	47	1	the	the	DET
cana-5491	47	2	hurwitzlerch	hurwitzlerch	PROPN
cana-5491	47	3	zeta	zeta	PROPN
cana-5491	47	4	function	function	NOUN
cana-5491	47	5	defined	define	VERB
cana-5491	47	6	by	by	ADP
cana-5491	47	7	power	power	NOUN
cana-5491	47	8	series	series	NOUN
cana-5491	47	9	(	(	PUNCT
cana-5491	47	10	7	7	NUM
cana-5491	47	11	)	)	PUNCT
cana-5491	47	12	has	have	VERB
cana-5491	47	13	efficiency	efficiency	NOUN
cana-5491	47	14	as	as	ADP
cana-5491	47	15	solution	solution	NOUN
cana-5491	47	16	of	of	ADP
cana-5491	47	17	fractional	fractional	ADJ
cana-5491	47	18	order	order	NOUN
cana-5491	47	19	differential	differential	ADJ
cana-5491	47	20	and	and	CCONJ
cana-5491	47	21	integral	integral	ADJ
cana-5491	47	22	equations	equation	NOUN
cana-5491	47	23	and	and	CCONJ
cana-5491	47	24	thus	thus	ADV
cana-5491	47	25	have	have	VERB
cana-5491	47	26	important	important	ADJ
cana-5491	47	27	role	role	NOUN
cana-5491	47	28	of	of	ADP
cana-5491	47	29	the	the	DET
cana-5491	47	30	fractional	fractional	ADJ
cana-5491	47	31	calculus	calculus	NOUN
cana-5491	47	32	theory	theory	NOUN
cana-5491	47	33	and	and	CCONJ
cana-5491	47	34	applications	application	NOUN
cana-5491	47	35	.	.	PUNCT
cana-5491	48	1	in	in	ADP
cana-5491	48	2	this	this	DET
cana-5491	48	3	section	section	NOUN
cana-5491	48	4	,	,	PUNCT
cana-5491	48	5	we	we	PRON
cana-5491	48	6	consider	consider	VERB
cana-5491	48	7	few	few	ADJ
cana-5491	48	8	examples	example	NOUN
cana-5491	48	9	that	that	PRON
cana-5491	48	10	demonstrate	demonstrate	VERB
cana-5491	48	11	the	the	DET
cana-5491	48	12	performance	performance	NOUN
cana-5491	48	13	and	and	CCONJ
cana-5491	48	14	efficiency	efficiency	NOUN
cana-5491	48	15	of	of	ADP
cana-5491	48	16	hurwitz	hurwitz	PROPN
cana-5491	48	17	-	-	PUNCT
cana-5491	48	18	lerch	lerch	PROPN
cana-5491	48	19	zeta	zeta	PROPN
cana-5491	48	20	function	function	NOUN
cana-5491	48	21	for	for	ADP
cana-5491	48	22	solving	solve	VERB
cana-5491	48	23	linear	linear	ADJ
cana-5491	48	24	fractional	fractional	ADJ
cana-5491	48	25	differential	differential	ADJ
cana-5491	48	26	equations	equation	NOUN
cana-5491	48	27	with	with	ADP
cana-5491	48	28	fractional	fractional	ADJ
cana-5491	48	29	derivatives	derivative	NOUN
cana-5491	48	30	.	.	PUNCT
cana-5491	49	1	the	the	DET
cana-5491	49	2	hurwitz	hurwitz	PROPN
cana-5491	49	3	-	-	PUNCT
cana-5491	49	4	lerch	lerch	PROPN
cana-5491	49	5	zeta	zeta	PROPN
cana-5491	49	6	function	function	PROPN
cana-5491	49	7	suggests	suggest	VERB
cana-5491	49	8	that	that	SCONJ
cana-5491	49	9	the	the	DET
cana-5491	49	10	linear	linear	ADJ
cana-5491	49	11	term	term	NOUN
cana-5491	49	12	$	$	SYM
cana-5491	49	13	y(x)$	y(x)$	NOUN
cana-5491	49	14	is	be	AUX
cana-5491	49	15	decomposed	decompose	VERB
cana-5491	49	16	by	by	ADP
cana-5491	49	17	an	an	DET
cana-5491	49	18	power	power	NOUN
cana-5491	49	19	series	series	NOUN
cana-5491	49	20	:	:	PUNCT
cana-5491	49	21	𝑓(𝑤	𝑓(𝑤	NUM
cana-5491	49	22	)	)	PUNCT
cana-5491	49	23	=	=	SYM
cana-5491	49	24	𝜑𝜃1,𝜃2,𝜃3	𝜑𝜃1,𝜃2,𝜃3	PROPN
cana-5491	49	25	𝛽,𝛾,𝛿	𝛽,𝛾,𝛿	NUM
cana-5491	49	26	(	(	PUNCT
cana-5491	49	27	𝑤	𝑤	PROPN
cana-5491	49	28	,	,	PUNCT
cana-5491	49	29	𝑠	𝑠	PROPN
cana-5491	49	30	,	,	PUNCT
cana-5491	49	31	𝑝	𝑝	NOUN
cana-5491	49	32	)	)	PUNCT
cana-5491	49	33	=	=	SYM
cana-5491	49	34	∑	∑	PROPN
cana-5491	49	35	(	(	PUNCT
cana-5491	49	36	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	49	37	(	(	PUNCT
cana-5491	49	38	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	49	39	(	(	PUNCT
cana-5491	49	40	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	49	41	(	(	PUNCT
cana-5491	49	42	𝑟+𝑝)𝑠	𝑟+𝑝)𝑠	NUM
cana-5491	49	43	(	(	PUNCT
cana-5491	49	44	𝐴𝑤𝛼)𝑟	𝐴𝑤𝛼)𝑟	PROPN
cana-5491	49	45	(	(	PUNCT
cana-5491	49	46	8)∞	8)∞	NUM
cana-5491	49	47	𝑟=0	𝑟=0	SYM
cana-5491	49	48	𝑓(𝑤	𝑓(𝑤	PROPN
cana-5491	49	49	)	)	PUNCT
cana-5491	49	50	=	=	SYM
cana-5491	50	1	1	1	NUM
cana-5491	50	2	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	50	3	+	+	CCONJ
cana-5491	50	4	(	(	PUNCT
cana-5491	50	5	𝜃1)𝛽	𝜃1)𝛽	NOUN
cana-5491	50	6	(	(	PUNCT
cana-5491	50	7	𝜃3)𝛿	𝜃3)𝛿	NOUN
cana-5491	50	8	(	(	PUNCT
cana-5491	50	9	𝜃2)𝛾	𝜃2)𝛾	NOUN
cana-5491	50	10	(	(	PUNCT
cana-5491	50	11	1	1	NUM
cana-5491	50	12	+	+	NUM
cana-5491	50	13	𝑝)𝑠	𝑝)𝑠	X
cana-5491	50	14	(	(	PUNCT
cana-5491	50	15	𝐴𝑤𝛼)1	𝐴𝑤𝛼)1	NOUN
cana-5491	50	16	+	+	X
cana-5491	50	17	.	.	PUNCT
cana-5491	50	18	.	.	PUNCT
cana-5491	50	19	.	.	PUNCT
cana-5491	50	20	.	.	PUNCT
cana-5491	51	1	(	(	PUNCT
cana-5491	51	2	9	9	X
cana-5491	51	3	)	)	PUNCT
cana-5491	51	4	theorem	theorem	NOUN
cana-5491	51	5	1	1	NUM
cana-5491	51	6	.	.	PUNCT
cana-5491	52	1	the	the	DET
cana-5491	52	2	following	follow	VERB
cana-5491	52	3	derivative	derivative	ADJ
cana-5491	52	4	formula	formula	NOUN
cana-5491	52	5	holds	hold	VERB
cana-5491	52	6	:	:	PUNCT
cana-5491	52	7	𝐷𝛼𝑓(𝑤	𝐷𝛼𝑓(𝑤	PROPN
cana-5491	52	8	)	)	PUNCT
cana-5491	53	1	=	=	PUNCT
cana-5491	53	2	∑	∑	PUNCT
cana-5491	53	3	(	(	PUNCT
cana-5491	53	4	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	53	5	(	(	PUNCT
cana-5491	53	6	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	53	7	(	(	PUNCT
cana-5491	53	8	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	53	9	(	(	PUNCT
cana-5491	53	10	𝑟+𝑝)𝑠	𝑟+𝑝)𝑠	NOUN
cana-5491	53	11	𝐴𝑟𝛤(𝑟𝛼+1	𝐴𝑟𝛤(𝑟𝛼+1	NOUN
cana-5491	53	12	)	)	PUNCT
cana-5491	53	13	𝛤(𝛼(𝑟−1)+1	𝛤(𝛼(𝑟−1)+1	NOUN
cana-5491	53	14	)	)	PUNCT
cana-5491	53	15	𝑤𝛼(𝑟−1)∞	𝑤𝛼(𝑟−1)∞	PROPN
cana-5491	53	16	𝑟=1	𝑟=1	NUM
cana-5491	53	17	(	(	PUNCT
cana-5491	53	18	10	10	NUM
cana-5491	53	19	)	)	PUNCT
cana-5491	53	20	proof	proof	NOUN
cana-5491	53	21	.	.	PUNCT
cana-5491	54	1	from	from	ADP
cana-5491	54	2	(	(	PUNCT
cana-5491	54	3	1	1	NUM
cana-5491	54	4	)	)	PUNCT
cana-5491	54	5	and	and	CCONJ
cana-5491	54	6	(	(	PUNCT
cana-5491	54	7	4	4	NUM
cana-5491	54	8	)	)	PUNCT
cana-5491	54	9	,	,	PUNCT
cana-5491	54	10	we	we	PRON
cana-5491	54	11	have	have	AUX
cana-5491	54	12	𝐷𝛼𝑓(𝑤	𝐷𝛼𝑓(𝑤	VERB
cana-5491	54	13	)	)	PUNCT
cana-5491	54	14	=	=	SYM
cana-5491	55	1	1	1	NUM
cana-5491	55	2	𝛤(𝑙	𝛤(𝑙	NOUN
cana-5491	55	3	−	−	NOUN
cana-5491	55	4	𝛼	𝛼	NOUN
cana-5491	55	5	)	)	PUNCT
cana-5491	55	6	∫(𝑤	∫(𝑤	PROPN
cana-5491	55	7	−	−	PROPN
cana-5491	55	8	𝑢)𝑙−𝛼−1	𝑢)𝑙−𝛼−1	PROPN
cana-5491	55	9	𝑤	𝑤	ADP
cana-5491	55	10	0	0	NUM
cana-5491	55	11	∑	∑	PRON
cana-5491	55	12	(	(	PUNCT
cana-5491	55	13	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	55	14	(	(	PUNCT
cana-5491	55	15	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	55	16	(	(	PUNCT
cana-5491	55	17	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	55	18	(	(	PUNCT
cana-5491	55	19	𝑟	𝑟	NOUN
cana-5491	55	20	+	+	SYM
cana-5491	55	21	𝑝)𝑠	𝑝)𝑠	X
cana-5491	56	1	𝐴𝛼𝐷𝑡𝑢𝑟𝛼𝑑	𝐴𝛼𝐷𝑡𝑢𝑟𝛼𝑑	PROPN
cana-5491	56	2	∞	∞	PROPN
cana-5491	56	3	𝑟=1	𝑟=1	PROPN
cana-5491	56	4	=	=	SYM
cana-5491	56	5	1	1	NUM
cana-5491	56	6	𝛤(𝑙	𝛤(𝑙	NOUN
cana-5491	56	7	−	−	NOUN
cana-5491	56	8	𝛼	𝛼	NOUN
cana-5491	56	9	)	)	PUNCT
cana-5491	56	10	∫(𝑤	∫(𝑤	PROPN
cana-5491	56	11	−	−	PROPN
cana-5491	56	12	𝑢)𝑙−𝛼−1	𝑢)𝑙−𝛼−1	PROPN
cana-5491	56	13	𝑤	𝑤	ADP
cana-5491	56	14	0	0	NUM
cana-5491	56	15	∑	∑	PUNCT
cana-5491	56	16	(	(	PUNCT
cana-5491	56	17	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	X
cana-5491	56	18	(	(	PUNCT
cana-5491	56	19	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	56	20	𝐴𝛼	𝐴𝛼	PROPN
cana-5491	56	21	(	(	PUNCT
cana-5491	56	22	𝑟	𝑟	NOUN
cana-5491	56	23	+	+	CCONJ
cana-5491	56	24	𝑝)𝑠	𝑝)𝑠	X
cana-5491	56	25	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	56	26	+	+	NOUN
cana-5491	56	27	1	1	X
cana-5491	56	28	)	)	PUNCT
cana-5491	56	29	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	56	30	−	−	NOUN
cana-5491	56	31	𝑙	𝑙	PROPN
cana-5491	56	32	+	+	NUM
cana-5491	56	33	1	1	NUM
cana-5491	56	34	)	)	PUNCT
cana-5491	56	35	𝑢𝑟𝛼−𝑙𝑑𝑢	𝑢𝑟𝛼−𝑙𝑑𝑢	X
cana-5491	56	36	∞	∞	X
cana-5491	56	37	𝑟=1	𝑟=1	X
cana-5491	56	38	=	=	SYM
cana-5491	56	39	1	1	NUM
cana-5491	56	40	𝛤(𝑙	𝛤(𝑙	NOUN
cana-5491	56	41	−	−	NOUN
cana-5491	56	42	𝛼	𝛼	NOUN
cana-5491	56	43	)	)	PUNCT
cana-5491	56	44	∑	∑	PUNCT
cana-5491	56	45	(	(	PUNCT
cana-5491	56	46	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	X
cana-5491	56	47	(	(	PUNCT
cana-5491	56	48	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	56	49	𝐴𝛼	𝐴𝛼	PROPN
cana-5491	56	50	(	(	PUNCT
cana-5491	56	51	𝑟	𝑟	NOUN
cana-5491	56	52	+	+	CCONJ
cana-5491	56	53	𝑝)𝑠	𝑝)𝑠	X
cana-5491	56	54	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	56	55	+	+	NOUN
cana-5491	56	56	1	1	X
cana-5491	56	57	)	)	PUNCT
cana-5491	56	58	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	56	59	−	−	NOUN
cana-5491	56	60	𝑙	𝑙	PROPN
cana-5491	56	61	+	+	NUM
cana-5491	56	62	1	1	NUM
cana-5491	56	63	)	)	PUNCT
cana-5491	56	64	∞	∞	PROPN
cana-5491	56	65	𝑟=1	𝑟=1	PROPN
cana-5491	57	1	∫(𝑤	∫(𝑤	PROPN
cana-5491	57	2	−	−	PROPN
cana-5491	57	3	𝑢)𝑙−𝛼−1	𝑢)𝑙−𝛼−1	PROPN
cana-5491	57	4	𝑤	𝑤	ADP
cana-5491	57	5	0	0	NUM
cana-5491	57	6	𝑢𝑟𝛼−𝑙𝑑𝑢	𝑢𝑟𝛼−𝑙𝑑𝑢	PROPN
cana-5491	57	7	=	=	SYM
cana-5491	57	8	1	1	NUM
cana-5491	57	9	𝛤(𝑙	𝛤(𝑙	NOUN
cana-5491	57	10	−	−	NOUN
cana-5491	57	11	𝛼	𝛼	NOUN
cana-5491	57	12	)	)	PUNCT
cana-5491	57	13	∑	∑	PUNCT
cana-5491	57	14	(	(	PUNCT
cana-5491	57	15	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	X
cana-5491	57	16	(	(	PUNCT
cana-5491	57	17	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	57	18	𝐴𝛼	𝐴𝛼	PROPN
cana-5491	57	19	(	(	PUNCT
cana-5491	57	20	𝑟	𝑟	NOUN
cana-5491	57	21	+	+	CCONJ
cana-5491	57	22	𝑝)𝑠	𝑝)𝑠	X
cana-5491	57	23	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	57	24	+	+	NOUN
cana-5491	57	25	1	1	X
cana-5491	57	26	)	)	PUNCT
cana-5491	57	27	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	57	28	−	−	NOUN
cana-5491	57	29	𝑙	𝑙	PROPN
cana-5491	57	30	+	+	NUM
cana-5491	57	31	1	1	NUM
cana-5491	57	32	)	)	PUNCT
cana-5491	57	33	∞	∞	PROPN
cana-5491	57	34	𝑟=1	𝑟=1	PROPN
cana-5491	57	35	∫	∫	PROPN
cana-5491	57	36	𝑤𝑙−𝛼−1(1	𝑤𝑙−𝛼−1(1	PUNCT
cana-5491	57	37	−	−	PROPN
cana-5491	57	38	𝑢	𝑢	PROPN
cana-5491	57	39	𝑤	𝑤	NOUN
cana-5491	57	40	)	)	PUNCT
cana-5491	57	41	𝑙−𝛼−1	𝑙−𝛼−1	NOUN
cana-5491	57	42	𝑤	𝑤	ADP
cana-5491	57	43	0	0	NUM
cana-5491	57	44	𝑢𝑟𝛼−𝑙𝑑𝑢	𝑢𝑟𝛼−𝑙𝑑𝑢	PROPN
cana-5491	57	45	now	now	ADV
cana-5491	57	46	let	let	VERB
cana-5491	57	47	𝑢	𝑢	NOUN
cana-5491	57	48	𝑤	𝑤	ADP
cana-5491	57	49	=	=	PRON
cana-5491	57	50	𝑣	𝑣	PRON
cana-5491	57	51	𝑑𝑢	𝑑𝑢	X
cana-5491	58	1	=	=	X
cana-5491	58	2	𝑤𝑑𝑣.	𝑤𝑑𝑣.	X
cana-5491	58	3	then	then	ADV
cana-5491	58	4	=	=	SYM
cana-5491	59	1	1	1	NUM
cana-5491	59	2	𝛤(𝑙	𝛤(𝑙	NOUN
cana-5491	59	3	−	−	NOUN
cana-5491	59	4	𝛼	𝛼	NOUN
cana-5491	59	5	)	)	PUNCT
cana-5491	59	6	∑	∑	PUNCT
cana-5491	59	7	(	(	PUNCT
cana-5491	59	8	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	𝜃1)𝑟𝛽(𝜃2)𝑟𝛾	X
cana-5491	59	9	(	(	PUNCT
cana-5491	59	10	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	59	11	𝐴𝛼	𝐴𝛼	PROPN
cana-5491	59	12	(	(	PUNCT
cana-5491	59	13	𝑟	𝑟	NOUN
cana-5491	59	14	+	+	CCONJ
cana-5491	59	15	𝑝)𝑠	𝑝)𝑠	X
cana-5491	59	16	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	59	17	+	+	NOUN
cana-5491	59	18	1	1	X
cana-5491	59	19	)	)	PUNCT
cana-5491	59	20	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	59	21	−	−	NOUN
cana-5491	59	22	𝑙	𝑙	PROPN
cana-5491	59	23	+	+	NUM
cana-5491	59	24	1	1	NUM
cana-5491	59	25	)	)	PUNCT
cana-5491	59	26	∞	∞	PROPN
cana-5491	59	27	𝑟=1	𝑟=1	PROPN
cana-5491	59	28	𝑤𝛼(𝑟−1	𝑤𝛼(𝑟−1	PROPN
cana-5491	59	29	)	)	PUNCT
cana-5491	59	30	𝛤(𝑟𝛼	𝛤(𝑟𝛼	NOUN
cana-5491	59	31	−	−	NOUN
cana-5491	59	32	𝑙	𝑙	PROPN
cana-5491	59	33	+	+	NUM
cana-5491	59	34	1)𝛤(𝑙	1)𝛤(𝑙	NUM
cana-5491	59	35	−	−	NUM
cana-5491	59	36	𝛼	𝛼	NOUN
cana-5491	59	37	)	)	PUNCT
cana-5491	59	38	𝛤(𝛼(𝑟	𝛤(𝛼(𝑟	PUNCT
cana-5491	60	1	−	−	PROPN
cana-5491	60	2	1	1	X
cana-5491	60	3	)	)	PUNCT
cana-5491	60	4	+	+	CCONJ
cana-5491	60	5	1	1	X
cana-5491	60	6	)	)	PUNCT
cana-5491	60	7	on	on	ADP
cana-5491	60	8	solving	solve	VERB
cana-5491	60	9	we	we	PRON
cana-5491	60	10	get	get	VERB
cana-5491	60	11	the	the	DET
cana-5491	60	12	desired	desire	VERB
cana-5491	60	13	result	result	NOUN
cana-5491	60	14	(	(	PUNCT
cana-5491	60	15	10	10	NUM
cana-5491	60	16	)	)	PUNCT
cana-5491	60	17	.	.	PUNCT
cana-5491	61	1	theorem	theorem	NOUN
cana-5491	61	2	2	2	NUM
cana-5491	61	3	.	.	PUNCT
cana-5491	62	1	the	the	DET
cana-5491	62	2	following	follow	VERB
cana-5491	62	3	derivative	derivative	ADJ
cana-5491	62	4	formula	formula	NOUN
cana-5491	62	5	holds	hold	VERB
cana-5491	62	6	:	:	PUNCT
cana-5491	62	7	communications	communication	NOUN
cana-5491	62	8	on	on	ADP
cana-5491	62	9	applied	apply	VERB
cana-5491	62	10	nonlinear	nonlinear	ADJ
cana-5491	62	11	analysis	analysis	NOUN
cana-5491	62	12	issn	issn	NOUN
cana-5491	62	13	:	:	PUNCT
cana-5491	62	14	1074	1074	NUM
cana-5491	62	15	-	-	PUNCT
cana-5491	62	16	133x	133x	NUM
cana-5491	62	17	vol	vol	VERB
cana-5491	62	18	32	32	NUM
cana-5491	62	19	no	no	NOUN
cana-5491	62	20	.	.	PUNCT
cana-5491	63	1	10s	10	NOUN
cana-5491	63	2	(	(	PUNCT
cana-5491	63	3	2025	2025	NUM
cana-5491	63	4	)	)	PUNCT
cana-5491	63	5	2437	2437	NUM
cana-5491	63	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	63	7	𝐷2𝛼𝑓(𝑤	𝐷2𝛼𝑓(𝑤	PROPN
cana-5491	63	8	)	)	PUNCT
cana-5491	63	9	=	=	PUNCT
cana-5491	63	10	∑	∑	PUNCT
cana-5491	63	11	(	(	PUNCT
cana-5491	63	12	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	63	13	(	(	PUNCT
cana-5491	63	14	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	63	15	(	(	PUNCT
cana-5491	63	16	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	63	17	(	(	PUNCT
cana-5491	63	18	𝑟+𝑝)𝑠	𝑟+𝑝)𝑠	NOUN
cana-5491	63	19	𝐴𝑟𝛤(𝑟𝛼+1	𝐴𝑟𝛤(𝑟𝛼+1	NOUN
cana-5491	63	20	)	)	PUNCT
cana-5491	63	21	𝛤(𝛼(𝑟−2)+1	𝛤(𝛼(𝑟−2)+1	NOUN
cana-5491	63	22	)	)	PUNCT
cana-5491	63	23	𝑤𝛼(𝑟−2)∞	𝑤𝛼(𝑟−2)∞	NOUN
cana-5491	63	24	𝑟=2	𝑟=2	PROPN
cana-5491	63	25	.	.	PUNCT
cana-5491	64	1	(	(	PUNCT
cana-5491	64	2	11	11	NUM
cana-5491	64	3	)	)	PUNCT
cana-5491	64	4	similarly	similarly	ADV
cana-5491	64	5	,	,	PUNCT
cana-5491	64	6	we	we	PRON
cana-5491	64	7	can	can	AUX
cana-5491	64	8	proof	proof	VERB
cana-5491	64	9	the	the	DET
cana-5491	64	10	above	above	ADJ
cana-5491	64	11	result	result	NOUN
cana-5491	64	12	as	as	ADP
cana-5491	64	13	a	a	DET
cana-5491	64	14	proof	proof	NOUN
cana-5491	64	15	of	of	ADP
cana-5491	64	16	theorem	theorem	NOUN
cana-5491	64	17	1	1	NUM
cana-5491	64	18	.	.	NOUN
cana-5491	64	19	3	3	NUM
cana-5491	64	20	.	.	NOUN
cana-5491	64	21	numerical	numerical	ADJ
cana-5491	64	22	applications	application	NOUN
cana-5491	64	23	in	in	ADP
cana-5491	64	24	this	this	DET
cana-5491	64	25	section	section	NOUN
cana-5491	64	26	,	,	PUNCT
cana-5491	64	27	we	we	PRON
cana-5491	64	28	consider	consider	VERB
cana-5491	64	29	few	few	ADJ
cana-5491	64	30	examples	example	NOUN
cana-5491	64	31	that	that	PRON
cana-5491	64	32	demonstrate	demonstrate	VERB
cana-5491	64	33	the	the	DET
cana-5491	64	34	extended	extended	ADJ
cana-5491	64	35	hurwitz	hurwitz	PROPN
cana-5491	64	36	-	-	PUNCT
cana-5491	64	37	lerch	lerch	PROPN
cana-5491	64	38	zeta	zeta	PROPN
cana-5491	64	39	function	function	NOUN
cana-5491	64	40	for	for	ADP
cana-5491	64	41	solving	solve	VERB
cana-5491	64	42	linear	linear	ADJ
cana-5491	64	43	differential	differential	ADJ
cana-5491	64	44	equation	equation	NOUN
cana-5491	64	45	with	with	ADP
cana-5491	64	46	fractional	fractional	ADJ
cana-5491	64	47	derivative	derivative	NOUN
cana-5491	64	48	.	.	PUNCT
cana-5491	65	1	example	example	NOUN
cana-5491	66	1	1	1	NUM
cana-5491	66	2	.	.	PUNCT
cana-5491	66	3	the	the	DET
cana-5491	66	4	solution	solution	NOUN
cana-5491	66	5	of	of	ADP
cana-5491	66	6	following	follow	VERB
cana-5491	66	7	fractional	fractional	ADJ
cana-5491	66	8	differential	differential	NOUN
cana-5491	66	9	equation	equation	NOUN
cana-5491	66	10	𝐷𝛼𝑓(𝑤	𝐷𝛼𝑓(𝑤	PROPN
cana-5491	66	11	)	)	PUNCT
cana-5491	67	1	−	−	PROPN
cana-5491	67	2	𝐶𝑓(𝑤	𝐶𝑓(𝑤	ADJ
cana-5491	67	3	)	)	PUNCT
cana-5491	67	4	=	=	SYM
cana-5491	67	5	0	0	X
cana-5491	67	6	.	.	PUNCT
cana-5491	68	1	by	by	ADP
cana-5491	68	2	equation	equation	NOUN
cana-5491	68	3	(	(	PUNCT
cana-5491	68	4	4	4	NUM
cana-5491	68	5	)	)	PUNCT
cana-5491	68	6	and	and	CCONJ
cana-5491	68	7	theorem	theorem	VERB
cana-5491	68	8	(	(	PUNCT
cana-5491	68	9	1	1	NUM
cana-5491	68	10	)	)	PUNCT
cana-5491	68	11	∑	∑	PRON
cana-5491	68	12	(	(	PUNCT
cana-5491	68	13	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	68	14	(	(	PUNCT
cana-5491	68	15	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	68	16	(	(	PUNCT
cana-5491	68	17	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	68	18	(	(	PUNCT
cana-5491	68	19	𝑟	𝑟	NOUN
cana-5491	68	20	+	+	SYM
cana-5491	68	21	𝑝)𝑠	𝑝)𝑠	X
cana-5491	68	22	𝐴𝑟𝛤(𝑟𝛼	𝐴𝑟𝛤(𝑟𝛼	ADJ
cana-5491	68	23	+	+	CCONJ
cana-5491	68	24	1	1	NUM
cana-5491	68	25	)	)	PUNCT
cana-5491	68	26	𝛤(𝛼(𝑟	𝛤(𝛼(𝑟	NOUN
cana-5491	68	27	−	−	NOUN
cana-5491	68	28	1	1	X
cana-5491	68	29	)	)	PUNCT
cana-5491	68	30	+	+	CCONJ
cana-5491	68	31	1	1	X
cana-5491	68	32	)	)	PUNCT
cana-5491	68	33	𝑤𝛼(𝑟−1	𝑤𝛼(𝑟−1	NUM
cana-5491	68	34	)	)	PUNCT
cana-5491	68	35	∞	∞	PROPN
cana-5491	69	1	𝑟=1	𝑟=1	PRON
cana-5491	69	2	−	−	PROPN
cana-5491	69	3	𝐶	𝐶	PROPN
cana-5491	69	4	∑	∑	PROPN
cana-5491	69	5	(	(	PUNCT
cana-5491	69	6	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	69	7	(	(	PUNCT
cana-5491	69	8	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	69	9	(	(	PUNCT
cana-5491	69	10	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	69	11	(	(	PUNCT
cana-5491	69	12	𝑟	𝑟	NOUN
cana-5491	69	13	+	+	CCONJ
cana-5491	69	14	𝑝)𝑠	𝑝)𝑠	X
cana-5491	69	15	(	(	PUNCT
cana-5491	69	16	𝐴𝑤𝛼)𝑟	𝐴𝑤𝛼)𝑟	NOUN
cana-5491	69	17	=	=	SYM
cana-5491	69	18	0	0	NUM
cana-5491	69	19	∞	∞	NUM
cana-5491	69	20	𝑟=0	𝑟=0	PUNCT
cana-5491	69	21	replace	replace	VERB
cana-5491	69	22	r	r	NOUN
cana-5491	69	23	by	by	ADP
cana-5491	69	24	r+1	r+1	PROPN
cana-5491	69	25	in	in	ADP
cana-5491	69	26	first	first	ADJ
cana-5491	69	27	summation	summation	NOUN
cana-5491	69	28	∑	∑	PROPN
cana-5491	69	29	(	(	PUNCT
cana-5491	69	30	𝜃1)(𝑟+1)𝛽	𝜃1)(𝑟+1)𝛽	PROPN
cana-5491	69	31	(	(	PUNCT
cana-5491	69	32	𝜃3)(𝑟+1)𝛿	𝜃3)(𝑟+1)𝛿	PROPN
cana-5491	69	33	(	(	PUNCT
cana-5491	69	34	𝜃2)(𝑟+1)𝛾	𝜃2)(𝑟+1)𝛾	NOUN
cana-5491	69	35	(	(	PUNCT
cana-5491	69	36	𝑟	𝑟	NOUN
cana-5491	69	37	+	+	CCONJ
cana-5491	69	38	1	1	NUM
cana-5491	69	39	+	+	NUM
cana-5491	69	40	𝑝)𝑠	𝑝)𝑠	NOUN
cana-5491	69	41	𝐴𝑟+1𝛤((𝑟	𝐴𝑟+1𝛤((𝑟	NOUN
cana-5491	69	42	+	+	CCONJ
cana-5491	69	43	1)𝛼	1)𝛼	NUM
cana-5491	69	44	+	+	CCONJ
cana-5491	69	45	1	1	NUM
cana-5491	69	46	)	)	PUNCT
cana-5491	69	47	𝛤(𝛼𝑟	𝛤(𝛼𝑟	NOUN
cana-5491	70	1	+	+	CCONJ
cana-5491	70	2	1	1	X
cana-5491	70	3	)	)	PUNCT
cana-5491	70	4	𝑤𝛼𝑟	𝑤𝛼𝑟	NOUN
cana-5491	70	5	∞	∞	NUM
cana-5491	70	6	𝑟=0	𝑟=0	PUNCT
cana-5491	70	7	−	−	PROPN
cana-5491	70	8	𝐶	𝐶	PROPN
cana-5491	70	9	∑	∑	PROPN
cana-5491	70	10	(	(	PUNCT
cana-5491	70	11	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	70	12	(	(	PUNCT
cana-5491	70	13	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	70	14	(	(	PUNCT
cana-5491	70	15	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	70	16	(	(	PUNCT
cana-5491	70	17	𝑟	𝑟	NOUN
cana-5491	70	18	+	+	CCONJ
cana-5491	70	19	𝑝)𝑠	𝑝)𝑠	X
cana-5491	70	20	(	(	PUNCT
cana-5491	70	21	𝐴𝑤𝛼)𝑟	𝐴𝑤𝛼)𝑟	NOUN
cana-5491	70	22	=	=	SYM
cana-5491	70	23	0	0	NUM
cana-5491	70	24	∞	∞	NUM
cana-5491	70	25	𝑟=0	𝑟=0	PUNCT
cana-5491	71	1	∑	∑	PUNCT
cana-5491	71	2	[	[	PUNCT
cana-5491	71	3	(	(	PUNCT
cana-5491	71	4	𝜃1)(𝑟+1)𝛽	𝜃1)(𝑟+1)𝛽	NOUN
cana-5491	71	5	(	(	PUNCT
cana-5491	71	6	𝜃3)(𝑟+1)𝛿	𝜃3)(𝑟+1)𝛿	PROPN
cana-5491	71	7	(	(	PUNCT
cana-5491	71	8	𝜃2)(𝑟+1)𝛾	𝜃2)(𝑟+1)𝛾	NOUN
cana-5491	71	9	(	(	PUNCT
cana-5491	71	10	𝑟	𝑟	NOUN
cana-5491	71	11	+	+	CCONJ
cana-5491	71	12	1	1	NUM
cana-5491	71	13	+	+	CCONJ
cana-5491	71	14	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	71	15	𝐴1𝛤((𝑟	𝐴1𝛤((𝑟	NOUN
cana-5491	71	16	+	+	CCONJ
cana-5491	71	17	1)𝛼	1)𝛼	NUM
cana-5491	71	18	+	+	CCONJ
cana-5491	71	19	1	1	NUM
cana-5491	71	20	)	)	PUNCT
cana-5491	71	21	𝛤(𝛼𝑟	𝛤(𝛼𝑟	NOUN
cana-5491	71	22	+	+	CCONJ
cana-5491	71	23	1	1	X
cana-5491	71	24	)	)	PUNCT
cana-5491	71	25	𝑤𝛼𝑟	𝑤𝛼𝑟	NOUN
cana-5491	71	26	∞	∞	NUM
cana-5491	71	27	𝑟=0	𝑟=0	PUNCT
cana-5491	71	28	−	−	PROPN
cana-5491	71	29	𝐶	𝐶	PROPN
cana-5491	71	30	∑	∑	PROPN
cana-5491	71	31	(	(	PUNCT
cana-5491	71	32	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	71	33	(	(	PUNCT
cana-5491	71	34	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	71	35	(	(	PUNCT
cana-5491	71	36	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	71	37	(	(	PUNCT
cana-5491	71	38	𝑟	𝑟	NOUN
cana-5491	71	39	+	+	SYM
cana-5491	71	40	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	72	1	]	]	X
cana-5491	72	2	𝐴𝑟𝑤𝑟𝛼	𝐴𝑟𝑤𝑟𝛼	PROPN
cana-5491	72	3	=	=	PUNCT
cana-5491	72	4	0	0	NUM
cana-5491	72	5	∞	∞	NUM
cana-5491	72	6	𝑟=0	𝑟=0	PUNCT
cana-5491	72	7	now	now	ADV
cana-5491	72	8	equating	equate	VERB
cana-5491	72	9	to	to	ADP
cana-5491	72	10	zero	zero	NUM
cana-5491	72	11	the	the	DET
cana-5491	72	12	coefficient	coefficient	NOUN
cana-5491	72	13	of	of	ADP
cana-5491	72	14	𝑤𝑟𝛼	𝑤𝑟𝛼	PROPN
cana-5491	72	15	,	,	PUNCT
cana-5491	72	16	we	we	PRON
cana-5491	72	17	get	get	VERB
cana-5491	72	18	(	(	PUNCT
cana-5491	72	19	𝜃1)(𝑟+1)𝛽(𝜃2)(𝑟+1)𝛾	𝜃1)(𝑟+1)𝛽(𝜃2)(𝑟+1)𝛾	NUM
cana-5491	72	20	(	(	PUNCT
cana-5491	72	21	𝜃3)(𝑟+1)𝛿	𝜃3)(𝑟+1)𝛿	PROPN
cana-5491	72	22	𝐴	𝐴	PROPN
cana-5491	72	23	(	(	PUNCT
cana-5491	72	24	𝑟	𝑟	NOUN
cana-5491	72	25	+	+	CCONJ
cana-5491	72	26	1	1	NUM
cana-5491	72	27	+	+	CCONJ
cana-5491	72	28	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	72	29	𝛤((𝑟	𝛤((𝑟	NOUN
cana-5491	72	30	+	+	CCONJ
cana-5491	72	31	1)𝛼	1)𝛼	NUM
cana-5491	72	32	+	+	CCONJ
cana-5491	72	33	1	1	NUM
cana-5491	72	34	)	)	PUNCT
cana-5491	72	35	𝛤(𝛼𝑟	𝛤(𝛼𝑟	NOUN
cana-5491	73	1	+	+	CCONJ
cana-5491	73	2	1	1	X
cana-5491	73	3	)	)	PUNCT
cana-5491	73	4	=	=	SYM
cana-5491	73	5	𝐶	𝐶	PROPN
cana-5491	73	6	(	(	PUNCT
cana-5491	73	7	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	73	8	(	(	PUNCT
cana-5491	73	9	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	73	10	(	(	PUNCT
cana-5491	73	11	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	73	12	(	(	PUNCT
cana-5491	73	13	𝑝	𝑝	PROPN
cana-5491	73	14	+	+	CCONJ
cana-5491	73	15	𝑟)𝑠	𝑟)𝑠	PUNCT
cana-5491	73	16	𝑎𝑡	𝑎𝑡	ADP
cana-5491	73	17	𝑟	𝑟	X
cana-5491	73	18	=	=	SYM
cana-5491	73	19	0	0	NUM
cana-5491	73	20	,	,	PUNCT
cana-5491	73	21	(	(	PUNCT
cana-5491	73	22	𝜃1)𝛽(𝜃2)𝛾	𝜃1)𝛽(𝜃2)𝛾	NOUN
cana-5491	73	23	(	(	PUNCT
cana-5491	73	24	𝜃3)𝛿(𝑝	𝜃3)𝛿(𝑝	NOUN
cana-5491	73	25	+	+	NOUN
cana-5491	73	26	1)𝑠	1)𝑠	NUM
cana-5491	73	27	𝛤(𝛼	𝛤(𝛼	X
cana-5491	73	28	+	+	ADJ
cana-5491	73	29	1	1	NUM
cana-5491	73	30	)	)	PUNCT
cana-5491	73	31	𝛤(1	𝛤(1	NUM
cana-5491	73	32	)	)	PUNCT
cana-5491	73	33	𝐴	𝐴	NOUN
cana-5491	73	34	=	=	SYM
cana-5491	73	35	𝐶	𝐶	PROPN
cana-5491	73	36	1	1	NUM
cana-5491	73	37	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	73	38	(	(	PUNCT
cana-5491	73	39	𝜃1)𝛽(𝜃2)𝛾	𝜃1)𝛽(𝜃2)𝛾	NOUN
cana-5491	73	40	(	(	PUNCT
cana-5491	73	41	𝜃3)𝛿(𝑝	𝜃3)𝛿(𝑝	VERB
cana-5491	73	42	+	+	PROPN
cana-5491	73	43	1)𝑠	1)𝑠	NUM
cana-5491	73	44	𝐴	𝐴	NOUN
cana-5491	73	45	=	=	SYM
cana-5491	73	46	𝐶	𝐶	PROPN
cana-5491	73	47	1	1	NUM
cana-5491	73	48	𝑝𝑠𝛤(𝛼	𝑝𝑠𝛤(𝛼	INTJ
cana-5491	73	49	+	+	NOUN
cana-5491	73	50	1	1	X
cana-5491	73	51	)	)	PUNCT
cana-5491	73	52	𝑎𝑡	𝑎𝑡	ADP
cana-5491	73	53	𝑟	𝑟	NOUN
cana-5491	73	54	=	=	SYM
cana-5491	73	55	1	1	NUM
cana-5491	73	56	,	,	PUNCT
cana-5491	73	57	(	(	PUNCT
cana-5491	73	58	𝜃1)2𝛽(𝜃2)2𝛾	𝜃1)2𝛽(𝜃2)2𝛾	X
cana-5491	73	59	(	(	PUNCT
cana-5491	73	60	𝜃3)2𝛿(𝑝	𝜃3)2𝛿(𝑝	PROPN
cana-5491	73	61	+	+	X
cana-5491	73	62	2)𝑠	2)𝑠	NOUN
cana-5491	73	63	𝛤(2𝛼	𝛤(2𝛼	VERB
cana-5491	73	64	+	+	CCONJ
cana-5491	73	65	1	1	NUM
cana-5491	73	66	)	)	PUNCT
cana-5491	73	67	𝛤(𝛼	𝛤(𝛼	PRON
cana-5491	73	68	+	+	ADJ
cana-5491	73	69	1	1	X
cana-5491	73	70	)	)	PUNCT
cana-5491	73	71	𝐴	𝐴	PROPN
cana-5491	73	72	=	=	PROPN
cana-5491	73	73	𝐶	𝐶	PROPN
cana-5491	73	74	(	(	PUNCT
cana-5491	73	75	𝜃1)𝛽(𝜃2)𝛾	𝜃1)𝛽(𝜃2)𝛾	NOUN
cana-5491	73	76	(	(	PUNCT
cana-5491	73	77	𝜃3)𝛿(𝑝	𝜃3)𝛿(𝑝	NOUN
cana-5491	73	78	+	+	ADJ
cana-5491	73	79	1)𝑠	1)𝑠	NUM
cana-5491	73	80	(	(	PUNCT
cana-5491	73	81	𝜃1)2𝛽(𝜃2)2𝛾	𝜃1)2𝛽(𝜃2)2𝛾	X
cana-5491	73	82	(	(	PUNCT
cana-5491	73	83	𝜃3)2𝛿(𝑝	𝜃3)2𝛿(𝑝	PROPN
cana-5491	73	84	+	+	X
cana-5491	73	85	2)𝑠	2)𝑠	NOUN
cana-5491	73	86	𝛤(2𝛼	𝛤(2𝛼	VERB
cana-5491	73	87	+	+	CCONJ
cana-5491	73	88	1	1	NUM
cana-5491	73	89	)	)	PUNCT
cana-5491	73	90	𝛤(𝛼	𝛤(𝛼	PRON
cana-5491	73	91	+	+	NUM
cana-5491	73	92	1	1	X
cana-5491	73	93	)	)	PUNCT
cana-5491	73	94	𝐴𝐴	𝐴𝐴	PROPN
cana-5491	73	95	=	=	SYM
cana-5491	73	96	𝐶	𝐶	PROPN
cana-5491	73	97	(	(	PUNCT
cana-5491	73	98	𝜃1)𝛽(𝜃2)𝛾	𝜃1)𝛽(𝜃2)𝛾	NOUN
cana-5491	73	99	(	(	PUNCT
cana-5491	73	100	𝜃3)𝛿(𝑝	𝜃3)𝛿(𝑝	VERB
cana-5491	73	101	+	+	PROPN
cana-5491	73	102	1)𝑠	1)𝑠	PROPN
cana-5491	73	103	𝐴	𝐴	PROPN
cana-5491	73	104	(	(	PUNCT
cana-5491	73	105	𝜃1)2𝛽(𝜃2)2𝛾	𝜃1)2𝛽(𝜃2)2𝛾	X
cana-5491	73	106	(	(	PUNCT
cana-5491	73	107	𝜃3)2𝛿(𝑝	𝜃3)2𝛿(𝑝	PROPN
cana-5491	73	108	+	+	X
cana-5491	73	109	2)𝑠	2)𝑠	NOUN
cana-5491	73	110	𝛤(2𝛼	𝛤(2𝛼	VERB
cana-5491	73	111	+	+	CCONJ
cana-5491	73	112	1	1	NUM
cana-5491	73	113	)	)	PUNCT
cana-5491	74	1	𝛤(𝛼	𝛤(𝛼	PRON
cana-5491	75	1	+	+	NUM
cana-5491	75	2	1	1	X
cana-5491	75	3	)	)	PUNCT
cana-5491	75	4	𝐴𝐴	𝐴𝐴	PROPN
cana-5491	75	5	=	=	SYM
cana-5491	75	6	𝐶𝐶	𝐶𝐶	PROPN
cana-5491	75	7	1	1	NUM
cana-5491	75	8	𝑝𝑠𝛤(𝛼	𝑝𝑠𝛤(𝛼	NOUN
cana-5491	76	1	+	+	NOUN
cana-5491	76	2	1	1	X
cana-5491	76	3	)	)	PUNCT
cana-5491	76	4	=	=	VERB
cana-5491	76	5	𝐶2	𝐶2	ADJ
cana-5491	77	1	1	1	NUM
cana-5491	78	1	𝑝𝑠𝛤(𝛼	𝑝𝑠𝛤(𝛼	INTJ
cana-5491	78	2	+	+	NOUN
cana-5491	78	3	1	1	X
cana-5491	78	4	)	)	PUNCT
cana-5491	78	5	𝑡ℎ𝑢𝑠	𝑡ℎ𝑢𝑠	NOUN
cana-5491	78	6	(	(	PUNCT
cana-5491	78	7	𝜃1)2𝛽(𝜃2)2𝛾	𝜃1)2𝛽(𝜃2)2𝛾	X
cana-5491	78	8	(	(	PUNCT
cana-5491	78	9	𝜃3)2𝛿(𝑝	𝜃3)2𝛿(𝑝	PROPN
cana-5491	78	10	+	+	SYM
cana-5491	78	11	2)𝑠	2)𝑠	NOUN
cana-5491	78	12	𝐴2	𝐴2	NOUN
cana-5491	78	13	=	=	SYM
cana-5491	78	14	𝐶2	𝐶2	PROPN
cana-5491	78	15	1	1	NUM
cana-5491	78	16	𝑝𝑠𝛤(2𝛼	𝑝𝑠𝛤(2𝛼	ADJ
cana-5491	79	1	+	+	CCONJ
cana-5491	79	2	1	1	X
cana-5491	79	3	)	)	PUNCT
cana-5491	79	4	𝑎𝑡	𝑎𝑡	ADP
cana-5491	79	5	𝑟	𝑟	NOUN
cana-5491	79	6	=	=	SYM
cana-5491	79	7	2	2	NUM
cana-5491	79	8	,	,	PUNCT
cana-5491	79	9	(	(	PUNCT
cana-5491	79	10	𝜃1)3𝛽(𝜃2)3𝛾	𝜃1)3𝛽(𝜃2)3𝛾	X
cana-5491	79	11	(	(	PUNCT
cana-5491	79	12	𝜃3)3𝛿(𝑝	𝜃3)3𝛿(𝑝	X
cana-5491	79	13	+	+	NUM
cana-5491	79	14	3)𝑠	3)𝑠	NUM
cana-5491	79	15	𝛤(3𝛼	𝛤(3𝛼	NOUN
cana-5491	79	16	+	+	CCONJ
cana-5491	79	17	1	1	NUM
cana-5491	79	18	)	)	PUNCT
cana-5491	79	19	2𝛤(𝛼	2𝛤(𝛼	NUM
cana-5491	80	1	+	+	CCONJ
cana-5491	80	2	1	1	X
cana-5491	80	3	)	)	PUNCT
cana-5491	80	4	𝐴	𝐴	PROPN
cana-5491	80	5	=	=	PROPN
cana-5491	80	6	𝐶	𝐶	PROPN
cana-5491	80	7	(	(	PUNCT
cana-5491	80	8	𝜃1)2𝛽(𝜃2)2𝛾	𝜃1)2𝛽(𝜃2)2𝛾	X
cana-5491	80	9	(	(	PUNCT
cana-5491	80	10	𝜃3)2𝛿(𝑝	𝜃3)2𝛿(𝑝	PROPN
cana-5491	80	11	+	+	SYM
cana-5491	80	12	2)𝑠	2)𝑠	NOUN
cana-5491	80	13	communications	communication	NOUN
cana-5491	80	14	on	on	ADP
cana-5491	80	15	applied	apply	VERB
cana-5491	80	16	nonlinear	nonlinear	ADJ
cana-5491	80	17	analysis	analysis	NOUN
cana-5491	80	18	issn	issn	NOUN
cana-5491	80	19	:	:	PUNCT
cana-5491	80	20	1074	1074	NUM
cana-5491	80	21	-	-	PUNCT
cana-5491	80	22	133x	133x	NUM
cana-5491	80	23	vol	vol	VERB
cana-5491	80	24	32	32	NUM
cana-5491	80	25	no	no	NOUN
cana-5491	80	26	.	.	PUNCT
cana-5491	81	1	10s	10	NOUN
cana-5491	81	2	(	(	PUNCT
cana-5491	81	3	2025	2025	NUM
cana-5491	81	4	)	)	PUNCT
cana-5491	81	5	2438	2438	NUM
cana-5491	81	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	81	7	𝑜𝑛	𝑜𝑛	NOUN
cana-5491	81	8	𝑠𝑜𝑙𝑣𝑖𝑛𝑔	𝑠𝑜𝑙𝑣𝑖𝑛𝑔	NOUN
cana-5491	81	9	(	(	PUNCT
cana-5491	81	10	𝜃1)3𝛽(𝜃2)3𝛾(𝜃3)3𝛿(𝑝	𝜃1)3𝛽(𝜃2)3𝛾(𝜃3)3𝛿(𝑝	PROPN
cana-5491	81	11	+	+	PROPN
cana-5491	81	12	3)𝑠	3)𝑠	NUM
cana-5491	81	13	𝐴	𝐴	NOUN
cana-5491	81	14	=	=	SYM
cana-5491	81	15	𝐶3	𝐶3	PROPN
cana-5491	81	16	1	1	NUM
cana-5491	81	17	𝑝𝑠𝛤(3𝛼	𝑝𝑠𝛤(3𝛼	PROPN
cana-5491	81	18	+	+	CCONJ
cana-5491	81	19	1	1	NUM
cana-5491	81	20	)	)	PUNCT
cana-5491	81	21	substituting	substitute	VERB
cana-5491	81	22	these	these	DET
cana-5491	81	23	values	value	NOUN
cana-5491	81	24	in	in	ADP
cana-5491	81	25	equation	equation	NOUN
cana-5491	81	26	(	(	PUNCT
cana-5491	81	27	5	5	NUM
cana-5491	81	28	)	)	PUNCT
cana-5491	81	29	,	,	PUNCT
cana-5491	81	30	we	we	PRON
cana-5491	81	31	get	get	VERB
cana-5491	81	32	𝑓(𝑤	𝑓(𝑤	NOUN
cana-5491	81	33	)	)	PUNCT
cana-5491	81	34	=	=	SYM
cana-5491	82	1	1	1	NUM
cana-5491	82	2	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	82	3	+	+	SYM
cana-5491	82	4	𝐶	𝐶	PROPN
cana-5491	82	5	1	1	NUM
cana-5491	82	6	𝑝𝑠𝛤(𝛼	𝑝𝑠𝛤(𝛼	NOUN
cana-5491	82	7	+	+	NOUN
cana-5491	82	8	1	1	X
cana-5491	82	9	)	)	PUNCT
cana-5491	82	10	𝑤𝛼	𝑤𝛼	NOUN
cana-5491	83	1	+	+	CCONJ
cana-5491	83	2	𝐶2	𝐶2	PROPN
cana-5491	83	3	1	1	NUM
cana-5491	83	4	𝑝𝑠𝛤(2𝛼	𝑝𝑠𝛤(2𝛼	ADJ
cana-5491	83	5	+	+	CCONJ
cana-5491	83	6	1	1	NUM
cana-5491	83	7	)	)	PUNCT
cana-5491	83	8	𝑤2𝛼	𝑤2𝛼	NOUN
cana-5491	83	9	+	+	CCONJ
cana-5491	83	10	⋯	⋯	NOUN
cana-5491	83	11	(	(	PUNCT
cana-5491	83	12	10	10	NUM
cana-5491	83	13	)	)	PUNCT
cana-5491	83	14	example	example	NOUN
cana-5491	84	1	2	2	NUM
cana-5491	84	2	.	.	PUNCT
cana-5491	85	1	again	again	ADV
cana-5491	85	2	we	we	PRON
cana-5491	85	3	take	take	VERB
cana-5491	85	4	a	a	DET
cana-5491	85	5	fractional	fractional	ADJ
cana-5491	85	6	differential	differential	NOUN
cana-5491	85	7	equation	equation	NOUN
cana-5491	85	8	𝐷2𝛼𝑓(𝑤	𝐷2𝛼𝑓(𝑤	PROPN
cana-5491	85	9	)	)	PUNCT
cana-5491	85	10	−	−	PROPN
cana-5491	86	1	𝐵𝑓(𝑤	𝐵𝑓(𝑤	SYM
cana-5491	86	2	)	)	PUNCT
cana-5491	86	3	=	=	SYM
cana-5491	86	4	0	0	PUNCT
cana-5491	86	5	(	(	PUNCT
cana-5491	86	6	11	11	NUM
cana-5491	86	7	)	)	PUNCT
cana-5491	86	8	from	from	ADP
cana-5491	86	9	eq	eq	ADP
cana-5491	86	10	.	.	PUNCT
cana-5491	87	1	(	(	PUNCT
cana-5491	87	2	4	4	NUM
cana-5491	87	3	)	)	PUNCT
cana-5491	87	4	and	and	CCONJ
cana-5491	87	5	theorem	theorem	VERB
cana-5491	87	6	(	(	PUNCT
cana-5491	87	7	2	2	NUM
cana-5491	87	8	)	)	PUNCT
cana-5491	87	9	,	,	PUNCT
cana-5491	87	10	we	we	PRON
cana-5491	87	11	get	get	VERB
cana-5491	87	12	∑	∑	PUNCT
cana-5491	87	13	(	(	PUNCT
cana-5491	87	14	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	87	15	(	(	PUNCT
cana-5491	87	16	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	87	17	(	(	PUNCT
cana-5491	87	18	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	87	19	(	(	PUNCT
cana-5491	87	20	𝑟	𝑟	NOUN
cana-5491	87	21	+	+	SYM
cana-5491	87	22	𝑝)𝑠	𝑝)𝑠	X
cana-5491	88	1	𝐴𝑟𝛤(𝑟𝛼	𝐴𝑟𝛤(𝑟𝛼	ADJ
cana-5491	89	1	+	+	CCONJ
cana-5491	89	2	1	1	NUM
cana-5491	89	3	)	)	PUNCT
cana-5491	89	4	𝛤(𝛼(𝑟	𝛤(𝛼(𝑟	NOUN
cana-5491	89	5	−	−	PROPN
cana-5491	89	6	2	2	X
cana-5491	89	7	)	)	PUNCT
cana-5491	89	8	+	+	CCONJ
cana-5491	89	9	1	1	X
cana-5491	89	10	)	)	PUNCT
cana-5491	89	11	𝑤𝛼(𝑟−2	𝑤𝛼(𝑟−2	PROPN
cana-5491	89	12	)	)	PUNCT
cana-5491	89	13	∞	∞	NUM
cana-5491	89	14	𝑟=2	𝑟=2	NUM
cana-5491	89	15	−	−	NOUN
cana-5491	89	16	𝐵	𝐵	NOUN
cana-5491	89	17	∑	∑	PUNCT
cana-5491	89	18	(	(	PUNCT
cana-5491	89	19	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	89	20	(	(	PUNCT
cana-5491	89	21	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	89	22	(	(	PUNCT
cana-5491	89	23	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	89	24	(	(	PUNCT
cana-5491	89	25	𝑟	𝑟	NOUN
cana-5491	89	26	+	+	SYM
cana-5491	89	27	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	89	28	𝐴𝑟(𝑤𝛼)𝑟	𝐴𝑟(𝑤𝛼)𝑟	NOUN
cana-5491	89	29	=	=	SYM
cana-5491	89	30	0	0	NUM
cana-5491	89	31	∞	∞	NUM
cana-5491	89	32	𝑟=0	𝑟=0	PUNCT
cana-5491	89	33	replace	replace	VERB
cana-5491	89	34	r	r	NOUN
cana-5491	89	35	by	by	ADP
cana-5491	89	36	r+2	r+2	NUM
cana-5491	89	37	in	in	ADP
cana-5491	89	38	the	the	DET
cana-5491	89	39	above	above	ADJ
cana-5491	89	40	equation	equation	NOUN
cana-5491	89	41	(	(	PUNCT
cana-5491	89	42	only	only	ADV
cana-5491	89	43	in	in	ADP
cana-5491	89	44	first	first	ADJ
cana-5491	89	45	summation	summation	NOUN
cana-5491	89	46	)	)	PUNCT
cana-5491	89	47	then	then	ADV
cana-5491	89	48	∑	∑	INTJ
cana-5491	89	49	(	(	PUNCT
cana-5491	89	50	𝜃1)(𝑟+2)𝛽	𝜃1)(𝑟+2)𝛽	NOUN
cana-5491	89	51	(	(	PUNCT
cana-5491	89	52	𝜃3)(𝑟+2)𝛿	𝜃3)(𝑟+2)𝛿	PROPN
cana-5491	89	53	(	(	PUNCT
cana-5491	89	54	𝜃2)(𝑟+2)𝛾	𝜃2)(𝑟+2)𝛾	NUM
cana-5491	89	55	(	(	PUNCT
cana-5491	89	56	𝑟	𝑟	NOUN
cana-5491	89	57	+	+	CCONJ
cana-5491	89	58	2	2	NUM
cana-5491	89	59	+	+	CCONJ
cana-5491	89	60	𝑝)𝑠	𝑝)𝑠	NOUN
cana-5491	89	61	𝐴𝑟+2𝛤((𝑟	𝐴𝑟+2𝛤((𝑟	NOUN
cana-5491	90	1	+	+	NOUN
cana-5491	90	2	2)𝛼	2)𝛼	NUM
cana-5491	90	3	+	+	NOUN
cana-5491	90	4	1	1	X
cana-5491	90	5	)	)	PUNCT
cana-5491	90	6	𝛤(𝛼𝑟	𝛤(𝛼𝑟	NOUN
cana-5491	90	7	+	+	CCONJ
cana-5491	90	8	1	1	X
cana-5491	90	9	)	)	PUNCT
cana-5491	90	10	𝑤𝛼𝑟	𝑤𝛼𝑟	NOUN
cana-5491	90	11	∞	∞	NUM
cana-5491	90	12	𝑟=0	𝑟=0	PUNCT
cana-5491	90	13	−	−	PROPN
cana-5491	90	14	𝐵	𝐵	NOUN
cana-5491	90	15	∑	∑	PUNCT
cana-5491	90	16	(	(	PUNCT
cana-5491	90	17	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	90	18	(	(	PUNCT
cana-5491	90	19	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	90	20	(	(	PUNCT
cana-5491	90	21	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	90	22	(	(	PUNCT
cana-5491	90	23	𝑟	𝑟	NOUN
cana-5491	90	24	+	+	SYM
cana-5491	90	25	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	90	26	𝐴𝑟(𝑤𝛼)𝑟	𝐴𝑟(𝑤𝛼)𝑟	NOUN
cana-5491	90	27	=	=	SYM
cana-5491	90	28	0	0	NUM
cana-5491	90	29	∞	∞	NUM
cana-5491	90	30	𝑟=0	𝑟=0	PUNCT
cana-5491	90	31	∑	∑	PUNCT
cana-5491	90	32	[	[	PUNCT
cana-5491	90	33	(	(	PUNCT
cana-5491	90	34	𝜃1)(𝑟+2)𝛽	𝜃1)(𝑟+2)𝛽	NOUN
cana-5491	90	35	(	(	PUNCT
cana-5491	90	36	𝜃3)(𝑟+2)𝛿	𝜃3)(𝑟+2)𝛿	PROPN
cana-5491	90	37	(	(	PUNCT
cana-5491	90	38	𝜃2)(𝑟+2)𝛾	𝜃2)(𝑟+2)𝛾	NUM
cana-5491	90	39	(	(	PUNCT
cana-5491	90	40	𝑟	𝑟	NOUN
cana-5491	90	41	+	+	CCONJ
cana-5491	90	42	2	2	NUM
cana-5491	90	43	+	+	CCONJ
cana-5491	90	44	𝑝)𝑠	𝑝)𝑠	X
cana-5491	91	1	𝐴2𝛤((𝑟	𝐴2𝛤((𝑟	ADV
cana-5491	91	2	+	+	CCONJ
cana-5491	91	3	2)𝛼	2)𝛼	NUM
cana-5491	91	4	+	+	NOUN
cana-5491	91	5	1	1	X
cana-5491	91	6	)	)	PUNCT
cana-5491	91	7	𝛤(𝛼𝑟	𝛤(𝛼𝑟	NOUN
cana-5491	91	8	+	+	CCONJ
cana-5491	91	9	1	1	X
cana-5491	91	10	)	)	PUNCT
cana-5491	91	11	𝐴2	𝐴2	NOUN
cana-5491	91	12	∞	∞	NUM
cana-5491	91	13	𝑟=0	𝑟=0	PUNCT
cana-5491	91	14	−	−	PROPN
cana-5491	91	15	𝐵	𝐵	PROPN
cana-5491	91	16	(	(	PUNCT
cana-5491	91	17	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	91	18	(	(	PUNCT
cana-5491	91	19	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	91	20	(	(	PUNCT
cana-5491	91	21	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	91	22	(	(	PUNCT
cana-5491	91	23	𝑟	𝑟	NOUN
cana-5491	91	24	+	+	SYM
cana-5491	91	25	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	91	26	]	]	X
cana-5491	91	27	𝐴𝑟(𝑤𝛼)𝑟	𝐴𝑟(𝑤𝛼)𝑟	NOUN
cana-5491	91	28	=	=	SYM
cana-5491	91	29	0	0	NUM
cana-5491	91	30	equating	equate	VERB
cana-5491	91	31	to	to	ADP
cana-5491	91	32	zero	zero	NUM
cana-5491	91	33	the	the	DET
cana-5491	91	34	coefficient	coefficient	NOUN
cana-5491	91	35	of	of	ADP
cana-5491	91	36	𝑤𝑟𝛼	𝑤𝑟𝛼	PROPN
cana-5491	91	37	in	in	ADP
cana-5491	91	38	the	the	DET
cana-5491	91	39	above	above	ADJ
cana-5491	91	40	equation	equation	NOUN
cana-5491	91	41	(	(	PUNCT
cana-5491	91	42	𝜃1)(𝑟+2)𝛽	𝜃1)(𝑟+2)𝛽	NOUN
cana-5491	91	43	(	(	PUNCT
cana-5491	91	44	𝜃3)(𝑟+2)𝛿	𝜃3)(𝑟+2)𝛿	PROPN
cana-5491	91	45	(	(	PUNCT
cana-5491	91	46	𝜃2)(𝑟+2)𝛾	𝜃2)(𝑟+2)𝛾	NUM
cana-5491	91	47	(	(	PUNCT
cana-5491	91	48	𝑟	𝑟	NOUN
cana-5491	91	49	+	+	CCONJ
cana-5491	91	50	2	2	NUM
cana-5491	91	51	+	+	CCONJ
cana-5491	91	52	𝑝)𝑠	𝑝)𝑠	NOUN
cana-5491	91	53	𝛤((𝑟	𝛤((𝑟	NOUN
cana-5491	91	54	+	+	CCONJ
cana-5491	91	55	2)𝛼	2)𝛼	NUM
cana-5491	92	1	+	+	NOUN
cana-5491	92	2	1	1	X
cana-5491	92	3	)	)	PUNCT
cana-5491	92	4	𝛤(𝛼𝑟	𝛤(𝛼𝑟	NOUN
cana-5491	93	1	+	+	CCONJ
cana-5491	93	2	1	1	X
cana-5491	93	3	)	)	PUNCT
cana-5491	93	4	𝐴2	𝐴2	NOUN
cana-5491	93	5	−	−	PROPN
cana-5491	93	6	𝐵	𝐵	PROPN
cana-5491	93	7	(	(	PUNCT
cana-5491	93	8	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	93	9	(	(	PUNCT
cana-5491	93	10	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	93	11	(	(	PUNCT
cana-5491	93	12	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	93	13	(	(	PUNCT
cana-5491	93	14	𝑟	𝑟	NOUN
cana-5491	93	15	+	+	SYM
cana-5491	93	16	𝑝)𝑠	𝑝)𝑠	SYM
cana-5491	93	17	=	=	SYM
cana-5491	93	18	0	0	NUM
cana-5491	94	1	(	(	PUNCT
cana-5491	94	2	𝜃1)(𝑟+2)𝛽	𝜃1)(𝑟+2)𝛽	NOUN
cana-5491	94	3	(	(	PUNCT
cana-5491	94	4	𝜃3)(𝑟+2)𝛿	𝜃3)(𝑟+2)𝛿	PROPN
cana-5491	94	5	(	(	PUNCT
cana-5491	94	6	𝜃2)(𝑟+2)𝛾	𝜃2)(𝑟+2)𝛾	NUM
cana-5491	94	7	(	(	PUNCT
cana-5491	94	8	𝑟	𝑟	NOUN
cana-5491	94	9	+	+	CCONJ
cana-5491	94	10	2	2	NUM
cana-5491	94	11	+	+	CCONJ
cana-5491	94	12	𝑝)𝑠	𝑝)𝑠	NOUN
cana-5491	94	13	𝛤((𝑟	𝛤((𝑟	NOUN
cana-5491	94	14	+	+	CCONJ
cana-5491	94	15	2)𝛼	2)𝛼	NUM
cana-5491	94	16	+	+	NOUN
cana-5491	94	17	1	1	X
cana-5491	94	18	)	)	PUNCT
cana-5491	94	19	𝛤(𝛼𝑟	𝛤(𝛼𝑟	NOUN
cana-5491	95	1	+	+	CCONJ
cana-5491	95	2	1	1	X
cana-5491	95	3	)	)	PUNCT
cana-5491	95	4	𝐴2	𝐴2	NOUN
cana-5491	95	5	=	=	SYM
cana-5491	95	6	𝐵	𝐵	PROPN
cana-5491	95	7	(	(	PUNCT
cana-5491	95	8	𝜃1)𝑟𝛽	𝜃1)𝑟𝛽	PROPN
cana-5491	95	9	(	(	PUNCT
cana-5491	95	10	𝜃3)𝑟𝛿	𝜃3)𝑟𝛿	NOUN
cana-5491	95	11	(	(	PUNCT
cana-5491	95	12	𝜃2)𝑟𝛾	𝜃2)𝑟𝛾	NOUN
cana-5491	95	13	(	(	PUNCT
cana-5491	95	14	𝑟	𝑟	NOUN
cana-5491	95	15	+	+	SYM
cana-5491	95	16	𝑝)𝑠	𝑝)𝑠	ADV
cana-5491	95	17	𝑎𝑡	𝑎𝑡	ADP
cana-5491	95	18	𝑟	𝑟	NOUN
cana-5491	95	19	=	=	SYM
cana-5491	95	20	0	0	NUM
cana-5491	95	21	,	,	PUNCT
cana-5491	95	22	(	(	PUNCT
cana-5491	95	23	𝜃1)2𝛽	𝜃1)2𝛽	X
cana-5491	95	24	(	(	PUNCT
cana-5491	95	25	𝜃3)2𝛿	𝜃3)2𝛿	NOUN
cana-5491	95	26	(	(	PUNCT
cana-5491	95	27	𝜃2)2𝛾	𝜃2)2𝛾	NOUN
cana-5491	95	28	(	(	PUNCT
cana-5491	95	29	2	2	NUM
cana-5491	95	30	+	+	CCONJ
cana-5491	95	31	𝑝)𝑠	𝑝)𝑠	X
cana-5491	95	32	𝛤(2𝛼	𝛤(2𝛼	X
cana-5491	95	33	+	+	CCONJ
cana-5491	95	34	1	1	NUM
cana-5491	95	35	)	)	PUNCT
cana-5491	95	36	𝛤(1	𝛤(1	NUM
cana-5491	95	37	)	)	PUNCT
cana-5491	95	38	𝐴2	𝐴2	NOUN
cana-5491	95	39	=	=	SYM
cana-5491	95	40	𝐵	𝐵	PROPN
cana-5491	95	41	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	95	42	(	(	PUNCT
cana-5491	95	43	𝜃1)2𝛽	𝜃1)2𝛽	PROPN
cana-5491	95	44	(	(	PUNCT
cana-5491	95	45	𝜃3)2𝛿	𝜃3)2𝛿	NOUN
cana-5491	95	46	(	(	PUNCT
cana-5491	95	47	𝜃2)2𝛾	𝜃2)2𝛾	NOUN
cana-5491	95	48	(	(	PUNCT
cana-5491	95	49	2	2	NUM
cana-5491	95	50	+	+	NUM
cana-5491	95	51	𝑝)𝑠	𝑝)𝑠	X
cana-5491	95	52	𝐴2	𝐴2	NOUN
cana-5491	95	53	=	=	SYM
cana-5491	95	54	𝐵	𝐵	PROPN
cana-5491	95	55	𝑝𝑠𝛤(2𝛼	𝑝𝑠𝛤(2𝛼	NOUN
cana-5491	96	1	+	+	CCONJ
cana-5491	96	2	1	1	X
cana-5491	96	3	)	)	PUNCT
cana-5491	96	4	𝑎𝑡	𝑎𝑡	ADP
cana-5491	96	5	𝑟	𝑟	NOUN
cana-5491	96	6	=	=	SYM
cana-5491	96	7	1	1	NUM
cana-5491	96	8	,	,	PUNCT
cana-5491	96	9	(	(	PUNCT
cana-5491	96	10	𝜃1)3𝛽	𝜃1)3𝛽	ADP
cana-5491	96	11	(	(	PUNCT
cana-5491	96	12	𝜃3)3𝛿	𝜃3)3𝛿	NOUN
cana-5491	96	13	(	(	PUNCT
cana-5491	96	14	𝜃2)3𝛾	𝜃2)3𝛾	PROPN
cana-5491	96	15	(	(	PUNCT
cana-5491	96	16	3	3	NUM
cana-5491	96	17	+	+	CCONJ
cana-5491	96	18	𝑝)𝑠	𝑝)𝑠	X
cana-5491	96	19	𝐴3	𝐴3	PROPN
cana-5491	96	20	=	=	SYM
cana-5491	96	21	𝐴𝐵	𝐴𝐵	PROPN
cana-5491	96	22	(	(	PUNCT
cana-5491	96	23	𝜃1)𝛽(𝜃2)𝛾	𝜃1)𝛽(𝜃2)𝛾	NOUN
cana-5491	96	24	(	(	PUNCT
cana-5491	96	25	𝜃3)𝛿(𝑝	𝜃3)𝛿(𝑝	NOUN
cana-5491	96	26	+	+	NOUN
cana-5491	96	27	1)𝑠	1)𝑠	NUM
cana-5491	96	28	𝑎𝑡	𝑎𝑡	ADP
cana-5491	96	29	𝑟	𝑟	NOUN
cana-5491	96	30	=	=	SYM
cana-5491	96	31	2	2	NUM
cana-5491	96	32	,	,	PUNCT
cana-5491	96	33	(	(	PUNCT
cana-5491	96	34	𝜃1)4𝛽	𝜃1)4𝛽	NOUN
cana-5491	96	35	(	(	PUNCT
cana-5491	96	36	𝜃3)4𝛿	𝜃3)4𝛿	PROPN
cana-5491	96	37	(	(	PUNCT
cana-5491	96	38	𝜃2)4𝛾	𝜃2)4𝛾	X
cana-5491	96	39	(	(	PUNCT
cana-5491	96	40	4	4	NUM
cana-5491	96	41	+	+	CCONJ
cana-5491	96	42	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	96	43	𝛤(4𝛼	𝛤(4𝛼	NOUN
cana-5491	96	44	+	+	CCONJ
cana-5491	96	45	1	1	X
cana-5491	96	46	)	)	PUNCT
cana-5491	96	47	𝛤(2𝛼	𝛤(2𝛼	PROPN
cana-5491	96	48	)	)	PUNCT
cana-5491	97	1	𝐴2	𝐴2	PROPN
cana-5491	97	2	=	=	SYM
cana-5491	97	3	𝐵	𝐵	PROPN
cana-5491	97	4	(	(	PUNCT
cana-5491	97	5	𝜃1)2𝛽(𝜃2)2𝛾	𝜃1)2𝛽(𝜃2)2𝛾	X
cana-5491	97	6	(	(	PUNCT
cana-5491	97	7	𝜃3)2𝛿(𝑝	𝜃3)2𝛿(𝑝	PROPN
cana-5491	97	8	+	+	SYM
cana-5491	97	9	2)𝑠	2)𝑠	NOUN
cana-5491	97	10	multiplying	multiply	VERB
cana-5491	97	11	above	above	ADP
cana-5491	97	12	equation	equation	NOUN
cana-5491	97	13	by	by	ADP
cana-5491	97	14	a2	a2	PROPN
cana-5491	97	15	(	(	PUNCT
cana-5491	97	16	𝜃1)4𝛽	𝜃1)4𝛽	PROPN
cana-5491	97	17	(	(	PUNCT
cana-5491	97	18	𝜃3)4𝛿	𝜃3)4𝛿	PROPN
cana-5491	97	19	(	(	PUNCT
cana-5491	97	20	𝜃2)4𝛾	𝜃2)4𝛾	X
cana-5491	97	21	(	(	PUNCT
cana-5491	97	22	4	4	NUM
cana-5491	97	23	+	+	CCONJ
cana-5491	97	24	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	97	25	𝛤(4𝛼	𝛤(4𝛼	NOUN
cana-5491	97	26	+	+	CCONJ
cana-5491	97	27	1	1	X
cana-5491	97	28	)	)	PUNCT
cana-5491	97	29	𝛤(2𝛼	𝛤(2𝛼	PROPN
cana-5491	97	30	)	)	PUNCT
cana-5491	97	31	𝐴4	𝐴4	NOUN
cana-5491	97	32	=	=	SYM
cana-5491	97	33	𝐵	𝐵	PROPN
cana-5491	97	34	(	(	PUNCT
cana-5491	97	35	𝜃1)2𝛽(𝜃2)2𝛾	𝜃1)2𝛽(𝜃2)2𝛾	X
cana-5491	97	36	(	(	PUNCT
cana-5491	97	37	𝜃3)2𝛿(𝑝	𝜃3)2𝛿(𝑝	PROPN
cana-5491	97	38	+	+	SYM
cana-5491	97	39	2)𝑠	2)𝑠	NOUN
cana-5491	97	40	𝐴2	𝐴2	NOUN
cana-5491	97	41	communications	communication	NOUN
cana-5491	97	42	on	on	ADP
cana-5491	97	43	applied	apply	VERB
cana-5491	97	44	nonlinear	nonlinear	ADJ
cana-5491	97	45	analysis	analysis	NOUN
cana-5491	97	46	issn	issn	NOUN
cana-5491	97	47	:	:	PUNCT
cana-5491	97	48	1074	1074	NUM
cana-5491	97	49	-	-	PUNCT
cana-5491	97	50	133x	133x	NUM
cana-5491	97	51	vol	vol	VERB
cana-5491	97	52	32	32	NUM
cana-5491	97	53	no	no	NOUN
cana-5491	97	54	.	.	PUNCT
cana-5491	98	1	10s	10	NOUN
cana-5491	98	2	(	(	PUNCT
cana-5491	98	3	2025	2025	NUM
cana-5491	98	4	)	)	PUNCT
cana-5491	98	5	2439	2439	NUM
cana-5491	98	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	98	7	(	(	PUNCT
cana-5491	98	8	𝜃1)4𝛽	𝜃1)4𝛽	NOUN
cana-5491	98	9	(	(	PUNCT
cana-5491	98	10	𝜃3)4𝛿	𝜃3)4𝛿	PROPN
cana-5491	98	11	(	(	PUNCT
cana-5491	98	12	𝜃2)4𝛾	𝜃2)4𝛾	X
cana-5491	98	13	(	(	PUNCT
cana-5491	98	14	4	4	NUM
cana-5491	98	15	+	+	CCONJ
cana-5491	98	16	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	98	17	𝛤(4𝛼	𝛤(4𝛼	NOUN
cana-5491	98	18	+	+	CCONJ
cana-5491	98	19	1	1	X
cana-5491	98	20	)	)	PUNCT
cana-5491	98	21	𝛤(2𝛼	𝛤(2𝛼	PROPN
cana-5491	98	22	)	)	PUNCT
cana-5491	98	23	𝐴4	𝐴4	NOUN
cana-5491	98	24	=	=	PUNCT
cana-5491	98	25	𝐵2	𝐵2	PROPN
cana-5491	98	26	1	1	NUM
cana-5491	98	27	𝑝𝑠𝛤(4𝛼	𝑝𝑠𝛤(4𝛼	NOUN
cana-5491	98	28	+	+	CCONJ
cana-5491	98	29	1	1	X
cana-5491	98	30	)	)	PUNCT
cana-5491	98	31	now	now	ADV
cana-5491	98	32	put	put	VERB
cana-5491	98	33	these	these	DET
cana-5491	98	34	values	value	NOUN
cana-5491	98	35	in	in	ADP
cana-5491	98	36	equation	equation	NOUN
cana-5491	98	37	(	(	PUNCT
cana-5491	98	38	5	5	NUM
cana-5491	98	39	)	)	PUNCT
cana-5491	98	40	then	then	ADV
cana-5491	98	41	,	,	PUNCT
cana-5491	98	42	𝑓(𝑤	𝑓(𝑤	PROPN
cana-5491	98	43	)	)	PUNCT
cana-5491	98	44	=	=	SYM
cana-5491	99	1	1	1	NUM
cana-5491	99	2	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	99	3	+	+	NUM
cana-5491	99	4	(	(	PUNCT
cana-5491	99	5	𝜃1)𝛽(𝜃2)𝛾	𝜃1)𝛽(𝜃2)𝛾	NOUN
cana-5491	99	6	(	(	PUNCT
cana-5491	99	7	𝜃3)𝛿(𝑝	𝜃3)𝛿(𝑝	VERB
cana-5491	99	8	+	+	SYM
cana-5491	99	9	1)𝑠	1)𝑠	NUM
cana-5491	99	10	𝐴𝑤𝛼	𝐴𝑤𝛼	PROPN
cana-5491	99	11	+	+	CCONJ
cana-5491	99	12	𝐵	𝐵	NOUN
cana-5491	99	13	1	1	NUM
cana-5491	99	14	𝑝𝑠𝛤(2𝛼	𝑝𝑠𝛤(2𝛼	NOUN
cana-5491	99	15	+	+	CCONJ
cana-5491	99	16	1	1	NUM
cana-5491	99	17	)	)	PUNCT
cana-5491	99	18	𝑤2𝛼	𝑤2𝛼	NOUN
cana-5491	99	19	+	+	CCONJ
cana-5491	99	20	𝐵	𝐵	NOUN
cana-5491	99	21	(	(	PUNCT
cana-5491	99	22	𝜃1)𝛽(𝜃2)𝛾	𝜃1)𝛽(𝜃2)𝛾	NOUN
cana-5491	99	23	(	(	PUNCT
cana-5491	99	24	𝜃3)𝛿(𝑝	𝜃3)𝛿(𝑝	VERB
cana-5491	99	25	+	+	ADJ
cana-5491	99	26	1)𝑠	1)𝑠	NUM
cana-5491	99	27	𝐴𝑤3𝛼	𝐴𝑤3𝛼	NOUN
cana-5491	99	28	+	+	CCONJ
cana-5491	99	29	𝐵2	𝐵2	NOUN
cana-5491	99	30	1	1	NUM
cana-5491	99	31	𝑝𝑠𝛤(4𝛼	𝑝𝑠𝛤(4𝛼	NOUN
cana-5491	99	32	+	+	CCONJ
cana-5491	99	33	1	1	NUM
cana-5491	99	34	)	)	PUNCT
cana-5491	99	35	𝑤4𝛼	𝑤4𝛼	PUNCT
cana-5491	99	36	+	+	SYM
cana-5491	99	37	⋯	⋯	PROPN
cana-5491	99	38	(	(	PUNCT
cana-5491	99	39	12	12	NUM
cana-5491	99	40	)	)	PUNCT
cana-5491	99	41	example	example	NOUN
cana-5491	99	42	3	3	X
cana-5491	99	43	.	.	PUNCT
cana-5491	99	44	consider	consider	VERB
cana-5491	99	45	fractional	fractional	ADJ
cana-5491	99	46	differential	differential	ADJ
cana-5491	99	47	equation	equation	NOUN
cana-5491	99	48	d2αf(w	d2αf(w	PROPN
cana-5491	99	49	)	)	PUNCT
cana-5491	100	1	+	+	CCONJ
cana-5491	100	2	dαf(w	dαf(w	PROPN
cana-5491	100	3	)	)	PUNCT
cana-5491	100	4	−	−	PROPN
cana-5491	100	5	3f(w	3f(w	PROPN
cana-5491	100	6	)	)	PUNCT
cana-5491	100	7	=	=	SYM
cana-5491	101	1	0	0	X
cana-5491	101	2	.	.	PUNCT
cana-5491	102	1	(	(	PUNCT
cana-5491	102	2	13	13	NUM
cana-5491	102	3	)	)	PUNCT
cana-5491	102	4	then	then	ADV
cana-5491	102	5	by	by	ADP
cana-5491	102	6	equation	equation	NOUN
cana-5491	102	7	(	(	PUNCT
cana-5491	102	8	4	4	NUM
cana-5491	102	9	)	)	PUNCT
cana-5491	102	10	,	,	PUNCT
cana-5491	102	11	theorem	theorem	ADJ
cana-5491	102	12	(	(	PUNCT
cana-5491	102	13	1	1	NUM
cana-5491	102	14	)	)	PUNCT
cana-5491	102	15	and	and	CCONJ
cana-5491	102	16	theorem	theorem	VERB
cana-5491	102	17	(	(	PUNCT
cana-5491	102	18	2	2	NUM
cana-5491	102	19	)	)	PUNCT
cana-5491	102	20	,	,	PUNCT
cana-5491	102	21	∑	∑	PROPN
cana-5491	102	22	(	(	PUNCT
cana-5491	102	23	𝜃1)𝑘𝛽	𝜃1)𝑘𝛽	PROPN
cana-5491	102	24	(	(	PUNCT
cana-5491	102	25	𝜃3)𝑘𝛿	𝜃3)𝑘𝛿	PROPN
cana-5491	102	26	(	(	PUNCT
cana-5491	102	27	𝜃2)𝑘𝛾	𝜃2)𝑘𝛾	PROPN
cana-5491	102	28	(	(	PUNCT
cana-5491	102	29	𝑘	𝑘	X
cana-5491	102	30	+	+	CCONJ
cana-5491	102	31	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	102	32	𝐴𝑘𝛤(𝑘𝛼	𝐴𝑘𝛤(𝑘𝛼	NOUN
cana-5491	102	33	+	+	NOUN
cana-5491	102	34	1	1	NUM
cana-5491	102	35	)	)	PUNCT
cana-5491	102	36	𝛤(𝛼(𝑘	𝛤(𝛼(𝑘	PUNCT
cana-5491	103	1	−	−	NOUN
cana-5491	103	2	2	2	NUM
cana-5491	103	3	)	)	PUNCT
cana-5491	103	4	+	+	CCONJ
cana-5491	103	5	1	1	X
cana-5491	103	6	)	)	PUNCT
cana-5491	103	7	𝑤𝛼(𝑘−2	𝑤𝛼(𝑘−2	NUM
cana-5491	103	8	)	)	PUNCT
cana-5491	103	9	∞	∞	NUM
cana-5491	103	10	𝑘=2	𝑘=2	PROPN
cana-5491	104	1	+	+	CCONJ
cana-5491	104	2	∑	∑	PUNCT
cana-5491	104	3	(	(	PUNCT
cana-5491	104	4	𝜃1)𝑘𝛽	𝜃1)𝑘𝛽	PROPN
cana-5491	104	5	(	(	PUNCT
cana-5491	104	6	𝜃3)𝑘𝛿	𝜃3)𝑘𝛿	PROPN
cana-5491	104	7	(	(	PUNCT
cana-5491	104	8	𝜃2)𝑘𝛾	𝜃2)𝑘𝛾	PROPN
cana-5491	104	9	(	(	PUNCT
cana-5491	104	10	𝑘	𝑘	X
cana-5491	104	11	+	+	CCONJ
cana-5491	104	12	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	104	13	𝐴𝑘𝛤(𝑘𝛼	𝐴𝑘𝛤(𝑘𝛼	NOUN
cana-5491	104	14	+	+	NOUN
cana-5491	104	15	1	1	NUM
cana-5491	104	16	)	)	PUNCT
cana-5491	104	17	𝛤(𝛼(𝑘	𝛤(𝛼(𝑘	PUNCT
cana-5491	105	1	−	−	NOUN
cana-5491	105	2	1	1	X
cana-5491	105	3	)	)	PUNCT
cana-5491	105	4	+	+	CCONJ
cana-5491	105	5	1	1	X
cana-5491	105	6	)	)	PUNCT
cana-5491	105	7	𝑤𝛼(𝑘−1	𝑤𝛼(𝑘−1	PROPN
cana-5491	105	8	)	)	PUNCT
cana-5491	105	9	∞	∞	PROPN
cana-5491	106	1	𝑘=1	𝑘=1	NOUN
cana-5491	106	2	−	−	NOUN
cana-5491	106	3	3	3	NUM
cana-5491	106	4	∑	∑	PUNCT
cana-5491	106	5	(	(	PUNCT
cana-5491	106	6	𝜃1)𝑘𝛽	𝜃1)𝑘𝛽	PROPN
cana-5491	106	7	(	(	PUNCT
cana-5491	106	8	𝜃3)𝑘𝛿	𝜃3)𝑘𝛿	PROPN
cana-5491	106	9	(	(	PUNCT
cana-5491	106	10	𝜃2)𝑘𝛾	𝜃2)𝑘𝛾	PROPN
cana-5491	106	11	(	(	PUNCT
cana-5491	106	12	𝑘	𝑘	PROPN
cana-5491	106	13	+	+	CCONJ
cana-5491	106	14	𝑝)𝑠	𝑝)𝑠	X
cana-5491	106	15	𝐴𝑘(𝑤𝛼)𝑘	𝐴𝑘(𝑤𝛼)𝑘	ADJ
cana-5491	106	16	=	=	SYM
cana-5491	106	17	0	0	NUM
cana-5491	106	18	∞	∞	NUM
cana-5491	107	1	𝑘=0	𝑘=0	VERB
cana-5491	107	2	replacing	replace	VERB
cana-5491	107	3	k	k	X
cana-5491	107	4	by	by	ADP
cana-5491	107	5	k+2	k+2	PROPN
cana-5491	107	6	in	in	ADP
cana-5491	107	7	the	the	DET
cana-5491	107	8	first	first	ADJ
cana-5491	107	9	summation	summation	NOUN
cana-5491	107	10	and	and	CCONJ
cana-5491	107	11	k	k	X
cana-5491	107	12	by	by	ADP
cana-5491	107	13	k+1	k+1	X
cana-5491	107	14	in	in	ADP
cana-5491	107	15	second	second	ADJ
cana-5491	107	16	summation	summation	NOUN
cana-5491	107	17	respectively	respectively	ADV
cana-5491	107	18	,	,	PUNCT
cana-5491	107	19	∑	∑	ADP
cana-5491	107	20	(	(	PUNCT
cana-5491	107	21	𝜃1)(𝑘+2)𝛽	𝜃1)(𝑘+2)𝛽	NOUN
cana-5491	107	22	(	(	PUNCT
cana-5491	107	23	𝜃3)(𝑘+2)𝛿	𝜃3)(𝑘+2)𝛿	NOUN
cana-5491	107	24	(	(	PUNCT
cana-5491	107	25	𝜃2)(𝑘+2)𝛾	𝜃2)(𝑘+2)𝛾	PROPN
cana-5491	107	26	(	(	PUNCT
cana-5491	107	27	(	(	PUNCT
cana-5491	107	28	𝑘	𝑘	X
cana-5491	107	29	+	+	NOUN
cana-5491	107	30	2	2	NUM
cana-5491	107	31	)	)	PUNCT
cana-5491	107	32	+	+	PUNCT
cana-5491	107	33	𝑝)𝑠	𝑝)𝑠	X
cana-5491	107	34	𝐴𝑘+2𝛤((𝑘	𝐴𝑘+2𝛤((𝑘	X
cana-5491	108	1	+	+	CCONJ
cana-5491	108	2	2)𝛼	2)𝛼	NUM
cana-5491	108	3	+	+	NOUN
cana-5491	108	4	1	1	NUM
cana-5491	108	5	)	)	PUNCT
cana-5491	108	6	𝛤(𝛼(𝑘	𝛤(𝛼(𝑘	X
cana-5491	108	7	)	)	PUNCT
cana-5491	109	1	+	+	CCONJ
cana-5491	109	2	1	1	X
cana-5491	109	3	)	)	PUNCT
cana-5491	109	4	𝑤𝛼𝑘	𝑤𝛼𝑘	NOUN
cana-5491	109	5	∞	∞	NUM
cana-5491	109	6	𝑘=0	𝑘=0	PROPN
cana-5491	110	1	+	+	CCONJ
cana-5491	110	2	∑	∑	PROPN
cana-5491	110	3	(	(	PUNCT
cana-5491	110	4	𝜃1)(𝑘+1)𝛽	𝜃1)(𝑘+1)𝛽	PROPN
cana-5491	110	5	(	(	PUNCT
cana-5491	110	6	𝜃3)(𝑘+1)𝛿	𝜃3)(𝑘+1)𝛿	PROPN
cana-5491	110	7	(	(	PUNCT
cana-5491	110	8	𝜃2)(𝑘+1)𝛾	𝜃2)(𝑘+1)𝛾	NOUN
cana-5491	110	9	(	(	PUNCT
cana-5491	110	10	𝑘	𝑘	PROPN
cana-5491	110	11	+	+	NOUN
cana-5491	110	12	1	1	NUM
cana-5491	110	13	+	+	CCONJ
cana-5491	110	14	𝑝)𝑠	𝑝)𝑠	X
cana-5491	110	15	𝐴𝑘+1𝛤((𝑘	𝐴𝑘+1𝛤((𝑘	NOUN
cana-5491	110	16	+	+	CCONJ
cana-5491	110	17	1)𝛼	1)𝛼	NUM
cana-5491	110	18	+	+	CCONJ
cana-5491	110	19	1	1	X
cana-5491	110	20	)	)	PUNCT
cana-5491	110	21	𝛤(𝛼𝑘	𝛤(𝛼𝑘	NOUN
cana-5491	110	22	+	+	NOUN
cana-5491	110	23	1	1	NUM
cana-5491	110	24	)	)	PUNCT
cana-5491	110	25	𝑤𝛼𝑘	𝑤𝛼𝑘	NUM
cana-5491	110	26	∞	∞	NUM
cana-5491	110	27	𝑘=0	𝑘=0	ADP
cana-5491	110	28	−	−	PROPN
cana-5491	110	29	3	3	NUM
cana-5491	110	30	∑	∑	PUNCT
cana-5491	110	31	(	(	PUNCT
cana-5491	110	32	𝜃1)𝑘𝛽	𝜃1)𝑘𝛽	PROPN
cana-5491	110	33	(	(	PUNCT
cana-5491	110	34	𝜃3)𝑘𝛿	𝜃3)𝑘𝛿	PROPN
cana-5491	110	35	(	(	PUNCT
cana-5491	110	36	𝜃2)𝑘𝛾	𝜃2)𝑘𝛾	PROPN
cana-5491	110	37	(	(	PUNCT
cana-5491	110	38	𝑘	𝑘	PROPN
cana-5491	110	39	+	+	CCONJ
cana-5491	110	40	𝑝)𝑠	𝑝)𝑠	X
cana-5491	111	1	𝐴𝑘(𝑤𝛼)𝑘	𝐴𝑘(𝑤𝛼)𝑘	ADJ
cana-5491	111	2	=	=	SYM
cana-5491	111	3	0	0	NUM
cana-5491	111	4	∞	∞	NUM
cana-5491	111	5	𝑘=0	𝑘=0	ADP
cana-5491	111	6	∑	∑	PROPN
cana-5491	111	7	[	[	PUNCT
cana-5491	111	8	(	(	PUNCT
cana-5491	111	9	𝜃1)(𝑘+2)𝛽	𝜃1)(𝑘+2)𝛽	NOUN
cana-5491	111	10	(	(	PUNCT
cana-5491	111	11	𝜃3)(𝑘+2)𝛿	𝜃3)(𝑘+2)𝛿	NOUN
cana-5491	111	12	(	(	PUNCT
cana-5491	111	13	𝜃2)(𝑘+2)𝛾	𝜃2)(𝑘+2)𝛾	PROPN
cana-5491	111	14	(	(	PUNCT
cana-5491	111	15	(	(	PUNCT
cana-5491	111	16	𝑘	𝑘	X
cana-5491	111	17	+	+	NOUN
cana-5491	111	18	2	2	NUM
cana-5491	111	19	)	)	PUNCT
cana-5491	111	20	+	+	NUM
cana-5491	111	21	𝑝)𝑠	𝑝)𝑠	X
cana-5491	111	22	𝐴2𝛤((𝑘	𝐴2𝛤((𝑘	NOUN
cana-5491	112	1	+	+	CCONJ
cana-5491	112	2	2)𝛼	2)𝛼	NUM
cana-5491	112	3	+	+	NOUN
cana-5491	112	4	1	1	NUM
cana-5491	112	5	)	)	PUNCT
cana-5491	112	6	𝛤(𝛼(𝑘	𝛤(𝛼(𝑘	X
cana-5491	112	7	)	)	PUNCT
cana-5491	113	1	+	+	CCONJ
cana-5491	113	2	1	1	X
cana-5491	113	3	)	)	PUNCT
cana-5491	113	4	∞	∞	NUM
cana-5491	113	5	𝑘=0	𝑘=0	PROPN
cana-5491	114	1	+	+	CCONJ
cana-5491	114	2	(	(	PUNCT
cana-5491	114	3	𝜃1)(𝑘+1)𝛽	𝜃1)(𝑘+1)𝛽	NOUN
cana-5491	114	4	(	(	PUNCT
cana-5491	114	5	𝜃3)(𝑘+1)𝛿	𝜃3)(𝑘+1)𝛿	PROPN
cana-5491	114	6	(	(	PUNCT
cana-5491	114	7	𝜃2)(𝑘+1)𝛾	𝜃2)(𝑘+1)𝛾	NOUN
cana-5491	114	8	(	(	PUNCT
cana-5491	114	9	𝑘	𝑘	PROPN
cana-5491	114	10	+	+	ADJ
cana-5491	114	11	1	1	NUM
cana-5491	114	12	+	+	CCONJ
cana-5491	114	13	𝑝)𝑠	𝑝)𝑠	X
cana-5491	114	14	𝐴1𝛤((𝑘	𝐴1𝛤((𝑘	PROPN
cana-5491	115	1	+	+	CCONJ
cana-5491	115	2	1)𝛼	1)𝛼	NUM
cana-5491	115	3	+	+	CCONJ
cana-5491	115	4	1	1	X
cana-5491	115	5	)	)	PUNCT
cana-5491	115	6	𝛤(𝛼𝑘	𝛤(𝛼𝑘	NOUN
cana-5491	115	7	+	+	NOUN
cana-5491	115	8	1	1	NUM
cana-5491	115	9	)	)	PUNCT
cana-5491	115	10	−	−	NOUN
cana-5491	115	11	3	3	NUM
cana-5491	115	12	(	(	PUNCT
cana-5491	115	13	𝜃1)𝑘𝛽	𝜃1)𝑘𝛽	PROPN
cana-5491	115	14	(	(	PUNCT
cana-5491	115	15	𝜃3)𝑘𝛿	𝜃3)𝑘𝛿	PROPN
cana-5491	115	16	(	(	PUNCT
cana-5491	115	17	𝜃2)𝑘𝛾	𝜃2)𝑘𝛾	PROPN
cana-5491	115	18	(	(	PUNCT
cana-5491	115	19	𝑘	𝑘	PROPN
cana-5491	115	20	+	+	NUM
cana-5491	115	21	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	115	22	]	]	X
cana-5491	115	23	𝐴𝑘(𝑤𝛼)𝑘	𝐴𝑘(𝑤𝛼)𝑘	X
cana-5491	115	24	=	=	SYM
cana-5491	115	25	0	0	NUM
cana-5491	115	26	now	now	ADV
cana-5491	115	27	equating	equate	VERB
cana-5491	115	28	to	to	ADP
cana-5491	115	29	zero	zero	NUM
cana-5491	115	30	the	the	DET
cana-5491	115	31	coefficient	coefficient	NOUN
cana-5491	115	32	of	of	ADP
cana-5491	115	33	𝑤𝑘𝛼	𝑤𝑘𝛼	INTJ
cana-5491	115	34	(	(	PUNCT
cana-5491	115	35	𝜃1)(𝑘+2)𝛽	𝜃1)(𝑘+2)𝛽	NOUN
cana-5491	115	36	(	(	PUNCT
cana-5491	115	37	𝜃3)(𝑘+2)𝛿	𝜃3)(𝑘+2)𝛿	NOUN
cana-5491	115	38	(	(	PUNCT
cana-5491	115	39	𝜃2)(𝑘+2)𝛾	𝜃2)(𝑘+2)𝛾	PROPN
cana-5491	115	40	(	(	PUNCT
cana-5491	115	41	(	(	PUNCT
cana-5491	115	42	𝑘	𝑘	X
cana-5491	115	43	+	+	NOUN
cana-5491	115	44	2	2	NUM
cana-5491	115	45	)	)	PUNCT
cana-5491	116	1	+	+	NUM
cana-5491	116	2	𝑝)𝑠	𝑝)𝑠	X
cana-5491	116	3	𝐴2𝛤((𝑘	𝐴2𝛤((𝑘	NOUN
cana-5491	117	1	+	+	CCONJ
cana-5491	117	2	2)𝛼	2)𝛼	NUM
cana-5491	117	3	+	+	NOUN
cana-5491	117	4	1	1	NUM
cana-5491	117	5	)	)	PUNCT
cana-5491	117	6	𝛤(𝛼(𝑘	𝛤(𝛼(𝑘	X
cana-5491	117	7	)	)	PUNCT
cana-5491	118	1	+	+	CCONJ
cana-5491	119	1	1	1	X
cana-5491	119	2	)	)	PUNCT
cana-5491	119	3	+	+	CCONJ
cana-5491	119	4	(	(	PUNCT
cana-5491	119	5	𝜃1)(𝑘+1)𝛽	𝜃1)(𝑘+1)𝛽	NOUN
cana-5491	119	6	(	(	PUNCT
cana-5491	119	7	𝜃3)(𝑘+1)𝛿	𝜃3)(𝑘+1)𝛿	PROPN
cana-5491	119	8	(	(	PUNCT
cana-5491	119	9	𝜃2)(𝑘+1)𝛾	𝜃2)(𝑘+1)𝛾	NOUN
cana-5491	119	10	(	(	PUNCT
cana-5491	119	11	𝑘	𝑘	PROPN
cana-5491	119	12	+	+	ADJ
cana-5491	119	13	1	1	NUM
cana-5491	119	14	+	+	CCONJ
cana-5491	119	15	𝑝)𝑠	𝑝)𝑠	X
cana-5491	119	16	𝐴1𝛤((𝑘	𝐴1𝛤((𝑘	PROPN
cana-5491	119	17	+	+	CCONJ
cana-5491	119	18	1)𝛼	1)𝛼	NUM
cana-5491	119	19	+	+	CCONJ
cana-5491	119	20	1	1	X
cana-5491	119	21	)	)	PUNCT
cana-5491	119	22	𝛤(𝛼𝑘	𝛤(𝛼𝑘	NOUN
cana-5491	119	23	+	+	NOUN
cana-5491	119	24	1	1	NUM
cana-5491	119	25	)	)	PUNCT
cana-5491	119	26	−	−	NOUN
cana-5491	119	27	3	3	NUM
cana-5491	119	28	(	(	PUNCT
cana-5491	119	29	𝜃1)𝑘𝛽	𝜃1)𝑘𝛽	PROPN
cana-5491	119	30	(	(	PUNCT
cana-5491	119	31	𝜃3)𝑘𝛿	𝜃3)𝑘𝛿	PROPN
cana-5491	119	32	(	(	PUNCT
cana-5491	119	33	𝜃2)𝑘𝛾	𝜃2)𝑘𝛾	PROPN
cana-5491	119	34	(	(	PUNCT
cana-5491	119	35	𝑘	𝑘	X
cana-5491	119	36	+	+	X
cana-5491	119	37	𝑝)𝑠	𝑝)𝑠	NOUN
cana-5491	120	1	=	=	SYM
cana-5491	120	2	0	0	NUM
cana-5491	121	1	𝑎𝑡	𝑎𝑡	ADP
cana-5491	121	2	𝑘	𝑘	X
cana-5491	121	3	=	=	SYM
cana-5491	121	4	0	0	PROPN
cana-5491	121	5	,	,	PUNCT
cana-5491	121	6	(	(	PUNCT
cana-5491	121	7	𝜃1)2𝛽	𝜃1)2𝛽	X
cana-5491	121	8	(	(	PUNCT
cana-5491	121	9	𝜃3)2𝛿	𝜃3)2𝛿	NOUN
cana-5491	121	10	(	(	PUNCT
cana-5491	121	11	𝜃2)2𝛾	𝜃2)2𝛾	NOUN
cana-5491	121	12	(	(	PUNCT
cana-5491	121	13	2	2	NUM
cana-5491	121	14	+	+	NUM
cana-5491	121	15	𝑝)𝑠	𝑝)𝑠	PUNCT
cana-5491	121	16	𝐴2𝛤(2𝛼	𝐴2𝛤(2𝛼	PUNCT
cana-5491	121	17	+	+	ADJ
cana-5491	121	18	1	1	NUM
cana-5491	121	19	)	)	PUNCT
cana-5491	121	20	𝛤(1	𝛤(1	NUM
cana-5491	121	21	)	)	PUNCT
cana-5491	122	1	+	+	CCONJ
cana-5491	122	2	(	(	PUNCT
cana-5491	122	3	𝜃1)𝛽	𝜃1)𝛽	NOUN
cana-5491	122	4	(	(	PUNCT
cana-5491	122	5	𝜃3)𝛿	𝜃3)𝛿	NOUN
cana-5491	122	6	(	(	PUNCT
cana-5491	122	7	𝜃2)𝛾	𝜃2)𝛾	NOUN
cana-5491	122	8	(	(	PUNCT
cana-5491	122	9	1	1	NUM
cana-5491	122	10	+	+	NUM
cana-5491	122	11	𝑝)𝑠	𝑝)𝑠	X
cana-5491	122	12	𝐴1𝛤(𝛼	𝐴1𝛤(𝛼	PROPN
cana-5491	123	1	+	+	CCONJ
cana-5491	123	2	1	1	NUM
cana-5491	123	3	)	)	PUNCT
cana-5491	123	4	𝛤(1	𝛤(1	NUM
cana-5491	123	5	)	)	PUNCT
cana-5491	123	6	−	−	NOUN
cana-5491	123	7	3	3	NUM
cana-5491	123	8	1	1	NUM
cana-5491	123	9	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	123	10	=	=	SYM
cana-5491	123	11	0	0	NUM
cana-5491	123	12	communications	communication	NOUN
cana-5491	123	13	on	on	ADP
cana-5491	123	14	applied	apply	VERB
cana-5491	123	15	nonlinear	nonlinear	ADJ
cana-5491	123	16	analysis	analysis	NOUN
cana-5491	123	17	issn	issn	NOUN
cana-5491	123	18	:	:	PUNCT
cana-5491	123	19	1074	1074	NUM
cana-5491	123	20	-	-	PUNCT
cana-5491	123	21	133x	133x	NUM
cana-5491	123	22	vol	vol	VERB
cana-5491	123	23	32	32	NUM
cana-5491	123	24	no	no	NOUN
cana-5491	123	25	.	.	PUNCT
cana-5491	124	1	10s	10	NOUN
cana-5491	124	2	(	(	PUNCT
cana-5491	124	3	2025	2025	NUM
cana-5491	124	4	)	)	PUNCT
cana-5491	124	5	2440	2440	NUM
cana-5491	124	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	125	1	𝑎𝑡	𝑎𝑡	ADP
cana-5491	125	2	𝑘	𝑘	X
cana-5491	125	3	=	=	NOUN
cana-5491	125	4	1	1	NUM
cana-5491	125	5	,	,	PUNCT
cana-5491	125	6	(	(	PUNCT
cana-5491	125	7	𝜃1)3𝛽	𝜃1)3𝛽	ADP
cana-5491	125	8	(	(	PUNCT
cana-5491	125	9	𝜃3)3𝛿	𝜃3)3𝛿	NOUN
cana-5491	125	10	(	(	PUNCT
cana-5491	125	11	𝜃2)3𝛾	𝜃2)3𝛾	PROPN
cana-5491	125	12	(	(	PUNCT
cana-5491	125	13	3	3	NUM
cana-5491	125	14	+	+	CCONJ
cana-5491	125	15	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	125	16	𝛤(3𝛼	𝛤(3𝛼	NOUN
cana-5491	125	17	+	+	PROPN
cana-5491	125	18	1	1	NUM
cana-5491	125	19	)	)	PUNCT
cana-5491	125	20	𝛤(𝛼	𝛤(𝛼	PRON
cana-5491	125	21	+	+	ADJ
cana-5491	125	22	1	1	X
cana-5491	125	23	)	)	PUNCT
cana-5491	125	24	𝐴3	𝐴3	PROPN
cana-5491	125	25	+	+	CCONJ
cana-5491	125	26	(	(	PUNCT
cana-5491	125	27	𝜃1)2𝛽	𝜃1)2𝛽	X
cana-5491	125	28	(	(	PUNCT
cana-5491	125	29	𝜃3)2𝛿	𝜃3)2𝛿	NOUN
cana-5491	125	30	(	(	PUNCT
cana-5491	125	31	𝜃2)2𝛾	𝜃2)2𝛾	NOUN
cana-5491	125	32	(	(	PUNCT
cana-5491	125	33	2	2	NUM
cana-5491	125	34	+	+	CCONJ
cana-5491	125	35	𝑝)𝑠	𝑝)𝑠	X
cana-5491	125	36	𝛤(2𝛼	𝛤(2𝛼	X
cana-5491	126	1	+	+	PUNCT
cana-5491	126	2	1	1	X
cana-5491	126	3	)	)	PUNCT
cana-5491	126	4	𝛤(𝛼	𝛤(𝛼	PRON
cana-5491	126	5	+	+	NOUN
cana-5491	126	6	1	1	X
cana-5491	126	7	)	)	PUNCT
cana-5491	126	8	𝐴2	𝐴2	NOUN
cana-5491	126	9	−	−	PROPN
cana-5491	126	10	3	3	NUM
cana-5491	126	11	(	(	PUNCT
cana-5491	126	12	𝜃1)𝛽	𝜃1)𝛽	NOUN
cana-5491	126	13	(	(	PUNCT
cana-5491	126	14	𝜃3)𝛿	𝜃3)𝛿	NOUN
cana-5491	126	15	(	(	PUNCT
cana-5491	126	16	𝜃2)𝛾	𝜃2)𝛾	NOUN
cana-5491	126	17	(	(	PUNCT
cana-5491	126	18	1	1	NUM
cana-5491	126	19	+	+	CCONJ
cana-5491	126	20	𝑝)𝑠	𝑝)𝑠	X
cana-5491	126	21	𝐴	𝐴	NOUN
cana-5491	126	22	=	=	PUNCT
cana-5491	127	1	0	0	NUM
cana-5491	128	1	𝑎𝑡	𝑎𝑡	ADP
cana-5491	128	2	𝑘	𝑘	X
cana-5491	128	3	=	=	SYM
cana-5491	128	4	2	2	NUM
cana-5491	128	5	,	,	PUNCT
cana-5491	128	6	(	(	PUNCT
cana-5491	128	7	𝜃1)4𝛽	𝜃1)4𝛽	NOUN
cana-5491	128	8	(	(	PUNCT
cana-5491	128	9	𝜃3)4𝛿	𝜃3)4𝛿	PROPN
cana-5491	128	10	(	(	PUNCT
cana-5491	128	11	𝜃2)4𝛾	𝜃2)4𝛾	X
cana-5491	128	12	(	(	PUNCT
cana-5491	128	13	4	4	NUM
cana-5491	128	14	+	+	CCONJ
cana-5491	128	15	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	128	16	𝛤(4𝛼	𝛤(4𝛼	NOUN
cana-5491	128	17	+	+	CCONJ
cana-5491	128	18	1	1	X
cana-5491	128	19	)	)	PUNCT
cana-5491	128	20	𝛤(2𝛼	𝛤(2𝛼	VERB
cana-5491	129	1	+	+	PUNCT
cana-5491	129	2	1	1	X
cana-5491	129	3	)	)	PUNCT
cana-5491	129	4	𝐴2	𝐴2	NOUN
cana-5491	129	5	+	+	CCONJ
cana-5491	129	6	(	(	PUNCT
cana-5491	129	7	𝜃1)2𝛽	𝜃1)2𝛽	X
cana-5491	129	8	(	(	PUNCT
cana-5491	129	9	𝜃3)2𝛿	𝜃3)2𝛿	NOUN
cana-5491	129	10	(	(	PUNCT
cana-5491	129	11	𝜃2)2𝛾	𝜃2)2𝛾	NOUN
cana-5491	129	12	(	(	PUNCT
cana-5491	129	13	3	3	NUM
cana-5491	129	14	+	+	CCONJ
cana-5491	129	15	𝑝)𝑠	𝑝)𝑠	ADJ
cana-5491	129	16	𝛤(3𝛼	𝛤(3𝛼	NOUN
cana-5491	129	17	+	+	NOUN
cana-5491	129	18	1	1	NUM
cana-5491	129	19	)	)	PUNCT
cana-5491	129	20	𝛤(2𝛼	𝛤(2𝛼	VERB
cana-5491	130	1	+	+	CCONJ
cana-5491	130	2	1	1	X
cana-5491	130	3	)	)	PUNCT
cana-5491	130	4	𝐴1	𝐴1	PROPN
cana-5491	130	5	−	−	PROPN
cana-5491	130	6	3	3	NUM
cana-5491	130	7	(	(	PUNCT
cana-5491	130	8	𝜃1)𝛽	𝜃1)𝛽	NOUN
cana-5491	130	9	(	(	PUNCT
cana-5491	130	10	𝜃3)𝛿	𝜃3)𝛿	NOUN
cana-5491	130	11	(	(	PUNCT
cana-5491	130	12	𝜃2)𝛾	𝜃2)𝛾	NOUN
cana-5491	130	13	(	(	PUNCT
cana-5491	130	14	2	2	NUM
cana-5491	130	15	+	+	CCONJ
cana-5491	130	16	𝑝)𝑠	𝑝)𝑠	X
cana-5491	130	17	=	=	SYM
cana-5491	130	18	0	0	NUM
cana-5491	131	1	and	and	CCONJ
cana-5491	131	2	so	so	ADV
cana-5491	131	3	on	on	ADV
cana-5491	131	4	.	.	PUNCT
cana-5491	132	1	by	by	ADP
cana-5491	132	2	equation	equation	NOUN
cana-5491	132	3	(	(	PUNCT
cana-5491	132	4	5	5	NUM
cana-5491	132	5	)	)	PUNCT
cana-5491	132	6	,	,	PUNCT
cana-5491	132	7	we	we	PRON
cana-5491	132	8	get	get	VERB
cana-5491	132	9	following	follow	VERB
cana-5491	132	10	solution	solution	NOUN
cana-5491	132	11	𝑓(𝑤	𝑓(𝑤	PROPN
cana-5491	132	12	)	)	PUNCT
cana-5491	132	13	=	=	SYM
cana-5491	133	1	1	1	NUM
cana-5491	133	2	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	133	3	+	+	CCONJ
cana-5491	133	4	𝐵𝑤𝛼	𝐵𝑤𝛼	PROPN
cana-5491	133	5	+	+	CCONJ
cana-5491	133	6	1	1	NUM
cana-5491	133	7	𝛤(2𝛼	𝛤(2𝛼	ADJ
cana-5491	133	8	+	+	CCONJ
cana-5491	133	9	1	1	X
cana-5491	133	10	)	)	PUNCT
cana-5491	133	11	[	[	PUNCT
cana-5491	133	12	3	3	NUM
cana-5491	133	13	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	133	14	−	−	PROPN
cana-5491	133	15	𝐵𝛤(𝛼	𝐵𝛤(𝛼	PROPN
cana-5491	134	1	+	+	CCONJ
cana-5491	134	2	1	1	NUM
cana-5491	134	3	)	)	PUNCT
cana-5491	134	4	]	]	PUNCT
cana-5491	135	1	+	+	CCONJ
cana-5491	135	2	1	1	NUM
cana-5491	135	3	𝛤(3𝛼	𝛤(3𝛼	NOUN
cana-5491	135	4	+	+	NOUN
cana-5491	135	5	1	1	NUM
cana-5491	135	6	)	)	PUNCT
cana-5491	136	1	[	[	X
cana-5491	136	2	−	−	X
cana-5491	136	3	3	3	NUM
cana-5491	136	4	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	136	5	+	+	NOUN
cana-5491	136	6	4𝐵𝛤(𝛼	4𝐵𝛤(𝛼	NUM
cana-5491	137	1	+	+	CCONJ
cana-5491	137	2	1	1	NUM
cana-5491	137	3	)	)	PUNCT
cana-5491	137	4	]	]	PUNCT
cana-5491	138	1	+	+	CCONJ
cana-5491	138	2	1	1	NUM
cana-5491	138	3	𝛤(4𝛼	𝛤(4𝛼	NOUN
cana-5491	138	4	+	+	NOUN
cana-5491	138	5	1	1	X
cana-5491	138	6	)	)	PUNCT
cana-5491	138	7	[	[	PUNCT
cana-5491	138	8	12	12	NUM
cana-5491	138	9	𝑝𝑠	𝑝𝑠	CCONJ
cana-5491	138	10	−	−	PROPN
cana-5491	138	11	5𝐵𝛤(𝛼	5𝐵𝛤(𝛼	PROPN
cana-5491	139	1	+	+	CCONJ
cana-5491	139	2	1	1	NUM
cana-5491	139	3	)	)	PUNCT
cana-5491	139	4	]	]	PUNCT
cana-5491	140	1	+	+	CCONJ
cana-5491	140	2	⋯	⋯	ADP
cana-5491	140	3	4	4	NUM
cana-5491	140	4	.	.	PUNCT
cana-5491	140	5	conclusion	conclusion	NOUN
cana-5491	140	6	we	we	PRON
cana-5491	140	7	compute	compute	VERB
cana-5491	140	8	caputo	caputo	PROPN
cana-5491	140	9	derivative	derivative	ADJ
cana-5491	140	10	formula	formula	NOUN
cana-5491	140	11	of	of	ADP
cana-5491	140	12	the	the	DET
cana-5491	140	13	extended	extended	ADJ
cana-5491	140	14	hurwitz	hurwitz	PROPN
cana-5491	140	15	-	-	PUNCT
cana-5491	140	16	lerch	lerch	PROPN
cana-5491	140	17	zeta	zeta	PROPN
cana-5491	140	18	function	function	PROPN
cana-5491	140	19	.	.	PUNCT
cana-5491	141	1	moreover	moreover	ADV
cana-5491	141	2	,	,	PUNCT
cana-5491	141	3	we	we	PRON
cana-5491	141	4	obtained	obtain	VERB
cana-5491	141	5	the	the	DET
cana-5491	141	6	solution	solution	NOUN
cana-5491	141	7	of	of	ADP
cana-5491	141	8	fractional	fractional	ADJ
cana-5491	141	9	differential	differential	ADJ
cana-5491	141	10	equation	equation	NOUN
cana-5491	141	11	involving	involve	VERB
cana-5491	141	12	hurwitz	hurwitz	PROPN
cana-5491	141	13	-	-	PUNCT
cana-5491	141	14	lerch	lerch	PROPN
cana-5491	141	15	zeta	zeta	PROPN
cana-5491	141	16	function	function	PROPN
cana-5491	141	17	.	.	PUNCT
cana-5491	142	1	on	on	ADP
cana-5491	142	2	the	the	DET
cana-5491	142	3	basis	basis	NOUN
cana-5491	142	4	of	of	ADP
cana-5491	142	5	the	the	DET
cana-5491	142	6	above	above	ADJ
cana-5491	142	7	result	result	NOUN
cana-5491	142	8	.	.	PUNCT
cana-5491	143	1	we	we	PRON
cana-5491	143	2	should	should	AUX
cana-5491	143	3	be	be	AUX
cana-5491	143	4	able	able	ADJ
cana-5491	143	5	to	to	PART
cana-5491	143	6	solve	solve	VERB
cana-5491	143	7	fractional	fractional	ADJ
cana-5491	143	8	differential	differential	NOUN
cana-5491	143	9	equations	equation	NOUN
cana-5491	143	10	involving	involve	VERB
cana-5491	143	11	other	other	ADJ
cana-5491	143	12	special	special	ADJ
cana-5491	143	13	functions	function	NOUN
cana-5491	143	14	,	,	PUNCT
cana-5491	143	15	such	such	ADJ
cana-5491	143	16	as	as	ADP
cana-5491	143	17	mittag	mittag	ADJ
cana-5491	143	18	-	-	PUNCT
cana-5491	143	19	leffler	leffler	NOUN
cana-5491	143	20	functions	function	NOUN
cana-5491	143	21	,	,	PUNCT
cana-5491	143	22	hypergeometric	hypergeometric	ADJ
cana-5491	143	23	polynomials	polynomial	NOUN
cana-5491	143	24	,	,	PUNCT
cana-5491	143	25	and	and	CCONJ
cana-5491	143	26	jacobi	jacobi	PROPN
cana-5491	143	27	polynomials	polynomial	NOUN
cana-5491	143	28	.	.	PUNCT
cana-5491	144	1	acknowledgement	acknowledgement	NOUN
cana-5491	144	2	:	:	PUNCT
cana-5491	144	3	all	all	DET
cana-5491	144	4	authors	author	NOUN
cana-5491	144	5	would	would	AUX
cana-5491	144	6	like	like	VERB
cana-5491	144	7	to	to	ADP
cana-5491	144	8	thanks	thanks	NUM
cana-5491	144	9	integral	integral	ADJ
cana-5491	144	10	university	university	NOUN
cana-5491	144	11	,	,	PUNCT
cana-5491	144	12	lucknow	lucknow	PROPN
cana-5491	144	13	,	,	PUNCT
cana-5491	144	14	india	india	PROPN
cana-5491	144	15	for	for	ADP
cana-5491	144	16	providing	provide	VERB
cana-5491	144	17	the	the	DET
cana-5491	144	18	manuscript	manuscript	NOUN
cana-5491	144	19	(	(	PUNCT
cana-5491	144	20	mcn	mcn	PROPN
cana-5491	144	21	):	):	PUNCT
cana-5491	144	22	iu	iu	PROPN
cana-5491	144	23	/	/	SYM
cana-5491	144	24	r&d/2024	r&d/2024	NOUN
cana-5491	144	25	-	-	PUNCT
cana-5491	144	26	mcn0003211	mcn0003211	NOUN
cana-5491	144	27	for	for	ADP
cana-5491	144	28	this	this	DET
cana-5491	144	29	work	work	NOUN
cana-5491	144	30	.	.	PUNCT
cana-5491	145	1	conflict	conflict	NOUN
cana-5491	145	2	of	of	ADP
cana-5491	145	3	interest	interest	NOUN
cana-5491	145	4	:	:	PUNCT
cana-5491	145	5	the	the	DET
cana-5491	145	6	authors	author	NOUN
cana-5491	145	7	declare	declare	VERB
cana-5491	145	8	that	that	SCONJ
cana-5491	145	9	there	there	PRON
cana-5491	145	10	is	be	VERB
cana-5491	145	11	no	no	DET
cana-5491	145	12	conflict	conflict	NOUN
cana-5491	145	13	of	of	ADP
cana-5491	145	14	interest	interest	NOUN
cana-5491	145	15	.	.	PUNCT
cana-5491	146	1	references	reference	NOUN
cana-5491	146	2	[	[	X
cana-5491	146	3	1	1	NUM
cana-5491	146	4	]	]	X
cana-5491	146	5	e.d	e.d	PROPN
cana-5491	146	6	.	.	PROPN
cana-5491	146	7	,rainville,:special	,rainville,:special	ADJ
cana-5491	146	8	functions	function	NOUN
cana-5491	146	9	,	,	PUNCT
cana-5491	146	10	the	the	DET
cana-5491	146	11	macmillan	macmillan	PROPN
cana-5491	146	12	company	company	PROPN
cana-5491	146	13	,	,	PUNCT
cana-5491	146	14	new	new	PROPN
cana-5491	146	15	york	york	PROPN
cana-5491	146	16	,	,	PUNCT
cana-5491	146	17	2013	2013	NUM
cana-5491	146	18	.	.	PUNCT
cana-5491	147	1	[	[	X
cana-5491	147	2	2	2	X
cana-5491	147	3	]	]	X
cana-5491	147	4	e.	e.	PROPN
cana-5491	147	5	abuteen	abuteen	PROPN
cana-5491	147	6	,	,	PUNCT
cana-5491	147	7	solving	solve	VERB
cana-5491	147	8	fractional	fractional	ADJ
cana-5491	147	9	riccati	riccati	PROPN
cana-5491	147	10	differential	differential	NOUN
cana-5491	147	11	equation	equation	NOUN
cana-5491	147	12	with	with	ADP
cana-5491	147	13	caputofabrizio	caputofabrizio	PROPN
cana-5491	147	14	fractional	fractional	ADJ
cana-5491	147	15	derivative	derivative	ADJ
cana-5491	147	16	,	,	PUNCT
cana-5491	147	17	european	european	ADJ
cana-5491	147	18	journal	journal	NOUN
cana-5491	147	19	of	of	ADP
cana-5491	147	20	pure	pure	ADJ
cana-5491	147	21	and	and	CCONJ
cana-5491	147	22	applied	apply	VERB
cana-5491	147	23	mathematics,17(1	mathematics,17(1	PROPN
cana-5491	147	24	)	)	PUNCT
cana-5491	147	25	,	,	PUNCT
cana-5491	147	26	2024	2024	NUM
cana-5491	147	27	,	,	PUNCT
cana-5491	147	28	communications	communication	NOUN
cana-5491	147	29	on	on	ADP
cana-5491	147	30	applied	apply	VERB
cana-5491	147	31	nonlinear	nonlinear	ADJ
cana-5491	147	32	analysis	analysis	NOUN
cana-5491	147	33	issn	issn	NOUN
cana-5491	147	34	:	:	PUNCT
cana-5491	147	35	1074	1074	NUM
cana-5491	147	36	-	-	PUNCT
cana-5491	147	37	133x	133x	NUM
cana-5491	147	38	vol	vol	VERB
cana-5491	147	39	32	32	NUM
cana-5491	147	40	no	no	NOUN
cana-5491	147	41	.	.	PUNCT
cana-5491	148	1	10s	10	NOUN
cana-5491	148	2	(	(	PUNCT
cana-5491	148	3	2025	2025	NUM
cana-5491	148	4	)	)	PUNCT
cana-5491	148	5	2441	2441	NUM
cana-5491	148	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5491	148	7	372	372	NUM
cana-5491	148	8	-	-	SYM
cana-5491	148	9	384	384	NUM
cana-5491	148	10	[	[	X
cana-5491	148	11	3	3	NUM
cana-5491	148	12	]	]	PUNCT
cana-5491	148	13	m.	m.	NOUN
cana-5491	148	14	garg	garg	PROPN
cana-5491	148	15	,	,	PUNCT
cana-5491	148	16	k.	k.	PROPN
cana-5491	148	17	jain	jain	PROPN
cana-5491	148	18	and	and	CCONJ
cana-5491	148	19	s.l	s.l	PROPN
cana-5491	148	20	.	.	PROPN
cana-5491	148	21	kalla	kalla	PROPN
cana-5491	148	22	,	,	PUNCT
cana-5491	148	23	a	a	DET
cana-5491	148	24	further	further	ADJ
cana-5491	148	25	study	study	NOUN
cana-5491	148	26	of	of	ADP
cana-5491	148	27	general	general	PROPN
cana-5491	148	28	hurwitz	hurwitz	PROPN
cana-5491	148	29	-	-	PUNCT
cana-5491	148	30	lerch	lerch	PROPN
cana-5491	148	31	zeta	zeta	PROPN
cana-5491	148	32	function	function	PROPN
cana-5491	148	33	,	,	PUNCT
cana-5491	148	34	algebras	algebras	PROPN
cana-5491	148	35	groups	group	NOUN
cana-5491	148	36	geom	geom	PROPN
cana-5491	148	37	,	,	PUNCT
cana-5491	148	38	,	,	PUNCT
cana-5491	148	39	25	25	NUM
cana-5491	148	40	,	,	PUNCT
cana-5491	148	41	2008	2008	NUM
cana-5491	148	42	,	,	PUNCT
cana-5491	148	43	311	311	NUM
cana-5491	148	44	-	-	SYM
cana-5491	148	45	319	319	NUM
cana-5491	148	46	.	.	PUNCT
cana-5491	149	1	[	[	X
cana-5491	149	2	4	4	X
cana-5491	149	3	]	]	PUNCT
cana-5491	149	4	m.	m.	NOUN
cana-5491	149	5	ghayasuddin	ghayasuddin	PROPN
cana-5491	149	6	,	,	PUNCT
cana-5491	149	7	n.	n.	PROPN
cana-5491	149	8	u.	u.	PROPN
cana-5491	149	9	khan	khan	PROPN
cana-5491	149	10	,	,	PUNCT
cana-5491	149	11	w.	w.	PROPN
cana-5491	149	12	a.	a.	PROPN
cana-5491	149	13	khan	khan	PROPN
cana-5491	149	14	,	,	PUNCT
cana-5491	149	15	m.	m.	NOUN
cana-5491	149	16	ahmad	ahmad	PROPN
cana-5491	149	17	,	,	PUNCT
cana-5491	149	18	a	a	DET
cana-5491	149	19	note	note	NOUN
cana-5491	149	20	on	on	ADP
cana-5491	149	21	extended	extended	ADJ
cana-5491	149	22	hurwitz	hurwitz	PROPN
cana-5491	149	23	lerch	lerch	PROPN
cana-5491	149	24	zeta	zeta	PROPN
cana-5491	149	25	function	function	PROPN
cana-5491	149	26	,	,	PUNCT
cana-5491	149	27	italian	italian	ADJ
cana-5491	149	28	journal	journal	NOUN
cana-5491	149	29	of	of	ADP
cana-5491	149	30	pure	pure	ADJ
cana-5491	149	31	and	and	CCONJ
cana-5491	149	32	applied	applied	ADJ
cana-5491	149	33	mathematics	mathematic	NOUN
cana-5491	149	34	,	,	PUNCT
cana-5491	149	35	49	49	NUM
cana-5491	149	36	,	,	PUNCT
cana-5491	149	37	2023	2023	NUM
cana-5491	149	38	,	,	PUNCT
cana-5491	149	39	82–89	82–89	NUM
cana-5491	149	40	.	.	PUNCT
cana-5491	150	1	[	[	X
cana-5491	150	2	5	5	NUM
cana-5491	150	3	]	]	PUNCT
cana-5491	150	4	m.	m.	NOUN
cana-5491	150	5	kamarujjama	kamarujjama	PROPN
cana-5491	150	6	,	,	PUNCT
cana-5491	150	7	o.	o.	PROPN
cana-5491	150	8	khan	khan	PROPN
cana-5491	150	9	,	,	PUNCT
cana-5491	150	10	computation	computation	NOUN
cana-5491	150	11	of	of	ADP
cana-5491	150	12	new	new	ADJ
cana-5491	150	13	class	class	NOUN
cana-5491	150	14	of	of	ADP
cana-5491	150	15	integrals	integral	NOUN
cana-5491	150	16	involving	involve	VERB
cana-5491	150	17	generalized	generalized	ADJ
cana-5491	150	18	galue	galue	ADJ
cana-5491	150	19	type	type	NOUN
cana-5491	150	20	struve	struve	PROPN
cana-5491	150	21	function	function	NOUN
cana-5491	150	22	,	,	PUNCT
cana-5491	150	23	j.	j.	PROPN
cana-5491	150	24	comput	comput	PROPN
cana-5491	150	25	.	.	PUNCT
cana-5491	151	1	appl	appl	PROPN
cana-5491	151	2	.	.	PROPN
cana-5491	151	3	math	math	PROPN
cana-5491	151	4	.	.	PUNCT
cana-5491	152	1	,	,	PUNCT
cana-5491	152	2	351	351	NUM
cana-5491	152	3	(	(	PUNCT
cana-5491	152	4	2019	2019	NUM
cana-5491	152	5	)	)	PUNCT
cana-5491	152	6	228	228	NUM
cana-5491	152	7	-	-	SYM
cana-5491	152	8	236	236	NUM
cana-5491	152	9	.	.	PUNCT
cana-5491	153	1	[	[	X
cana-5491	153	2	6	6	NUM
cana-5491	153	3	]	]	PUNCT
cana-5491	153	4	m.	m.	NOUN
cana-5491	153	5	,	,	PUNCT
cana-5491	153	6	saigo	saigo	NOUN
cana-5491	153	7	,	,	PUNCT
cana-5491	153	8	:	:	PUNCT
cana-5491	153	9	a	a	DET
cana-5491	153	10	remark	remark	NOUN
cana-5491	153	11	on	on	ADP
cana-5491	153	12	integral	integral	ADJ
cana-5491	153	13	operators	operator	NOUN
cana-5491	153	14	involving	involve	VERB
cana-5491	153	15	the	the	DET
cana-5491	153	16	gauss	gauss	ADJ
cana-5491	153	17	hypergeometric	hypergeometric	ADJ
cana-5491	153	18	functions	function	NOUN
cana-5491	153	19	,	,	PUNCT
cana-5491	153	20	math	math	NOUN
cana-5491	153	21	.	.	PUNCT
cana-5491	153	22	rep	rep	PROPN
cana-5491	153	23	.	.	PROPN
cana-5491	153	24	coll	coll	PROPN
cana-5491	153	25	.	.	PUNCT
cana-5491	154	1	gen	gen	PROPN
cana-5491	154	2	.	.	PROPN
cana-5491	154	3	educ	educ	PROPN
cana-5491	154	4	.	.	PUNCT
cana-5491	155	1	kyushu	kyushu	PROPN
cana-5491	155	2	univ	univ	PROPN
cana-5491	155	3	.	.	PROPN
cana-5491	156	1	11	11	NUM
cana-5491	156	2	(	(	PUNCT
cana-5491	156	3	1978	1978	NUM
cana-5491	156	4	)	)	PUNCT
cana-5491	156	5	,	,	PUNCT
cana-5491	156	6	135	135	NUM
cana-5491	156	7	-	-	SYM
cana-5491	156	8	143	143	NUM
cana-5491	156	9	.	.	PUNCT
cana-5491	157	1	[	[	X
cana-5491	157	2	7	7	X
cana-5491	157	3	]	]	PUNCT
cana-5491	157	4	m.	m.	NOUN
cana-5491	157	5	j.	j.	PROPN
cana-5491	157	6	s.	s.	PROPN
cana-5491	157	7	shahwan	shahwan	PROPN
cana-5491	157	8	,	,	PUNCT
cana-5491	157	9	m.	m.	NOUN
cana-5491	157	10	g.	g.	PROPN
cana-5491	157	11	bin	bin	PROPN
cana-5491	157	12	-	-	PROPN
cana-5491	157	13	saad	saad	PROPN
cana-5491	157	14	,	,	PUNCT
cana-5491	157	15	m.	m.	NOUN
cana-5491	157	16	a.	a.	PROPN
cana-5491	157	17	pathan	pathan	PROPN
cana-5491	157	18	,	,	PUNCT
cana-5491	157	19	on	on	ADP
cana-5491	157	20	extended	extended	ADJ
cana-5491	157	21	hurwitz	hurwitz	PROPN
cana-5491	157	22	-	-	PUNCT
cana-5491	157	23	lerch	lerch	PROPN
cana-5491	157	24	zeta	zeta	PROPN
cana-5491	157	25	function	function	PROPN
cana-5491	157	26	,	,	PUNCT
cana-5491	157	27	kyungpook	kyungpook	PROPN
cana-5491	157	28	math	math	NOUN
cana-5491	157	29	.	.	PUNCT
cana-5491	158	1	j.	j.	PROPN
cana-5491	158	2	63	63	NUM
cana-5491	158	3	,	,	PUNCT
cana-5491	158	4	2023	2023	NUM
cana-5491	158	5	,	,	PUNCT
cana-5491	158	6	485	485	NUM
cana-5491	158	7	-	-	SYM
cana-5491	158	8	506	506	NUM
cana-5491	158	9	.	.	PUNCT
cana-5491	159	1	[	[	X
cana-5491	159	2	8	8	NUM
cana-5491	159	3	]	]	X
cana-5491	159	4	n.	n.	PROPN
cana-5491	159	5	u.	u.	PROPN
cana-5491	159	6	khan	khan	PROPN
cana-5491	159	7	,	,	PUNCT
cana-5491	159	8	m.	m.	PROPN
cana-5491	159	9	iqbal	iqbal	PROPN
cana-5491	159	10	khan	khan	PROPN
cana-5491	159	11	,	,	PUNCT
cana-5491	159	12	o.	o.	PROPN
cana-5491	159	13	khan	khan	PROPN
cana-5491	159	14	,	,	PUNCT
cana-5491	159	15	certain	certain	ADJ
cana-5491	159	16	finite	finite	ADJ
cana-5491	159	17	integrals	integral	NOUN
cana-5491	159	18	involving	involve	VERB
cana-5491	159	19	generalized	generalize	VERB
cana-5491	159	20	wright	wright	PROPN
cana-5491	159	21	function	function	NOUN
cana-5491	159	22	,	,	PUNCT
cana-5491	159	23	advanced	advanced	ADJ
cana-5491	159	24	mathematical	mathematical	ADJ
cana-5491	159	25	models	model	NOUN
cana-5491	159	26	&	&	CCONJ
cana-5491	159	27	applications	application	NOUN
cana-5491	159	28	,	,	PUNCT
cana-5491	159	29	6	6	NUM
cana-5491	159	30	,	,	PUNCT
cana-5491	159	31	(	(	PUNCT
cana-5491	159	32	3	3	NUM
cana-5491	159	33	)	)	PUNCT
cana-5491	159	34	,	,	PUNCT
cana-5491	159	35	(	(	PUNCT
cana-5491	159	36	2021	2021	NUM
cana-5491	159	37	)	)	PUNCT
cana-5491	159	38	,	,	PUNCT
cana-5491	159	39	292	292	NUM
cana-5491	159	40	-	-	SYM
cana-5491	159	41	301	301	NUM
cana-5491	159	42	[	[	X
cana-5491	159	43	9	9	NUM
cana-5491	159	44	]	]	X
cana-5491	159	45	n.	n.	PROPN
cana-5491	159	46	khan	khan	PROPN
cana-5491	159	47	,	,	PUNCT
cana-5491	159	48	s.	s.	PROPN
cana-5491	159	49	husain	husain	PROPN
cana-5491	159	50	,	,	PUNCT
cana-5491	159	51	o.	o.	PROPN
cana-5491	159	52	khan	khan	PROPN
cana-5491	159	53	,	,	PUNCT
cana-5491	159	54	a	a	DET
cana-5491	159	55	novel	novel	ADJ
cana-5491	159	56	kind	kind	NOUN
cana-5491	159	57	of	of	ADP
cana-5491	159	58	beta	beta	ADJ
cana-5491	159	59	logarithmic	logarithmic	ADJ
cana-5491	159	60	function	function	NOUN
cana-5491	159	61	and	and	CCONJ
cana-5491	159	62	their	their	PRON
cana-5491	159	63	properties	property	NOUN
cana-5491	159	64	,	,	PUNCT
cana-5491	159	65	hacet	hacet	PROPN
cana-5491	159	66	.	.	PUNCT
cana-5491	160	1	j.	j.	PROPN
cana-5491	160	2	math	math	PROPN
cana-5491	160	3	.	.	PUNCT
cana-5491	161	1	stat	stat	PROPN
cana-5491	161	2	.	.	PUNCT
cana-5491	162	1	52	52	NUM
cana-5491	162	2	(	(	PUNCT
cana-5491	162	3	4	4	NUM
cana-5491	162	4	)	)	PUNCT
cana-5491	162	5	(	(	PUNCT
cana-5491	162	6	2023	2023	NUM
cana-5491	162	7	)	)	PUNCT
cana-5491	162	8	,	,	PUNCT
cana-5491	162	9	945–955	945–955	NUM
cana-5491	162	10	[	[	X
cana-5491	162	11	10	10	NUM
cana-5491	162	12	]	]	X
cana-5491	162	13	o.	o.	PROPN
cana-5491	162	14	khan	khan	PROPN
cana-5491	162	15	,	,	PUNCT
cana-5491	162	16	m.	m.	NOUN
cana-5491	162	17	kamarujjama	kamarujjama	PROPN
cana-5491	162	18	,	,	PUNCT
cana-5491	162	19	n.u	n.u	PROPN
cana-5491	162	20	.	.	PROPN
cana-5491	162	21	khan	khan	PROPN
cana-5491	162	22	,	,	PUNCT
cana-5491	162	23	d.	d.	PROPN
cana-5491	162	24	baleanu	baleanu	PROPN
cana-5491	162	25	,	,	PUNCT
cana-5491	162	26	k.s	k.s	PROPN
cana-5491	162	27	.	.	PROPN
cana-5491	162	28	nisar	nisar	PROPN
cana-5491	162	29	,	,	PUNCT
cana-5491	162	30	computable	computable	ADJ
cana-5491	162	31	solution	solution	NOUN
cana-5491	162	32	of	of	ADP
cana-5491	162	33	fractional	fractional	ADJ
cana-5491	162	34	kinetic	kinetic	ADJ
cana-5491	162	35	equations	equation	NOUN
cana-5491	162	36	using	use	VERB
cana-5491	162	37	mathieu	mathieu	NOUN
cana-5491	162	38	-	-	PUNCT
cana-5491	162	39	type	type	NOUN
cana-5491	162	40	series	series	NOUN
cana-5491	162	41	,	,	PUNCT
cana-5491	162	42	adv	adv	PROPN
cana-5491	162	43	.	.	PROPN
cana-5491	162	44	differ	differ	VERB
cana-5491	162	45	equ	equ	PROPN
cana-5491	162	46	.	.	PROPN
cana-5491	162	47	2019	2019	NUM
cana-5491	162	48	,	,	PUNCT
cana-5491	162	49	234	234	NUM
cana-5491	162	50	(	(	PUNCT
cana-5491	162	51	2019	2019	NUM
cana-5491	162	52	)	)	PUNCT
cana-5491	162	53	.	.	PUNCT
cana-5491	163	1	[	[	X
cana-5491	163	2	11	11	NUM
cana-5491	163	3	]	]	PUNCT
cana-5491	163	4	r.	r.	PROPN
cana-5491	163	5	k.	k.	PROPN
cana-5491	163	6	parmar	parmar	PROPN
cana-5491	163	7	and	and	CCONJ
cana-5491	163	8	r.	r.	PROPN
cana-5491	163	9	k	k	PROPN
cana-5491	163	10	raina	raina	PROPN
cana-5491	163	11	,	,	PUNCT
cana-5491	163	12	on	on	ADP
cana-5491	163	13	a	a	DET
cana-5491	163	14	certain	certain	ADJ
cana-5491	163	15	extension	extension	NOUN
cana-5491	163	16	of	of	ADP
cana-5491	163	17	the	the	DET
cana-5491	163	18	hurwitz	hurwitz	PROPN
cana-5491	163	19	-	-	PUNCT
cana-5491	163	20	lerch	lerch	PROPN
cana-5491	163	21	zeta	zeta	PROPN
cana-5491	163	22	function	function	PROPN
cana-5491	163	23	,	,	PUNCT
cana-5491	163	24	annals	annal	NOUN
cana-5491	163	25	of	of	ADP
cana-5491	163	26	west	west	PROPN
cana-5491	163	27	university	university	PROPN
cana-5491	163	28	of	of	ADP
cana-5491	163	29	timisoara	timisoara	PROPN
cana-5491	163	30	-	-	PUNCT
cana-5491	163	31	mathematics	mathematic	NOUN
cana-5491	163	32	and	and	CCONJ
cana-5491	163	33	computer	computer	NOUN
cana-5491	163	34	science	science	NOUN
cana-5491	163	35	,	,	PUNCT
cana-5491	163	36	52	52	NUM
cana-5491	163	37	(	(	PUNCT
cana-5491	163	38	2	2	NUM
cana-5491	163	39	)	)	PUNCT
cana-5491	163	40	,	,	PUNCT
cana-5491	163	41	2014	2014	NUM
cana-5491	163	42	,	,	PUNCT
cana-5491	163	43	157	157	NUM
cana-5491	163	44	-	-	SYM
cana-5491	163	45	170	170	NUM
cana-5491	163	46	.	.	PUNCT
cana-5491	164	1	[	[	X
cana-5491	164	2	12	12	NUM
cana-5491	164	3	]	]	X
cana-5491	164	4	r.k	r.k	PROPN
cana-5491	164	5	.	.	PROPN
cana-5491	164	6	raina	raina	PROPN
cana-5491	164	7	and	and	CCONJ
cana-5491	164	8	p.k	p.k	PROPN
cana-5491	164	9	.	.	PROPN
cana-5491	164	10	chhajed	chhajed	NOUN
cana-5491	164	11	,	,	PUNCT
cana-5491	164	12	certain	certain	ADJ
cana-5491	164	13	results	result	NOUN
cana-5491	164	14	involving	involve	VERB
cana-5491	164	15	a	a	DET
cana-5491	164	16	class	class	NOUN
cana-5491	164	17	of	of	ADP
cana-5491	164	18	functions	function	NOUN
cana-5491	164	19	associated	associate	VERB
cana-5491	164	20	with	with	ADP
cana-5491	164	21	the	the	DET
cana-5491	164	22	hurwitz	hurwitz	PROPN
cana-5491	164	23	zeta	zeta	PROPN
cana-5491	164	24	function	function	PROPN
cana-5491	164	25	,	,	PUNCT
cana-5491	164	26	acta	acta	PROPN
cana-5491	164	27	mathematica	mathematica	PROPN
cana-5491	164	28	universitatis	universitatis	PROPN
cana-5491	164	29	comenianae	comenianae	PROPN
cana-5491	164	30	,	,	PUNCT
cana-5491	164	31	new	new	ADJ
cana-5491	164	32	series	series	NOUN
cana-5491	164	33	,	,	PUNCT
cana-5491	164	34	73(1	73(1	NUM
cana-5491	164	35	)	)	PUNCT
cana-5491	164	36	,	,	PUNCT
cana-5491	164	37	2004	2004	NUM
cana-5491	164	38	,	,	PUNCT
cana-5491	164	39	89	89	NUM
cana-5491	164	40	-	-	SYM
cana-5491	164	41	100	100	NUM
cana-5491	164	42	.	.	PUNCT
cana-5491	165	1	[	[	X
cana-5491	165	2	13	13	NUM
cana-5491	165	3	]	]	X
cana-5491	165	4	s.d	s.d	PROPN
cana-5491	165	5	.	.	PROPN
cana-5491	165	6	lin	lin	PROPN
cana-5491	165	7	and	and	CCONJ
cana-5491	165	8	h.m	h.m	PROPN
cana-5491	165	9	.	.	PROPN
cana-5491	165	10	srivastava	srivastava	PROPN
cana-5491	165	11	,	,	PUNCT
cana-5491	165	12	some	some	DET
cana-5491	165	13	families	family	NOUN
cana-5491	165	14	of	of	ADP
cana-5491	165	15	the	the	DET
cana-5491	165	16	hurwitz	hurwitz	PROPN
cana-5491	165	17	–	–	PUNCT
cana-5491	165	18	lerch	lerch	PROPN
cana-5491	165	19	zeta	zeta	PROPN
cana-5491	165	20	functions	function	NOUN
cana-5491	165	21	and	and	CCONJ
cana-5491	165	22	associated	associate	VERB
cana-5491	165	23	fractional	fractional	ADJ
cana-5491	165	24	derivative	derivative	ADJ
cana-5491	165	25	and	and	CCONJ
cana-5491	165	26	other	other	ADJ
cana-5491	165	27	integral	integral	ADJ
cana-5491	165	28	representations	representation	NOUN
cana-5491	165	29	,	,	PUNCT
cana-5491	165	30	applied	apply	VERB
cana-5491	165	31	mathematics	mathematic	NOUN
cana-5491	165	32	and	and	CCONJ
cana-5491	165	33	computation	computation	NOUN
cana-5491	165	34	,	,	PUNCT
cana-5491	165	35	154	154	NUM
cana-5491	165	36	(	(	PUNCT
cana-5491	165	37	3),2004,725	3),2004,725	NUM
cana-5491	165	38	-	-	NUM
cana-5491	165	39	733	733	NUM
cana-5491	165	40	.	.	PUNCT
cana-5491	166	1	[	[	X
cana-5491	166	2	14	14	NUM
cana-5491	166	3	]	]	X
cana-5491	166	4	s.p	s.p	PROPN
cana-5491	166	5	.	.	PUNCT
cana-5491	166	6	goyal	goyal	PROPN
cana-5491	166	7	and	and	CCONJ
cana-5491	166	8	r.k	r.k	PROPN
cana-5491	166	9	.	.	PROPN
cana-5491	166	10	laddha	laddha	PROPN
cana-5491	166	11	,	,	PUNCT
cana-5491	166	12	on	on	ADP
cana-5491	166	13	the	the	DET
cana-5491	166	14	generalized	generalize	VERB
cana-5491	166	15	riemann	riemann	PROPN
cana-5491	166	16	zeta	zeta	PROPN
cana-5491	166	17	functions	function	NOUN
cana-5491	166	18	and	and	CCONJ
cana-5491	166	19	the	the	DET
cana-5491	166	20	generalized	generalized	ADJ
cana-5491	166	21	lambert	lambert	PROPN
cana-5491	166	22	transform	transform	NOUN
cana-5491	166	23	,	,	PUNCT
cana-5491	166	24	ganita	ganita	NOUN
cana-5491	166	25	sandesh	sandesh	NOUN
cana-5491	166	26	,	,	PUNCT
cana-5491	166	27	11(2	11(2	NOUN
cana-5491	166	28	)	)	PUNCT
cana-5491	166	29	,	,	PUNCT
cana-5491	166	30	1997	1997	NUM
cana-5491	166	31	,	,	PUNCT
cana-5491	166	32	99	99	NUM
cana-5491	166	33	-	-	SYM
cana-5491	166	34	108	108	NUM
cana-5491	166	35	.	.	PUNCT
cana-5491	167	1	[	[	X
cana-5491	167	2	15	15	NUM
cana-5491	167	3	]	]	X
cana-5491	167	4	s.z.rida	s.z.rida	PROPN
cana-5491	167	5	,	,	PUNCT
cana-5491	167	6	a.a.m.arafa	a.a.m.arafa	PROPN
cana-5491	167	7	,	,	PUNCT
cana-5491	167	8	new	new	ADJ
cana-5491	167	9	method	method	NOUN
cana-5491	167	10	for	for	ADP
cana-5491	167	11	solving	solve	VERB
cana-5491	167	12	linear	linear	ADJ
cana-5491	167	13	fractional	fractional	ADJ
cana-5491	167	14	differential	differential	NOUN
cana-5491	167	15	equations	equation	NOUN
cana-5491	167	16	,	,	PUNCT
cana-5491	167	17	int	int	NOUN
cana-5491	167	18	..	..	PUNCT
cana-5491	167	19	j.	j.	PROPN
cana-5491	167	20	differential	differential	PROPN
cana-5491	167	21	equation	equation	NOUN
cana-5491	167	22	,	,	PUNCT
cana-5491	167	23	2011	2011	NUM
cana-5491	167	24	,	,	PUNCT
cana-5491	167	25	article	article	NOUN
cana-5491	167	26	id-814132	id-814132	PROPN
cana-5491	167	27	,	,	PUNCT
cana-5491	167	28	1	1	NUM
cana-5491	167	29	-	-	SYM
cana-5491	167	30	8	8	NUM
cana-5491	167	31	.	.	PUNCT
cana-5491	168	1	[	[	X
cana-5491	168	2	16	16	NUM
cana-5491	168	3	]	]	PUNCT
cana-5491	168	4	s.kumar	s.kumar	PROPN
cana-5491	168	5	,	,	PUNCT
cana-5491	168	6	o.khan	o.khan	ADV
cana-5491	168	7	,	,	PUNCT
cana-5491	168	8	n.u.khan	n.u.khan	ADJ
cana-5491	168	9	,	,	PUNCT
cana-5491	168	10	solution	solution	NOUN
cana-5491	168	11	of	of	ADP
cana-5491	168	12	fractional	fractional	ADJ
cana-5491	168	13	differential	differential	ADJ
cana-5491	168	14	equation	equation	NOUN
cana-5491	168	15	involving	involve	VERB
cana-5491	168	16	certain	certain	ADJ
cana-5491	168	17	special	special	ADJ
cana-5491	168	18	functions	function	NOUN
cana-5491	168	19	,	,	PUNCT
cana-5491	168	20	journal	journal	NOUN
cana-5491	168	21	of	of	ADP
cana-5491	168	22	xidian	xidian	PROPN
cana-5491	168	23	university	university	PROPN
cana-5491	168	24	,	,	PUNCT
cana-5491	168	25	18	18	NUM
cana-5491	168	26	(	(	PUNCT
cana-5491	168	27	6	6	NUM
cana-5491	168	28	)	)	PUNCT
cana-5491	168	29	,	,	PUNCT
cana-5491	168	30	2024	2024	NUM
cana-5491	168	31	,	,	PUNCT
cana-5491	168	32	88	88	NUM
cana-5491	168	33	-	-	SYM
cana-5491	168	34	94	94	NUM
cana-5491	168	35	..	..	PUNCT