id	sid	tid	token	lemma	pos
cana-5994	1	1	some	some	DET
cana-5994	1	2	properties	property	NOUN
cana-5994	1	3	of	of	ADP
cana-5994	1	4	differential	differential	ADJ
cana-5994	1	5	equations	equation	NOUN
cana-5994	1	6	of	of	ADP
cana-5994	1	7	higher	high	ADJ
cana-5994	1	8	-	-	PUNCT
cana-5994	1	9	order	order	NOUN
cana-5994	1	10	q	q	ADJ
cana-5994	1	11	-	-	PUNCT
cana-5994	1	12	frobenius	frobenius	ADJ
cana-5994	1	13	-	-	PUNCT
cana-5994	1	14	tangent	tangent	NOUN
cana-5994	1	15	polynomials	polynomial	NOUN
cana-5994	1	16	idrees	idree	NOUN
cana-5994	1	17	ahmad	ahmad	PROPN
cana-5994	1	18	khan	khan	PROPN
cana-5994	1	19	and	and	CCONJ
cana-5994	1	20	sumit	sumit	PROPN
cana-5994	1	21	kumar	kumar	PROPN
cana-5994	1	22	abstract	abstract	PROPN
cana-5994	1	23	.	.	PUNCT
cana-5994	2	1	the	the	DET
cana-5994	2	2	classical	classical	ADJ
cana-5994	2	3	q	q	ADJ
cana-5994	2	4	-	-	PUNCT
cana-5994	2	5	frobenius	frobenius	ADJ
cana-5994	2	6	-	-	PUNCT
cana-5994	2	7	tangent	tangent	NOUN
cana-5994	2	8	polynomials	polynomial	NOUN
cana-5994	2	9	dealt	deal	VERB
cana-5994	2	10	within	within	ADP
cana-5994	2	11	this	this	DET
cana-5994	2	12	paper	paper	NOUN
cana-5994	2	13	contains	contain	VERB
cana-5994	2	14	many	many	ADJ
cana-5994	2	15	application	application	NOUN
cana-5994	2	16	in	in	ADP
cana-5994	2	17	various	various	ADJ
cana-5994	2	18	areas	area	NOUN
cana-5994	2	19	.	.	PUNCT
cana-5994	3	1	we	we	PRON
cana-5994	3	2	construct	construct	VERB
cana-5994	3	3	new	new	ADJ
cana-5994	3	4	types	type	NOUN
cana-5994	3	5	of	of	ADP
cana-5994	3	6	differential	differential	ADJ
cana-5994	3	7	equations	equation	NOUN
cana-5994	3	8	for	for	ADP
cana-5994	3	9	q	q	ADJ
cana-5994	3	10	-	-	PUNCT
cana-5994	3	11	frobenius	frobenius	ADJ
cana-5994	3	12	-	-	PUNCT
cana-5994	3	13	tangent	tangent	NOUN
cana-5994	3	14	polynomials	polynomial	NOUN
cana-5994	3	15	using	use	VERB
cana-5994	3	16	q	q	NOUN
cana-5994	3	17	-	-	PUNCT
cana-5994	3	18	derivatives	derivative	NOUN
cana-5994	3	19	,	,	PUNCT
cana-5994	3	20	find	find	VERB
cana-5994	3	21	some	some	DET
cana-5994	3	22	properties	property	NOUN
cana-5994	3	23	and	and	CCONJ
cana-5994	3	24	several	several	ADJ
cana-5994	3	25	difference	difference	NOUN
cana-5994	3	26	equations	equation	NOUN
cana-5994	3	27	of	of	ADP
cana-5994	3	28	these	these	DET
cana-5994	3	29	polynomials	polynomial	NOUN
cana-5994	3	30	.	.	PUNCT
cana-5994	4	1	1	1	X
cana-5994	4	2	.	.	X
cana-5994	4	3	introduction	introduction	NOUN
cana-5994	4	4	in	in	ADP
cana-5994	4	5	recent	recent	ADJ
cana-5994	4	6	years	year	NOUN
cana-5994	4	7	,	,	PUNCT
cana-5994	4	8	numerous	numerous	ADJ
cana-5994	4	9	researchers	researcher	NOUN
cana-5994	4	10	have	have	AUX
cana-5994	4	11	explored	explore	VERB
cana-5994	4	12	the	the	DET
cana-5994	4	13	bernoulli	bernoulli	PROPN
cana-5994	4	14	,	,	PUNCT
cana-5994	4	15	euler	euler	PROPN
cana-5994	4	16	,	,	PUNCT
cana-5994	4	17	genocchi	genocchi	PROPN
cana-5994	4	18	,	,	PUNCT
cana-5994	4	19	frobenius	frobenius	NOUN
cana-5994	4	20	-	-	PUNCT
cana-5994	4	21	euler	euler	NOUN
cana-5994	4	22	,	,	PUNCT
cana-5994	4	23	and	and	CCONJ
cana-5994	4	24	tangent	tangent	NOUN
cana-5994	4	25	polynomials	polynomial	NOUN
cana-5994	4	26	in	in	ADP
cana-5994	4	27	their	their	PRON
cana-5994	4	28	classical	classical	ADJ
cana-5994	4	29	,	,	PUNCT
cana-5994	4	30	generalized	generalized	ADJ
cana-5994	4	31	,	,	PUNCT
cana-5994	4	32	and	and	CCONJ
cana-5994	4	33	unified	unified	ADJ
cana-5994	4	34	forms	form	NOUN
cana-5994	4	35	to	to	PART
cana-5994	4	36	investigate	investigate	VERB
cana-5994	4	37	their	their	PRON
cana-5994	4	38	properties	property	NOUN
cana-5994	4	39	,	,	PUNCT
cana-5994	4	40	relationships	relationship	NOUN
cana-5994	4	41	,	,	PUNCT
cana-5994	4	42	and	and	CCONJ
cana-5994	4	43	applications	application	NOUN
cana-5994	4	44	[	[	X
cana-5994	4	45	5	5	NUM
cana-5994	4	46	,	,	PUNCT
cana-5994	4	47	6	6	NUM
cana-5994	4	48	,	,	PUNCT
cana-5994	4	49	7	7	NUM
cana-5994	4	50	,	,	PUNCT
cana-5994	4	51	9	9	NUM
cana-5994	4	52	,	,	PUNCT
cana-5994	4	53	10	10	NUM
cana-5994	4	54	]	]	PUNCT
cana-5994	4	55	.	.	PUNCT
cana-5994	5	1	building	build	VERB
cana-5994	5	2	on	on	ADP
cana-5994	5	3	these	these	DET
cana-5994	5	4	studies	study	NOUN
cana-5994	5	5	,	,	PUNCT
cana-5994	5	6	jackson	jackson	PROPN
cana-5994	5	7	introduced	introduce	VERB
cana-5994	5	8	the	the	DET
cana-5994	5	9	q	q	NOUN
cana-5994	5	10	-	-	PUNCT
cana-5994	5	11	bernoulli	bernoulli	NOUN
cana-5994	5	12	,	,	PUNCT
cana-5994	5	13	q	q	NOUN
cana-5994	5	14	-	-	PUNCT
cana-5994	5	15	euler	euler	NOUN
cana-5994	5	16	,	,	PUNCT
cana-5994	5	17	and	and	CCONJ
cana-5994	5	18	q	q	X
cana-5994	5	19	-	-	PUNCT
cana-5994	5	20	genocchi	genocchi	NOUN
cana-5994	5	21	polynomials	polynomial	VERB
cana-5994	5	22	[	[	X
cana-5994	5	23	1	1	NUM
cana-5994	5	24	,	,	PUNCT
cana-5994	5	25	2	2	NUM
cana-5994	5	26	]	]	PUNCT
cana-5994	5	27	,	,	PUNCT
cana-5994	5	28	while	while	SCONJ
cana-5994	5	29	kang	kang	PROPN
cana-5994	6	1	[	[	X
cana-5994	6	2	6	6	NUM
cana-5994	6	3	,	,	PUNCT
cana-5994	6	4	8	8	NUM
cana-5994	6	5	]	]	PUNCT
cana-5994	6	6	and	and	CCONJ
cana-5994	6	7	kang	kang	PROPN
cana-5994	6	8	and	and	CCONJ
cana-5994	6	9	kim	kim	PROPN
cana-5994	7	1	[	[	X
cana-5994	7	2	7	7	NUM
cana-5994	7	3	]	]	PUNCT
cana-5994	7	4	examined	examine	VERB
cana-5994	7	5	generalized	generalized	ADJ
cana-5994	7	6	q	q	ADJ
cana-5994	7	7	-	-	PUNCT
cana-5994	7	8	tangent	tangent	ADJ
cana-5994	7	9	polynomials	polynomial	NOUN
cana-5994	7	10	.	.	PUNCT
cana-5994	8	1	kang	kang	PROPN
cana-5994	8	2	and	and	CCONJ
cana-5994	8	3	khan	khan	PROPN
cana-5994	9	1	[	[	X
cana-5994	9	2	5	5	NUM
cana-5994	9	3	]	]	PUNCT
cana-5994	9	4	studied	study	VERB
cana-5994	9	5	q	q	ADJ
cana-5994	9	6	-	-	PUNCT
cana-5994	9	7	frobenius	frobenius	ADJ
cana-5994	9	8	-	-	PUNCT
cana-5994	9	9	euler	euler	NOUN
cana-5994	9	10	polynomials	polynomial	NOUN
cana-5994	9	11	,	,	PUNCT
cana-5994	9	12	nisar	nisar	PROPN
cana-5994	9	13	et	et	PROPN
cana-5994	9	14	al	al	PROPN
cana-5994	9	15	.	.	PUNCT
cana-5994	10	1	[	[	X
cana-5994	10	2	13	13	NUM
cana-5994	10	3	]	]	PUNCT
cana-5994	10	4	introduced	introduce	VERB
cana-5994	10	5	q	q	ADJ
cana-5994	10	6	-	-	PUNCT
cana-5994	10	7	frobenius	frobenius	ADJ
cana-5994	10	8	-	-	PUNCT
cana-5994	10	9	tangent	tangent	NOUN
cana-5994	10	10	polynomials	polynomial	NOUN
cana-5994	10	11	,	,	PUNCT
cana-5994	10	12	and	and	CCONJ
cana-5994	10	13	ryoo	ryoo	NOUN
cana-5994	10	14	and	and	CCONJ
cana-5994	10	15	kang	kang	PROPN
cana-5994	11	1	[	[	X
cana-5994	11	2	15	15	NUM
cana-5994	11	3	,	,	PUNCT
cana-5994	11	4	16	16	NUM
cana-5994	11	5	]	]	PUNCT
cana-5994	11	6	investigated	investigate	VERB
cana-5994	11	7	the	the	DET
cana-5994	11	8	q	q	ADJ
cana-5994	11	9	-	-	PUNCT
cana-5994	11	10	differential	differential	ADJ
cana-5994	11	11	equation	equation	NOUN
cana-5994	11	12	forms	form	NOUN
cana-5994	11	13	of	of	ADP
cana-5994	11	14	euler	euler	PROPN
cana-5994	11	15	and	and	CCONJ
cana-5994	11	16	genocchi	genocchi	PROPN
cana-5994	11	17	polynomials	polynomial	NOUN
cana-5994	11	18	.	.	PUNCT
cana-5994	12	1	these	these	DET
cana-5994	12	2	works	work	NOUN
cana-5994	12	3	have	have	AUX
cana-5994	12	4	uncovered	uncover	VERB
cana-5994	12	5	numerous	numerous	ADJ
cana-5994	12	6	properties	property	NOUN
cana-5994	12	7	,	,	PUNCT
cana-5994	12	8	relationships	relationship	NOUN
cana-5994	12	9	,	,	PUNCT
cana-5994	12	10	and	and	CCONJ
cana-5994	12	11	applications	application	NOUN
cana-5994	12	12	in	in	ADP
cana-5994	12	13	fields	field	NOUN
cana-5994	12	14	such	such	ADJ
cana-5994	12	15	as	as	ADP
cana-5994	12	16	umbral	umbral	ADJ
cana-5994	12	17	calculus	calculus	NOUN
cana-5994	12	18	,	,	PUNCT
cana-5994	12	19	p	p	NOUN
cana-5994	12	20	-	-	PUNCT
cana-5994	12	21	adic	adic	ADJ
cana-5994	12	22	analysis	analysis	NOUN
cana-5994	12	23	,	,	PUNCT
cana-5994	12	24	and	and	CCONJ
cana-5994	12	25	combinatorics	combinatoric	NOUN
cana-5994	12	26	.	.	PUNCT
cana-5994	13	1	let	let	VERB
cana-5994	13	2	σ	σ	NUM
cana-5994	13	3	∈	∈	PROPN
cana-5994	13	4	r	r	NOUN
cana-5994	13	5	,	,	PUNCT
cana-5994	13	6	p(ψ	p(ψ	NOUN
cana-5994	13	7	)	)	PUNCT
cana-5994	13	8	and	and	CCONJ
cana-5994	13	9	g(ψ	g(ψ	PROPN
cana-5994	13	10	)	)	PUNCT
cana-5994	13	11	are	be	AUX
cana-5994	13	12	continuous	continuous	ADJ
cana-5994	13	13	function	function	NOUN
cana-5994	13	14	,	,	PUNCT
cana-5994	13	15	the	the	DET
cana-5994	13	16	equation	equation	NOUN
cana-5994	13	17	of	of	ADP
cana-5994	13	18	bernoulli	bernoulli	NOUN
cana-5994	13	19	polynomials	polynomial	NOUN
cana-5994	13	20	as	as	ADP
cana-5994	13	21	dϕ	dϕ	ADP
cana-5994	13	22	dψ	dψ	NOUN
cana-5994	14	1	+	+	CCONJ
cana-5994	14	2	p(ψ)ϕ−	p(ψ)ϕ−	PROPN
cana-5994	14	3	g(ψ)ϕσ	g(ψ)ϕσ	PROPN
cana-5994	14	4	=	=	SYM
cana-5994	14	5	0	0	PROPN
cana-5994	14	6	,	,	PUNCT
cana-5994	14	7	(	(	PUNCT
cana-5994	14	8	1.1	1.1	NUM
cana-5994	14	9	)	)	PUNCT
cana-5994	14	10	let	let	VERB
cana-5994	14	11	σ	σ	NOUN
cana-5994	14	12	=	=	SYM
cana-5994	14	13	0	0	NUM
cana-5994	14	14	,	,	PUNCT
cana-5994	14	15	we	we	PRON
cana-5994	14	16	will	will	AUX
cana-5994	14	17	get	get	VERB
cana-5994	14	18	linear	linear	ADJ
cana-5994	14	19	equation	equation	NOUN
cana-5994	14	20	and	and	CCONJ
cana-5994	14	21	it	it	PRON
cana-5994	14	22	is	be	AUX
cana-5994	14	23	not	not	PART
cana-5994	14	24	nonlinear	nonlinear	ADJ
cana-5994	14	25	equation	equation	NOUN
cana-5994	14	26	.	.	PUNCT
cana-5994	15	1	if	if	SCONJ
cana-5994	15	2	η	η	PROPN
cana-5994	15	3	=	=	SYM
cana-5994	15	4	ϕ1−σ	ϕ1−σ	PROPN
cana-5994	15	5	in	in	ADP
cana-5994	15	6	(	(	PUNCT
cana-5994	15	7	1.1	1.1	NUM
cana-5994	15	8	)	)	PUNCT
cana-5994	15	9	,	,	PUNCT
cana-5994	15	10	we	we	PRON
cana-5994	15	11	get	get	VERB
cana-5994	15	12	differential	differential	ADJ
cana-5994	15	13	equation	equation	NOUN
cana-5994	15	14	of	of	ADP
cana-5994	15	15	bernoulli	bernoulli	NOUN
cana-5994	15	16	polynomials	polynomial	NOUN
cana-5994	15	17	.	.	PUNCT
cana-5994	16	1	dη	dη	ADP
cana-5994	16	2	dψ	dψ	INTJ
cana-5994	17	1	+	+	CCONJ
cana-5994	17	2	(	(	PUNCT
cana-5994	17	3	1−	1−	NUM
cana-5994	17	4	σ)p(ψ)η	σ)p(ψ)η	NOUN
cana-5994	17	5	=	=	SYM
cana-5994	17	6	(	(	PUNCT
cana-5994	17	7	1−	1−	NUM
cana-5994	17	8	σ)g(ψ	σ)g(ψ	PROPN
cana-5994	17	9	)	)	PUNCT
cana-5994	17	10	,	,	PUNCT
cana-5994	17	11	putting	put	VERB
cana-5994	17	12	σ	σ	NOUN
cana-5994	17	13	=	=	SYM
cana-5994	17	14	0	0	NUM
cana-5994	17	15	,	,	PUNCT
cana-5994	17	16	the	the	DET
cana-5994	17	17	equation	equation	NOUN
cana-5994	17	18	(	(	PUNCT
cana-5994	17	19	1.1	1.1	NUM
cana-5994	17	20	)	)	PUNCT
cana-5994	17	21	gives	give	VERB
cana-5994	17	22	the	the	DET
cana-5994	17	23	differential	differential	ADJ
cana-5994	17	24	equation	equation	NOUN
cana-5994	17	25	of	of	ADP
cana-5994	17	26	the	the	DET
cana-5994	17	27	frobeniustangent	frobeniustangent	ADJ
cana-5994	17	28	polynomials	polynomial	NOUN
cana-5994	17	29	as	as	SCONJ
cana-5994	17	30	follows	follow	VERB
cana-5994	17	31	.	.	PUNCT
cana-5994	18	1	d	d	X
cana-5994	18	2	dψ	dψ	PROPN
cana-5994	18	3	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	18	4	;	;	PUNCT
cana-5994	18	5	η	η	PROPN
cana-5994	18	6	)	)	PUNCT
cana-5994	18	7	+	+	CCONJ
cana-5994	18	8	1	1	NUM
cana-5994	18	9	1−	1−	NUM
cana-5994	18	10	η	η	PROPN
cana-5994	18	11	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	18	12	;	;	PUNCT
cana-5994	18	13	η	η	NOUN
cana-5994	18	14	)	)	PUNCT
cana-5994	18	15	+	+	CCONJ
cana-5994	18	16	1	1	NUM
cana-5994	18	17	1−	1−	NUM
cana-5994	18	18	η	η	PROPN
cana-5994	18	19	ft0(ψ	ft0(ψ	PROPN
cana-5994	18	20	;	;	PUNCT
cana-5994	18	21	η)−	η)−	PROPN
cana-5994	18	22	ψυ	ψυ	ADP
cana-5994	18	23	=	=	SYM
cana-5994	18	24	0	0	PROPN
cana-5994	18	25	,	,	PUNCT
cana-5994	18	26	(	(	PUNCT
cana-5994	18	27	1.2	1.2	NUM
cana-5994	18	28	)	)	PUNCT
cana-5994	18	29	where	where	SCONJ
cana-5994	18	30	ftυ(ψ	ftυ(ψ	NOUN
cana-5994	18	31	;	;	PUNCT
cana-5994	18	32	η	η	X
cana-5994	18	33	)	)	PUNCT
cana-5994	18	34	is	be	AUX
cana-5994	18	35	the	the	DET
cana-5994	18	36	frobenius	frobenius	ADJ
cana-5994	18	37	-	-	PUNCT
cana-5994	18	38	tangent	tangent	NOUN
cana-5994	18	39	polynomials	polynomial	NOUN
cana-5994	18	40	are	be	AUX
cana-5994	18	41	as	as	SCONJ
cana-5994	18	42	follows	follow	VERB
cana-5994	18	43	∞∑	∞∑	NUM
cana-5994	18	44	υ=0	υ=0	X
cana-5994	18	45	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	18	46	;	;	PUNCT
cana-5994	18	47	η	η	NOUN
cana-5994	18	48	)	)	PUNCT
cana-5994	18	49	φυ	φυ	ADP
cana-5994	18	50	υ	υ	NOUN
cana-5994	18	51	!	!	PUNCT
cana-5994	18	52	=	=	SYM
cana-5994	19	1	1−	1−	NUM
cana-5994	19	2	η	η	X
cana-5994	19	3	e(1−η)φ	e(1−η)φ	ADP
cana-5994	19	4	−	−	PROPN
cana-5994	19	5	η	η	PROPN
cana-5994	19	6	eψφ	eψφ	PROPN
cana-5994	19	7	.	.	PUNCT
cana-5994	20	1	(	(	PUNCT
cana-5994	20	2	1.3	1.3	NUM
cana-5994	20	3	)	)	PUNCT
cana-5994	20	4	the	the	DET
cana-5994	20	5	corresponding	correspond	VERB
cana-5994	20	6	frobenius	frobenius	NOUN
cana-5994	20	7	-	-	PUNCT
cana-5994	20	8	tangent	tangent	NOUN
cana-5994	20	9	numbers	number	NOUN
cana-5994	20	10	have	have	AUX
cana-5994	20	11	also	also	ADV
cana-5994	20	12	been	be	AUX
cana-5994	20	13	derived	derive	VERB
cana-5994	20	14	by	by	ADP
cana-5994	20	15	ftυ(η	ftυ(η	NOUN
cana-5994	20	16	)	)	PUNCT
cana-5994	20	17	=	=	NOUN
cana-5994	20	18	ftυ(0	ftυ(0	NOUN
cana-5994	20	19	;	;	PUNCT
cana-5994	20	20	η	η	PROPN
cana-5994	20	21	)	)	PUNCT
cana-5994	20	22	.	.	PUNCT
cana-5994	21	1	2010	2010	NUM
cana-5994	21	2	mathematics	mathematic	NOUN
cana-5994	21	3	subject	subject	NOUN
cana-5994	21	4	classification	classification	NOUN
cana-5994	21	5	.	.	PUNCT
cana-5994	22	1	81p15	81p15	NUM
cana-5994	22	2	,	,	PUNCT
cana-5994	22	3	11b83	11b83	NUM
cana-5994	22	4	;	;	PUNCT
cana-5994	22	5	33b10	33b10	NUM
cana-5994	22	6	;	;	PUNCT
cana-5994	22	7	34a34	34a34	NUM
cana-5994	22	8	.	.	PUNCT
cana-5994	23	1	key	key	ADJ
cana-5994	23	2	words	word	NOUN
cana-5994	23	3	and	and	CCONJ
cana-5994	23	4	phrases	phrase	NOUN
cana-5994	23	5	.	.	PUNCT
cana-5994	24	1	q	q	X
cana-5994	24	2	-	-	PUNCT
cana-5994	24	3	numbers	number	NOUN
cana-5994	24	4	,	,	PUNCT
cana-5994	24	5	q	q	ADJ
cana-5994	24	6	-	-	NOUN
cana-5994	24	7	derivative	derivative	ADJ
cana-5994	24	8	,	,	PUNCT
cana-5994	24	9	q	q	ADJ
cana-5994	24	10	-	-	PUNCT
cana-5994	24	11	frobenius	frobenius	ADJ
cana-5994	24	12	-	-	PUNCT
cana-5994	24	13	tangent	tangent	NOUN
cana-5994	24	14	polynomials	polynomial	NOUN
cana-5994	24	15	,	,	PUNCT
cana-5994	24	16	differential	differential	ADJ
cana-5994	24	17	equation	equation	NOUN
cana-5994	24	18	.	.	PUNCT
cana-5994	25	1	communications	communication	NOUN
cana-5994	25	2	on	on	ADP
cana-5994	25	3	applied	apply	VERB
cana-5994	25	4	nonlinear	nonlinear	ADJ
cana-5994	25	5	analysis	analysis	NOUN
cana-5994	25	6	issn	issn	NOUN
cana-5994	25	7	:	:	PUNCT
cana-5994	25	8	1074	1074	NUM
cana-5994	25	9	-	-	PUNCT
cana-5994	25	10	133x	133x	NUM
cana-5994	25	11	vol	vol	NOUN
cana-5994	25	12	32	32	NUM
cana-5994	25	13	no	no	NOUN
cana-5994	25	14	.	.	PUNCT
cana-5994	26	1	9s	9s	NUM
cana-5994	26	2	(	(	PUNCT
cana-5994	26	3	2025	2025	NUM
cana-5994	26	4	)	)	PUNCT
cana-5994	26	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	26	6	3218	3218	NUM
cana-5994	26	7	article	article	NOUN
cana-5994	26	8	history	history	NOUN
cana-5994	26	9	:	:	PUNCT
cana-5994	26	10	received	receive	VERB
cana-5994	26	11	:	:	PUNCT
cana-5994	26	12	21	21	NUM
cana-5994	26	13	-	-	SYM
cana-5994	26	14	02	02	NUM
cana-5994	26	15	-	-	PUNCT
cana-5994	26	16	2025	2025	NUM
cana-5994	26	17	revised	revise	VERB
cana-5994	26	18	:	:	PUNCT
cana-5994	26	19	25	25	NUM
cana-5994	26	20	-	-	PUNCT
cana-5994	26	21	03	03	NUM
cana-5994	26	22	-	-	PUNCT
cana-5994	26	23	2025	2025	NUM
cana-5994	26	24	accepted	accept	VERB
cana-5994	26	25	:	:	PUNCT
cana-5994	26	26	29	29	NUM
cana-5994	26	27	-	-	SYM
cana-5994	26	28	04	04	NUM
cana-5994	26	29	-	-	PUNCT
cana-5994	26	30	2025	2025	NUM
cana-5994	26	31	2	2	NUM
cana-5994	26	32	idrees	idree	NOUN
cana-5994	26	33	ahmad	ahmad	PROPN
cana-5994	26	34	khan	khan	PROPN
cana-5994	26	35	and	and	CCONJ
cana-5994	26	36	sumit	sumit	PROPN
cana-5994	26	37	kumar	kumar	PROPN
cana-5994	26	38	by	by	ADP
cana-5994	26	39	the	the	DET
cana-5994	26	40	above	above	ADJ
cana-5994	26	41	method	method	NOUN
cana-5994	26	42	,	,	PUNCT
cana-5994	26	43	the	the	DET
cana-5994	26	44	first	first	ADJ
cana-5994	26	45	order	order	NOUN
cana-5994	26	46	differential	differential	ADJ
cana-5994	26	47	equation	equation	NOUN
cana-5994	26	48	of	of	ADP
cana-5994	26	49	q	q	ADJ
cana-5994	26	50	-	-	PUNCT
cana-5994	26	51	bernoulli	bernoulli	NOUN
cana-5994	26	52	differential	differential	NOUN
cana-5994	26	53	can	can	AUX
cana-5994	26	54	written	write	VERB
cana-5994	26	55	as	as	ADP
cana-5994	26	56	dqϕ+	dqϕ+	PROPN
cana-5994	26	57	p(ψ)ϕ−	p(ψ)ϕ−	PROPN
cana-5994	26	58	g(ψ)ϕσ	g(ψ)ϕσ	PROPN
cana-5994	26	59	=	=	NOUN
cana-5994	26	60	0	0	NUM
cana-5994	26	61	in	in	ADP
cana-5994	26	62	q	q	NOUN
cana-5994	26	63	-	-	NOUN
cana-5994	26	64	calculus	calculus	NOUN
cana-5994	26	65	.	.	PUNCT
cana-5994	27	1	again	again	ADV
cana-5994	27	2	,	,	PUNCT
cana-5994	27	3	if	if	SCONJ
cana-5994	27	4	σ	σ	PROPN
cana-5994	27	5	=	=	SYM
cana-5994	27	6	0	0	NUM
cana-5994	27	7	,	,	PUNCT
cana-5994	27	8	the	the	DET
cana-5994	27	9	equation	equation	NOUN
cana-5994	27	10	(	(	PUNCT
cana-5994	27	11	1.1	1.1	NUM
cana-5994	27	12	)	)	PUNCT
cana-5994	27	13	will	will	AUX
cana-5994	27	14	give	give	VERB
cana-5994	27	15	the	the	DET
cana-5994	27	16	first	first	ADJ
cana-5994	27	17	order	order	NOUN
cana-5994	27	18	q	q	ADJ
cana-5994	27	19	-	-	PUNCT
cana-5994	27	20	differential	differential	ADJ
cana-5994	27	21	equation	equation	NOUN
cana-5994	27	22	of	of	ADP
cana-5994	27	23	q	q	ADJ
cana-5994	27	24	-	-	PUNCT
cana-5994	27	25	frobenius	frobenius	ADJ
cana-5994	27	26	-	-	PUNCT
cana-5994	27	27	tangent	tangent	NOUN
cana-5994	27	28	polynomials	polynomial	NOUN
cana-5994	27	29	as	as	ADP
cana-5994	27	30	d	d	X
cana-5994	27	31	(	(	PUNCT
cana-5994	27	32	1	1	NUM
cana-5994	27	33	)	)	PUNCT
cana-5994	27	34	q	q	NOUN
cana-5994	27	35	,	,	PUNCT
cana-5994	27	36	ψftυ	ψftυ	NOUN
cana-5994	27	37	,	,	PUNCT
cana-5994	27	38	q(ψ	q(ψ	NUM
cana-5994	27	39	;	;	PUNCT
cana-5994	27	40	η	η	X
cana-5994	27	41	)	)	PUNCT
cana-5994	28	1	+	+	CCONJ
cana-5994	28	2	(	(	PUNCT
cana-5994	28	3	1−	1−	NUM
cana-5994	28	4	η)−1	η)−1	NOUN
cana-5994	28	5	(	(	PUNCT
cana-5994	28	6	ft0,q(ψ	ft0,q(ψ	X
cana-5994	28	7	;	;	PUNCT
cana-5994	28	8	η	η	X
cana-5994	28	9	)	)	PUNCT
cana-5994	28	10	+	+	NUM
cana-5994	28	11	ftυ	ftυ	NOUN
cana-5994	28	12	,	,	PUNCT
cana-5994	28	13	q(ψ	q(ψ	NOUN
cana-5994	28	14	;	;	PUNCT
cana-5994	28	15	η))−	η))−	NOUN
cana-5994	28	16	ψυ	ψυ	ADP
cana-5994	28	17	=	=	SYM
cana-5994	28	18	0	0	PROPN
cana-5994	28	19	,	,	PUNCT
cana-5994	28	20	(	(	PUNCT
cana-5994	28	21	1.4	1.4	NUM
cana-5994	28	22	)	)	PUNCT
cana-5994	28	23	and	and	CCONJ
cana-5994	28	24	dq	dq	PROPN
cana-5994	28	25	is	be	AUX
cana-5994	28	26	called	call	VERB
cana-5994	28	27	the	the	DET
cana-5994	28	28	q	q	NOUN
cana-5994	28	29	-	-	ADJ
cana-5994	28	30	derivative	derivative	ADJ
cana-5994	28	31	and	and	CCONJ
cana-5994	28	32	ftυ	ftυ	NOUN
cana-5994	28	33	,	,	PUNCT
cana-5994	28	34	q(ψ	q(ψ	NUM
cana-5994	28	35	;	;	PUNCT
cana-5994	28	36	η	η	X
cana-5994	28	37	)	)	PUNCT
cana-5994	28	38	is	be	AUX
cana-5994	28	39	the	the	DET
cana-5994	28	40	q	q	ADJ
cana-5994	28	41	-	-	PUNCT
cana-5994	28	42	frobenius	frobenius	ADJ
cana-5994	28	43	-	-	PUNCT
cana-5994	28	44	tangent	tangent	NOUN
cana-5994	28	45	polynomials	polynomial	NOUN
cana-5994	28	46	.	.	PUNCT
cana-5994	29	1	let	let	VERB
cana-5994	29	2	υ	υ	PRON
cana-5994	29	3	∈	∈	PROPN
cana-5994	29	4	z0	z0	PROPN
cana-5994	29	5	and	and	CCONJ
cana-5994	29	6	η	η	PROPN
cana-5994	29	7	∈	∈	PROPN
cana-5994	29	8	z	z	PROPN
cana-5994	29	9	,	,	PUNCT
cana-5994	29	10	the	the	DET
cana-5994	29	11	q	q	ADJ
cana-5994	29	12	-	-	PUNCT
cana-5994	29	13	frobenius	frobenius	ADJ
cana-5994	29	14	-	-	PUNCT
cana-5994	29	15	tangent	tangent	NOUN
cana-5994	29	16	polynomials	polynomial	NOUN
cana-5994	29	17	are	be	AUX
cana-5994	29	18	defined	define	VERB
cana-5994	29	19	by	by	ADP
cana-5994	29	20	(	(	PUNCT
cana-5994	29	21	see	see	VERB
cana-5994	29	22	[	[	X
cana-5994	29	23	7	7	NUM
cana-5994	29	24	]	]	SYM
cana-5994	29	25	)	)	PUNCT
cana-5994	30	1	1−	1−	NUM
cana-5994	30	2	η	η	X
cana-5994	30	3	eq((1−	eq((1−	X
cana-5994	30	4	η)φ)−	η)φ)−	PRON
cana-5994	30	5	η	η	X
cana-5994	30	6	eq(ψφ	eq(ψφ	PROPN
cana-5994	30	7	)	)	PUNCT
cana-5994	30	8	=	=	PUNCT
cana-5994	31	1	∞∑	∞∑	NUM
cana-5994	31	2	υ=0	υ=0	NUM
cana-5994	31	3	ftυ	ftυ	NOUN
cana-5994	31	4	,	,	PUNCT
cana-5994	31	5	q(ψ	q(ψ	NUM
cana-5994	31	6	;	;	PUNCT
cana-5994	31	7	η	η	X
cana-5994	31	8	)	)	PUNCT
cana-5994	31	9	φυ	φυ	ADP
cana-5994	32	1	[	[	X
cana-5994	33	1	υ]q	υ]q	NOUN
cana-5994	33	2	!	!	PUNCT
cana-5994	33	3	.	.	PUNCT
cana-5994	34	1	(	(	PUNCT
cana-5994	34	2	1.5	1.5	NUM
cana-5994	34	3	)	)	PUNCT
cana-5994	34	4	the	the	DET
cana-5994	34	5	corresponding	corresponding	ADJ
cana-5994	34	6	q	q	ADJ
cana-5994	34	7	-	-	PUNCT
cana-5994	34	8	frobenius	frobenius	ADJ
cana-5994	34	9	-	-	PUNCT
cana-5994	34	10	tangent	tangent	NOUN
cana-5994	34	11	numbers	number	NOUN
cana-5994	34	12	have	have	AUX
cana-5994	34	13	also	also	ADV
cana-5994	34	14	been	be	AUX
cana-5994	34	15	derived	derive	VERB
cana-5994	34	16	by	by	ADP
cana-5994	34	17	ftυ	ftυ	NOUN
cana-5994	34	18	,	,	PUNCT
cana-5994	34	19	q(η	q(η	PROPN
cana-5994	34	20	)	)	PUNCT
cana-5994	34	21	=	=	SYM
cana-5994	34	22	ftυ	ftυ	NOUN
cana-5994	34	23	,	,	PUNCT
cana-5994	34	24	q(0	q(0	PROPN
cana-5994	34	25	;	;	PUNCT
cana-5994	34	26	η	η	PROPN
cana-5994	34	27	)	)	PUNCT
cana-5994	34	28	.	.	PUNCT
cana-5994	35	1	it	it	PRON
cana-5994	35	2	is	be	AUX
cana-5994	35	3	worthy	worthy	ADJ
cana-5994	35	4	note	note	NOUN
cana-5994	35	5	that	that	SCONJ
cana-5994	35	6	if	if	SCONJ
cana-5994	35	7	q	q	NOUN
cana-5994	35	8	→	→	SYM
cana-5994	35	9	1	1	NUM
cana-5994	35	10	then	then	ADV
cana-5994	35	11	(	(	PUNCT
cana-5994	35	12	1.5	1.5	NUM
cana-5994	35	13	)	)	PUNCT
cana-5994	35	14	becomes	become	VERB
cana-5994	35	15	(	(	PUNCT
cana-5994	35	16	1.2	1.2	NUM
cana-5994	35	17	)	)	PUNCT
cana-5994	35	18	.	.	PUNCT
cana-5994	36	1	the	the	DET
cana-5994	36	2	main	main	ADJ
cana-5994	36	3	purpose	purpose	NOUN
cana-5994	36	4	of	of	ADP
cana-5994	36	5	this	this	DET
cana-5994	36	6	paper	paper	NOUN
cana-5994	36	7	is	be	AUX
cana-5994	36	8	to	to	PART
cana-5994	36	9	establish	establish	VERB
cana-5994	36	10	higher	high	ADJ
cana-5994	36	11	-	-	PUNCT
cana-5994	36	12	order	order	NOUN
cana-5994	36	13	differential	differential	ADJ
cana-5994	36	14	equations	equation	NOUN
cana-5994	36	15	for	for	ADP
cana-5994	36	16	the	the	DET
cana-5994	36	17	q	q	ADJ
cana-5994	36	18	-	-	PUNCT
cana-5994	36	19	frobenius	frobenius	ADJ
cana-5994	36	20	-	-	PUNCT
cana-5994	36	21	tangent	tangent	NOUN
cana-5994	36	22	polynomials	polynomial	NOUN
cana-5994	36	23	as	as	SCONJ
cana-5994	36	24	defined	define	VERB
cana-5994	36	25	by	by	ADP
cana-5994	36	26	(	(	PUNCT
cana-5994	36	27	1.5	1.5	NUM
cana-5994	36	28	)	)	PUNCT
cana-5994	36	29	.	.	PUNCT
cana-5994	37	1	building	build	VERB
cana-5994	37	2	on	on	ADP
cana-5994	37	3	this	this	DET
cana-5994	37	4	concept	concept	NOUN
cana-5994	37	5	,	,	PUNCT
cana-5994	37	6	we	we	PRON
cana-5994	37	7	will	will	AUX
cana-5994	37	8	apply	apply	VERB
cana-5994	37	9	the	the	DET
cana-5994	37	10	theory	theory	NOUN
cana-5994	37	11	of	of	ADP
cana-5994	37	12	q	q	NOUN
cana-5994	37	13	-	-	NOUN
cana-5994	37	14	calculus	calculus	NOUN
cana-5994	37	15	throughout	throughout	ADP
cana-5994	37	16	the	the	DET
cana-5994	37	17	paper	paper	NOUN
cana-5994	37	18	.	.	PUNCT
cana-5994	38	1	let	let	VERB
cana-5994	38	2	us	we	PRON
cana-5994	38	3	begin	begin	VERB
cana-5994	38	4	by	by	ADP
cana-5994	38	5	introducing	introduce	VERB
cana-5994	38	6	some	some	DET
cana-5994	38	7	definitions	definition	NOUN
cana-5994	38	8	from	from	ADP
cana-5994	38	9	q	q	ADJ
cana-5994	38	10	-	-	PUNCT
cana-5994	38	11	calculus	calculus	NOUN
cana-5994	38	12	theory	theory	NOUN
cana-5994	38	13	.	.	PUNCT
cana-5994	39	1	the	the	DET
cana-5994	39	2	shifted	shift	VERB
cana-5994	39	3	factorial	factorial	NOUN
cana-5994	39	4	(	(	PUNCT
cana-5994	39	5	ϕ)υ	ϕ)υ	X
cana-5994	39	6	in	in	ADP
cana-5994	39	7	term	term	NOUN
cana-5994	39	8	of	of	ADP
cana-5994	39	9	q	q	NOUN
cana-5994	39	10	-	-	PUNCT
cana-5994	39	11	analogue	analogue	NOUN
cana-5994	39	12	is	be	AUX
cana-5994	39	13	given	give	VERB
cana-5994	39	14	by	by	ADP
cana-5994	39	15	[	[	X
cana-5994	39	16	3	3	NUM
cana-5994	39	17	,	,	PUNCT
cana-5994	39	18	4	4	NUM
cana-5994	39	19	]	]	PUNCT
cana-5994	39	20	(	(	PUNCT
cana-5994	39	21	ϕ	ϕ	NOUN
cana-5994	39	22	;	;	PUNCT
cana-5994	39	23	q)0	q)0	PROPN
cana-5994	39	24	=	=	SYM
cana-5994	39	25	1	1	NUM
cana-5994	39	26	,	,	PUNCT
cana-5994	39	27	(	(	PUNCT
cana-5994	39	28	ϕ	ϕ	NOUN
cana-5994	39	29	;	;	PUNCT
cana-5994	39	30	q)υ	q)υ	X
cana-5994	39	31	=	=	SYM
cana-5994	39	32	υ−1∏	υ−1∏	VERB
cana-5994	39	33	σ=0	σ=0	NOUN
cana-5994	39	34	(	(	PUNCT
cana-5994	39	35	1−	1−	NUM
cana-5994	39	36	qσϕ	qσϕ	NOUN
cana-5994	39	37	)	)	PUNCT
cana-5994	39	38	,	,	PUNCT
cana-5994	39	39	υ	υ	PROPN
cana-5994	39	40	∈	∈	PROPN
cana-5994	39	41	n.	n.	NOUN
cana-5994	39	42	the	the	DET
cana-5994	39	43	factorial	factorial	ADJ
cana-5994	39	44	function	function	NOUN
cana-5994	39	45	in	in	ADP
cana-5994	39	46	q	q	ADJ
cana-5994	39	47	-	-	PUNCT
cana-5994	39	48	calculus	calculus	NOUN
cana-5994	39	49	theory	theory	NOUN
cana-5994	39	50	given	give	VERB
cana-5994	39	51	by	by	ADP
cana-5994	39	52	[	[	X
cana-5994	39	53	ϕ]q	ϕ]q	X
cana-5994	39	54	=	=	SYM
cana-5994	39	55	1−	1−	NUM
cana-5994	39	56	qϕ	qϕ	NOUN
cana-5994	39	57	1−	1−	NUM
cana-5994	39	58	q	q	NOUN
cana-5994	39	59	,	,	PUNCT
cana-5994	39	60	q	q	NOUN
cana-5994	39	61	∈	∈	NOUN
cana-5994	39	62	c−	c−	NOUN
cana-5994	39	63	{	{	PUNCT
cana-5994	39	64	1};ϕ	1};ϕ	NUM
cana-5994	39	65	∈	∈	NOUN
cana-5994	39	66	c	c	NOUN
cana-5994	39	67	,	,	PUNCT
cana-5994	39	68	[	[	X
cana-5994	39	69	υ]q	υ]q	NOUN
cana-5994	39	70	!	!	PUNCT
cana-5994	39	71	=	=	PUNCT
cana-5994	40	1	υ∏	υ∏	X
cana-5994	40	2	σ=1	σ=1	PUNCT
cana-5994	41	1	[	[	X
cana-5994	41	2	σ]q	σ]q	X
cana-5994	41	3	=	=	PUNCT
cana-5994	42	1	[	[	X
cana-5994	42	2	1]q[2]q	1]q[2]q	NUM
cana-5994	42	3	·	·	PUNCT
cana-5994	42	4	·	·	PUNCT
cana-5994	42	5	·	·	PUNCT
cana-5994	43	1	[	[	X
cana-5994	43	2	υ]q	υ]q	NOUN
cana-5994	43	3	=	=	PUNCT
cana-5994	43	4	(	(	PUNCT
cana-5994	43	5	q	q	NOUN
cana-5994	43	6	;	;	PUNCT
cana-5994	43	7	q)υ	q)υ	X
cana-5994	43	8	(	(	PUNCT
cana-5994	43	9	1−	1−	NUM
cana-5994	43	10	q)υ	q)υ	NOUN
cana-5994	43	11	,	,	PUNCT
cana-5994	43	12	q	q	PROPN
cana-5994	43	13	̸=	̸=	PROPN
cana-5994	43	14	1	1	NUM
cana-5994	43	15	;	;	PUNCT
cana-5994	43	16	υ	υ	PROPN
cana-5994	43	17	∈	∈	PROPN
cana-5994	43	18	n	n	CCONJ
cana-5994	43	19	,	,	PUNCT
cana-5994	43	20	[	[	X
cana-5994	43	21	0]q	0]q	NOUN
cana-5994	43	22	!	!	PUNCT
cana-5994	43	23	=	=	SYM
cana-5994	43	24	1	1	NUM
cana-5994	43	25	,	,	PUNCT
cana-5994	43	26	q	q	PUNCT
cana-5994	43	27	∈	∈	PROPN
cana-5994	43	28	c	c	NOUN
cana-5994	43	29	;	;	PUNCT
cana-5994	43	30	0	0	PUNCT
cana-5994	43	31	<	<	X
cana-5994	43	32	q	q	X
cana-5994	43	33	<	<	X
cana-5994	43	34	1	1	NUM
cana-5994	43	35	.	.	PUNCT
cana-5994	44	1	the	the	DET
cana-5994	44	2	definition	definition	NOUN
cana-5994	44	3	q	q	ADJ
cana-5994	44	4	-	-	PUNCT
cana-5994	44	5	binomial	binomial	ADJ
cana-5994	44	6	coefficient	coefficient	NOUN
cana-5994	44	7	of	of	ADP
cana-5994	44	8	gauss	gauss	NOUN
cana-5994	44	9	(	(	PUNCT
cana-5994	44	10	υ	υ	NOUN
cana-5994	44	11	ψ	ψ	NOUN
cana-5994	44	12	)	)	PUNCT
cana-5994	44	13	q	q	PUNCT
cana-5994	44	14	is	be	AUX
cana-5994	44	15	given	give	VERB
cana-5994	44	16	by	by	ADP
cana-5994	44	17	(	(	PUNCT
cana-5994	44	18	υ	υ	NOUN
cana-5994	44	19	ψ	ψ	NOUN
cana-5994	44	20	)	)	PUNCT
cana-5994	44	21	q	q	NOUN
cana-5994	45	1	=	=	PUNCT
cana-5994	46	1	[	[	X
cana-5994	46	2	υ]q	υ]q	NOUN
cana-5994	46	3	!	!	PUNCT
cana-5994	47	1	[	[	X
cana-5994	47	2	ν]q![υ	ν]q![υ	NOUN
cana-5994	47	3	−	−	NOUN
cana-5994	47	4	ν]q	ν]q	NOUN
cana-5994	47	5	!	!	PUNCT
cana-5994	47	6	=	=	PUNCT
cana-5994	48	1	(	(	PUNCT
cana-5994	48	2	q	q	NOUN
cana-5994	48	3	;	;	PUNCT
cana-5994	48	4	q)υ	q)υ	X
cana-5994	48	5	(	(	PUNCT
cana-5994	48	6	q	q	NOUN
cana-5994	48	7	;	;	PUNCT
cana-5994	48	8	q)ν(q	q)ν(q	NOUN
cana-5994	48	9	;	;	PUNCT
cana-5994	48	10	q)υ−ν	q)υ−ν	NOUN
cana-5994	48	11	,	,	PUNCT
cana-5994	48	12	ν	ν	X
cana-5994	48	13	=	=	SYM
cana-5994	48	14	0	0	NUM
cana-5994	48	15	,	,	PUNCT
cana-5994	48	16	1	1	NUM
cana-5994	48	17	,	,	PUNCT
cana-5994	48	18	·	·	PUNCT
cana-5994	48	19	·	·	PUNCT
cana-5994	48	20	·	·	PUNCT
cana-5994	48	21	,	,	PUNCT
cana-5994	48	22	υ	υ	X
cana-5994	48	23	.	.	PUNCT
cana-5994	49	1	the	the	DET
cana-5994	49	2	function	function	NOUN
cana-5994	49	3	(	(	PUNCT
cana-5994	49	4	ψ	ψ	X
cana-5994	49	5	+	+	X
cana-5994	49	6	ϕ)υq	ϕ)υq	PROPN
cana-5994	49	7	is	be	AUX
cana-5994	49	8	given	give	VERB
cana-5994	49	9	by	by	ADP
cana-5994	49	10	(	(	PUNCT
cana-5994	49	11	ψ	ψ	X
cana-5994	49	12	+	+	X
cana-5994	49	13	ϕ)υq	ϕ)υq	PROPN
cana-5994	49	14	=	=	SYM
cana-5994	49	15	υ∑	υ∑	ADJ
cana-5994	49	16	ν=0	ν=0	PRON
cana-5994	49	17	(	(	PUNCT
cana-5994	49	18	υ	υ	NOUN
cana-5994	49	19	ν	ν	NOUN
cana-5994	49	20	)	)	PUNCT
cana-5994	49	21	q	q	PROPN
cana-5994	49	22	qν(ν−1)/2ψυ−νϕν	qν(ν−1)/2ψυ−νϕν	NOUN
cana-5994	49	23	,	,	PUNCT
cana-5994	49	24	υ	υ	PROPN
cana-5994	49	25	∈	∈	PROPN
cana-5994	49	26	n0	n0	PROPN
cana-5994	49	27	.	.	PUNCT
cana-5994	50	1	(	(	PUNCT
cana-5994	50	2	1.6	1.6	NUM
cana-5994	50	3	)	)	PUNCT
cana-5994	50	4	the	the	DET
cana-5994	50	5	definition	definition	NOUN
cana-5994	50	6	of	of	ADP
cana-5994	50	7	exponential	exponential	ADJ
cana-5994	50	8	function	function	NOUN
cana-5994	50	9	in	in	ADP
cana-5994	50	10	q	q	ADJ
cana-5994	50	11	-	-	PUNCT
cana-5994	50	12	calculus	calculus	NOUN
cana-5994	50	13	theory	theory	NOUN
cana-5994	50	14	is	be	AUX
cana-5994	50	15	given	give	VERB
cana-5994	50	16	by	by	ADP
cana-5994	50	17	eq(ψ	eq(ψ	NOUN
cana-5994	50	18	)	)	PUNCT
cana-5994	50	19	=	=	PUNCT
cana-5994	51	1	∞∑	∞∑	NUM
cana-5994	51	2	υ=0	υ=0	PUNCT
cana-5994	51	3	ψυ	ψυ	ADP
cana-5994	52	1	[	[	X
cana-5994	52	2	υ]q	υ]q	NOUN
cana-5994	52	3	!	!	PUNCT
cana-5994	53	1	=	=	SYM
cana-5994	53	2	1	1	NUM
cana-5994	53	3	(	(	PUNCT
cana-5994	53	4	(	(	PUNCT
cana-5994	53	5	1−	1−	NUM
cana-5994	53	6	q)ψ	q)ψ	X
cana-5994	53	7	;	;	PUNCT
cana-5994	53	8	q)∞	q)∞	INTJ
cana-5994	53	9	,	,	PUNCT
cana-5994	53	10	0	0	NUM
cana-5994	54	1	<	<	X
cana-5994	54	2	|	|	ADV
cana-5994	54	3	q	q	NOUN
cana-5994	54	4	|	|	X
cana-5994	54	5	<	<	X
cana-5994	54	6	1	1	NUM
cana-5994	54	7	;	;	PUNCT
cana-5994	54	8	|	|	ADV
cana-5994	54	9	ψ	ψ	X
cana-5994	54	10	|<|	|<|	NOUN
cana-5994	54	11	1−	1−	NUM
cana-5994	54	12	q	q	NOUN
cana-5994	54	13	|−1	|−1	NUM
cana-5994	54	14	,	,	PUNCT
cana-5994	54	15	(	(	PUNCT
cana-5994	54	16	1.7	1.7	NUM
cana-5994	54	17	)	)	PUNCT
cana-5994	54	18	for	for	ADP
cana-5994	54	19	ψ	ψ	X
cana-5994	54	20	̸=	̸=	PROPN
cana-5994	54	21	0	0	NUM
cana-5994	54	22	,	,	PUNCT
cana-5994	54	23	the	the	DET
cana-5994	54	24	definition	definition	NOUN
cana-5994	54	25	of	of	ADP
cana-5994	54	26	q	q	ADJ
cana-5994	54	27	-	-	ADJ
cana-5994	54	28	derivative	derivative	ADJ
cana-5994	54	29	dq	dq	NOUN
cana-5994	54	30	,	,	PUNCT
cana-5994	54	31	ψf(ψ	ψf(ψ	NUM
cana-5994	54	32	)	)	PUNCT
cana-5994	54	33	as	as	ADP
cana-5994	54	34	dq	dq	PROPN
cana-5994	54	35	,	,	PUNCT
cana-5994	54	36	ψf(ψ	ψf(ψ	NOUN
cana-5994	54	37	)	)	PUNCT
cana-5994	54	38	=	=	SYM
cana-5994	54	39	dqf(ψ	dqf(ψ	NOUN
cana-5994	54	40	)	)	PUNCT
cana-5994	54	41	=	=	SYM
cana-5994	54	42	f(ψ)−	f(ψ)−	PROPN
cana-5994	54	43	f(qψ	f(qψ	PROPN
cana-5994	54	44	)	)	PUNCT
cana-5994	54	45	(	(	PUNCT
cana-5994	54	46	1−	1−	NUM
cana-5994	54	47	q)ψ	q)ψ	NUM
cana-5994	54	48	,	,	PUNCT
cana-5994	54	49	(	(	PUNCT
cana-5994	54	50	1.8	1.8	NUM
cana-5994	54	51	)	)	PUNCT
cana-5994	54	52	and	and	CCONJ
cana-5994	54	53	dqf(0	dqf(0	NOUN
cana-5994	54	54	)	)	PUNCT
cana-5994	55	1	=	=	PUNCT
cana-5994	56	1	f	f	X
cana-5994	56	2	′	′	NUM
cana-5994	57	1	(	(	PUNCT
cana-5994	57	2	0	0	NUM
cana-5994	57	3	)	)	PUNCT
cana-5994	57	4	.	.	PUNCT
cana-5994	58	1	communications	communication	NOUN
cana-5994	58	2	on	on	ADP
cana-5994	58	3	applied	apply	VERB
cana-5994	58	4	nonlinear	nonlinear	ADJ
cana-5994	58	5	analysis	analysis	NOUN
cana-5994	58	6	issn	issn	NOUN
cana-5994	58	7	:	:	PUNCT
cana-5994	58	8	1074	1074	NUM
cana-5994	58	9	-	-	PUNCT
cana-5994	58	10	133x	133x	NUM
cana-5994	58	11	vol	vol	NOUN
cana-5994	58	12	32	32	NUM
cana-5994	58	13	no	no	NOUN
cana-5994	58	14	.	.	PUNCT
cana-5994	59	1	9s	9s	NUM
cana-5994	59	2	(	(	PUNCT
cana-5994	59	3	2025	2025	NUM
cana-5994	59	4	)	)	PUNCT
cana-5994	60	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	60	2	3219	3219	NUM
cana-5994	61	1	some	some	DET
cana-5994	61	2	properties	property	NOUN
cana-5994	61	3	of	of	ADP
cana-5994	61	4	differential	differential	ADJ
cana-5994	61	5	equations	equation	NOUN
cana-5994	61	6	of	of	ADP
cana-5994	61	7	higher	high	ADJ
cana-5994	61	8	-	-	PUNCT
cana-5994	61	9	order	order	NOUN
cana-5994	61	10	3	3	NUM
cana-5994	61	11	here	here	ADV
cana-5994	61	12	the	the	DET
cana-5994	61	13	function	function	NOUN
cana-5994	61	14	f	f	PROPN
cana-5994	61	15	is	be	AUX
cana-5994	61	16	differentiable	differentiable	ADJ
cana-5994	61	17	at	at	ADP
cana-5994	61	18	zero	zero	NUM
cana-5994	61	19	,	,	PUNCT
cana-5994	61	20	and	and	CCONJ
cana-5994	61	21	it	it	PRON
cana-5994	61	22	is	be	AUX
cana-5994	61	23	obvious	obvious	ADJ
cana-5994	61	24	thatdqψ	thatdqψ	ADP
cana-5994	61	25	υ	υ	NOUN
cana-5994	61	26	=	=	PUNCT
cana-5994	62	1	[	[	X
cana-5994	62	2	υ]qψ	υ]qψ	NUM
cana-5994	62	3	υ−1	υ−1	PROPN
cana-5994	62	4	.	.	PUNCT
cana-5994	63	1	let	let	VERB
cana-5994	63	2	us	we	PRON
cana-5994	63	3	point	point	VERB
cana-5994	63	4	out	out	ADP
cana-5994	63	5	that	that	SCONJ
cana-5994	63	6	d	d	NOUN
cana-5994	63	7	(	(	PUNCT
cana-5994	63	8	ν	ν	NOUN
cana-5994	63	9	)	)	PUNCT
cana-5994	63	10	q	q	NOUN
cana-5994	63	11	,	,	PUNCT
cana-5994	63	12	ψf(ψ	ψf(ψ	NOUN
cana-5994	63	13	)	)	PUNCT
cana-5994	63	14	converges	converge	NOUN
cana-5994	63	15	to	to	ADP
cana-5994	63	16	f	f	PROPN
cana-5994	63	17	(	(	PUNCT
cana-5994	63	18	ν)(ψ	ν)(ψ	PROPN
cana-5994	63	19	)	)	PUNCT
cana-5994	63	20	as	as	SCONJ
cana-5994	63	21	q	q	PROPN
cana-5994	63	22	goes	go	VERB
cana-5994	63	23	to	to	ADP
cana-5994	63	24	1	1	NUM
cana-5994	63	25	.	.	PUNCT
cana-5994	63	26	by(1.8	by(1.8	NOUN
cana-5994	63	27	)	)	PUNCT
cana-5994	63	28	,	,	PUNCT
cana-5994	63	29	the	the	DET
cana-5994	63	30	some	some	DET
cana-5994	63	31	formulae	formulae	NOUN
cana-5994	63	32	of	of	ADP
cana-5994	63	33	q	q	NOUN
cana-5994	63	34	-	-	NOUN
cana-5994	63	35	derivative	derivative	ADJ
cana-5994	63	36	.	.	PUNCT
cana-5994	64	1	(	(	PUNCT
cana-5994	64	2	i	i	NOUN
cana-5994	64	3	)	)	PUNCT
cana-5994	64	4	dq	dq	PROPN
cana-5994	64	5	(	(	PUNCT
cana-5994	64	6	f(ψ)g(ψ	f(ψ)g(ψ	NOUN
cana-5994	64	7	)	)	PUNCT
cana-5994	64	8	)	)	PUNCT
cana-5994	65	1	=	=	SYM
cana-5994	65	2	q(ψ)dqf(ψ	q(ψ)dqf(ψ	NOUN
cana-5994	65	3	)	)	PUNCT
cana-5994	65	4	+	+	CCONJ
cana-5994	65	5	f(qψ)dqg(ψ	f(qψ)dqg(ψ	X
cana-5994	65	6	)	)	PUNCT
cana-5994	65	7	=	=	SYM
cana-5994	65	8	f(ψ)dqg(ψ	f(ψ)dqg(ψ	PROPN
cana-5994	65	9	)	)	PUNCT
cana-5994	65	10	+	+	NUM
cana-5994	65	11	g(qψ)dqf(ψ	g(qψ)dqf(ψ	NOUN
cana-5994	65	12	)	)	PUNCT
cana-5994	65	13	,	,	PUNCT
cana-5994	65	14	(	(	PUNCT
cana-5994	65	15	ii	ii	NOUN
cana-5994	65	16	)	)	PUNCT
cana-5994	65	17	dq	dq	PROPN
cana-5994	65	18	(	(	PUNCT
cana-5994	65	19	f(ψ	f(ψ	NOUN
cana-5994	65	20	)	)	PUNCT
cana-5994	65	21	g(ψ	g(ψ	PROPN
cana-5994	65	22	)	)	PUNCT
cana-5994	65	23	)	)	PUNCT
cana-5994	66	1	=	=	PUNCT
cana-5994	66	2	g(qψ)dqf(ψ)−	g(qψ)dqf(ψ)−	X
cana-5994	66	3	f(qψ)dqg(ψ	f(qψ)dqg(ψ	PROPN
cana-5994	66	4	)	)	PUNCT
cana-5994	66	5	g(ψ)g(qψ	g(ψ)g(qψ	NOUN
cana-5994	66	6	)	)	PUNCT
cana-5994	66	7	=	=	SYM
cana-5994	66	8	g(ψ)dqf(ψ)−	g(ψ)dqf(ψ)−	NOUN
cana-5994	66	9	f(ψ)dqg(ψ	f(ψ)dqg(ψ	PROPN
cana-5994	66	10	)	)	PUNCT
cana-5994	66	11	g(ψ)g(qψ	g(ψ)g(qψ	PROPN
cana-5994	66	12	)	)	PUNCT
cana-5994	66	13	,	,	PUNCT
cana-5994	66	14	(	(	PUNCT
cana-5994	66	15	iii	iii	X
cana-5994	66	16	)	)	PUNCT
cana-5994	66	17	for	for	ADP
cana-5994	66	18	any	any	DET
cana-5994	66	19	constant	constant	ADJ
cana-5994	66	20	a	a	PRON
cana-5994	66	21	and	and	CCONJ
cana-5994	66	22	b	b	NOUN
cana-5994	66	23	,	,	PUNCT
cana-5994	66	24	dq	dq	PROPN
cana-5994	66	25	(	(	PUNCT
cana-5994	66	26	af(ψ	af(ψ	NOUN
cana-5994	66	27	)	)	PUNCT
cana-5994	66	28	+	+	CCONJ
cana-5994	66	29	bg(ψ	bg(ψ	NOUN
cana-5994	66	30	)	)	PUNCT
cana-5994	66	31	)	)	PUNCT
cana-5994	67	1	=	=	SYM
cana-5994	67	2	adqf(ψ	adqf(ψ	NOUN
cana-5994	67	3	)	)	PUNCT
cana-5994	67	4	+	+	NUM
cana-5994	67	5	bdqg(ψ	bdqg(ψ	PROPN
cana-5994	67	6	)	)	PUNCT
cana-5994	67	7	.	.	PUNCT
cana-5994	68	1	the	the	DET
cana-5994	68	2	q	q	ADJ
cana-5994	68	3	-	-	PUNCT
cana-5994	68	4	bernoulli	bernoulli	NOUN
cana-5994	68	5	bυ	bυ	NOUN
cana-5994	68	6	,	,	PUNCT
cana-5994	68	7	q(ψ	q(ψ	NOUN
cana-5994	68	8	)	)	PUNCT
cana-5994	68	9	,	,	PUNCT
cana-5994	68	10	the	the	DET
cana-5994	68	11	q	q	NOUN
cana-5994	68	12	-	-	PUNCT
cana-5994	68	13	euler	euler	NOUN
cana-5994	68	14	eυ	eυ	PROPN
cana-5994	68	15	,	,	PUNCT
cana-5994	68	16	q(ψ	q(ψ	NOUN
cana-5994	68	17	)	)	PUNCT
cana-5994	68	18	and	and	CCONJ
cana-5994	68	19	q	q	ADJ
cana-5994	68	20	-	-	PUNCT
cana-5994	68	21	genocchi	genocchi	PROPN
cana-5994	68	22	polynomials	polynomial	NOUN
cana-5994	68	23	gυ	gυ	VERB
cana-5994	68	24	,	,	PUNCT
cana-5994	68	25	q(ψ	q(ψ	NUM
cana-5994	68	26	)	)	PUNCT
cana-5994	69	1	are	be	AUX
cana-5994	69	2	defined	define	VERB
cana-5994	69	3	by	by	ADP
cana-5994	69	4	(	(	PUNCT
cana-5994	69	5	see	see	VERB
cana-5994	69	6	[	[	X
cana-5994	69	7	11	11	NUM
cana-5994	69	8	,	,	PUNCT
cana-5994	69	9	12	12	NUM
cana-5994	69	10	,	,	PUNCT
cana-5994	69	11	14	14	NUM
cana-5994	69	12	]	]	PUNCT
cana-5994	69	13	):	):	PUNCT
cana-5994	69	14	φ	φ	PROPN
cana-5994	69	15	eq(φ)−	eq(φ)−	PROPN
cana-5994	69	16	1	1	NUM
cana-5994	69	17	eq(ψφ	eq(ψφ	ADJ
cana-5994	69	18	)	)	PUNCT
cana-5994	69	19	=	=	PUNCT
cana-5994	70	1	∞∑	∞∑	NUM
cana-5994	70	2	υ=0	υ=0	PUNCT
cana-5994	70	3	bυ	bυ	NOUN
cana-5994	70	4	,	,	PUNCT
cana-5994	70	5	q(ψ	q(ψ	NOUN
cana-5994	70	6	)	)	PUNCT
cana-5994	70	7	φυ	φυ	ADP
cana-5994	71	1	[	[	X
cana-5994	71	2	υ]q	υ]q	NOUN
cana-5994	71	3	!	!	PUNCT
cana-5994	72	1	(	(	PUNCT
cana-5994	72	2	|	|	ADV
cana-5994	72	3	φ	φ	NUM
cana-5994	72	4	|	|	NOUN
cana-5994	72	5	<	<	X
cana-5994	72	6	2π	2π	NOUN
cana-5994	72	7	)	)	PUNCT
cana-5994	72	8	,	,	PUNCT
cana-5994	72	9	(	(	PUNCT
cana-5994	72	10	1.9	1.9	NUM
cana-5994	72	11	)	)	PUNCT
cana-5994	72	12	2	2	NUM
cana-5994	72	13	eq(φ	eq(φ	X
cana-5994	72	14	)	)	PUNCT
cana-5994	73	1	+	+	CCONJ
cana-5994	73	2	1	1	NUM
cana-5994	73	3	eq(ψφ	eq(ψφ	ADJ
cana-5994	73	4	)	)	PUNCT
cana-5994	73	5	=	=	PUNCT
cana-5994	74	1	∞∑	∞∑	PRON
cana-5994	74	2	υ=0	υ=0	PUNCT
cana-5994	74	3	eυ	eυ	ADJ
cana-5994	74	4	,	,	PUNCT
cana-5994	74	5	q(ψ	q(ψ	NOUN
cana-5994	74	6	)	)	PUNCT
cana-5994	74	7	φυ	φυ	ADP
cana-5994	75	1	[	[	X
cana-5994	75	2	υ]q	υ]q	NOUN
cana-5994	75	3	!	!	PUNCT
cana-5994	76	1	(	(	PUNCT
cana-5994	76	2	|	|	ADV
cana-5994	76	3	φ	φ	NUM
cana-5994	76	4	|	|	NOUN
cana-5994	76	5	<	<	X
cana-5994	76	6	π	π	PROPN
cana-5994	76	7	)	)	PUNCT
cana-5994	76	8	,	,	PUNCT
cana-5994	76	9	(	(	PUNCT
cana-5994	76	10	1.10	1.10	NUM
cana-5994	76	11	)	)	PUNCT
cana-5994	76	12	2φ	2φ	NUM
cana-5994	76	13	eq(φ	eq(φ	NUM
cana-5994	76	14	)	)	PUNCT
cana-5994	77	1	+	+	CCONJ
cana-5994	77	2	1	1	NUM
cana-5994	77	3	eq(ψφ	eq(ψφ	ADJ
cana-5994	77	4	)	)	PUNCT
cana-5994	77	5	=	=	PUNCT
cana-5994	78	1	∞∑	∞∑	NUM
cana-5994	78	2	υ=0	υ=0	PUNCT
cana-5994	78	3	gυ	gυ	ADJ
cana-5994	78	4	,	,	PUNCT
cana-5994	78	5	q(ψ	q(ψ	NOUN
cana-5994	78	6	)	)	PUNCT
cana-5994	78	7	φυ	φυ	ADP
cana-5994	78	8	[	[	X
cana-5994	78	9	υ]q	υ]q	NOUN
cana-5994	78	10	!	!	PUNCT
cana-5994	79	1	(	(	PUNCT
cana-5994	79	2	|	|	ADV
cana-5994	79	3	ψ	ψ	X
cana-5994	79	4	|	|	ADV
cana-5994	79	5	<	<	X
cana-5994	79	6	π	π	PROPN
cana-5994	79	7	)	)	PUNCT
cana-5994	79	8	,	,	PUNCT
cana-5994	79	9	(	(	PUNCT
cana-5994	79	10	1.11	1.11	NUM
cana-5994	79	11	)	)	PUNCT
cana-5994	79	12	respectively	respectively	ADV
cana-5994	79	13	.	.	PUNCT
cana-5994	80	1	clearly	clearly	ADV
cana-5994	80	2	,	,	PUNCT
cana-5994	80	3	we	we	PRON
cana-5994	80	4	have	have	VERB
cana-5994	80	5	bυ	bυ	NOUN
cana-5994	80	6	,	,	PUNCT
cana-5994	80	7	q	q	NOUN
cana-5994	80	8	=	=	SYM
cana-5994	80	9	bυ	bυ	NOUN
cana-5994	80	10	,	,	PUNCT
cana-5994	80	11	q(0),eυ	q(0),eυ	PROPN
cana-5994	80	12	,	,	PUNCT
cana-5994	80	13	q	q	X
cana-5994	80	14	=	=	SYM
cana-5994	80	15	eυ	eυ	PROPN
cana-5994	80	16	,	,	PUNCT
cana-5994	80	17	q(0),gυ	q(0),gυ	PROPN
cana-5994	80	18	,	,	PUNCT
cana-5994	80	19	q	q	NOUN
cana-5994	80	20	=	=	PUNCT
cana-5994	80	21	gυ	gυ	PROPN
cana-5994	80	22	,	,	PUNCT
cana-5994	80	23	q(0	q(0	PROPN
cana-5994	80	24	)	)	PUNCT
cana-5994	80	25	.	.	PUNCT
cana-5994	81	1	the	the	DET
cana-5994	81	2	main	main	ADJ
cana-5994	81	3	purpose	purpose	NOUN
cana-5994	81	4	of	of	ADP
cana-5994	81	5	this	this	DET
cana-5994	81	6	paper	paper	NOUN
cana-5994	81	7	,	,	PUNCT
cana-5994	81	8	we	we	PRON
cana-5994	81	9	find	find	VERB
cana-5994	81	10	some	some	DET
cana-5994	81	11	differential	differential	ADJ
cana-5994	81	12	equation	equation	NOUN
cana-5994	81	13	for	for	ADP
cana-5994	81	14	q	q	NOUN
cana-5994	81	15	-	-	PUNCT
cana-5994	81	16	analogue	analogue	NOUN
cana-5994	81	17	of	of	ADP
cana-5994	81	18	frobenius	frobenius	ADJ
cana-5994	81	19	-	-	PUNCT
cana-5994	81	20	tangent	tangent	NOUN
cana-5994	81	21	numbers	number	NOUN
cana-5994	81	22	and	and	CCONJ
cana-5994	81	23	polynomials	polynomial	NOUN
cana-5994	81	24	.	.	PUNCT
cana-5994	82	1	based	base	VERB
cana-5994	82	2	on	on	ADP
cana-5994	82	3	these	these	DET
cana-5994	82	4	polynomials	polynomial	NOUN
cana-5994	82	5	,	,	PUNCT
cana-5994	82	6	we	we	PRON
cana-5994	82	7	construct	construct	VERB
cana-5994	82	8	some	some	DET
cana-5994	82	9	differential	differential	ADJ
cana-5994	82	10	equation	equation	NOUN
cana-5994	82	11	of	of	ADP
cana-5994	82	12	these	these	DET
cana-5994	82	13	polynomials	polynomial	NOUN
cana-5994	82	14	.	.	PUNCT
cana-5994	83	1	also	also	ADV
cana-5994	83	2	,	,	PUNCT
cana-5994	83	3	we	we	PRON
cana-5994	83	4	derive	derive	VERB
cana-5994	83	5	differential	differential	ADJ
cana-5994	83	6	equations	equation	NOUN
cana-5994	83	7	associated	associate	VERB
cana-5994	83	8	with	with	ADP
cana-5994	83	9	symmetric	symmetric	ADJ
cana-5994	83	10	properties	property	NOUN
cana-5994	83	11	.	.	PUNCT
cana-5994	84	1	2	2	X
cana-5994	84	2	.	.	X
cana-5994	84	3	differential	differential	ADJ
cana-5994	84	4	equations	equation	NOUN
cana-5994	84	5	of	of	ADP
cana-5994	84	6	q	q	NOUN
cana-5994	84	7	-	-	PUNCT
cana-5994	84	8	analogue	analogue	NOUN
cana-5994	84	9	of	of	ADP
cana-5994	84	10	frobenius	frobenius	ADJ
cana-5994	84	11	-	-	PUNCT
cana-5994	84	12	tangent	tangent	NOUN
cana-5994	84	13	polynomials	polynomial	NOUN
cana-5994	84	14	this	this	DET
cana-5994	84	15	section	section	NOUN
cana-5994	84	16	is	be	AUX
cana-5994	84	17	dedicated	dedicate	VERB
cana-5994	84	18	to	to	ADP
cana-5994	84	19	deriving	derive	VERB
cana-5994	84	20	some	some	DET
cana-5994	84	21	fundamental	fundamental	ADJ
cana-5994	84	22	higher	high	ADJ
cana-5994	84	23	-	-	PUNCT
cana-5994	84	24	order	order	NOUN
cana-5994	84	25	q	q	ADJ
cana-5994	84	26	-	-	PUNCT
cana-5994	84	27	differential	differential	ADJ
cana-5994	84	28	equations	equation	NOUN
cana-5994	84	29	for	for	ADP
cana-5994	84	30	q	q	ADJ
cana-5994	84	31	-	-	PUNCT
cana-5994	84	32	frobenius	frobenius	ADJ
cana-5994	84	33	-	-	PUNCT
cana-5994	84	34	tangent	tangent	NOUN
cana-5994	84	35	polynomials	polynomial	NOUN
cana-5994	84	36	through	through	ADP
cana-5994	84	37	the	the	DET
cana-5994	84	38	use	use	NOUN
cana-5994	84	39	of	of	ADP
cana-5994	84	40	q	q	ADJ
cana-5994	84	41	-	-	PUNCT
cana-5994	84	42	calculus	calculus	NOUN
cana-5994	84	43	theory	theory	NOUN
cana-5994	84	44	.	.	PUNCT
cana-5994	85	1	through	through	ADP
cana-5994	85	2	the	the	DET
cana-5994	85	3	application	application	NOUN
cana-5994	85	4	of	of	ADP
cana-5994	85	5	q	q	NOUN
cana-5994	85	6	-	-	PUNCT
cana-5994	85	7	derivatives	derivative	NOUN
cana-5994	85	8	,	,	PUNCT
cana-5994	85	9	we	we	PRON
cana-5994	85	10	will	will	AUX
cana-5994	85	11	obtain	obtain	VERB
cana-5994	85	12	several	several	ADJ
cana-5994	85	13	related	relate	VERB
cana-5994	85	14	differential	differential	ADJ
cana-5994	85	15	equations	equation	NOUN
cana-5994	85	16	that	that	PRON
cana-5994	85	17	connect	connect	VERB
cana-5994	85	18	to	to	ADP
cana-5994	85	19	the	the	DET
cana-5994	85	20	q	q	NOUN
cana-5994	85	21	-	-	PUNCT
cana-5994	85	22	analogue	analogue	NOUN
cana-5994	85	23	of	of	ADP
cana-5994	85	24	frobenius	frobenius	ADJ
cana-5994	85	25	-	-	PUNCT
cana-5994	85	26	tangent	tangent	NOUN
cana-5994	85	27	polynomials	polynomial	NOUN
cana-5994	85	28	,	,	PUNCT
cana-5994	85	29	based	base	VERB
cana-5994	85	30	on	on	ADP
cana-5994	85	31	definition	definition	NOUN
cana-5994	85	32	(	(	PUNCT
cana-5994	85	33	1.5	1.5	NUM
cana-5994	85	34	)	)	PUNCT
cana-5994	85	35	.	.	PUNCT
cana-5994	86	1	furthermore	furthermore	ADV
cana-5994	86	2	,	,	PUNCT
cana-5994	86	3	we	we	PRON
cana-5994	86	4	will	will	AUX
cana-5994	86	5	establish	establish	VERB
cana-5994	86	6	a	a	DET
cana-5994	86	7	q	q	ADJ
cana-5994	86	8	-	-	PUNCT
cana-5994	86	9	differential	differential	ADJ
cana-5994	86	10	equation	equation	NOUN
cana-5994	86	11	that	that	PRON
cana-5994	86	12	captures	capture	VERB
cana-5994	86	13	the	the	DET
cana-5994	86	14	symmetric	symmetric	ADJ
cana-5994	86	15	property	property	NOUN
cana-5994	86	16	of	of	ADP
cana-5994	86	17	these	these	DET
cana-5994	86	18	polynomials	polynomial	NOUN
cana-5994	86	19	via	via	ADP
cana-5994	86	20	q	q	NOUN
cana-5994	86	21	-	-	PUNCT
cana-5994	86	22	derivatives	derivative	NOUN
cana-5994	86	23	.	.	PUNCT
cana-5994	87	1	theorem	theorem	VERB
cana-5994	87	2	2.1	2.1	NUM
cana-5994	87	3	.	.	PUNCT
cana-5994	88	1	a	a	DET
cana-5994	88	2	solutions	solution	NOUN
cana-5994	88	3	of	of	ADP
cana-5994	88	4	the	the	DET
cana-5994	88	5	following	follow	VERB
cana-5994	88	6	differential	differential	ADJ
cana-5994	88	7	equation	equation	NOUN
cana-5994	88	8	(	(	PUNCT
cana-5994	88	9	i)ftυ−ν	i)ftυ−ν	NOUN
cana-5994	88	10	,	,	PUNCT
cana-5994	88	11	q(ψ	q(ψ	NUM
cana-5994	88	12	;	;	PUNCT
cana-5994	88	13	η	η	X
cana-5994	88	14	)	)	PUNCT
cana-5994	88	15	=	=	PUNCT
cana-5994	89	1	[	[	X
cana-5994	89	2	υ	υ	NOUN
cana-5994	89	3	−	−	NOUN
cana-5994	89	4	ν]q	ν]q	NOUN
cana-5994	89	5	!	!	PUNCT
cana-5994	90	1	[	[	X
cana-5994	90	2	υ]q	υ]q	NOUN
cana-5994	90	3	!	!	PUNCT
cana-5994	91	1	d	d	NOUN
cana-5994	91	2	(	(	PUNCT
cana-5994	91	3	ν	ν	NOUN
cana-5994	91	4	)	)	PUNCT
cana-5994	91	5	q	q	NOUN
cana-5994	91	6	,	,	PUNCT
cana-5994	91	7	ψftυ	ψftυ	NOUN
cana-5994	91	8	,	,	PUNCT
cana-5994	91	9	q(ψ	q(ψ	NUM
cana-5994	91	10	;	;	PUNCT
cana-5994	91	11	η	η	NOUN
cana-5994	91	12	)	)	PUNCT
cana-5994	91	13	,	,	PUNCT
cana-5994	91	14	(	(	PUNCT
cana-5994	91	15	2.1	2.1	NUM
cana-5994	91	16	)	)	PUNCT
cana-5994	91	17	(	(	PUNCT
cana-5994	91	18	ii)ftυ−ν	ii)ftυ−ν	PROPN
cana-5994	91	19	,	,	PUNCT
cana-5994	91	20	q(q−1ψ	q(q−1ψ	NOUN
cana-5994	91	21	;	;	PUNCT
cana-5994	91	22	η	η	NOUN
cana-5994	91	23	)	)	PUNCT
cana-5994	91	24	=	=	SYM
cana-5994	91	25	qν	qν	PART
cana-5994	91	26	[	[	X
cana-5994	91	27	υ	υ	NOUN
cana-5994	91	28	−	−	NOUN
cana-5994	91	29	ν]q	ν]q	NOUN
cana-5994	91	30	!	!	PUNCT
cana-5994	92	1	[	[	X
cana-5994	92	2	υ]q	υ]q	NOUN
cana-5994	92	3	!	!	PUNCT
cana-5994	93	1	d	d	NOUN
cana-5994	93	2	(	(	PUNCT
cana-5994	93	3	ν	ν	NOUN
cana-5994	93	4	)	)	PUNCT
cana-5994	93	5	q	q	NOUN
cana-5994	93	6	,	,	PUNCT
cana-5994	93	7	ψftυ	ψftυ	NOUN
cana-5994	93	8	,	,	PUNCT
cana-5994	93	9	q(q	q(q	PROPN
cana-5994	93	10	−1ψ	−1ψ	PROPN
cana-5994	93	11	;	;	PUNCT
cana-5994	93	12	η	η	PROPN
cana-5994	93	13	)	)	PUNCT
cana-5994	93	14	.	.	PUNCT
cana-5994	94	1	(	(	PUNCT
cana-5994	94	2	2.2	2.2	NUM
cana-5994	94	3	)	)	PUNCT
cana-5994	94	4	communications	communication	NOUN
cana-5994	94	5	on	on	ADP
cana-5994	94	6	applied	apply	VERB
cana-5994	94	7	nonlinear	nonlinear	ADJ
cana-5994	94	8	analysis	analysis	NOUN
cana-5994	94	9	issn	issn	NOUN
cana-5994	94	10	:	:	PUNCT
cana-5994	94	11	1074	1074	NUM
cana-5994	94	12	-	-	PUNCT
cana-5994	94	13	133x	133x	NUM
cana-5994	94	14	vol	vol	NOUN
cana-5994	94	15	32	32	NUM
cana-5994	94	16	no	no	NOUN
cana-5994	94	17	.	.	PUNCT
cana-5994	95	1	9s	9s	NUM
cana-5994	95	2	(	(	PUNCT
cana-5994	95	3	2025	2025	NUM
cana-5994	95	4	)	)	PUNCT
cana-5994	95	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	95	6	3220	3220	NUM
cana-5994	95	7	4	4	NUM
cana-5994	95	8	idrees	idree	NOUN
cana-5994	95	9	ahmad	ahmad	PROPN
cana-5994	95	10	khan	khan	PROPN
cana-5994	95	11	and	and	CCONJ
cana-5994	95	12	sumit	sumit	PROPN
cana-5994	95	13	kumar	kumar	PROPN
cana-5994	95	14	proof	proof	PROPN
cana-5994	95	15	.	.	PUNCT
cana-5994	96	1	using	use	VERB
cana-5994	96	2	(	(	PUNCT
cana-5994	96	3	1.5	1.5	NUM
cana-5994	96	4	)	)	PUNCT
cana-5994	96	5	and	and	CCONJ
cana-5994	96	6	(	(	PUNCT
cana-5994	96	7	1.8	1.8	NUM
cana-5994	96	8	)	)	PUNCT
cana-5994	96	9	,	,	PUNCT
cana-5994	96	10	we	we	PRON
cana-5994	96	11	note	note	VERB
cana-5994	96	12	that	that	SCONJ
cana-5994	97	1	d	d	NOUN
cana-5994	97	2	(	(	PUNCT
cana-5994	97	3	1	1	NUM
cana-5994	97	4	)	)	PUNCT
cana-5994	97	5	q	q	NOUN
cana-5994	97	6	,	,	PUNCT
cana-5994	97	7	ψ	ψ	VERB
cana-5994	97	8	∞∑	∞∑	PRON
cana-5994	97	9	υ=0	υ=0	PUNCT
cana-5994	97	10	tυ	tυ	ADP
cana-5994	97	11	,	,	PUNCT
cana-5994	97	12	q(ψ	q(ψ	NUM
cana-5994	97	13	;	;	PUNCT
cana-5994	97	14	η	η	X
cana-5994	97	15	)	)	PUNCT
cana-5994	97	16	φυ	φυ	ADP
cana-5994	98	1	[	[	X
cana-5994	99	1	υ]q	υ]q	NOUN
cana-5994	99	2	!	!	PUNCT
cana-5994	99	3	=	=	SYM
cana-5994	100	1	1−	1−	NUM
cana-5994	100	2	η	η	X
cana-5994	100	3	eq((1−	eq((1−	X
cana-5994	100	4	η)φ)−	η)φ)−	PRON
cana-5994	100	5	η	η	X
cana-5994	100	6	d	d	X
cana-5994	100	7	(	(	PUNCT
cana-5994	100	8	1	1	NUM
cana-5994	100	9	)	)	PUNCT
cana-5994	100	10	q	q	NOUN
cana-5994	100	11	,	,	PUNCT
cana-5994	100	12	ψeq(ψφ	ψeq(ψφ	NOUN
cana-5994	100	13	)	)	PUNCT
cana-5994	100	14	=	=	PUNCT
cana-5994	101	1	∞∑	∞∑	PRON
cana-5994	101	2	υ=0	υ=0	PUNCT
cana-5994	101	3	[	[	X
cana-5994	101	4	υ]qtυ−1,q(ψ	υ]qtυ−1,q(ψ	X
cana-5994	101	5	;	;	PUNCT
cana-5994	101	6	η	η	X
cana-5994	101	7	)	)	PUNCT
cana-5994	101	8	φυ	φυ	ADP
cana-5994	101	9	[	[	X
cana-5994	101	10	υ]q	υ]q	NOUN
cana-5994	101	11	!	!	PUNCT
cana-5994	101	12	.	.	PUNCT
cana-5994	102	1	(	(	PUNCT
cana-5994	102	2	2.3	2.3	NUM
cana-5994	102	3	)	)	PUNCT
cana-5994	102	4	from	from	ADP
cana-5994	102	5	the	the	DET
cana-5994	102	6	equation	equation	NOUN
cana-5994	102	7	(	(	PUNCT
cana-5994	102	8	2.3	2.3	NUM
cana-5994	102	9	)	)	PUNCT
cana-5994	102	10	,	,	PUNCT
cana-5994	102	11	we	we	PRON
cana-5994	102	12	get	get	VERB
cana-5994	102	13	d	d	X
cana-5994	102	14	(	(	PUNCT
cana-5994	102	15	1	1	NUM
cana-5994	102	16	)	)	PUNCT
cana-5994	102	17	q	q	NOUN
cana-5994	102	18	,	,	PUNCT
cana-5994	102	19	ψtυ	ψtυ	NOUN
cana-5994	102	20	,	,	PUNCT
cana-5994	102	21	q(ψ	q(ψ	NUM
cana-5994	102	22	;	;	PUNCT
cana-5994	102	23	η	η	X
cana-5994	102	24	)	)	PUNCT
cana-5994	102	25	=	=	PUNCT
cana-5994	103	1	[	[	X
cana-5994	103	2	υ]qtυ−1,q(ψ	υ]qtυ−1,q(ψ	X
cana-5994	103	3	;	;	PUNCT
cana-5994	103	4	η	η	NOUN
cana-5994	103	5	)	)	PUNCT
cana-5994	103	6	.	.	PUNCT
cana-5994	104	1	in	in	ADP
cana-5994	104	2	similar	similar	ADJ
cana-5994	104	3	method	method	NOUN
cana-5994	104	4	,	,	PUNCT
cana-5994	104	5	we	we	PRON
cana-5994	104	6	find	find	VERB
cana-5994	104	7	d	d	X
cana-5994	104	8	(	(	PUNCT
cana-5994	104	9	2	2	NUM
cana-5994	104	10	)	)	PUNCT
cana-5994	104	11	q	q	NOUN
cana-5994	104	12	,	,	PUNCT
cana-5994	104	13	ψtυ	ψtυ	NOUN
cana-5994	104	14	,	,	PUNCT
cana-5994	104	15	q(ψ	q(ψ	NUM
cana-5994	104	16	;	;	PUNCT
cana-5994	104	17	η	η	X
cana-5994	104	18	)	)	PUNCT
cana-5994	104	19	=	=	PUNCT
cana-5994	105	1	[	[	X
cana-5994	105	2	υ]q[υ	υ]q[υ	X
cana-5994	105	3	−	−	PROPN
cana-5994	105	4	1]qtυ−2,q(ψ;u	1]qtυ−2,q(ψ;u	NUM
cana-5994	105	5	)	)	PUNCT
cana-5994	105	6	.	.	PUNCT
cana-5994	106	1	therefore	therefore	ADV
cana-5994	106	2	,	,	PUNCT
cana-5994	106	3	we	we	PRON
cana-5994	106	4	have	have	VERB
cana-5994	106	5	d	d	X
cana-5994	106	6	(	(	PUNCT
cana-5994	106	7	ν	ν	NOUN
cana-5994	106	8	)	)	PUNCT
cana-5994	106	9	q	q	NOUN
cana-5994	106	10	,	,	PUNCT
cana-5994	106	11	ψtυ	ψtυ	NOUN
cana-5994	106	12	,	,	PUNCT
cana-5994	106	13	q(x	q(x	PROPN
cana-5994	106	14	;	;	PUNCT
cana-5994	106	15	η	η	X
cana-5994	106	16	)	)	PUNCT
cana-5994	106	17	=	=	PUNCT
cana-5994	107	1	[	[	X
cana-5994	107	2	υ]q[υ	υ]q[υ	NOUN
cana-5994	107	3	−	−	PROPN
cana-5994	107	4	1]q	1]q	PROPN
cana-5994	107	5	·	·	PUNCT
cana-5994	107	6	·	·	PUNCT
cana-5994	107	7	·	·	PUNCT
cana-5994	108	1	[	[	X
cana-5994	108	2	υ	υ	X
cana-5994	108	3	−	−	PROPN
cana-5994	108	4	(	(	PUNCT
cana-5994	108	5	ν	ν	X
cana-5994	108	6	−	−	PROPN
cana-5994	108	7	1)]qtυ−1,q(ψ	1)]qtυ−1,q(ψ	NUM
cana-5994	108	8	;	;	PUNCT
cana-5994	108	9	η	η	PROPN
cana-5994	108	10	)	)	PUNCT
cana-5994	108	11	.	.	PUNCT
cana-5994	109	1	hence	hence	ADV
cana-5994	109	2	,	,	PUNCT
cana-5994	109	3	we	we	PRON
cana-5994	109	4	find	find	VERB
cana-5994	109	5	the	the	DET
cana-5994	109	6	desired	desire	VERB
cana-5994	109	7	result	result	NOUN
cana-5994	109	8	at	at	ADP
cana-5994	109	9	once	once	ADV
cana-5994	109	10	.	.	PUNCT
cana-5994	110	1	(	(	PUNCT
cana-5994	110	2	ii	ii	NOUN
cana-5994	110	3	)	)	PUNCT
cana-5994	110	4	similarly	similarly	ADV
cana-5994	110	5	,	,	PUNCT
cana-5994	110	6	we	we	PRON
cana-5994	110	7	can	can	AUX
cana-5994	110	8	proof	proof	NOUN
cana-5994	110	9	of	of	ADP
cana-5994	110	10	theorem	theorem	ADJ
cana-5994	110	11	2.1	2.1	NUM
cana-5994	110	12	(	(	PUNCT
cana-5994	110	13	ii	ii	NOUN
cana-5994	110	14	)	)	PUNCT
cana-5994	110	15	,	,	PUNCT
cana-5994	110	16	so	so	ADV
cana-5994	110	17	we	we	PRON
cana-5994	110	18	omit	omit	VERB
cana-5994	110	19	the	the	DET
cana-5994	110	20	proof	proof	NOUN
cana-5994	110	21	.	.	PUNCT
cana-5994	111	1	□	□	PUNCT
cana-5994	111	2	theorem	theorem	VERB
cana-5994	111	3	2.2	2.2	NUM
cana-5994	111	4	.	.	PUNCT
cana-5994	112	1	the	the	DET
cana-5994	112	2	differential	differential	ADJ
cana-5994	112	3	equation	equation	NOUN
cana-5994	112	4	of	of	ADP
cana-5994	112	5	the	the	DET
cana-5994	112	6	q	q	ADJ
cana-5994	112	7	-	-	PUNCT
cana-5994	112	8	frobenius	frobenius	ADJ
cana-5994	112	9	tangent	tangent	NOUN
cana-5994	112	10	polynomials	polynomial	NOUN
cana-5994	112	11	as	as	SCONJ
cana-5994	112	12	follows	follow	VERB
cana-5994	112	13	(	(	PUNCT
cana-5994	112	14	1−	1−	NUM
cana-5994	112	15	η)υ−1	η)υ−1	PROPN
cana-5994	113	1	[	[	X
cana-5994	113	2	υ]q	υ]q	NOUN
cana-5994	113	3	!	!	PUNCT
cana-5994	114	1	d	d	NOUN
cana-5994	114	2	(	(	PUNCT
cana-5994	114	3	υ	υ	NOUN
cana-5994	114	4	)	)	PUNCT
cana-5994	114	5	q	q	NOUN
cana-5994	114	6	,	,	PUNCT
cana-5994	114	7	ψftυ	ψftυ	NOUN
cana-5994	114	8	,	,	PUNCT
cana-5994	114	9	q(ψ	q(ψ	NUM
cana-5994	114	10	;	;	PUNCT
cana-5994	114	11	η)+	η)+	NOUN
cana-5994	114	12	(	(	PUNCT
cana-5994	114	13	1−	1−	NUM
cana-5994	114	14	η)υ−2	η)υ−2	PROPN
cana-5994	114	15	[	[	X
cana-5994	114	16	υ	υ	X
cana-5994	114	17	−	−	NOUN
cana-5994	114	18	1]q	1]q	NUM
cana-5994	114	19	!	!	PUNCT
cana-5994	115	1	d	d	X
cana-5994	115	2	(	(	PUNCT
cana-5994	115	3	υ−1	υ−1	PROPN
cana-5994	115	4	)	)	PUNCT
cana-5994	115	5	q	q	NOUN
cana-5994	115	6	,	,	PUNCT
cana-5994	115	7	ψ	ψ	NOUN
cana-5994	115	8	ftυ	ftυ	NOUN
cana-5994	115	9	,	,	PUNCT
cana-5994	115	10	q(ψ	q(ψ	NUM
cana-5994	115	11	;	;	PUNCT
cana-5994	115	12	η)+	η)+	NOUN
cana-5994	115	13	(	(	PUNCT
cana-5994	115	14	1−	1−	NUM
cana-5994	115	15	η)υ−3	η)υ−3	NOUN
cana-5994	116	1	[	[	X
cana-5994	116	2	υ	υ	X
cana-5994	116	3	−	−	NOUN
cana-5994	116	4	2]q	2]q	NUM
cana-5994	116	5	!	!	PUNCT
cana-5994	117	1	d	d	X
cana-5994	117	2	(	(	PUNCT
cana-5994	117	3	υ−2	υ−2	NOUN
cana-5994	117	4	)	)	PUNCT
cana-5994	117	5	q	q	NOUN
cana-5994	117	6	,	,	PUNCT
cana-5994	117	7	ψ	ψ	NOUN
cana-5994	117	8	ftυ	ftυ	NOUN
cana-5994	117	9	,	,	PUNCT
cana-5994	117	10	q(ψ	q(ψ	NUM
cana-5994	117	11	;	;	PUNCT
cana-5994	117	12	η	η	X
cana-5994	117	13	)	)	PUNCT
cana-5994	117	14	+	+	CCONJ
cana-5994	117	15	·	·	PUNCT
cana-5994	117	16	·	·	PUNCT
cana-5994	117	17	·	·	PUNCT
cana-5994	117	18	+	+	NUM
cana-5994	117	19	(	(	PUNCT
cana-5994	117	20	1−	1−	NUM
cana-5994	117	21	η)3	η)3	NOUN
cana-5994	117	22	[	[	X
cana-5994	117	23	4]q	4]q	NOUN
cana-5994	117	24	!	!	PUNCT
cana-5994	118	1	d	d	X
cana-5994	118	2	(	(	PUNCT
cana-5994	118	3	4	4	NUM
cana-5994	118	4	)	)	PUNCT
cana-5994	118	5	q	q	NOUN
cana-5994	118	6	,	,	PUNCT
cana-5994	118	7	ψftυ	ψftυ	NOUN
cana-5994	118	8	,	,	PUNCT
cana-5994	118	9	q(ψ	q(ψ	NUM
cana-5994	118	10	;	;	PUNCT
cana-5994	118	11	η	η	X
cana-5994	118	12	)	)	PUNCT
cana-5994	118	13	+	+	CCONJ
cana-5994	118	14	(	(	PUNCT
cana-5994	118	15	1−	1−	NUM
cana-5994	118	16	η)2	η)2	NOUN
cana-5994	118	17	[	[	X
cana-5994	118	18	3]q	3]q	NUM
cana-5994	118	19	!	!	PUNCT
cana-5994	119	1	d	d	NOUN
cana-5994	119	2	(	(	PUNCT
cana-5994	119	3	3	3	NUM
cana-5994	119	4	)	)	PUNCT
cana-5994	119	5	q	q	NOUN
cana-5994	119	6	,	,	PUNCT
cana-5994	119	7	ψftυ	ψftυ	NOUN
cana-5994	119	8	,	,	PUNCT
cana-5994	119	9	q(ψ	q(ψ	NUM
cana-5994	119	10	;	;	PUNCT
cana-5994	119	11	η	η	X
cana-5994	119	12	)	)	PUNCT
cana-5994	119	13	+	+	CCONJ
cana-5994	119	14	(	(	PUNCT
cana-5994	119	15	1−	1−	NUM
cana-5994	119	16	η	η	NOUN
cana-5994	119	17	)	)	PUNCT
cana-5994	120	1	[	[	X
cana-5994	120	2	2]q	2]q	NUM
cana-5994	120	3	!	!	PUNCT
cana-5994	121	1	d	d	X
cana-5994	121	2	(	(	PUNCT
cana-5994	121	3	2	2	NUM
cana-5994	121	4	)	)	PUNCT
cana-5994	121	5	q	q	NOUN
cana-5994	121	6	,	,	PUNCT
cana-5994	121	7	ψftυ	ψftυ	NOUN
cana-5994	121	8	,	,	PUNCT
cana-5994	121	9	q(ψ	q(ψ	NOUN
cana-5994	121	10	;	;	PUNCT
cana-5994	121	11	η)+d	η)+d	PROPN
cana-5994	121	12	(	(	PUNCT
cana-5994	121	13	1	1	NUM
cana-5994	121	14	)	)	PUNCT
cana-5994	121	15	q	q	NOUN
cana-5994	121	16	,	,	PUNCT
cana-5994	121	17	ψftυ	ψftυ	NOUN
cana-5994	121	18	,	,	PUNCT
cana-5994	121	19	q(ψ	q(ψ	NOUN
cana-5994	121	20	;	;	PUNCT
cana-5994	121	21	η)+(1−η)−1(ft0,q(ψ	η)+(1−η)−1(ft0,q(ψ	NUM
cana-5994	121	22	;	;	PUNCT
cana-5994	121	23	η)−ηftυ	η)−ηftυ	NOUN
cana-5994	121	24	,	,	PUNCT
cana-5994	121	25	q(ψ	q(ψ	NUM
cana-5994	121	26	;	;	PUNCT
cana-5994	121	27	η)−ψυ	η)−ψυ	X
cana-5994	121	28	=	=	SYM
cana-5994	121	29	0	0	X
cana-5994	121	30	.	.	PUNCT
cana-5994	121	31	proof	proof	NOUN
cana-5994	121	32	.	.	PUNCT
cana-5994	122	1	by	by	ADP
cana-5994	122	2	using	use	VERB
cana-5994	122	3	(	(	PUNCT
cana-5994	122	4	2.1	2.1	NUM
cana-5994	122	5	)	)	PUNCT
cana-5994	122	6	,	,	PUNCT
cana-5994	122	7	we	we	PRON
cana-5994	122	8	see	see	VERB
cana-5994	122	9	that	that	SCONJ
cana-5994	122	10	(	(	PUNCT
cana-5994	122	11	1−	1−	NUM
cana-5994	122	12	η)eq(ψφ	η)eq(ψφ	NUM
cana-5994	122	13	)	)	PUNCT
cana-5994	122	14	=	=	PUNCT
cana-5994	123	1	∞∑	∞∑	NUM
cana-5994	123	2	υ=0	υ=0	NUM
cana-5994	123	3	ftυ	ftυ	NOUN
cana-5994	123	4	,	,	PUNCT
cana-5994	123	5	q(ψ	q(ψ	NUM
cana-5994	123	6	;	;	PUNCT
cana-5994	123	7	η	η	X
cana-5994	123	8	)	)	PUNCT
cana-5994	123	9	φυ	φυ	ADP
cana-5994	124	1	[	[	X
cana-5994	124	2	υ]q	υ]q	NOUN
cana-5994	124	3	!	!	PUNCT
cana-5994	125	1	(	(	PUNCT
cana-5994	125	2	eq((1−	eq((1−	X
cana-5994	125	3	η)φ)−	η)φ)−	NUM
cana-5994	125	4	η	η	NOUN
cana-5994	125	5	)	)	PUNCT
cana-5994	125	6	=	=	PUNCT
cana-5994	126	1	∞∑	∞∑	NUM
cana-5994	126	2	υ=0	υ=0	NUM
cana-5994	126	3	ftυ	ftυ	NOUN
cana-5994	126	4	,	,	PUNCT
cana-5994	126	5	q(ψ	q(ψ	NUM
cana-5994	126	6	;	;	PUNCT
cana-5994	126	7	η	η	X
cana-5994	126	8	)	)	PUNCT
cana-5994	126	9	φυ	φυ	ADP
cana-5994	127	1	[	[	X
cana-5994	127	2	υ]q	υ]q	NOUN
cana-5994	127	3	!	!	PUNCT
cana-5994	128	1	(	(	PUNCT
cana-5994	128	2	∞∑	∞∑	NUM
cana-5994	128	3	ν=0	ν=0	X
cana-5994	128	4	(	(	PUNCT
cana-5994	128	5	1−	1−	NUM
cana-5994	128	6	η)ν	η)ν	NOUN
cana-5994	128	7	φν	φν	PROPN
cana-5994	129	1	[	[	X
cana-5994	129	2	ν]q	ν]q	NOUN
cana-5994	129	3	!	!	PUNCT
cana-5994	130	1	−	−	PUNCT
cana-5994	130	2	η	η	PROPN
cana-5994	130	3	)	)	PUNCT
cana-5994	130	4	=	=	PUNCT
cana-5994	131	1	∞∑	∞∑	NUM
cana-5994	131	2	υ=0	υ=0	PUNCT
cana-5994	131	3	(	(	PUNCT
cana-5994	131	4	υ∑	υ∑	VERB
cana-5994	131	5	ν=0	ν=0	PROPN
cana-5994	131	6	(	(	PUNCT
cana-5994	131	7	υ	υ	NOUN
cana-5994	131	8	ν	ν	NOUN
cana-5994	131	9	)	)	PUNCT
cana-5994	131	10	q	q	PROPN
cana-5994	132	1	(	(	PUNCT
cana-5994	132	2	1−	1−	NUM
cana-5994	132	3	u)νftυ−ν	u)νftυ−ν	NOUN
cana-5994	132	4	,	,	PUNCT
cana-5994	132	5	q(ψ	q(ψ	NOUN
cana-5994	132	6	;	;	PUNCT
cana-5994	132	7	η)−	η)−	PROPN
cana-5994	132	8	ηftυ	ηftυ	NOUN
cana-5994	132	9	,	,	PUNCT
cana-5994	132	10	q(ψ	q(ψ	NUM
cana-5994	132	11	;	;	PUNCT
cana-5994	132	12	η	η	NOUN
cana-5994	132	13	)	)	PUNCT
cana-5994	132	14	)	)	PUNCT
cana-5994	132	15	φυ	φυ	ADP
cana-5994	133	1	[	[	X
cana-5994	133	2	υ]q	υ]q	NOUN
cana-5994	133	3	!	!	PUNCT
cana-5994	133	4	.	.	PUNCT
cana-5994	134	1	(	(	PUNCT
cana-5994	134	2	2.4	2.4	NUM
cana-5994	134	3	)	)	PUNCT
cana-5994	134	4	and	and	CCONJ
cana-5994	134	5	(	(	PUNCT
cana-5994	134	6	1−	1−	NUM
cana-5994	134	7	η)eq(ψφ	η)eq(ψφ	NUM
cana-5994	134	8	)	)	PUNCT
cana-5994	134	9	=	=	PUNCT
cana-5994	134	10	(	(	PUNCT
cana-5994	134	11	1−	1−	NUM
cana-5994	134	12	η	η	NOUN
cana-5994	134	13	)	)	PUNCT
cana-5994	134	14	∞∑	∞∑	NUM
cana-5994	134	15	υ=0	υ=0	X
cana-5994	134	16	ψυ	ψυ	ADP
cana-5994	134	17	φυ	φυ	ADP
cana-5994	134	18	[	[	X
cana-5994	134	19	υ]q	υ]q	NOUN
cana-5994	134	20	!	!	PUNCT
cana-5994	134	21	.	.	PUNCT
cana-5994	135	1	(	(	PUNCT
cana-5994	135	2	2.5	2.5	NUM
cana-5994	135	3	)	)	PUNCT
cana-5994	135	4	therefore	therefore	ADV
cana-5994	135	5	,	,	PUNCT
cana-5994	135	6	by	by	ADP
cana-5994	135	7	(	(	PUNCT
cana-5994	135	8	2.4	2.4	NUM
cana-5994	135	9	)	)	PUNCT
cana-5994	135	10	and	and	CCONJ
cana-5994	135	11	(	(	PUNCT
cana-5994	135	12	2.5	2.5	NUM
cana-5994	135	13	)	)	PUNCT
cana-5994	135	14	,	,	PUNCT
cana-5994	135	15	we	we	PRON
cana-5994	135	16	get	get	VERB
cana-5994	135	17	υ∑	υ∑	VERB
cana-5994	136	1	ν=0	ν=0	PRON
cana-5994	136	2	(	(	PUNCT
cana-5994	136	3	υ	υ	NOUN
cana-5994	136	4	ν	ν	NOUN
cana-5994	136	5	)	)	PUNCT
cana-5994	136	6	q	q	PROPN
cana-5994	137	1	(	(	PUNCT
cana-5994	137	2	1−	1−	NUM
cana-5994	137	3	η)νftυ−ν	η)νftυ−ν	NOUN
cana-5994	137	4	,	,	PUNCT
cana-5994	137	5	q(ψ	q(ψ	NOUN
cana-5994	137	6	;	;	PUNCT
cana-5994	137	7	η	η	NOUN
cana-5994	137	8	)	)	PUNCT
cana-5994	137	9	=	=	SYM
cana-5994	137	10	(	(	PUNCT
cana-5994	137	11	1−	1−	NUM
cana-5994	137	12	η)ψυ	η)ψυ	PROPN
cana-5994	137	13	.	.	PROPN
cana-5994	137	14	(	(	PUNCT
cana-5994	137	15	2.6	2.6	NUM
cana-5994	137	16	)	)	PUNCT
cana-5994	137	17	taking	take	VERB
cana-5994	137	18	the	the	DET
cana-5994	137	19	ν	ν	NOUN
cana-5994	137	20	−	−	X
cana-5994	137	21	th	th	X
cana-5994	137	22	derivative	derivative	NOUN
cana-5994	137	23	of	of	ADP
cana-5994	137	24	above	above	ADP
cana-5994	137	25	equation	equation	NOUN
cana-5994	137	26	,	,	PUNCT
cana-5994	137	27	we	we	PRON
cana-5994	137	28	obtain	obtain	VERB
cana-5994	137	29	υ∑	υ∑	ADJ
cana-5994	137	30	ν=0	ν=0	PROPN
cana-5994	137	31	(	(	PUNCT
cana-5994	137	32	1−	1−	NUM
cana-5994	137	33	η)ν−1	η)ν−1	NOUN
cana-5994	138	1	[	[	X
cana-5994	138	2	ν]q	ν]q	NOUN
cana-5994	138	3	!	!	PUNCT
cana-5994	139	1	d	d	NOUN
cana-5994	139	2	(	(	PUNCT
cana-5994	139	3	ν	ν	NOUN
cana-5994	139	4	)	)	PUNCT
cana-5994	139	5	q	q	NOUN
cana-5994	139	6	,	,	PUNCT
cana-5994	139	7	ψftυ	ψftυ	NOUN
cana-5994	139	8	,	,	PUNCT
cana-5994	139	9	q(ψ	q(ψ	NUM
cana-5994	139	10	;	;	PUNCT
cana-5994	139	11	η	η	NOUN
cana-5994	139	12	)	)	PUNCT
cana-5994	139	13	=	=	SYM
cana-5994	139	14	η(1−	η(1−	PROPN
cana-5994	139	15	η)−1	η)−1	NOUN
cana-5994	139	16	ftυ	ftυ	NOUN
cana-5994	139	17	,	,	PUNCT
cana-5994	139	18	q(ψ	q(ψ	NUM
cana-5994	139	19	;	;	PUNCT
cana-5994	139	20	η	η	X
cana-5994	139	21	)	)	PUNCT
cana-5994	139	22	+	+	CCONJ
cana-5994	139	23	(	(	PUNCT
cana-5994	139	24	1−	1−	NUM
cana-5994	139	25	η)ψυ	η)ψυ	PROPN
cana-5994	139	26	=	=	SYM
cana-5994	139	27	0	0	X
cana-5994	139	28	.	.	PUNCT
cana-5994	140	1	hence	hence	ADV
cana-5994	140	2	,	,	PUNCT
cana-5994	140	3	we	we	PRON
cana-5994	140	4	find	find	VERB
cana-5994	140	5	the	the	DET
cana-5994	140	6	desired	desire	VERB
cana-5994	140	7	result	result	NOUN
cana-5994	140	8	at	at	ADP
cana-5994	140	9	once	once	ADV
cana-5994	140	10	.	.	PUNCT
cana-5994	141	1	□	□	PUNCT
cana-5994	141	2	communications	communication	NOUN
cana-5994	141	3	on	on	ADP
cana-5994	141	4	applied	apply	VERB
cana-5994	141	5	nonlinear	nonlinear	ADJ
cana-5994	141	6	analysis	analysis	NOUN
cana-5994	141	7	issn	issn	NOUN
cana-5994	141	8	:	:	PUNCT
cana-5994	141	9	1074	1074	NUM
cana-5994	141	10	-	-	PUNCT
cana-5994	141	11	133x	133x	NUM
cana-5994	141	12	vol	vol	NOUN
cana-5994	141	13	32	32	NUM
cana-5994	141	14	no	no	NOUN
cana-5994	141	15	.	.	PUNCT
cana-5994	142	1	9s	9s	NUM
cana-5994	142	2	(	(	PUNCT
cana-5994	142	3	2025	2025	NUM
cana-5994	142	4	)	)	PUNCT
cana-5994	142	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	142	6	3221	3221	NUM
cana-5994	143	1	some	some	DET
cana-5994	143	2	properties	property	NOUN
cana-5994	143	3	of	of	ADP
cana-5994	143	4	differential	differential	ADJ
cana-5994	143	5	equations	equation	NOUN
cana-5994	143	6	of	of	ADP
cana-5994	143	7	higher	high	ADJ
cana-5994	143	8	-	-	PUNCT
cana-5994	143	9	order	order	NOUN
cana-5994	143	10	5	5	NUM
cana-5994	143	11	corollary	corollary	ADJ
cana-5994	143	12	2.1	2.1	NUM
cana-5994	143	13	.	.	PUNCT
cana-5994	144	1	as	as	ADP
cana-5994	144	2	q	q	NOUN
cana-5994	144	3	approaches	approach	NOUN
cana-5994	144	4	1	1	NUM
cana-5994	144	5	in	in	ADP
cana-5994	144	6	theorem	theorem	NOUN
cana-5994	144	7	2.2	2.2	NUM
cana-5994	144	8	,	,	PUNCT
cana-5994	144	9	we	we	PRON
cana-5994	144	10	derive	derive	VERB
cana-5994	144	11	.	.	PUNCT
cana-5994	145	1	(	(	PUNCT
cana-5994	145	2	1−	1−	NUM
cana-5994	145	3	η)υ−1	η)υ−1	PROPN
cana-5994	145	4	υ	υ	PROPN
cana-5994	145	5	!	!	NOUN
cana-5994	145	6	dυ	dυ	ADP
cana-5994	145	7	dψυ	dψυ	PROPN
cana-5994	145	8	ftυ(ψ	ftυ(ψ	NOUN
cana-5994	145	9	;	;	PUNCT
cana-5994	145	10	η)+	η)+	NOUN
cana-5994	145	11	(	(	PUNCT
cana-5994	145	12	1−	1−	NUM
cana-5994	145	13	η)υ−2	η)υ−2	PROPN
cana-5994	145	14	[	[	X
cana-5994	145	15	υ	υ	NOUN
cana-5994	145	16	−	−	PROPN
cana-5994	145	17	1	1	NUM
cana-5994	145	18	]	]	PUNCT
cana-5994	145	19	!	!	PUNCT
cana-5994	146	1	dυ−1	dυ−1	PROPN
cana-5994	146	2	dψυ−1	dψυ−1	PROPN
cana-5994	146	3	f	f	PROPN
cana-5994	146	4	tυ(ψ	tυ(ψ	PROPN
cana-5994	146	5	;	;	PUNCT
cana-5994	146	6	η)+	η)+	NOUN
cana-5994	146	7	(	(	PUNCT
cana-5994	146	8	1−	1−	NUM
cana-5994	146	9	η)υ−3	η)υ−3	NOUN
cana-5994	147	1	[	[	X
cana-5994	147	2	υ	υ	X
cana-5994	147	3	−	−	PROPN
cana-5994	147	4	2	2	NUM
cana-5994	147	5	]	]	PUNCT
cana-5994	147	6	!	!	PUNCT
cana-5994	148	1	dυ−2	dυ−2	PROPN
cana-5994	148	2	dψυ−2	dψυ−2	PROPN
cana-5994	148	3	f	f	PROPN
cana-5994	148	4	tυ(ψ	tυ(ψ	NUM
cana-5994	148	5	;	;	PUNCT
cana-5994	148	6	η	η	X
cana-5994	148	7	)	)	PUNCT
cana-5994	148	8	+	+	CCONJ
cana-5994	148	9	·	·	PUNCT
cana-5994	148	10	·	·	PUNCT
cana-5994	148	11	·	·	PUNCT
cana-5994	148	12	+	+	NUM
cana-5994	148	13	(	(	PUNCT
cana-5994	148	14	1−	1−	NUM
cana-5994	148	15	η)3	η)3	NOUN
cana-5994	148	16	4	4	NUM
cana-5994	148	17	!	!	PUNCT
cana-5994	148	18	d4	d4	PROPN
cana-5994	148	19	dψ4	dψ4	PROPN
cana-5994	148	20	f	f	PROPN
cana-5994	148	21	tυ(ψ	tυ(ψ	NUM
cana-5994	148	22	;	;	PUNCT
cana-5994	148	23	η	η	X
cana-5994	148	24	)	)	PUNCT
cana-5994	148	25	+	+	CCONJ
cana-5994	148	26	(	(	PUNCT
cana-5994	148	27	1−	1−	NUM
cana-5994	148	28	η)2	η)2	NOUN
cana-5994	148	29	3	3	NUM
cana-5994	148	30	!	!	PUNCT
cana-5994	148	31	d3	d3	PROPN
cana-5994	148	32	dψ3	dψ3	PROPN
cana-5994	148	33	f	f	PROPN
cana-5994	148	34	tυ(ψ	tυ(ψ	NUM
cana-5994	148	35	;	;	PUNCT
cana-5994	148	36	η	η	X
cana-5994	148	37	)	)	PUNCT
cana-5994	148	38	+	+	CCONJ
cana-5994	148	39	(	(	PUNCT
cana-5994	148	40	1−	1−	NUM
cana-5994	148	41	η	η	NOUN
cana-5994	148	42	)	)	PUNCT
cana-5994	148	43	2	2	NUM
cana-5994	148	44	!	!	X
cana-5994	148	45	d2	d2	PROPN
cana-5994	148	46	dψ2	dψ2	PROPN
cana-5994	148	47	f	f	PROPN
cana-5994	148	48	tυ(ψ	tυ(ψ	PROPN
cana-5994	148	49	;	;	PUNCT
cana-5994	148	50	η)+	η)+	NOUN
cana-5994	148	51	d	d	X
cana-5994	148	52	dψ	dψ	X
cana-5994	148	53	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	148	54	;	;	PUNCT
cana-5994	148	55	η)+	η)+	NOUN
cana-5994	148	56	(	(	PUNCT
cana-5994	148	57	1−	1−	NUM
cana-5994	148	58	η)−1(ft0(ψ	η)−1(ft0(ψ	NOUN
cana-5994	148	59	;	;	PUNCT
cana-5994	148	60	η)−	η)−	PROPN
cana-5994	148	61	ηftυ(ψ	ηftυ(ψ	NUM
cana-5994	148	62	;	;	PUNCT
cana-5994	148	63	η)−ψυ	η)−ψυ	PROPN
cana-5994	148	64	=	=	SYM
cana-5994	148	65	0	0	X
cana-5994	148	66	.	.	PUNCT
cana-5994	148	67	theorem	theorem	VERB
cana-5994	148	68	2.3	2.3	NUM
cana-5994	148	69	.	.	PUNCT
cana-5994	149	1	let	let	VERB
cana-5994	149	2	υ	υ	PRON
cana-5994	149	3	≥	≥	NOUN
cana-5994	149	4	0	0	NUM
cana-5994	149	5	.	.	PUNCT
cana-5994	150	1	then	then	ADV
cana-5994	150	2	ftυ	ftυ	NOUN
cana-5994	150	3	,	,	PUNCT
cana-5994	150	4	q(1	q(1	PROPN
cana-5994	150	5	;	;	PUNCT
cana-5994	150	6	η)−	η)−	PROPN
cana-5994	150	7	ηtυ	ηtυ	NOUN
cana-5994	150	8	,	,	PUNCT
cana-5994	150	9	q(η	q(η	PROPN
cana-5994	150	10	)	)	PUNCT
cana-5994	151	1	[	[	X
cana-5994	151	2	υ]q	υ]q	NOUN
cana-5994	151	3	!	!	PUNCT
cana-5994	152	1	d	d	NOUN
cana-5994	152	2	(	(	PUNCT
cana-5994	152	3	υ	υ	NOUN
cana-5994	152	4	)	)	PUNCT
cana-5994	152	5	q	q	NOUN
cana-5994	152	6	,	,	PUNCT
cana-5994	152	7	ψftυ	ψftυ	NOUN
cana-5994	152	8	,	,	PUNCT
cana-5994	152	9	q(ψ	q(ψ	NUM
cana-5994	152	10	;	;	PUNCT
cana-5994	152	11	η)+	η)+	NOUN
cana-5994	152	12	ftυ−1,q(1	ftυ−1,q(1	NOUN
cana-5994	152	13	;	;	PUNCT
cana-5994	152	14	η)−	η)−	PROPN
cana-5994	152	15	ηftυ−1,q(η	ηftυ−1,q(η	NOUN
cana-5994	152	16	)	)	PUNCT
cana-5994	153	1	[	[	X
cana-5994	153	2	υ	υ	X
cana-5994	153	3	−	−	NOUN
cana-5994	153	4	1]q	1]q	NUM
cana-5994	153	5	!	!	PUNCT
cana-5994	154	1	d	d	X
cana-5994	154	2	(	(	PUNCT
cana-5994	154	3	υ−1	υ−1	PROPN
cana-5994	154	4	)	)	PUNCT
cana-5994	154	5	q	q	NOUN
cana-5994	154	6	,	,	PUNCT
cana-5994	154	7	ψ	ψ	NOUN
cana-5994	154	8	ftυ	ftυ	NOUN
cana-5994	154	9	,	,	PUNCT
cana-5994	154	10	q(ψ	q(ψ	NUM
cana-5994	154	11	;	;	PUNCT
cana-5994	154	12	η)+	η)+	NOUN
cana-5994	154	13	·	·	PUNCT
cana-5994	154	14	·	·	PUNCT
cana-5994	154	15	·	·	PUNCT
cana-5994	154	16	+	+	CCONJ
cana-5994	154	17	ft2,q(1	ft2,q(1	ADV
cana-5994	154	18	;	;	PUNCT
cana-5994	154	19	η)−	η)−	PROPN
cana-5994	154	20	ηft2,q(η	ηft2,q(η	NOUN
cana-5994	154	21	)	)	PUNCT
cana-5994	155	1	[	[	X
cana-5994	155	2	2]q	2]q	NUM
cana-5994	155	3	!	!	PUNCT
cana-5994	156	1	d	d	X
cana-5994	156	2	(	(	PUNCT
cana-5994	156	3	2	2	NUM
cana-5994	156	4	)	)	PUNCT
cana-5994	156	5	q	q	NOUN
cana-5994	156	6	,	,	PUNCT
cana-5994	156	7	ψftυ	ψftυ	NOUN
cana-5994	156	8	,	,	PUNCT
cana-5994	156	9	q(1	q(1	PROPN
cana-5994	156	10	;	;	PUNCT
cana-5994	156	11	η)+(ft1,q(1	η)+(ft1,q(1	NOUN
cana-5994	156	12	;	;	PUNCT
cana-5994	156	13	η)−ηft1,q(η))d	η)−ηft1,q(η))d	NUM
cana-5994	156	14	(	(	PUNCT
cana-5994	156	15	1	1	NUM
cana-5994	156	16	)	)	PUNCT
cana-5994	156	17	q	q	NOUN
cana-5994	156	18	,	,	PUNCT
cana-5994	156	19	ψftυ	ψftυ	NOUN
cana-5994	156	20	,	,	PUNCT
cana-5994	156	21	q(ψ	q(ψ	NUM
cana-5994	156	22	;	;	PUNCT
cana-5994	156	23	η	η	X
cana-5994	156	24	)	)	PUNCT
cana-5994	156	25	+	+	ADJ
cana-5994	156	26	(	(	PUNCT
cana-5994	156	27	ft0,q(1	ft0,q(1	NOUN
cana-5994	156	28	;	;	PUNCT
cana-5994	156	29	η)−	η)−	PROPN
cana-5994	156	30	ηft0,q(η)−	ηft0,q(η)−	NOUN
cana-5994	156	31	(	(	PUNCT
cana-5994	156	32	1−	1−	NUM
cana-5994	156	33	η))ftυ	η))ftυ	NOUN
cana-5994	156	34	,	,	PUNCT
cana-5994	156	35	q(ψ	q(ψ	NUM
cana-5994	156	36	;	;	PUNCT
cana-5994	156	37	η	η	X
cana-5994	156	38	)	)	PUNCT
cana-5994	156	39	=	=	SYM
cana-5994	156	40	0	0	X
cana-5994	156	41	.	.	PUNCT
cana-5994	156	42	proof	proof	NOUN
cana-5994	156	43	.	.	PUNCT
cana-5994	157	1	from	from	ADP
cana-5994	157	2	(	(	PUNCT
cana-5994	157	3	2.1	2.1	NUM
cana-5994	157	4	)	)	PUNCT
cana-5994	157	5	,	,	PUNCT
cana-5994	157	6	we	we	PRON
cana-5994	157	7	have	have	VERB
cana-5994	157	8	∞∑	∞∑	NUM
cana-5994	157	9	υ=0	υ=0	NUM
cana-5994	157	10	ftυ	ftυ	NOUN
cana-5994	157	11	,	,	PUNCT
cana-5994	157	12	q(ψ;u	q(ψ;u	PUNCT
cana-5994	157	13	)	)	PUNCT
cana-5994	157	14	φυ	φυ	ADP
cana-5994	158	1	[	[	X
cana-5994	158	2	υ]q	υ]q	NOUN
cana-5994	158	3	!	!	PUNCT
cana-5994	158	4	=	=	SYM
cana-5994	159	1	1−	1−	NUM
cana-5994	159	2	η	η	X
cana-5994	159	3	eq((1−	eq((1−	X
cana-5994	159	4	η)φ)−	η)φ)−	PRON
cana-5994	159	5	η	η	X
cana-5994	159	6	eq(ψφ	eq(ψφ	PROPN
cana-5994	159	7	)	)	PUNCT
cana-5994	159	8	=	=	SYM
cana-5994	159	9	1	1	NUM
cana-5994	159	10	1−	1−	NUM
cana-5994	159	11	η	η	PROPN
cana-5994	159	12	(	(	PUNCT
cana-5994	159	13	1−	1−	NUM
cana-5994	159	14	η	η	X
cana-5994	159	15	eq((1−	eq((1−	X
cana-5994	159	16	η)φ)−	η)φ)−	PRON
cana-5994	159	17	η	η	X
cana-5994	159	18	eq((1−	eq((1−	X
cana-5994	159	19	η)φ)−	η)φ)−	PUNCT
cana-5994	159	20	η	η	PROPN
cana-5994	159	21	1−	1−	NUM
cana-5994	159	22	η	η	PROPN
cana-5994	159	23	eq((1−	eq((1−	X
cana-5994	159	24	η)φ)−	η)φ)−	X
cana-5994	159	25	η	η	NOUN
cana-5994	159	26	)	)	PUNCT
cana-5994	159	27	1−	1−	NUM
cana-5994	159	28	η	η	PROPN
cana-5994	159	29	eq((1−	eq((1−	X
cana-5994	159	30	η)φ)−	η)φ)−	PRON
cana-5994	159	31	η	η	X
cana-5994	159	32	eq(ψφ	eq(ψφ	PROPN
cana-5994	159	33	)	)	PUNCT
cana-5994	159	34	.	.	PUNCT
cana-5994	160	1	(	(	PUNCT
cana-5994	160	2	1−η	1−η	NUM
cana-5994	160	3	)	)	PUNCT
cana-5994	160	4	∞∑	∞∑	PRON
cana-5994	160	5	υ=0	υ=0	ADJ
cana-5994	160	6	ftυ	ftυ	NOUN
cana-5994	160	7	,	,	PUNCT
cana-5994	160	8	q(ψ	q(ψ	NUM
cana-5994	160	9	;	;	PUNCT
cana-5994	160	10	η	η	X
cana-5994	160	11	)	)	PUNCT
cana-5994	160	12	φυ	φυ	ADP
cana-5994	161	1	[	[	X
cana-5994	162	1	υ]q	υ]q	NOUN
cana-5994	162	2	!	!	PUNCT
cana-5994	162	3	=	=	NOUN
cana-5994	163	1	∞∑	∞∑	PRON
cana-5994	163	2	υ=0	υ=0	PUNCT
cana-5994	163	3	(	(	PUNCT
cana-5994	163	4	υ∑	υ∑	VERB
cana-5994	163	5	ν=0	ν=0	PROPN
cana-5994	163	6	(	(	PUNCT
cana-5994	163	7	υ	υ	NOUN
cana-5994	163	8	ν	ν	NOUN
cana-5994	163	9	)	)	PUNCT
cana-5994	163	10	q	q	PROPN
cana-5994	164	1	(	(	PUNCT
cana-5994	164	2	ftν	ftν	NOUN
cana-5994	164	3	,	,	PUNCT
cana-5994	164	4	q(1	q(1	PROPN
cana-5994	164	5	;	;	PUNCT
cana-5994	164	6	η)−	η)−	PROPN
cana-5994	164	7	ηftν	ηftν	NOUN
cana-5994	164	8	,	,	PUNCT
cana-5994	164	9	q(η	q(η	PROPN
cana-5994	164	10	)	)	PUNCT
cana-5994	164	11	)	)	PUNCT
cana-5994	164	12	ftυ−ν	ftυ−ν	NOUN
cana-5994	164	13	,	,	PUNCT
cana-5994	164	14	q(ψ	q(ψ	NUM
cana-5994	164	15	;	;	PUNCT
cana-5994	164	16	η	η	NOUN
cana-5994	164	17	)	)	PUNCT
cana-5994	164	18	)	)	PUNCT
cana-5994	164	19	φυ	φυ	ADP
cana-5994	165	1	[	[	X
cana-5994	165	2	υ]q	υ]q	X
cana-5994	165	3	!	!	PUNCT
cana-5994	166	1	υ∑	υ∑	ADJ
cana-5994	167	1	ν=0	ν=0	PRON
cana-5994	167	2	(	(	PUNCT
cana-5994	167	3	υ	υ	NOUN
cana-5994	167	4	ν	ν	NOUN
cana-5994	167	5	)	)	PUNCT
cana-5994	167	6	q	q	PROPN
cana-5994	167	7	(	(	PUNCT
cana-5994	167	8	ftν	ftν	NOUN
cana-5994	167	9	,	,	PUNCT
cana-5994	167	10	q(1	q(1	PROPN
cana-5994	167	11	;	;	PUNCT
cana-5994	167	12	η)−	η)−	PROPN
cana-5994	167	13	ηftν	ηftν	NOUN
cana-5994	167	14	,	,	PUNCT
cana-5994	167	15	q(η	q(η	PROPN
cana-5994	167	16	)	)	PUNCT
cana-5994	167	17	)	)	PUNCT
cana-5994	167	18	ftυ−ν	ftυ−ν	NOUN
cana-5994	167	19	,	,	PUNCT
cana-5994	167	20	q(ψ	q(ψ	NOUN
cana-5994	167	21	;	;	PUNCT
cana-5994	167	22	η)−	η)−	PROPN
cana-5994	167	23	(	(	PUNCT
cana-5994	167	24	1−	1−	NUM
cana-5994	167	25	η)ftυ	η)ftυ	NOUN
cana-5994	167	26	,	,	PUNCT
cana-5994	167	27	q(ψ	q(ψ	NUM
cana-5994	167	28	;	;	PUNCT
cana-5994	167	29	η	η	X
cana-5994	167	30	)	)	PUNCT
cana-5994	167	31	=	=	SYM
cana-5994	167	32	0	0	X
cana-5994	167	33	.	.	PUNCT
cana-5994	167	34	(	(	PUNCT
cana-5994	167	35	2.7	2.7	NUM
cana-5994	167	36	)	)	PUNCT
cana-5994	167	37	replacing	replace	VERB
cana-5994	167	38	ftυ−ν	ftυ−ν	NOUN
cana-5994	167	39	,	,	PUNCT
cana-5994	167	40	q(ψ	q(ψ	NUM
cana-5994	167	41	;	;	PUNCT
cana-5994	167	42	η	η	NOUN
cana-5994	167	43	)	)	PUNCT
cana-5994	167	44	with	with	ADP
cana-5994	167	45	d(ν	d(ν	PROPN
cana-5994	167	46	)	)	PUNCT
cana-5994	167	47	q	q	NOUN
cana-5994	167	48	,	,	PUNCT
cana-5994	167	49	ψftυ	ψftυ	NOUN
cana-5994	167	50	,	,	PUNCT
cana-5994	167	51	q(ψ	q(ψ	NUM
cana-5994	167	52	;	;	PUNCT
cana-5994	167	53	η	η	X
cana-5994	167	54	)	)	PUNCT
cana-5994	167	55	in	in	ADP
cana-5994	167	56	equation	equation	NOUN
cana-5994	167	57	(	(	PUNCT
cana-5994	167	58	2.7	2.7	NUM
cana-5994	167	59	)	)	PUNCT
cana-5994	167	60	,	,	PUNCT
cana-5994	167	61	we	we	PRON
cana-5994	167	62	have	have	AUX
cana-5994	167	63	υ∑	υ∑	VERB
cana-5994	167	64	ν=0	ν=0	PROPN
cana-5994	167	65	(	(	PUNCT
cana-5994	167	66	ftν	ftν	NOUN
cana-5994	167	67	,	,	PUNCT
cana-5994	167	68	q(1	q(1	PROPN
cana-5994	167	69	;	;	PUNCT
cana-5994	167	70	η)−	η)−	PROPN
cana-5994	167	71	ηftν	ηftν	NOUN
cana-5994	167	72	,	,	PUNCT
cana-5994	167	73	q(η	q(η	PROPN
cana-5994	167	74	)	)	PUNCT
cana-5994	167	75	)	)	PUNCT
cana-5994	168	1	[	[	X
cana-5994	168	2	ν]q	ν]q	NOUN
cana-5994	168	3	!	!	PUNCT
cana-5994	169	1	d	d	NOUN
cana-5994	169	2	(	(	PUNCT
cana-5994	169	3	ν	ν	NOUN
cana-5994	169	4	)	)	PUNCT
cana-5994	169	5	q	q	NOUN
cana-5994	169	6	,	,	PUNCT
cana-5994	169	7	ψftυ	ψftυ	NOUN
cana-5994	169	8	,	,	PUNCT
cana-5994	169	9	q(ψ	q(ψ	NOUN
cana-5994	169	10	;	;	PUNCT
cana-5994	169	11	η)−	η)−	PROPN
cana-5994	169	12	(	(	PUNCT
cana-5994	169	13	1−	1−	NUM
cana-5994	169	14	η)ftυ	η)ftυ	NOUN
cana-5994	169	15	,	,	PUNCT
cana-5994	169	16	q(ψ	q(ψ	NUM
cana-5994	169	17	;	;	PUNCT
cana-5994	169	18	η	η	X
cana-5994	169	19	)	)	PUNCT
cana-5994	169	20	=	=	SYM
cana-5994	169	21	0	0	X
cana-5994	169	22	.	.	PUNCT
cana-5994	170	1	hence	hence	ADV
cana-5994	170	2	,	,	PUNCT
cana-5994	170	3	we	we	PRON
cana-5994	170	4	find	find	VERB
cana-5994	170	5	the	the	DET
cana-5994	170	6	desired	desire	VERB
cana-5994	170	7	result	result	NOUN
cana-5994	170	8	at	at	ADP
cana-5994	170	9	once	once	ADV
cana-5994	170	10	.	.	PUNCT
cana-5994	171	1	□	□	PUNCT
cana-5994	171	2	corollary	corollary	ADJ
cana-5994	171	3	2.2	2.2	NUM
cana-5994	171	4	.	.	PUNCT
cana-5994	172	1	as	as	ADP
cana-5994	172	2	q	q	NOUN
cana-5994	172	3	approaches	approach	NOUN
cana-5994	172	4	1	1	NUM
cana-5994	172	5	in	in	ADP
cana-5994	172	6	theorem	theorem	ADJ
cana-5994	172	7	2.3	2.3	NUM
cana-5994	172	8	,	,	PUNCT
cana-5994	172	9	we	we	PRON
cana-5994	172	10	derive	derive	VERB
cana-5994	172	11	ftυ(1	ftυ(1	NOUN
cana-5994	172	12	;	;	PUNCT
cana-5994	172	13	η)−	η)−	PROPN
cana-5994	172	14	ηtυ(η	ηtυ(η	PROPN
cana-5994	172	15	)	)	PUNCT
cana-5994	172	16	υ	υ	NOUN
cana-5994	172	17	!	!	PUNCT
cana-5994	172	18	dυ	dυ	PROPN
cana-5994	172	19	dxυ	dxυ	PROPN
cana-5994	172	20	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	172	21	;	;	PUNCT
cana-5994	172	22	η	η	PROPN
cana-5994	172	23	)	)	PUNCT
cana-5994	172	24	+	+	CCONJ
cana-5994	172	25	ftυ−1(1	ftυ−1(1	ADJ
cana-5994	172	26	;	;	PUNCT
cana-5994	172	27	η)−	η)−	PROPN
cana-5994	172	28	ηftυ−1(η	ηftυ−1(η	NOUN
cana-5994	172	29	)	)	PUNCT
cana-5994	172	30	(	(	PUNCT
cana-5994	172	31	υ	υ	NOUN
cana-5994	172	32	−	−	PROPN
cana-5994	172	33	1	1	NUM
cana-5994	172	34	)	)	PUNCT
cana-5994	172	35	!	!	PUNCT
cana-5994	173	1	dυ−1	dυ−1	PROPN
cana-5994	173	2	dψυ−1	dψυ−1	PROPN
cana-5994	173	3	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	173	4	;	;	PUNCT
cana-5994	173	5	η)+	η)+	NOUN
cana-5994	173	6	·	·	PUNCT
cana-5994	173	7	·	·	PUNCT
cana-5994	173	8	·	·	PUNCT
cana-5994	174	1	+	+	NUM
cana-5994	174	2	ft2(1	ft2(1	NOUN
cana-5994	174	3	;	;	PUNCT
cana-5994	174	4	η)−	η)−	PROPN
cana-5994	174	5	ηft2(η	ηft2(η	NOUN
cana-5994	174	6	)	)	PUNCT
cana-5994	174	7	2	2	NUM
cana-5994	174	8	!	!	X
cana-5994	174	9	d2	d2	PROPN
cana-5994	174	10	dψ2	dψ2	PROPN
cana-5994	174	11	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	174	12	;	;	PUNCT
cana-5994	174	13	η	η	X
cana-5994	174	14	)	)	PUNCT
cana-5994	174	15	+	+	CCONJ
cana-5994	174	16	(	(	PUNCT
cana-5994	174	17	ft1(1	ft1(1	X
cana-5994	174	18	;	;	PUNCT
cana-5994	174	19	η)−	η)−	PROPN
cana-5994	174	20	ηt1(η	ηt1(η	PROPN
cana-5994	174	21	)	)	PUNCT
cana-5994	174	22	)	)	PUNCT
cana-5994	175	1	d	d	NOUN
cana-5994	175	2	dψ	dψ	PROPN
cana-5994	175	3	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	175	4	;	;	PUNCT
cana-5994	175	5	η	η	PROPN
cana-5994	175	6	)	)	PUNCT
cana-5994	175	7	+	+	PROPN
cana-5994	175	8	(	(	PUNCT
cana-5994	175	9	ft0(1	ft0(1	NOUN
cana-5994	175	10	;	;	PUNCT
cana-5994	175	11	η)−	η)−	NOUN
cana-5994	175	12	ηft0(η)−	ηft0(η)−	NOUN
cana-5994	175	13	(	(	PUNCT
cana-5994	175	14	1−	1−	NUM
cana-5994	175	15	η))ftυ(ψ	η))ftυ(ψ	PROPN
cana-5994	175	16	;	;	PUNCT
cana-5994	175	17	η	η	X
cana-5994	175	18	)	)	PUNCT
cana-5994	175	19	=	=	SYM
cana-5994	175	20	0	0	X
cana-5994	175	21	.	.	PUNCT
cana-5994	175	22	theorem	theorem	VERB
cana-5994	175	23	2.4	2.4	NUM
cana-5994	175	24	.	.	PUNCT
cana-5994	176	1	let	let	VERB
cana-5994	176	2	υ	υ	PRON
cana-5994	176	3	≥	≥	NOUN
cana-5994	176	4	0	0	NUM
cana-5994	176	5	.	.	PUNCT
cana-5994	177	1	then	then	ADV
cana-5994	177	2	υ−1∑	υ−1∑	ADJ
cana-5994	177	3	ν=0	ν=0	PROPN
cana-5994	177	4	(	(	PUNCT
cana-5994	177	5	1−	1−	NUM
cana-5994	177	6	η)υ−ν−1	η)υ−ν−1	PROPN
cana-5994	177	7	ftν	ftν	NOUN
cana-5994	177	8	,	,	PUNCT
cana-5994	177	9	q(η	q(η	PROPN
cana-5994	177	10	)	)	PUNCT
cana-5994	178	1	[	[	X
cana-5994	178	2	υ	υ	X
cana-5994	178	3	−	−	NOUN
cana-5994	178	4	ν	ν	NOUN
cana-5994	178	5	−	−	PROPN
cana-5994	178	6	1]q![ν]q	1]q![ν]q	NUM
cana-5994	178	7	!	!	PUNCT
cana-5994	179	1	d	d	X
cana-5994	179	2	(	(	PUNCT
cana-5994	179	3	υ−1	υ−1	PROPN
cana-5994	179	4	)	)	PUNCT
cana-5994	179	5	q	q	NOUN
cana-5994	179	6	,	,	PUNCT
cana-5994	179	7	ψ	ψ	X
cana-5994	179	8	ftυ−1,q(ψ	ftυ−1,q(ψ	NOUN
cana-5994	179	9	;	;	PUNCT
cana-5994	179	10	η)+	η)+	X
cana-5994	179	11	υ−2∑	υ−2∑	ADV
cana-5994	179	12	ν=0	ν=0	PROPN
cana-5994	179	13	(	(	PUNCT
cana-5994	179	14	1−	1−	NUM
cana-5994	179	15	η)υ−ν−2qftν	η)υ−ν−2qftν	PROPN
cana-5994	179	16	,	,	PUNCT
cana-5994	179	17	q(η	q(η	PROPN
cana-5994	179	18	)	)	PUNCT
cana-5994	180	1	[	[	X
cana-5994	180	2	υ	υ	X
cana-5994	180	3	−	−	NOUN
cana-5994	180	4	ν	ν	NOUN
cana-5994	180	5	−	−	PROPN
cana-5994	180	6	2]q![ν]q	2]q![ν]q	NUM
cana-5994	180	7	!	!	PUNCT
cana-5994	181	1	d	d	X
cana-5994	181	2	(	(	PUNCT
cana-5994	181	3	υ−2	υ−2	NOUN
cana-5994	181	4	)	)	PUNCT
cana-5994	181	5	q	q	NOUN
cana-5994	181	6	,	,	PUNCT
cana-5994	181	7	ψ	ψ	X
cana-5994	181	8	ftυ−1,q(ψ	ftυ−1,q(ψ	NOUN
cana-5994	181	9	;	;	PUNCT
cana-5994	181	10	η)+	η)+	X
cana-5994	181	11	·	·	PUNCT
cana-5994	181	12	·	·	PUNCT
cana-5994	181	13	·	·	PUNCT
cana-5994	182	1	+	+	NUM
cana-5994	182	2	2∑	2∑	X
cana-5994	182	3	ν=0	ν=0	PRON
cana-5994	182	4	(	(	PUNCT
cana-5994	182	5	1−	1−	NUM
cana-5994	182	6	η)2−νqυ−3	η)2−νqυ−3	NOUN
cana-5994	182	7	ftν	ftν	NOUN
cana-5994	182	8	,	,	PUNCT
cana-5994	182	9	q(η	q(η	PROPN
cana-5994	182	10	)	)	PUNCT
cana-5994	183	1	[	[	X
cana-5994	183	2	2−	2−	NUM
cana-5994	183	3	ν]q![ν]q	ν]q![ν]q	NOUN
cana-5994	183	4	!	!	PUNCT
cana-5994	184	1	d	d	X
cana-5994	184	2	(	(	PUNCT
cana-5994	184	3	2	2	NUM
cana-5994	184	4	)	)	PUNCT
cana-5994	184	5	q	q	NOUN
cana-5994	184	6	,	,	PUNCT
cana-5994	184	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	184	8	;	;	PUNCT
cana-5994	184	9	η)+	η)+	VERB
cana-5994	184	10	1∑	1∑	PROPN
cana-5994	184	11	ν=0	ν=0	PROPN
cana-5994	184	12	(	(	PUNCT
cana-5994	184	13	1−	1−	NUM
cana-5994	184	14	η)1−νqυ−2	η)1−νqυ−2	PROPN
cana-5994	184	15	ftν	ftν	NOUN
cana-5994	184	16	,	,	PUNCT
cana-5994	184	17	q(η	q(η	PROPN
cana-5994	184	18	)	)	PUNCT
cana-5994	185	1	[	[	X
cana-5994	185	2	1−	1−	NUM
cana-5994	185	3	ν]q![ν]q	ν]q![ν]q	NOUN
cana-5994	185	4	!	!	PUNCT
cana-5994	186	1	d	d	X
cana-5994	186	2	(	(	PUNCT
cana-5994	186	3	1	1	NUM
cana-5994	186	4	)	)	PUNCT
cana-5994	186	5	q	q	NOUN
cana-5994	186	6	,	,	PUNCT
cana-5994	186	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	186	8	;	;	PUNCT
cana-5994	186	9	η	η	X
cana-5994	186	10	)	)	PUNCT
cana-5994	187	1	+	+	CCONJ
cana-5994	187	2	(	(	PUNCT
cana-5994	187	3	qυ−1	qυ−1	PROPN
cana-5994	187	4	ft0,q(η)−	ft0,q(η)−	PROPN
cana-5994	187	5	qυψ	qυψ	PROPN
cana-5994	187	6	)	)	PUNCT
cana-5994	188	1	ftυ−1,q(ψ	ftυ−1,q(ψ	PROPN
cana-5994	188	2	;	;	PUNCT
cana-5994	188	3	η	η	X
cana-5994	188	4	)	)	PUNCT
cana-5994	188	5	+	+	NUM
cana-5994	188	6	ftυ	ftυ	NOUN
cana-5994	188	7	,	,	PUNCT
cana-5994	188	8	q(qψ	q(qψ	NUM
cana-5994	188	9	;	;	PUNCT
cana-5994	188	10	η	η	PROPN
cana-5994	188	11	)	)	PUNCT
cana-5994	188	12	=	=	SYM
cana-5994	188	13	0	0	X
cana-5994	188	14	.	.	PUNCT
cana-5994	188	15	communications	communication	NOUN
cana-5994	188	16	on	on	ADP
cana-5994	188	17	applied	apply	VERB
cana-5994	188	18	nonlinear	nonlinear	ADJ
cana-5994	188	19	analysis	analysis	NOUN
cana-5994	188	20	issn	issn	NOUN
cana-5994	188	21	:	:	PUNCT
cana-5994	188	22	1074	1074	NUM
cana-5994	188	23	-	-	PUNCT
cana-5994	188	24	133x	133x	NUM
cana-5994	188	25	vol	vol	NOUN
cana-5994	188	26	32	32	NUM
cana-5994	188	27	no	no	NOUN
cana-5994	188	28	.	.	PUNCT
cana-5994	189	1	9s	9s	NUM
cana-5994	189	2	(	(	PUNCT
cana-5994	189	3	2025	2025	NUM
cana-5994	189	4	)	)	PUNCT
cana-5994	189	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	189	6	3222	3222	NUM
cana-5994	189	7	6	6	NUM
cana-5994	189	8	idrees	idree	NOUN
cana-5994	189	9	ahmad	ahmad	PROPN
cana-5994	189	10	khan	khan	PROPN
cana-5994	189	11	and	and	CCONJ
cana-5994	189	12	sumit	sumit	PROPN
cana-5994	189	13	kumar	kumar	PROPN
cana-5994	189	14	proof	proof	PROPN
cana-5994	189	15	.	.	PUNCT
cana-5994	190	1	let	let	VERB
cana-5994	190	2	qψ	qψ	PROPN
cana-5994	190	3	→	→	SYM
cana-5994	190	4	ψ	ψ	X
cana-5994	190	5	in	in	ADP
cana-5994	190	6	(	(	PUNCT
cana-5994	190	7	1.5	1.5	NUM
cana-5994	190	8	)	)	PUNCT
cana-5994	190	9	,	,	PUNCT
cana-5994	190	10	we	we	PRON
cana-5994	190	11	have	have	VERB
cana-5994	190	12	dq	dq	PROPN
cana-5994	190	13	,	,	PUNCT
cana-5994	190	14	φ	φ	X
cana-5994	190	15	∞∑	∞∑	NUM
cana-5994	190	16	υ=0	υ=0	ADJ
cana-5994	190	17	ftυ	ftυ	NOUN
cana-5994	190	18	,	,	PUNCT
cana-5994	190	19	q(qψ	q(qψ	NUM
cana-5994	190	20	;	;	PUNCT
cana-5994	190	21	η	η	PROPN
cana-5994	190	22	)	)	PUNCT
cana-5994	190	23	φυ	φυ	ADP
cana-5994	191	1	[	[	X
cana-5994	192	1	υ]q	υ]q	NOUN
cana-5994	192	2	!	!	PUNCT
cana-5994	192	3	=	=	SYM
cana-5994	193	1	eq(qφψ)dq	eq(qφψ)dq	PROPN
cana-5994	193	2	,	,	PUNCT
cana-5994	193	3	φ	φ	PROPN
cana-5994	193	4	(	(	PUNCT
cana-5994	193	5	1−	1−	NUM
cana-5994	193	6	η	η	X
cana-5994	193	7	eq((1−	eq((1−	X
cana-5994	193	8	η)φ)−	η)φ)−	X
cana-5994	193	9	η	η	NOUN
cana-5994	193	10	)	)	PUNCT
cana-5994	193	11	+	+	CCONJ
cana-5994	193	12	1−	1−	NUM
cana-5994	193	13	η	η	X
cana-5994	193	14	eq((1−	eq((1−	X
cana-5994	193	15	η)φ)−	η)φ)−	PRON
cana-5994	193	16	η	η	X
cana-5994	193	17	dq	dq	PROPN
cana-5994	193	18	,	,	PUNCT
cana-5994	193	19	φeq(qφψ	φeq(qφψ	PROPN
cana-5994	193	20	)	)	PUNCT
cana-5994	193	21	=	=	PUNCT
cana-5994	194	1	∞∑	∞∑	NUM
cana-5994	194	2	υ=0	υ=0	PUNCT
cana-5994	194	3	qυftυ	qυftυ	NOUN
cana-5994	194	4	,	,	PUNCT
cana-5994	194	5	q(ψ	q(ψ	NUM
cana-5994	194	6	;	;	PUNCT
cana-5994	194	7	η	η	X
cana-5994	194	8	)	)	PUNCT
cana-5994	194	9	φυ	φυ	ADP
cana-5994	194	10	[	[	X
cana-5994	194	11	υ]q	υ]q	NOUN
cana-5994	194	12	!	!	PUNCT
cana-5994	194	13	(	(	PUNCT
cana-5994	195	1	qψ	qψ	ADP
cana-5994	195	2	−	−	VERB
cana-5994	195	3	∞∑	∞∑	NOUN
cana-5994	195	4	υ=0	υ=0	X
cana-5994	195	5	(	(	PUNCT
cana-5994	195	6	υ∑	υ∑	VERB
cana-5994	195	7	ν=0	ν=0	PROPN
cana-5994	195	8	(	(	PUNCT
cana-5994	195	9	υ	υ	NOUN
cana-5994	195	10	ν	ν	NOUN
cana-5994	195	11	)	)	PUNCT
cana-5994	195	12	q	q	PROPN
cana-5994	196	1	(	(	PUNCT
cana-5994	196	2	1−	1−	NUM
cana-5994	196	3	η)υ−νftν	η)υ−νftν	NOUN
cana-5994	196	4	,	,	PUNCT
cana-5994	196	5	q(η	q(η	PROPN
cana-5994	196	6	)	)	PUNCT
cana-5994	196	7	)	)	PUNCT
cana-5994	197	1	φυ	φυ	ADP
cana-5994	197	2	[	[	X
cana-5994	197	3	υ]q	υ]q	NOUN
cana-5994	197	4	!	!	PUNCT
cana-5994	197	5	)	)	PUNCT
cana-5994	198	1	=	=	PUNCT
cana-5994	199	1	∞∑	∞∑	PRON
cana-5994	199	2	υ=0	υ=0	PUNCT
cana-5994	199	3	(	(	PUNCT
cana-5994	199	4	qυ+1ψftυ	qυ+1ψftυ	NOUN
cana-5994	199	5	,	,	PUNCT
cana-5994	199	6	q(η)−	q(η)−	NOUN
cana-5994	199	7	υ∑	υ∑	ADJ
cana-5994	199	8	θ=0	θ=0	PROPN
cana-5994	199	9	θ∑	θ∑	ADP
cana-5994	199	10	ν=0	ν=0	PRON
cana-5994	199	11	(	(	PUNCT
cana-5994	199	12	υ	υ	NOUN
cana-5994	199	13	θ	θ	NOUN
cana-5994	199	14	)	)	PUNCT
cana-5994	199	15	q	q	PROPN
cana-5994	200	1	(	(	PUNCT
cana-5994	200	2	θ	θ	NOUN
cana-5994	200	3	ν	ν	NOUN
cana-5994	200	4	)	)	PUNCT
cana-5994	200	5	q	q	PROPN
cana-5994	201	1	(	(	PUNCT
cana-5994	201	2	1−	1−	NUM
cana-5994	201	3	η)θ−νqυ−θftν	η)θ−νqυ−θftν	PROPN
cana-5994	201	4	,	,	PUNCT
cana-5994	201	5	q(η)ftυ−θ	q(η)ftυ−θ	NOUN
cana-5994	201	6	,	,	PUNCT
cana-5994	201	7	q(ψ	q(ψ	NUM
cana-5994	201	8	;	;	PUNCT
cana-5994	201	9	η	η	NOUN
cana-5994	201	10	)	)	PUNCT
cana-5994	201	11	)	)	PUNCT
cana-5994	202	1	φυ	φυ	ADP
cana-5994	203	1	[	[	X
cana-5994	203	2	υ]q	υ]q	X
cana-5994	203	3	!	!	PUNCT
cana-5994	204	1	(	(	PUNCT
cana-5994	204	2	2.8	2.8	NUM
cana-5994	204	3	)	)	PUNCT
cana-5994	204	4	dq	dq	PROPN
cana-5994	204	5	,	,	PUNCT
cana-5994	204	6	φ	φ	X
cana-5994	204	7	∞∑	∞∑	NUM
cana-5994	204	8	υ=0	υ=0	ADJ
cana-5994	204	9	ftυ	ftυ	NOUN
cana-5994	204	10	,	,	PUNCT
cana-5994	204	11	q(qψ	q(qψ	NUM
cana-5994	204	12	;	;	PUNCT
cana-5994	204	13	η	η	PROPN
cana-5994	204	14	)	)	PUNCT
cana-5994	204	15	φυ	φυ	ADP
cana-5994	205	1	[	[	X
cana-5994	206	1	υ]q	υ]q	NOUN
cana-5994	206	2	!	!	PUNCT
cana-5994	206	3	=	=	NOUN
cana-5994	207	1	∞∑	∞∑	PRON
cana-5994	207	2	υ=0	υ=0	PUNCT
cana-5994	208	1	[	[	X
cana-5994	208	2	υ]qq	υ]qq	X
cana-5994	208	3	υψftυ−1,q(ψ	υψftυ−1,q(ψ	NUM
cana-5994	208	4	;	;	PUNCT
cana-5994	208	5	η	η	X
cana-5994	208	6	)	)	PUNCT
cana-5994	208	7	φυ	φυ	ADP
cana-5994	208	8	[	[	X
cana-5994	208	9	υ]q	υ]q	NOUN
cana-5994	208	10	!	!	PUNCT
cana-5994	209	1	−	−	PUNCT
cana-5994	210	1	∞∑	∞∑	PRON
cana-5994	210	2	υ=0	υ=0	PUNCT
cana-5994	210	3	[	[	X
cana-5994	210	4	υ]q	υ]q	NOUN
cana-5994	210	5	(	(	PUNCT
cana-5994	210	6	υ−1∑	υ−1∑	NUM
cana-5994	210	7	θ=0	θ=0	X
cana-5994	210	8	θ∑	θ∑	ADP
cana-5994	210	9	ν=0	ν=0	PRON
cana-5994	210	10	(	(	PUNCT
cana-5994	210	11	υ	υ	NOUN
cana-5994	210	12	−	−	PROPN
cana-5994	210	13	1	1	NUM
cana-5994	210	14	θ	θ	NOUN
cana-5994	210	15	)	)	PUNCT
cana-5994	210	16	q	q	NOUN
cana-5994	210	17	(	(	PUNCT
cana-5994	210	18	θ	θ	NOUN
cana-5994	210	19	ν	ν	NOUN
cana-5994	210	20	)	)	PUNCT
cana-5994	210	21	q	q	PROPN
cana-5994	210	22	(	(	PUNCT
cana-5994	210	23	1−	1−	NUM
cana-5994	210	24	η)θ−νqυ−θ−1	η)θ−νqυ−θ−1	PROPN
cana-5994	210	25	ftν	ftν	NOUN
cana-5994	210	26	,	,	PUNCT
cana-5994	210	27	q(η)ftυ−θ−1,q(ψ	q(η)ftυ−θ−1,q(ψ	NUM
cana-5994	210	28	;	;	PUNCT
cana-5994	210	29	η	η	NOUN
cana-5994	210	30	)	)	PUNCT
cana-5994	210	31	)	)	PUNCT
cana-5994	210	32	φυ	φυ	ADP
cana-5994	211	1	[	[	X
cana-5994	211	2	υ]q	υ]q	NOUN
cana-5994	211	3	!	!	PUNCT
cana-5994	211	4	.	.	PUNCT
cana-5994	212	1	(	(	PUNCT
cana-5994	212	2	2.9	2.9	NUM
cana-5994	212	3	)	)	PUNCT
cana-5994	212	4	on	on	ADP
cana-5994	212	5	the	the	DET
cana-5994	212	6	other	other	ADJ
cana-5994	212	7	hand	hand	NOUN
cana-5994	212	8	,	,	PUNCT
cana-5994	212	9	we	we	PRON
cana-5994	212	10	have	have	VERB
cana-5994	212	11	φdq	φdq	NOUN
cana-5994	212	12	,	,	PUNCT
cana-5994	213	1	φ	φ	NUM
cana-5994	213	2	∞∑	∞∑	NUM
cana-5994	213	3	υ=0	υ=0	ADJ
cana-5994	213	4	ftυ	ftυ	NOUN
cana-5994	213	5	,	,	PUNCT
cana-5994	213	6	q(qψ	q(qψ	NUM
cana-5994	213	7	;	;	PUNCT
cana-5994	213	8	η	η	PROPN
cana-5994	213	9	)	)	PUNCT
cana-5994	213	10	φυ	φυ	ADP
cana-5994	214	1	[	[	X
cana-5994	215	1	υ]q	υ]q	NOUN
cana-5994	215	2	!	!	PUNCT
cana-5994	215	3	=	=	NOUN
cana-5994	216	1	∞∑	∞∑	PRON
cana-5994	216	2	υ=0	υ=0	PUNCT
cana-5994	216	3	[	[	X
cana-5994	216	4	υ]qftυ	υ]qftυ	NOUN
cana-5994	216	5	,	,	PUNCT
cana-5994	216	6	q(qψ	q(qψ	NUM
cana-5994	216	7	;	;	PUNCT
cana-5994	216	8	η	η	PROPN
cana-5994	216	9	)	)	PUNCT
cana-5994	216	10	φυ	φυ	ADP
cana-5994	216	11	[	[	X
cana-5994	216	12	υ]q	υ]q	NOUN
cana-5994	216	13	!	!	PUNCT
cana-5994	216	14	.	.	PUNCT
cana-5994	217	1	(	(	PUNCT
cana-5994	217	2	2.10	2.10	NUM
cana-5994	217	3	)	)	PUNCT
cana-5994	217	4	by	by	ADP
cana-5994	217	5	(	(	PUNCT
cana-5994	217	6	2.9	2.9	NUM
cana-5994	217	7	)	)	PUNCT
cana-5994	217	8	and	and	CCONJ
cana-5994	217	9	(	(	PUNCT
cana-5994	217	10	2.10	2.10	NUM
cana-5994	217	11	)	)	PUNCT
cana-5994	217	12	,	,	PUNCT
cana-5994	217	13	we	we	PRON
cana-5994	217	14	have	have	VERB
cana-5994	217	15	υ−1∑	υ−1∑	NOUN
cana-5994	217	16	θ=0	θ=0	NOUN
cana-5994	217	17	θ∑	θ∑	ADP
cana-5994	217	18	ν=0	ν=0	PRON
cana-5994	217	19	(	(	PUNCT
cana-5994	217	20	υ	υ	NOUN
cana-5994	217	21	−	−	PROPN
cana-5994	217	22	1	1	NUM
cana-5994	217	23	θ	θ	NOUN
cana-5994	217	24	)	)	PUNCT
cana-5994	218	1	q	q	NOUN
cana-5994	219	1	(	(	PUNCT
cana-5994	219	2	θ	θ	NOUN
cana-5994	219	3	ν	ν	NOUN
cana-5994	219	4	)	)	PUNCT
cana-5994	219	5	q	q	PROPN
cana-5994	220	1	(	(	PUNCT
cana-5994	220	2	1−	1−	NUM
cana-5994	220	3	η)θ−νqυ−θ−1	η)θ−νqυ−θ−1	PROPN
cana-5994	220	4	ftν	ftν	NOUN
cana-5994	220	5	,	,	PUNCT
cana-5994	220	6	q(η)ftυ−θ−1,q(ψ	q(η)ftυ−θ−1,q(ψ	NUM
cana-5994	220	7	;	;	PUNCT
cana-5994	220	8	η	η	NOUN
cana-5994	220	9	)	)	PUNCT
cana-5994	220	10	=	=	SYM
cana-5994	220	11	qυψftυ−1,q(ψ	qυψftυ−1,q(ψ	NOUN
cana-5994	220	12	;	;	PUNCT
cana-5994	220	13	η)−	η)−	PROPN
cana-5994	220	14	ftυ	ftυ	NOUN
cana-5994	220	15	,	,	PUNCT
cana-5994	220	16	q(qψ	q(qψ	NUM
cana-5994	220	17	;	;	PUNCT
cana-5994	220	18	η	η	PROPN
cana-5994	220	19	)	)	PUNCT
cana-5994	220	20	.	.	PUNCT
cana-5994	221	1	(	(	PUNCT
cana-5994	221	2	2.11	2.11	NUM
cana-5994	221	3	)	)	PUNCT
cana-5994	221	4	in	in	ADP
cana-5994	221	5	theorem	theorem	ADJ
cana-5994	221	6	2.1	2.1	NUM
cana-5994	221	7	(	(	PUNCT
cana-5994	221	8	i	i	NOUN
cana-5994	221	9	)	)	PUNCT
cana-5994	221	10	,	,	PUNCT
cana-5994	221	11	we	we	PRON
cana-5994	221	12	get	get	VERB
cana-5994	221	13	ftυ−ν−1,q(ψ	ftυ−ν−1,q(ψ	NOUN
cana-5994	221	14	;	;	PUNCT
cana-5994	221	15	η	η	X
cana-5994	221	16	)	)	PUNCT
cana-5994	221	17	=	=	PUNCT
cana-5994	222	1	[	[	X
cana-5994	222	2	υ	υ	X
cana-5994	222	3	−	−	NOUN
cana-5994	222	4	ν	ν	NOUN
cana-5994	222	5	−	−	NOUN
cana-5994	222	6	1]q	1]q	NUM
cana-5994	222	7	!	!	PUNCT
cana-5994	223	1	[	[	X
cana-5994	223	2	υ	υ	X
cana-5994	223	3	−	−	NOUN
cana-5994	223	4	1]q	1]q	NUM
cana-5994	223	5	!	!	PUNCT
cana-5994	224	1	d	d	X
cana-5994	224	2	(	(	PUNCT
cana-5994	224	3	ν	ν	NOUN
cana-5994	224	4	)	)	PUNCT
cana-5994	224	5	q	q	NOUN
cana-5994	224	6	,	,	PUNCT
cana-5994	224	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	224	8	;	;	PUNCT
cana-5994	224	9	η	η	PROPN
cana-5994	224	10	)	)	PUNCT
cana-5994	224	11	.	.	PUNCT
cana-5994	225	1	(	(	PUNCT
cana-5994	225	2	2.12	2.12	NUM
cana-5994	225	3	)	)	PUNCT
cana-5994	225	4	υ−1∑	υ−1∑	NOUN
cana-5994	225	5	θ=0	θ=0	X
cana-5994	225	6	θ∑	θ∑	ADP
cana-5994	225	7	ν=0	ν=0	PRON
cana-5994	225	8	(	(	PUNCT
cana-5994	225	9	υ	υ	NOUN
cana-5994	225	10	−	−	PROPN
cana-5994	225	11	1	1	NUM
cana-5994	225	12	θ	θ	NOUN
cana-5994	225	13	)	)	PUNCT
cana-5994	225	14	q	q	NOUN
cana-5994	226	1	(	(	PUNCT
cana-5994	226	2	θ	θ	NOUN
cana-5994	226	3	ν	ν	NOUN
cana-5994	226	4	)	)	PUNCT
cana-5994	226	5	q	q	PROPN
cana-5994	227	1	(	(	PUNCT
cana-5994	227	2	1−	1−	NUM
cana-5994	227	3	η)θ−νqυ−θ−1	η)θ−νqυ−θ−1	PROPN
cana-5994	227	4	ftν	ftν	NOUN
cana-5994	227	5	,	,	PUNCT
cana-5994	227	6	q(η)ftυ−l−1,q(ψ	q(η)ftυ−l−1,q(ψ	NUM
cana-5994	227	7	;	;	PUNCT
cana-5994	227	8	η	η	PROPN
cana-5994	227	9	)	)	PUNCT
cana-5994	227	10	=	=	PUNCT
cana-5994	227	11	υ−1∑	υ−1∑	NUM
cana-5994	227	12	θ=0	θ=0	X
cana-5994	227	13	θ∑	θ∑	ADP
cana-5994	227	14	ν=0	ν=0	PROPN
cana-5994	227	15	(	(	PUNCT
cana-5994	227	16	1−	1−	NUM
cana-5994	227	17	η)θ−νqυ−θ−1	η)θ−νqυ−θ−1	PROPN
cana-5994	227	18	ftν	ftν	NOUN
cana-5994	227	19	,	,	PUNCT
cana-5994	227	20	q(η	q(η	PROPN
cana-5994	227	21	)	)	PUNCT
cana-5994	228	1	[	[	X
cana-5994	228	2	θ	θ	X
cana-5994	228	3	−	−	NOUN
cana-5994	228	4	ν]q![ν]q	ν]q![ν]q	NOUN
cana-5994	228	5	!	!	PUNCT
cana-5994	229	1	d	d	X
cana-5994	229	2	(	(	PUNCT
cana-5994	229	3	θ	θ	NOUN
cana-5994	229	4	)	)	PUNCT
cana-5994	229	5	q	q	NOUN
cana-5994	229	6	,	,	PUNCT
cana-5994	229	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	229	8	;	;	PUNCT
cana-5994	229	9	η	η	PROPN
cana-5994	229	10	)	)	PUNCT
cana-5994	229	11	.	.	PUNCT
cana-5994	230	1	(	(	PUNCT
cana-5994	230	2	2.13	2.13	NUM
cana-5994	230	3	)	)	PUNCT
cana-5994	230	4	therefore	therefore	ADV
cana-5994	230	5	,	,	PUNCT
cana-5994	230	6	we	we	PRON
cana-5994	230	7	acquire	acquire	VERB
cana-5994	230	8	at	at	ADP
cana-5994	230	9	the	the	DET
cana-5994	230	10	desired	desire	VERB
cana-5994	230	11	result	result	NOUN
cana-5994	230	12	.	.	PUNCT
cana-5994	231	1	□	□	PUNCT
cana-5994	231	2	corollary	corollary	ADJ
cana-5994	231	3	2.3	2.3	NUM
cana-5994	231	4	.	.	PUNCT
cana-5994	232	1	as	as	ADP
cana-5994	232	2	q	q	NOUN
cana-5994	232	3	approaches	approach	NOUN
cana-5994	232	4	1	1	NUM
cana-5994	232	5	in	in	ADP
cana-5994	232	6	theorem	theorem	ADJ
cana-5994	232	7	2.4	2.4	NUM
cana-5994	232	8	,	,	PUNCT
cana-5994	232	9	we	we	PRON
cana-5994	232	10	derive	derive	VERB
cana-5994	232	11	υ−1∑	υ−1∑	ADJ
cana-5994	232	12	ν=0	ν=0	NOUN
cana-5994	232	13	(	(	PUNCT
cana-5994	232	14	1−	1−	NUM
cana-5994	232	15	η)υ−ν−1	η)υ−ν−1	PROPN
cana-5994	232	16	ftν(η	ftν(η	NOUN
cana-5994	232	17	)	)	PUNCT
cana-5994	233	1	[	[	X
cana-5994	233	2	υ	υ	X
cana-5994	233	3	−	−	NOUN
cana-5994	233	4	ν	ν	NOUN
cana-5994	233	5	−	−	NOUN
cana-5994	233	6	1]![ν	1]![ν	NUM
cana-5994	233	7	]	]	PUNCT
cana-5994	233	8	!	!	PUNCT
cana-5994	234	1	d(υ−1	d(υ−1	PROPN
cana-5994	234	2	)	)	PUNCT
cana-5994	234	3	dψυ−1	dψυ−1	PROPN
cana-5994	234	4	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	234	5	;	;	PUNCT
cana-5994	234	6	η)+	η)+	VERB
cana-5994	234	7	υ−2∑	υ−2∑	ADV
cana-5994	234	8	ν=0	ν=0	PROPN
cana-5994	234	9	(	(	PUNCT
cana-5994	234	10	1−	1−	NUM
cana-5994	234	11	η)υ−ν−2	η)υ−ν−2	PROPN
cana-5994	234	12	ftν(η	ftν(η	NOUN
cana-5994	234	13	)	)	PUNCT
cana-5994	235	1	[	[	X
cana-5994	235	2	υ	υ	X
cana-5994	235	3	−	−	NOUN
cana-5994	235	4	ν	ν	NOUN
cana-5994	235	5	−	−	NOUN
cana-5994	235	6	2]![ν	2]![ν	NUM
cana-5994	235	7	]	]	X
cana-5994	235	8	!	!	PUNCT
cana-5994	236	1	d(υ−2	d(υ−2	ADJ
cana-5994	236	2	)	)	PUNCT
cana-5994	236	3	dψυ−2	dψυ−2	NOUN
cana-5994	236	4	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	236	5	;	;	PUNCT
cana-5994	236	6	η)+	η)+	X
cana-5994	236	7	·	·	PUNCT
cana-5994	236	8	·	·	PUNCT
cana-5994	236	9	·	·	PUNCT
cana-5994	237	1	+	+	NUM
cana-5994	237	2	2∑	2∑	X
cana-5994	237	3	ν=0	ν=0	PRON
cana-5994	237	4	(	(	PUNCT
cana-5994	237	5	1−	1−	NUM
cana-5994	237	6	η)2−νftν(η	η)2−νftν(η	NOUN
cana-5994	237	7	)	)	PUNCT
cana-5994	238	1	[	[	X
cana-5994	238	2	2−	2−	NUM
cana-5994	238	3	ν]![ν	ν]![ν	PROPN
cana-5994	238	4	]	]	PUNCT
cana-5994	238	5	!	!	PUNCT
cana-5994	239	1	d2	d2	PROPN
cana-5994	239	2	dψ2	dψ2	PROPN
cana-5994	239	3	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	239	4	;	;	PUNCT
cana-5994	239	5	η	η	X
cana-5994	239	6	)	)	PUNCT
cana-5994	240	1	+	+	CCONJ
cana-5994	240	2	θ∑	θ∑	ADP
cana-5994	240	3	ν=0	ν=0	PRON
cana-5994	240	4	(	(	PUNCT
cana-5994	240	5	1−	1−	NUM
cana-5994	240	6	η)θ−νftν(η	η)θ−νftν(η	NOUN
cana-5994	240	7	)	)	PUNCT
cana-5994	241	1	[	[	X
cana-5994	241	2	θ	θ	X
cana-5994	241	3	−	−	PROPN
cana-5994	241	4	ν]![ν	ν]![ν	PROPN
cana-5994	241	5	]	]	PUNCT
cana-5994	241	6	!	!	PUNCT
cana-5994	242	1	d	d	X
cana-5994	242	2	dψ	dψ	NOUN
cana-5994	242	3	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	242	4	;	;	PUNCT
cana-5994	242	5	η	η	X
cana-5994	242	6	)	)	PUNCT
cana-5994	242	7	+	+	CCONJ
cana-5994	242	8	(	(	PUNCT
cana-5994	242	9	ft0(η)−	ft0(η)−	NOUN
cana-5994	242	10	ψ	ψ	NOUN
cana-5994	242	11	)	)	PUNCT
cana-5994	242	12	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	242	13	;	;	PUNCT
cana-5994	242	14	η	η	X
cana-5994	242	15	)	)	PUNCT
cana-5994	242	16	+	+	X
cana-5994	242	17	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	242	18	;	;	PUNCT
cana-5994	242	19	η	η	NOUN
cana-5994	242	20	)	)	PUNCT
cana-5994	242	21	=	=	SYM
cana-5994	242	22	0	0	X
cana-5994	242	23	.	.	PUNCT
cana-5994	242	24	theorem	theorem	VERB
cana-5994	242	25	2.5	2.5	NUM
cana-5994	242	26	.	.	PUNCT
cana-5994	243	1	let	let	VERB
cana-5994	243	2	υ	υ	PRON
cana-5994	243	3	≥	≥	NOUN
cana-5994	243	4	0	0	NUM
cana-5994	243	5	.	.	PUNCT
cana-5994	244	1	then	then	ADV
cana-5994	244	2	υ−1∑	υ−1∑	ADJ
cana-5994	244	3	ν=0	ν=0	PROPN
cana-5994	244	4	(	(	PUNCT
cana-5994	244	5	1−	1−	NUM
cana-5994	244	6	η)υ−1hν	η)υ−1hν	PROPN
cana-5994	244	7	,	,	PUNCT
cana-5994	244	8	q(η	q(η	PROPN
cana-5994	244	9	)	)	PUNCT
cana-5994	245	1	[	[	X
cana-5994	245	2	υ	υ	X
cana-5994	245	3	−	−	NOUN
cana-5994	245	4	ν	ν	NOUN
cana-5994	245	5	−	−	PROPN
cana-5994	245	6	1]q![ν]q	1]q![ν]q	NUM
cana-5994	245	7	!	!	PUNCT
cana-5994	246	1	d	d	X
cana-5994	246	2	(	(	PUNCT
cana-5994	246	3	υ−1	υ−1	PROPN
cana-5994	246	4	)	)	PUNCT
cana-5994	246	5	q	q	NOUN
cana-5994	246	6	,	,	PUNCT
cana-5994	246	7	ψ	ψ	X
cana-5994	246	8	ftυ−1,q(ψ	ftυ−1,q(ψ	NOUN
cana-5994	246	9	;	;	PUNCT
cana-5994	246	10	η)+	η)+	X
cana-5994	246	11	υ−2∑	υ−2∑	ADV
cana-5994	246	12	ν=0	ν=0	PROPN
cana-5994	246	13	(	(	PUNCT
cana-5994	246	14	1−	1−	NUM
cana-5994	246	15	η)υ−2qhν	η)υ−2qhν	NOUN
cana-5994	246	16	,	,	PUNCT
cana-5994	246	17	q(η	q(η	PROPN
cana-5994	246	18	)	)	PUNCT
cana-5994	247	1	[	[	X
cana-5994	247	2	υ	υ	X
cana-5994	247	3	−	−	NOUN
cana-5994	247	4	ν	ν	NOUN
cana-5994	247	5	−	−	PROPN
cana-5994	247	6	2]q![ν]q	2]q![ν]q	NUM
cana-5994	247	7	!	!	PUNCT
cana-5994	248	1	d	d	X
cana-5994	248	2	(	(	PUNCT
cana-5994	248	3	υ−2	υ−2	NOUN
cana-5994	248	4	)	)	PUNCT
cana-5994	248	5	q	q	NOUN
cana-5994	248	6	,	,	PUNCT
cana-5994	248	7	ψ	ψ	X
cana-5994	248	8	ftυ−1,q(ψ	ftυ−1,q(ψ	NOUN
cana-5994	248	9	;	;	PUNCT
cana-5994	248	10	η)+	η)+	X
cana-5994	248	11	·	·	PUNCT
cana-5994	248	12	·	·	PUNCT
cana-5994	248	13	·	·	PUNCT
cana-5994	249	1	communications	communication	NOUN
cana-5994	249	2	on	on	ADP
cana-5994	249	3	applied	apply	VERB
cana-5994	249	4	nonlinear	nonlinear	ADJ
cana-5994	249	5	analysis	analysis	NOUN
cana-5994	249	6	issn	issn	NOUN
cana-5994	249	7	:	:	PUNCT
cana-5994	249	8	1074	1074	NUM
cana-5994	249	9	-	-	PUNCT
cana-5994	249	10	133x	133x	NUM
cana-5994	249	11	vol	vol	NOUN
cana-5994	249	12	32	32	NUM
cana-5994	249	13	no	no	NOUN
cana-5994	249	14	.	.	PUNCT
cana-5994	250	1	9s	9s	NUM
cana-5994	250	2	(	(	PUNCT
cana-5994	250	3	2025	2025	NUM
cana-5994	250	4	)	)	PUNCT
cana-5994	251	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	251	2	3223	3223	NUM
cana-5994	252	1	some	some	DET
cana-5994	252	2	properties	property	NOUN
cana-5994	252	3	of	of	ADP
cana-5994	252	4	differential	differential	ADJ
cana-5994	252	5	equations	equation	NOUN
cana-5994	252	6	of	of	ADP
cana-5994	252	7	higher	high	ADJ
cana-5994	252	8	-	-	PUNCT
cana-5994	252	9	order	order	NOUN
cana-5994	252	10	7	7	NUM
cana-5994	252	11	+	+	NUM
cana-5994	252	12	2∑	2∑	NUM
cana-5994	252	13	ν=0	ν=0	PRON
cana-5994	252	14	(	(	PUNCT
cana-5994	252	15	1−	1−	NUM
cana-5994	252	16	η)2qυ−3hν	η)2qυ−3hν	NOUN
cana-5994	252	17	,	,	PUNCT
cana-5994	252	18	q(η	q(η	PROPN
cana-5994	252	19	)	)	PUNCT
cana-5994	253	1	[	[	X
cana-5994	253	2	2−	2−	NUM
cana-5994	253	3	ν]q![ν]q	ν]q![ν]q	NOUN
cana-5994	253	4	!	!	PUNCT
cana-5994	254	1	d	d	X
cana-5994	254	2	(	(	PUNCT
cana-5994	254	3	2	2	NUM
cana-5994	254	4	)	)	PUNCT
cana-5994	254	5	q	q	NOUN
cana-5994	254	6	,	,	PUNCT
cana-5994	254	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	254	8	;	;	PUNCT
cana-5994	254	9	η)+	η)+	VERB
cana-5994	254	10	1∑	1∑	PROPN
cana-5994	254	11	ν=0	ν=0	PROPN
cana-5994	254	12	(	(	PUNCT
cana-5994	254	13	1−	1−	NUM
cana-5994	254	14	η)qυ−2hν	η)qυ−2hν	PROPN
cana-5994	254	15	,	,	PUNCT
cana-5994	254	16	q(η	q(η	PROPN
cana-5994	254	17	)	)	PUNCT
cana-5994	255	1	[	[	X
cana-5994	255	2	1−	1−	NUM
cana-5994	255	3	ν]q![ν]q	ν]q![ν]q	NOUN
cana-5994	255	4	!	!	PUNCT
cana-5994	256	1	d	d	X
cana-5994	256	2	(	(	PUNCT
cana-5994	256	3	1	1	NUM
cana-5994	256	4	)	)	PUNCT
cana-5994	256	5	q	q	NOUN
cana-5994	256	6	,	,	PUNCT
cana-5994	256	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	256	8	;	;	PUNCT
cana-5994	256	9	η	η	X
cana-5994	256	10	)	)	PUNCT
cana-5994	256	11	+	+	CCONJ
cana-5994	256	12	(	(	PUNCT
cana-5994	256	13	q−1h0,q	q−1h0,q	PROPN
cana-5994	256	14	−	−	PROPN
cana-5994	256	15	ψ	ψ	NOUN
cana-5994	256	16	)	)	PUNCT
cana-5994	256	17	qυftυ−1,q(ψ	qυftυ−1,q(ψ	PROPN
cana-5994	256	18	;	;	PUNCT
cana-5994	256	19	η	η	PROPN
cana-5994	256	20	)	)	PUNCT
cana-5994	256	21	+	+	NUM
cana-5994	256	22	ftυ	ftυ	NOUN
cana-5994	256	23	,	,	PUNCT
cana-5994	256	24	q(qψ	q(qψ	NUM
cana-5994	256	25	;	;	PUNCT
cana-5994	256	26	η	η	PROPN
cana-5994	256	27	)	)	PUNCT
cana-5994	256	28	=	=	SYM
cana-5994	256	29	0	0	X
cana-5994	256	30	.	.	PUNCT
cana-5994	256	31	proof	proof	NOUN
cana-5994	256	32	.	.	PUNCT
cana-5994	257	1	using	use	VERB
cana-5994	257	2	(	(	PUNCT
cana-5994	257	3	1.5	1.5	NUM
cana-5994	257	4	)	)	PUNCT
cana-5994	257	5	,	,	PUNCT
cana-5994	257	6	we	we	PRON
cana-5994	257	7	have	have	VERB
cana-5994	257	8	dq	dq	PROPN
cana-5994	257	9	,	,	PUNCT
cana-5994	257	10	φ	φ	X
cana-5994	257	11	∞∑	∞∑	NUM
cana-5994	257	12	υ=0	υ=0	ADJ
cana-5994	257	13	ftυ	ftυ	NOUN
cana-5994	257	14	,	,	PUNCT
cana-5994	257	15	q(qψ	q(qψ	NUM
cana-5994	257	16	;	;	PUNCT
cana-5994	257	17	η	η	PROPN
cana-5994	257	18	)	)	PUNCT
cana-5994	257	19	φυ	φυ	ADP
cana-5994	258	1	[	[	X
cana-5994	259	1	υ]q	υ]q	NOUN
cana-5994	259	2	!	!	PUNCT
cana-5994	259	3	=	=	NOUN
cana-5994	260	1	∞∑	∞∑	PRON
cana-5994	260	2	υ=0	υ=0	PUNCT
cana-5994	260	3	qυftυ	qυftυ	NOUN
cana-5994	260	4	,	,	PUNCT
cana-5994	260	5	q(ψ	q(ψ	NUM
cana-5994	260	6	;	;	PUNCT
cana-5994	260	7	η	η	X
cana-5994	260	8	)	)	PUNCT
cana-5994	260	9	φυ	φυ	ADP
cana-5994	260	10	[	[	X
cana-5994	260	11	υ]q	υ]q	NOUN
cana-5994	260	12	!	!	PUNCT
cana-5994	260	13	(	(	PUNCT
cana-5994	260	14	qψ	qψ	ADP
cana-5994	260	15	−	−	VERB
cana-5994	260	16	∞∑	∞∑	NOUN
cana-5994	260	17	υ=0	υ=0	X
cana-5994	260	18	(	(	PUNCT
cana-5994	260	19	1−	1−	NUM
cana-5994	260	20	η)υhυ	η)υhυ	NOUN
cana-5994	260	21	,	,	PUNCT
cana-5994	260	22	q(η	q(η	PROPN
cana-5994	260	23	)	)	PUNCT
cana-5994	260	24	φυ	φυ	ADP
cana-5994	260	25	[	[	X
cana-5994	260	26	υ]q	υ]q	NOUN
cana-5994	260	27	!	!	PUNCT
cana-5994	261	1	∞∑	∞∑	PRON
cana-5994	261	2	υ=0	υ=0	X
cana-5994	261	3	(	(	PUNCT
cana-5994	261	4	1−	1−	NUM
cana-5994	261	5	η)υ	η)υ	ADV
cana-5994	261	6	φυ	φυ	ADP
cana-5994	261	7	[	[	X
cana-5994	261	8	υ]q	υ]q	NOUN
cana-5994	261	9	!	!	PUNCT
cana-5994	261	10	)	)	PUNCT
cana-5994	262	1	=	=	PUNCT
cana-5994	263	1	∞∑	∞∑	PRON
cana-5994	263	2	υ=0	υ=0	PUNCT
cana-5994	263	3	(	(	PUNCT
cana-5994	263	4	qυ+1ψftυ	qυ+1ψftυ	NOUN
cana-5994	263	5	,	,	PUNCT
cana-5994	263	6	q(ψ	q(ψ	NUM
cana-5994	263	7	;	;	PUNCT
cana-5994	263	8	η)−	η)−	PROPN
cana-5994	263	9	υ∑	υ∑	PROPN
cana-5994	263	10	θ=0	θ=0	PROPN
cana-5994	263	11	θ∑	θ∑	ADP
cana-5994	263	12	ν=0	ν=0	PRON
cana-5994	263	13	(	(	PUNCT
cana-5994	263	14	υ	υ	NOUN
cana-5994	263	15	θ	θ	NOUN
cana-5994	263	16	)	)	PUNCT
cana-5994	263	17	q	q	PROPN
cana-5994	264	1	(	(	PUNCT
cana-5994	264	2	θ	θ	NOUN
cana-5994	264	3	ν	ν	NOUN
cana-5994	264	4	)	)	PUNCT
cana-5994	264	5	q	q	PROPN
cana-5994	265	1	(	(	PUNCT
cana-5994	265	2	1−	1−	NUM
cana-5994	265	3	η)θqυ−θhν	η)θqυ−θhν	PROPN
cana-5994	265	4	,	,	PUNCT
cana-5994	265	5	q(η)ftυ−θ	q(η)ftυ−θ	NOUN
cana-5994	265	6	,	,	PUNCT
cana-5994	265	7	q(ψ	q(ψ	NUM
cana-5994	265	8	;	;	PUNCT
cana-5994	265	9	η	η	NOUN
cana-5994	265	10	)	)	PUNCT
cana-5994	265	11	)	)	PUNCT
cana-5994	266	1	φυ	φυ	ADP
cana-5994	267	1	[	[	X
cana-5994	267	2	υ]q	υ]q	NOUN
cana-5994	267	3	!	!	PUNCT
cana-5994	267	4	.	.	PUNCT
cana-5994	268	1	therefore	therefore	ADV
cana-5994	268	2	,	,	PUNCT
cana-5994	268	3	we	we	PRON
cana-5994	268	4	have	have	VERB
cana-5994	268	5	υ−1∑	υ−1∑	NOUN
cana-5994	268	6	θ=0	θ=0	NOUN
cana-5994	268	7	θ∑	θ∑	ADP
cana-5994	268	8	ν=0	ν=0	PROPN
cana-5994	268	9	(	(	PUNCT
cana-5994	268	10	1−	1−	NUM
cana-5994	268	11	η)θqυ−θ−1hν	η)θqυ−θ−1hν	NOUN
cana-5994	268	12	,	,	PUNCT
cana-5994	268	13	q(η	q(η	PROPN
cana-5994	268	14	)	)	PUNCT
cana-5994	269	1	[	[	X
cana-5994	269	2	θ	θ	X
cana-5994	269	3	−	−	NOUN
cana-5994	269	4	ν]q![ν]q	ν]q![ν]q	NOUN
cana-5994	269	5	!	!	PUNCT
cana-5994	270	1	d	d	X
cana-5994	270	2	(	(	PUNCT
cana-5994	270	3	θ	θ	NOUN
cana-5994	270	4	)	)	PUNCT
cana-5994	270	5	q	q	NOUN
cana-5994	270	6	,	,	PUNCT
cana-5994	270	7	ψftυ−θ	ψftυ−θ	NOUN
cana-5994	270	8	,	,	PUNCT
cana-5994	270	9	q(ψ	q(ψ	NOUN
cana-5994	270	10	;	;	PUNCT
cana-5994	270	11	η)−qυψftυ−θ	η)−qυψftυ−θ	PROPN
cana-5994	270	12	,	,	PUNCT
cana-5994	270	13	q(ψ	q(ψ	NOUN
cana-5994	270	14	;	;	PUNCT
cana-5994	270	15	η)+ftυ	η)+ftυ	NOUN
cana-5994	270	16	,	,	PUNCT
cana-5994	270	17	q(qψ	q(qψ	NUM
cana-5994	270	18	;	;	PUNCT
cana-5994	270	19	η	η	PROPN
cana-5994	270	20	)	)	PUNCT
cana-5994	270	21	=	=	SYM
cana-5994	270	22	0	0	NUM
cana-5994	270	23	,	,	PUNCT
cana-5994	270	24	(	(	PUNCT
cana-5994	270	25	2.14	2.14	NUM
cana-5994	270	26	)	)	PUNCT
cana-5994	270	27	which	which	PRON
cana-5994	270	28	is	be	AUX
cana-5994	270	29	the	the	DET
cana-5994	270	30	desired	desire	VERB
cana-5994	270	31	result	result	NOUN
cana-5994	270	32	.	.	PUNCT
cana-5994	271	1	□	□	PUNCT
cana-5994	271	2	corollary	corollary	ADJ
cana-5994	271	3	2.4	2.4	NUM
cana-5994	271	4	.	.	PUNCT
cana-5994	272	1	as	as	ADP
cana-5994	272	2	q	q	NOUN
cana-5994	272	3	approaches	approach	NOUN
cana-5994	272	4	1	1	NUM
cana-5994	272	5	in	in	ADP
cana-5994	272	6	theorem	theorem	ADJ
cana-5994	272	7	2.5	2.5	NUM
cana-5994	272	8	,	,	PUNCT
cana-5994	272	9	we	we	PRON
cana-5994	272	10	derive	derive	VERB
cana-5994	272	11	υ−1∑	υ−1∑	ADJ
cana-5994	272	12	ν=0	ν=0	PROPN
cana-5994	272	13	(	(	PUNCT
cana-5994	272	14	1−	1−	NUM
cana-5994	272	15	η)υ−1hν(η	η)υ−1hν(η	NOUN
cana-5994	272	16	)	)	PUNCT
cana-5994	273	1	[	[	X
cana-5994	273	2	υ	υ	X
cana-5994	273	3	−	−	NOUN
cana-5994	273	4	ν	ν	NOUN
cana-5994	273	5	−	−	NOUN
cana-5994	273	6	1]![ν	1]![ν	NUM
cana-5994	273	7	]	]	PUNCT
cana-5994	273	8	!	!	PUNCT
cana-5994	274	1	d	d	X
cana-5994	274	2	(	(	PUNCT
cana-5994	274	3	υ−1	υ−1	PROPN
cana-5994	274	4	)	)	PUNCT
cana-5994	274	5	ψ	ψ	NOUN
cana-5994	274	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	274	7	;	;	PUNCT
cana-5994	274	8	η)+	η)+	VERB
cana-5994	274	9	υ−2∑	υ−2∑	ADV
cana-5994	274	10	ν=0	ν=0	PROPN
cana-5994	274	11	(	(	PUNCT
cana-5994	274	12	1−	1−	NUM
cana-5994	274	13	η)υ−2hν(η	η)υ−2hν(η	NOUN
cana-5994	274	14	)	)	PUNCT
cana-5994	275	1	[	[	X
cana-5994	275	2	υ	υ	X
cana-5994	275	3	−	−	NOUN
cana-5994	275	4	ν	ν	NOUN
cana-5994	275	5	−	−	NOUN
cana-5994	275	6	2]![ν	2]![ν	NUM
cana-5994	275	7	]	]	X
cana-5994	275	8	!	!	PUNCT
cana-5994	276	1	d	d	X
cana-5994	276	2	(	(	PUNCT
cana-5994	276	3	υ−2	υ−2	NOUN
cana-5994	276	4	)	)	PUNCT
cana-5994	276	5	ψ	ψ	NOUN
cana-5994	276	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	276	7	;	;	PUNCT
cana-5994	276	8	η)+	η)+	X
cana-5994	276	9	·	·	PUNCT
cana-5994	276	10	·	·	PUNCT
cana-5994	276	11	·	·	PUNCT
cana-5994	277	1	+	+	NUM
cana-5994	277	2	2∑	2∑	X
cana-5994	277	3	ν=0	ν=0	PRON
cana-5994	277	4	(	(	PUNCT
cana-5994	277	5	1−	1−	NUM
cana-5994	277	6	η)2hν(η	η)2hν(η	NOUN
cana-5994	277	7	)	)	PUNCT
cana-5994	278	1	[	[	X
cana-5994	278	2	2−	2−	NUM
cana-5994	278	3	ν]![ν	ν]![ν	PROPN
cana-5994	278	4	]	]	PUNCT
cana-5994	278	5	!	!	PUNCT
cana-5994	279	1	d	d	X
cana-5994	279	2	(	(	PUNCT
cana-5994	279	3	2	2	NUM
cana-5994	279	4	)	)	PUNCT
cana-5994	279	5	ψ	ψ	NOUN
cana-5994	279	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	279	7	;	;	PUNCT
cana-5994	279	8	η	η	X
cana-5994	279	9	)	)	PUNCT
cana-5994	280	1	+	+	CCONJ
cana-5994	280	2	1∑	1∑	NUM
cana-5994	280	3	ν=0	ν=0	PROPN
cana-5994	280	4	(	(	PUNCT
cana-5994	280	5	1−	1−	NUM
cana-5994	280	6	η)hν(η	η)hν(η	NOUN
cana-5994	280	7	)	)	PUNCT
cana-5994	281	1	[	[	X
cana-5994	281	2	1−	1−	NUM
cana-5994	281	3	ν]!ν	ν]!ν	NOUN
cana-5994	281	4	!	!	PUNCT
cana-5994	282	1	d	d	X
cana-5994	282	2	(	(	PUNCT
cana-5994	282	3	1	1	NUM
cana-5994	282	4	)	)	PUNCT
cana-5994	282	5	ψ	ψ	NOUN
cana-5994	282	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	282	7	;	;	PUNCT
cana-5994	282	8	η	η	X
cana-5994	282	9	)	)	PUNCT
cana-5994	283	1	+	+	CCONJ
cana-5994	283	2	(	(	PUNCT
cana-5994	283	3	h0	h0	NOUN
cana-5994	283	4	−	−	PROPN
cana-5994	283	5	ψ	ψ	SYM
cana-5994	283	6	)	)	PUNCT
cana-5994	283	7	ftυ−θ(ψ	ftυ−θ(ψ	PROPN
cana-5994	283	8	;	;	PUNCT
cana-5994	283	9	η	η	PROPN
cana-5994	283	10	)	)	PUNCT
cana-5994	283	11	+	+	X
cana-5994	283	12	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	283	13	;	;	PUNCT
cana-5994	283	14	η	η	NOUN
cana-5994	283	15	)	)	PUNCT
cana-5994	283	16	=	=	SYM
cana-5994	283	17	0	0	X
cana-5994	283	18	.	.	PUNCT
cana-5994	283	19	theorem	theorem	VERB
cana-5994	283	20	2.6	2.6	NUM
cana-5994	283	21	.	.	PUNCT
cana-5994	284	1	let	let	VERB
cana-5994	284	2	υ	υ	PRON
cana-5994	284	3	≥	≥	NOUN
cana-5994	284	4	0	0	NUM
cana-5994	284	5	.	.	PUNCT
cana-5994	285	1	then	then	ADV
cana-5994	285	2	ftυ−1,q(1−	ftυ−1,q(1−	VERB
cana-5994	285	3	η	η	PROPN
cana-5994	285	4	)	)	PUNCT
cana-5994	286	1	[	[	X
cana-5994	286	2	υ	υ	X
cana-5994	286	3	−	−	NOUN
cana-5994	286	4	1]q	1]q	NUM
cana-5994	286	5	!	!	PUNCT
cana-5994	287	1	d	d	X
cana-5994	287	2	(	(	PUNCT
cana-5994	287	3	υ−1	υ−1	PROPN
cana-5994	287	4	)	)	PUNCT
cana-5994	287	5	q	q	NOUN
cana-5994	287	6	,	,	PUNCT
cana-5994	287	7	ψ	ψ	X
cana-5994	287	8	ftυ−1,q(ψ	ftυ−1,q(ψ	NOUN
cana-5994	287	9	;	;	PUNCT
cana-5994	287	10	η	η	X
cana-5994	287	11	)	)	PUNCT
cana-5994	287	12	+	+	CCONJ
cana-5994	287	13	qftυ−2,q(1−	qftυ−2,q(1−	PROPN
cana-5994	287	14	η	η	PROPN
cana-5994	287	15	)	)	PUNCT
cana-5994	288	1	[	[	X
cana-5994	288	2	υ	υ	X
cana-5994	288	3	−	−	NOUN
cana-5994	288	4	2]q	2]q	NUM
cana-5994	288	5	!	!	PUNCT
cana-5994	289	1	d	d	X
cana-5994	289	2	(	(	PUNCT
cana-5994	289	3	υ−2	υ−2	NOUN
cana-5994	289	4	)	)	PUNCT
cana-5994	289	5	q	q	NOUN
cana-5994	289	6	,	,	PUNCT
cana-5994	289	7	ψ	ψ	X
cana-5994	289	8	ftυ−1,q(ψ	ftυ−1,q(ψ	NOUN
cana-5994	289	9	;	;	PUNCT
cana-5994	289	10	η	η	X
cana-5994	289	11	)	)	PUNCT
cana-5994	289	12	+	+	CCONJ
cana-5994	289	13	·	·	PUNCT
cana-5994	289	14	·	·	PUNCT
cana-5994	289	15	·	·	PUNCT
cana-5994	290	1	+	+	CCONJ
cana-5994	290	2	qυ−4	qυ−4	PROPN
cana-5994	290	3	ft3,q(1−	ft3,q(1−	PROPN
cana-5994	290	4	η	η	PROPN
cana-5994	290	5	)	)	PUNCT
cana-5994	290	6	[	[	X
cana-5994	290	7	3]q	3]q	NUM
cana-5994	290	8	!	!	PUNCT
cana-5994	291	1	d	d	NOUN
cana-5994	291	2	(	(	PUNCT
cana-5994	291	3	3	3	NUM
cana-5994	291	4	)	)	PUNCT
cana-5994	291	5	q	q	NOUN
cana-5994	291	6	,	,	PUNCT
cana-5994	291	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	291	8	;	;	PUNCT
cana-5994	291	9	η	η	PROPN
cana-5994	291	10	)	)	PUNCT
cana-5994	291	11	+	+	NUM
cana-5994	291	12	qυ−3	qυ−3	PROPN
cana-5994	291	13	ft2,q(1−	ft2,q(1−	PROPN
cana-5994	291	14	η	η	PROPN
cana-5994	291	15	)	)	PUNCT
cana-5994	292	1	[	[	X
cana-5994	292	2	2]q	2]q	NUM
cana-5994	292	3	!	!	PUNCT
cana-5994	293	1	d	d	X
cana-5994	293	2	(	(	PUNCT
cana-5994	293	3	2	2	NUM
cana-5994	293	4	)	)	PUNCT
cana-5994	293	5	q	q	NOUN
cana-5994	293	6	,	,	PUNCT
cana-5994	293	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	293	8	;	;	PUNCT
cana-5994	293	9	η	η	PROPN
cana-5994	293	10	)	)	PUNCT
cana-5994	293	11	+	+	PROPN
cana-5994	293	12	qυ−2	qυ−2	PROPN
cana-5994	293	13	ft1,q(1−η)d(1	ft1,q(1−η)d(1	PROPN
cana-5994	293	14	)	)	PUNCT
cana-5994	293	15	q	q	NOUN
cana-5994	293	16	,	,	PUNCT
cana-5994	293	17	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	293	18	;	;	PUNCT
cana-5994	293	19	η)+	η)+	NOUN
cana-5994	293	20	(	(	PUNCT
cana-5994	293	21	q−1	q−1	PROPN
cana-5994	293	22	ft0,q(1−	ft0,q(1−	VERB
cana-5994	293	23	η)−	η)−	PROPN
cana-5994	293	24	ψ	ψ	NOUN
cana-5994	293	25	)	)	PUNCT
cana-5994	293	26	qυftυ−1,q(ψ	qυftυ−1,q(ψ	PROPN
cana-5994	293	27	;	;	PUNCT
cana-5994	293	28	η)+ftυ	η)+ftυ	NOUN
cana-5994	293	29	,	,	PUNCT
cana-5994	293	30	q(qψ	q(qψ	NUM
cana-5994	293	31	;	;	PUNCT
cana-5994	293	32	η	η	PROPN
cana-5994	293	33	)	)	PUNCT
cana-5994	293	34	=	=	SYM
cana-5994	293	35	0	0	X
cana-5994	293	36	.	.	PUNCT
cana-5994	293	37	proof	proof	NOUN
cana-5994	293	38	.	.	PUNCT
cana-5994	294	1	by	by	ADP
cana-5994	294	2	using	use	VERB
cana-5994	294	3	(	(	PUNCT
cana-5994	294	4	1.5	1.5	NUM
cana-5994	294	5	)	)	PUNCT
cana-5994	294	6	,	,	PUNCT
cana-5994	294	7	(	(	PUNCT
cana-5994	294	8	1.8	1.8	NUM
cana-5994	294	9	)	)	PUNCT
cana-5994	294	10	and	and	CCONJ
cana-5994	294	11	(	(	PUNCT
cana-5994	294	12	2.8	2.8	NUM
cana-5994	294	13	)	)	PUNCT
cana-5994	294	14	,	,	PUNCT
cana-5994	294	15	we	we	PRON
cana-5994	294	16	have	have	VERB
cana-5994	294	17	dq	dq	PROPN
cana-5994	294	18	,	,	PUNCT
cana-5994	294	19	φ	φ	X
cana-5994	294	20	∞∑	∞∑	NUM
cana-5994	294	21	υ=0	υ=0	ADJ
cana-5994	294	22	ftυ	ftυ	NOUN
cana-5994	294	23	,	,	PUNCT
cana-5994	294	24	q(qψ	q(qψ	NUM
cana-5994	294	25	;	;	PUNCT
cana-5994	294	26	η	η	PROPN
cana-5994	294	27	)	)	PUNCT
cana-5994	294	28	φυ	φυ	ADP
cana-5994	295	1	[	[	X
cana-5994	296	1	υ]q	υ]q	NOUN
cana-5994	296	2	!	!	PUNCT
cana-5994	296	3	=	=	NOUN
cana-5994	297	1	∞∑	∞∑	PRON
cana-5994	297	2	υ=0	υ=0	PUNCT
cana-5994	297	3	(	(	PUNCT
cana-5994	297	4	qυ+1ψftυ	qυ+1ψftυ	NOUN
cana-5994	297	5	,	,	PUNCT
cana-5994	297	6	q(ψ	q(ψ	NUM
cana-5994	297	7	;	;	PUNCT
cana-5994	297	8	η)−	η)−	PROPN
cana-5994	297	9	υ∑	υ∑	NOUN
cana-5994	297	10	ν=0	ν=0	PROPN
cana-5994	297	11	(	(	PUNCT
cana-5994	297	12	υ	υ	NOUN
cana-5994	297	13	ν	ν	NOUN
cana-5994	297	14	)	)	PUNCT
cana-5994	297	15	q	q	PROPN
cana-5994	297	16	qυ−νftυ	qυ−νftυ	NOUN
cana-5994	297	17	,	,	PUNCT
cana-5994	297	18	q(1−	q(1−	NOUN
cana-5994	297	19	η)ftυ−ν	η)ftυ−ν	NUM
cana-5994	297	20	,	,	PUNCT
cana-5994	297	21	q(ψ	q(ψ	NUM
cana-5994	297	22	;	;	PUNCT
cana-5994	297	23	η	η	NOUN
cana-5994	297	24	)	)	PUNCT
cana-5994	297	25	)	)	PUNCT
cana-5994	297	26	φυ	φυ	ADP
cana-5994	298	1	[	[	X
cana-5994	298	2	υ]q	υ]q	NOUN
cana-5994	298	3	!	!	PUNCT
cana-5994	298	4	.	.	PUNCT
cana-5994	299	1	(	(	PUNCT
cana-5994	299	2	2.15	2.15	NUM
cana-5994	299	3	)	)	PUNCT
cana-5994	299	4	on	on	ADP
cana-5994	299	5	multiplying	multiply	VERB
cana-5994	299	6	φ	φ	PROPN
cana-5994	299	7	in	in	ADP
cana-5994	299	8	the	the	DET
cana-5994	299	9	above	above	ADJ
cana-5994	299	10	equation	equation	NOUN
cana-5994	299	11	,	,	PUNCT
cana-5994	299	12	we	we	PRON
cana-5994	299	13	get	get	VERB
cana-5994	299	14	φdq	φdq	NOUN
cana-5994	299	15	,	,	PUNCT
cana-5994	299	16	φ	φ	NUM
cana-5994	299	17	∞∑	∞∑	NUM
cana-5994	299	18	υ=0	υ=0	ADJ
cana-5994	299	19	ftυ	ftυ	NOUN
cana-5994	299	20	,	,	PUNCT
cana-5994	299	21	q(qψ	q(qψ	NUM
cana-5994	299	22	;	;	PUNCT
cana-5994	299	23	η	η	PROPN
cana-5994	299	24	)	)	PUNCT
cana-5994	299	25	φυ	φυ	ADP
cana-5994	300	1	[	[	X
cana-5994	301	1	υ]q	υ]q	NOUN
cana-5994	301	2	!	!	PUNCT
cana-5994	301	3	=	=	NOUN
cana-5994	302	1	∞∑	∞∑	PRON
cana-5994	302	2	υ=0	υ=0	PUNCT
cana-5994	303	1	[	[	X
cana-5994	303	2	υ]qq	υ]qq	PROPN
cana-5994	303	3	υψftυ−1,q(q	υψftυ−1,q(q	PROPN
cana-5994	303	4	−1ψ	−1ψ	PROPN
cana-5994	303	5	;	;	PUNCT
cana-5994	303	6	η	η	X
cana-5994	303	7	)	)	PUNCT
cana-5994	303	8	φυ	φυ	ADP
cana-5994	304	1	[	[	X
cana-5994	304	2	υ]q	υ]q	X
cana-5994	304	3	!	!	PUNCT
cana-5994	304	4	communications	communication	NOUN
cana-5994	304	5	on	on	ADP
cana-5994	304	6	applied	apply	VERB
cana-5994	304	7	nonlinear	nonlinear	ADJ
cana-5994	304	8	analysis	analysis	NOUN
cana-5994	304	9	issn	issn	NOUN
cana-5994	304	10	:	:	PUNCT
cana-5994	304	11	1074	1074	NUM
cana-5994	304	12	-	-	PUNCT
cana-5994	304	13	133x	133x	NUM
cana-5994	304	14	vol	vol	NOUN
cana-5994	304	15	32	32	NUM
cana-5994	304	16	no	no	NOUN
cana-5994	304	17	.	.	PUNCT
cana-5994	305	1	9s	9s	NUM
cana-5994	305	2	(	(	PUNCT
cana-5994	305	3	2025	2025	NUM
cana-5994	305	4	)	)	PUNCT
cana-5994	305	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	305	6	3224	3224	NUM
cana-5994	305	7	8	8	NUM
cana-5994	305	8	idrees	idree	NOUN
cana-5994	305	9	ahmad	ahmad	PROPN
cana-5994	305	10	khan	khan	PROPN
cana-5994	305	11	and	and	CCONJ
cana-5994	305	12	sumit	sumit	PROPN
cana-5994	305	13	kumar	kumar	PROPN
cana-5994	305	14	−	−	PROPN
cana-5994	306	1	∞∑	∞∑	PROPN
cana-5994	306	2	υ=0	υ=0	PUNCT
cana-5994	306	3	[	[	X
cana-5994	306	4	υ]q	υ]q	NOUN
cana-5994	306	5	υ−1∑	υ−1∑	ADJ
cana-5994	306	6	ν=0	ν=0	NOUN
cana-5994	306	7	(	(	PUNCT
cana-5994	306	8	υ	υ	NOUN
cana-5994	306	9	−	−	PROPN
cana-5994	306	10	1	1	NUM
cana-5994	306	11	ν	ν	NOUN
cana-5994	306	12	)	)	PUNCT
cana-5994	306	13	q	q	PROPN
cana-5994	306	14	qυ−ν−1	qυ−ν−1	PROPN
cana-5994	306	15	ftν	ftν	NOUN
cana-5994	306	16	,	,	PUNCT
cana-5994	306	17	q(1−	q(1−	NOUN
cana-5994	306	18	η)ftυ−ν−1,q(ψ	η)ftυ−ν−1,q(ψ	NUM
cana-5994	306	19	;	;	PUNCT
cana-5994	306	20	η	η	X
cana-5994	306	21	)	)	PUNCT
cana-5994	306	22	φυ	φυ	ADP
cana-5994	307	1	[	[	X
cana-5994	308	1	υ]q	υ]q	NOUN
cana-5994	308	2	!	!	PUNCT
cana-5994	308	3	.	.	PUNCT
cana-5994	309	1	(	(	PUNCT
cana-5994	309	2	2.16	2.16	NUM
cana-5994	309	3	)	)	PUNCT
cana-5994	309	4	by	by	ADP
cana-5994	309	5	(	(	PUNCT
cana-5994	309	6	2.15	2.15	NUM
cana-5994	309	7	)	)	PUNCT
cana-5994	309	8	and	and	CCONJ
cana-5994	309	9	(	(	PUNCT
cana-5994	309	10	2.16	2.16	NUM
cana-5994	309	11	)	)	PUNCT
cana-5994	309	12	,	,	PUNCT
cana-5994	309	13	we	we	PRON
cana-5994	309	14	attain	attain	VERB
cana-5994	309	15	υ−1∑	υ−1∑	ADJ
cana-5994	309	16	ν=0	ν=0	NOUN
cana-5994	309	17	(	(	PUNCT
cana-5994	309	18	υ	υ	NOUN
cana-5994	309	19	−	−	PROPN
cana-5994	309	20	1	1	NUM
cana-5994	309	21	ν	ν	NOUN
cana-5994	309	22	)	)	PUNCT
cana-5994	309	23	q	q	PROPN
cana-5994	309	24	qυ−ν−1	qυ−ν−1	PROPN
cana-5994	309	25	ftν	ftν	NOUN
cana-5994	309	26	,	,	PUNCT
cana-5994	309	27	q(1−	q(1−	NOUN
cana-5994	309	28	η)ftυ−ν−1,q(ψ	η)ftυ−ν−1,q(ψ	NUM
cana-5994	309	29	;	;	PUNCT
cana-5994	309	30	η	η	PROPN
cana-5994	309	31	)	)	PUNCT
cana-5994	309	32	=	=	SYM
cana-5994	309	33	qυψftυ−1,q(q	qυψftυ−1,q(q	PROPN
cana-5994	309	34	−1ψ	−1ψ	PROPN
cana-5994	309	35	;	;	PUNCT
cana-5994	309	36	η)−	η)−	PROPN
cana-5994	309	37	qυψftυ−1,q(ψ	qυψftυ−1,q(ψ	NOUN
cana-5994	309	38	;	;	PUNCT
cana-5994	309	39	η)−	η)−	PROPN
cana-5994	309	40	ftυ	ftυ	NOUN
cana-5994	309	41	,	,	PUNCT
cana-5994	309	42	q(qψ	q(qψ	NUM
cana-5994	309	43	;	;	PUNCT
cana-5994	309	44	η	η	PROPN
cana-5994	309	45	)	)	PUNCT
cana-5994	309	46	.	.	PUNCT
cana-5994	310	1	(	(	PUNCT
cana-5994	310	2	2.17	2.17	NUM
cana-5994	310	3	)	)	PUNCT
cana-5994	310	4	applying	apply	VERB
cana-5994	310	5	a	a	DET
cana-5994	310	6	relation	relation	NOUN
cana-5994	310	7	between	between	ADP
cana-5994	310	8	dυ	dυ	NOUN
cana-5994	310	9	q	q	PROPN
cana-5994	310	10	,	,	PUNCT
cana-5994	310	11	ψftυ	ψftυ	NOUN
cana-5994	310	12	,	,	PUNCT
cana-5994	310	13	q(ψ	q(ψ	NUM
cana-5994	310	14	;	;	PUNCT
cana-5994	310	15	η	η	X
cana-5994	310	16	)	)	PUNCT
cana-5994	310	17	and	and	CCONJ
cana-5994	310	18	ftυ	ftυ	NOUN
cana-5994	310	19	,	,	PUNCT
cana-5994	310	20	q(ψ	q(ψ	NUM
cana-5994	310	21	;	;	PUNCT
cana-5994	310	22	η	η	X
cana-5994	310	23	)	)	PUNCT
cana-5994	310	24	in	in	ADP
cana-5994	310	25	the	the	DET
cana-5994	310	26	left	left	ADJ
cana-5994	310	27	-	-	PUNCT
cana-5994	310	28	hand	hand	NOUN
cana-5994	310	29	side	side	NOUN
cana-5994	310	30	of	of	ADP
cana-5994	310	31	(	(	PUNCT
cana-5994	310	32	2.17	2.17	NUM
cana-5994	310	33	)	)	PUNCT
cana-5994	310	34	,	,	PUNCT
cana-5994	310	35	we	we	PRON
cana-5994	310	36	obtain	obtain	VERB
cana-5994	310	37	υ−1∑	υ−1∑	ADJ
cana-5994	310	38	ν=0	ν=0	NOUN
cana-5994	310	39	(	(	PUNCT
cana-5994	310	40	υ	υ	NOUN
cana-5994	310	41	−	−	PROPN
cana-5994	310	42	1	1	NUM
cana-5994	310	43	ν	ν	NOUN
cana-5994	310	44	)	)	PUNCT
cana-5994	310	45	q	q	PROPN
cana-5994	310	46	qυ−ν−1	qυ−ν−1	PROPN
cana-5994	310	47	ftν	ftν	NOUN
cana-5994	310	48	,	,	PUNCT
cana-5994	310	49	q(1−	q(1−	NOUN
cana-5994	310	50	η)ftυ−ν−1,q(ψ	η)ftυ−ν−1,q(ψ	NUM
cana-5994	310	51	;	;	PUNCT
cana-5994	310	52	η	η	X
cana-5994	310	53	)	)	PUNCT
cana-5994	310	54	=	=	SYM
cana-5994	310	55	υ−1∑	υ−1∑	ADJ
cana-5994	310	56	ν=0	ν=0	PROPN
cana-5994	310	57	qυ−ν−1	qυ−ν−1	NOUN
cana-5994	310	58	ftν	ftν	NOUN
cana-5994	310	59	,	,	PUNCT
cana-5994	310	60	q(1−	q(1−	PROPN
cana-5994	310	61	η	η	NOUN
cana-5994	310	62	)	)	PUNCT
cana-5994	311	1	[	[	X
cana-5994	311	2	ν]q	ν]q	NOUN
cana-5994	311	3	!	!	PUNCT
cana-5994	312	1	d	d	NOUN
cana-5994	312	2	(	(	PUNCT
cana-5994	312	3	ν	ν	NOUN
cana-5994	312	4	)	)	PUNCT
cana-5994	312	5	q	q	NOUN
cana-5994	312	6	,	,	PUNCT
cana-5994	312	7	ψftυ−1,q(ψ	ψftυ−1,q(ψ	PROPN
cana-5994	312	8	;	;	PUNCT
cana-5994	312	9	η	η	PROPN
cana-5994	312	10	)	)	PUNCT
cana-5994	312	11	.	.	PUNCT
cana-5994	313	1	(	(	PUNCT
cana-5994	313	2	2.18	2.18	NUM
cana-5994	313	3	)	)	PUNCT
cana-5994	313	4	hence	hence	ADV
cana-5994	313	5	,	,	PUNCT
cana-5994	313	6	complete	complete	VERB
cana-5994	313	7	the	the	DET
cana-5994	313	8	proof	proof	NOUN
cana-5994	313	9	.	.	PUNCT
cana-5994	314	1	□	□	PUNCT
cana-5994	314	2	corollary	corollary	ADJ
cana-5994	314	3	2.5	2.5	NUM
cana-5994	314	4	.	.	PUNCT
cana-5994	315	1	as	as	ADP
cana-5994	315	2	q	q	NOUN
cana-5994	315	3	approaches	approach	NOUN
cana-5994	315	4	1	1	NUM
cana-5994	315	5	in	in	ADP
cana-5994	315	6	theorem	theorem	NOUN
cana-5994	315	7	2.6	2.6	NUM
cana-5994	315	8	,	,	PUNCT
cana-5994	315	9	we	we	PRON
cana-5994	315	10	derive	derive	VERB
cana-5994	315	11	ftυ−1(1−	ftυ−1(1−	PROPN
cana-5994	315	12	η	η	PROPN
cana-5994	315	13	)	)	PUNCT
cana-5994	316	1	[	[	X
cana-5994	316	2	υ	υ	X
cana-5994	316	3	−	−	NOUN
cana-5994	316	4	1	1	NUM
cana-5994	316	5	]	]	PUNCT
cana-5994	316	6	!	!	PUNCT
cana-5994	317	1	d	d	X
cana-5994	317	2	(	(	PUNCT
cana-5994	317	3	υ−1	υ−1	PROPN
cana-5994	317	4	)	)	PUNCT
cana-5994	317	5	ψ	ψ	NOUN
cana-5994	317	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	317	7	;	;	PUNCT
cana-5994	317	8	η	η	X
cana-5994	317	9	)	)	PUNCT
cana-5994	317	10	+	+	NUM
cana-5994	317	11	ftυ−2(1−	ftυ−2(1−	PROPN
cana-5994	317	12	η	η	PROPN
cana-5994	317	13	)	)	PUNCT
cana-5994	318	1	[	[	X
cana-5994	318	2	υ	υ	X
cana-5994	318	3	−	−	NOUN
cana-5994	318	4	2	2	NUM
cana-5994	318	5	]	]	PUNCT
cana-5994	318	6	!	!	PUNCT
cana-5994	319	1	d	d	X
cana-5994	319	2	(	(	PUNCT
cana-5994	319	3	υ−2	υ−2	NOUN
cana-5994	319	4	)	)	PUNCT
cana-5994	319	5	ψ	ψ	NOUN
cana-5994	319	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	319	7	;	;	PUNCT
cana-5994	319	8	η	η	X
cana-5994	319	9	)	)	PUNCT
cana-5994	319	10	+	+	CCONJ
cana-5994	319	11	·	·	PUNCT
cana-5994	319	12	·	·	PUNCT
cana-5994	319	13	·	·	PUNCT
cana-5994	320	1	+	+	CCONJ
cana-5994	320	2	ft3(1−	ft3(1−	PROPN
cana-5994	320	3	η	η	PROPN
cana-5994	320	4	)	)	PUNCT
cana-5994	321	1	[	[	X
cana-5994	321	2	3	3	NUM
cana-5994	321	3	]	]	PUNCT
cana-5994	321	4	!	!	PUNCT
cana-5994	322	1	d	d	X
cana-5994	322	2	(	(	PUNCT
cana-5994	322	3	3	3	NUM
cana-5994	322	4	)	)	PUNCT
cana-5994	322	5	ψ	ψ	NOUN
cana-5994	322	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	322	7	;	;	PUNCT
cana-5994	322	8	η	η	X
cana-5994	322	9	)	)	PUNCT
cana-5994	322	10	+	+	CCONJ
cana-5994	322	11	ft2(1−	ft2(1−	PROPN
cana-5994	322	12	η	η	NOUN
cana-5994	322	13	)	)	PUNCT
cana-5994	322	14	2	2	NUM
cana-5994	322	15	!	!	X
cana-5994	323	1	d	d	NOUN
cana-5994	323	2	(	(	PUNCT
cana-5994	323	3	2	2	NUM
cana-5994	323	4	)	)	PUNCT
cana-5994	323	5	ψ	ψ	NOUN
cana-5994	323	6	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	323	7	;	;	PUNCT
cana-5994	323	8	η	η	X
cana-5994	323	9	)	)	PUNCT
cana-5994	323	10	+	+	ADJ
cana-5994	323	11	ft1(1−	ft1(1−	X
cana-5994	323	12	η)d	η)d	NOUN
cana-5994	323	13	(	(	PUNCT
cana-5994	323	14	1	1	NUM
cana-5994	323	15	)	)	PUNCT
cana-5994	323	16	ψ	ψ	NOUN
cana-5994	323	17	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	323	18	;	;	PUNCT
cana-5994	323	19	η	η	X
cana-5994	323	20	)	)	PUNCT
cana-5994	324	1	+	+	CCONJ
cana-5994	324	2	(	(	PUNCT
cana-5994	324	3	ft0(1−	ft0(1−	PUNCT
cana-5994	324	4	η)−	η)−	PROPN
cana-5994	324	5	ψ	ψ	NOUN
cana-5994	324	6	)	)	PUNCT
cana-5994	324	7	ftυ−1(ψ	ftυ−1(ψ	NOUN
cana-5994	324	8	;	;	PUNCT
cana-5994	324	9	η	η	X
cana-5994	324	10	)	)	PUNCT
cana-5994	324	11	+	+	X
cana-5994	324	12	ftυ(ψ	ftυ(ψ	PROPN
cana-5994	324	13	;	;	PUNCT
cana-5994	324	14	η	η	NOUN
cana-5994	324	15	)	)	PUNCT
cana-5994	324	16	=	=	SYM
cana-5994	324	17	0	0	X
cana-5994	324	18	.	.	PUNCT
cana-5994	324	19	theorem	theorem	VERB
cana-5994	324	20	2.7	2.7	NUM
cana-5994	324	21	.	.	PUNCT
cana-5994	325	1	let	let	VERB
cana-5994	325	2	υ	υ	PRON
cana-5994	325	3	≥	≥	NOUN
cana-5994	325	4	0	0	NUM
cana-5994	325	5	.	.	PUNCT
cana-5994	326	1	then	then	ADV
cana-5994	326	2	ftν	ftν	NOUN
cana-5994	326	3	,	,	PUNCT
cana-5994	326	4	q(b−1ζ	q(b−1ζ	PROPN
cana-5994	326	5	;	;	PUNCT
cana-5994	326	6	η	η	PROPN
cana-5994	326	7	)	)	PUNCT
cana-5994	327	1	[	[	X
cana-5994	327	2	υ]q	υ]q	NOUN
cana-5994	327	3	!	!	PUNCT
cana-5994	328	1	d	d	NOUN
cana-5994	328	2	(	(	PUNCT
cana-5994	328	3	υ	υ	NOUN
cana-5994	328	4	)	)	PUNCT
cana-5994	328	5	q	q	NOUN
cana-5994	328	6	,	,	PUNCT
cana-5994	328	7	ψftυ	ψftυ	VERB
cana-5994	328	8	,	,	PUNCT
cana-5994	328	9	q(a−1ψ	q(a−1ψ	PROPN
cana-5994	328	10	;	;	PUNCT
cana-5994	328	11	η	η	X
cana-5994	328	12	)	)	PUNCT
cana-5994	328	13	+	+	CCONJ
cana-5994	328	14	b−1	b−1	PROPN
cana-5994	328	15	ftυ−1,q(b	ftυ−1,q(b	PROPN
cana-5994	328	16	−1ζ	−1ζ	PROPN
cana-5994	328	17	;	;	PUNCT
cana-5994	328	18	η	η	PROPN
cana-5994	328	19	)	)	PUNCT
cana-5994	329	1	[	[	X
cana-5994	329	2	υ	υ	X
cana-5994	329	3	−	−	NOUN
cana-5994	329	4	1]q	1]q	NUM
cana-5994	329	5	!	!	PUNCT
cana-5994	330	1	d	d	X
cana-5994	330	2	(	(	PUNCT
cana-5994	330	3	υ−1	υ−1	PROPN
cana-5994	330	4	)	)	PUNCT
cana-5994	330	5	q	q	NOUN
cana-5994	330	6	,	,	PUNCT
cana-5994	330	7	ψ	ψ	X
cana-5994	330	8	ftυ	ftυ	NOUN
cana-5994	330	9	,	,	PUNCT
cana-5994	330	10	q(a−1ψ	q(a−1ψ	PROPN
cana-5994	330	11	;	;	PUNCT
cana-5994	330	12	η	η	X
cana-5994	330	13	)	)	PUNCT
cana-5994	330	14	+	+	CCONJ
cana-5994	330	15	·	·	PUNCT
cana-5994	330	16	·	·	PUNCT
cana-5994	330	17	·	·	PUNCT
cana-5994	331	1	+	+	X
cana-5994	331	2	b1−υft1,q(b	b1−υft1,q(b	PROPN
cana-5994	331	3	−1ζ	−1ζ	PROPN
cana-5994	331	4	;	;	PUNCT
cana-5994	331	5	η)d	η)d	X
cana-5994	331	6	(	(	PUNCT
cana-5994	331	7	1	1	NUM
cana-5994	331	8	)	)	PUNCT
cana-5994	331	9	q	q	NOUN
cana-5994	331	10	,	,	PUNCT
cana-5994	331	11	ψftυ	ψftυ	VERB
cana-5994	331	12	,	,	PUNCT
cana-5994	331	13	q(a−1ψ	q(a−1ψ	PROPN
cana-5994	331	14	;	;	PUNCT
cana-5994	331	15	η	η	X
cana-5994	331	16	)	)	PUNCT
cana-5994	331	17	+	+	CCONJ
cana-5994	331	18	b−υft0,q(b	b−υft0,q(b	PROPN
cana-5994	331	19	−1ζ	−1ζ	PROPN
cana-5994	331	20	;	;	PUNCT
cana-5994	331	21	η)ftυ	η)ftυ	NOUN
cana-5994	331	22	,	,	PUNCT
cana-5994	331	23	q(a−1ψ	q(a−1ψ	AUX
cana-5994	331	24	;	;	PUNCT
cana-5994	331	25	η	η	NOUN
cana-5994	331	26	)	)	PUNCT
cana-5994	331	27	=	=	SYM
cana-5994	331	28	ftν	ftν	NOUN
cana-5994	331	29	,	,	PUNCT
cana-5994	331	30	q(a−1ζ	q(a−1ζ	PROPN
cana-5994	331	31	;	;	PUNCT
cana-5994	331	32	η	η	PROPN
cana-5994	331	33	)	)	PUNCT
cana-5994	332	1	[	[	X
cana-5994	332	2	υ]q	υ]q	NOUN
cana-5994	332	3	!	!	PUNCT
cana-5994	333	1	d	d	NOUN
cana-5994	333	2	(	(	PUNCT
cana-5994	333	3	υ	υ	NOUN
cana-5994	333	4	)	)	PUNCT
cana-5994	333	5	q	q	NOUN
cana-5994	333	6	,	,	PUNCT
cana-5994	333	7	ψftυ	ψftυ	NOUN
cana-5994	333	8	,	,	PUNCT
cana-5994	333	9	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	333	10	;	;	PUNCT
cana-5994	333	11	η)+	η)+	PROPN
cana-5994	333	12	a−1	a−1	PROPN
cana-5994	333	13	ftυ−1,q(a	ftυ−1,q(a	PROPN
cana-5994	333	14	−1ζ	−1ζ	PROPN
cana-5994	333	15	;	;	PUNCT
cana-5994	333	16	η	η	PROPN
cana-5994	333	17	)	)	PUNCT
cana-5994	334	1	[	[	X
cana-5994	334	2	υ	υ	X
cana-5994	334	3	−	−	NOUN
cana-5994	334	4	1]q	1]q	NUM
cana-5994	334	5	!	!	PUNCT
cana-5994	335	1	d	d	X
cana-5994	335	2	(	(	PUNCT
cana-5994	335	3	υ−1	υ−1	PROPN
cana-5994	335	4	)	)	PUNCT
cana-5994	335	5	q	q	NOUN
cana-5994	335	6	,	,	PUNCT
cana-5994	335	7	ψ	ψ	X
cana-5994	335	8	ftυ	ftυ	NOUN
cana-5994	335	9	,	,	PUNCT
cana-5994	335	10	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	335	11	;	;	PUNCT
cana-5994	335	12	η)+	η)+	X
cana-5994	335	13	·	·	PUNCT
cana-5994	335	14	·	·	PUNCT
cana-5994	335	15	·	·	PUNCT
cana-5994	336	1	+	+	X
cana-5994	336	2	a1−υft1,q(a	a1−υft1,q(a	PROPN
cana-5994	336	3	−1ζ	−1ζ	PROPN
cana-5994	336	4	;	;	PUNCT
cana-5994	336	5	η)d	η)d	X
cana-5994	336	6	(	(	PUNCT
cana-5994	336	7	1	1	NUM
cana-5994	336	8	)	)	PUNCT
cana-5994	336	9	q	q	NOUN
cana-5994	336	10	,	,	PUNCT
cana-5994	336	11	ψftυ	ψftυ	NOUN
cana-5994	336	12	,	,	PUNCT
cana-5994	336	13	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	336	14	;	;	PUNCT
cana-5994	336	15	η	η	NOUN
cana-5994	336	16	)	)	PUNCT
cana-5994	337	1	+	+	PROPN
cana-5994	337	2	a−υft0,q(a	a−υft0,q(a	PROPN
cana-5994	337	3	−1ζ	−1ζ	PROPN
cana-5994	337	4	;	;	PUNCT
cana-5994	337	5	η)ftυ	η)ftυ	NOUN
cana-5994	337	6	,	,	PUNCT
cana-5994	337	7	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	337	8	;	;	PUNCT
cana-5994	337	9	η	η	NOUN
cana-5994	337	10	)	)	PUNCT
cana-5994	337	11	.	.	PUNCT
cana-5994	338	1	proof	proof	NOUN
cana-5994	338	2	.	.	PUNCT
cana-5994	339	1	let	let	VERB
cana-5994	339	2	a(φ	a(φ	NUM
cana-5994	339	3	)	)	PUNCT
cana-5994	339	4	=	=	PUNCT
cana-5994	340	1	(	(	PUNCT
cana-5994	340	2	1−	1−	NUM
cana-5994	340	3	η)2eq(ab(ψ	η)2eq(ab(ψ	PROPN
cana-5994	340	4	+	+	CCONJ
cana-5994	340	5	η)φ	η)φ	X
cana-5994	340	6	)	)	PUNCT
cana-5994	340	7	(	(	PUNCT
cana-5994	340	8	eq((1−	eq((1−	PROPN
cana-5994	340	9	η)aφ)−	η)aφ)−	DET
cana-5994	340	10	η)(eq((1−	η)(eq((1−	NUM
cana-5994	340	11	η)bφ)−	η)bφ)−	PROPN
cana-5994	340	12	η	η	NOUN
cana-5994	340	13	)	)	PUNCT
cana-5994	340	14	.	.	PUNCT
cana-5994	341	1	using	use	VERB
cana-5994	341	2	the	the	DET
cana-5994	341	3	definition	definition	NOUN
cana-5994	341	4	(	(	PUNCT
cana-5994	341	5	1.5	1.5	NUM
cana-5994	341	6	)	)	PUNCT
cana-5994	341	7	and	and	CCONJ
cana-5994	341	8	cauchy	cauchy	PROPN
cana-5994	341	9	products	product	NOUN
cana-5994	341	10	,	,	PUNCT
cana-5994	341	11	then	then	ADV
cana-5994	341	12	a(φ	a(φ	NUM
cana-5994	341	13	)	)	PUNCT
cana-5994	341	14	=	=	PUNCT
cana-5994	342	1	∞∑	∞∑	NUM
cana-5994	342	2	υ=0	υ=0	PUNCT
cana-5994	342	3	(	(	PUNCT
cana-5994	342	4	υ∑	υ∑	VERB
cana-5994	342	5	ν=0	ν=0	PROPN
cana-5994	342	6	(	(	PUNCT
cana-5994	342	7	υ	υ	NOUN
cana-5994	342	8	ν	ν	NOUN
cana-5994	342	9	)	)	PUNCT
cana-5994	342	10	q	q	PROPN
cana-5994	342	11	aυ−νbνftν	aυ−νbνftν	NOUN
cana-5994	342	12	,	,	PUNCT
cana-5994	342	13	q(b−1ζ	q(b−1ζ	PROPN
cana-5994	342	14	;	;	PUNCT
cana-5994	342	15	η)ftυ−ν	η)ftυ−ν	X
cana-5994	342	16	,	,	PUNCT
cana-5994	342	17	q(a−1ψ	q(a−1ψ	PROPN
cana-5994	342	18	;	;	PUNCT
cana-5994	342	19	η	η	NOUN
cana-5994	342	20	)	)	PUNCT
cana-5994	342	21	)	)	PUNCT
cana-5994	342	22	φυ	φυ	ADP
cana-5994	343	1	[	[	X
cana-5994	343	2	υ]q	υ]q	NOUN
cana-5994	343	3	!	!	PUNCT
cana-5994	343	4	.	.	PUNCT
cana-5994	344	1	(	(	PUNCT
cana-5994	344	2	2.19	2.19	NUM
cana-5994	344	3	)	)	PUNCT
cana-5994	344	4	similarly	similarly	ADV
cana-5994	344	5	,	,	PUNCT
cana-5994	344	6	we	we	PRON
cana-5994	344	7	have	have	VERB
cana-5994	344	8	a(φ	a(φ	NUM
cana-5994	344	9	)	)	PUNCT
cana-5994	344	10	=	=	PUNCT
cana-5994	345	1	∞∑	∞∑	NUM
cana-5994	345	2	υ=0	υ=0	PUNCT
cana-5994	345	3	(	(	PUNCT
cana-5994	345	4	υ∑	υ∑	VERB
cana-5994	345	5	ν=0	ν=0	PROPN
cana-5994	345	6	(	(	PUNCT
cana-5994	345	7	υ	υ	NOUN
cana-5994	345	8	ν	ν	NOUN
cana-5994	345	9	)	)	PUNCT
cana-5994	345	10	q	q	PROPN
cana-5994	345	11	bυ−νaνftν	bυ−νaνftν	PROPN
cana-5994	345	12	,	,	PUNCT
cana-5994	345	13	q(a−1ζ	q(a−1ζ	PROPN
cana-5994	345	14	;	;	PUNCT
cana-5994	345	15	η)ftυ−ν	η)ftυ−ν	X
cana-5994	345	16	,	,	PUNCT
cana-5994	345	17	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	345	18	;	;	PUNCT
cana-5994	345	19	η	η	NOUN
cana-5994	345	20	)	)	PUNCT
cana-5994	345	21	)	)	PUNCT
cana-5994	345	22	φυ	φυ	ADP
cana-5994	346	1	[	[	X
cana-5994	346	2	υ]q	υ]q	NOUN
cana-5994	346	3	!	!	PUNCT
cana-5994	346	4	.	.	PUNCT
cana-5994	347	1	(	(	PUNCT
cana-5994	347	2	2.20	2.20	NUM
cana-5994	347	3	)	)	PUNCT
cana-5994	347	4	by	by	ADP
cana-5994	347	5	(	(	PUNCT
cana-5994	347	6	2.19	2.19	NUM
cana-5994	347	7	)	)	PUNCT
cana-5994	347	8	and	and	CCONJ
cana-5994	347	9	(	(	PUNCT
cana-5994	347	10	2.20	2.20	NUM
cana-5994	347	11	)	)	PUNCT
cana-5994	347	12	,	,	PUNCT
cana-5994	347	13	we	we	PRON
cana-5994	347	14	have	have	AUX
cana-5994	347	15	υ∑	υ∑	VERB
cana-5994	347	16	ν=0	ν=0	NOUN
cana-5994	347	17	(	(	PUNCT
cana-5994	347	18	υ	υ	NOUN
cana-5994	347	19	ν	ν	NOUN
cana-5994	347	20	)	)	PUNCT
cana-5994	347	21	q	q	PROPN
cana-5994	347	22	aυ−νbνftν	aυ−νbνftν	NOUN
cana-5994	347	23	,	,	PUNCT
cana-5994	347	24	q(b−1ζ	q(b−1ζ	PROPN
cana-5994	347	25	;	;	PUNCT
cana-5994	347	26	η)ftυ−ν	η)ftυ−ν	X
cana-5994	347	27	,	,	PUNCT
cana-5994	347	28	q(a−1ψ	q(a−1ψ	PROPN
cana-5994	347	29	;	;	PUNCT
cana-5994	347	30	η	η	NOUN
cana-5994	347	31	)	)	PUNCT
cana-5994	347	32	communications	communication	NOUN
cana-5994	347	33	on	on	ADP
cana-5994	347	34	applied	apply	VERB
cana-5994	347	35	nonlinear	nonlinear	ADJ
cana-5994	347	36	analysis	analysis	NOUN
cana-5994	347	37	issn	issn	NOUN
cana-5994	347	38	:	:	PUNCT
cana-5994	347	39	1074	1074	NUM
cana-5994	347	40	-	-	PUNCT
cana-5994	347	41	133x	133x	NUM
cana-5994	347	42	vol	vol	NOUN
cana-5994	347	43	32	32	NUM
cana-5994	347	44	no	no	NOUN
cana-5994	347	45	.	.	PUNCT
cana-5994	348	1	9s	9s	NUM
cana-5994	348	2	(	(	PUNCT
cana-5994	348	3	2025	2025	NUM
cana-5994	348	4	)	)	PUNCT
cana-5994	349	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	349	2	3225	3225	NUM
cana-5994	350	1	some	some	DET
cana-5994	350	2	properties	property	NOUN
cana-5994	350	3	of	of	ADP
cana-5994	350	4	differential	differential	ADJ
cana-5994	350	5	equations	equation	NOUN
cana-5994	350	6	of	of	ADP
cana-5994	350	7	higher	high	ADJ
cana-5994	350	8	-	-	PUNCT
cana-5994	350	9	order	order	NOUN
cana-5994	350	10	9	9	NUM
cana-5994	350	11	=	=	SYM
cana-5994	350	12	υ∑	υ∑	ADJ
cana-5994	350	13	ν=0	ν=0	PRON
cana-5994	350	14	(	(	PUNCT
cana-5994	350	15	υ	υ	NOUN
cana-5994	350	16	ν	ν	NOUN
cana-5994	350	17	)	)	PUNCT
cana-5994	350	18	q	q	PROPN
cana-5994	350	19	bυ−νaνftν	bυ−νaνftν	PROPN
cana-5994	350	20	,	,	PUNCT
cana-5994	350	21	q(a−1ζ	q(a−1ζ	PROPN
cana-5994	350	22	;	;	PUNCT
cana-5994	350	23	η)ftυ−ν	η)ftυ−ν	X
cana-5994	350	24	,	,	PUNCT
cana-5994	350	25	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	350	26	;	;	PUNCT
cana-5994	350	27	η	η	NOUN
cana-5994	350	28	)	)	PUNCT
cana-5994	350	29	.	.	PUNCT
cana-5994	351	1	(	(	PUNCT
cana-5994	351	2	2.21	2.21	NUM
cana-5994	351	3	)	)	PUNCT
cana-5994	351	4	applying	apply	VERB
cana-5994	351	5	a	a	DET
cana-5994	351	6	relation	relation	NOUN
cana-5994	351	7	between	between	ADP
cana-5994	351	8	d	d	PROPN
cana-5994	351	9	(	(	PUNCT
cana-5994	351	10	υ	υ	NOUN
cana-5994	351	11	)	)	PUNCT
cana-5994	351	12	q	q	NOUN
cana-5994	351	13	,	,	PUNCT
cana-5994	351	14	ψftυ	ψftυ	NOUN
cana-5994	351	15	,	,	PUNCT
cana-5994	351	16	q(ψ	q(ψ	NUM
cana-5994	351	17	;	;	PUNCT
cana-5994	351	18	η	η	X
cana-5994	351	19	)	)	PUNCT
cana-5994	351	20	and	and	CCONJ
cana-5994	351	21	ftυ	ftυ	NOUN
cana-5994	351	22	,	,	PUNCT
cana-5994	351	23	q(ψ	q(ψ	NUM
cana-5994	351	24	;	;	PUNCT
cana-5994	351	25	η	η	X
cana-5994	351	26	)	)	PUNCT
cana-5994	351	27	in	in	ADP
cana-5994	351	28	(	(	PUNCT
cana-5994	351	29	2.21	2.21	NUM
cana-5994	351	30	)	)	PUNCT
cana-5994	351	31	,	,	PUNCT
cana-5994	351	32	we	we	PRON
cana-5994	351	33	have	have	VERB
cana-5994	351	34	bν−υftν	bν−υftν	NOUN
cana-5994	351	35	,	,	PUNCT
cana-5994	351	36	q(b−1ζ	q(b−1ζ	PROPN
cana-5994	351	37	;	;	PUNCT
cana-5994	351	38	η	η	PROPN
cana-5994	351	39	)	)	PUNCT
cana-5994	352	1	[	[	X
cana-5994	352	2	ν]q	ν]q	NOUN
cana-5994	352	3	!	!	PUNCT
cana-5994	353	1	d	d	NOUN
cana-5994	353	2	(	(	PUNCT
cana-5994	353	3	ν	ν	NOUN
cana-5994	353	4	)	)	PUNCT
cana-5994	353	5	q	q	NOUN
cana-5994	353	6	,	,	PUNCT
cana-5994	353	7	ψftυ−ν	ψftυ−ν	PROPN
cana-5994	353	8	,	,	PUNCT
cana-5994	353	9	q(a−1ψ	q(a−1ψ	ADP
cana-5994	353	10	;	;	PUNCT
cana-5994	353	11	η	η	NOUN
cana-5994	353	12	)	)	PUNCT
cana-5994	353	13	=	=	SYM
cana-5994	353	14	aν−υftν	aν−υftν	NOUN
cana-5994	353	15	,	,	PUNCT
cana-5994	353	16	q(a−1ζ	q(a−1ζ	PROPN
cana-5994	353	17	;	;	PUNCT
cana-5994	353	18	η	η	X
cana-5994	353	19	)	)	PUNCT
cana-5994	354	1	[	[	X
cana-5994	354	2	ν]q	ν]q	NOUN
cana-5994	354	3	!	!	PUNCT
cana-5994	355	1	d	d	NOUN
cana-5994	355	2	(	(	PUNCT
cana-5994	355	3	ν	ν	NOUN
cana-5994	355	4	)	)	PUNCT
cana-5994	355	5	q	q	NOUN
cana-5994	355	6	,	,	PUNCT
cana-5994	355	7	ψftυ−ν	ψftυ−ν	PROPN
cana-5994	355	8	,	,	PUNCT
cana-5994	355	9	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	355	10	;	;	PUNCT
cana-5994	355	11	η	η	NOUN
cana-5994	355	12	)	)	PUNCT
cana-5994	355	13	.	.	PUNCT
cana-5994	356	1	hence	hence	ADV
cana-5994	356	2	,	,	PUNCT
cana-5994	356	3	complete	complete	VERB
cana-5994	356	4	the	the	DET
cana-5994	356	5	proof	proof	NOUN
cana-5994	356	6	.	.	PUNCT
cana-5994	357	1	□	□	PUNCT
cana-5994	357	2	corollary	corollary	ADJ
cana-5994	357	3	2.6	2.6	NUM
cana-5994	357	4	.	.	PUNCT
cana-5994	358	1	letting	let	VERB
cana-5994	358	2	a	a	DET
cana-5994	358	3	=	=	SYM
cana-5994	358	4	1	1	NUM
cana-5994	358	5	in	in	ADP
cana-5994	358	6	theorem	theorem	ADJ
cana-5994	358	7	2.7	2.7	NUM
cana-5994	358	8	,	,	PUNCT
cana-5994	358	9	we	we	PRON
cana-5994	358	10	have	have	VERB
cana-5994	358	11	ftν	ftν	NOUN
cana-5994	358	12	,	,	PUNCT
cana-5994	358	13	q(b−1ζ	q(b−1ζ	PROPN
cana-5994	358	14	;	;	PUNCT
cana-5994	358	15	η	η	PROPN
cana-5994	358	16	)	)	PUNCT
cana-5994	359	1	[	[	X
cana-5994	359	2	υ]q	υ]q	NOUN
cana-5994	359	3	!	!	PUNCT
cana-5994	360	1	d	d	NOUN
cana-5994	360	2	(	(	PUNCT
cana-5994	360	3	υ	υ	NOUN
cana-5994	360	4	)	)	PUNCT
cana-5994	360	5	q	q	NOUN
cana-5994	360	6	,	,	PUNCT
cana-5994	360	7	ψftυ	ψftυ	NOUN
cana-5994	360	8	,	,	PUNCT
cana-5994	360	9	q(ψ	q(ψ	NUM
cana-5994	360	10	;	;	PUNCT
cana-5994	360	11	η	η	X
cana-5994	360	12	)	)	PUNCT
cana-5994	360	13	+	+	CCONJ
cana-5994	360	14	b−1	b−1	PROPN
cana-5994	360	15	ftυ−1,q(b	ftυ−1,q(b	PROPN
cana-5994	360	16	−1ζ	−1ζ	PROPN
cana-5994	360	17	;	;	PUNCT
cana-5994	360	18	η	η	PROPN
cana-5994	360	19	)	)	PUNCT
cana-5994	361	1	[	[	X
cana-5994	361	2	υ	υ	X
cana-5994	361	3	−	−	NOUN
cana-5994	361	4	1]q	1]q	NUM
cana-5994	361	5	!	!	PUNCT
cana-5994	362	1	d	d	X
cana-5994	362	2	(	(	PUNCT
cana-5994	362	3	υ−1	υ−1	PROPN
cana-5994	362	4	)	)	PUNCT
cana-5994	362	5	q	q	NOUN
cana-5994	362	6	,	,	PUNCT
cana-5994	362	7	ψ	ψ	NOUN
cana-5994	362	8	ftυ	ftυ	NOUN
cana-5994	362	9	,	,	PUNCT
cana-5994	362	10	q(ψ	q(ψ	NUM
cana-5994	362	11	;	;	PUNCT
cana-5994	362	12	η	η	X
cana-5994	362	13	)	)	PUNCT
cana-5994	362	14	+	+	CCONJ
cana-5994	362	15	·	·	PUNCT
cana-5994	362	16	·	·	PUNCT
cana-5994	362	17	·	·	PUNCT
cana-5994	363	1	+	+	X
cana-5994	363	2	b1−υft1,q(b	b1−υft1,q(b	PROPN
cana-5994	363	3	−1ζ	−1ζ	PROPN
cana-5994	363	4	;	;	PUNCT
cana-5994	363	5	η)d	η)d	X
cana-5994	363	6	(	(	PUNCT
cana-5994	363	7	1	1	NUM
cana-5994	363	8	)	)	PUNCT
cana-5994	363	9	q	q	NOUN
cana-5994	363	10	,	,	PUNCT
cana-5994	363	11	ψftυ	ψftυ	NOUN
cana-5994	363	12	,	,	PUNCT
cana-5994	363	13	q(ψ	q(ψ	NUM
cana-5994	363	14	;	;	PUNCT
cana-5994	363	15	η	η	X
cana-5994	363	16	)	)	PUNCT
cana-5994	363	17	+	+	CCONJ
cana-5994	363	18	b−υft0,q(b	b−υft0,q(b	PROPN
cana-5994	363	19	−1ζ	−1ζ	PROPN
cana-5994	363	20	;	;	PUNCT
cana-5994	363	21	η)ftυ	η)ftυ	NOUN
cana-5994	363	22	,	,	PUNCT
cana-5994	363	23	q(ψ	q(ψ	NUM
cana-5994	363	24	;	;	PUNCT
cana-5994	363	25	η	η	NOUN
cana-5994	363	26	)	)	PUNCT
cana-5994	363	27	=	=	SYM
cana-5994	363	28	ftν	ftν	NOUN
cana-5994	363	29	,	,	PUNCT
cana-5994	363	30	q(ζ	q(ζ	NOUN
cana-5994	363	31	;	;	PUNCT
cana-5994	363	32	η	η	PROPN
cana-5994	363	33	)	)	PUNCT
cana-5994	364	1	[	[	X
cana-5994	364	2	υ]q	υ]q	NOUN
cana-5994	364	3	!	!	PUNCT
cana-5994	365	1	d	d	NOUN
cana-5994	365	2	(	(	PUNCT
cana-5994	365	3	υ	υ	NOUN
cana-5994	365	4	)	)	PUNCT
cana-5994	365	5	q	q	NOUN
cana-5994	365	6	,	,	PUNCT
cana-5994	365	7	ψftυ	ψftυ	NOUN
cana-5994	365	8	,	,	PUNCT
cana-5994	365	9	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	365	10	;	;	PUNCT
cana-5994	365	11	η	η	NOUN
cana-5994	365	12	)	)	PUNCT
cana-5994	365	13	+	+	CCONJ
cana-5994	365	14	a−1	a−1	PROPN
cana-5994	365	15	ftυ−1,q(ζ	ftυ−1,q(ζ	PROPN
cana-5994	365	16	;	;	PUNCT
cana-5994	365	17	η	η	X
cana-5994	365	18	)	)	PUNCT
cana-5994	366	1	[	[	X
cana-5994	366	2	υ	υ	X
cana-5994	366	3	−	−	NOUN
cana-5994	366	4	1]q	1]q	NUM
cana-5994	366	5	!	!	PUNCT
cana-5994	367	1	d	d	X
cana-5994	367	2	(	(	PUNCT
cana-5994	367	3	υ−1	υ−1	PROPN
cana-5994	367	4	)	)	PUNCT
cana-5994	367	5	q	q	NOUN
cana-5994	367	6	,	,	PUNCT
cana-5994	367	7	ψ	ψ	X
cana-5994	367	8	ftυ	ftυ	NOUN
cana-5994	367	9	,	,	PUNCT
cana-5994	367	10	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	367	11	;	;	PUNCT
cana-5994	367	12	η	η	NOUN
cana-5994	367	13	)	)	PUNCT
cana-5994	367	14	+	+	CCONJ
cana-5994	367	15	·	·	PUNCT
cana-5994	367	16	·	·	PUNCT
cana-5994	367	17	·	·	PUNCT
cana-5994	368	1	+	+	PUNCT
cana-5994	368	2	ft1,q(ζ	ft1,q(ζ	NOUN
cana-5994	368	3	;	;	PUNCT
cana-5994	368	4	η)d	η)d	NOUN
cana-5994	368	5	(	(	PUNCT
cana-5994	368	6	1	1	NUM
cana-5994	368	7	)	)	PUNCT
cana-5994	368	8	q	q	NOUN
cana-5994	368	9	,	,	PUNCT
cana-5994	368	10	ψftυ	ψftυ	NOUN
cana-5994	368	11	,	,	PUNCT
cana-5994	368	12	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	368	13	;	;	PUNCT
cana-5994	368	14	η	η	NOUN
cana-5994	368	15	)	)	PUNCT
cana-5994	368	16	+	+	CCONJ
cana-5994	368	17	ft0,q(ζ	ft0,q(ζ	NOUN
cana-5994	368	18	;	;	PUNCT
cana-5994	368	19	η)ftυ	η)ftυ	NOUN
cana-5994	368	20	,	,	PUNCT
cana-5994	368	21	q(b−1ψ	q(b−1ψ	NOUN
cana-5994	368	22	;	;	PUNCT
cana-5994	368	23	η	η	NOUN
cana-5994	368	24	)	)	PUNCT
cana-5994	368	25	.	.	PUNCT
cana-5994	369	1	corollary	corollary	ADJ
cana-5994	369	2	2.7	2.7	NUM
cana-5994	369	3	.	.	PUNCT
cana-5994	370	1	as	as	ADP
cana-5994	370	2	q	q	NOUN
cana-5994	370	3	approaches	approach	NOUN
cana-5994	370	4	1	1	NUM
cana-5994	370	5	in	in	ADP
cana-5994	370	6	theorem	theorem	ADJ
cana-5994	370	7	2.7	2.7	NUM
cana-5994	370	8	,	,	PUNCT
cana-5994	370	9	we	we	PRON
cana-5994	370	10	derive	derive	VERB
cana-5994	370	11	ftν(b−1ζ	ftν(b−1ζ	NOUN
cana-5994	370	12	;	;	PUNCT
cana-5994	370	13	η	η	X
cana-5994	370	14	)	)	PUNCT
cana-5994	371	1	[	[	X
cana-5994	371	2	υ	υ	X
cana-5994	371	3	]	]	X
cana-5994	371	4	!	!	PUNCT
cana-5994	372	1	d	d	NOUN
cana-5994	372	2	(	(	PUNCT
cana-5994	372	3	υ	υ	NOUN
cana-5994	372	4	)	)	PUNCT
cana-5994	372	5	ψ	ψ	NOUN
cana-5994	372	6	ftυ(a−1ψ	ftυ(a−1ψ	PROPN
cana-5994	372	7	;	;	PUNCT
cana-5994	372	8	η	η	X
cana-5994	372	9	)	)	PUNCT
cana-5994	373	1	+	+	CCONJ
cana-5994	373	2	b−1	b−1	PROPN
cana-5994	373	3	ftυ−1(b	ftυ−1(b	PROPN
cana-5994	373	4	−1ζ	−1ζ	PROPN
cana-5994	373	5	;	;	PUNCT
cana-5994	373	6	η	η	PROPN
cana-5994	373	7	)	)	PUNCT
cana-5994	374	1	[	[	X
cana-5994	374	2	υ	υ	X
cana-5994	374	3	−	−	NOUN
cana-5994	374	4	1	1	NUM
cana-5994	374	5	]	]	PUNCT
cana-5994	374	6	!	!	PUNCT
cana-5994	375	1	d	d	X
cana-5994	375	2	(	(	PUNCT
cana-5994	375	3	υ−1	υ−1	PROPN
cana-5994	375	4	)	)	PUNCT
cana-5994	375	5	ψ	ψ	X
cana-5994	375	6	ftυ(a−1ψ	ftυ(a−1ψ	NOUN
cana-5994	375	7	;	;	PUNCT
cana-5994	375	8	η	η	X
cana-5994	375	9	)	)	PUNCT
cana-5994	375	10	+	+	CCONJ
cana-5994	375	11	·	·	PUNCT
cana-5994	375	12	·	·	PUNCT
cana-5994	375	13	·	·	PUNCT
cana-5994	376	1	+	+	NUM
cana-5994	376	2	b1−υft1(b	b1−υft1(b	PROPN
cana-5994	376	3	−1ζ	−1ζ	PROPN
cana-5994	376	4	;	;	PUNCT
cana-5994	376	5	η)d	η)d	NOUN
cana-5994	376	6	(	(	PUNCT
cana-5994	376	7	1	1	NUM
cana-5994	376	8	)	)	PUNCT
cana-5994	376	9	ψ	ψ	NOUN
cana-5994	376	10	ftυ(a−1ψ	ftυ(a−1ψ	PROPN
cana-5994	376	11	;	;	PUNCT
cana-5994	376	12	η	η	X
cana-5994	376	13	)	)	PUNCT
cana-5994	376	14	+	+	PROPN
cana-5994	376	15	b−υft0(b	b−υft0(b	PROPN
cana-5994	376	16	−1ζ	−1ζ	PROPN
cana-5994	376	17	;	;	PUNCT
cana-5994	376	18	η)ftυ(a−1ψ	η)ftυ(a−1ψ	NUM
cana-5994	376	19	;	;	PUNCT
cana-5994	376	20	η	η	NOUN
cana-5994	376	21	)	)	PUNCT
cana-5994	376	22	=	=	SYM
cana-5994	376	23	ftν(a−1ζ	ftν(a−1ζ	PROPN
cana-5994	376	24	;	;	PUNCT
cana-5994	376	25	η	η	X
cana-5994	376	26	)	)	PUNCT
cana-5994	377	1	[	[	X
cana-5994	377	2	υ	υ	X
cana-5994	377	3	]	]	X
cana-5994	377	4	!	!	PUNCT
cana-5994	378	1	d	d	NOUN
cana-5994	378	2	(	(	PUNCT
cana-5994	378	3	υ	υ	NOUN
cana-5994	378	4	)	)	PUNCT
cana-5994	378	5	ψ	ψ	NOUN
cana-5994	378	6	ftυ(b−1ψ	ftυ(b−1ψ	PROPN
cana-5994	378	7	;	;	PUNCT
cana-5994	378	8	η	η	NOUN
cana-5994	378	9	)	)	PUNCT
cana-5994	378	10	+	+	CCONJ
cana-5994	378	11	a−1	a−1	PROPN
cana-5994	378	12	ftυ−1(a	ftυ−1(a	PUNCT
cana-5994	379	1	−1ζ	−1ζ	PROPN
cana-5994	379	2	;	;	PUNCT
cana-5994	379	3	η	η	PROPN
cana-5994	379	4	)	)	PUNCT
cana-5994	380	1	[	[	X
cana-5994	380	2	υ	υ	X
cana-5994	380	3	−	−	NOUN
cana-5994	380	4	1	1	NUM
cana-5994	380	5	]	]	PUNCT
cana-5994	380	6	!	!	PUNCT
cana-5994	381	1	d	d	X
cana-5994	381	2	(	(	PUNCT
cana-5994	381	3	υ−1	υ−1	PROPN
cana-5994	381	4	)	)	PUNCT
cana-5994	381	5	ψ	ψ	PROPN
cana-5994	381	6	ftυ(b−1ψ	ftυ(b−1ψ	PROPN
cana-5994	381	7	;	;	PUNCT
cana-5994	381	8	η	η	NOUN
cana-5994	381	9	)	)	PUNCT
cana-5994	381	10	+	+	CCONJ
cana-5994	381	11	·	·	PUNCT
cana-5994	381	12	·	·	PUNCT
cana-5994	381	13	·	·	PUNCT
cana-5994	382	1	+	+	PUNCT
cana-5994	382	2	a1−υft1(a	a1−υft1(a	PROPN
cana-5994	382	3	−1ζ	−1ζ	PROPN
cana-5994	382	4	;	;	PUNCT
cana-5994	382	5	η)d	η)d	X
cana-5994	382	6	(	(	PUNCT
cana-5994	382	7	1	1	NUM
cana-5994	382	8	)	)	PUNCT
cana-5994	382	9	ψ	ψ	PRON
cana-5994	382	10	ftυ(b−1ψ	ftυ(b−1ψ	PROPN
cana-5994	382	11	;	;	PUNCT
cana-5994	382	12	η	η	PROPN
cana-5994	382	13	)	)	PUNCT
cana-5994	382	14	+	+	PROPN
cana-5994	382	15	a−υft0(a	a−υft0(a	PROPN
cana-5994	382	16	−1ζ	−1ζ	PROPN
cana-5994	382	17	;	;	PUNCT
cana-5994	382	18	η)ftυ(b−1ψ	η)ftυ(b−1ψ	PROPN
cana-5994	382	19	;	;	PUNCT
cana-5994	382	20	η	η	NOUN
cana-5994	382	21	)	)	PUNCT
cana-5994	382	22	.	.	PUNCT
cana-5994	383	1	3	3	X
cana-5994	383	2	.	.	X
cana-5994	383	3	conclusion	conclusion	NOUN
cana-5994	383	4	we	we	PRON
cana-5994	383	5	have	have	AUX
cana-5994	383	6	constructed	construct	VERB
cana-5994	383	7	the	the	DET
cana-5994	383	8	q	q	NOUN
cana-5994	383	9	-	-	PUNCT
cana-5994	383	10	analogue	analogue	NOUN
cana-5994	383	11	of	of	ADP
cana-5994	383	12	frobenius	frobenius	ADJ
cana-5994	383	13	-	-	PUNCT
cana-5994	383	14	tangent	tangent	NOUN
cana-5994	383	15	polynomials	polynomial	NOUN
cana-5994	383	16	and	and	CCONJ
cana-5994	383	17	numbers	number	NOUN
cana-5994	383	18	,	,	PUNCT
cana-5994	383	19	and	and	CCONJ
cana-5994	383	20	derived	derive	VERB
cana-5994	383	21	several	several	ADJ
cana-5994	383	22	differential	differential	ADJ
cana-5994	383	23	equations	equation	NOUN
cana-5994	383	24	with	with	ADP
cana-5994	383	25	these	these	DET
cana-5994	383	26	polynomials	polynomial	NOUN
cana-5994	383	27	as	as	ADP
cana-5994	383	28	solutions	solution	NOUN
cana-5994	383	29	.	.	PUNCT
cana-5994	384	1	additionally	additionally	ADV
cana-5994	384	2	,	,	PUNCT
cana-5994	384	3	we	we	PRON
cana-5994	384	4	identified	identify	VERB
cana-5994	384	5	differential	differential	ADJ
cana-5994	384	6	equations	equation	NOUN
cana-5994	384	7	that	that	PRON
cana-5994	384	8	combine	combine	VERB
cana-5994	384	9	q	q	ADJ
cana-5994	384	10	-	-	PUNCT
cana-5994	384	11	frobenius	frobenius	ADJ
cana-5994	384	12	and	and	CCONJ
cana-5994	384	13	qtangent	qtangent	ADJ
cana-5994	384	14	polynomials	polynomial	NOUN
cana-5994	384	15	.	.	PUNCT
cana-5994	385	1	several	several	ADJ
cana-5994	385	2	properties	property	NOUN
cana-5994	385	3	of	of	ADP
cana-5994	385	4	the	the	DET
cana-5994	385	5	q	q	NOUN
cana-5994	385	6	-	-	PUNCT
cana-5994	385	7	analogue	analogue	NOUN
cana-5994	385	8	of	of	ADP
cana-5994	385	9	frobenius	frobenius	ADJ
cana-5994	385	10	-	-	PUNCT
cana-5994	385	11	tangent	tangent	NOUN
cana-5994	385	12	polynomials	polynomial	NOUN
cana-5994	385	13	and	and	CCONJ
cana-5994	385	14	numbers	number	NOUN
cana-5994	385	15	were	be	AUX
cana-5994	385	16	also	also	ADV
cana-5994	385	17	established	establish	VERB
cana-5994	385	18	.	.	PUNCT
cana-5994	386	1	the	the	DET
cana-5994	386	2	results	result	NOUN
cana-5994	386	3	obtained	obtain	VERB
cana-5994	386	4	in	in	ADP
cana-5994	386	5	this	this	DET
cana-5994	386	6	paper	paper	NOUN
cana-5994	386	7	are	be	AUX
cana-5994	386	8	broadly	broadly	ADV
cana-5994	386	9	general	general	ADJ
cana-5994	386	10	and	and	CCONJ
cana-5994	386	11	may	may	AUX
cana-5994	386	12	lead	lead	VERB
cana-5994	386	13	to	to	ADP
cana-5994	386	14	potential	potential	ADJ
cana-5994	386	15	applications	application	NOUN
cana-5994	386	16	in	in	ADP
cana-5994	386	17	the	the	DET
cana-5994	386	18	theory	theory	NOUN
cana-5994	386	19	of	of	ADP
cana-5994	386	20	special	special	ADJ
cana-5994	386	21	functions	function	NOUN
cana-5994	386	22	.	.	PUNCT
cana-5994	387	1	moreover	moreover	ADV
cana-5994	387	2	,	,	PUNCT
cana-5994	387	3	the	the	DET
cana-5994	387	4	main	main	ADJ
cana-5994	387	5	findings	finding	NOUN
cana-5994	387	6	are	be	AUX
cana-5994	387	7	significant	significant	ADJ
cana-5994	387	8	,	,	PUNCT
cana-5994	387	9	as	as	SCONJ
cana-5994	387	10	they	they	PRON
cana-5994	387	11	allow	allow	VERB
cana-5994	387	12	us	we	PRON
cana-5994	387	13	to	to	PART
cana-5994	387	14	deduce	deduce	VERB
cana-5994	387	15	important	important	ADJ
cana-5994	387	16	integral	integral	ADJ
cana-5994	387	17	formulas	formula	NOUN
cana-5994	387	18	for	for	ADP
cana-5994	387	19	specific	specific	ADJ
cana-5994	387	20	parameter	parameter	NOUN
cana-5994	387	21	values	value	NOUN
cana-5994	387	22	,	,	PUNCT
cana-5994	387	23	which	which	PRON
cana-5994	387	24	could	could	AUX
cana-5994	387	25	be	be	AUX
cana-5994	387	26	particularly	particularly	ADV
cana-5994	387	27	useful	useful	ADJ
cana-5994	387	28	in	in	ADP
cana-5994	387	29	laser	laser	NOUN
cana-5994	387	30	technology	technology	NOUN
cana-5994	387	31	.	.	PUNCT
cana-5994	388	1	acknowledgement	acknowledgement	NOUN
cana-5994	388	2	authors	author	NOUN
cana-5994	388	3	are	be	AUX
cana-5994	388	4	thankful	thankful	ADJ
cana-5994	388	5	to	to	ADP
cana-5994	388	6	integral	integral	ADJ
cana-5994	388	7	university	university	NOUN
cana-5994	388	8	,	,	PUNCT
cana-5994	388	9	lucknow	lucknow	NOUN
cana-5994	388	10	for	for	ADP
cana-5994	388	11	providing	provide	VERB
cana-5994	388	12	manuscript	manuscript	NOUN
cana-5994	388	13	communication	communication	NOUN
cana-5994	388	14	number	number	NOUN
cana-5994	388	15	(	(	PUNCT
cana-5994	388	16	mcn	mcn	PROPN
cana-5994	388	17	):	):	PUNCT
cana-5994	388	18	iu	iu	PROPN
cana-5994	388	19	/	/	SYM
cana-5994	388	20	r	r	NOUN
cana-5994	388	21	d/2024	d/2024	NOUN
cana-5994	388	22	-	-	PUNCT
cana-5994	388	23	mcn0003247	mcn0003247	NOUN
cana-5994	388	24	references	reference	NOUN
cana-5994	388	25	[	[	X
cana-5994	388	26	1	1	NUM
cana-5994	388	27	]	]	X
cana-5994	388	28	alshejari	alshejari	NOUN
cana-5994	388	29	,	,	PUNCT
cana-5994	388	30	a	a	PRON
cana-5994	388	31	,	,	PUNCT
cana-5994	388	32	khan	khan	PROPN
cana-5994	388	33	,	,	PUNCT
cana-5994	388	34	w.	w.	PROPN
cana-5994	388	35	a	a	PRON
cana-5994	388	36	,	,	PUNCT
cana-5994	388	37	duran	duran	PROPN
cana-5994	388	38	,	,	PUNCT
cana-5994	388	39	u	u	NOUN
cana-5994	388	40	,	,	PUNCT
cana-5994	388	41	ryoo	ryoo	NOUN
cana-5994	388	42	,	,	PUNCT
cana-5994	388	43	c.	c.	PROPN
cana-5994	388	44	s.	s.	PROPN
cana-5994	388	45	a	a	DET
cana-5994	388	46	study	study	NOUN
cana-5994	388	47	on	on	ADP
cana-5994	388	48	differential	differential	ADJ
cana-5994	388	49	equations	equation	NOUN
cana-5994	388	50	associated	associate	VERB
cana-5994	388	51	with	with	ADP
cana-5994	388	52	(	(	PUNCT
cana-5994	388	53	q	q	NOUN
cana-5994	388	54	,	,	PUNCT
cana-5994	388	55	h)-frobenius	h)-frobenius	NOUN
cana-5994	388	56	-	-	PUNCT
cana-5994	388	57	genocchi	genocchi	NOUN
cana-5994	388	58	polynomials	polynomial	NOUN
cana-5994	388	59	.	.	PUNCT
cana-5994	389	1	journal	journal	PROPN
cana-5994	389	2	of	of	ADP
cana-5994	389	3	mathematics	mathematic	NOUN
cana-5994	389	4	and	and	CCONJ
cana-5994	389	5	computer	computer	NOUN
cana-5994	389	6	science	science	NOUN
cana-5994	389	7	,	,	PUNCT
cana-5994	389	8	2025	2025	NUM
cana-5994	389	9	.	.	PUNCT
cana-5994	390	1	in	in	ADP
cana-5994	390	2	press	press	NOUN
cana-5994	390	3	.	.	PUNCT
cana-5994	391	1	communications	communication	NOUN
cana-5994	391	2	on	on	ADP
cana-5994	391	3	applied	apply	VERB
cana-5994	391	4	nonlinear	nonlinear	ADJ
cana-5994	391	5	analysis	analysis	NOUN
cana-5994	391	6	issn	issn	NOUN
cana-5994	391	7	:	:	PUNCT
cana-5994	391	8	1074	1074	NUM
cana-5994	391	9	-	-	PUNCT
cana-5994	391	10	133x	133x	NUM
cana-5994	391	11	vol	vol	NOUN
cana-5994	391	12	32	32	NUM
cana-5994	391	13	no	no	NOUN
cana-5994	391	14	.	.	PUNCT
cana-5994	392	1	9s	9s	NUM
cana-5994	392	2	(	(	PUNCT
cana-5994	392	3	2025	2025	NUM
cana-5994	392	4	)	)	PUNCT
cana-5994	393	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	393	2	3226	3226	NUM
cana-5994	393	3	10	10	NUM
cana-5994	393	4	idrees	idree	NOUN
cana-5994	393	5	ahmad	ahmad	PROPN
cana-5994	393	6	khan	khan	PROPN
cana-5994	393	7	and	and	CCONJ
cana-5994	393	8	sumit	sumit	PROPN
cana-5994	393	9	kumar	kumar	PROPN
cana-5994	393	10	[	[	X
cana-5994	393	11	2	2	NUM
cana-5994	393	12	]	]	X
cana-5994	393	13	aledamat	aledamat	NOUN
cana-5994	393	14	,	,	PUNCT
cana-5994	393	15	a	a	PRON
cana-5994	393	16	,	,	PUNCT
cana-5994	393	17	khan	khan	PROPN
cana-5994	393	18	,	,	PUNCT
cana-5994	393	19	w.	w.	PROPN
cana-5994	393	20	a	a	PRON
cana-5994	393	21	,	,	PUNCT
cana-5994	393	22	duran	duran	PROPN
cana-5994	393	23	,	,	PUNCT
cana-5994	393	24	u	u	NOUN
cana-5994	393	25	,	,	PUNCT
cana-5994	393	26	kirmani	kirmani	PROPN
cana-5994	393	27	,	,	PUNCT
cana-5994	393	28	s.	s.	PROPN
cana-5994	393	29	a.	a.	PROPN
cana-5994	393	30	k	k	PROPN
cana-5994	393	31	,	,	PUNCT
cana-5994	393	32	ryoo	ryoo	NOUN
cana-5994	393	33	,	,	PUNCT
cana-5994	393	34	c.	c.	PROPN
cana-5994	393	35	s.	s.	PROPN
cana-5994	394	1	a	a	DET
cana-5994	394	2	study	study	NOUN
cana-5994	394	3	on	on	ADP
cana-5994	394	4	differential	differential	ADJ
cana-5994	394	5	equations	equation	NOUN
cana-5994	394	6	associated	associate	VERB
cana-5994	394	7	with	with	ADP
cana-5994	394	8	(	(	PUNCT
cana-5994	394	9	q	q	NOUN
cana-5994	394	10	,	,	PUNCT
cana-5994	394	11	h)-frobenius	h)-frobenius	NOUN
cana-5994	394	12	-	-	PUNCT
cana-5994	394	13	euler	euler	NOUN
cana-5994	394	14	polynomials	polynomial	NOUN
cana-5994	394	15	.	.	PUNCT
cana-5994	395	1	journal	journal	NOUN
cana-5994	395	2	of	of	ADP
cana-5994	395	3	mathematics	mathematic	NOUN
cana-5994	395	4	and	and	CCONJ
cana-5994	395	5	computer	computer	NOUN
cana-5994	395	6	science	science	NOUN
cana-5994	395	7	,	,	PUNCT
cana-5994	395	8	2025	2025	NUM
cana-5994	395	9	,	,	PUNCT
cana-5994	395	10	36(3	36(3	NUM
cana-5994	395	11	)	)	PUNCT
cana-5994	395	12	,	,	PUNCT
cana-5994	395	13	386	386	NUM
cana-5994	395	14	-	-	SYM
cana-5994	395	15	398	398	NUM
cana-5994	395	16	.	.	PUNCT
cana-5994	396	1	[	[	X
cana-5994	396	2	3	3	NUM
cana-5994	396	3	]	]	X
cana-5994	396	4	jackson	jackson	PROPN
cana-5994	396	5	,	,	PUNCT
cana-5994	396	6	h.f	h.f	PROPN
cana-5994	396	7	.	.	PROPN
cana-5994	396	8	q	q	ADJ
cana-5994	396	9	-	-	PUNCT
cana-5994	396	10	difference	difference	NOUN
cana-5994	396	11	equations	equation	NOUN
cana-5994	396	12	.	.	PUNCT
cana-5994	397	1	am	be	AUX
cana-5994	397	2	.	.	PUNCT
cana-5994	398	1	j.	j.	PROPN
cana-5994	398	2	math	math	PROPN
cana-5994	398	3	.	.	PUNCT
cana-5994	399	1	1910	1910	NUM
cana-5994	399	2	,	,	PUNCT
cana-5994	399	3	32	32	NUM
cana-5994	399	4	,	,	PUNCT
cana-5994	399	5	305	305	NUM
cana-5994	399	6	-	-	SYM
cana-5994	399	7	314	314	NUM
cana-5994	399	8	.	.	PUNCT
cana-5994	400	1	[	[	X
cana-5994	400	2	4	4	NUM
cana-5994	400	3	]	]	X
cana-5994	400	4	jackson	jackson	PROPN
cana-5994	400	5	,	,	PUNCT
cana-5994	400	6	h.f	h.f	PROPN
cana-5994	400	7	.	.	PROPN
cana-5994	400	8	on	on	ADP
cana-5994	400	9	q	q	NOUN
cana-5994	400	10	-	-	PUNCT
cana-5994	400	11	functions	function	NOUN
cana-5994	400	12	and	and	CCONJ
cana-5994	400	13	a	a	DET
cana-5994	400	14	certain	certain	ADJ
cana-5994	400	15	difference	difference	NOUN
cana-5994	400	16	operator	operator	NOUN
cana-5994	400	17	.	.	PUNCT
cana-5994	401	1	trans	trans	PROPN
cana-5994	401	2	.	.	PUNCT
cana-5994	401	3	r.	r.	PROPN
cana-5994	401	4	soc	soc	PROPN
cana-5994	401	5	.	.	PUNCT
cana-5994	402	1	edinb	edinb	PROPN
cana-5994	402	2	.	.	PUNCT
cana-5994	403	1	2013	2013	NUM
cana-5994	403	2	,	,	PUNCT
cana-5994	403	3	46	46	NUM
cana-5994	403	4	,	,	PUNCT
cana-5994	403	5	253	253	NUM
cana-5994	403	6	-	-	SYM
cana-5994	403	7	281	281	NUM
cana-5994	403	8	.	.	PUNCT
cana-5994	404	1	[	[	X
cana-5994	404	2	5	5	NUM
cana-5994	404	3	]	]	X
cana-5994	404	4	kang	kang	PROPN
cana-5994	404	5	,	,	PUNCT
cana-5994	404	6	j.y	j.y	PROPN
cana-5994	404	7	.	.	PROPN
cana-5994	404	8	;	;	PUNCT
cana-5994	404	9	khan	khan	PROPN
cana-5994	404	10	,	,	PUNCT
cana-5994	404	11	w.a	w.a	PROPN
cana-5994	404	12	.	.	PROPN
cana-5994	404	13	a	a	DET
cana-5994	404	14	new	new	ADJ
cana-5994	404	15	class	class	NOUN
cana-5994	404	16	of	of	ADP
cana-5994	404	17	q	q	ADJ
cana-5994	404	18	-	-	ADJ
cana-5994	404	19	hermite	hermite	ADJ
cana-5994	404	20	based	base	VERB
cana-5994	404	21	apostol	apostol	NOUN
cana-5994	404	22	type	type	PROPN
cana-5994	404	23	frobenius	frobenius	NOUN
cana-5994	404	24	genocchi	genocchi	NOUN
cana-5994	404	25	polynomials	polynomial	NOUN
cana-5994	404	26	.	.	PUNCT
cana-5994	405	1	communication	communication	NOUN
cana-5994	405	2	of	of	ADP
cana-5994	405	3	the	the	DET
cana-5994	405	4	korean	korean	ADJ
cana-5994	405	5	mathematical	mathematical	ADJ
cana-5994	405	6	society	society	NOUN
cana-5994	405	7	.	.	PUNCT
cana-5994	406	1	2020	2020	NUM
cana-5994	406	2	,	,	PUNCT
cana-5994	406	3	35(3	35(3	NUM
cana-5994	406	4	)	)	PUNCT
cana-5994	406	5	,	,	PUNCT
cana-5994	406	6	759	759	NUM
cana-5994	406	7	-	-	SYM
cana-5994	406	8	771	771	NUM
cana-5994	406	9	.	.	PUNCT
cana-5994	407	1	[	[	X
cana-5994	407	2	6	6	NUM
cana-5994	407	3	]	]	X
cana-5994	407	4	kang	kang	PROPN
cana-5994	407	5	,	,	PUNCT
cana-5994	407	6	j.y	j.y	PROPN
cana-5994	407	7	.	.	PROPN
cana-5994	407	8	properties	property	NOUN
cana-5994	407	9	of	of	ADP
cana-5994	407	10	differential	differential	ADJ
cana-5994	407	11	equations	equation	NOUN
cana-5994	407	12	related	relate	VERB
cana-5994	407	13	to	to	PART
cana-5994	407	14	degenerate	degenerate	VERB
cana-5994	407	15	q	q	ADJ
cana-5994	407	16	-	-	PUNCT
cana-5994	407	17	tangent	tangent	ADJ
cana-5994	407	18	numbers	number	NOUN
cana-5994	407	19	and	and	CCONJ
cana-5994	407	20	polynomials	polynomial	NOUN
cana-5994	407	21	.	.	PUNCT
cana-5994	407	22	symmetry	symmetry	NOUN
cana-5994	407	23	.	.	PUNCT
cana-5994	408	1	2023	2023	NUM
cana-5994	408	2	,	,	PUNCT
cana-5994	408	3	15	15	NUM
cana-5994	408	4	,	,	PUNCT
cana-5994	408	5	874	874	NUM
cana-5994	408	6	.	.	PUNCT
cana-5994	409	1	[	[	X
cana-5994	409	2	7	7	X
cana-5994	409	3	]	]	X
cana-5994	409	4	kang	kang	PROPN
cana-5994	409	5	,	,	PUNCT
cana-5994	409	6	j.y	j.y	PROPN
cana-5994	409	7	.	.	PROPN
cana-5994	409	8	;	;	PUNCT
cana-5994	409	9	kim	kim	PROPN
cana-5994	409	10	,	,	PUNCT
cana-5994	409	11	y.g	y.g	PROPN
cana-5994	409	12	.	.	PUNCT
cana-5994	410	1	some	some	DET
cana-5994	410	2	properties	property	NOUN
cana-5994	410	3	and	and	CCONJ
cana-5994	410	4	identities	identity	NOUN
cana-5994	410	5	of	of	ADP
cana-5994	410	6	q	q	NOUN
cana-5994	410	7	-	-	PUNCT
cana-5994	410	8	frobenius	frobenius	ADJ
cana-5994	410	9	-	-	PUNCT
cana-5994	410	10	tangent	tangent	NOUN
cana-5994	410	11	polynomials	polynomial	NOUN
cana-5994	410	12	.	.	PUNCT
cana-5994	411	1	j.	j.	PROPN
cana-5994	411	2	appl	appl	PROPN
cana-5994	411	3	.	.	PUNCT
cana-5994	412	1	pure	pure	ADJ
cana-5994	412	2	math	math	NOUN
cana-5994	412	3	.	.	PUNCT
cana-5994	413	1	2020	2020	NUM
cana-5994	413	2	,	,	PUNCT
cana-5994	413	3	(	(	PUNCT
cana-5994	413	4	3	3	NUM
cana-5994	413	5	-	-	SYM
cana-5994	413	6	4	4	NUM
cana-5994	413	7	)	)	PUNCT
cana-5994	413	8	,	,	PUNCT
cana-5994	413	9	127	127	NUM
cana-5994	413	10	-	-	SYM
cana-5994	413	11	137	137	NUM
cana-5994	413	12	.	.	PUNCT
cana-5994	414	1	[	[	X
cana-5994	414	2	8	8	NUM
cana-5994	414	3	]	]	X
cana-5994	414	4	kang	kang	PROPN
cana-5994	414	5	,	,	PUNCT
cana-5994	414	6	j.y	j.y	PROPN
cana-5994	414	7	.	.	PUNCT
cana-5994	414	8	the	the	DET
cana-5994	414	9	forms	form	NOUN
cana-5994	414	10	and	and	CCONJ
cana-5994	414	11	properties	property	NOUN
cana-5994	414	12	of	of	ADP
cana-5994	414	13	differential	differential	ADJ
cana-5994	414	14	equations	equation	NOUN
cana-5994	414	15	of	of	ADP
cana-5994	414	16	higher	high	ADJ
cana-5994	414	17	order	order	NOUN
cana-5994	414	18	for	for	ADP
cana-5994	414	19	q	q	ADJ
cana-5994	414	20	-	-	PUNCT
cana-5994	414	21	tangent	tangent	ADJ
cana-5994	414	22	polynomials	polynomial	NOUN
cana-5994	414	23	.	.	PUNCT
cana-5994	415	1	j.	j.	PROPN
cana-5994	415	2	appl	appl	PROPN
cana-5994	415	3	.	.	PROPN
cana-5994	415	4	math	math	PROPN
cana-5994	415	5	.	.	PUNCT
cana-5994	416	1	2022	2022	NUM
cana-5994	416	2	,	,	PUNCT
cana-5994	416	3	40(5	40(5	NOUN
cana-5994	416	4	-	-	SYM
cana-5994	416	5	6	6	NUM
cana-5994	416	6	)	)	PUNCT
cana-5994	416	7	,	,	PUNCT
cana-5994	416	8	1117	1117	NUM
cana-5994	416	9	-	-	SYM
cana-5994	416	10	1128	1128	NUM
cana-5994	416	11	.	.	PUNCT
cana-5994	417	1	[	[	X
cana-5994	417	2	9	9	NUM
cana-5994	417	3	]	]	X
cana-5994	417	4	khan	khan	PROPN
cana-5994	417	5	,	,	PUNCT
cana-5994	417	6	w.	w.	PROPN
cana-5994	417	7	a	a	PROPN
cana-5994	417	8	,	,	PUNCT
cana-5994	417	9	yadav	yadav	PROPN
cana-5994	417	10	,	,	PUNCT
cana-5994	417	11	v.	v.	ADP
cana-5994	417	12	a	a	DET
cana-5994	417	13	study	study	NOUN
cana-5994	417	14	on	on	ADP
cana-5994	417	15	q	q	ADJ
cana-5994	417	16	-	-	PUNCT
cana-5994	417	17	analogue	analogue	NOUN
cana-5994	417	18	of	of	ADP
cana-5994	417	19	degenerate	degenerate	ADJ
cana-5994	417	20	changhee	changhee	NOUN
cana-5994	417	21	numbers	number	NOUN
cana-5994	417	22	and	and	CCONJ
cana-5994	417	23	polynomials	polynomial	NOUN
cana-5994	417	24	.	.	PUNCT
cana-5994	418	1	southeast	southeast	ADJ
cana-5994	418	2	asian	asian	PROPN
cana-5994	418	3	journal	journal	PROPN
cana-5994	418	4	of	of	ADP
cana-5994	418	5	mathematics	mathematics	PROPN
cana-5994	418	6	and	and	CCONJ
cana-5994	418	7	mathematical	mathematical	ADJ
cana-5994	418	8	sciences	science	NOUN
cana-5994	418	9	,	,	PUNCT
cana-5994	418	10	2023	2023	NUM
cana-5994	418	11	,	,	PUNCT
cana-5994	418	12	19(1	19(1	NUM
cana-5994	418	13	)	)	PUNCT
cana-5994	418	14	,	,	PUNCT
cana-5994	418	15	29	29	NUM
cana-5994	418	16	-	-	SYM
cana-5994	418	17	42	42	NUM
cana-5994	418	18	.	.	PUNCT
cana-5994	419	1	[	[	X
cana-5994	419	2	10	10	NUM
cana-5994	419	3	]	]	X
cana-5994	419	4	khan	khan	PROPN
cana-5994	419	5	,	,	PUNCT
cana-5994	419	6	w.	w.	PROPN
cana-5994	419	7	a.	a.	PROPN
cana-5994	419	8	a	a	DET
cana-5994	419	9	study	study	NOUN
cana-5994	419	10	on	on	ADP
cana-5994	419	11	q	q	ADJ
cana-5994	419	12	-	-	PUNCT
cana-5994	419	13	analogue	analogue	NOUN
cana-5994	419	14	of	of	ADP
cana-5994	419	15	degenerate	degenerate	ADJ
cana-5994	419	16	1/2	1/2	NUM
cana-5994	419	17	-	-	PUNCT
cana-5994	419	18	changhee	changhee	NOUN
cana-5994	419	19	numbers	number	NOUN
cana-5994	419	20	and	and	CCONJ
cana-5994	419	21	polynomials	polynomial	NOUN
cana-5994	419	22	.	.	PUNCT
cana-5994	420	1	southeast	southeast	ADJ
cana-5994	420	2	asian	asian	PROPN
cana-5994	420	3	journal	journal	PROPN
cana-5994	420	4	of	of	ADP
cana-5994	420	5	mathematics	mathematics	PROPN
cana-5994	420	6	and	and	CCONJ
cana-5994	420	7	mathematical	mathematical	ADJ
cana-5994	420	8	sciences	science	NOUN
cana-5994	420	9	,	,	PUNCT
cana-5994	420	10	2022	2022	NUM
cana-5994	420	11	,	,	PUNCT
cana-5994	420	12	18(2	18(2	NUM
cana-5994	420	13	)	)	PUNCT
cana-5994	420	14	,	,	PUNCT
cana-5994	420	15	1	1	NUM
cana-5994	420	16	-	-	SYM
cana-5994	420	17	12	12	NUM
cana-5994	420	18	.	.	PUNCT
cana-5994	421	1	[	[	X
cana-5994	421	2	11	11	NUM
cana-5994	421	3	]	]	X
cana-5994	421	4	mahmudov	mahmudov	X
cana-5994	421	5	,	,	PUNCT
cana-5994	421	6	n.i	n.i	PROPN
cana-5994	421	7	.	.	PROPN
cana-5994	421	8	q	q	PROPN
cana-5994	421	9	-	-	PUNCT
cana-5994	421	10	analogues	analogue	NOUN
cana-5994	421	11	of	of	ADP
cana-5994	421	12	the	the	DET
cana-5994	421	13	bernoulli	bernoulli	PROPN
cana-5994	421	14	and	and	CCONJ
cana-5994	421	15	genocchi	genocchi	PROPN
cana-5994	421	16	polynomials	polynomial	NOUN
cana-5994	421	17	and	and	CCONJ
cana-5994	421	18	the	the	DET
cana-5994	421	19	srivastava	srivastava	NOUN
cana-5994	421	20	-	-	PUNCT
cana-5994	421	21	pinter	pinter	NOUN
cana-5994	421	22	addition	addition	NOUN
cana-5994	421	23	theorems	theorem	NOUN
cana-5994	421	24	.	.	PUNCT
cana-5994	422	1	discrete	discrete	VERB
cana-5994	422	2	and	and	CCONJ
cana-5994	422	3	dynamics	dynamic	NOUN
cana-5994	422	4	in	in	ADP
cana-5994	422	5	nature	nature	NOUN
cana-5994	422	6	and	and	CCONJ
cana-5994	422	7	soc	soc	NOUN
cana-5994	422	8	.	.	PUNCT
cana-5994	423	1	article	article	NOUN
cana-5994	423	2	number	number	NOUN
cana-5994	423	3	169348	169348	NUM
cana-5994	423	4	,	,	PUNCT
cana-5994	423	5	2012	2012	NUM
cana-5994	423	6	.	.	PUNCT
cana-5994	424	1	doi:10.1155/2012/169348	doi:10.1155/2012/169348	NOUN
cana-5994	424	2	.	.	PUNCT
cana-5994	425	1	[	[	X
cana-5994	425	2	12	12	NUM
cana-5994	425	3	]	]	PUNCT
cana-5994	425	4	mahmudov	mahmudov	X
cana-5994	425	5	,	,	PUNCT
cana-5994	425	6	n.i	n.i	PROPN
cana-5994	425	7	.	.	PROPN
cana-5994	425	8	on	on	ADP
cana-5994	425	9	a	a	DET
cana-5994	425	10	class	class	NOUN
cana-5994	425	11	of	of	ADP
cana-5994	425	12	q	q	NOUN
cana-5994	425	13	-	-	PUNCT
cana-5994	425	14	bernoulli	bernoulli	NOUN
cana-5994	425	15	and	and	CCONJ
cana-5994	425	16	q	q	NOUN
cana-5994	425	17	-	-	PUNCT
cana-5994	425	18	euler	euler	NOUN
cana-5994	425	19	polynomials	polynomial	NOUN
cana-5994	425	20	.	.	PUNCT
cana-5994	426	1	adv	adv	INTJ
cana-5994	426	2	.	.	PUNCT
cana-5994	427	1	in	in	ADP
cana-5994	427	2	diff	diff	PROPN
cana-5994	427	3	.	.	PUNCT
cana-5994	428	1	equa	equa	NOUN
cana-5994	428	2	.	.	PUNCT
cana-5994	429	1	2013	2013	NUM
cana-5994	429	2	.	.	PUNCT
cana-5994	430	1	doi:10.1186/1687	doi:10.1186/1687	ADJ
cana-5994	430	2	-	-	PUNCT
cana-5994	430	3	1847	1847	NUM
cana-5994	430	4	-	-	PUNCT
cana-5994	430	5	2013	2013	NUM
cana-5994	430	6	-	-	PUNCT
cana-5994	430	7	1	1	NUM
cana-5994	430	8	.	.	PUNCT
cana-5994	431	1	[	[	X
cana-5994	431	2	13	13	NUM
cana-5994	431	3	]	]	X
cana-5994	431	4	nisar	nisar	PROPN
cana-5994	431	5	,	,	PUNCT
cana-5994	431	6	k.s	k.s	PROPN
cana-5994	431	7	.	.	PROPN
cana-5994	431	8	;	;	PUNCT
cana-5994	431	9	khan	khan	PROPN
cana-5994	431	10	,	,	PUNCT
cana-5994	431	11	w.a	w.a	PROPN
cana-5994	431	12	.	.	PROPN
cana-5994	431	13	notes	note	NOUN
cana-5994	431	14	on	on	ADP
cana-5994	431	15	q	q	ADJ
cana-5994	431	16	-	-	ADJ
cana-5994	431	17	hermite	hermite	ADJ
cana-5994	431	18	based	base	VERB
cana-5994	431	19	unified	unified	ADJ
cana-5994	431	20	apostol	apostol	NOUN
cana-5994	431	21	type	type	NOUN
cana-5994	431	22	polynomials	polynomial	NOUN
cana-5994	431	23	.	.	PUNCT
cana-5994	432	1	journal	journal	NOUN
cana-5994	432	2	of	of	ADP
cana-5994	432	3	interdisciplinary	interdisciplinary	ADJ
cana-5994	432	4	mathematics	mathematic	NOUN
cana-5994	432	5	.	.	PUNCT
cana-5994	433	1	2019	2019	NUM
cana-5994	433	2	,	,	PUNCT
cana-5994	433	3	22(7	22(7	NUM
cana-5994	433	4	)	)	PUNCT
cana-5994	433	5	,	,	PUNCT
cana-5994	433	6	11851203	11851203	NUM
cana-5994	433	7	.	.	PUNCT
cana-5994	434	1	[	[	X
cana-5994	434	2	14	14	NUM
cana-5994	434	3	]	]	X
cana-5994	434	4	nadeem	nadeem	PROPN
cana-5994	434	5	,	,	PUNCT
cana-5994	434	6	m	m	PROPN
cana-5994	434	7	,	,	PUNCT
cana-5994	434	8	khan	khan	PROPN
cana-5994	434	9	,	,	PUNCT
cana-5994	434	10	w.	w.	PROPN
cana-5994	434	11	a.	a.	PROPN
cana-5994	434	12	symmetric	symmetric	PROPN
cana-5994	434	13	identities	identity	NOUN
cana-5994	434	14	for	for	ADP
cana-5994	434	15	degenerate	degenerate	ADJ
cana-5994	434	16	q	q	ADJ
cana-5994	434	17	-	-	ADJ
cana-5994	434	18	genocchi	genocchi	ADJ
cana-5994	434	19	numbers	number	NOUN
cana-5994	434	20	and	and	CCONJ
cana-5994	434	21	polynomials	polynomial	NOUN
cana-5994	434	22	.	.	PUNCT
cana-5994	435	1	southeast	southeast	ADJ
cana-5994	435	2	asian	asian	PROPN
cana-5994	435	3	journal	journal	PROPN
cana-5994	435	4	of	of	ADP
cana-5994	435	5	mathematics	mathematics	PROPN
cana-5994	435	6	and	and	CCONJ
cana-5994	435	7	mathematical	mathematical	ADJ
cana-5994	435	8	sciences	science	NOUN
cana-5994	435	9	,	,	PUNCT
cana-5994	435	10	2023	2023	NUM
cana-5994	435	11	,	,	PUNCT
cana-5994	435	12	19(1	19(1	NUM
cana-5994	435	13	)	)	PUNCT
cana-5994	435	14	,	,	PUNCT
cana-5994	435	15	17	17	NUM
cana-5994	435	16	-	-	SYM
cana-5994	435	17	28	28	NUM
cana-5994	435	18	.	.	PUNCT
cana-5994	436	1	[	[	X
cana-5994	436	2	15	15	NUM
cana-5994	436	3	]	]	X
cana-5994	436	4	ryoo	ryoo	NOUN
cana-5994	436	5	,	,	PUNCT
cana-5994	436	6	c.s	c.s	PROPN
cana-5994	436	7	.	.	PROPN
cana-5994	436	8	;	;	PUNCT
cana-5994	436	9	kang	kang	PROPN
cana-5994	436	10	,	,	PUNCT
cana-5994	436	11	j.y	j.y	PROPN
cana-5994	436	12	.	.	PROPN
cana-5994	436	13	various	various	ADJ
cana-5994	436	14	types	type	NOUN
cana-5994	436	15	of	of	ADP
cana-5994	436	16	q	q	ADJ
cana-5994	436	17	-	-	PUNCT
cana-5994	436	18	differential	differential	ADJ
cana-5994	436	19	equations	equation	NOUN
cana-5994	436	20	of	of	ADP
cana-5994	436	21	higherorder	higherorder	NOUN
cana-5994	436	22	for	for	ADP
cana-5994	436	23	q	q	NOUN
cana-5994	436	24	-	-	PUNCT
cana-5994	436	25	euler	euler	NOUN
cana-5994	436	26	and	and	CCONJ
cana-5994	436	27	q	q	ADJ
cana-5994	436	28	-	-	PUNCT
cana-5994	436	29	genocchi	genocchi	NOUN
cana-5994	436	30	polynomials	polynomial	NOUN
cana-5994	436	31	.	.	PUNCT
cana-5994	437	1	mathematics	mathematic	NOUN
cana-5994	437	2	2022	2022	NUM
cana-5994	437	3	,	,	PUNCT
cana-5994	437	4	10	10	NUM
cana-5994	437	5	,	,	PUNCT
cana-5994	437	6	1181	1181	NUM
cana-5994	437	7	.	.	PUNCT
cana-5994	438	1	[	[	X
cana-5994	438	2	16	16	NUM
cana-5994	438	3	]	]	PUNCT
cana-5994	438	4	ryoo	ryoo	NOUN
cana-5994	438	5	,	,	PUNCT
cana-5994	438	6	c.s	c.s	PROPN
cana-5994	438	7	.	.	PROPN
cana-5994	438	8	;	;	PUNCT
cana-5994	438	9	kang	kang	PROPN
cana-5994	438	10	,	,	PUNCT
cana-5994	438	11	j.y	j.y	PROPN
cana-5994	438	12	.	.	PROPN
cana-5994	438	13	properties	property	NOUN
cana-5994	438	14	of	of	ADP
cana-5994	438	15	q	q	ADJ
cana-5994	438	16	-	-	PUNCT
cana-5994	438	17	differential	differential	ADJ
cana-5994	438	18	equations	equation	NOUN
cana-5994	438	19	of	of	ADP
cana-5994	438	20	higher	high	ADJ
cana-5994	438	21	order	order	NOUN
cana-5994	438	22	and	and	CCONJ
cana-5994	438	23	visualization	visualization	NOUN
cana-5994	438	24	of	of	ADP
cana-5994	438	25	fractal	fractal	NOUN
cana-5994	438	26	using	use	VERB
cana-5994	438	27	q	q	ADJ
cana-5994	438	28	-	-	PUNCT
cana-5994	438	29	bernoulli	bernoulli	NOUN
cana-5994	438	30	polynomials	polynomial	NOUN
cana-5994	438	31	.	.	PUNCT
cana-5994	439	1	mathematics	mathematic	NOUN
cana-5994	439	2	2022	2022	NUM
cana-5994	439	3	,	,	PUNCT
cana-5994	439	4	6	6	NUM
cana-5994	439	5	,	,	PUNCT
cana-5994	439	6	296	296	NUM
cana-5994	439	7	.	.	PUNCT
cana-5994	440	1	department	department	NOUN
cana-5994	440	2	of	of	ADP
cana-5994	440	3	mathematics	mathematics	PROPN
cana-5994	440	4	and	and	CCONJ
cana-5994	440	5	statistics	statistic	NOUN
cana-5994	440	6	,	,	PUNCT
cana-5994	440	7	faculty	faculty	NOUN
cana-5994	440	8	of	of	ADP
cana-5994	440	9	science	science	NOUN
cana-5994	440	10	,	,	PUNCT
cana-5994	440	11	integral	integral	ADJ
cana-5994	440	12	university	university	NOUN
cana-5994	440	13	,	,	PUNCT
cana-5994	440	14	lucknow226026	lucknow226026	PROPN
cana-5994	440	15	,	,	PUNCT
cana-5994	440	16	india	india	PROPN
cana-5994	440	17	email	email	NOUN
cana-5994	440	18	address	address	NOUN
cana-5994	440	19	:	:	PUNCT
cana-5994	440	20	khanidrees077@gmail.com	khanidrees077@gmail.com	X
cana-5994	440	21	department	department	PROPN
cana-5994	440	22	of	of	ADP
cana-5994	440	23	mathematics	mathematics	PROPN
cana-5994	440	24	and	and	CCONJ
cana-5994	440	25	statistics	statistic	NOUN
cana-5994	440	26	,	,	PUNCT
cana-5994	440	27	faculty	faculty	NOUN
cana-5994	440	28	of	of	ADP
cana-5994	440	29	science	science	NOUN
cana-5994	440	30	,	,	PUNCT
cana-5994	440	31	integral	integral	ADJ
cana-5994	440	32	university	university	NOUN
cana-5994	440	33	,	,	PUNCT
cana-5994	440	34	lucknow226026	lucknow226026	PROPN
cana-5994	440	35	,	,	PUNCT
cana-5994	440	36	india	india	PROPN
cana-5994	440	37	email	email	NOUN
cana-5994	440	38	address	address	NOUN
cana-5994	440	39	:	:	PUNCT
cana-5994	440	40	lect.sumit@gmail.com	lect.sumit@gmail.com	X
cana-5994	440	41	communications	communication	NOUN
cana-5994	440	42	on	on	ADP
cana-5994	440	43	applied	apply	VERB
cana-5994	440	44	nonlinear	nonlinear	ADJ
cana-5994	440	45	analysis	analysis	NOUN
cana-5994	440	46	issn	issn	NOUN
cana-5994	440	47	:	:	PUNCT
cana-5994	440	48	1074	1074	NUM
cana-5994	440	49	-	-	PUNCT
cana-5994	440	50	133x	133x	NUM
cana-5994	440	51	vol	vol	NOUN
cana-5994	440	52	32	32	NUM
cana-5994	440	53	no	no	NOUN
cana-5994	440	54	.	.	PUNCT
cana-5994	441	1	9s	9s	NUM
cana-5994	441	2	(	(	PUNCT
cana-5994	441	3	2025	2025	NUM
cana-5994	441	4	)	)	PUNCT
cana-5994	441	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-5994	441	6	3227	3227	NUM