id	sid	tid	token	lemma	pos
cana-2799	1	1	communications	communication	NOUN
cana-2799	1	2	on	on	ADP
cana-2799	1	3	applied	apply	VERB
cana-2799	1	4	nonlinear	nonlinear	ADJ
cana-2799	1	5	analysis	analysis	NOUN
cana-2799	1	6	issn	issn	NOUN
cana-2799	1	7	:	:	PUNCT
cana-2799	1	8	1074	1074	NUM
cana-2799	1	9	-	-	PUNCT
cana-2799	1	10	133x	133x	NUM
cana-2799	1	11	vol	vol	NOUN
cana-2799	1	12	32	32	NUM
cana-2799	1	13	no	no	NOUN
cana-2799	1	14	.	.	PUNCT
cana-2799	2	1	4s	4s	NUM
cana-2799	2	2	(	(	PUNCT
cana-2799	2	3	2025	2025	NUM
cana-2799	2	4	)	)	PUNCT
cana-2799	2	5	258	258	NUM
cana-2799	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	2	7	extorial	extorial	ADJ
cana-2799	2	8	in	in	ADP
cana-2799	2	9	rl	rl	ADP
cana-2799	2	10	circuits	circuit	NOUN
cana-2799	2	11	and	and	CCONJ
cana-2799	2	12	heat	heat	NOUN
cana-2799	2	13	flows	flow	VERB
cana-2799	2	14	t.	t.	PROPN
cana-2799	2	15	sathinathan1	sathinathan1	PROPN
cana-2799	2	16	*	*	PROPN
cana-2799	2	17	,	,	PUNCT
cana-2799	2	18	s.	s.	PROPN
cana-2799	2	19	john	john	PROPN
cana-2799	2	20	borg2	borg2	PROPN
cana-2799	2	21	,	,	PUNCT
cana-2799	2	22	g.	g.	PROPN
cana-2799	2	23	britto	britto	PROPN
cana-2799	2	24	antony	antony	PROPN
cana-2799	2	25	xavier3	xavier3	PROPN
cana-2799	3	1	1,2,3department	1,2,3department	NUM
cana-2799	3	2	of	of	ADP
cana-2799	3	3	mathematics	mathematic	NOUN
cana-2799	3	4	,	,	PUNCT
cana-2799	3	5	sacred	sacred	ADJ
cana-2799	3	6	heart	heart	NOUN
cana-2799	3	7	college	college	NOUN
cana-2799	3	8	(	(	PUNCT
cana-2799	3	9	autonomous	autonomous	ADJ
cana-2799	3	10	)	)	PUNCT
cana-2799	3	11	,	,	PUNCT
cana-2799	3	12	tirupattur	tirupattur	PROPN
cana-2799	3	13	635601	635601	NUM
cana-2799	3	14	,	,	PUNCT
cana-2799	3	15	tamilnadu	tamilnadu	NOUN
cana-2799	3	16	,	,	PUNCT
cana-2799	3	17	india	india	PROPN
cana-2799	3	18	.	.	PUNCT
cana-2799	4	1	∗corresponding	∗corresponde	VERB
cana-2799	4	2	author	author	NOUN
cana-2799	4	3	:	:	PUNCT
cana-2799	4	4	sathithoma@gmail.com	sathithoma@gmail.com	X
cana-2799	4	5	article	article	NOUN
cana-2799	4	6	history	history	NOUN
cana-2799	4	7	:	:	PUNCT
cana-2799	4	8	received	receive	VERB
cana-2799	4	9	:	:	PUNCT
cana-2799	4	10	21	21	NUM
cana-2799	4	11	-	-	SYM
cana-2799	4	12	09	09	NUM
cana-2799	4	13	-	-	PUNCT
cana-2799	4	14	2024	2024	NUM
cana-2799	4	15	revised	revise	VERB
cana-2799	4	16	:	:	PUNCT
cana-2799	4	17	24	24	NUM
cana-2799	4	18	-	-	SYM
cana-2799	4	19	11	11	NUM
cana-2799	4	20	-	-	PUNCT
cana-2799	4	21	2024	2024	NUM
cana-2799	4	22	accepted	accept	VERB
cana-2799	4	23	:	:	PUNCT
cana-2799	4	24	05	05	NUM
cana-2799	4	25	-	-	SYM
cana-2799	4	26	12	12	NUM
cana-2799	4	27	-	-	PUNCT
cana-2799	4	28	2024	2024	NUM
cana-2799	4	29	abstract	abstract	NOUN
cana-2799	4	30	:	:	PUNCT
cana-2799	4	31	the	the	DET
cana-2799	4	32	newly	newly	ADV
cana-2799	4	33	defined	define	VERB
cana-2799	4	34	ℓ-extorial	ℓ-extorial	ADJ
cana-2799	4	35	function	function	NOUN
cana-2799	4	36	was	be	AUX
cana-2799	4	37	used	use	VERB
cana-2799	4	38	in	in	ADP
cana-2799	4	39	this	this	DET
cana-2799	4	40	paper	paper	NOUN
cana-2799	4	41	to	to	PART
cana-2799	4	42	generate	generate	VERB
cana-2799	4	43	the	the	DET
cana-2799	4	44	solution	solution	NOUN
cana-2799	4	45	of	of	ADP
cana-2799	4	46	the	the	DET
cana-2799	4	47	rl	rl	PROPN
cana-2799	4	48	circuit	circuit	NOUN
cana-2799	4	49	.	.	PUNCT
cana-2799	5	1	when	when	SCONJ
cana-2799	5	2	an	an	DET
cana-2799	5	3	inductor	inductor	NOUN
cana-2799	5	4	and	and	CCONJ
cana-2799	5	5	a	a	DET
cana-2799	5	6	resistor	resistor	NOUN
cana-2799	5	7	are	be	AUX
cana-2799	5	8	linked	link	VERB
cana-2799	5	9	across	across	ADP
cana-2799	5	10	a	a	DET
cana-2799	5	11	voltage	voltage	NOUN
cana-2799	5	12	source	source	NOUN
cana-2799	5	13	,	,	PUNCT
cana-2799	5	14	the	the	DET
cana-2799	5	15	resulting	result	VERB
cana-2799	5	16	circuit	circuit	NOUN
cana-2799	5	17	is	be	AUX
cana-2799	5	18	known	know	VERB
cana-2799	5	19	as	as	ADP
cana-2799	5	20	an	an	DET
cana-2799	5	21	rl	rl	NOUN
cana-2799	5	22	circuit	circuit	NOUN
cana-2799	5	23	.	.	PUNCT
cana-2799	6	1	depending	depend	VERB
cana-2799	6	2	on	on	ADP
cana-2799	6	3	how	how	SCONJ
cana-2799	6	4	the	the	DET
cana-2799	6	5	resistor	resistor	NOUN
cana-2799	6	6	and	and	CCONJ
cana-2799	6	7	inductor	inductor	NOUN
cana-2799	6	8	are	be	AUX
cana-2799	6	9	connected	connect	VERB
cana-2799	6	10	,	,	PUNCT
cana-2799	6	11	this	this	DET
cana-2799	6	12	circuit	circuit	NOUN
cana-2799	6	13	may	may	AUX
cana-2799	6	14	be	be	AUX
cana-2799	6	15	in	in	ADP
cana-2799	6	16	series	series	NOUN
cana-2799	6	17	or	or	CCONJ
cana-2799	6	18	parallel	parallel	NOUN
cana-2799	6	19	.	.	PUNCT
cana-2799	7	1	one	one	NUM
cana-2799	7	2	of	of	ADP
cana-2799	7	3	the	the	DET
cana-2799	7	4	solutions	solution	NOUN
cana-2799	7	5	to	to	PART
cana-2799	7	6	rl	rl	VERB
cana-2799	7	7	’s	’s	PART
cana-2799	7	8	mathematical	mathematical	ADJ
cana-2799	7	9	difference	difference	NOUN
cana-2799	7	10	problem	problem	NOUN
cana-2799	7	11	is	be	AUX
cana-2799	7	12	the	the	DET
cana-2799	7	13	extorial	extorial	ADJ
cana-2799	7	14	function	function	NOUN
cana-2799	7	15	.	.	PUNCT
cana-2799	8	1	to	to	PART
cana-2799	8	2	solve	solve	VERB
cana-2799	8	3	the	the	DET
cana-2799	8	4	rl	rl	NOUN
cana-2799	8	5	circuit	circuit	NOUN
cana-2799	8	6	,	,	PUNCT
cana-2799	8	7	we	we	PRON
cana-2799	8	8	thus	thus	ADV
cana-2799	8	9	develop	develop	VERB
cana-2799	8	10	the	the	DET
cana-2799	8	11	theory	theory	NOUN
cana-2799	8	12	of	of	ADP
cana-2799	8	13	the	the	DET
cana-2799	8	14	extorial	extorial	ADJ
cana-2799	8	15	function	function	NOUN
cana-2799	8	16	and	and	CCONJ
cana-2799	8	17	employ	employ	VERB
cana-2799	8	18	it	it	PRON
cana-2799	8	19	.	.	PUNCT
cana-2799	9	1	keywords	keyword	NOUN
cana-2799	9	2	:	:	PUNCT
cana-2799	9	3	ℓ-extorial	ℓ-extorial	ADJ
cana-2799	9	4	function	function	NOUN
cana-2799	9	5	,	,	PUNCT
cana-2799	9	6	ℓ-delta	ℓ-delta	NOUN
cana-2799	9	7	operator	operator	NOUN
cana-2799	9	8	,	,	PUNCT
cana-2799	9	9	rl	rl	NOUN
cana-2799	9	10	circuit	circuit	NOUN
cana-2799	9	11	,	,	PUNCT
cana-2799	9	12	heat	heat	NOUN
cana-2799	9	13	equation	equation	NOUN
cana-2799	9	14	.	.	PUNCT
cana-2799	10	1	1	1	X
cana-2799	10	2	.	.	X
cana-2799	10	3	introduction	introduction	NOUN
cana-2799	10	4	a	a	DET
cana-2799	10	5	difference	difference	NOUN
cana-2799	10	6	equation	equation	NOUN
cana-2799	10	7	is	be	AUX
cana-2799	10	8	an	an	DET
cana-2799	10	9	equation	equation	NOUN
cana-2799	10	10	that	that	PRON
cana-2799	10	11	contains	contain	VERB
cana-2799	10	12	sequence	sequence	NOUN
cana-2799	10	13	differences	difference	NOUN
cana-2799	10	14	.	.	PUNCT
cana-2799	11	1	there	there	PRON
cana-2799	11	2	are	be	VERB
cana-2799	11	3	various	various	ADJ
cana-2799	11	4	types	type	NOUN
cana-2799	11	5	of	of	ADP
cana-2799	11	6	difference	difference	NOUN
cana-2799	11	7	equations	equation	NOUN
cana-2799	11	8	namely	namely	ADV
cana-2799	11	9	ordinary	ordinary	ADJ
cana-2799	11	10	,	,	PUNCT
cana-2799	11	11	delay	delay	NOUN
cana-2799	11	12	,	,	PUNCT
cana-2799	11	13	advanced	advanced	ADJ
cana-2799	11	14	,	,	PUNCT
cana-2799	11	15	neutral	neutral	ADJ
cana-2799	11	16	,	,	PUNCT
cana-2799	11	17	quasilinear	quasilinear	NOUN
cana-2799	11	18	,	,	PUNCT
cana-2799	11	19	half	half	ADJ
cana-2799	11	20	linear	linear	ADJ
cana-2799	11	21	,	,	PUNCT
cana-2799	11	22	etc	etc	X
cana-2799	11	23	.	.	X
cana-2799	12	1	these	these	DET
cana-2799	12	2	equations	equation	NOUN
cana-2799	12	3	occur	occur	VERB
cana-2799	12	4	in	in	ADP
cana-2799	12	5	numerous	numerous	ADJ
cana-2799	12	6	settings	setting	NOUN
cana-2799	12	7	and	and	CCONJ
cana-2799	12	8	forms	form	NOUN
cana-2799	12	9	,	,	PUNCT
cana-2799	12	10	both	both	PRON
cana-2799	12	11	in	in	ADP
cana-2799	12	12	mathematics	mathematic	NOUN
cana-2799	12	13	itself	itself	PRON
cana-2799	12	14	and	and	CCONJ
cana-2799	12	15	its	its	PRON
cana-2799	12	16	applications	application	NOUN
cana-2799	12	17	to	to	ADP
cana-2799	12	18	biology	biology	NOUN
cana-2799	12	19	,	,	PUNCT
cana-2799	12	20	computer	computer	NOUN
cana-2799	12	21	science	science	NOUN
cana-2799	12	22	,	,	PUNCT
cana-2799	12	23	digital	digital	ADJ
cana-2799	12	24	signal	signal	NOUN
cana-2799	12	25	processing	processing	NOUN
cana-2799	12	26	,	,	PUNCT
cana-2799	12	27	economics	economic	NOUN
cana-2799	12	28	,	,	PUNCT
cana-2799	12	29	statistics	statistic	NOUN
cana-2799	12	30	and	and	CCONJ
cana-2799	12	31	other	other	ADJ
cana-2799	12	32	fields	field	NOUN
cana-2799	12	33	.	.	PUNCT
cana-2799	13	1	the	the	DET
cana-2799	13	2	fractional	fractional	ADJ
cana-2799	13	3	sum	sum	NOUN
cana-2799	13	4	of	of	ADP
cana-2799	13	5	a	a	DET
cana-2799	13	6	function	function	NOUN
cana-2799	13	7	𝑓	𝑓	PRON
cana-2799	13	8	(	(	PUNCT
cana-2799	13	9	or	or	CCONJ
cana-2799	13	10	𝜈𝑡ℎ	𝜈𝑡ℎ	INTJ
cana-2799	13	11	order	order	NOUN
cana-2799	13	12	delta	delta	NOUN
cana-2799	13	13	integration	integration	NOUN
cana-2799	13	14	)	)	PUNCT
cana-2799	13	15	is	be	AUX
cana-2799	13	16	defined	define	VERB
cana-2799	13	17	by	by	ADP
cana-2799	13	18	(	(	PUNCT
cana-2799	13	19	δ𝑎	δ𝑎	NOUN
cana-2799	13	20	−𝜈𝑢)(𝜅	−𝜈𝑢)(𝜅	ADJ
cana-2799	13	21	)	)	PUNCT
cana-2799	13	22	=	=	SYM
cana-2799	13	23	1	1	NUM
cana-2799	13	24	γ(𝜈	γ(𝜈	NUM
cana-2799	13	25	)	)	PUNCT
cana-2799	13	26	∑𝜅−𝜈	∑𝜅−𝜈	PROPN
cana-2799	13	27	𝑠=𝑎	𝑠=𝑎	PROPN
cana-2799	13	28	γ(𝜅−𝑠	γ(𝜅−𝑠	PROPN
cana-2799	13	29	)	)	PUNCT
cana-2799	13	30	γ(𝜅−𝑠−(𝜈−1	γ(𝜅−𝑠−(𝜈−1	PROPN
cana-2799	13	31	)	)	PUNCT
cana-2799	13	32	)	)	PUNCT
cana-2799	14	1	𝑢(𝑠	𝑢(𝑠	NOUN
cana-2799	14	2	)	)	PUNCT
cana-2799	14	3	,	,	PUNCT
cana-2799	14	4	(	(	PUNCT
cana-2799	14	5	1	1	X
cana-2799	14	6	)	)	PUNCT
cana-2799	14	7	where	where	SCONJ
cana-2799	14	8	𝜈	𝜈	X
cana-2799	14	9	>	>	X
cana-2799	14	10	0	0	PROPN
cana-2799	14	11	,	,	PUNCT
cana-2799	14	12	𝑓	𝑓	PRON
cana-2799	14	13	is	be	AUX
cana-2799	14	14	defined	define	VERB
cana-2799	14	15	for	for	ADP
cana-2799	14	16	𝑠	𝑠	PROPN
cana-2799	14	17	=	=	SYM
cana-2799	14	18	𝑎	𝑎	NOUN
cana-2799	14	19	𝑚𝑜𝑑(1	𝑚𝑜𝑑(1	NOUN
cana-2799	14	20	)	)	PUNCT
cana-2799	14	21	and	and	CCONJ
cana-2799	14	22	δ	δ	PROPN
cana-2799	14	23	−𝜈𝑓	−𝜈𝑓	PROPN
cana-2799	14	24	is	be	AUX
cana-2799	14	25	defined	define	VERB
cana-2799	14	26	for	for	ADP
cana-2799	14	27	𝜅	𝜅	NOUN
cana-2799	14	28	=	=	SYM
cana-2799	14	29	𝑎	𝑎	PROPN
cana-2799	14	30	+	+	X
cana-2799	14	31	𝜈	𝜈	X
cana-2799	14	32	𝑚𝑜𝑑(1	𝑚𝑜𝑑(1	ADJ
cana-2799	14	33	)	)	PUNCT
cana-2799	14	34	.	.	PUNCT
cana-2799	15	1	the	the	DET
cana-2799	15	2	basic	basic	ADJ
cana-2799	15	3	theory	theory	NOUN
cana-2799	15	4	of	of	ADP
cana-2799	15	5	difference	difference	NOUN
cana-2799	15	6	equations	equation	NOUN
cana-2799	15	7	is	be	AUX
cana-2799	15	8	based	base	VERB
cana-2799	15	9	on	on	ADP
cana-2799	15	10	the	the	DET
cana-2799	15	11	difference	difference	NOUN
cana-2799	15	12	operator	operator	NOUN
cana-2799	15	13	δ	δ	PROPN
cana-2799	15	14	defined	define	VERB
cana-2799	15	15	as	as	ADP
cana-2799	15	16	δ𝑢(𝜅	δ𝑢(𝜅	NUM
cana-2799	15	17	)	)	PUNCT
cana-2799	15	18	=	=	PUNCT
cana-2799	16	1	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	16	2	+	+	CCONJ
cana-2799	16	3	1	1	X
cana-2799	16	4	)	)	PUNCT
cana-2799	16	5	−	−	PROPN
cana-2799	16	6	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	16	7	)	)	PUNCT
cana-2799	16	8	,	,	PUNCT
cana-2799	16	9	where	where	SCONJ
cana-2799	16	10	{	{	PUNCT
cana-2799	16	11	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	16	12	)	)	PUNCT
cana-2799	16	13	}	}	PUNCT
cana-2799	16	14	is	be	AUX
cana-2799	16	15	a	a	DET
cana-2799	16	16	sequence	sequence	NOUN
cana-2799	16	17	or	or	CCONJ
cana-2799	16	18	a	a	DET
cana-2799	16	19	function	function	NOUN
cana-2799	16	20	of	of	ADP
cana-2799	16	21	𝜅	𝜅	NUM
cana-2799	16	22	of	of	ADP
cana-2799	16	23	numbers	number	NOUN
cana-2799	16	24	.	.	PUNCT
cana-2799	17	1	many	many	ADJ
cana-2799	17	2	authors	author	NOUN
cana-2799	17	3	(	(	PUNCT
cana-2799	17	4	[	[	X
cana-2799	17	5	7],[9	7],[9	NUM
cana-2799	17	6	]	]	PUNCT
cana-2799	17	7	)	)	PUNCT
cana-2799	17	8	have	have	AUX
cana-2799	17	9	suggested	suggest	VERB
cana-2799	17	10	the	the	DET
cana-2799	17	11	definition	definition	NOUN
cana-2799	17	12	of	of	ADP
cana-2799	17	13	generalized	generalized	ADJ
cana-2799	17	14	difference	difference	NOUN
cana-2799	17	15	operator	operator	NOUN
cana-2799	17	16	δℓ	δℓ	NOUN
cana-2799	17	17	on	on	ADP
cana-2799	17	18	real	real	ADV
cana-2799	17	19	valued	value	VERB
cana-2799	17	20	function	function	NOUN
cana-2799	17	21	u	u	NOUN
cana-2799	17	22	defined	define	VERB
cana-2799	17	23	on	on	ADP
cana-2799	17	24	ℝ	ℝ	PROPN
cana-2799	17	25	=	=	PUNCT
cana-2799	17	26	(	(	PUNCT
cana-2799	17	27	−∞,∞	−∞,∞	NOUN
cana-2799	17	28	)	)	PUNCT
cana-2799	17	29	as	as	ADP
cana-2799	17	30	δℓ𝑢(𝜅	δℓ𝑢(𝜅	PROPN
cana-2799	17	31	)	)	PUNCT
cana-2799	17	32	=	=	PUNCT
cana-2799	17	33	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	17	34	+	+	CCONJ
cana-2799	17	35	ℓ	ℓ	X
cana-2799	17	36	)	)	PUNCT
cana-2799	17	37	−	−	PROPN
cana-2799	17	38	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	17	39	)	)	PUNCT
cana-2799	17	40	,	,	PUNCT
cana-2799	17	41	𝜅	𝜅	PROPN
cana-2799	17	42	∈	∈	PROPN
cana-2799	17	43	ℝ	ℝ	PROPN
cana-2799	17	44	,	,	PUNCT
cana-2799	17	45	ℓ	ℓ	PROPN
cana-2799	17	46	>	>	X
cana-2799	17	47	0	0	PROPN
cana-2799	17	48	.	.	PUNCT
cana-2799	18	1	(	(	PUNCT
cana-2799	18	2	2	2	X
cana-2799	18	3	)	)	PUNCT
cana-2799	18	4	e.	e.	PROPN
cana-2799	18	5	thandapani	thandapani	PROPN
cana-2799	18	6	,	,	PUNCT
cana-2799	18	7	m.maria	m.maria	NOUN
cana-2799	18	8	susai	susai	PROPN
cana-2799	18	9	manuel	manuel	NOUN
cana-2799	18	10	,	,	PUNCT
cana-2799	18	11	g.b.a	g.b.a	NOUN
cana-2799	18	12	xavier	xavier	NOUN
cana-2799	19	1	[	[	X
cana-2799	19	2	8	8	NUM
cana-2799	19	3	]	]	PUNCT
cana-2799	19	4	considered	consider	VERB
cana-2799	19	5	the	the	DET
cana-2799	19	6	definition	definition	NOUN
cana-2799	19	7	of	of	ADP
cana-2799	19	8	δℓ	δℓ	NOUN
cana-2799	19	9	as	as	SCONJ
cana-2799	19	10	given	give	VERB
cana-2799	19	11	in	in	ADP
cana-2799	19	12	(	(	PUNCT
cana-2799	19	13	2	2	NUM
cana-2799	19	14	)	)	PUNCT
cana-2799	19	15	and	and	CCONJ
cana-2799	19	16	developed	develop	VERB
cana-2799	19	17	the	the	DET
cana-2799	19	18	theory	theory	NOUN
cana-2799	19	19	of	of	ADP
cana-2799	19	20	difference	difference	NOUN
cana-2799	19	21	equations	equation	NOUN
cana-2799	19	22	in	in	ADP
cana-2799	19	23	a	a	DET
cana-2799	19	24	different	different	ADJ
cana-2799	19	25	direction	direction	NOUN
cana-2799	19	26	.	.	PUNCT
cana-2799	20	1	if	if	SCONJ
cana-2799	20	2	there	there	PRON
cana-2799	20	3	exists	exist	VERB
cana-2799	20	4	a	a	DET
cana-2799	20	5	function	function	NOUN
cana-2799	20	6	v	v	ADP
cana-2799	20	7	such	such	ADJ
cana-2799	20	8	that	that	DET
cana-2799	20	9	δℓ𝑣(𝜅	δℓ𝑣(𝜅	PROPN
cana-2799	20	10	)	)	PUNCT
cana-2799	20	11	=	=	SYM
cana-2799	20	12	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	20	13	)	)	PUNCT
cana-2799	20	14	,	,	PUNCT
cana-2799	20	15	then	then	ADV
cana-2799	20	16	we	we	PRON
cana-2799	20	17	call	call	VERB
cana-2799	20	18	this	this	DET
cana-2799	20	19	function	function	NOUN
cana-2799	20	20	𝑣	𝑣	ADP
cana-2799	20	21	as	as	ADP
cana-2799	20	22	δℓ	δℓ	ADP
cana-2799	20	23	−1𝑣.	−1𝑣.	X
cana-2799	20	24	hence	hence	ADV
cana-2799	20	25	,	,	PUNCT
cana-2799	20	26	for	for	ADP
cana-2799	20	27	𝜅	𝜅	DET
cana-2799	20	28	∈	∈	NOUN
cana-2799	20	29	ℝ	ℝ	NOUN
cana-2799	20	30	=	=	SYM
cana-2799	20	31	∪	∪	NOUN
cana-2799	20	32	0≤𝑗<ℓ	0≤𝑗<ℓ	PROPN
cana-2799	20	33	ℕℓ(𝑗	ℕℓ(𝑗	NOUN
cana-2799	20	34	)	)	PUNCT
cana-2799	20	35	,	,	PUNCT
cana-2799	20	36	if	if	SCONJ
cana-2799	20	37	δℓ𝑣(𝜅	δℓ𝑣(𝜅	PROPN
cana-2799	20	38	)	)	PUNCT
cana-2799	20	39	=	=	SYM
cana-2799	20	40	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	20	41	)	)	PUNCT
cana-2799	20	42	,	,	PUNCT
cana-2799	20	43	then	then	ADV
cana-2799	20	44	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	20	45	)	)	PUNCT
cana-2799	20	46	=	=	SYM
cana-2799	20	47	δℓ	δℓ	NOUN
cana-2799	20	48	−1𝑢(𝜅	−1𝑢(𝜅	NOUN
cana-2799	20	49	)	)	PUNCT
cana-2799	21	1	+	+	CCONJ
cana-2799	21	2	𝑐𝑗	𝑐𝑗	INTJ
cana-2799	21	3	,	,	PUNCT
cana-2799	21	4	(	(	PUNCT
cana-2799	21	5	3	3	X
cana-2799	21	6	)	)	PUNCT
cana-2799	21	7	where	where	SCONJ
cana-2799	21	8	𝑐𝑗	𝑐𝑗	NOUN
cana-2799	21	9	is	be	AUX
cana-2799	21	10	constant	constant	ADJ
cana-2799	21	11	for	for	ADP
cana-2799	21	12	all	all	DET
cana-2799	21	13	𝜅	𝜅	PRON
cana-2799	21	14	in	in	ADP
cana-2799	21	15	each	each	DET
cana-2799	21	16	ℕℓ(𝑗	ℕℓ(𝑗	NOUN
cana-2799	21	17	)	)	PUNCT
cana-2799	21	18	=	=	PRON
cana-2799	21	19	{	{	PUNCT
cana-2799	21	20	𝑗	𝑗	PROPN
cana-2799	21	21	,	,	PUNCT
cana-2799	21	22	𝑗	𝑗	PROPN
cana-2799	21	23	+	+	NUM
cana-2799	21	24	ℓ	ℓ	NUM
cana-2799	21	25	,	,	PUNCT
cana-2799	21	26	𝑗	𝑗	NOUN
cana-2799	21	27	+	+	NUM
cana-2799	21	28	2ℓ	2ℓ	NOUN
cana-2799	21	29	,	,	PUNCT
cana-2799	21	30	…	…	PUNCT
cana-2799	21	31	}	}	PUNCT
cana-2799	21	32	,	,	PUNCT
cana-2799	21	33	𝑗	𝑗	PROPN
cana-2799	21	34	=	=	X
cana-2799	21	35	𝜅	𝜅	X
cana-2799	21	36	−	−	PROPN
cana-2799	21	37	[	[	PUNCT
cana-2799	21	38	𝜅	𝜅	PROPN
cana-2799	21	39	ℓ	ℓ	PROPN
cana-2799	21	40	]	]	PUNCT
cana-2799	21	41	ℓ.	ℓ.	NOUN
cana-2799	21	42	communications	communication	NOUN
cana-2799	21	43	on	on	ADP
cana-2799	21	44	applied	apply	VERB
cana-2799	21	45	nonlinear	nonlinear	ADJ
cana-2799	21	46	analysis	analysis	NOUN
cana-2799	21	47	issn	issn	NOUN
cana-2799	21	48	:	:	PUNCT
cana-2799	21	49	1074	1074	NUM
cana-2799	21	50	-	-	PUNCT
cana-2799	21	51	133x	133x	NUM
cana-2799	21	52	vol	vol	NOUN
cana-2799	21	53	32	32	NUM
cana-2799	21	54	no	no	NOUN
cana-2799	21	55	.	.	PUNCT
cana-2799	22	1	4s	4s	NUM
cana-2799	22	2	(	(	PUNCT
cana-2799	22	3	2025	2025	NUM
cana-2799	22	4	)	)	PUNCT
cana-2799	22	5	259	259	NUM
cana-2799	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	22	7	in	in	ADP
cana-2799	22	8	1989	1989	NUM
cana-2799	22	9	,	,	PUNCT
cana-2799	22	10	miller	miller	PROPN
cana-2799	22	11	and	and	CCONJ
cana-2799	22	12	rose	rise	VERB
cana-2799	22	13	introduced	introduce	VERB
cana-2799	22	14	the	the	DET
cana-2799	22	15	discrete	discrete	ADJ
cana-2799	22	16	analogue	analogue	NOUN
cana-2799	22	17	of	of	ADP
cana-2799	22	18	the	the	DET
cana-2799	22	19	riemann	riemann	PROPN
cana-2799	22	20	-	-	PUNCT
cana-2799	22	21	liouville	liouville	VERB
cana-2799	22	22	fractional	fractional	ADJ
cana-2799	22	23	derivative	derivative	NOUN
cana-2799	22	24	and	and	CCONJ
cana-2799	22	25	proved	prove	VERB
cana-2799	22	26	some	some	DET
cana-2799	22	27	properties	property	NOUN
cana-2799	22	28	of	of	ADP
cana-2799	22	29	the	the	DET
cana-2799	22	30	fractional	fractional	ADJ
cana-2799	22	31	difference	difference	NOUN
cana-2799	22	32	operator	operator	NOUN
cana-2799	22	33	.	.	PUNCT
cana-2799	23	1	in	in	ADP
cana-2799	23	2	1984	1984	NUM
cana-2799	23	3	,	,	PUNCT
cana-2799	23	4	jerzy	jerzy	X
cana-2799	23	5	popenda	popenda	NOUN
cana-2799	23	6	[	[	X
cana-2799	23	7	4	4	X
cana-2799	23	8	]	]	PUNCT
cana-2799	23	9	introduced	introduce	VERB
cana-2799	23	10	a	a	DET
cana-2799	23	11	particular	particular	ADJ
cana-2799	23	12	type	type	NOUN
cana-2799	23	13	of	of	ADP
cana-2799	23	14	difference	difference	NOUN
cana-2799	23	15	operator	operator	NOUN
cana-2799	23	16	on	on	ADP
cana-2799	23	17	𝑢	𝑢	NOUN
cana-2799	23	18	as	as	ADP
cana-2799	23	19	δ𝛼𝑢(𝜅	δ𝛼𝑢(𝜅	NOUN
cana-2799	23	20	)	)	PUNCT
cana-2799	23	21	=	=	PUNCT
cana-2799	23	22	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	23	23	+	+	CCONJ
cana-2799	23	24	1	1	X
cana-2799	23	25	)	)	PUNCT
cana-2799	23	26	−	−	NOUN
cana-2799	23	27	𝛼𝑢(𝜅	𝛼𝑢(𝜅	NUM
cana-2799	23	28	)	)	PUNCT
cana-2799	23	29	,	,	PUNCT
cana-2799	23	30	in	in	ADP
cana-2799	23	31	2011	2011	NUM
cana-2799	23	32	,	,	PUNCT
cana-2799	23	33	m.maria	m.maria	NOUN
cana-2799	23	34	susai	susai	NOUN
cana-2799	23	35	manuel	manuel	NOUN
cana-2799	23	36	,	,	PUNCT
cana-2799	23	37	et.al	et.al	PROPN
cana-2799	23	38	,	,	PUNCT
cana-2799	23	39	[	[	X
cana-2799	23	40	5	5	NUM
cana-2799	23	41	]	]	PUNCT
cana-2799	23	42	extended	extend	VERB
cana-2799	23	43	the	the	DET
cana-2799	23	44	operator	operator	NOUN
cana-2799	23	45	δ𝛼	δ𝛼	ADP
cana-2799	23	46	to	to	PART
cana-2799	23	47	generalized	generalize	VERB
cana-2799	23	48	𝛼	𝛼	PRON
cana-2799	23	49	−difference	−difference	NOUN
cana-2799	23	50	operator	operator	NOUN
cana-2799	23	51	as	as	ADP
cana-2799	23	52	δ	δ	PROPN
cana-2799	23	53	𝛼(ℓ	𝛼(ℓ	PROPN
cana-2799	23	54	)	)	PUNCT
cana-2799	23	55	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	23	56	)	)	PUNCT
cana-2799	23	57	=	=	SYM
cana-2799	24	1	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	24	2	+	+	CCONJ
cana-2799	24	3	ℓ	ℓ	NOUN
cana-2799	24	4	)	)	PUNCT
cana-2799	24	5	−	−	NOUN
cana-2799	24	6	𝛼𝑣(𝜅	𝛼𝑣(𝜅	NUM
cana-2799	24	7	)	)	PUNCT
cana-2799	24	8	for	for	ADP
cana-2799	24	9	real	real	ADV
cana-2799	24	10	valued	value	VERB
cana-2799	24	11	function	function	NOUN
cana-2799	24	12	𝑣.	𝑣.	NOUN
cana-2799	24	13	in	in	ADP
cana-2799	24	14	2014	2014	NUM
cana-2799	24	15	,	,	PUNCT
cana-2799	24	16	the	the	DET
cana-2799	24	17	authors	author	NOUN
cana-2799	24	18	in	in	ADP
cana-2799	24	19	[	[	X
cana-2799	24	20	1	1	NUM
cana-2799	24	21	]	]	PUNCT
cana-2799	24	22	have	have	AUX
cana-2799	24	23	applied	apply	VERB
cana-2799	24	24	the	the	DET
cana-2799	24	25	q	q	ADJ
cana-2799	24	26	-	-	PUNCT
cana-2799	24	27	difference	difference	NOUN
cana-2799	24	28	operator	operator	NOUN
cana-2799	24	29	defined	define	VERB
cana-2799	24	30	by	by	ADP
cana-2799	24	31	δ𝑞𝑣(𝜅	δ𝑞𝑣(𝜅	PROPN
cana-2799	24	32	)	)	PUNCT
cana-2799	24	33	=	=	PUNCT
cana-2799	25	1	𝑣(𝑞𝜅	𝑣(𝑞𝜅	NOUN
cana-2799	25	2	)	)	PUNCT
cana-2799	25	3	−	−	PROPN
cana-2799	25	4	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	25	5	)	)	PUNCT
cana-2799	25	6	and	and	CCONJ
cana-2799	25	7	delta	delta	NOUN
cana-2799	25	8	operator	operator	NOUN
cana-2799	25	9	δ	δ	PROPN
cana-2799	25	10	𝜅(ℓ	𝜅(ℓ	NOUN
cana-2799	25	11	)	)	PUNCT
cana-2799	25	12	with	with	ADP
cana-2799	25	13	variable	variable	ADJ
cana-2799	25	14	coefficients	coefficient	NOUN
cana-2799	25	15	defined	define	VERB
cana-2799	25	16	by	by	ADP
cana-2799	25	17	δ	δ	PROPN
cana-2799	25	18	𝜅(ℓ	𝜅(ℓ	ADJ
cana-2799	25	19	)	)	PUNCT
cana-2799	25	20	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	25	21	)	)	PUNCT
cana-2799	25	22	=	=	SYM
cana-2799	26	1	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	26	2	+	+	CCONJ
cana-2799	26	3	ℓ	ℓ	X
cana-2799	26	4	)	)	PUNCT
cana-2799	26	5	−	−	NOUN
cana-2799	26	6	𝜅𝑣(𝜅	𝜅𝑣(𝜅	NUM
cana-2799	26	7	)	)	PUNCT
cana-2799	26	8	,	,	PUNCT
cana-2799	26	9	ℓ	ℓ	PROPN
cana-2799	26	10	≠	≠	PROPN
cana-2799	26	11	0	0	NUM
cana-2799	26	12	∈	∈	PROPN
cana-2799	26	13	ℝ.	ℝ.	PROPN
cana-2799	26	14	also	also	ADV
cana-2799	26	15	the	the	DET
cana-2799	26	16	generalized	generalized	ADJ
cana-2799	26	17	difference	difference	NOUN
cana-2799	26	18	operator	operator	NOUN
cana-2799	26	19	with	with	ADP
cana-2799	26	20	𝑛-shift	𝑛-shift	PROPN
cana-2799	26	21	values	value	NOUN
cana-2799	26	22	𝑙	𝑙	X
cana-2799	27	1	=	=	SYM
cana-2799	28	1	(	(	PUNCT
cana-2799	28	2	ℓ1	ℓ1	NOUN
cana-2799	28	3	,	,	PUNCT
cana-2799	28	4	ℓ2	ℓ2	NOUN
cana-2799	28	5	,	,	PUNCT
cana-2799	28	6	ℓ3	ℓ3	PROPN
cana-2799	28	7	,	,	PUNCT
cana-2799	28	8	.	.	PUNCT
cana-2799	28	9	.	.	PUNCT
cana-2799	29	1	.	.	PUNCT
cana-2799	30	1	,	,	PUNCT
cana-2799	30	2	ℓ𝑛	ℓ𝑛	NOUN
cana-2799	30	3	)	)	PUNCT
cana-2799	30	4	≠	≠	PROPN
cana-2799	30	5	0	0	NUM
cana-2799	30	6	on	on	ADP
cana-2799	30	7	a	a	DET
cana-2799	30	8	real	real	ADV
cana-2799	30	9	valued	value	VERB
cana-2799	30	10	function	function	NOUN
cana-2799	30	11	𝑣	𝑣	ADP
cana-2799	30	12	:	:	PUNCT
cana-2799	30	13	ℝ𝑛	ℝ𝑛	X
cana-2799	30	14	→	→	SYM
cana-2799	30	15	ℝ	ℝ	PROPN
cana-2799	30	16	is	be	AUX
cana-2799	30	17	defined	define	VERB
cana-2799	30	18	as	as	ADP
cana-2799	30	19	δ	δ	PROPN
cana-2799	30	20	(	(	PUNCT
cana-2799	30	21	ℓ	ℓ	NOUN
cana-2799	30	22	)	)	PUNCT
cana-2799	30	23	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	30	24	)	)	PUNCT
cana-2799	31	1	=	=	PUNCT
cana-2799	31	2	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	31	3	+	+	CCONJ
cana-2799	31	4	ℓ1	ℓ1	NOUN
cana-2799	31	5	,	,	PUNCT
cana-2799	31	6	𝜅2	𝜅2	NOUN
cana-2799	31	7	+	+	CCONJ
cana-2799	31	8	ℓ2	ℓ2	NOUN
cana-2799	31	9	,	,	PUNCT
cana-2799	31	10	.	.	PUNCT
cana-2799	31	11	.	.	PUNCT
cana-2799	32	1	.	.	PUNCT
cana-2799	33	1	,	,	PUNCT
cana-2799	33	2	𝜅𝑛	𝜅𝑛	ADP
cana-2799	33	3	+	+	ADJ
cana-2799	33	4	ℓ𝑛	ℓ𝑛	NOUN
cana-2799	33	5	)	)	PUNCT
cana-2799	33	6	−	−	NOUN
cana-2799	33	7	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	33	8	,	,	PUNCT
cana-2799	33	9	𝜅2	𝜅2	NOUN
cana-2799	33	10	,	,	PUNCT
cana-2799	33	11	.	.	PUNCT
cana-2799	33	12	.	.	PUNCT
cana-2799	34	1	.	.	PUNCT
cana-2799	35	1	,	,	PUNCT
cana-2799	35	2	𝜅𝑛	𝜅𝑛	PROPN
cana-2799	35	3	)	)	PUNCT
cana-2799	35	4	.	.	PUNCT
cana-2799	36	1	(	(	PUNCT
cana-2799	36	2	4	4	X
cana-2799	36	3	)	)	PUNCT
cana-2799	36	4	this	this	DET
cana-2799	36	5	operator	operator	NOUN
cana-2799	36	6	δ	δ	PROPN
cana-2799	36	7	(	(	PUNCT
cana-2799	36	8	ℓ	ℓ	NOUN
cana-2799	36	9	)	)	PUNCT
cana-2799	36	10	becomes	become	VERB
cana-2799	36	11	generalized	generalized	ADJ
cana-2799	36	12	partial	partial	ADJ
cana-2799	36	13	difference	difference	NOUN
cana-2799	36	14	operator	operator	NOUN
cana-2799	36	15	if	if	SCONJ
cana-2799	36	16	some	some	PRON
cana-2799	36	17	ℓ𝑖	ℓ𝑖	VERB
cana-2799	36	18	=	=	NOUN
cana-2799	36	19	0	0	NUM
cana-2799	36	20	.	.	PUNCT
cana-2799	37	1	the	the	DET
cana-2799	37	2	equations	equation	NOUN
cana-2799	37	3	involving	involve	VERB
cana-2799	37	4	δ	δ	PROPN
cana-2799	37	5	(	(	PUNCT
cana-2799	37	6	ℓ	ℓ	NOUN
cana-2799	37	7	)	)	PUNCT
cana-2799	37	8	with	with	ADP
cana-2799	37	9	atleast	atleast	ADJ
cana-2799	37	10	one	one	NUM
cana-2799	37	11	ℓ𝑖	ℓ𝑖	VERB
cana-2799	37	12	=	=	SYM
cana-2799	37	13	0	0	NUM
cana-2799	37	14	is	be	AUX
cana-2799	37	15	called	call	VERB
cana-2799	37	16	generalized	generalized	ADJ
cana-2799	37	17	partial	partial	ADJ
cana-2799	37	18	difference	difference	NOUN
cana-2799	37	19	equation	equation	NOUN
cana-2799	37	20	.	.	PUNCT
cana-2799	38	1	for	for	ADP
cana-2799	38	2	one	one	NUM
cana-2799	38	3	shift	shift	NOUN
cana-2799	38	4	value	value	NOUN
cana-2799	38	5	,	,	PUNCT
cana-2799	38	6	we	we	PRON
cana-2799	38	7	take	take	VERB
cana-2799	38	8	δ(ℓ	δ(ℓ	NOUN
cana-2799	38	9	)	)	PUNCT
cana-2799	38	10	as	as	ADP
cana-2799	38	11	δℓ.	δℓ.	NUM
cana-2799	38	12	by	by	ADP
cana-2799	38	13	defining	define	VERB
cana-2799	38	14	the	the	DET
cana-2799	38	15	inverse	inverse	NOUN
cana-2799	38	16	δℓ	δℓ	NOUN
cana-2799	38	17	−1	−1	NOUN
cana-2799	38	18	,	,	PUNCT
cana-2799	38	19	many	many	ADJ
cana-2799	38	20	interesting	interesting	ADJ
cana-2799	38	21	results	result	NOUN
cana-2799	38	22	on	on	ADP
cana-2799	38	23	sum	sum	NOUN
cana-2799	38	24	of	of	ADP
cana-2799	38	25	partial	partial	ADJ
cana-2799	38	26	sums	sum	NOUN
cana-2799	38	27	of	of	ADP
cana-2799	38	28	higher	high	ADJ
cana-2799	38	29	power	power	NOUN
cana-2799	38	30	of	of	ADP
cana-2799	38	31	arithmetic	arithmetic	ADJ
cana-2799	38	32	and	and	CCONJ
cana-2799	38	33	geometric	geometric	ADJ
cana-2799	38	34	functions	function	NOUN
cana-2799	38	35	and	and	CCONJ
cana-2799	38	36	applications	application	NOUN
cana-2799	38	37	in	in	ADP
cana-2799	38	38	numerical	numerical	ADJ
cana-2799	38	39	methods	method	NOUN
cana-2799	38	40	(	(	PUNCT
cana-2799	38	41	see	see	VERB
cana-2799	38	42	[	[	X
cana-2799	38	43	8	8	NUM
cana-2799	38	44	,	,	PUNCT
cana-2799	38	45	6	6	NUM
cana-2799	38	46	,	,	PUNCT
cana-2799	38	47	2	2	NUM
cana-2799	38	48	,	,	PUNCT
cana-2799	38	49	3	3	NUM
cana-2799	38	50	,	,	PUNCT
cana-2799	38	51	10	10	NUM
cana-2799	38	52	,	,	PUNCT
cana-2799	38	53	11	11	NUM
cana-2799	38	54	,	,	PUNCT
cana-2799	38	55	12	12	NUM
cana-2799	38	56	,	,	PUNCT
cana-2799	38	57	13	13	NUM
cana-2799	38	58	]	]	PUNCT
cana-2799	38	59	)	)	PUNCT
cana-2799	38	60	are	be	AUX
cana-2799	38	61	obtained	obtain	VERB
cana-2799	38	62	.	.	PUNCT
cana-2799	39	1	the	the	DET
cana-2799	39	2	difference	difference	NOUN
cana-2799	39	3	operator	operator	NOUN
cana-2799	39	4	defined	define	VERB
cana-2799	39	5	in	in	ADP
cana-2799	39	6	(	(	PUNCT
cana-2799	39	7	2	2	NUM
cana-2799	39	8	)	)	PUNCT
cana-2799	39	9	becomes	become	VERB
cana-2799	39	10	the	the	DET
cana-2799	39	11	usual	usual	ADJ
cana-2799	39	12	difference	difference	NOUN
cana-2799	39	13	operator	operator	NOUN
cana-2799	39	14	δ	δ	PROPN
cana-2799	39	15	when	when	SCONJ
cana-2799	39	16	ℓ	ℓ	PROPN
cana-2799	39	17	=	=	SYM
cana-2799	39	18	1	1	X
cana-2799	39	19	.	.	PUNCT
cana-2799	40	1	we	we	PRON
cana-2799	40	2	obtain	obtain	VERB
cana-2799	40	3	several	several	ADJ
cana-2799	40	4	results	result	NOUN
cana-2799	40	5	on	on	ADP
cana-2799	40	6	factorial	factorial	ADJ
cana-2799	40	7	function	function	NOUN
cana-2799	40	8	by	by	ADP
cana-2799	40	9	applying	apply	VERB
cana-2799	40	10	δℓ	δℓ	ADP
cana-2799	40	11	−1	−1	NOUN
cana-2799	40	12	.	.	PUNCT
cana-2799	41	1	the	the	DET
cana-2799	41	2	fractional	fractional	ADJ
cana-2799	41	3	sum	sum	NOUN
cana-2799	41	4	of	of	ADP
cana-2799	41	5	a	a	DET
cana-2799	41	6	function	function	NOUN
cana-2799	41	7	𝑓	𝑓	PRON
cana-2799	41	8	(	(	PUNCT
cana-2799	41	9	or	or	CCONJ
cana-2799	41	10	𝜈𝑡ℎ	𝜈𝑡ℎ	INTJ
cana-2799	41	11	order	order	NOUN
cana-2799	41	12	delta	delta	NOUN
cana-2799	41	13	integration	integration	NOUN
cana-2799	41	14	)	)	PUNCT
cana-2799	41	15	is	be	AUX
cana-2799	41	16	defined	define	VERB
cana-2799	41	17	by	by	ADP
cana-2799	41	18	(	(	PUNCT
cana-2799	41	19	δ𝑎	δ𝑎	NOUN
cana-2799	41	20	−𝜈𝑢)(𝜅	−𝜈𝑢)(𝜅	ADJ
cana-2799	41	21	)	)	PUNCT
cana-2799	41	22	=	=	SYM
cana-2799	41	23	1	1	NUM
cana-2799	41	24	γ(𝜈	γ(𝜈	NUM
cana-2799	41	25	)	)	PUNCT
cana-2799	41	26	∑𝜅−𝜈	∑𝜅−𝜈	PROPN
cana-2799	41	27	𝑠=𝑎	𝑠=𝑎	PROPN
cana-2799	41	28	γ(𝜅−𝑠	γ(𝜅−𝑠	PROPN
cana-2799	41	29	)	)	PUNCT
cana-2799	41	30	γ(𝜅−𝑠−(𝜈−1	γ(𝜅−𝑠−(𝜈−1	PROPN
cana-2799	41	31	)	)	PUNCT
cana-2799	41	32	)	)	PUNCT
cana-2799	42	1	𝑢(𝑠	𝑢(𝑠	NOUN
cana-2799	42	2	)	)	PUNCT
cana-2799	42	3	,	,	PUNCT
cana-2799	42	4	(	(	PUNCT
cana-2799	42	5	1	1	X
cana-2799	42	6	)	)	PUNCT
cana-2799	42	7	where	where	SCONJ
cana-2799	42	8	𝜈	𝜈	X
cana-2799	42	9	>	>	X
cana-2799	42	10	0	0	PROPN
cana-2799	42	11	,	,	PUNCT
cana-2799	42	12	𝑓	𝑓	PRON
cana-2799	42	13	is	be	AUX
cana-2799	42	14	defined	define	VERB
cana-2799	42	15	for	for	ADP
cana-2799	42	16	𝑠	𝑠	PROPN
cana-2799	42	17	=	=	SYM
cana-2799	42	18	𝑎	𝑎	NOUN
cana-2799	42	19	𝑚𝑜𝑑(1	𝑚𝑜𝑑(1	NOUN
cana-2799	42	20	)	)	PUNCT
cana-2799	42	21	and	and	CCONJ
cana-2799	42	22	δ	δ	PROPN
cana-2799	42	23	−𝜈𝑓	−𝜈𝑓	PROPN
cana-2799	42	24	is	be	AUX
cana-2799	42	25	defined	define	VERB
cana-2799	42	26	for	for	ADP
cana-2799	42	27	𝜅	𝜅	NOUN
cana-2799	42	28	=	=	SYM
cana-2799	42	29	𝑎	𝑎	PROPN
cana-2799	42	30	+	+	X
cana-2799	42	31	𝜈	𝜈	X
cana-2799	42	32	𝑚𝑜𝑑(1	𝑚𝑜𝑑(1	ADJ
cana-2799	42	33	)	)	PUNCT
cana-2799	42	34	.	.	PUNCT
cana-2799	43	1	the	the	DET
cana-2799	43	2	basic	basic	ADJ
cana-2799	43	3	theory	theory	NOUN
cana-2799	43	4	of	of	ADP
cana-2799	43	5	difference	difference	NOUN
cana-2799	43	6	equations	equation	NOUN
cana-2799	43	7	is	be	AUX
cana-2799	43	8	based	base	VERB
cana-2799	43	9	on	on	ADP
cana-2799	43	10	the	the	DET
cana-2799	43	11	difference	difference	NOUN
cana-2799	43	12	operator	operator	NOUN
cana-2799	43	13	δ	δ	PROPN
cana-2799	43	14	defined	define	VERB
cana-2799	43	15	as	as	ADP
cana-2799	43	16	δ𝑢(𝜅	δ𝑢(𝜅	NUM
cana-2799	43	17	)	)	PUNCT
cana-2799	43	18	=	=	PUNCT
cana-2799	44	1	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	44	2	+	+	CCONJ
cana-2799	44	3	1	1	X
cana-2799	44	4	)	)	PUNCT
cana-2799	44	5	−	−	PROPN
cana-2799	44	6	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	44	7	)	)	PUNCT
cana-2799	44	8	,	,	PUNCT
cana-2799	44	9	where	where	SCONJ
cana-2799	44	10	{	{	PUNCT
cana-2799	44	11	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	44	12	)	)	PUNCT
cana-2799	44	13	}	}	PUNCT
cana-2799	44	14	is	be	AUX
cana-2799	44	15	a	a	DET
cana-2799	44	16	sequence	sequence	NOUN
cana-2799	44	17	or	or	CCONJ
cana-2799	44	18	a	a	DET
cana-2799	44	19	function	function	NOUN
cana-2799	44	20	of	of	ADP
cana-2799	44	21	𝜅	𝜅	NUM
cana-2799	44	22	of	of	ADP
cana-2799	44	23	numbers	number	NOUN
cana-2799	44	24	.	.	PUNCT
cana-2799	45	1	many	many	ADJ
cana-2799	45	2	authors	author	NOUN
cana-2799	45	3	(	(	PUNCT
cana-2799	45	4	[	[	X
cana-2799	45	5	7],[9	7],[9	NUM
cana-2799	45	6	]	]	PUNCT
cana-2799	45	7	)	)	PUNCT
cana-2799	45	8	have	have	AUX
cana-2799	45	9	suggested	suggest	VERB
cana-2799	45	10	the	the	DET
cana-2799	45	11	definition	definition	NOUN
cana-2799	45	12	of	of	ADP
cana-2799	45	13	generalized	generalized	ADJ
cana-2799	45	14	difference	difference	NOUN
cana-2799	45	15	operator	operator	NOUN
cana-2799	45	16	δℓ	δℓ	NOUN
cana-2799	45	17	on	on	ADP
cana-2799	45	18	real	real	ADV
cana-2799	45	19	valued	value	VERB
cana-2799	45	20	function	function	NOUN
cana-2799	45	21	u	u	NOUN
cana-2799	45	22	defined	define	VERB
cana-2799	45	23	on	on	ADP
cana-2799	45	24	ℝ	ℝ	PROPN
cana-2799	45	25	=	=	PUNCT
cana-2799	45	26	(	(	PUNCT
cana-2799	45	27	−∞,∞	−∞,∞	NOUN
cana-2799	45	28	)	)	PUNCT
cana-2799	45	29	as	as	ADP
cana-2799	45	30	δℓ𝑢(𝜅	δℓ𝑢(𝜅	PROPN
cana-2799	45	31	)	)	PUNCT
cana-2799	45	32	=	=	PUNCT
cana-2799	45	33	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	45	34	+	+	CCONJ
cana-2799	45	35	ℓ	ℓ	X
cana-2799	45	36	)	)	PUNCT
cana-2799	45	37	−	−	PROPN
cana-2799	45	38	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	45	39	)	)	PUNCT
cana-2799	45	40	,	,	PUNCT
cana-2799	45	41	𝜅	𝜅	PROPN
cana-2799	45	42	∈	∈	PROPN
cana-2799	45	43	ℝ	ℝ	PROPN
cana-2799	45	44	,	,	PUNCT
cana-2799	45	45	ℓ	ℓ	PROPN
cana-2799	45	46	>	>	X
cana-2799	45	47	0	0	PROPN
cana-2799	45	48	.	.	PUNCT
cana-2799	46	1	(	(	PUNCT
cana-2799	46	2	2	2	X
cana-2799	46	3	)	)	PUNCT
cana-2799	46	4	e.	e.	PROPN
cana-2799	46	5	thandapani	thandapani	PROPN
cana-2799	46	6	,	,	PUNCT
cana-2799	46	7	m.maria	m.maria	NOUN
cana-2799	46	8	susai	susai	PROPN
cana-2799	46	9	manuel	manuel	NOUN
cana-2799	46	10	,	,	PUNCT
cana-2799	46	11	g.b.a	g.b.a	NOUN
cana-2799	46	12	xavier	xavier	NOUN
cana-2799	47	1	[	[	X
cana-2799	47	2	8	8	NUM
cana-2799	47	3	]	]	PUNCT
cana-2799	47	4	considered	consider	VERB
cana-2799	47	5	the	the	DET
cana-2799	47	6	definition	definition	NOUN
cana-2799	47	7	of	of	ADP
cana-2799	47	8	δℓ	δℓ	NOUN
cana-2799	47	9	as	as	SCONJ
cana-2799	47	10	given	give	VERB
cana-2799	47	11	in	in	ADP
cana-2799	47	12	(	(	PUNCT
cana-2799	47	13	2	2	NUM
cana-2799	47	14	)	)	PUNCT
cana-2799	47	15	and	and	CCONJ
cana-2799	47	16	developed	develop	VERB
cana-2799	47	17	the	the	DET
cana-2799	47	18	theory	theory	NOUN
cana-2799	47	19	of	of	ADP
cana-2799	47	20	difference	difference	NOUN
cana-2799	47	21	equations	equation	NOUN
cana-2799	47	22	in	in	ADP
cana-2799	47	23	a	a	DET
cana-2799	47	24	different	different	ADJ
cana-2799	47	25	direction	direction	NOUN
cana-2799	47	26	.	.	PUNCT
cana-2799	48	1	if	if	SCONJ
cana-2799	48	2	there	there	PRON
cana-2799	48	3	exists	exist	VERB
cana-2799	48	4	a	a	DET
cana-2799	48	5	function	function	NOUN
cana-2799	48	6	v	v	ADP
cana-2799	48	7	such	such	ADJ
cana-2799	48	8	that	that	DET
cana-2799	48	9	δℓ𝑣(𝜅	δℓ𝑣(𝜅	PROPN
cana-2799	48	10	)	)	PUNCT
cana-2799	48	11	=	=	SYM
cana-2799	48	12	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	48	13	)	)	PUNCT
cana-2799	48	14	,	,	PUNCT
cana-2799	48	15	then	then	ADV
cana-2799	48	16	we	we	PRON
cana-2799	48	17	call	call	VERB
cana-2799	48	18	this	this	DET
cana-2799	48	19	function	function	NOUN
cana-2799	48	20	𝑣	𝑣	ADP
cana-2799	48	21	as	as	ADP
cana-2799	48	22	δℓ	δℓ	ADP
cana-2799	48	23	−1𝑣.	−1𝑣.	X
cana-2799	48	24	hence	hence	ADV
cana-2799	48	25	,	,	PUNCT
cana-2799	48	26	for	for	ADP
cana-2799	48	27	𝜅	𝜅	DET
cana-2799	48	28	∈	∈	NOUN
cana-2799	48	29	ℝ	ℝ	NOUN
cana-2799	48	30	=	=	SYM
cana-2799	48	31	∪	∪	NOUN
cana-2799	48	32	0≤𝑗<ℓ	0≤𝑗<ℓ	PROPN
cana-2799	48	33	ℕℓ(𝑗	ℕℓ(𝑗	NOUN
cana-2799	48	34	)	)	PUNCT
cana-2799	48	35	,	,	PUNCT
cana-2799	48	36	if	if	SCONJ
cana-2799	48	37	δℓ𝑣(𝜅	δℓ𝑣(𝜅	PROPN
cana-2799	48	38	)	)	PUNCT
cana-2799	48	39	=	=	SYM
cana-2799	48	40	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	48	41	)	)	PUNCT
cana-2799	48	42	,	,	PUNCT
cana-2799	48	43	then	then	ADV
cana-2799	48	44	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	48	45	)	)	PUNCT
cana-2799	48	46	=	=	SYM
cana-2799	48	47	δℓ	δℓ	NOUN
cana-2799	48	48	−1𝑢(𝜅	−1𝑢(𝜅	NOUN
cana-2799	48	49	)	)	PUNCT
cana-2799	49	1	+	+	CCONJ
cana-2799	49	2	𝑐𝑗	𝑐𝑗	INTJ
cana-2799	49	3	,	,	PUNCT
cana-2799	49	4	(	(	PUNCT
cana-2799	49	5	3	3	X
cana-2799	49	6	)	)	PUNCT
cana-2799	49	7	where	where	SCONJ
cana-2799	49	8	𝑐𝑗	𝑐𝑗	NOUN
cana-2799	49	9	is	be	AUX
cana-2799	49	10	constant	constant	ADJ
cana-2799	49	11	for	for	ADP
cana-2799	49	12	all	all	DET
cana-2799	49	13	𝜅	𝜅	PRON
cana-2799	49	14	in	in	ADP
cana-2799	49	15	each	each	DET
cana-2799	49	16	ℕℓ(𝑗	ℕℓ(𝑗	NOUN
cana-2799	49	17	)	)	PUNCT
cana-2799	49	18	=	=	PRON
cana-2799	49	19	{	{	PUNCT
cana-2799	49	20	𝑗	𝑗	PROPN
cana-2799	49	21	,	,	PUNCT
cana-2799	49	22	𝑗	𝑗	PROPN
cana-2799	49	23	+	+	NUM
cana-2799	49	24	ℓ	ℓ	NUM
cana-2799	49	25	,	,	PUNCT
cana-2799	49	26	𝑗	𝑗	NOUN
cana-2799	49	27	+	+	NUM
cana-2799	49	28	2ℓ	2ℓ	NOUN
cana-2799	49	29	,	,	PUNCT
cana-2799	49	30	…	…	PUNCT
cana-2799	49	31	}	}	PUNCT
cana-2799	49	32	,	,	PUNCT
cana-2799	49	33	𝑗	𝑗	PROPN
cana-2799	49	34	=	=	X
cana-2799	49	35	𝜅	𝜅	X
cana-2799	49	36	−	−	PROPN
cana-2799	49	37	[	[	PUNCT
cana-2799	49	38	𝜅	𝜅	PROPN
cana-2799	49	39	ℓ	ℓ	PROPN
cana-2799	49	40	]	]	PUNCT
cana-2799	49	41	ℓ.	ℓ.	NOUN
cana-2799	49	42	in	in	ADP
cana-2799	49	43	1989	1989	NUM
cana-2799	49	44	,	,	PUNCT
cana-2799	49	45	miller	miller	PROPN
cana-2799	49	46	and	and	CCONJ
cana-2799	49	47	rose	rise	VERB
cana-2799	49	48	introduced	introduce	VERB
cana-2799	49	49	the	the	DET
cana-2799	49	50	discrete	discrete	ADJ
cana-2799	49	51	analogue	analogue	NOUN
cana-2799	49	52	of	of	ADP
cana-2799	49	53	the	the	DET
cana-2799	49	54	riemann	riemann	PROPN
cana-2799	49	55	-	-	PUNCT
cana-2799	49	56	liouville	liouville	VERB
cana-2799	49	57	fractional	fractional	ADJ
cana-2799	49	58	derivative	derivative	NOUN
cana-2799	49	59	and	and	CCONJ
cana-2799	49	60	proved	prove	VERB
cana-2799	49	61	some	some	DET
cana-2799	49	62	properties	property	NOUN
cana-2799	49	63	of	of	ADP
cana-2799	49	64	the	the	DET
cana-2799	49	65	fractional	fractional	ADJ
cana-2799	49	66	difference	difference	NOUN
cana-2799	49	67	operator	operator	NOUN
cana-2799	49	68	.	.	PUNCT
cana-2799	50	1	in	in	ADP
cana-2799	50	2	1984	1984	NUM
cana-2799	50	3	,	,	PUNCT
cana-2799	50	4	jerzy	jerzy	X
cana-2799	50	5	popenda	popenda	NOUN
cana-2799	50	6	[	[	X
cana-2799	50	7	4	4	X
cana-2799	50	8	]	]	PUNCT
cana-2799	50	9	introduced	introduce	VERB
cana-2799	50	10	a	a	DET
cana-2799	50	11	particular	particular	ADJ
cana-2799	50	12	type	type	NOUN
cana-2799	50	13	of	of	ADP
cana-2799	50	14	difference	difference	NOUN
cana-2799	50	15	operator	operator	NOUN
cana-2799	50	16	on	on	ADP
cana-2799	50	17	𝑢	𝑢	NOUN
cana-2799	50	18	as	as	ADP
cana-2799	50	19	δ𝛼𝑢(𝜅	δ𝛼𝑢(𝜅	NOUN
cana-2799	50	20	)	)	PUNCT
cana-2799	50	21	=	=	PUNCT
cana-2799	50	22	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	50	23	+	+	CCONJ
cana-2799	50	24	1	1	X
cana-2799	50	25	)	)	PUNCT
cana-2799	50	26	−	−	NOUN
cana-2799	50	27	𝛼𝑢(𝜅	𝛼𝑢(𝜅	NUM
cana-2799	50	28	)	)	PUNCT
cana-2799	50	29	,	,	PUNCT
cana-2799	50	30	in	in	ADP
cana-2799	50	31	2011	2011	NUM
cana-2799	50	32	,	,	PUNCT
cana-2799	50	33	m.maria	m.maria	NOUN
cana-2799	50	34	susai	susai	NOUN
cana-2799	50	35	manuel	manuel	NOUN
cana-2799	50	36	,	,	PUNCT
cana-2799	50	37	et.al	et.al	PROPN
cana-2799	50	38	,	,	PUNCT
cana-2799	50	39	[	[	X
cana-2799	50	40	5	5	NUM
cana-2799	50	41	]	]	PUNCT
cana-2799	50	42	extended	extend	VERB
cana-2799	50	43	the	the	DET
cana-2799	50	44	operator	operator	NOUN
cana-2799	50	45	δ𝛼	δ𝛼	ADP
cana-2799	50	46	to	to	PART
cana-2799	50	47	generalized	generalize	VERB
cana-2799	50	48	𝛼	𝛼	DET
cana-2799	50	49	−	−	NOUN
cana-2799	50	50	difference	difference	NOUN
cana-2799	50	51	operator	operator	NOUN
cana-2799	50	52	as	as	SCONJ
cana-2799	50	53	communications	communication	NOUN
cana-2799	50	54	on	on	ADP
cana-2799	50	55	applied	apply	VERB
cana-2799	50	56	nonlinear	nonlinear	ADJ
cana-2799	50	57	analysis	analysis	NOUN
cana-2799	50	58	issn	issn	NOUN
cana-2799	50	59	:	:	PUNCT
cana-2799	50	60	1074	1074	NUM
cana-2799	50	61	-	-	PUNCT
cana-2799	50	62	133x	133x	NUM
cana-2799	50	63	vol	vol	NOUN
cana-2799	50	64	32	32	NUM
cana-2799	50	65	no	no	NOUN
cana-2799	50	66	.	.	PUNCT
cana-2799	51	1	4s	4s	NUM
cana-2799	51	2	(	(	PUNCT
cana-2799	51	3	2025	2025	NUM
cana-2799	51	4	)	)	PUNCT
cana-2799	51	5	260	260	NUM
cana-2799	51	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	51	7	δ	δ	PROPN
cana-2799	51	8	𝛼(ℓ	𝛼(ℓ	ADJ
cana-2799	51	9	)	)	PUNCT
cana-2799	51	10	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	51	11	)	)	PUNCT
cana-2799	52	1	=	=	SYM
cana-2799	52	2	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	52	3	+	+	CCONJ
cana-2799	52	4	ℓ	ℓ	NOUN
cana-2799	52	5	)	)	PUNCT
cana-2799	52	6	−	−	NOUN
cana-2799	52	7	𝛼𝑣(𝜅	𝛼𝑣(𝜅	NUM
cana-2799	52	8	)	)	PUNCT
cana-2799	52	9	for	for	ADP
cana-2799	52	10	real	real	ADV
cana-2799	52	11	valued	value	VERB
cana-2799	52	12	function	function	NOUN
cana-2799	52	13	𝑣.	𝑣.	NOUN
cana-2799	52	14	in	in	ADP
cana-2799	52	15	2014	2014	NUM
cana-2799	52	16	,	,	PUNCT
cana-2799	52	17	the	the	DET
cana-2799	52	18	authors	author	NOUN
cana-2799	52	19	in	in	ADP
cana-2799	52	20	[	[	X
cana-2799	52	21	1	1	NUM
cana-2799	52	22	]	]	PUNCT
cana-2799	52	23	have	have	AUX
cana-2799	52	24	applied	apply	VERB
cana-2799	52	25	the	the	DET
cana-2799	52	26	q	q	ADJ
cana-2799	52	27	-	-	PUNCT
cana-2799	52	28	difference	difference	NOUN
cana-2799	52	29	operator	operator	NOUN
cana-2799	52	30	defined	define	VERB
cana-2799	52	31	by	by	ADP
cana-2799	52	32	δ𝑞𝑣(𝜅	δ𝑞𝑣(𝜅	PROPN
cana-2799	52	33	)	)	PUNCT
cana-2799	52	34	=	=	PUNCT
cana-2799	53	1	𝑣(𝑞𝜅	𝑣(𝑞𝜅	NOUN
cana-2799	53	2	)	)	PUNCT
cana-2799	53	3	−	−	PROPN
cana-2799	53	4	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	53	5	)	)	PUNCT
cana-2799	53	6	and	and	CCONJ
cana-2799	53	7	delta	delta	NOUN
cana-2799	53	8	operator	operator	NOUN
cana-2799	53	9	δ	δ	PROPN
cana-2799	53	10	𝜅(ℓ	𝜅(ℓ	NOUN
cana-2799	53	11	)	)	PUNCT
cana-2799	53	12	with	with	ADP
cana-2799	53	13	variable	variable	ADJ
cana-2799	53	14	coefficients	coefficient	NOUN
cana-2799	53	15	defined	define	VERB
cana-2799	53	16	by	by	ADP
cana-2799	53	17	δ	δ	PROPN
cana-2799	53	18	𝜅(ℓ	𝜅(ℓ	ADJ
cana-2799	53	19	)	)	PUNCT
cana-2799	53	20	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	53	21	)	)	PUNCT
cana-2799	53	22	=	=	SYM
cana-2799	54	1	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	54	2	+	+	CCONJ
cana-2799	54	3	ℓ	ℓ	X
cana-2799	54	4	)	)	PUNCT
cana-2799	54	5	−	−	NOUN
cana-2799	54	6	𝜅𝑣(𝜅	𝜅𝑣(𝜅	NUM
cana-2799	54	7	)	)	PUNCT
cana-2799	54	8	,	,	PUNCT
cana-2799	54	9	ℓ	ℓ	PROPN
cana-2799	54	10	≠	≠	PROPN
cana-2799	54	11	0	0	NUM
cana-2799	54	12	∈	∈	PROPN
cana-2799	54	13	ℝ.	ℝ.	PROPN
cana-2799	54	14	also	also	ADV
cana-2799	54	15	the	the	DET
cana-2799	54	16	generalized	generalized	ADJ
cana-2799	54	17	difference	difference	NOUN
cana-2799	54	18	operator	operator	NOUN
cana-2799	54	19	with	with	ADP
cana-2799	54	20	𝑛-shift	𝑛-shift	PROPN
cana-2799	54	21	values	value	NOUN
cana-2799	54	22	𝑙	𝑙	X
cana-2799	55	1	=	=	SYM
cana-2799	56	1	(	(	PUNCT
cana-2799	56	2	ℓ1	ℓ1	NOUN
cana-2799	56	3	,	,	PUNCT
cana-2799	56	4	ℓ2	ℓ2	NOUN
cana-2799	56	5	,	,	PUNCT
cana-2799	56	6	ℓ3	ℓ3	PROPN
cana-2799	56	7	,	,	PUNCT
cana-2799	56	8	.	.	PUNCT
cana-2799	56	9	.	.	PUNCT
cana-2799	57	1	.	.	PUNCT
cana-2799	58	1	,	,	PUNCT
cana-2799	58	2	ℓ𝑛	ℓ𝑛	NOUN
cana-2799	58	3	)	)	PUNCT
cana-2799	58	4	≠	≠	PROPN
cana-2799	58	5	0	0	NUM
cana-2799	58	6	on	on	ADP
cana-2799	58	7	a	a	DET
cana-2799	58	8	real	real	ADV
cana-2799	58	9	valued	value	VERB
cana-2799	58	10	function	function	NOUN
cana-2799	58	11	𝑣	𝑣	ADP
cana-2799	58	12	:	:	PUNCT
cana-2799	58	13	ℝ𝑛	ℝ𝑛	X
cana-2799	58	14	→	→	SYM
cana-2799	58	15	ℝ	ℝ	PROPN
cana-2799	58	16	is	be	AUX
cana-2799	58	17	defined	define	VERB
cana-2799	58	18	as	as	ADP
cana-2799	58	19	δ	δ	PROPN
cana-2799	58	20	(	(	PUNCT
cana-2799	58	21	ℓ	ℓ	NOUN
cana-2799	58	22	)	)	PUNCT
cana-2799	58	23	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	58	24	)	)	PUNCT
cana-2799	59	1	=	=	PUNCT
cana-2799	59	2	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	59	3	+	+	CCONJ
cana-2799	59	4	ℓ1	ℓ1	NOUN
cana-2799	59	5	,	,	PUNCT
cana-2799	59	6	𝜅2	𝜅2	NOUN
cana-2799	59	7	+	+	CCONJ
cana-2799	59	8	ℓ2	ℓ2	NOUN
cana-2799	59	9	,	,	PUNCT
cana-2799	59	10	.	.	PUNCT
cana-2799	59	11	.	.	PUNCT
cana-2799	60	1	.	.	PUNCT
cana-2799	61	1	,	,	PUNCT
cana-2799	61	2	𝜅𝑛	𝜅𝑛	ADP
cana-2799	61	3	+	+	ADJ
cana-2799	61	4	ℓ𝑛	ℓ𝑛	NOUN
cana-2799	61	5	)	)	PUNCT
cana-2799	61	6	−	−	NOUN
cana-2799	61	7	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	61	8	,	,	PUNCT
cana-2799	61	9	𝜅2	𝜅2	NOUN
cana-2799	61	10	,	,	PUNCT
cana-2799	61	11	.	.	PUNCT
cana-2799	61	12	.	.	PUNCT
cana-2799	62	1	.	.	PUNCT
cana-2799	63	1	,	,	PUNCT
cana-2799	63	2	𝜅𝑛	𝜅𝑛	PROPN
cana-2799	63	3	)	)	PUNCT
cana-2799	63	4	.	.	PUNCT
cana-2799	64	1	(	(	PUNCT
cana-2799	64	2	4	4	X
cana-2799	64	3	)	)	PUNCT
cana-2799	64	4	this	this	DET
cana-2799	64	5	operator	operator	NOUN
cana-2799	64	6	δ	δ	PROPN
cana-2799	64	7	(	(	PUNCT
cana-2799	64	8	ℓ	ℓ	NOUN
cana-2799	64	9	)	)	PUNCT
cana-2799	64	10	becomes	become	VERB
cana-2799	64	11	generalized	generalized	ADJ
cana-2799	64	12	partial	partial	ADJ
cana-2799	64	13	difference	difference	NOUN
cana-2799	64	14	operator	operator	NOUN
cana-2799	64	15	if	if	SCONJ
cana-2799	64	16	some	some	PRON
cana-2799	64	17	ℓ𝑖	ℓ𝑖	VERB
cana-2799	64	18	=	=	NOUN
cana-2799	64	19	0	0	NUM
cana-2799	64	20	.	.	PUNCT
cana-2799	65	1	the	the	DET
cana-2799	65	2	equations	equation	NOUN
cana-2799	65	3	involving	involve	VERB
cana-2799	65	4	δ	δ	PROPN
cana-2799	65	5	(	(	PUNCT
cana-2799	65	6	ℓ	ℓ	NOUN
cana-2799	65	7	)	)	PUNCT
cana-2799	65	8	with	with	ADP
cana-2799	65	9	atleast	atleast	ADJ
cana-2799	65	10	one	one	NUM
cana-2799	65	11	ℓ𝑖	ℓ𝑖	VERB
cana-2799	65	12	=	=	SYM
cana-2799	65	13	0	0	NUM
cana-2799	65	14	is	be	AUX
cana-2799	65	15	called	call	VERB
cana-2799	65	16	generalized	generalized	ADJ
cana-2799	65	17	partial	partial	ADJ
cana-2799	65	18	difference	difference	NOUN
cana-2799	65	19	equation	equation	NOUN
cana-2799	65	20	.	.	PUNCT
cana-2799	66	1	for	for	ADP
cana-2799	66	2	one	one	NUM
cana-2799	66	3	shift	shift	NOUN
cana-2799	66	4	value	value	NOUN
cana-2799	66	5	,	,	PUNCT
cana-2799	66	6	we	we	PRON
cana-2799	66	7	take	take	VERB
cana-2799	66	8	δ(ℓ	δ(ℓ	NOUN
cana-2799	66	9	)	)	PUNCT
cana-2799	66	10	as	as	ADP
cana-2799	66	11	δℓ.	δℓ.	NUM
cana-2799	66	12	by	by	ADP
cana-2799	66	13	defining	define	VERB
cana-2799	66	14	the	the	DET
cana-2799	66	15	inverse	inverse	NOUN
cana-2799	66	16	δℓ	δℓ	NOUN
cana-2799	66	17	−1	−1	NOUN
cana-2799	66	18	,	,	PUNCT
cana-2799	66	19	many	many	ADJ
cana-2799	66	20	interesting	interesting	ADJ
cana-2799	66	21	results	result	NOUN
cana-2799	66	22	on	on	ADP
cana-2799	66	23	sum	sum	NOUN
cana-2799	66	24	of	of	ADP
cana-2799	66	25	partial	partial	ADJ
cana-2799	66	26	sums	sum	NOUN
cana-2799	66	27	of	of	ADP
cana-2799	66	28	higher	high	ADJ
cana-2799	66	29	power	power	NOUN
cana-2799	66	30	of	of	ADP
cana-2799	66	31	arithmetic	arithmetic	ADJ
cana-2799	66	32	and	and	CCONJ
cana-2799	66	33	geometric	geometric	ADJ
cana-2799	66	34	functions	function	NOUN
cana-2799	66	35	and	and	CCONJ
cana-2799	66	36	applications	application	NOUN
cana-2799	66	37	in	in	ADP
cana-2799	66	38	numerical	numerical	ADJ
cana-2799	66	39	methods	method	NOUN
cana-2799	66	40	(	(	PUNCT
cana-2799	66	41	see	see	VERB
cana-2799	66	42	[	[	X
cana-2799	66	43	8	8	NUM
cana-2799	66	44	,	,	PUNCT
cana-2799	66	45	6	6	NUM
cana-2799	66	46	,	,	PUNCT
cana-2799	66	47	2	2	NUM
cana-2799	66	48	,	,	PUNCT
cana-2799	66	49	3	3	NUM
cana-2799	66	50	,	,	PUNCT
cana-2799	66	51	10	10	NUM
cana-2799	66	52	,	,	PUNCT
cana-2799	66	53	11	11	NUM
cana-2799	66	54	,	,	PUNCT
cana-2799	66	55	12	12	NUM
cana-2799	66	56	,	,	PUNCT
cana-2799	66	57	13	13	NUM
cana-2799	66	58	]	]	PUNCT
cana-2799	66	59	)	)	PUNCT
cana-2799	66	60	are	be	AUX
cana-2799	66	61	obtained	obtain	VERB
cana-2799	66	62	.	.	PUNCT
cana-2799	67	1	the	the	DET
cana-2799	67	2	difference	difference	NOUN
cana-2799	67	3	operator	operator	NOUN
cana-2799	67	4	defined	define	VERB
cana-2799	67	5	in	in	ADP
cana-2799	67	6	(	(	PUNCT
cana-2799	67	7	2	2	NUM
cana-2799	67	8	)	)	PUNCT
cana-2799	67	9	becomes	become	VERB
cana-2799	67	10	the	the	DET
cana-2799	67	11	usual	usual	ADJ
cana-2799	67	12	difference	difference	NOUN
cana-2799	67	13	operator	operator	NOUN
cana-2799	67	14	δ	δ	PROPN
cana-2799	67	15	when	when	SCONJ
cana-2799	67	16	ℓ	ℓ	PROPN
cana-2799	67	17	=	=	SYM
cana-2799	67	18	1	1	X
cana-2799	67	19	.	.	PUNCT
cana-2799	68	1	we	we	PRON
cana-2799	68	2	obtain	obtain	VERB
cana-2799	68	3	several	several	ADJ
cana-2799	68	4	results	result	NOUN
cana-2799	68	5	on	on	ADP
cana-2799	68	6	factorial	factorial	ADJ
cana-2799	68	7	function	function	NOUN
cana-2799	68	8	by	by	ADP
cana-2799	68	9	applying	apply	VERB
cana-2799	68	10	δℓ	δℓ	ADP
cana-2799	68	11	−1	−1	NOUN
cana-2799	68	12	.	.	PUNCT
cana-2799	69	1	2	2	X
cana-2799	69	2	.	.	X
cana-2799	69	3	the	the	DET
cana-2799	69	4	𝓵	𝓵	ADJ
cana-2799	69	5	extorial	extorial	ADJ
cana-2799	69	6	function	function	NOUN
cana-2799	69	7	and	and	CCONJ
cana-2799	69	8	its	its	PRON
cana-2799	69	9	properties	property	NOUN
cana-2799	69	10	the	the	DET
cana-2799	69	11	ℓ-extorial	ℓ-extorial	ADJ
cana-2799	69	12	function	function	NOUN
cana-2799	69	13	is	be	AUX
cana-2799	69	14	arrived	arrive	VERB
cana-2799	69	15	by	by	ADP
cana-2799	69	16	replacing	replace	VERB
cana-2799	69	17	the	the	DET
cana-2799	69	18	polynomial	polynomial	NOUN
cana-2799	69	19	𝜅𝑛	𝜅𝑛	ADP
cana-2799	69	20	by	by	ADP
cana-2799	69	21	polynomial	polynomial	ADJ
cana-2799	69	22	factorial	factorial	NOUN
cana-2799	69	23	function	function	NOUN
cana-2799	69	24	𝜅ℓ	𝜅ℓ	X
cana-2799	69	25	(	(	PUNCT
cana-2799	69	26	𝑛	𝑛	PROPN
cana-2799	69	27	)	)	PUNCT
cana-2799	69	28	in	in	ADP
cana-2799	69	29	the	the	DET
cana-2799	69	30	exponential	exponential	ADJ
cana-2799	69	31	function	function	NOUN
cana-2799	69	32	𝑒𝜅.	𝑒𝜅.	VERB
cana-2799	69	33	the	the	DET
cana-2799	69	34	formal	formal	ADJ
cana-2799	69	35	definition	definition	NOUN
cana-2799	69	36	of	of	ADP
cana-2799	69	37	extorial	extorial	ADJ
cana-2799	69	38	function	function	NOUN
cana-2799	69	39	is	be	AUX
cana-2799	69	40	given	give	VERB
cana-2799	69	41	below	below	ADV
cana-2799	69	42	.	.	PUNCT
cana-2799	70	1	definition	definition	NOUN
cana-2799	70	2	2.1	2.1	NUM
cana-2799	70	3	.	.	PUNCT
cana-2799	71	1	the	the	DET
cana-2799	71	2	ℓ-extorial	ℓ-extorial	ADJ
cana-2799	71	3	function	function	NOUN
cana-2799	71	4	denoted	denote	VERB
cana-2799	71	5	as	as	ADP
cana-2799	71	6	𝑒(𝜅ℓ	𝑒(𝜅ℓ	NOUN
cana-2799	71	7	(	(	PUNCT
cana-2799	71	8	𝑛	𝑛	NOUN
cana-2799	71	9	)	)	PUNCT
cana-2799	71	10	)	)	PUNCT
cana-2799	71	11	is	be	AUX
cana-2799	71	12	defined	define	VERB
cana-2799	71	13	as	as	ADP
cana-2799	71	14	𝑒(𝜅ℓ	𝑒(𝜅ℓ	NOUN
cana-2799	71	15	(	(	PUNCT
cana-2799	71	16	𝑛	𝑛	NOUN
cana-2799	71	17	)	)	PUNCT
cana-2799	71	18	)	)	PUNCT
cana-2799	72	1	=	=	SYM
cana-2799	72	2	1	1	NUM
cana-2799	72	3	+	+	NUM
cana-2799	72	4	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	72	5	(	(	PUNCT
cana-2799	72	6	𝑛	𝑛	NOUN
cana-2799	72	7	)	)	PUNCT
cana-2799	72	8	1	1	NUM
cana-2799	72	9	!	!	PUNCT
cana-2799	73	1	+	+	CCONJ
cana-2799	73	2	𝜅ℓ	𝜅ℓ	X
cana-2799	73	3	(	(	PUNCT
cana-2799	73	4	2𝑛	2𝑛	NUM
cana-2799	73	5	)	)	PUNCT
cana-2799	73	6	2	2	NUM
cana-2799	73	7	!	!	PUNCT
cana-2799	74	1	+	+	CCONJ
cana-2799	74	2	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	74	3	(	(	PUNCT
cana-2799	74	4	3𝑛	3𝑛	NUM
cana-2799	74	5	)	)	PUNCT
cana-2799	74	6	3	3	NUM
cana-2799	74	7	!	!	PUNCT
cana-2799	75	1	+	+	CCONJ
cana-2799	75	2	⋯	⋯	X
cana-2799	75	3	+	+	CCONJ
cana-2799	75	4	∞	∞	PROPN
cana-2799	75	5	,	,	PUNCT
cana-2799	75	6	(	(	PUNCT
cana-2799	75	7	5	5	NUM
cana-2799	75	8	)	)	PUNCT
cana-2799	75	9	where	where	SCONJ
cana-2799	75	10	|ℓ|	|ℓ|	VERB
cana-2799	75	11	≤	≤	NUM
cana-2799	75	12	1	1	NUM
cana-2799	75	13	and	and	CCONJ
cana-2799	75	14	𝑛	𝑛	NOUN
cana-2799	75	15	,	,	PUNCT
cana-2799	75	16	𝜅	𝜅	PRON
cana-2799	75	17	∈	∈	PROPN
cana-2799	75	18	ℝ.	ℝ.	PROPN
cana-2799	75	19	definition	definition	NOUN
cana-2799	75	20	2.2	2.2	NUM
cana-2799	75	21	.	.	PUNCT
cana-2799	76	1	for	for	ADP
cana-2799	76	2	ℓ	ℓ	PROPN
cana-2799	76	3	∈	∈	PROPN
cana-2799	76	4	(	(	PUNCT
cana-2799	76	5	−1,1	−1,1	NOUN
cana-2799	76	6	)	)	PUNCT
cana-2799	76	7	and	and	CCONJ
cana-2799	76	8	𝜅	𝜅	PRON
cana-2799	76	9	∈	∈	PROPN
cana-2799	76	10	ℝ	ℝ	PROPN
cana-2799	76	11	,	,	PUNCT
cana-2799	76	12	the	the	DET
cana-2799	76	13	𝑛𝑡ℎ	𝑛𝑡ℎ	NOUN
cana-2799	76	14	order	order	VERB
cana-2799	76	15	ℓ-extorial	ℓ-extorial	ADJ
cana-2799	76	16	function	function	NOUN
cana-2799	76	17	denoted	denote	VERB
cana-2799	76	18	as	as	ADP
cana-2799	76	19	𝑒𝑛(𝜅ℓ	𝑒𝑛(𝜅ℓ	PROPN
cana-2799	76	20	)	)	PUNCT
cana-2799	76	21	is	be	AUX
cana-2799	76	22	defined	define	VERB
cana-2799	76	23	as	as	ADP
cana-2799	76	24	𝑒𝑛(𝜅ℓ	𝑒𝑛(𝜅ℓ	PROPN
cana-2799	76	25	)	)	PUNCT
cana-2799	76	26	=	=	SYM
cana-2799	77	1	1	1	NUM
cana-2799	77	2	+	+	NUM
cana-2799	77	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	77	4	(	(	PUNCT
cana-2799	77	5	𝑛	𝑛	NOUN
cana-2799	77	6	)	)	PUNCT
cana-2799	77	7	𝑛	𝑛	NOUN
cana-2799	77	8	!	!	PUNCT
cana-2799	78	1	+	+	CCONJ
cana-2799	78	2	𝜅ℓ	𝜅ℓ	X
cana-2799	78	3	(	(	PUNCT
cana-2799	78	4	2𝑛	2𝑛	NUM
cana-2799	78	5	)	)	PUNCT
cana-2799	78	6	(	(	PUNCT
cana-2799	78	7	2𝑛	2𝑛	NUM
cana-2799	78	8	)	)	PUNCT
cana-2799	78	9	!	!	PUNCT
cana-2799	79	1	+	+	CCONJ
cana-2799	79	2	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	79	3	(	(	PUNCT
cana-2799	79	4	3𝑛	3𝑛	NUM
cana-2799	79	5	)	)	PUNCT
cana-2799	79	6	(	(	PUNCT
cana-2799	79	7	3𝑛	3𝑛	NUM
cana-2799	79	8	)	)	PUNCT
cana-2799	79	9	!	!	PUNCT
cana-2799	80	1	+	+	CCONJ
cana-2799	80	2	⋯	⋯	X
cana-2799	80	3	+	+	CCONJ
cana-2799	80	4	∞.	∞.	PROPN
cana-2799	80	5	(	(	PUNCT
cana-2799	80	6	6	6	NUM
cana-2799	80	7	)	)	PUNCT
cana-2799	80	8	from	from	ADP
cana-2799	80	9	the	the	DET
cana-2799	80	10	definition	definition	NOUN
cana-2799	80	11	of	of	ADP
cana-2799	80	12	extorial	extorial	ADJ
cana-2799	80	13	function	function	NOUN
cana-2799	80	14	,	,	PUNCT
cana-2799	80	15	we	we	PRON
cana-2799	80	16	obtain	obtain	VERB
cana-2799	80	17	following	follow	VERB
cana-2799	80	18	lemma	lemma	PROPN
cana-2799	80	19	.	.	PUNCT
cana-2799	81	1	lemma	lemma	PROPN
cana-2799	81	2	2.3	2.3	NUM
cana-2799	81	3	.	.	PUNCT
cana-2799	82	1	for	for	ADP
cana-2799	82	2	any	any	DET
cana-2799	82	3	real	real	ADJ
cana-2799	82	4	𝜅	𝜅	NOUN
cana-2799	82	5	and	and	CCONJ
cana-2799	82	6	ℓ	ℓ	NOUN
cana-2799	82	7	,	,	PUNCT
cana-2799	82	8	𝑛	𝑛	PROPN
cana-2799	82	9	∈	∈	PROPN
cana-2799	82	10	ℕ	ℕ	PROPN
cana-2799	82	11	,	,	PUNCT
cana-2799	82	12	we	we	PRON
cana-2799	82	13	have	have	VERB
cana-2799	82	14	(	(	PUNCT
cana-2799	82	15	i	i	NOUN
cana-2799	82	16	)	)	PUNCT
cana-2799	82	17	𝑒𝑛(−𝜅ℓ	𝑒𝑛(−𝜅ℓ	NUM
cana-2799	82	18	)	)	PUNCT
cana-2799	82	19	=	=	SYM
cana-2799	83	1	𝑒𝑛(𝜅−ℓ	𝑒𝑛(𝜅−ℓ	X
cana-2799	83	2	)	)	PUNCT
cana-2799	83	3	if	if	SCONJ
cana-2799	83	4	n	n	NUM
cana-2799	83	5	𝑖𝑠	𝑖𝑠	ADP
cana-2799	83	6	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-2799	83	7	&	&	CCONJ
cana-2799	83	8	1	1	NUM
cana-2799	83	9	−	−	PROPN
cana-2799	83	10	𝜅(−ℓ	𝜅(−ℓ	NOUN
cana-2799	83	11	)	)	PUNCT
cana-2799	83	12	(	(	PUNCT
cana-2799	83	13	𝑛	𝑛	NOUN
cana-2799	83	14	)	)	PUNCT
cana-2799	83	15	𝑛	𝑛	NOUN
cana-2799	83	16	!	!	PUNCT
cana-2799	84	1	+	+	NUM
cana-2799	84	2	𝜅(−ℓ	𝜅(−ℓ	NOUN
cana-2799	84	3	)	)	PUNCT
cana-2799	84	4	(	(	PUNCT
cana-2799	84	5	2𝑛	2𝑛	NUM
cana-2799	84	6	)	)	PUNCT
cana-2799	84	7	(	(	PUNCT
cana-2799	84	8	2𝑛	2𝑛	NUM
cana-2799	84	9	)	)	PUNCT
cana-2799	84	10	!	!	PUNCT
cana-2799	85	1	−	−	NOUN
cana-2799	85	2	𝜅(−ℓ	𝜅(−ℓ	SYM
cana-2799	85	3	)	)	PUNCT
cana-2799	85	4	(	(	PUNCT
cana-2799	85	5	3𝑛	3𝑛	NUM
cana-2799	85	6	)	)	PUNCT
cana-2799	85	7	(	(	PUNCT
cana-2799	85	8	3𝑛	3𝑛	NUM
cana-2799	85	9	)	)	PUNCT
cana-2799	85	10	!	!	PUNCT
cana-2799	86	1	+	+	CCONJ
cana-2799	86	2	⋯	⋯	VERB
cana-2799	86	3	if	if	SCONJ
cana-2799	86	4	n	n	PROPN
cana-2799	86	5	𝑖𝑠	𝑖𝑠	ADV
cana-2799	86	6	𝑜𝑑𝑑	𝑜𝑑𝑑	NOUN
cana-2799	86	7	and	and	CCONJ
cana-2799	86	8	(	(	PUNCT
cana-2799	86	9	ii	ii	NOUN
cana-2799	86	10	)	)	PUNCT
cana-2799	86	11	𝑒𝑛(−𝜅(−ℓ	𝑒𝑛(−𝜅(−ℓ	NOUN
cana-2799	86	12	)	)	PUNCT
cana-2799	86	13	)	)	PUNCT
cana-2799	87	1	=	=	SYM
cana-2799	87	2	𝑒𝑛(𝜅ℓ	𝑒𝑛(𝜅ℓ	ADJ
cana-2799	87	3	)	)	PUNCT
cana-2799	87	4	if	if	SCONJ
cana-2799	87	5	n	n	NUM
cana-2799	87	6	𝑖𝑠	𝑖𝑠	ADP
cana-2799	87	7	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-2799	87	8	&	&	CCONJ
cana-2799	87	9	1	1	NUM
cana-2799	87	10	−	−	PROPN
cana-2799	87	11	(	(	PUNCT
cana-2799	87	12	𝜅)ℓ	𝜅)ℓ	ADJ
cana-2799	87	13	(	(	PUNCT
cana-2799	87	14	𝑛	𝑛	NOUN
cana-2799	87	15	)	)	PUNCT
cana-2799	87	16	𝑛	𝑛	NOUN
cana-2799	87	17	!	!	PUNCT
cana-2799	88	1	+	+	CCONJ
cana-2799	88	2	(	(	PUNCT
cana-2799	88	3	𝜅)ℓ	𝜅)ℓ	VERB
cana-2799	88	4	(	(	PUNCT
cana-2799	88	5	2𝑛	2𝑛	NUM
cana-2799	88	6	)	)	PUNCT
cana-2799	88	7	2𝑛	2𝑛	NUM
cana-2799	88	8	!	!	PUNCT
cana-2799	89	1	−	−	PROPN
cana-2799	89	2	(	(	PUNCT
cana-2799	89	3	𝜅)ℓ	𝜅)ℓ	VERB
cana-2799	89	4	(	(	PUNCT
cana-2799	89	5	3𝑛	3𝑛	NUM
cana-2799	89	6	)	)	PUNCT
cana-2799	89	7	3𝑛	3𝑛	NUM
cana-2799	89	8	!	!	PUNCT
cana-2799	90	1	+	+	CCONJ
cana-2799	90	2	⋯	⋯	VERB
cana-2799	90	3	if	if	SCONJ
cana-2799	90	4	n	n	PROPN
cana-2799	90	5	𝑖𝑠	𝑖𝑠	ADV
cana-2799	90	6	𝑜𝑑𝑑	𝑜𝑑𝑑	PROPN
cana-2799	90	7	lemma	lemma	PROPN
cana-2799	90	8	2.4	2.4	NUM
cana-2799	90	9	.	.	PUNCT
cana-2799	91	1	let	let	VERB
cana-2799	91	2	𝜅	𝜅	PRON
cana-2799	91	3	∈	∈	PROPN
cana-2799	91	4	ℝ	ℝ	PROPN
cana-2799	91	5	and	and	CCONJ
cana-2799	91	6	𝑛	𝑛	NOUN
cana-2799	91	7	,	,	PUNCT
cana-2799	91	8	ℓ	ℓ	PROPN
cana-2799	91	9	∈	∈	PROPN
cana-2799	91	10	ℕ.	ℕ.	PROPN
cana-2799	91	11	then	then	ADV
cana-2799	91	12	,	,	PUNCT
cana-2799	91	13	we	we	PRON
cana-2799	91	14	have	have	VERB
cana-2799	91	15	δℓ𝑒𝑛(𝜅ℓ	δℓ𝑒𝑛(𝜅ℓ	ADJ
cana-2799	91	16	)	)	PUNCT
cana-2799	91	17	=	=	PUNCT
cana-2799	92	1	ℓ∑∞	ℓ∑∞	NOUN
cana-2799	92	2	𝑚=1	𝑚=1	X
cana-2799	92	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	92	4	(	(	PUNCT
cana-2799	92	5	𝑚𝑛−1	𝑚𝑛−1	PROPN
cana-2799	92	6	)	)	PUNCT
cana-2799	92	7	(	(	PUNCT
cana-2799	92	8	𝑚𝑛−1	𝑚𝑛−1	NOUN
cana-2799	92	9	)	)	PUNCT
cana-2799	92	10	!	!	PUNCT
cana-2799	93	1	,	,	PUNCT
cana-2799	93	2	𝑛𝑚	𝑛𝑚	ADP
cana-2799	93	3	≠	≠	PROPN
cana-2799	93	4	1	1	NUM
cana-2799	93	5	.	.	PUNCT
cana-2799	93	6	communications	communication	NOUN
cana-2799	93	7	on	on	ADP
cana-2799	93	8	applied	apply	VERB
cana-2799	93	9	nonlinear	nonlinear	ADJ
cana-2799	93	10	analysis	analysis	NOUN
cana-2799	93	11	issn	issn	NOUN
cana-2799	93	12	:	:	PUNCT
cana-2799	93	13	1074	1074	NUM
cana-2799	93	14	-	-	PUNCT
cana-2799	93	15	133x	133x	NUM
cana-2799	93	16	vol	vol	NOUN
cana-2799	93	17	32	32	NUM
cana-2799	93	18	no	no	NOUN
cana-2799	93	19	.	.	PUNCT
cana-2799	94	1	4s	4s	NUM
cana-2799	94	2	(	(	PUNCT
cana-2799	94	3	2025	2025	NUM
cana-2799	94	4	)	)	PUNCT
cana-2799	94	5	261	261	NUM
cana-2799	94	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	94	7	proof	proof	NOUN
cana-2799	94	8	.	.	PUNCT
cana-2799	95	1	we	we	PRON
cana-2799	95	2	shall	shall	AUX
cana-2799	95	3	prove	prove	VERB
cana-2799	95	4	this	this	PRON
cana-2799	95	5	by	by	ADP
cana-2799	95	6	induction	induction	NOUN
cana-2799	95	7	method	method	NOUN
cana-2799	95	8	𝑒2(𝜅ℓ	𝑒2(𝜅ℓ	PROPN
cana-2799	95	9	)	)	PUNCT
cana-2799	95	10	=	=	SYM
cana-2799	95	11	1	1	NUM
cana-2799	95	12	+	+	NUM
cana-2799	95	13	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	95	14	(	(	PUNCT
cana-2799	95	15	2	2	NUM
cana-2799	95	16	)	)	PUNCT
cana-2799	95	17	2	2	NUM
cana-2799	95	18	!	!	PUNCT
cana-2799	96	1	+	+	CCONJ
cana-2799	96	2	𝜅ℓ	𝜅ℓ	X
cana-2799	96	3	(	(	PUNCT
cana-2799	96	4	4	4	NUM
cana-2799	96	5	)	)	PUNCT
cana-2799	96	6	4	4	NUM
cana-2799	96	7	!	!	PUNCT
cana-2799	97	1	+	+	CCONJ
cana-2799	97	2	𝜅ℓ	𝜅ℓ	X
cana-2799	97	3	(	(	PUNCT
cana-2799	97	4	6	6	NUM
cana-2799	97	5	)	)	PUNCT
cana-2799	97	6	6	6	NUM
cana-2799	97	7	!	!	PUNCT
cana-2799	98	1	+	+	CCONJ
cana-2799	98	2	⋯	⋯	VERB
cana-2799	98	3	+	+	CCONJ
cana-2799	98	4	∞	∞	NUM
cana-2799	98	5	δℓ𝑒2(𝜅ℓ	δℓ𝑒2(𝜅ℓ	PROPN
cana-2799	98	6	)	)	PUNCT
cana-2799	99	1	=	=	SYM
cana-2799	99	2	δℓ	δℓ	NOUN
cana-2799	99	3	𝜅ℓ	𝜅ℓ	X
cana-2799	99	4	(	(	PUNCT
cana-2799	99	5	2	2	NUM
cana-2799	99	6	)	)	PUNCT
cana-2799	99	7	2	2	NUM
cana-2799	99	8	!	!	PUNCT
cana-2799	100	1	+	+	CCONJ
cana-2799	100	2	δℓ	δℓ	NOUN
cana-2799	100	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	100	4	(	(	PUNCT
cana-2799	100	5	4	4	NUM
cana-2799	100	6	)	)	PUNCT
cana-2799	100	7	4	4	NUM
cana-2799	100	8	!	!	PUNCT
cana-2799	101	1	+	+	CCONJ
cana-2799	101	2	δℓ	δℓ	NOUN
cana-2799	101	3	𝜅ℓ	𝜅ℓ	X
cana-2799	101	4	(	(	PUNCT
cana-2799	101	5	6	6	NUM
cana-2799	101	6	)	)	PUNCT
cana-2799	101	7	6	6	NUM
cana-2799	101	8	!	!	PUNCT
cana-2799	102	1	+	+	CCONJ
cana-2799	102	2	⋯	⋯	VERB
cana-2799	102	3	+	+	CCONJ
cana-2799	102	4	∞	∞	PROPN
cana-2799	102	5	=	=	SYM
cana-2799	102	6	ℓ	ℓ	PROPN
cana-2799	102	7	[	[	PUNCT
cana-2799	102	8	𝜅ℓ	𝜅ℓ	X
cana-2799	102	9	(	(	PUNCT
cana-2799	102	10	1	1	NUM
cana-2799	102	11	)	)	PUNCT
cana-2799	102	12	1	1	NUM
cana-2799	102	13	!	!	PUNCT
cana-2799	103	1	+	+	CCONJ
cana-2799	103	2	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	103	3	(	(	PUNCT
cana-2799	103	4	3	3	NUM
cana-2799	103	5	)	)	PUNCT
cana-2799	103	6	3	3	NUM
cana-2799	103	7	!	!	PUNCT
cana-2799	104	1	+	+	CCONJ
cana-2799	104	2	𝜅ℓ	𝜅ℓ	X
cana-2799	104	3	(	(	PUNCT
cana-2799	104	4	5	5	NUM
cana-2799	104	5	)	)	PUNCT
cana-2799	104	6	5	5	NUM
cana-2799	104	7	!	!	PUNCT
cana-2799	105	1	+	+	CCONJ
cana-2799	105	2	⋯	⋯	ADP
cana-2799	105	3	]	]	PUNCT
cana-2799	105	4	𝑒3(𝜅ℓ	𝑒3(𝜅ℓ	PROPN
cana-2799	105	5	)	)	PUNCT
cana-2799	105	6	=	=	SYM
cana-2799	105	7	1	1	NUM
cana-2799	105	8	+	+	NUM
cana-2799	105	9	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	105	10	(	(	PUNCT
cana-2799	105	11	3	3	NUM
cana-2799	105	12	)	)	PUNCT
cana-2799	105	13	3	3	NUM
cana-2799	105	14	!	!	PUNCT
cana-2799	106	1	+	+	CCONJ
cana-2799	106	2	𝜅ℓ	𝜅ℓ	X
cana-2799	106	3	(	(	PUNCT
cana-2799	106	4	6	6	NUM
cana-2799	106	5	)	)	PUNCT
cana-2799	106	6	6	6	NUM
cana-2799	106	7	!	!	PUNCT
cana-2799	107	1	+	+	CCONJ
cana-2799	107	2	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	107	3	(	(	PUNCT
cana-2799	107	4	9	9	NUM
cana-2799	107	5	)	)	PUNCT
cana-2799	107	6	9	9	NUM
cana-2799	107	7	!	!	PUNCT
cana-2799	108	1	+	+	CCONJ
cana-2799	108	2	⋯	⋯	VERB
cana-2799	108	3	+	+	CCONJ
cana-2799	108	4	∞	∞	PROPN
cana-2799	108	5	δℓ𝑒3(𝜅ℓ	δℓ𝑒3(𝜅ℓ	PROPN
cana-2799	108	6	)	)	PUNCT
cana-2799	109	1	=	=	PUNCT
cana-2799	109	2	δℓ	δℓ	NOUN
cana-2799	109	3	𝜅ℓ	𝜅ℓ	X
cana-2799	109	4	(	(	PUNCT
cana-2799	109	5	3	3	NUM
cana-2799	109	6	)	)	PUNCT
cana-2799	109	7	3	3	NUM
cana-2799	109	8	!	!	PUNCT
cana-2799	110	1	+	+	CCONJ
cana-2799	110	2	δℓ	δℓ	NOUN
cana-2799	110	3	𝜅ℓ	𝜅ℓ	X
cana-2799	110	4	(	(	PUNCT
cana-2799	110	5	6	6	NUM
cana-2799	110	6	)	)	PUNCT
cana-2799	110	7	6	6	NUM
cana-2799	110	8	!	!	PUNCT
cana-2799	111	1	+	+	CCONJ
cana-2799	111	2	δℓ	δℓ	NOUN
cana-2799	111	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	111	4	(	(	PUNCT
cana-2799	111	5	9	9	NUM
cana-2799	111	6	)	)	PUNCT
cana-2799	111	7	9	9	NUM
cana-2799	111	8	!	!	PUNCT
cana-2799	112	1	+	+	CCONJ
cana-2799	112	2	⋯	⋯	VERB
cana-2799	112	3	+	+	CCONJ
cana-2799	112	4	∞	∞	PROPN
cana-2799	112	5	=	=	SYM
cana-2799	112	6	ℓ	ℓ	PROPN
cana-2799	112	7	[	[	PUNCT
cana-2799	112	8	𝜅ℓ	𝜅ℓ	X
cana-2799	112	9	(	(	PUNCT
cana-2799	112	10	2	2	NUM
cana-2799	112	11	)	)	PUNCT
cana-2799	112	12	2	2	NUM
cana-2799	112	13	!	!	PUNCT
cana-2799	113	1	+	+	CCONJ
cana-2799	113	2	𝜅ℓ	𝜅ℓ	X
cana-2799	113	3	(	(	PUNCT
cana-2799	113	4	5	5	NUM
cana-2799	113	5	)	)	PUNCT
cana-2799	113	6	5	5	NUM
cana-2799	113	7	!	!	PUNCT
cana-2799	114	1	+	+	CCONJ
cana-2799	114	2	𝜅ℓ	𝜅ℓ	X
cana-2799	114	3	(	(	PUNCT
cana-2799	114	4	8)	8)	NUM
cana-2799	114	5	8	8	NUM
cana-2799	114	6	!	!	PUNCT
cana-2799	115	1	+	+	CCONJ
cana-2799	115	2	⋯	⋯	X
cana-2799	115	3	]	]	PUNCT
cana-2799	115	4	in	in	ADP
cana-2799	115	5	general	general	ADJ
cana-2799	115	6	,	,	PUNCT
cana-2799	115	7	we	we	PRON
cana-2799	115	8	find	find	VERB
cana-2799	115	9	for	for	ADP
cana-2799	115	10	𝑛	𝑛	PROPN
cana-2799	115	11	≥	≥	NUM
cana-2799	115	12	1	1	NUM
cana-2799	115	13	δℓ𝑒𝑛(𝜅ℓ	δℓ𝑒𝑛(𝜅ℓ	ADJ
cana-2799	115	14	)	)	PUNCT
cana-2799	115	15	=	=	SYM
cana-2799	115	16	ℓ	ℓ	PROPN
cana-2799	115	17	[	[	PUNCT
cana-2799	115	18	𝜅ℓ	𝜅ℓ	X
cana-2799	115	19	(	(	PUNCT
cana-2799	115	20	𝑛−1	𝑛−1	NUM
cana-2799	115	21	)	)	PUNCT
cana-2799	115	22	(	(	PUNCT
cana-2799	115	23	𝑛−1	𝑛−1	NOUN
cana-2799	115	24	)	)	PUNCT
cana-2799	115	25	!	!	PUNCT
cana-2799	116	1	+	+	CCONJ
cana-2799	116	2	𝜅ℓ	𝜅ℓ	X
cana-2799	116	3	(	(	PUNCT
cana-2799	116	4	2𝑛−1	2𝑛−1	NUM
cana-2799	116	5	)	)	PUNCT
cana-2799	116	6	(	(	PUNCT
cana-2799	116	7	2𝑛−1	2𝑛−1	NOUN
cana-2799	116	8	)	)	PUNCT
cana-2799	116	9	!	!	PUNCT
cana-2799	117	1	+	+	CCONJ
cana-2799	117	2	𝜅ℓ	𝜅ℓ	X
cana-2799	117	3	(	(	PUNCT
cana-2799	117	4	3𝑛−1	3𝑛−1	NUM
cana-2799	117	5	)	)	PUNCT
cana-2799	117	6	(	(	PUNCT
cana-2799	117	7	3𝑛−1	3𝑛−1	NUM
cana-2799	117	8	)	)	PUNCT
cana-2799	117	9	!	!	PUNCT
cana-2799	118	1	+	+	CCONJ
cana-2799	118	2	⋯	⋯	ADP
cana-2799	118	3	]	]	PUNCT
cana-2799	118	4	=	=	PUNCT
cana-2799	118	5	ℓ∑∞	ℓ∑∞	NOUN
cana-2799	118	6	𝑚=1	𝑚=1	X
cana-2799	118	7	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	118	8	(	(	PUNCT
cana-2799	118	9	𝑚𝑛−1	𝑚𝑛−1	PROPN
cana-2799	118	10	)	)	PUNCT
cana-2799	118	11	(	(	PUNCT
cana-2799	118	12	𝑚𝑛−1	𝑚𝑛−1	NOUN
cana-2799	118	13	)	)	PUNCT
cana-2799	118	14	!	!	PUNCT
cana-2799	119	1	lemma	lemma	PROPN
cana-2799	119	2	2.5	2.5	NUM
cana-2799	119	3	.	.	PUNCT
cana-2799	120	1	for	for	ADP
cana-2799	120	2	any	any	DET
cana-2799	120	3	positive	positive	ADJ
cana-2799	120	4	integer	integer	NOUN
cana-2799	120	5	𝑚	𝑚	NOUN
cana-2799	120	6	,	,	PUNCT
cana-2799	120	7	we	we	PRON
cana-2799	120	8	have	have	AUX
cana-2799	120	9	δℓ	δℓ	VERB
cana-2799	120	10	𝑚𝑒𝑚(𝜅ℓ	𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	120	11	)	)	PUNCT
cana-2799	120	12	=	=	SYM
cana-2799	120	13	ℓ	ℓ	NOUN
cana-2799	120	14	𝑚𝑒𝑚(𝜅ℓ	𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	120	15	)	)	PUNCT
cana-2799	120	16	.	.	PUNCT
cana-2799	121	1	proof	proof	NOUN
cana-2799	121	2	.	.	PUNCT
cana-2799	122	1	δℓ𝑒1(𝜅ℓ	δℓ𝑒1(𝜅ℓ	X
cana-2799	122	2	)	)	PUNCT
cana-2799	122	3	=	=	SYM
cana-2799	122	4	0	0	PUNCT
cana-2799	123	1	+	+	CCONJ
cana-2799	123	2	δℓ	δℓ	NOUN
cana-2799	123	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	123	4	(	(	PUNCT
cana-2799	123	5	1	1	NUM
cana-2799	123	6	)	)	PUNCT
cana-2799	123	7	1	1	NUM
cana-2799	123	8	!	!	PUNCT
cana-2799	124	1	+	+	CCONJ
cana-2799	124	2	δℓ	δℓ	NOUN
cana-2799	124	3	𝜅ℓ	𝜅ℓ	X
cana-2799	124	4	(	(	PUNCT
cana-2799	124	5	2	2	NUM
cana-2799	124	6	)	)	PUNCT
cana-2799	124	7	2	2	NUM
cana-2799	124	8	!	!	PUNCT
cana-2799	125	1	+	+	CCONJ
cana-2799	125	2	δℓ	δℓ	NOUN
cana-2799	125	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	125	4	(	(	PUNCT
cana-2799	125	5	3	3	NUM
cana-2799	125	6	)	)	PUNCT
cana-2799	125	7	3	3	NUM
cana-2799	125	8	!	!	PUNCT
cana-2799	126	1	+	+	NUM
cana-2799	126	2	⋯	⋯	NOUN
cana-2799	126	3	=	=	SYM
cana-2799	126	4	ℓ𝑒1(𝜅ℓ	ℓ𝑒1(𝜅ℓ	X
cana-2799	126	5	)	)	PUNCT
cana-2799	126	6	.	.	PUNCT
cana-2799	127	1	δℓ𝑒2(𝜅ℓ	δℓ𝑒2(𝜅ℓ	PUNCT
cana-2799	127	2	)	)	PUNCT
cana-2799	128	1	=	=	SYM
cana-2799	128	2	0	0	PUNCT
cana-2799	129	1	+	+	CCONJ
cana-2799	129	2	δℓ	δℓ	NOUN
cana-2799	129	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	129	4	(	(	PUNCT
cana-2799	129	5	2	2	NUM
cana-2799	129	6	)	)	PUNCT
cana-2799	129	7	2	2	NUM
cana-2799	129	8	!	!	PUNCT
cana-2799	130	1	+	+	CCONJ
cana-2799	130	2	δℓ	δℓ	NOUN
cana-2799	130	3	𝜅ℓ	𝜅ℓ	NOUN
cana-2799	130	4	(	(	PUNCT
cana-2799	130	5	4	4	NUM
cana-2799	130	6	)	)	PUNCT
cana-2799	130	7	4	4	NUM
cana-2799	130	8	!	!	PUNCT
cana-2799	131	1	+	+	CCONJ
cana-2799	131	2	δℓ	δℓ	NOUN
cana-2799	131	3	𝜅ℓ	𝜅ℓ	X
cana-2799	131	4	(	(	PUNCT
cana-2799	131	5	6	6	NUM
cana-2799	131	6	)	)	PUNCT
cana-2799	131	7	6	6	NUM
cana-2799	131	8	!	!	PUNCT
cana-2799	132	1	+	+	CCONJ
cana-2799	132	2	⋯	⋯	NOUN
cana-2799	132	3	=	=	SYM
cana-2799	132	4	2ℓ𝜅ℓ(1	2ℓ𝜅ℓ(1	NUM
cana-2799	132	5	)	)	PUNCT
cana-2799	132	6	2	2	NUM
cana-2799	132	7	!	!	PUNCT
cana-2799	133	1	+	+	CCONJ
cana-2799	133	2	4ℓ𝜅ℓ(3	4ℓ𝜅ℓ(3	NUM
cana-2799	133	3	)	)	PUNCT
cana-2799	133	4	4	4	NUM
cana-2799	133	5	!	!	PUNCT
cana-2799	134	1	+	+	CCONJ
cana-2799	134	2	6ℓ𝜅ℓ(5	6ℓ𝜅ℓ(5	NUM
cana-2799	134	3	)	)	PUNCT
cana-2799	134	4	6	6	NUM
cana-2799	134	5	!	!	PUNCT
cana-2799	135	1	+	+	CCONJ
cana-2799	135	2	⋯	⋯	ADP
cana-2799	135	3	δℓ	δℓ	ADJ
cana-2799	135	4	2𝑒2(𝜅ℓ	2𝑒2(𝜅ℓ	NUM
cana-2799	135	5	)	)	PUNCT
cana-2799	136	1	=	=	SYM
cana-2799	136	2	2ℓ(ℓ𝜅ℓ	2ℓ(ℓ𝜅ℓ	NUM
cana-2799	136	3	(	(	PUNCT
cana-2799	136	4	0	0	NUM
cana-2799	136	5	)	)	PUNCT
cana-2799	136	6	)	)	PUNCT
cana-2799	137	1	2	2	X
cana-2799	137	2	!	!	PUNCT
cana-2799	138	1	+	+	CCONJ
cana-2799	138	2	4ℓ(3ℓ𝜅ℓ	4ℓ(3ℓ𝜅ℓ	PRON
cana-2799	138	3	(	(	PUNCT
cana-2799	138	4	2	2	NUM
cana-2799	138	5	)	)	PUNCT
cana-2799	138	6	)	)	PUNCT
cana-2799	139	1	4	4	NUM
cana-2799	139	2	!	!	PUNCT
cana-2799	140	1	+	+	CCONJ
cana-2799	140	2	6ℓ(5ℓ𝜅ℓ	6ℓ(5ℓ𝜅ℓ	NUM
cana-2799	140	3	(	(	PUNCT
cana-2799	140	4	4	4	NUM
cana-2799	140	5	)	)	PUNCT
cana-2799	140	6	)	)	PUNCT
cana-2799	140	7	6	6	NUM
cana-2799	140	8	!	!	PUNCT
cana-2799	141	1	+	+	CCONJ
cana-2799	141	2	⋯	⋯	VERB
cana-2799	141	3	=	=	SYM
cana-2799	141	4	ℓ	ℓ	PROPN
cana-2799	141	5	2𝑒2(𝜅ℓ	2𝑒2(𝜅ℓ	NUM
cana-2799	141	6	)	)	PUNCT
cana-2799	141	7	,	,	PUNCT
cana-2799	141	8	which	which	PRON
cana-2799	141	9	yields	yield	VERB
cana-2799	141	10	δℓ	δℓ	VERB
cana-2799	141	11	𝑚𝑒𝑚(𝜅ℓ	𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	141	12	)	)	PUNCT
cana-2799	141	13	=	=	SYM
cana-2799	141	14	ℓ	ℓ	NOUN
cana-2799	141	15	𝑚𝑒𝑚(𝜅ℓ	𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	141	16	)	)	PUNCT
cana-2799	141	17	.	.	PUNCT
cana-2799	142	1	lemma	lemma	PROPN
cana-2799	142	2	2.6	2.6	NUM
cana-2799	142	3	.	.	PUNCT
cana-2799	143	1	for	for	ADP
cana-2799	143	2	positive	positive	ADJ
cana-2799	143	3	𝑚	𝑚	NOUN
cana-2799	143	4	and	and	CCONJ
cana-2799	143	5	real	real	ADJ
cana-2799	143	6	𝜅	𝜅	NOUN
cana-2799	143	7	,	,	PUNCT
cana-2799	143	8	we	we	PRON
cana-2799	143	9	have	have	AUX
cana-2799	143	10	δℓ	δℓ	VERB
cana-2799	143	11	(	(	PUNCT
cana-2799	143	12	−𝑚	−𝑚	ADJ
cana-2799	143	13	)	)	PUNCT
cana-2799	143	14	𝑒𝑚(𝜅ℓ	𝑒𝑚(𝜅ℓ	NOUN
cana-2799	143	15	)	)	PUNCT
cana-2799	143	16	=	=	SYM
cana-2799	143	17	𝑒𝑚(𝜅ℓ	𝑒𝑚(𝜅ℓ	NOUN
cana-2799	143	18	)	)	PUNCT
cana-2799	143	19	ℓ𝑚	ℓ𝑚	NOUN
cana-2799	143	20	,	,	PUNCT
cana-2799	143	21	ℓ	ℓ	PROPN
cana-2799	143	22	∈	∈	PROPN
cana-2799	143	23	ℕ.	ℕ.	PROPN
cana-2799	143	24	proof	proof	NOUN
cana-2799	143	25	.	.	PUNCT
cana-2799	144	1	from	from	ADP
cana-2799	144	2	the	the	DET
cana-2799	144	3	lemma	lemma	PROPN
cana-2799	144	4	5	5	NUM
cana-2799	144	5	,	,	PUNCT
cana-2799	144	6	we	we	PRON
cana-2799	144	7	find	find	VERB
cana-2799	144	8	δℓ	δℓ	ADP
cana-2799	144	9	𝑚𝑒𝑚(𝜅ℓ	𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	144	10	)	)	PUNCT
cana-2799	144	11	=	=	SYM
cana-2799	144	12	ℓ	ℓ	NOUN
cana-2799	144	13	𝑚𝑒𝑚(𝜅ℓ	𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	144	14	)	)	PUNCT
cana-2799	144	15	.	.	PUNCT
cana-2799	145	1	taking	take	VERB
cana-2799	145	2	δℓ	δℓ	ADV
cana-2799	145	3	−𝑚	−𝑚	ADV
cana-2799	145	4	on	on	ADP
cana-2799	145	5	both	both	DET
cana-2799	145	6	sides	side	NOUN
cana-2799	145	7	,	,	PUNCT
cana-2799	145	8	we	we	PRON
cana-2799	145	9	get	get	VERB
cana-2799	145	10	δℓ	δℓ	ADP
cana-2799	145	11	−𝑚(δℓ	−𝑚(δℓ	PROPN
cana-2799	145	12	𝑚𝑒𝑚(𝜅ℓ	𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	145	13	)	)	PUNCT
cana-2799	145	14	)	)	PUNCT
cana-2799	146	1	=	=	PRON
cana-2799	146	2	δℓ	δℓ	NOUN
cana-2799	146	3	−𝑚(ℓ𝑚𝑒𝑚(𝜅ℓ	−𝑚(ℓ𝑚𝑒𝑚(𝜅ℓ	NOUN
cana-2799	146	4	)	)	PUNCT
cana-2799	146	5	)	)	PUNCT
cana-2799	146	6	,	,	PUNCT
cana-2799	146	7	which	which	PRON
cana-2799	146	8	gives	give	VERB
cana-2799	146	9	δℓ	δℓ	ADP
cana-2799	146	10	(	(	PUNCT
cana-2799	146	11	−𝑚	−𝑚	ADJ
cana-2799	146	12	)	)	PUNCT
cana-2799	146	13	𝑒𝑚(𝜅ℓ	𝑒𝑚(𝜅ℓ	NOUN
cana-2799	146	14	)	)	PUNCT
cana-2799	146	15	=	=	SYM
cana-2799	146	16	𝑒𝑚(𝜅ℓ	𝑒𝑚(𝜅ℓ	NOUN
cana-2799	146	17	)	)	PUNCT
cana-2799	146	18	ℓ𝑚	ℓ𝑚	NOUN
cana-2799	146	19	.	.	PUNCT
cana-2799	147	1	definition	definition	NOUN
cana-2799	147	2	2.7	2.7	NUM
cana-2799	147	3	.	.	PUNCT
cana-2799	148	1	for	for	ADP
cana-2799	148	2	|ℓ|	|ℓ|	PRON
cana-2799	148	3	<	<	X
cana-2799	148	4	1	1	NUM
cana-2799	148	5	,	,	PUNCT
cana-2799	148	6	and	and	CCONJ
cana-2799	148	7	𝑛	𝑛	DET
cana-2799	148	8	∈	∈	PROPN
cana-2799	148	9	ℕ	ℕ	PROPN
cana-2799	148	10	,	,	PUNCT
cana-2799	148	11	𝑒(−𝑛)(𝑘ℓ	𝑒(−𝑛)(𝑘ℓ	PROPN
cana-2799	148	12	)	)	PUNCT
cana-2799	148	13	is	be	AUX
cana-2799	148	14	defined	define	VERB
cana-2799	148	15	as	as	ADP
cana-2799	148	16	𝑒(−𝑛)(𝜅ℓ	𝑒(−𝑛)(𝜅ℓ	NOUN
cana-2799	148	17	)	)	PUNCT
cana-2799	148	18	=	=	SYM
cana-2799	149	1	1	1	NUM
cana-2799	149	2	+	+	SYM
cana-2799	149	3	1	1	NUM
cana-2799	149	4	𝑛	𝑛	NOUN
cana-2799	149	5	!	!	NOUN
cana-2799	149	6	1	1	NUM
cana-2799	149	7	𝜅	𝜅	PROPN
cana-2799	149	8	ℓ	ℓ	PROPN
cana-2799	149	9	(	(	PUNCT
cana-2799	149	10	𝑛	𝑛	PROPN
cana-2799	149	11	)	)	PUNCT
cana-2799	149	12	+	+	CCONJ
cana-2799	149	13	1	1	NUM
cana-2799	149	14	(	(	PUNCT
cana-2799	149	15	2𝑛	2𝑛	NUM
cana-2799	149	16	)	)	PUNCT
cana-2799	149	17	!	!	PUNCT
cana-2799	150	1	1	1	NUM
cana-2799	150	2	𝜅	𝜅	NUM
cana-2799	150	3	ℓ	ℓ	PROPN
cana-2799	150	4	(	(	PUNCT
cana-2799	150	5	2𝑛	2𝑛	NUM
cana-2799	150	6	)	)	PUNCT
cana-2799	151	1	+	+	CCONJ
cana-2799	151	2	1	1	NUM
cana-2799	151	3	(	(	PUNCT
cana-2799	151	4	3𝑛	3𝑛	NUM
cana-2799	151	5	)	)	PUNCT
cana-2799	151	6	!	!	PUNCT
cana-2799	152	1	1	1	NUM
cana-2799	152	2	𝜅	𝜅	NUM
cana-2799	152	3	ℓ	ℓ	PROPN
cana-2799	152	4	(	(	PUNCT
cana-2799	152	5	3𝑛	3𝑛	NUM
cana-2799	152	6	)	)	PUNCT
cana-2799	153	1	+	+	CCONJ
cana-2799	153	2	⋯	⋯	VERB
cana-2799	153	3	+	+	CCONJ
cana-2799	153	4	∞.	∞.	PROPN
cana-2799	153	5	(	(	PUNCT
cana-2799	153	6	7	7	NUM
cana-2799	153	7	)	)	PUNCT
cana-2799	153	8	lemma	lemma	PROPN
cana-2799	153	9	2.8	2.8	NUM
cana-2799	153	10	.	.	PUNCT
cana-2799	154	1	for	for	ADP
cana-2799	154	2	ℓ	ℓ	PROPN
cana-2799	154	3	∈	∈	PROPN
cana-2799	154	4	(	(	PUNCT
cana-2799	154	5	−1,1	−1,1	NOUN
cana-2799	154	6	)	)	PUNCT
cana-2799	154	7	and	and	CCONJ
cana-2799	154	8	positive	positive	ADJ
cana-2799	154	9	𝜅	𝜅	NOUN
cana-2799	154	10	,	,	PUNCT
cana-2799	154	11	we	we	PRON
cana-2799	154	12	have	have	VERB
cana-2799	154	13	δℓ𝑒(−𝑛)(𝜅ℓ	δℓ𝑒(−𝑛)(𝜅ℓ	NOUN
cana-2799	154	14	)	)	PUNCT
cana-2799	155	1	=	=	PUNCT
cana-2799	155	2	−ℓ	−ℓ	NOUN
cana-2799	155	3	[	[	PUNCT
cana-2799	155	4	1	1	NUM
cana-2799	155	5	(	(	PUNCT
cana-2799	155	6	𝑛−1	𝑛−1	NUM
cana-2799	155	7	)	)	PUNCT
cana-2799	155	8	!	!	PUNCT
cana-2799	156	1	1	1	NUM
cana-2799	156	2	(	(	PUNCT
cana-2799	156	3	𝜅+ℓ	𝜅+ℓ	NUM
cana-2799	156	4	)	)	PUNCT
cana-2799	156	5	ℓ	ℓ	PROPN
cana-2799	156	6	(	(	PUNCT
cana-2799	156	7	𝑛+1	𝑛+1	PROPN
cana-2799	156	8	)	)	PUNCT
cana-2799	156	9	+	+	CCONJ
cana-2799	156	10	1	1	NUM
cana-2799	156	11	(	(	PUNCT
cana-2799	156	12	2𝑛−1	2𝑛−1	NOUN
cana-2799	156	13	)	)	PUNCT
cana-2799	156	14	!	!	PUNCT
cana-2799	157	1	1	1	NUM
cana-2799	157	2	(	(	PUNCT
cana-2799	157	3	𝜅+ℓ	𝜅+ℓ	NUM
cana-2799	157	4	)	)	PUNCT
cana-2799	157	5	ℓ	ℓ	PROPN
cana-2799	157	6	(	(	PUNCT
cana-2799	157	7	2𝑛+1	2𝑛+1	NUM
cana-2799	157	8	)	)	PUNCT
cana-2799	158	1	+	+	CCONJ
cana-2799	158	2	1	1	NUM
cana-2799	158	3	(	(	PUNCT
cana-2799	158	4	3𝑛−1	3𝑛−1	NUM
cana-2799	158	5	)	)	PUNCT
cana-2799	158	6	!	!	PUNCT
cana-2799	159	1	1	1	NUM
cana-2799	159	2	(	(	PUNCT
cana-2799	159	3	𝜅+ℓ	𝜅+ℓ	NUM
cana-2799	159	4	)	)	PUNCT
cana-2799	159	5	ℓ	ℓ	PROPN
cana-2799	159	6	(	(	PUNCT
cana-2799	159	7	3𝑛+1	3𝑛+1	NUM
cana-2799	159	8	)	)	PUNCT
cana-2799	160	1	+	+	CCONJ
cana-2799	160	2	⋯	⋯	X
cana-2799	160	3	]	]	PUNCT
cana-2799	160	4	proof	proof	NOUN
cana-2799	160	5	.	.	PUNCT
cana-2799	161	1	putting	put	VERB
cana-2799	161	2	𝑛	𝑛	PRON
cana-2799	161	3	=	=	SYM
cana-2799	161	4	1	1	NUM
cana-2799	161	5	in	in	ADP
cana-2799	161	6	(	(	PUNCT
cana-2799	161	7	7	7	NUM
cana-2799	161	8	)	)	PUNCT
cana-2799	161	9	,	,	PUNCT
cana-2799	161	10	we	we	PRON
cana-2799	161	11	get	get	VERB
cana-2799	161	12	communications	communication	NOUN
cana-2799	161	13	on	on	ADP
cana-2799	161	14	applied	apply	VERB
cana-2799	161	15	nonlinear	nonlinear	ADJ
cana-2799	161	16	analysis	analysis	NOUN
cana-2799	161	17	issn	issn	NOUN
cana-2799	161	18	:	:	PUNCT
cana-2799	161	19	1074	1074	NUM
cana-2799	161	20	-	-	PUNCT
cana-2799	161	21	133x	133x	NUM
cana-2799	161	22	vol	vol	NOUN
cana-2799	161	23	32	32	NUM
cana-2799	161	24	no	no	NOUN
cana-2799	161	25	.	.	PUNCT
cana-2799	162	1	4s	4s	NUM
cana-2799	162	2	(	(	PUNCT
cana-2799	162	3	2025	2025	NUM
cana-2799	162	4	)	)	PUNCT
cana-2799	162	5	262	262	NUM
cana-2799	162	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	162	7	𝑒(−1)(𝜅ℓ	𝑒(−1)(𝜅ℓ	NOUN
cana-2799	162	8	)	)	PUNCT
cana-2799	162	9	=	=	SYM
cana-2799	163	1	1	1	NUM
cana-2799	163	2	+	+	NUM
cana-2799	163	3	1	1	NUM
cana-2799	163	4	1	1	NUM
cana-2799	163	5	!	!	ADP
cana-2799	163	6	1	1	NUM
cana-2799	163	7	𝜅	𝜅	PROPN
cana-2799	163	8	ℓ	ℓ	PROPN
cana-2799	163	9	(	(	PUNCT
cana-2799	163	10	1	1	NUM
cana-2799	163	11	)	)	PUNCT
cana-2799	163	12	+	+	CCONJ
cana-2799	163	13	1	1	NUM
cana-2799	163	14	2	2	NUM
cana-2799	163	15	!	!	NOUN
cana-2799	163	16	1	1	NUM
cana-2799	163	17	𝜅	𝜅	PROPN
cana-2799	163	18	ℓ	ℓ	PROPN
cana-2799	163	19	(	(	PUNCT
cana-2799	163	20	2	2	NUM
cana-2799	163	21	)	)	PUNCT
cana-2799	163	22	+	+	CCONJ
cana-2799	163	23	1	1	NUM
cana-2799	163	24	3	3	NUM
cana-2799	163	25	!	!	NOUN
cana-2799	163	26	1	1	NUM
cana-2799	163	27	𝜅	𝜅	PROPN
cana-2799	163	28	ℓ	ℓ	PROPN
cana-2799	163	29	(	(	PUNCT
cana-2799	163	30	3	3	NUM
cana-2799	163	31	)	)	PUNCT
cana-2799	163	32	+	+	CCONJ
cana-2799	163	33	⋯	⋯	VERB
cana-2799	163	34	+	+	NOUN
cana-2799	163	35	∞δℓ𝑒(−1)(𝜅ℓ	∞δℓ𝑒(−1)(𝜅ℓ	NUM
cana-2799	163	36	)	)	PUNCT
cana-2799	163	37	=	=	SYM
cana-2799	164	1	1	1	NUM
cana-2799	164	2	+	+	CCONJ
cana-2799	164	3	δℓ	δℓ	VERB
cana-2799	164	4	1	1	NUM
cana-2799	164	5	1	1	NUM
cana-2799	164	6	!	!	ADP
cana-2799	164	7	1	1	NUM
cana-2799	164	8	𝜅	𝜅	PROPN
cana-2799	164	9	ℓ	ℓ	PROPN
cana-2799	164	10	(	(	PUNCT
cana-2799	164	11	1	1	NUM
cana-2799	164	12	)	)	PUNCT
cana-2799	164	13	+	+	CCONJ
cana-2799	164	14	δℓ	δℓ	VERB
cana-2799	164	15	1	1	NUM
cana-2799	164	16	2	2	NUM
cana-2799	164	17	!	!	NOUN
cana-2799	164	18	1	1	NUM
cana-2799	164	19	𝜅	𝜅	PROPN
cana-2799	164	20	ℓ	ℓ	PROPN
cana-2799	164	21	(	(	PUNCT
cana-2799	164	22	2	2	NUM
cana-2799	164	23	)	)	PUNCT
cana-2799	164	24	+	+	CCONJ
cana-2799	164	25	δℓ	δℓ	VERB
cana-2799	164	26	1	1	NUM
cana-2799	164	27	3	3	NUM
cana-2799	164	28	!	!	NOUN
cana-2799	164	29	1	1	NUM
cana-2799	164	30	𝜅	𝜅	PROPN
cana-2799	164	31	ℓ	ℓ	PROPN
cana-2799	164	32	(	(	PUNCT
cana-2799	164	33	3	3	NUM
cana-2799	164	34	)	)	PUNCT
cana-2799	164	35	+	+	CCONJ
cana-2799	164	36	⋯	⋯	VERB
cana-2799	164	37	+	+	CCONJ
cana-2799	164	38	∞	∞	NUM
cana-2799	164	39	=	=	NOUN
cana-2799	164	40	−ℓ	−ℓ	NOUN
cana-2799	164	41	[	[	PUNCT
cana-2799	164	42	1	1	NUM
cana-2799	164	43	(	(	PUNCT
cana-2799	164	44	𝜅	𝜅	X
cana-2799	164	45	+	+	NOUN
cana-2799	164	46	ℓ	ℓ	NOUN
cana-2799	164	47	)	)	PUNCT
cana-2799	164	48	ℓ	ℓ	PROPN
cana-2799	164	49	(	(	PUNCT
cana-2799	164	50	2	2	NUM
cana-2799	164	51	)	)	PUNCT
cana-2799	164	52	+	+	CCONJ
cana-2799	164	53	1	1	NUM
cana-2799	164	54	1	1	NUM
cana-2799	164	55	!	!	SYM
cana-2799	164	56	1	1	NUM
cana-2799	164	57	(	(	PUNCT
cana-2799	164	58	𝜅	𝜅	X
cana-2799	164	59	+	+	NOUN
cana-2799	164	60	ℓ	ℓ	NOUN
cana-2799	164	61	)	)	PUNCT
cana-2799	164	62	ℓ	ℓ	PROPN
cana-2799	164	63	(	(	PUNCT
cana-2799	164	64	3	3	NUM
cana-2799	164	65	)	)	PUNCT
cana-2799	164	66	+	+	CCONJ
cana-2799	164	67	1	1	NUM
cana-2799	164	68	2	2	NUM
cana-2799	164	69	!	!	SYM
cana-2799	164	70	1	1	NUM
cana-2799	164	71	(	(	PUNCT
cana-2799	164	72	𝜅	𝜅	X
cana-2799	164	73	+	+	NOUN
cana-2799	164	74	ℓ	ℓ	NOUN
cana-2799	164	75	)	)	PUNCT
cana-2799	164	76	ℓ	ℓ	PROPN
cana-2799	164	77	(	(	PUNCT
cana-2799	164	78	4	4	NUM
cana-2799	164	79	)	)	PUNCT
cana-2799	164	80	+	+	CCONJ
cana-2799	164	81	⋯	⋯	ADP
cana-2799	164	82	]	]	PUNCT
cana-2799	164	83	.	.	PUNCT
cana-2799	165	1	putting	put	VERB
cana-2799	165	2	𝑛	𝑛	PRON
cana-2799	165	3	=	=	SYM
cana-2799	165	4	2	2	NUM
cana-2799	165	5	in	in	ADP
cana-2799	165	6	(	(	PUNCT
cana-2799	165	7	7	7	NUM
cana-2799	165	8	)	)	PUNCT
cana-2799	165	9	,	,	PUNCT
cana-2799	165	10	we	we	PRON
cana-2799	165	11	get	get	VERB
cana-2799	165	12	𝑒(−2)(𝜅ℓ	𝑒(−2)(𝜅ℓ	NOUN
cana-2799	165	13	)	)	PUNCT
cana-2799	165	14	=	=	SYM
cana-2799	166	1	1	1	NUM
cana-2799	166	2	+	+	NUM
cana-2799	166	3	1	1	NUM
cana-2799	166	4	2	2	NUM
cana-2799	166	5	!	!	NOUN
cana-2799	166	6	1	1	NUM
cana-2799	166	7	𝜅	𝜅	PROPN
cana-2799	166	8	ℓ	ℓ	PROPN
cana-2799	166	9	(	(	PUNCT
cana-2799	166	10	2	2	NUM
cana-2799	166	11	)	)	PUNCT
cana-2799	166	12	+	+	CCONJ
cana-2799	166	13	1	1	NUM
cana-2799	166	14	4	4	NUM
cana-2799	166	15	!	!	ADP
cana-2799	166	16	1	1	NUM
cana-2799	166	17	𝜅	𝜅	PROPN
cana-2799	166	18	ℓ	ℓ	PROPN
cana-2799	166	19	(	(	PUNCT
cana-2799	166	20	4	4	NUM
cana-2799	166	21	)	)	PUNCT
cana-2799	166	22	+	+	CCONJ
cana-2799	166	23	1	1	NUM
cana-2799	166	24	6	6	NUM
cana-2799	166	25	!	!	ADP
cana-2799	166	26	1	1	NUM
cana-2799	166	27	𝜅	𝜅	PROPN
cana-2799	166	28	ℓ	ℓ	PROPN
cana-2799	166	29	(	(	PUNCT
cana-2799	166	30	6	6	NUM
cana-2799	166	31	)	)	PUNCT
cana-2799	166	32	+	+	CCONJ
cana-2799	166	33	⋯	⋯	VERB
cana-2799	166	34	+	+	CCONJ
cana-2799	166	35	∞δℓ𝑒(−2)(𝜅ℓ	∞δℓ𝑒(−2)(𝜅ℓ	X
cana-2799	166	36	)	)	PUNCT
cana-2799	166	37	=	=	SYM
cana-2799	167	1	1	1	NUM
cana-2799	167	2	+	+	CCONJ
cana-2799	167	3	δℓ	δℓ	VERB
cana-2799	167	4	1	1	NUM
cana-2799	167	5	2	2	NUM
cana-2799	167	6	!	!	NOUN
cana-2799	167	7	1	1	NUM
cana-2799	167	8	𝜅	𝜅	PROPN
cana-2799	167	9	ℓ	ℓ	PROPN
cana-2799	167	10	(	(	PUNCT
cana-2799	167	11	2	2	NUM
cana-2799	167	12	)	)	PUNCT
cana-2799	167	13	+	+	CCONJ
cana-2799	167	14	δℓ	δℓ	VERB
cana-2799	167	15	1	1	NUM
cana-2799	167	16	4	4	NUM
cana-2799	167	17	!	!	ADP
cana-2799	167	18	1	1	NUM
cana-2799	167	19	𝜅	𝜅	PROPN
cana-2799	167	20	ℓ	ℓ	PROPN
cana-2799	167	21	(	(	PUNCT
cana-2799	167	22	4	4	NUM
cana-2799	167	23	)	)	PUNCT
cana-2799	167	24	+	+	CCONJ
cana-2799	167	25	δℓ	δℓ	VERB
cana-2799	167	26	1	1	NUM
cana-2799	167	27	6	6	NUM
cana-2799	167	28	!	!	ADP
cana-2799	167	29	1	1	NUM
cana-2799	167	30	𝜅	𝜅	PROPN
cana-2799	167	31	ℓ	ℓ	PROPN
cana-2799	167	32	(	(	PUNCT
cana-2799	167	33	6	6	NUM
cana-2799	167	34	)	)	PUNCT
cana-2799	167	35	+	+	CCONJ
cana-2799	167	36	⋯	⋯	VERB
cana-2799	167	37	+	+	CCONJ
cana-2799	167	38	∞	∞	NUM
cana-2799	167	39	=	=	NOUN
cana-2799	167	40	−ℓ	−ℓ	NOUN
cana-2799	167	41	[	[	PUNCT
cana-2799	167	42	1	1	NUM
cana-2799	167	43	1	1	NUM
cana-2799	167	44	!	!	SYM
cana-2799	167	45	1	1	NUM
cana-2799	167	46	(	(	PUNCT
cana-2799	167	47	𝜅	𝜅	X
cana-2799	167	48	+	+	NOUN
cana-2799	167	49	ℓ	ℓ	NOUN
cana-2799	167	50	)	)	PUNCT
cana-2799	167	51	ℓ	ℓ	PROPN
cana-2799	167	52	(	(	PUNCT
cana-2799	167	53	3	3	NUM
cana-2799	167	54	)	)	PUNCT
cana-2799	167	55	+	+	CCONJ
cana-2799	167	56	1	1	NUM
cana-2799	167	57	3	3	NUM
cana-2799	167	58	!	!	SYM
cana-2799	167	59	1	1	NUM
cana-2799	167	60	(	(	PUNCT
cana-2799	167	61	𝜅	𝜅	X
cana-2799	167	62	+	+	NOUN
cana-2799	167	63	ℓ	ℓ	NOUN
cana-2799	167	64	)	)	PUNCT
cana-2799	167	65	ℓ	ℓ	PROPN
cana-2799	167	66	(	(	PUNCT
cana-2799	167	67	5	5	NUM
cana-2799	167	68	)	)	PUNCT
cana-2799	167	69	+	+	CCONJ
cana-2799	167	70	1	1	NUM
cana-2799	167	71	5	5	NUM
cana-2799	167	72	!	!	SYM
cana-2799	167	73	1	1	NUM
cana-2799	167	74	(	(	PUNCT
cana-2799	167	75	𝜅	𝜅	X
cana-2799	167	76	+	+	NOUN
cana-2799	167	77	ℓ	ℓ	NOUN
cana-2799	167	78	)	)	PUNCT
cana-2799	167	79	ℓ	ℓ	PROPN
cana-2799	167	80	(	(	PUNCT
cana-2799	167	81	7	7	NUM
cana-2799	167	82	)	)	PUNCT
cana-2799	167	83	+	+	CCONJ
cana-2799	167	84	⋯	⋯	ADP
cana-2799	167	85	]	]	PUNCT
cana-2799	167	86	.	.	PUNCT
cana-2799	168	1	putting	put	VERB
cana-2799	168	2	𝑛	𝑛	PRON
cana-2799	168	3	=	=	SYM
cana-2799	168	4	3	3	NUM
cana-2799	168	5	in	in	ADP
cana-2799	168	6	(	(	PUNCT
cana-2799	168	7	7	7	NUM
cana-2799	168	8	)	)	PUNCT
cana-2799	168	9	,	,	PUNCT
cana-2799	168	10	we	we	PRON
cana-2799	168	11	get	get	VERB
cana-2799	168	12	𝑒(−3)(𝜅ℓ	𝑒(−3)(𝜅ℓ	NUM
cana-2799	168	13	)	)	PUNCT
cana-2799	168	14	=	=	SYM
cana-2799	169	1	1	1	NUM
cana-2799	169	2	+	+	NUM
cana-2799	169	3	1	1	NUM
cana-2799	169	4	3	3	NUM
cana-2799	169	5	!	!	NOUN
cana-2799	169	6	1	1	NUM
cana-2799	169	7	𝜅	𝜅	PROPN
cana-2799	169	8	ℓ	ℓ	PROPN
cana-2799	169	9	(	(	PUNCT
cana-2799	169	10	3	3	NUM
cana-2799	169	11	)	)	PUNCT
cana-2799	169	12	+	+	CCONJ
cana-2799	169	13	1	1	NUM
cana-2799	169	14	6	6	NUM
cana-2799	169	15	!	!	ADP
cana-2799	169	16	1	1	NUM
cana-2799	169	17	𝜅	𝜅	PROPN
cana-2799	169	18	ℓ	ℓ	PROPN
cana-2799	169	19	(	(	PUNCT
cana-2799	169	20	6	6	NUM
cana-2799	169	21	)	)	PUNCT
cana-2799	169	22	+	+	CCONJ
cana-2799	170	1	1	1	NUM
cana-2799	170	2	9	9	NUM
cana-2799	170	3	!	!	SYM
cana-2799	170	4	1	1	NUM
cana-2799	170	5	𝜅	𝜅	PROPN
cana-2799	170	6	ℓ	ℓ	PROPN
cana-2799	170	7	(	(	PUNCT
cana-2799	170	8	9	9	NUM
cana-2799	170	9	)	)	PUNCT
cana-2799	170	10	+	+	CCONJ
cana-2799	170	11	⋯	⋯	VERB
cana-2799	170	12	+	+	NOUN
cana-2799	170	13	∞δℓ𝑒(−3)(𝜅ℓ	∞δℓ𝑒(−3)(𝜅ℓ	NOUN
cana-2799	170	14	)	)	PUNCT
cana-2799	170	15	=	=	SYM
cana-2799	171	1	1	1	NUM
cana-2799	171	2	+	+	CCONJ
cana-2799	171	3	δℓ	δℓ	VERB
cana-2799	171	4	1	1	NUM
cana-2799	171	5	3	3	NUM
cana-2799	171	6	!	!	NOUN
cana-2799	171	7	1	1	NUM
cana-2799	171	8	𝜅	𝜅	PROPN
cana-2799	171	9	ℓ	ℓ	PROPN
cana-2799	171	10	(	(	PUNCT
cana-2799	171	11	3	3	NUM
cana-2799	171	12	)	)	PUNCT
cana-2799	171	13	+	+	CCONJ
cana-2799	171	14	δℓ	δℓ	VERB
cana-2799	171	15	1	1	NUM
cana-2799	171	16	6	6	NUM
cana-2799	171	17	!	!	ADP
cana-2799	171	18	1	1	NUM
cana-2799	171	19	𝜅	𝜅	PROPN
cana-2799	171	20	ℓ	ℓ	PROPN
cana-2799	171	21	(	(	PUNCT
cana-2799	171	22	6	6	NUM
cana-2799	171	23	)	)	PUNCT
cana-2799	171	24	+	+	CCONJ
cana-2799	171	25	δℓ	δℓ	VERB
cana-2799	171	26	1	1	NUM
cana-2799	171	27	9	9	NUM
cana-2799	171	28	!	!	SYM
cana-2799	171	29	1	1	NUM
cana-2799	171	30	𝜅	𝜅	PROPN
cana-2799	171	31	ℓ	ℓ	PROPN
cana-2799	171	32	(	(	PUNCT
cana-2799	171	33	9	9	NUM
cana-2799	171	34	)	)	PUNCT
cana-2799	171	35	+	+	CCONJ
cana-2799	171	36	⋯	⋯	VERB
cana-2799	171	37	+	+	CCONJ
cana-2799	171	38	∞	∞	NUM
cana-2799	171	39	=	=	NOUN
cana-2799	171	40	−ℓ	−ℓ	NOUN
cana-2799	171	41	[	[	PUNCT
cana-2799	171	42	1	1	NUM
cana-2799	171	43	2	2	NUM
cana-2799	171	44	!	!	SYM
cana-2799	171	45	1	1	NUM
cana-2799	171	46	(	(	PUNCT
cana-2799	171	47	𝜅	𝜅	X
cana-2799	171	48	+	+	NOUN
cana-2799	171	49	ℓ	ℓ	NOUN
cana-2799	171	50	)	)	PUNCT
cana-2799	171	51	ℓ	ℓ	PROPN
cana-2799	171	52	(	(	PUNCT
cana-2799	171	53	4	4	NUM
cana-2799	171	54	)	)	PUNCT
cana-2799	171	55	+	+	CCONJ
cana-2799	171	56	1	1	NUM
cana-2799	171	57	5	5	NUM
cana-2799	171	58	!	!	SYM
cana-2799	171	59	1	1	NUM
cana-2799	171	60	(	(	PUNCT
cana-2799	171	61	𝜅	𝜅	X
cana-2799	171	62	+	+	NOUN
cana-2799	171	63	ℓ	ℓ	NOUN
cana-2799	171	64	)	)	PUNCT
cana-2799	171	65	ℓ	ℓ	PROPN
cana-2799	171	66	(	(	PUNCT
cana-2799	171	67	7	7	NUM
cana-2799	171	68	)	)	PUNCT
cana-2799	171	69	+	+	CCONJ
cana-2799	171	70	1	1	NUM
cana-2799	171	71	8	8	NUM
cana-2799	171	72	!	!	SYM
cana-2799	171	73	1	1	NUM
cana-2799	171	74	(	(	PUNCT
cana-2799	171	75	𝜅	𝜅	X
cana-2799	171	76	+	+	NOUN
cana-2799	171	77	ℓ	ℓ	NOUN
cana-2799	171	78	)	)	PUNCT
cana-2799	171	79	ℓ	ℓ	PROPN
cana-2799	171	80	(	(	PUNCT
cana-2799	171	81	10	10	NUM
cana-2799	171	82	)	)	PUNCT
cana-2799	171	83	+	+	CCONJ
cana-2799	171	84	⋯	⋯	ADP
cana-2799	171	85	]	]	PUNCT
cana-2799	171	86	.	.	PUNCT
cana-2799	172	1	in	in	ADP
cana-2799	172	2	general	general	ADJ
cana-2799	172	3	,	,	PUNCT
cana-2799	172	4	δℓ𝑒(−𝑛)(𝜅ℓ	δℓ𝑒(−𝑛)(𝜅ℓ	NOUN
cana-2799	172	5	)	)	PUNCT
cana-2799	172	6	=	=	PUNCT
cana-2799	172	7	−ℓ	−ℓ	NOUN
cana-2799	172	8	[	[	PUNCT
cana-2799	172	9	1	1	NUM
cana-2799	172	10	(	(	PUNCT
cana-2799	172	11	𝑛−1	𝑛−1	NUM
cana-2799	172	12	)	)	PUNCT
cana-2799	172	13	!	!	PUNCT
cana-2799	173	1	1	1	NUM
cana-2799	173	2	(	(	PUNCT
cana-2799	173	3	𝜅+ℓ	𝜅+ℓ	NUM
cana-2799	173	4	)	)	PUNCT
cana-2799	173	5	ℓ	ℓ	PROPN
cana-2799	173	6	(	(	PUNCT
cana-2799	173	7	𝑛+1	𝑛+1	PROPN
cana-2799	173	8	)	)	PUNCT
cana-2799	173	9	+	+	CCONJ
cana-2799	173	10	1	1	NUM
cana-2799	173	11	(	(	PUNCT
cana-2799	173	12	2𝑛−1	2𝑛−1	NOUN
cana-2799	173	13	)	)	PUNCT
cana-2799	173	14	!	!	PUNCT
cana-2799	174	1	1	1	NUM
cana-2799	174	2	(	(	PUNCT
cana-2799	174	3	𝜅+ℓ	𝜅+ℓ	NUM
cana-2799	174	4	)	)	PUNCT
cana-2799	174	5	ℓ	ℓ	PROPN
cana-2799	174	6	(	(	PUNCT
cana-2799	174	7	2𝑛+1	2𝑛+1	NUM
cana-2799	174	8	)	)	PUNCT
cana-2799	175	1	+	+	CCONJ
cana-2799	175	2	1	1	NUM
cana-2799	175	3	(	(	PUNCT
cana-2799	175	4	3𝑛−1	3𝑛−1	NUM
cana-2799	175	5	)	)	PUNCT
cana-2799	175	6	!	!	PUNCT
cana-2799	176	1	1	1	NUM
cana-2799	176	2	(	(	PUNCT
cana-2799	176	3	𝜅+ℓ	𝜅+ℓ	NUM
cana-2799	176	4	)	)	PUNCT
cana-2799	176	5	ℓ	ℓ	PROPN
cana-2799	176	6	(	(	PUNCT
cana-2799	176	7	3𝑛+1	3𝑛+1	NUM
cana-2799	176	8	)	)	PUNCT
cana-2799	177	1	+	+	CCONJ
cana-2799	177	2	⋯	⋯	ADP
cana-2799	177	3	]	]	PUNCT
cana-2799	177	4	.	.	PUNCT
cana-2799	178	1	3	3	X
cana-2799	178	2	.	.	X
cana-2799	178	3	current	current	ADJ
cana-2799	178	4	flows	flow	NOUN
cana-2799	178	5	in	in	ADP
cana-2799	178	6	rl	rl	PROPN
cana-2799	178	7	circuit	circuit	NOUN
cana-2799	178	8	consider	consider	VERB
cana-2799	178	9	a	a	DET
cana-2799	178	10	rl	rl	NOUN
cana-2799	178	11	circuit	circuit	NOUN
cana-2799	178	12	by	by	ADP
cana-2799	178	13	using	use	VERB
cana-2799	178	14	the	the	DET
cana-2799	178	15	kirchhoff	kirchhoff	NOUN
cana-2799	178	16	’s	’s	PART
cana-2799	178	17	circuit	circuit	NOUN
cana-2799	178	18	rule	rule	NOUN
cana-2799	178	19	.	.	PUNCT
cana-2799	179	1	the	the	DET
cana-2799	179	2	differential	differential	ADJ
cana-2799	179	3	equation	equation	NOUN
cana-2799	179	4	connecting	connect	VERB
cana-2799	179	5	voltage	voltage	NOUN
cana-2799	179	6	v	v	NOUN
cana-2799	179	7	,	,	PUNCT
cana-2799	179	8	resistance	resistance	NOUN
cana-2799	179	9	r	r	NOUN
cana-2799	179	10	,	,	PUNCT
cana-2799	179	11	current	current	ADJ
cana-2799	179	12	i	i	NOUN
cana-2799	179	13	and	and	CCONJ
cana-2799	179	14	induction	induction	NOUN
cana-2799	179	15	l	l	NOUN
cana-2799	179	16	in	in	ADP
cana-2799	179	17	series	series	NOUN
cana-2799	179	18	is	be	AUX
cana-2799	179	19	given	give	VERB
cana-2799	179	20	by	by	ADP
cana-2799	179	21	first	first	ADJ
cana-2799	179	22	order	order	NOUN
cana-2799	179	23	linear	linear	ADJ
cana-2799	179	24	difference	difference	NOUN
cana-2799	179	25	equation	equation	NOUN
cana-2799	179	26	𝑉	𝑉	NOUN
cana-2799	179	27	=	=	PUNCT
cana-2799	179	28	𝑅𝐼(𝜅	𝑅𝐼(𝜅	PROPN
cana-2799	179	29	)	)	PUNCT
cana-2799	180	1	+	+	CCONJ
cana-2799	180	2	𝐿	𝐿	PROPN
cana-2799	180	3	𝑑𝐼(𝜅	𝑑𝐼(𝜅	PROPN
cana-2799	180	4	)	)	PUNCT
cana-2799	180	5	𝑑𝜅	𝑑𝜅	INTJ
cana-2799	180	6	.	.	PUNCT
cana-2799	181	1	(	(	PUNCT
cana-2799	181	2	8)	8)	NUM
cana-2799	181	3	the	the	DET
cana-2799	181	4	discrete	discrete	ADJ
cana-2799	181	5	analogue	analogue	NOUN
cana-2799	181	6	of	of	ADP
cana-2799	181	7	(	(	PUNCT
cana-2799	181	8	8)	8)	NUM
cana-2799	181	9	is	be	AUX
cana-2799	181	10	assumed	assume	VERB
cana-2799	181	11	by	by	ADP
cana-2799	181	12	replacing	replace	VERB
cana-2799	181	13	𝑑𝐼(𝜅	𝑑𝐼(𝜅	PRON
cana-2799	181	14	)	)	PUNCT
cana-2799	181	15	=	=	SYM
cana-2799	182	1	δ𝐼(𝜅	δ𝐼(𝜅	PROPN
cana-2799	182	2	)	)	PUNCT
cana-2799	182	3	,	,	PUNCT
cana-2799	182	4	where	where	SCONJ
cana-2799	182	5	δ𝐼(𝜅	δ𝐼(𝜅	NOUN
cana-2799	182	6	)	)	PUNCT
cana-2799	182	7	=	=	PUNCT
cana-2799	183	1	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	183	2	+	+	NOUN
cana-2799	183	3	1	1	NUM
cana-2799	183	4	)	)	PUNCT
cana-2799	183	5	−	−	PROPN
cana-2799	183	6	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	183	7	)	)	PUNCT
cana-2799	183	8	and	and	CCONJ
cana-2799	183	9	𝑑𝜅	𝑑𝜅	ADP
cana-2799	183	10	=	=	SYM
cana-2799	183	11	1	1	NUM
cana-2799	183	12	in	in	ADP
cana-2799	183	13	(	(	PUNCT
cana-2799	183	14	8)	8)	NUM
cana-2799	183	15	.	.	PUNCT
cana-2799	184	1	the	the	DET
cana-2799	184	2	corresponding	corresponding	ADJ
cana-2799	184	3	difference	difference	NOUN
cana-2799	184	4	equation	equation	NOUN
cana-2799	184	5	for	for	ADP
cana-2799	184	6	the	the	DET
cana-2799	184	7	current	current	ADJ
cana-2799	184	8	flows	flow	NOUN
cana-2799	184	9	in	in	ADP
cana-2799	184	10	rl	rl	PROPN
cana-2799	184	11	series	series	PROPN
cana-2799	184	12	circuit	circuit	NOUN
cana-2799	184	13	in	in	ADP
cana-2799	184	14	the	the	DET
cana-2799	184	15	discrete	discrete	ADJ
cana-2799	184	16	case	case	NOUN
cana-2799	184	17	takes	take	VERB
cana-2799	184	18	the	the	DET
cana-2799	184	19	form	form	NOUN
cana-2799	184	20	,	,	PUNCT
cana-2799	184	21	at	at	ADP
cana-2799	184	22	time	time	NOUN
cana-2799	184	23	𝜅	𝜅	DET
cana-2799	184	24	𝑣(𝜅	𝑣(𝜅	NOUN
cana-2799	184	25	)	)	PUNCT
cana-2799	184	26	=	=	SYM
cana-2799	184	27	𝑅𝐼(𝜅	𝑅𝐼(𝜅	PROPN
cana-2799	184	28	)	)	PUNCT
cana-2799	185	1	+	+	PUNCT
cana-2799	185	2	𝐿δ𝐼(𝜅	𝐿δ𝐼(𝜅	NUM
cana-2799	185	3	)	)	PUNCT
cana-2799	185	4	.	.	PUNCT
cana-2799	186	1	(	(	PUNCT
cana-2799	186	2	9	9	X
cana-2799	186	3	)	)	PUNCT
cana-2799	186	4	communications	communication	NOUN
cana-2799	186	5	on	on	ADP
cana-2799	186	6	applied	apply	VERB
cana-2799	186	7	nonlinear	nonlinear	ADJ
cana-2799	186	8	analysis	analysis	NOUN
cana-2799	186	9	issn	issn	NOUN
cana-2799	186	10	:	:	PUNCT
cana-2799	186	11	1074	1074	NUM
cana-2799	186	12	-	-	PUNCT
cana-2799	186	13	133x	133x	NUM
cana-2799	186	14	vol	vol	NOUN
cana-2799	186	15	32	32	NUM
cana-2799	186	16	no	no	NOUN
cana-2799	186	17	.	.	PUNCT
cana-2799	187	1	4s	4s	NUM
cana-2799	187	2	(	(	PUNCT
cana-2799	187	3	2025	2025	NUM
cana-2799	187	4	)	)	PUNCT
cana-2799	187	5	263	263	NUM
cana-2799	187	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	187	7	due	due	ADP
cana-2799	187	8	to	to	ADP
cana-2799	187	9	the	the	DET
cana-2799	187	10	resistance	resistance	NOUN
cana-2799	187	11	of	of	ADP
cana-2799	187	12	conductor	conductor	NOUN
cana-2799	187	13	,	,	PUNCT
cana-2799	187	14	heat	heat	NOUN
cana-2799	187	15	temperature	temperature	NOUN
cana-2799	187	16	may	may	AUX
cana-2799	187	17	be	be	AUX
cana-2799	187	18	raised	raise	VERB
cana-2799	187	19	in	in	ADP
cana-2799	187	20	the	the	DET
cana-2799	187	21	rl	rl	PROPN
cana-2799	187	22	circuit	circuit	NOUN
cana-2799	187	23	.	.	PUNCT
cana-2799	188	1	in	in	ADP
cana-2799	188	2	that	that	DET
cana-2799	188	3	case	case	NOUN
cana-2799	188	4	,	,	PUNCT
cana-2799	188	5	we	we	PRON
cana-2799	188	6	need	need	VERB
cana-2799	188	7	to	to	PART
cana-2799	188	8	modify	modify	VERB
cana-2799	188	9	the	the	DET
cana-2799	188	10	difference	difference	NOUN
cana-2799	188	11	equation	equation	NOUN
cana-2799	188	12	(	(	PUNCT
cana-2799	188	13	9	9	NUM
cana-2799	188	14	)	)	PUNCT
cana-2799	188	15	.	.	PUNCT
cana-2799	189	1	in	in	ADP
cana-2799	189	2	that	that	DET
cana-2799	189	3	case	case	NOUN
cana-2799	189	4	the	the	DET
cana-2799	189	5	difference	difference	NOUN
cana-2799	189	6	equation	equation	NOUN
cana-2799	189	7	(	(	PUNCT
cana-2799	189	8	9	9	X
cana-2799	189	9	)	)	PUNCT
cana-2799	189	10	becomes	become	VERB
cana-2799	189	11	fractional	fractional	ADJ
cana-2799	189	12	difference	difference	NOUN
cana-2799	189	13	equation	equation	NOUN
cana-2799	189	14	as	as	ADP
cana-2799	189	15	𝑉	𝑉	PROPN
cana-2799	189	16	=	=	SYM
cana-2799	189	17	𝑅𝐼(𝜅	𝑅𝐼(𝜅	PROPN
cana-2799	189	18	)	)	PUNCT
cana-2799	190	1	+	+	CCONJ
cana-2799	191	1	𝐿δ	𝐿δ	NOUN
cana-2799	191	2	𝜈𝐼(𝜅	𝜈𝐼(𝜅	NOUN
cana-2799	191	3	)	)	PUNCT
cana-2799	191	4	,	,	PUNCT
cana-2799	191	5	(	(	PUNCT
cana-2799	191	6	0	0	X
cana-2799	191	7	<	<	X
cana-2799	191	8	𝜈	𝜈	X
cana-2799	191	9	<	<	X
cana-2799	191	10	1	1	NUM
cana-2799	191	11	)	)	PUNCT
cana-2799	191	12	.	.	PUNCT
cana-2799	192	1	(	(	PUNCT
cana-2799	192	2	10	10	NUM
cana-2799	192	3	)	)	PUNCT
cana-2799	192	4	which	which	PRON
cana-2799	192	5	can	can	AUX
cana-2799	192	6	be	be	AUX
cana-2799	192	7	expressed	express	VERB
cana-2799	192	8	as	as	ADP
cana-2799	192	9	𝑉(𝜅)−𝑅𝐼(𝜅	𝑉(𝜅)−𝑅𝐼(𝜅	ADJ
cana-2799	192	10	)	)	PUNCT
cana-2799	192	11	𝐿	𝐿	PROPN
cana-2799	192	12	=	=	PROPN
cana-2799	192	13	δ	δ	PROPN
cana-2799	192	14	𝜈𝐼(𝜅	𝜈𝐼(𝜅	PROPN
cana-2799	192	15	)	)	PUNCT
cana-2799	192	16	the	the	DET
cana-2799	192	17	corresponding	corresponding	ADJ
cana-2799	192	18	discrete	discrete	ADJ
cana-2799	192	19	integral	integral	ADJ
cana-2799	192	20	equations	equation	NOUN
cana-2799	192	21	δ	δ	PROPN
cana-2799	192	22	−𝜈	−𝜈	NOUN
cana-2799	192	23	(	(	PUNCT
cana-2799	192	24	𝑉(𝜅)−𝑅𝐼(𝜅	𝑉(𝜅)−𝑅𝐼(𝜅	NOUN
cana-2799	192	25	)	)	PUNCT
cana-2799	192	26	𝐿	𝐿	NOUN
cana-2799	192	27	)	)	PUNCT
cana-2799	192	28	=	=	SYM
cana-2799	192	29	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	192	30	)	)	PUNCT
cana-2799	192	31	by	by	ADP
cana-2799	192	32	applying	apply	VERB
cana-2799	192	33	𝛾𝑡ℎ	𝛾𝑡ℎ	PROPN
cana-2799	192	34	order	order	NOUN
cana-2799	192	35	delta	delta	NOUN
cana-2799	192	36	sum	sum	NOUN
cana-2799	192	37	given	give	VERB
cana-2799	192	38	by	by	ADP
cana-2799	192	39	(	(	PUNCT
cana-2799	192	40	1	1	NUM
cana-2799	192	41	)	)	PUNCT
cana-2799	192	42	,	,	PUNCT
cana-2799	192	43	1	1	NUM
cana-2799	192	44	𝐿γ(𝜈	𝐿γ(𝜈	NOUN
cana-2799	192	45	)	)	PUNCT
cana-2799	192	46	∑𝜅−𝜈	∑𝜅−𝜈	PROPN
cana-2799	192	47	𝑠=0	𝑠=0	PUNCT
cana-2799	192	48	γ(𝜅−𝑠	γ(𝜅−𝑠	PROPN
cana-2799	192	49	)	)	PUNCT
cana-2799	192	50	γ(𝜅−𝑠−(𝜈−1	γ(𝜅−𝑠−(𝜈−1	PROPN
cana-2799	192	51	)	)	PUNCT
cana-2799	192	52	)	)	PUNCT
cana-2799	192	53	(	(	PUNCT
cana-2799	192	54	𝑉(𝑠	𝑉(𝑠	NOUN
cana-2799	192	55	)	)	PUNCT
cana-2799	192	56	−	−	NOUN
cana-2799	192	57	𝑅𝐼(𝑠	𝑅𝐼(𝑠	NOUN
cana-2799	192	58	)	)	PUNCT
cana-2799	192	59	)	)	PUNCT
cana-2799	193	1	=	=	PUNCT
cana-2799	193	2	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	193	3	)	)	PUNCT
cana-2799	193	4	,	,	PUNCT
cana-2799	193	5	(	(	PUNCT
cana-2799	193	6	11	11	X
cana-2799	193	7	)	)	PUNCT
cana-2799	193	8	the	the	DET
cana-2799	193	9	corresponding	corresponding	ADJ
cana-2799	193	10	fractional	fractional	ADJ
cana-2799	193	11	difference	difference	NOUN
cana-2799	193	12	equation	equation	NOUN
cana-2799	193	13	for	for	ADP
cana-2799	193	14	de	de	NOUN
cana-2799	193	15	-	-	NOUN
cana-2799	193	16	energizing	energize	VERB
cana-2799	193	17	in	in	ADP
cana-2799	193	18	rl	rl	ADP
cana-2799	193	19	circuit	circuit	NOUN
cana-2799	193	20	is	be	AUX
cana-2799	193	21	obtained	obtain	VERB
cana-2799	193	22	by	by	ADP
cana-2799	193	23	putting	put	VERB
cana-2799	193	24	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	193	25	)	)	PUNCT
cana-2799	194	1	=	=	PUNCT
cana-2799	195	1	0	0	X
cana-2799	195	2	.	.	PUNCT
cana-2799	196	1	in	in	ADP
cana-2799	196	2	this	this	DET
cana-2799	196	3	case	case	NOUN
cana-2799	196	4	,	,	PUNCT
cana-2799	196	5	we	we	PRON
cana-2799	196	6	get	get	VERB
cana-2799	196	7	0	0	NUM
cana-2799	196	8	=	=	SYM
cana-2799	196	9	𝑅𝐼(𝜅	𝑅𝐼(𝜅	NUM
cana-2799	196	10	)	)	PUNCT
cana-2799	197	1	+	+	CCONJ
cana-2799	198	1	𝐿δ	𝐿δ	NOUN
cana-2799	198	2	𝜈𝐼(𝜅	𝜈𝐼(𝜅	NOUN
cana-2799	198	3	)	)	PUNCT
cana-2799	198	4	.	.	PUNCT
cana-2799	199	1	(	(	PUNCT
cana-2799	199	2	12	12	NUM
cana-2799	199	3	)	)	PUNCT
cana-2799	199	4	which	which	PRON
cana-2799	199	5	is	be	AUX
cana-2799	199	6	the	the	DET
cana-2799	199	7	same	same	ADJ
cana-2799	199	8	as	as	ADP
cana-2799	199	9	δ	δ	PROPN
cana-2799	199	10	𝜈𝐼(𝜅	𝜈𝐼(𝜅	PROPN
cana-2799	199	11	)	)	PUNCT
cana-2799	199	12	=	=	SYM
cana-2799	199	13	−	−	PROPN
cana-2799	199	14	𝑅𝐼(𝜅	𝑅𝐼(𝜅	ADJ
cana-2799	199	15	)	)	PUNCT
cana-2799	199	16	𝐿	𝐿	NOUN
cana-2799	199	17	.	.	PUNCT
cana-2799	200	1	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	200	2	)	)	PUNCT
cana-2799	200	3	=	=	SYM
cana-2799	200	4	δ	δ	PROPN
cana-2799	200	5	−𝜈	−𝜈	NOUN
cana-2799	200	6	(	(	PUNCT
cana-2799	200	7	−	−	PROPN
cana-2799	200	8	𝑅𝐼(𝜅	𝑅𝐼(𝜅	ADJ
cana-2799	200	9	)	)	PUNCT
cana-2799	200	10	𝐿	𝐿	NOUN
cana-2799	200	11	)	)	PUNCT
cana-2799	200	12	=	=	PUNCT
cana-2799	200	13	−	−	PROPN
cana-2799	200	14	𝑅	𝑅	PROPN
cana-2799	200	15	𝐿	𝐿	PROPN
cana-2799	200	16	δ	δ	PROPN
cana-2799	200	17	−𝜈𝐼(𝜅	−𝜈𝐼(𝜅	NOUN
cana-2799	200	18	)	)	PUNCT
cana-2799	200	19	by	by	ADP
cana-2799	200	20	applying	apply	VERB
cana-2799	200	21	fractional	fractional	ADJ
cana-2799	200	22	order	order	NOUN
cana-2799	200	23	delta	delta	NOUN
cana-2799	200	24	integration	integration	NOUN
cana-2799	200	25	(	(	PUNCT
cana-2799	200	26	1	1	NUM
cana-2799	200	27	)	)	PUNCT
cana-2799	200	28	,	,	PUNCT
cana-2799	200	29	we	we	PRON
cana-2799	200	30	obtain	obtain	VERB
cana-2799	200	31	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	200	32	)	)	PUNCT
cana-2799	200	33	=	=	SYM
cana-2799	200	34	𝑅	𝑅	PROPN
cana-2799	200	35	𝐿γ(𝜈	𝐿γ(𝜈	NOUN
cana-2799	200	36	)	)	PUNCT
cana-2799	200	37	∑𝜅−𝜈	∑𝜅−𝜈	PROPN
cana-2799	200	38	𝑠=0	𝑠=0	PUNCT
cana-2799	200	39	γ(𝜅−𝑠	γ(𝜅−𝑠	PROPN
cana-2799	200	40	)	)	PUNCT
cana-2799	200	41	γ(𝜅−𝑠−(𝜈−1	γ(𝜅−𝑠−(𝜈−1	PROPN
cana-2799	200	42	)	)	PUNCT
cana-2799	200	43	)	)	PUNCT
cana-2799	201	1	𝐼(𝑠	𝐼(𝑠	NUM
cana-2799	201	2	)	)	PUNCT
cana-2799	201	3	,	,	PUNCT
cana-2799	201	4	(	(	PUNCT
cana-2799	201	5	13	13	X
cana-2799	201	6	)	)	PUNCT
cana-2799	201	7	the	the	DET
cana-2799	201	8	solution	solution	NOUN
cana-2799	201	9	(	(	PUNCT
cana-2799	201	10	11	11	NUM
cana-2799	201	11	)	)	PUNCT
cana-2799	201	12	and	and	CCONJ
cana-2799	201	13	(	(	PUNCT
cana-2799	201	14	13	13	NUM
cana-2799	201	15	)	)	PUNCT
cana-2799	201	16	are	be	AUX
cana-2799	201	17	summation	summation	NOUN
cana-2799	201	18	forms	form	NOUN
cana-2799	201	19	.	.	PUNCT
cana-2799	202	1	through	through	ADP
cana-2799	202	2	our	our	PRON
cana-2799	202	3	research	research	NOUN
cana-2799	202	4	,	,	PUNCT
cana-2799	202	5	we	we	PRON
cana-2799	202	6	identifies	identify	VERB
cana-2799	202	7	that	that	SCONJ
cana-2799	202	8	these	these	DET
cana-2799	202	9	fractional	fractional	ADJ
cana-2799	202	10	difference	difference	NOUN
cana-2799	202	11	equations	equation	NOUN
cana-2799	202	12	have	have	VERB
cana-2799	202	13	exact	exact	ADJ
cana-2799	202	14	type	type	NOUN
cana-2799	202	15	solutions	solution	NOUN
cana-2799	202	16	,	,	PUNCT
cana-2799	202	17	when	when	SCONJ
cana-2799	202	18	the	the	DET
cana-2799	202	19	initial	initial	ADJ
cana-2799	202	20	time	time	NOUN
cana-2799	202	21	a	a	PRON
cana-2799	202	22	is	be	AUX
cana-2799	202	23	taken	take	VERB
cana-2799	202	24	as	as	ADP
cana-2799	202	25	zero	zero	NUM
cana-2799	202	26	.	.	PUNCT
cana-2799	203	1	we	we	PRON
cana-2799	203	2	obtain	obtain	VERB
cana-2799	203	3	exact	exact	ADJ
cana-2799	203	4	solution	solution	NOUN
cana-2799	203	5	for	for	ADP
cana-2799	203	6	the	the	DET
cana-2799	203	7	equations	equation	NOUN
cana-2799	203	8	(	(	PUNCT
cana-2799	203	9	9	9	NUM
cana-2799	203	10	)	)	PUNCT
cana-2799	203	11	and	and	CCONJ
cana-2799	203	12	(	(	PUNCT
cana-2799	203	13	10	10	NUM
cana-2799	203	14	)	)	PUNCT
cana-2799	203	15	using	use	VERB
cana-2799	203	16	our	our	PRON
cana-2799	203	17	newly	newly	ADV
cana-2799	203	18	defined	define	VERB
cana-2799	203	19	extorial	extorial	ADJ
cana-2799	203	20	functions	function	NOUN
cana-2799	203	21	.	.	PUNCT
cana-2799	204	1	4	4	X
cana-2799	204	2	.	.	NOUN
cana-2799	204	3	extorial	extorial	ADJ
cana-2799	204	4	type	type	NOUN
cana-2799	204	5	solution	solution	NOUN
cana-2799	204	6	of	of	ADP
cana-2799	204	7	rl	rl	NOUN
cana-2799	204	8	circuit	circuit	NOUN
cana-2799	204	9	in	in	ADP
cana-2799	204	10	this	this	DET
cana-2799	204	11	section	section	NOUN
cana-2799	204	12	,	,	PUNCT
cana-2799	204	13	we	we	PRON
cana-2799	204	14	find	find	VERB
cana-2799	204	15	solution	solution	NOUN
cana-2799	204	16	of	of	ADP
cana-2799	204	17	equation	equation	NOUN
cana-2799	204	18	(	(	PUNCT
cana-2799	204	19	9	9	NUM
cana-2799	204	20	)	)	PUNCT
cana-2799	204	21	after	after	ADP
cana-2799	204	22	arriving	arrive	VERB
cana-2799	204	23	at	at	ADP
cana-2799	204	24	some	some	DET
cana-2799	204	25	basic	basic	ADJ
cana-2799	204	26	results	result	NOUN
cana-2799	204	27	of	of	ADP
cana-2799	204	28	extorial	extorial	ADJ
cana-2799	204	29	functions	function	NOUN
cana-2799	204	30	.	.	PUNCT
cana-2799	205	1	this	this	DET
cana-2799	205	2	extorial	extorial	ADJ
cana-2799	205	3	function	function	NOUN
cana-2799	205	4	is	be	AUX
cana-2799	205	5	easily	easily	ADV
cana-2799	205	6	obtained	obtain	VERB
cana-2799	205	7	by	by	ADP
cana-2799	205	8	replacing	replace	VERB
cana-2799	205	9	polynomial	polynomial	ADJ
cana-2799	205	10	𝜅𝑛	𝜅𝑛	NOUN
cana-2799	205	11	into	into	ADP
cana-2799	205	12	factorial	factorial	ADJ
cana-2799	205	13	polynomial	polynomial	NOUN
cana-2799	205	14	in	in	ADP
cana-2799	205	15	the	the	DET
cana-2799	205	16	expansion	expansion	NOUN
cana-2799	205	17	of	of	ADP
cana-2799	205	18	exponential	exponential	ADJ
cana-2799	205	19	function	function	NOUN
cana-2799	205	20	𝑒𝜅.	𝑒𝜅.	NOUN
cana-2799	205	21	this	this	DET
cana-2799	205	22	function	function	NOUN
cana-2799	205	23	is	be	AUX
cana-2799	205	24	useful	useful	ADJ
cana-2799	205	25	to	to	PART
cana-2799	205	26	arrive	arrive	VERB
cana-2799	205	27	at	at	ADP
cana-2799	205	28	solutions	solution	NOUN
cana-2799	205	29	for	for	ADP
cana-2799	205	30	fractional	fractional	ADJ
cana-2799	205	31	difference	difference	NOUN
cana-2799	205	32	equation	equation	NOUN
cana-2799	205	33	.	.	PUNCT
cana-2799	206	1	consider	consider	VERB
cana-2799	206	2	the	the	DET
cana-2799	206	3	extorial	extorial	ADJ
cana-2799	206	4	function	function	NOUN
cana-2799	206	5	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	206	6	)	)	PUNCT
cana-2799	206	7	)	)	PUNCT
cana-2799	206	8	is	be	AUX
cana-2799	206	9	defined	define	VERB
cana-2799	206	10	by	by	ADP
cana-2799	206	11	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	206	12	)	)	PUNCT
cana-2799	206	13	)	)	PUNCT
cana-2799	207	1	=	=	SYM
cana-2799	207	2	1	1	NUM
cana-2799	207	3	+	+	CCONJ
cana-2799	207	4	(	(	PUNCT
cana-2799	207	5	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	207	6	(	(	PUNCT
cana-2799	207	7	1	1	NUM
cana-2799	207	8	)	)	PUNCT
cana-2799	207	9	1	1	NUM
cana-2799	207	10	!	!	PUNCT
cana-2799	208	1	+	+	CCONJ
cana-2799	208	2	(	(	PUNCT
cana-2799	208	3	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	208	4	(	(	PUNCT
cana-2799	208	5	2	2	NUM
cana-2799	208	6	)	)	PUNCT
cana-2799	208	7	2	2	NUM
cana-2799	208	8	!	!	PUNCT
cana-2799	209	1	+	+	CCONJ
cana-2799	209	2	⋯	⋯	VERB
cana-2799	209	3	+	+	CCONJ
cana-2799	209	4	∞	∞	NUM
cana-2799	209	5	=	=	SYM
cana-2799	209	6	∑∞	∑∞	NOUN
cana-2799	209	7	𝑟=0	𝑟=0	PUNCT
cana-2799	209	8	(	(	PUNCT
cana-2799	209	9	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	209	10	(	(	PUNCT
cana-2799	209	11	𝑟	𝑟	NOUN
cana-2799	209	12	)	)	PUNCT
cana-2799	209	13	𝑟	𝑟	NOUN
cana-2799	209	14	!	!	NOUN
cana-2799	209	15	,	,	PUNCT
cana-2799	209	16	(	(	PUNCT
cana-2799	209	17	14	14	NUM
cana-2799	209	18	)	)	PUNCT
cana-2799	209	19	where	where	SCONJ
cana-2799	209	20	(	(	PUNCT
cana-2799	209	21	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	209	22	(	(	PUNCT
cana-2799	209	23	𝑟	𝑟	NOUN
cana-2799	209	24	)	)	PUNCT
cana-2799	209	25	=	=	SYM
cana-2799	210	1	(	(	PUNCT
cana-2799	210	2	𝑚𝜅)(𝑚𝜅	𝑚𝜅)(𝑚𝜅	PUNCT
cana-2799	210	3	−	−	NOUN
cana-2799	210	4	𝑚	𝑚	NOUN
cana-2799	210	5	)	)	PUNCT
cana-2799	210	6	⋯	⋯	PROPN
cana-2799	210	7	(	(	PUNCT
cana-2799	210	8	𝑚𝜅	𝑚𝜅	PROPN
cana-2799	210	9	−	−	PROPN
cana-2799	210	10	(	(	PUNCT
cana-2799	210	11	𝑟	𝑟	NOUN
cana-2799	210	12	−	−	PROPN
cana-2799	210	13	1)𝑚	1)𝑚	NUM
cana-2799	210	14	)	)	PUNCT
cana-2799	210	15	for	for	ADP
cana-2799	210	16	positive	positive	ADJ
cana-2799	210	17	integer	integer	NOUN
cana-2799	210	18	𝑟	𝑟	NOUN
cana-2799	210	19	,	,	PUNCT
cana-2799	210	20	is	be	AUX
cana-2799	210	21	a	a	DET
cana-2799	210	22	falling	fall	VERB
cana-2799	210	23	polynomial	polynomial	ADJ
cana-2799	210	24	factorial	factorial	NOUN
cana-2799	210	25	.	.	PUNCT
cana-2799	211	1	in	in	ADP
cana-2799	211	2	general	general	ADJ
cana-2799	211	3	,	,	PUNCT
cana-2799	211	4	for	for	ADP
cana-2799	211	5	real	real	ADJ
cana-2799	211	6	index	index	NOUN
cana-2799	211	7	𝜈	𝜈	NOUN
cana-2799	211	8	,	,	PUNCT
cana-2799	211	9	we	we	PRON
cana-2799	211	10	have	have	VERB
cana-2799	211	11	communications	communication	NOUN
cana-2799	211	12	on	on	ADP
cana-2799	211	13	applied	apply	VERB
cana-2799	211	14	nonlinear	nonlinear	ADJ
cana-2799	211	15	analysis	analysis	NOUN
cana-2799	211	16	issn	issn	NOUN
cana-2799	211	17	:	:	PUNCT
cana-2799	211	18	1074	1074	NUM
cana-2799	211	19	-	-	PUNCT
cana-2799	211	20	133x	133x	NUM
cana-2799	211	21	vol	vol	NOUN
cana-2799	211	22	32	32	NUM
cana-2799	212	1	no	no	NOUN
cana-2799	212	2	.	.	PUNCT
cana-2799	213	1	4s	4s	NUM
cana-2799	213	2	(	(	PUNCT
cana-2799	213	3	2025	2025	NUM
cana-2799	213	4	)	)	PUNCT
cana-2799	213	5	264	264	NUM
cana-2799	213	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	213	7	𝑒(𝜈)((𝑚𝜅)(𝑚	𝑒(𝜈)((𝑚𝜅)(𝑚	PROPN
cana-2799	213	8	)	)	PUNCT
cana-2799	213	9	)	)	PUNCT
cana-2799	214	1	=	=	SYM
cana-2799	214	2	1	1	NUM
cana-2799	214	3	+	+	CCONJ
cana-2799	214	4	(	(	PUNCT
cana-2799	214	5	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	214	6	(	(	PUNCT
cana-2799	214	7	𝜈	𝜈	X
cana-2799	214	8	)	)	PUNCT
cana-2799	214	9	1	1	NUM
cana-2799	214	10	!	!	PUNCT
cana-2799	215	1	+	+	CCONJ
cana-2799	215	2	(	(	PUNCT
cana-2799	215	3	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	215	4	(	(	PUNCT
cana-2799	215	5	2𝜈	2𝜈	NOUN
cana-2799	215	6	)	)	PUNCT
cana-2799	215	7	2	2	NUM
cana-2799	215	8	!	!	PUNCT
cana-2799	216	1	+	+	CCONJ
cana-2799	216	2	⋯	⋯	VERB
cana-2799	216	3	+	+	CCONJ
cana-2799	216	4	∞	∞	NUM
cana-2799	216	5	=	=	SYM
cana-2799	216	6	∑∞	∑∞	NOUN
cana-2799	216	7	𝑟=0	𝑟=0	PUNCT
cana-2799	216	8	(	(	PUNCT
cana-2799	216	9	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	216	10	(	(	PUNCT
cana-2799	216	11	𝑟𝜈	𝑟𝜈	PROPN
cana-2799	216	12	)	)	PUNCT
cana-2799	216	13	𝑟	𝑟	NOUN
cana-2799	216	14	!	!	NOUN
cana-2799	216	15	,	,	PUNCT
cana-2799	216	16	(	(	PUNCT
cana-2799	216	17	15	15	NUM
cana-2799	216	18	)	)	PUNCT
cana-2799	216	19	where	where	SCONJ
cana-2799	216	20	(	(	PUNCT
cana-2799	216	21	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	216	22	(	(	PUNCT
cana-2799	216	23	𝑟𝜈	𝑟𝜈	PROPN
cana-2799	216	24	)	)	PUNCT
cana-2799	216	25	=	=	SYM
cana-2799	216	26	(	(	PUNCT
cana-2799	216	27	𝑚)(𝑟𝜈	𝑚)(𝑟𝜈	NOUN
cana-2799	216	28	)	)	PUNCT
cana-2799	216	29	γ(𝜅+1	γ(𝜅+1	PRON
cana-2799	216	30	)	)	PUNCT
cana-2799	217	1	γ(𝐾+1−𝑟𝜈	γ(𝐾+1−𝑟𝜈	ADJ
cana-2799	217	2	)	)	PUNCT
cana-2799	217	3	and	and	CCONJ
cana-2799	217	4	γ	γ	X
cana-2799	217	5	(	(	PUNCT
cana-2799	217	6	.	.	PUNCT
cana-2799	217	7	)	)	PUNCT
cana-2799	217	8	is	be	AUX
cana-2799	217	9	the	the	DET
cana-2799	217	10	gamma	gamma	PROPN
cana-2799	217	11	function	function	NOUN
cana-2799	217	12	.	.	PUNCT
cana-2799	218	1	lemma	lemma	PROPN
cana-2799	218	2	4.1	4.1	NUM
cana-2799	218	3	.	.	PUNCT
cana-2799	219	1	if	if	SCONJ
cana-2799	219	2	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	219	3	)	)	PUNCT
cana-2799	219	4	)	)	PUNCT
cana-2799	219	5	is	be	AUX
cana-2799	219	6	an	an	DET
cana-2799	219	7	extorial	extorial	ADJ
cana-2799	219	8	function	function	NOUN
cana-2799	219	9	,	,	PUNCT
cana-2799	219	10	then	then	ADV
cana-2799	219	11	we	we	PRON
cana-2799	219	12	have	have	VERB
cana-2799	219	13	δ𝑒1((𝑚𝜅)(𝑚	δ𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	219	14	)	)	PUNCT
cana-2799	219	15	)	)	PUNCT
cana-2799	220	1	=	=	SYM
cana-2799	220	2	(	(	PUNCT
cana-2799	220	3	𝑚)𝑒1((𝑚𝜅)(𝑚	𝑚)𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	220	4	)	)	PUNCT
cana-2799	220	5	)	)	PUNCT
cana-2799	220	6	.	.	PUNCT
cana-2799	221	1	(	(	PUNCT
cana-2799	221	2	16	16	X
cana-2799	221	3	)	)	PUNCT
cana-2799	221	4	proof	proof	NOUN
cana-2799	221	5	.	.	PUNCT
cana-2799	222	1	applying	apply	VERB
cana-2799	222	2	δ	δ	PROPN
cana-2799	222	3	on	on	ADP
cana-2799	222	4	the	the	DET
cana-2799	222	5	extorial	extorial	ADJ
cana-2799	222	6	function	function	NOUN
cana-2799	222	7	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	222	8	)	)	PUNCT
cana-2799	222	9	)	)	PUNCT
cana-2799	222	10	,	,	PUNCT
cana-2799	222	11	we	we	PRON
cana-2799	222	12	arrive	arrive	VERB
cana-2799	222	13	at	at	ADP
cana-2799	222	14	δ𝑒1((𝑚𝜅)(𝑚	δ𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	222	15	)	)	PUNCT
cana-2799	222	16	)	)	PUNCT
cana-2799	223	1	=	=	SYM
cana-2799	223	2	δ(1	δ(1	NOUN
cana-2799	223	3	)	)	PUNCT
cana-2799	224	1	+	+	NUM
cana-2799	224	2	δ	δ	PROPN
cana-2799	224	3	(	(	PUNCT
cana-2799	224	4	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	224	5	(	(	PUNCT
cana-2799	224	6	1	1	NUM
cana-2799	224	7	)	)	PUNCT
cana-2799	224	8	1	1	NUM
cana-2799	224	9	!	!	PUNCT
cana-2799	225	1	+	+	CCONJ
cana-2799	225	2	δ	δ	X
cana-2799	225	3	(	(	PUNCT
cana-2799	225	4	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	225	5	(	(	PUNCT
cana-2799	225	6	2	2	NUM
cana-2799	225	7	)	)	PUNCT
cana-2799	225	8	2	2	NUM
cana-2799	225	9	!	!	PUNCT
cana-2799	226	1	+	+	CCONJ
cana-2799	226	2	⋯	⋯	VERB
cana-2799	226	3	+	+	CCONJ
cana-2799	226	4	∞	∞	NUM
cana-2799	226	5	=	=	SYM
cana-2799	226	6	0	0	PUNCT
cana-2799	227	1	+	+	CCONJ
cana-2799	227	2	(	(	PUNCT
cana-2799	227	3	𝑚(𝜅+1))𝑚	𝑚(𝜅+1))𝑚	NOUN
cana-2799	227	4	(	(	PUNCT
cana-2799	227	5	1	1	NUM
cana-2799	227	6	)	)	PUNCT
cana-2799	227	7	1	1	NUM
cana-2799	227	8	!	!	PUNCT
cana-2799	228	1	+	+	CCONJ
cana-2799	228	2	(	(	PUNCT
cana-2799	228	3	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	228	4	(	(	PUNCT
cana-2799	228	5	1	1	NUM
cana-2799	228	6	)	)	PUNCT
cana-2799	228	7	1	1	NUM
cana-2799	228	8	!	!	PUNCT
cana-2799	229	1	+	+	CCONJ
cana-2799	229	2	1	1	NUM
cana-2799	229	3	2	2	NUM
cana-2799	229	4	!	!	PUNCT
cana-2799	230	1	[	[	X
cana-2799	230	2	(	(	PUNCT
cana-2799	230	3	𝑚(𝜅	𝑚(𝜅	PROPN
cana-2799	230	4	+	+	NUM
cana-2799	230	5	1))𝑚	1))𝑚	NUM
cana-2799	230	6	(	(	PUNCT
cana-2799	230	7	2	2	NUM
cana-2799	230	8	)	)	PUNCT
cana-2799	230	9	−	−	PROPN
cana-2799	230	10	(	(	PUNCT
cana-2799	230	11	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	230	12	(	(	PUNCT
cana-2799	230	13	2	2	NUM
cana-2799	230	14	)	)	PUNCT
cana-2799	230	15	]	]	PUNCT
cana-2799	231	1	+	+	CCONJ
cana-2799	231	2	⋯	⋯	VERB
cana-2799	231	3	+	+	CCONJ
cana-2799	231	4	∞	∞	NUM
cana-2799	231	5	=	=	SYM
cana-2799	231	6	(	(	PUNCT
cana-2799	231	7	𝑚	𝑚	NOUN
cana-2799	231	8	)	)	PUNCT
cana-2799	231	9	1	1	NUM
cana-2799	231	10	!	!	PUNCT
cana-2799	232	1	+	+	CCONJ
cana-2799	232	2	1	1	NUM
cana-2799	232	3	2	2	NUM
cana-2799	232	4	!	!	PUNCT
cana-2799	233	1	[	[	X
cana-2799	233	2	(	(	PUNCT
cana-2799	233	3	𝑚𝜅	𝑚𝜅	NOUN
cana-2799	233	4	+	+	X
cana-2799	233	5	𝑚)(𝑚𝜅	𝑚)(𝑚𝜅	NUM
cana-2799	233	6	)	)	PUNCT
cana-2799	233	7	−	−	PROPN
cana-2799	234	1	(	(	PUNCT
cana-2799	234	2	𝑚𝜅)(𝑚𝜅	𝑚𝜅)(𝑚𝜅	PUNCT
cana-2799	234	3	−	−	NOUN
cana-2799	234	4	𝑚	𝑚	NOUN
cana-2799	234	5	)	)	PUNCT
cana-2799	234	6	]	]	PUNCT
cana-2799	235	1	+	+	CCONJ
cana-2799	235	2	⋯	⋯	VERB
cana-2799	235	3	+	+	CCONJ
cana-2799	235	4	∞	∞	NUM
cana-2799	235	5	=	=	SYM
cana-2799	235	6	𝑚	𝑚	PROPN
cana-2799	235	7	+	+	PUNCT
cana-2799	235	8	2𝑚	2𝑚	NUM
cana-2799	235	9	2	2	NUM
cana-2799	235	10	!	!	PUNCT
cana-2799	236	1	(	(	PUNCT
cana-2799	236	2	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	236	3	(	(	PUNCT
cana-2799	236	4	1	1	NUM
cana-2799	236	5	)	)	PUNCT
cana-2799	236	6	+	+	NOUN
cana-2799	236	7	3𝑚	3𝑚	NUM
cana-2799	236	8	3	3	NUM
cana-2799	236	9	!	!	PUNCT
cana-2799	237	1	(	(	PUNCT
cana-2799	237	2	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	237	3	(	(	PUNCT
cana-2799	237	4	2	2	NUM
cana-2799	237	5	)	)	PUNCT
cana-2799	237	6	+	+	CCONJ
cana-2799	237	7	⋯	⋯	VERB
cana-2799	237	8	+	+	CCONJ
cana-2799	237	9	∞	∞	NUM
cana-2799	237	10	=	=	SYM
cana-2799	237	11	𝑚[1	𝑚[1	X
cana-2799	238	1	+	+	CCONJ
cana-2799	238	2	(	(	PUNCT
cana-2799	238	3	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	238	4	(	(	PUNCT
cana-2799	238	5	𝜅	𝜅	NOUN
cana-2799	238	6	)	)	PUNCT
cana-2799	238	7	1	1	NUM
cana-2799	238	8	!	!	PUNCT
cana-2799	239	1	+	+	CCONJ
cana-2799	239	2	(	(	PUNCT
cana-2799	239	3	𝑚𝜅)𝑚	𝑚𝜅)𝑚	PROPN
cana-2799	239	4	(	(	PUNCT
cana-2799	239	5	2	2	NUM
cana-2799	239	6	)	)	PUNCT
cana-2799	239	7	2	2	NUM
cana-2799	239	8	!	!	PUNCT
cana-2799	240	1	+	+	CCONJ
cana-2799	240	2	⋯	⋯	X
cana-2799	240	3	+	+	CCONJ
cana-2799	240	4	∞	∞	PROPN
cana-2799	240	5	]	]	PUNCT
cana-2799	240	6	δ𝑒(𝑚𝜅)𝑚	δ𝑒(𝑚𝜅)𝑚	NOUN
cana-2799	240	7	=	=	SYM
cana-2799	240	8	(	(	PUNCT
cana-2799	240	9	𝑚)𝑒1((𝑚𝜅)(𝑚	𝑚)𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	240	10	)	)	PUNCT
cana-2799	240	11	)	)	PUNCT
cana-2799	240	12	.	.	PUNCT
cana-2799	241	1	lemma	lemma	PROPN
cana-2799	241	2	4.2	4.2	NUM
cana-2799	241	3	.	.	PUNCT
cana-2799	242	1	the	the	DET
cana-2799	242	2	extorial	extorial	ADJ
cana-2799	242	3	function	function	NOUN
cana-2799	242	4	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	242	5	)	)	PUNCT
cana-2799	242	6	=	=	SYM
cana-2799	242	7	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	X
cana-2799	242	8	)	)	PUNCT
cana-2799	242	9	)	)	PUNCT
cana-2799	242	10	is	be	AUX
cana-2799	242	11	a	a	DET
cana-2799	242	12	solution	solution	NOUN
cana-2799	242	13	of	of	ADP
cana-2799	242	14	equation	equation	NOUN
cana-2799	242	15	(	(	PUNCT
cana-2799	242	16	𝐴δ	𝐴δ	PROPN
cana-2799	242	17	2	2	NUM
cana-2799	242	18	+	+	CCONJ
cana-2799	242	19	𝐵δ	𝐵δ	PROPN
cana-2799	242	20	+	+	NUM
cana-2799	242	21	𝐶)𝑢(𝜅	𝐶)𝑢(𝜅	NOUN
cana-2799	242	22	)	)	PUNCT
cana-2799	242	23	=	=	SYM
cana-2799	242	24	0	0	NUM
cana-2799	242	25	,	,	PUNCT
cana-2799	242	26	(	(	PUNCT
cana-2799	242	27	17	17	NUM
cana-2799	242	28	)	)	PUNCT
cana-2799	242	29	if	if	SCONJ
cana-2799	242	30	𝑚	𝑚	PROPN
cana-2799	242	31	is	be	AUX
cana-2799	242	32	a	a	DET
cana-2799	242	33	root	root	NOUN
cana-2799	242	34	of	of	ADP
cana-2799	242	35	the	the	DET
cana-2799	242	36	auxiliary	auxiliary	ADJ
cana-2799	242	37	equation	equation	NOUN
cana-2799	242	38	𝐴𝑚2	𝐴𝑚2	VERB
cana-2799	243	1	+	+	X
cana-2799	244	1	𝐵𝑚	𝐵𝑚	PROPN
cana-2799	244	2	+	+	CCONJ
cana-2799	244	3	𝐶	𝐶	PROPN
cana-2799	244	4	=	=	SYM
cana-2799	244	5	0	0	PROPN
cana-2799	244	6	.	.	PUNCT
cana-2799	244	7	proof	proof	NOUN
cana-2799	244	8	.	.	PUNCT
cana-2799	245	1	if	if	SCONJ
cana-2799	245	2	we	we	PRON
cana-2799	245	3	try	try	VERB
cana-2799	245	4	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	245	5	)	)	PUNCT
cana-2799	245	6	=	=	SYM
cana-2799	245	7	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	245	8	)	)	PUNCT
cana-2799	245	9	)	)	PUNCT
cana-2799	245	10	as	as	ADP
cana-2799	245	11	a	a	DET
cana-2799	245	12	solution	solution	NOUN
cana-2799	245	13	of	of	ADP
cana-2799	245	14	equation(17	equation(17	NOUN
cana-2799	245	15	)	)	PUNCT
cana-2799	245	16	,	,	PUNCT
cana-2799	245	17	then	then	ADV
cana-2799	245	18	it	it	PRON
cana-2799	245	19	should	should	AUX
cana-2799	245	20	satisfy	satisfy	VERB
cana-2799	245	21	the	the	DET
cana-2799	245	22	equation	equation	NOUN
cana-2799	245	23	𝐴δ	𝐴δ	PROPN
cana-2799	245	24	2𝑒1((𝑚𝜅)(𝑚	2𝑒1((𝑚𝜅)(𝑚	NUM
cana-2799	245	25	)	)	PUNCT
cana-2799	245	26	)	)	PUNCT
cana-2799	246	1	+	+	CCONJ
cana-2799	246	2	𝐵δ𝑒1((𝑚𝜅)(𝑚	𝐵δ𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	246	3	)	)	PUNCT
cana-2799	246	4	)	)	PUNCT
cana-2799	247	1	+	+	CCONJ
cana-2799	247	2	𝐶𝑒1((𝑚𝜅)(𝑚	𝐶𝑒1((𝑚𝜅)(𝑚	ADJ
cana-2799	247	3	)	)	PUNCT
cana-2799	247	4	)	)	PUNCT
cana-2799	248	1	=	=	PUNCT
cana-2799	248	2	0	0	X
cana-2799	248	3	.	.	PUNCT
cana-2799	249	1	(	(	PUNCT
cana-2799	249	2	18	18	NUM
cana-2799	249	3	)	)	PUNCT
cana-2799	249	4	by	by	ADP
cana-2799	249	5	linear	linear	ADJ
cana-2799	249	6	property	property	NOUN
cana-2799	249	7	of	of	ADP
cana-2799	249	8	δ	δ	PROPN
cana-2799	249	9	and	and	CCONJ
cana-2799	249	10	the	the	DET
cana-2799	249	11	expansion	expansion	NOUN
cana-2799	249	12	of	of	ADP
cana-2799	249	13	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	249	14	)	)	PUNCT
cana-2799	249	15	)	)	PUNCT
cana-2799	249	16	,	,	PUNCT
cana-2799	249	17	we	we	PRON
cana-2799	249	18	arrive	arrive	VERB
cana-2799	249	19	at	at	ADP
cana-2799	249	20	δℓ𝑒1((𝑚𝜅)(𝑚	δℓ𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	249	21	)	)	PUNCT
cana-2799	249	22	)	)	PUNCT
cana-2799	250	1	=	=	PUNCT
cana-2799	250	2	0	0	PUNCT
cana-2799	251	1	+	+	CCONJ
cana-2799	251	2	(	(	PUNCT
cana-2799	251	3	𝑚	𝑚	NOUN
cana-2799	251	4	)	)	PUNCT
cana-2799	251	5	(	(	PUNCT
cana-2799	251	6	𝑚𝜅)(𝑚	𝑚𝜅)(𝑚	PROPN
cana-2799	251	7	)	)	PUNCT
cana-2799	251	8	(	(	PUNCT
cana-2799	251	9	0	0	NUM
cana-2799	251	10	)	)	PUNCT
cana-2799	251	11	1	1	NUM
cana-2799	251	12	!	!	PUNCT
cana-2799	252	1	+	+	CCONJ
cana-2799	252	2	2𝑚(𝑚𝜅)(𝑚	2𝑚(𝑚𝜅)(𝑚	NUM
cana-2799	252	3	)	)	PUNCT
cana-2799	252	4	(	(	PUNCT
cana-2799	252	5	1	1	NUM
cana-2799	252	6	)	)	PUNCT
cana-2799	252	7	2	2	NUM
cana-2799	252	8	!	!	PUNCT
cana-2799	253	1	+	+	NUM
cana-2799	253	2	3𝑚(𝑚𝜅)(𝑚	3𝑚(𝑚𝜅)(𝑚	NUM
cana-2799	253	3	)	)	PUNCT
cana-2799	253	4	(	(	PUNCT
cana-2799	253	5	2	2	NUM
cana-2799	253	6	)	)	PUNCT
cana-2799	253	7	3	3	NUM
cana-2799	253	8	!	!	PUNCT
cana-2799	254	1	+	+	X
cana-2799	254	2	.	.	PUNCT
cana-2799	254	3	..	..	PUNCT
cana-2799	255	1	i.e	i.e	X
cana-2799	255	2	,	,	PUNCT
cana-2799	255	3	δℓ𝑒1((𝑚𝜅)(𝑚	δℓ𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	255	4	)	)	PUNCT
cana-2799	255	5	)	)	PUNCT
cana-2799	256	1	=	=	SYM
cana-2799	256	2	𝑚[1	𝑚[1	X
cana-2799	257	1	+	+	CCONJ
cana-2799	257	2	(	(	PUNCT
cana-2799	257	3	𝑚𝜅)(𝑚	𝑚𝜅)(𝑚	PROPN
cana-2799	257	4	)	)	PUNCT
cana-2799	257	5	(	(	PUNCT
cana-2799	257	6	1	1	X
cana-2799	257	7	)	)	PUNCT
cana-2799	257	8	1	1	NUM
cana-2799	257	9	!	!	PUNCT
cana-2799	258	1	+	+	CCONJ
cana-2799	258	2	(	(	PUNCT
cana-2799	258	3	𝑚𝜅)(𝑚	𝑚𝜅)(𝑚	PROPN
cana-2799	258	4	)	)	PUNCT
cana-2799	258	5	(	(	PUNCT
cana-2799	258	6	2	2	NUM
cana-2799	258	7	)	)	PUNCT
cana-2799	258	8	2	2	NUM
cana-2799	258	9	!	!	PUNCT
cana-2799	259	1	+	+	NOUN
cana-2799	259	2	.	.	PUNCT
cana-2799	259	3	.	.	PUNCT
cana-2799	259	4	.	.	PUNCT
cana-2799	260	1	]	]	PUNCT
cana-2799	261	1	=	=	PUNCT
cana-2799	261	2	𝑚𝑒1((𝑚𝜅)(𝑚	𝑚𝑒1((𝑚𝜅)(𝑚	PROPN
cana-2799	261	3	)	)	PUNCT
cana-2799	261	4	)	)	PUNCT
cana-2799	261	5	,	,	PUNCT
cana-2799	261	6	which	which	PRON
cana-2799	261	7	yields	yield	VERB
cana-2799	261	8	δℓ	δℓ	ADP
cana-2799	261	9	2𝑒1((𝑚𝜅)(𝑚	2𝑒1((𝑚𝜅)(𝑚	NUM
cana-2799	261	10	)	)	PUNCT
cana-2799	261	11	)	)	PUNCT
cana-2799	262	1	=	=	SYM
cana-2799	262	2	(	(	PUNCT
cana-2799	262	3	𝑚)δℓ𝑒1((𝑚𝜅)(𝑚	𝑚)δℓ𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	262	4	)	)	PUNCT
cana-2799	262	5	)	)	PUNCT
cana-2799	263	1	=	=	SYM
cana-2799	263	2	(	(	PUNCT
cana-2799	263	3	𝑚)2𝑒1((𝑚𝜅)(𝑚	𝑚)2𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	263	4	)	)	PUNCT
cana-2799	263	5	)	)	PUNCT
cana-2799	263	6	.	.	PUNCT
cana-2799	264	1	applying	apply	VERB
cana-2799	264	2	the	the	DET
cana-2799	264	3	values	value	NOUN
cana-2799	264	4	of	of	ADP
cana-2799	264	5	δ𝑒1((𝑚𝜅)(𝑚	δ𝑒1((𝑚𝜅)(𝑚	NOUN
cana-2799	264	6	)	)	PUNCT
cana-2799	264	7	)	)	PUNCT
cana-2799	264	8	and	and	CCONJ
cana-2799	264	9	δℓ	δℓ	ADP
cana-2799	264	10	2𝑒1((𝑚𝜅)(𝑚	2𝑒1((𝑚𝜅)(𝑚	NUM
cana-2799	264	11	)	)	PUNCT
cana-2799	264	12	)	)	PUNCT
cana-2799	264	13	in	in	ADP
cana-2799	264	14	(	(	PUNCT
cana-2799	264	15	18	18	NUM
cana-2799	264	16	)	)	PUNCT
cana-2799	264	17	,	,	PUNCT
cana-2799	264	18	we	we	PRON
cana-2799	264	19	obtain	obtain	VERB
cana-2799	264	20	(	(	PUNCT
cana-2799	264	21	𝐴𝑚2	𝐴𝑚2	X
cana-2799	264	22	+	+	X
cana-2799	264	23	𝐵𝑚	𝐵𝑚	NOUN
cana-2799	264	24	+	+	CCONJ
cana-2799	264	25	𝐶)𝑒1((𝑚𝜅)(𝑚	𝐶)𝑒1((𝑚𝜅)(𝑚	NUM
cana-2799	264	26	)	)	PUNCT
cana-2799	264	27	)	)	PUNCT
cana-2799	265	1	=	=	SYM
cana-2799	265	2	0	0	NUM
cana-2799	265	3	,	,	PUNCT
cana-2799	265	4	since	since	SCONJ
cana-2799	265	5	𝑒1((𝑚𝜅)𝑚	𝑒1((𝑚𝜅)𝑚	ADJ
cana-2799	265	6	)	)	PUNCT
cana-2799	265	7	≠	≠	PROPN
cana-2799	265	8	0	0	NUM
cana-2799	265	9	,	,	PUNCT
cana-2799	265	10	we	we	PRON
cana-2799	265	11	get	get	VERB
cana-2799	265	12	𝐴𝑚2	𝐴𝑚2	ADJ
cana-2799	266	1	+	+	CCONJ
cana-2799	267	1	𝐵𝑚	𝐵𝑚	PROPN
cana-2799	267	2	+	+	CCONJ
cana-2799	267	3	𝐶	𝐶	PROPN
cana-2799	267	4	=	=	SYM
cana-2799	267	5	0	0	PROPN
cana-2799	267	6	.	.	PUNCT
cana-2799	268	1	(	(	PUNCT
cana-2799	268	2	19	19	NUM
cana-2799	268	3	)	)	PUNCT
cana-2799	268	4	communications	communication	NOUN
cana-2799	268	5	on	on	ADP
cana-2799	268	6	applied	apply	VERB
cana-2799	268	7	nonlinear	nonlinear	ADJ
cana-2799	268	8	analysis	analysis	NOUN
cana-2799	268	9	issn	issn	NOUN
cana-2799	268	10	:	:	PUNCT
cana-2799	268	11	1074	1074	NUM
cana-2799	268	12	-	-	PUNCT
cana-2799	268	13	133x	133x	NUM
cana-2799	268	14	vol	vol	NOUN
cana-2799	268	15	32	32	NUM
cana-2799	268	16	no	no	NOUN
cana-2799	268	17	.	.	PUNCT
cana-2799	269	1	4s	4s	NUM
cana-2799	269	2	(	(	PUNCT
cana-2799	269	3	2025	2025	NUM
cana-2799	269	4	)	)	PUNCT
cana-2799	269	5	265	265	NUM
cana-2799	269	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	269	7	hence	hence	ADV
cana-2799	269	8	𝑢(𝜅	𝑢(𝜅	PROPN
cana-2799	269	9	)	)	PUNCT
cana-2799	269	10	=	=	SYM
cana-2799	269	11	𝑒1((𝑚𝜅)(𝑚	𝑒1((𝑚𝜅)(𝑚	X
cana-2799	269	12	)	)	PUNCT
cana-2799	269	13	)	)	PUNCT
cana-2799	269	14	is	be	AUX
cana-2799	269	15	a	a	DET
cana-2799	269	16	solution	solution	NOUN
cana-2799	269	17	of	of	ADP
cana-2799	269	18	(	(	PUNCT
cana-2799	269	19	17	17	NUM
cana-2799	269	20	)	)	PUNCT
cana-2799	269	21	when	when	SCONJ
cana-2799	269	22	𝑚	𝑚	PROPN
cana-2799	269	23	is	be	AUX
cana-2799	269	24	a	a	DET
cana-2799	269	25	root	root	NOUN
cana-2799	269	26	of	of	ADP
cana-2799	269	27	(	(	PUNCT
cana-2799	269	28	18	18	NUM
cana-2799	269	29	)	)	PUNCT
cana-2799	269	30	.	.	PUNCT
cana-2799	270	1	remark	remark	PROPN
cana-2799	270	2	4.3	4.3	NUM
cana-2799	270	3	.	.	PUNCT
cana-2799	271	1	the	the	DET
cana-2799	271	2	above	above	ADJ
cana-2799	271	3	lemma	lemma	PROPN
cana-2799	271	4	can	can	AUX
cana-2799	271	5	be	be	AUX
cana-2799	271	6	extended	extend	VERB
cana-2799	271	7	to	to	ADP
cana-2799	271	8	higher	high	ADJ
cana-2799	271	9	order	order	NOUN
cana-2799	271	10	linear	linear	ADJ
cana-2799	271	11	difference	difference	NOUN
cana-2799	271	12	equation	equation	NOUN
cana-2799	271	13	with	with	ADP
cana-2799	271	14	constant	constant	ADJ
cana-2799	271	15	coefficients	coefficient	NOUN
cana-2799	271	16	.	.	PUNCT
cana-2799	272	1	theorem	theorem	VERB
cana-2799	272	2	4.4	4.4	NUM
cana-2799	272	3	.	.	PUNCT
cana-2799	273	1	let	let	VERB
cana-2799	273	2	𝐼0	𝐼0	PROPN
cana-2799	273	3	be	be	AUX
cana-2799	273	4	initial	initial	ADJ
cana-2799	273	5	value	value	NOUN
cana-2799	273	6	of	of	ADP
cana-2799	273	7	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	273	8	)	)	PUNCT
cana-2799	273	9	and	and	CCONJ
cana-2799	274	1	𝜈	𝜈	X
cana-2799	274	2	=	=	SYM
cana-2799	274	3	1	1	X
cana-2799	274	4	.	.	PUNCT
cana-2799	275	1	the	the	DET
cana-2799	275	2	de	de	ADJ
cana-2799	275	3	-	-	ADJ
cana-2799	275	4	energizing	energizing	ADJ
cana-2799	275	5	difference	difference	NOUN
cana-2799	275	6	equation	equation	NOUN
cana-2799	275	7	(	(	PUNCT
cana-2799	275	8	12	12	NUM
cana-2799	275	9	)	)	PUNCT
cana-2799	275	10	for	for	ADP
cana-2799	275	11	𝜈	𝜈	X
cana-2799	275	12	=	=	SYM
cana-2799	275	13	1	1	NUM
cana-2799	275	14	has	have	VERB
cana-2799	275	15	a	a	DET
cana-2799	275	16	solution	solution	NOUN
cana-2799	275	17	of	of	ADP
cana-2799	275	18	the	the	DET
cana-2799	275	19	form	form	NOUN
cana-2799	275	20	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	275	21	)	)	PUNCT
cana-2799	275	22	=	=	NOUN
cana-2799	275	23	𝐼0𝑒1	𝐼0𝑒1	NOUN
cana-2799	275	24	(	(	PUNCT
cana-2799	275	25	(	(	PUNCT
cana-2799	275	26	−𝑅	−𝑅	PROPN
cana-2799	275	27	𝐿	𝐿	PROPN
cana-2799	275	28	𝜅	𝜅	PROPN
cana-2799	275	29	)	)	PUNCT
cana-2799	275	30	(	(	PUNCT
cana-2799	275	31	−𝑅	−𝑅	PROPN
cana-2799	275	32	𝐿	𝐿	PROPN
cana-2799	275	33	)	)	PUNCT
cana-2799	275	34	)	)	PUNCT
cana-2799	275	35	(	(	PUNCT
cana-2799	275	36	20	20	NUM
cana-2799	275	37	)	)	PUNCT
cana-2799	275	38	where	where	SCONJ
cana-2799	275	39	𝑒1	𝑒1	NOUN
cana-2799	275	40	denotes	denote	VERB
cana-2799	275	41	the	the	DET
cana-2799	275	42	extorial	extorial	ADJ
cana-2799	275	43	function	function	NOUN
cana-2799	275	44	.	.	PUNCT
cana-2799	276	1	proof	proof	NOUN
cana-2799	276	2	.	.	PUNCT
cana-2799	277	1	consider	consider	VERB
cana-2799	277	2	the	the	DET
cana-2799	277	3	first	first	ADJ
cana-2799	277	4	order	order	NOUN
cana-2799	277	5	difference	difference	NOUN
cana-2799	277	6	equation	equation	NOUN
cana-2799	277	7	𝐿δ𝐼(𝜅	𝐿δ𝐼(𝜅	PROPN
cana-2799	277	8	)	)	PUNCT
cana-2799	278	1	+	+	X
cana-2799	278	2	𝑅𝐼(𝜅	𝑅𝐼(𝜅	X
cana-2799	278	3	)	)	PUNCT
cana-2799	279	1	=	=	SYM
cana-2799	279	2	0	0	NUM
cana-2799	279	3	which	which	PRON
cana-2799	279	4	is	be	AUX
cana-2799	279	5	obtained	obtain	VERB
cana-2799	279	6	from	from	ADP
cana-2799	279	7	(	(	PUNCT
cana-2799	279	8	12	12	NUM
cana-2799	279	9	)	)	PUNCT
cana-2799	279	10	by	by	ADP
cana-2799	279	11	taking	take	VERB
cana-2799	279	12	𝜈	𝜈	X
cana-2799	279	13	=	=	SYM
cana-2799	279	14	1	1	NUM
cana-2799	279	15	and	and	CCONJ
cana-2799	279	16	its	its	PRON
cana-2799	279	17	auxillary	auxillary	PROPN
cana-2799	279	18	equation	equation	NOUN
cana-2799	279	19	ml+r=0	ml+r=0	NOUN
cana-2799	279	20	.	.	PUNCT
cana-2799	280	1	the	the	DET
cana-2799	280	2	auxiliary	auxiliary	ADJ
cana-2799	280	3	equation	equation	NOUN
cana-2799	280	4	𝑚𝐿	𝑚𝐿	VERB
cana-2799	280	5	+	+	PROPN
cana-2799	280	6	𝑅	𝑅	PROPN
cana-2799	280	7	=	=	SYM
cana-2799	280	8	0	0	PROPN
cana-2799	280	9	has	have	VERB
cana-2799	280	10	an	an	DET
cana-2799	280	11	unique	unique	ADJ
cana-2799	280	12	solution	solution	NOUN
cana-2799	280	13	𝑚	𝑚	NOUN
cana-2799	280	14	=	=	SYM
cana-2799	280	15	−𝑅	−𝑅	PROPN
cana-2799	280	16	𝐿	𝐿	PROPN
cana-2799	280	17	,	,	PUNCT
cana-2799	280	18	𝐿	𝐿	PROPN
cana-2799	280	19	≠	≠	PROPN
cana-2799	280	20	0	0	NUM
cana-2799	280	21	.	.	PUNCT
cana-2799	281	1	applying	apply	VERB
cana-2799	281	2	lemma	lemma	PROPN
cana-2799	281	3	4	4	NUM
cana-2799	281	4	for	for	ADP
cana-2799	281	5	first	first	ADJ
cana-2799	281	6	order	order	NOUN
cana-2799	281	7	difference	difference	NOUN
cana-2799	281	8	equation	equation	NOUN
cana-2799	281	9	by	by	ADP
cana-2799	281	10	taking	take	VERB
cana-2799	281	11	𝐴	𝐴	PROPN
cana-2799	281	12	=	=	SYM
cana-2799	281	13	0	0	NUM
cana-2799	281	14	,	,	PUNCT
cana-2799	281	15	𝐼(𝑡	𝐼(𝑡	NUM
cana-2799	281	16	)	)	PUNCT
cana-2799	281	17	=	=	SYM
cana-2799	281	18	𝐼0𝑒1	𝐼0𝑒1	NOUN
cana-2799	281	19	(	(	PUNCT
cana-2799	281	20	(	(	PUNCT
cana-2799	281	21	−𝑅	−𝑅	PROPN
cana-2799	281	22	𝐿	𝐿	PROPN
cana-2799	281	23	𝜅	𝜅	PROPN
cana-2799	281	24	)	)	PUNCT
cana-2799	281	25	(	(	PUNCT
cana-2799	281	26	−𝑅	−𝑅	PROPN
cana-2799	281	27	𝐿	𝐿	PROPN
cana-2799	281	28	)	)	PUNCT
cana-2799	281	29	)	)	PUNCT
cana-2799	281	30	,	,	PUNCT
cana-2799	281	31	which	which	PRON
cana-2799	281	32	is	be	AUX
cana-2799	281	33	a	a	DET
cana-2799	281	34	solution	solution	NOUN
cana-2799	281	35	of	of	ADP
cana-2799	281	36	the	the	DET
cana-2799	281	37	equation	equation	NOUN
cana-2799	281	38	(	(	PUNCT
cana-2799	281	39	12	12	NUM
cana-2799	281	40	)	)	PUNCT
cana-2799	281	41	for	for	ADP
cana-2799	281	42	𝜈	𝜈	X
cana-2799	281	43	=	=	SYM
cana-2799	281	44	1	1	X
cana-2799	281	45	.	.	PUNCT
cana-2799	281	46	theorem	theorem	VERB
cana-2799	281	47	4.5	4.5	NUM
cana-2799	281	48	.	.	PUNCT
cana-2799	282	1	for	for	ADP
cana-2799	282	2	𝜈	𝜈	X
cana-2799	282	3	=	=	SYM
cana-2799	282	4	1	1	NUM
cana-2799	282	5	,	,	PUNCT
cana-2799	282	6	the	the	DET
cana-2799	282	7	energizing	energize	VERB
cana-2799	282	8	difference	difference	NOUN
cana-2799	282	9	equation	equation	NOUN
cana-2799	282	10	(	(	PUNCT
cana-2799	282	11	10	10	NUM
cana-2799	282	12	)	)	PUNCT
cana-2799	282	13	has	have	VERB
cana-2799	282	14	a	a	DET
cana-2799	282	15	solution	solution	NOUN
cana-2799	282	16	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	282	17	)	)	PUNCT
cana-2799	282	18	=	=	SYM
cana-2799	282	19	𝑉	𝑉	PROPN
cana-2799	282	20	𝐿(𝑒𝑠−1)+𝑅	𝐿(𝑒𝑠−1)+𝑅	NOUN
cana-2799	282	21	+	+	CCONJ
cana-2799	282	22	𝐼0𝑒1	𝐼0𝑒1	NOUN
cana-2799	282	23	(	(	PUNCT
cana-2799	282	24	(	(	PUNCT
cana-2799	282	25	−𝑅	−𝑅	PROPN
cana-2799	282	26	𝐿	𝐿	PROPN
cana-2799	282	27	𝜅	𝜅	PROPN
cana-2799	282	28	)	)	PUNCT
cana-2799	282	29	(	(	PUNCT
cana-2799	282	30	−𝑅	−𝑅	PROPN
cana-2799	282	31	𝐿	𝐿	PROPN
cana-2799	282	32	)	)	PUNCT
cana-2799	282	33	)	)	PUNCT
cana-2799	282	34	,	,	PUNCT
cana-2799	282	35	(	(	PUNCT
cana-2799	282	36	21	21	NUM
cana-2799	282	37	)	)	PUNCT
cana-2799	282	38	where	where	SCONJ
cana-2799	282	39	s	s	NOUN
cana-2799	282	40	is	be	AUX
cana-2799	282	41	a	a	DET
cana-2799	282	42	constant	constant	ADJ
cana-2799	282	43	.	.	PUNCT
cana-2799	283	1	proof	proof	NOUN
cana-2799	283	2	.	.	PUNCT
cana-2799	284	1	let	let	VERB
cana-2799	284	2	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	284	3	)	)	PUNCT
cana-2799	284	4	=	=	PUNCT
cana-2799	284	5	𝑉	𝑉	PROPN
cana-2799	284	6	𝑐	𝑐	PROPN
cana-2799	284	7	𝑒𝑠𝜅	𝑒𝑠𝜅	NOUN
cana-2799	284	8	be	be	VERB
cana-2799	284	9	a	a	DET
cana-2799	284	10	solution	solution	NOUN
cana-2799	284	11	of	of	ADP
cana-2799	284	12	equation	equation	NOUN
cana-2799	284	13	(	(	PUNCT
cana-2799	284	14	10	10	NUM
cana-2799	284	15	)	)	PUNCT
cana-2799	284	16	for	for	ADP
cana-2799	284	17	𝜈	𝜈	X
cana-2799	284	18	=	=	SYM
cana-2799	284	19	1	1	NUM
cana-2799	284	20	,	,	PUNCT
cana-2799	284	21	where	where	SCONJ
cana-2799	284	22	c	c	NOUN
cana-2799	284	23	is	be	AUX
cana-2799	284	24	to	to	PART
cana-2799	284	25	be	be	AUX
cana-2799	284	26	determined	determine	VERB
cana-2799	284	27	.	.	PUNCT
cana-2799	285	1	since	since	SCONJ
cana-2799	285	2	𝑠	𝑠	PROPN
cana-2799	285	3	is	be	AUX
cana-2799	285	4	a	a	DET
cana-2799	285	5	constant	constant	ADJ
cana-2799	285	6	,	,	PUNCT
cana-2799	285	7	we	we	PRON
cana-2799	285	8	get	get	VERB
cana-2799	285	9	δ𝑒𝑠𝜅	δ𝑒𝑠𝜅	NOUN
cana-2799	285	10	=	=	SYM
cana-2799	285	11	𝑒𝑠(𝜅+1	𝑒𝑠(𝜅+1	X
cana-2799	285	12	)	)	PUNCT
cana-2799	285	13	−	−	PROPN
cana-2799	285	14	𝑒𝑠𝜅	𝑒𝑠𝜅	NOUN
cana-2799	285	15	=	=	SYM
cana-2799	285	16	(	(	PUNCT
cana-2799	285	17	𝑒𝑠	𝑒𝑠	INTJ
cana-2799	285	18	−	−	PROPN
cana-2799	285	19	1)𝑒𝑠𝜅	1)𝑒𝑠𝜅	NUM
cana-2799	285	20	and	and	CCONJ
cana-2799	285	21	δ𝐼(𝜅	δ𝐼(𝜅	NUM
cana-2799	285	22	)	)	PUNCT
cana-2799	285	23	=	=	SYM
cana-2799	285	24	𝑉	𝑉	NOUN
cana-2799	285	25	𝑐	𝑐	NOUN
cana-2799	285	26	δ𝑒𝑠𝜅	δ𝑒𝑠𝜅	NOUN
cana-2799	285	27	=	=	SYM
cana-2799	285	28	𝑉	𝑉	PROPN
cana-2799	285	29	𝑐	𝑐	PROPN
cana-2799	285	30	𝑒𝑠𝜅(𝑒𝑠	𝑒𝑠𝜅(𝑒𝑠	NOUN
cana-2799	285	31	−	−	PROPN
cana-2799	285	32	1	1	NUM
cana-2799	285	33	)	)	PUNCT
cana-2799	285	34	.	.	PUNCT
cana-2799	286	1	substituting	substitute	VERB
cana-2799	286	2	𝜈	𝜈	X
cana-2799	286	3	=	=	SYM
cana-2799	286	4	1	1	NUM
cana-2799	286	5	,	,	PUNCT
cana-2799	286	6	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	286	7	)	)	PUNCT
cana-2799	286	8	and	and	CCONJ
cana-2799	286	9	δ𝐼(𝜅	δ𝐼(𝜅	NUM
cana-2799	286	10	)	)	PUNCT
cana-2799	286	11	in	in	ADP
cana-2799	286	12	(	(	PUNCT
cana-2799	286	13	10	10	NUM
cana-2799	286	14	)	)	PUNCT
cana-2799	286	15	,	,	PUNCT
cana-2799	286	16	we	we	PRON
cana-2799	286	17	arrive	arrive	VERB
cana-2799	286	18	𝐼(𝜅)𝑅	𝐼(𝜅)𝑅	NOUN
cana-2799	286	19	+	+	X
cana-2799	286	20	𝐿δ𝐼(𝜅	𝐿δ𝐼(𝜅	NUM
cana-2799	286	21	)	)	PUNCT
cana-2799	286	22	=	=	SYM
cana-2799	286	23	𝑅	𝑅	PROPN
cana-2799	286	24	𝑉	𝑉	PROPN
cana-2799	286	25	𝑐	𝑐	PROPN
cana-2799	286	26	𝑒𝑠𝜅	𝑒𝑠𝜅	NOUN
cana-2799	286	27	+	+	CCONJ
cana-2799	286	28	𝐿	𝐿	PROPN
cana-2799	286	29	[	[	PUNCT
cana-2799	286	30	𝑉	𝑉	PROPN
cana-2799	286	31	𝑐	𝑐	PROPN
cana-2799	286	32	𝑒𝑠𝜅(𝑒𝑠	𝑒𝑠𝜅(𝑒𝑠	NOUN
cana-2799	286	33	−	−	PROPN
cana-2799	286	34	1	1	NUM
cana-2799	286	35	)	)	PUNCT
cana-2799	286	36	]	]	PUNCT
cana-2799	286	37	,	,	PUNCT
cana-2799	286	38	which	which	PRON
cana-2799	286	39	yields	yield	VERB
cana-2799	286	40	[	[	PUNCT
cana-2799	286	41	𝑅	𝑅	PROPN
cana-2799	286	42	+	+	PROPN
cana-2799	286	43	𝐿δ]𝐼	𝐿δ]𝐼	PROPN
cana-2799	286	44	=	=	SYM
cana-2799	286	45	𝑉	𝑉	PROPN
cana-2799	286	46	𝑐	𝑐	PROPN
cana-2799	287	1	[	[	X
cana-2799	287	2	𝐿(𝑒𝑠	𝐿(𝑒𝑠	ADJ
cana-2799	287	3	−	−	NOUN
cana-2799	287	4	1	1	NUM
cana-2799	287	5	+	+	CCONJ
cana-2799	287	6	𝑅)]𝑒𝑠𝜅.	𝑅)]𝑒𝑠𝜅.	NOUN
cana-2799	287	7	hence	hence	ADV
cana-2799	287	8	,	,	PUNCT
cana-2799	287	9	taking	take	VERB
cana-2799	287	10	𝑐	𝑐	NOUN
cana-2799	287	11	=	=	PRON
cana-2799	287	12	𝐿(𝑒𝑠	𝐿(𝑒𝑠	PROPN
cana-2799	287	13	−	−	PROPN
cana-2799	287	14	1	1	NUM
cana-2799	287	15	+	+	CCONJ
cana-2799	287	16	𝑅	𝑅	NOUN
cana-2799	287	17	)	)	PUNCT
cana-2799	287	18	,	,	PUNCT
cana-2799	287	19	we	we	PRON
cana-2799	287	20	find	find	VERB
cana-2799	287	21	a	a	DET
cana-2799	287	22	particular	particular	ADJ
cana-2799	287	23	solution	solution	NOUN
cana-2799	287	24	of	of	ADP
cana-2799	287	25	equation	equation	NOUN
cana-2799	287	26	(	(	PUNCT
cana-2799	287	27	10	10	NUM
cana-2799	287	28	)	)	PUNCT
cana-2799	287	29	when	when	SCONJ
cana-2799	287	30	𝜈	𝜈	X
cana-2799	287	31	=	=	SYM
cana-2799	287	32	1	1	NUM
cana-2799	287	33	,	,	PUNCT
cana-2799	287	34	as	as	ADP
cana-2799	287	35	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	287	36	)	)	PUNCT
cana-2799	287	37	=	=	SYM
cana-2799	287	38	𝑉	𝑉	PROPN
cana-2799	287	39	𝐿(𝑒𝑠−1+𝑅	𝐿(𝑒𝑠−1+𝑅	NOUN
cana-2799	287	40	)	)	PUNCT
cana-2799	287	41	𝑒𝑠𝜅	𝑒𝑠𝜅	NOUN
cana-2799	287	42	.	.	PUNCT
cana-2799	288	1	(	(	PUNCT
cana-2799	288	2	22	22	NUM
cana-2799	288	3	)	)	PUNCT
cana-2799	288	4	now	now	ADV
cana-2799	288	5	(	(	PUNCT
cana-2799	288	6	21	21	NUM
cana-2799	288	7	)	)	PUNCT
cana-2799	288	8	follows	follow	VERB
cana-2799	288	9	by	by	ADP
cana-2799	288	10	adding	add	VERB
cana-2799	288	11	(	(	PUNCT
cana-2799	288	12	20	20	NUM
cana-2799	288	13	)	)	PUNCT
cana-2799	288	14	and	and	CCONJ
cana-2799	288	15	(	(	PUNCT
cana-2799	288	16	22	22	NUM
cana-2799	288	17	)	)	PUNCT
cana-2799	288	18	the	the	DET
cana-2799	288	19	proof	proof	NOUN
cana-2799	288	20	is	be	AUX
cana-2799	288	21	complete	complete	ADJ
cana-2799	288	22	.	.	PUNCT
cana-2799	289	1	corollary	corollary	ADJ
cana-2799	289	2	4.6	4.6	NUM
cana-2799	289	3	.	.	PUNCT
cana-2799	290	1	if	if	SCONJ
cana-2799	290	2	𝐼0	𝐼0	PROPN
cana-2799	290	3	=	=	SYM
cana-2799	290	4	−𝑉	−𝑉	PROPN
cana-2799	290	5	𝑅	𝑅	PROPN
cana-2799	290	6	,	,	PUNCT
cana-2799	290	7	then	then	ADV
cana-2799	290	8	the	the	DET
cana-2799	290	9	extorial	extorial	ADJ
cana-2799	290	10	solution	solution	NOUN
cana-2799	290	11	of	of	ADP
cana-2799	290	12	difference	difference	NOUN
cana-2799	290	13	equation	equation	NOUN
cana-2799	290	14	(	(	PUNCT
cana-2799	290	15	9	9	NUM
cana-2799	290	16	)	)	PUNCT
cana-2799	290	17	of	of	ADP
cana-2799	290	18	the	the	DET
cana-2799	290	19	rl	rl	PROPN
cana-2799	290	20	circuit	circuit	NOUN
cana-2799	290	21	is	be	AUX
cana-2799	290	22	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	290	23	)	)	PUNCT
cana-2799	290	24	=	=	SYM
cana-2799	290	25	𝑉	𝑉	PROPN
cana-2799	290	26	𝑅	𝑅	PROPN
cana-2799	290	27	−	−	PROPN
cana-2799	290	28	𝑉	𝑉	PROPN
cana-2799	290	29	𝑅	𝑅	PROPN
cana-2799	290	30	𝑒1	𝑒1	NOUN
cana-2799	290	31	(	(	PUNCT
cana-2799	290	32	(	(	PUNCT
cana-2799	290	33	−𝑅	−𝑅	PROPN
cana-2799	290	34	𝐿	𝐿	PROPN
cana-2799	290	35	𝜅	𝜅	PROPN
cana-2799	290	36	)	)	PUNCT
cana-2799	290	37	(	(	PUNCT
cana-2799	290	38	−𝑅	−𝑅	PROPN
cana-2799	290	39	𝐿	𝐿	PROPN
cana-2799	290	40	)	)	PUNCT
cana-2799	290	41	)	)	PUNCT
cana-2799	290	42	.	.	PUNCT
cana-2799	291	1	proof	proof	NOUN
cana-2799	291	2	.	.	PUNCT
cana-2799	292	1	the	the	DET
cana-2799	292	2	proof	proof	NOUN
cana-2799	292	3	follows	follow	VERB
cana-2799	292	4	by	by	ADP
cana-2799	292	5	taking	take	VERB
cana-2799	292	6	𝑠	𝑠	PROPN
cana-2799	292	7	=	=	SYM
cana-2799	292	8	0	0	NUM
cana-2799	292	9	in	in	ADP
cana-2799	292	10	(	(	PUNCT
cana-2799	292	11	21	21	NUM
cana-2799	292	12	)	)	PUNCT
cana-2799	292	13	.	.	PUNCT
cana-2799	293	1	communications	communication	NOUN
cana-2799	293	2	on	on	ADP
cana-2799	293	3	applied	apply	VERB
cana-2799	293	4	nonlinear	nonlinear	ADJ
cana-2799	293	5	analysis	analysis	NOUN
cana-2799	293	6	issn	issn	NOUN
cana-2799	293	7	:	:	PUNCT
cana-2799	293	8	1074	1074	NUM
cana-2799	293	9	-	-	PUNCT
cana-2799	293	10	133x	133x	NUM
cana-2799	293	11	vol	vol	NOUN
cana-2799	293	12	32	32	NUM
cana-2799	293	13	no	no	NOUN
cana-2799	293	14	.	.	PUNCT
cana-2799	294	1	4s	4s	NUM
cana-2799	294	2	(	(	PUNCT
cana-2799	294	3	2025	2025	NUM
cana-2799	294	4	)	)	PUNCT
cana-2799	294	5	266	266	NUM
cana-2799	294	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	294	7	finding	find	VERB
cana-2799	294	8	the	the	DET
cana-2799	294	9	solutions	solution	NOUN
cana-2799	294	10	of	of	ADP
cana-2799	294	11	integer	integer	NOUN
cana-2799	294	12	order	order	NOUN
cana-2799	294	13	difference	difference	NOUN
cana-2799	294	14	equation	equation	NOUN
cana-2799	294	15	is	be	AUX
cana-2799	294	16	comparatively	comparatively	ADV
cana-2799	294	17	easier	easy	ADJ
cana-2799	294	18	than	than	ADP
cana-2799	294	19	the	the	DET
cana-2799	294	20	fractional	fractional	ADJ
cana-2799	294	21	order	order	NOUN
cana-2799	294	22	difference	difference	NOUN
cana-2799	294	23	equation	equation	NOUN
cana-2799	294	24	.	.	PUNCT
cana-2799	295	1	5	5	X
cana-2799	295	2	.	.	NOUN
cana-2799	295	3	extorial	extorial	ADJ
cana-2799	295	4	energizing	energize	VERB
cana-2799	295	5	for	for	ADP
cana-2799	295	6	rl	rl	ADP
cana-2799	295	7	circuit	circuit	NOUN
cana-2799	295	8	in	in	ADP
cana-2799	295	9	this	this	DET
cana-2799	295	10	section	section	NOUN
cana-2799	295	11	,	,	PUNCT
cana-2799	295	12	we	we	PRON
cana-2799	295	13	derive	derive	VERB
cana-2799	295	14	at	at	ADP
cana-2799	295	15	the	the	DET
cana-2799	295	16	solution	solution	NOUN
cana-2799	295	17	of	of	ADP
cana-2799	295	18	rl	rl	PROPN
cana-2799	295	19	circuit	circuit	NOUN
cana-2799	295	20	model	model	NOUN
cana-2799	295	21	with	with	ADP
cana-2799	295	22	extorial	extorial	ADJ
cana-2799	295	23	energizing	energizing	NOUN
cana-2799	295	24	.	.	PUNCT
cana-2799	296	1	here	here	ADV
cana-2799	296	2	,	,	PUNCT
cana-2799	296	3	we	we	PRON
cana-2799	296	4	deal	deal	VERB
cana-2799	296	5	with	with	ADP
cana-2799	296	6	fractional	fractional	ADJ
cana-2799	296	7	order	order	NOUN
cana-2799	296	8	difference	difference	NOUN
cana-2799	296	9	equation	equation	NOUN
cana-2799	296	10	also	also	ADV
cana-2799	296	11	.	.	PUNCT
cana-2799	297	1	theorem	theorem	VERB
cana-2799	297	2	5.1	5.1	NUM
cana-2799	297	3	.	.	PUNCT
cana-2799	298	1	the	the	DET
cana-2799	298	2	flow	flow	NOUN
cana-2799	298	3	of	of	ADP
cana-2799	298	4	current	current	NOUN
cana-2799	298	5	in	in	ADP
cana-2799	298	6	the	the	DET
cana-2799	298	7	rl	rl	PROPN
cana-2799	298	8	circuit	circuit	NOUN
cana-2799	298	9	creates	create	VERB
cana-2799	298	10	chaos	chaos	NOUN
cana-2799	298	11	due	due	ADP
cana-2799	298	12	to	to	PART
cana-2799	298	13	increase	increase	NOUN
cana-2799	298	14	of	of	ADP
cana-2799	298	15	temperature	temperature	NOUN
cana-2799	298	16	of	of	ADP
cana-2799	298	17	heat	heat	NOUN
cana-2799	298	18	.	.	PUNCT
cana-2799	299	1	in	in	ADP
cana-2799	299	2	this	this	DET
cana-2799	299	3	case	case	NOUN
cana-2799	299	4	,	,	PUNCT
cana-2799	299	5	the	the	DET
cana-2799	299	6	difference	difference	NOUN
cana-2799	299	7	equation	equation	NOUN
cana-2799	299	8	of	of	ADP
cana-2799	299	9	rl	rl	PROPN
cana-2799	299	10	circuit	circuit	NOUN
cana-2799	299	11	becomes	become	VERB
cana-2799	299	12	𝑉𝑒1((𝑠𝜅)𝑠	𝑉𝑒1((𝑠𝜅)𝑠	ADV
cana-2799	299	13	)	)	PUNCT
cana-2799	299	14	=	=	SYM
cana-2799	299	15	𝑅𝐼(𝜅	𝑅𝐼(𝜅	X
cana-2799	299	16	)	)	PUNCT
cana-2799	300	1	+	+	CCONJ
cana-2799	301	1	𝐿δ	𝐿δ	NOUN
cana-2799	301	2	𝜈𝐼(𝜅	𝜈𝐼(𝜅	NOUN
cana-2799	301	3	)	)	PUNCT
cana-2799	301	4	,	,	PUNCT
cana-2799	301	5	(	(	PUNCT
cana-2799	301	6	0	0	X
cana-2799	301	7	<	<	X
cana-2799	301	8	𝜈	𝜈	X
cana-2799	301	9	<	<	X
cana-2799	301	10	1	1	NUM
cana-2799	301	11	)	)	PUNCT
cana-2799	301	12	.	.	PUNCT
cana-2799	302	1	(	(	PUNCT
cana-2799	302	2	23	23	X
cana-2799	302	3	)	)	PUNCT
cana-2799	302	4	equation	equation	NOUN
cana-2799	302	5	(	(	PUNCT
cana-2799	302	6	23	23	NUM
cana-2799	302	7	)	)	PUNCT
cana-2799	302	8	is	be	AUX
cana-2799	302	9	𝜈𝑡ℎ	𝜈𝑡ℎ	PRON
cana-2799	302	10	order	order	NOUN
cana-2799	302	11	fractional	fractional	ADJ
cana-2799	302	12	difference	difference	NOUN
cana-2799	302	13	equation	equation	NOUN
cana-2799	302	14	.	.	PUNCT
cana-2799	303	1	when	when	SCONJ
cana-2799	303	2	there	there	PRON
cana-2799	303	3	is	be	VERB
cana-2799	303	4	no	no	DET
cana-2799	303	5	choas	choas	NOUN
cana-2799	303	6	in	in	ADP
cana-2799	303	7	rl	rl	PROPN
cana-2799	303	8	circuit	circuit	NOUN
cana-2799	303	9	,	,	PUNCT
cana-2799	303	10	the	the	DET
cana-2799	303	11	parameter	parameter	NOUN
cana-2799	303	12	𝜈	𝜈	X
cana-2799	303	13	takes	take	VERB
cana-2799	303	14	integer	integer	NOUN
cana-2799	303	15	value	value	NOUN
cana-2799	303	16	.	.	PUNCT
cana-2799	304	1	theorem	theorem	VERB
cana-2799	304	2	5.2	5.2	NUM
cana-2799	304	3	.	.	PUNCT
cana-2799	305	1	for	for	ADP
cana-2799	305	2	𝜈	𝜈	X
cana-2799	305	3	=	=	SYM
cana-2799	305	4	1	1	NUM
cana-2799	305	5	,	,	PUNCT
cana-2799	305	6	the	the	DET
cana-2799	305	7	difference	difference	NOUN
cana-2799	305	8	equation	equation	NOUN
cana-2799	305	9	(	(	PUNCT
cana-2799	305	10	23	23	NUM
cana-2799	305	11	)	)	PUNCT
cana-2799	305	12	has	have	VERB
cana-2799	305	13	a	a	DET
cana-2799	305	14	extorial	extorial	ADJ
cana-2799	305	15	solution	solution	NOUN
cana-2799	305	16	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	305	17	)	)	PUNCT
cana-2799	305	18	=	=	PUNCT
cana-2799	305	19	𝑉𝑒1((𝑠𝜅)𝑠ℓ	𝑉𝑒1((𝑠𝜅)𝑠ℓ	NUM
cana-2799	305	20	)	)	PUNCT
cana-2799	305	21	𝐿(𝑒𝑠−1)+𝑅	𝐿(𝑒𝑠−1)+𝑅	NOUN
cana-2799	305	22	+	+	CCONJ
cana-2799	305	23	𝐼0𝑒1	𝐼0𝑒1	NOUN
cana-2799	305	24	(	(	PUNCT
cana-2799	305	25	(	(	PUNCT
cana-2799	305	26	−𝑅	−𝑅	PROPN
cana-2799	305	27	𝐿	𝐿	PROPN
cana-2799	305	28	𝜅	𝜅	PROPN
cana-2799	305	29	)	)	PUNCT
cana-2799	305	30	(	(	PUNCT
cana-2799	305	31	−𝑅	−𝑅	PROPN
cana-2799	305	32	𝐿	𝐿	PROPN
cana-2799	305	33	)	)	PUNCT
cana-2799	305	34	)	)	PUNCT
cana-2799	305	35	.	.	PUNCT
cana-2799	306	1	(	(	PUNCT
cana-2799	306	2	24	24	NUM
cana-2799	306	3	)	)	PUNCT
cana-2799	306	4	proof	proof	NOUN
cana-2799	306	5	.	.	PUNCT
cana-2799	307	1	let	let	VERB
cana-2799	307	2	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	307	3	)	)	PUNCT
cana-2799	307	4	=	=	SYM
cana-2799	307	5	𝑉	𝑉	PROPN
cana-2799	307	6	𝑐	𝑐	PROPN
cana-2799	307	7	𝑒1((𝑠𝜅)𝑠	𝑒1((𝑠𝜅)𝑠	NOUN
cana-2799	307	8	)	)	PUNCT
cana-2799	307	9	be	be	AUX
cana-2799	307	10	a	a	DET
cana-2799	307	11	solution	solution	NOUN
cana-2799	307	12	of	of	ADP
cana-2799	307	13	equation	equation	NOUN
cana-2799	307	14	(	(	PUNCT
cana-2799	307	15	23	23	NUM
cana-2799	307	16	)	)	PUNCT
cana-2799	307	17	(	(	PUNCT
cana-2799	307	18	𝜈	𝜈	X
cana-2799	307	19	=	=	SYM
cana-2799	307	20	1	1	NUM
cana-2799	307	21	)	)	PUNCT
cana-2799	307	22	,	,	PUNCT
cana-2799	307	23	where	where	SCONJ
cana-2799	307	24	𝑐	𝑐	PROPN
cana-2799	307	25	is	be	AUX
cana-2799	307	26	to	to	PART
cana-2799	307	27	be	be	AUX
cana-2799	307	28	determined	determine	VERB
cana-2799	307	29	.	.	PUNCT
cana-2799	308	1	since	since	SCONJ
cana-2799	308	2	𝑠	𝑠	PROPN
cana-2799	308	3	is	be	AUX
cana-2799	308	4	a	a	DET
cana-2799	308	5	constant	constant	ADJ
cana-2799	308	6	,	,	PUNCT
cana-2799	308	7	from	from	ADP
cana-2799	308	8	(	(	PUNCT
cana-2799	308	9	24	24	NUM
cana-2799	308	10	)	)	PUNCT
cana-2799	308	11	,	,	PUNCT
cana-2799	308	12	we	we	PRON
cana-2799	308	13	get	get	VERB
cana-2799	308	14	δ𝑒1((𝑠𝜅)𝑠	δ𝑒1((𝑠𝜅)𝑠	ADJ
cana-2799	308	15	)	)	PUNCT
cana-2799	308	16	=	=	SYM
cana-2799	308	17	𝑒1((𝑠(𝜅	𝑒1((𝑠(𝜅	NOUN
cana-2799	308	18	+	+	X
cana-2799	309	1	1))(𝑠	1))(𝑠	NUM
cana-2799	309	2	)	)	PUNCT
cana-2799	309	3	)	)	PUNCT
cana-2799	310	1	−	−	PROPN
cana-2799	310	2	𝑒1((𝑠𝜅)𝑠	𝑒1((𝑠𝜅)𝑠	ADV
cana-2799	310	3	)	)	PUNCT
cana-2799	310	4	.	.	PUNCT
cana-2799	311	1	this	this	PRON
cana-2799	311	2	gives	give	VERB
cana-2799	311	3	δ𝐼(𝜅	δ𝐼(𝜅	PROPN
cana-2799	311	4	)	)	PUNCT
cana-2799	311	5	=	=	SYM
cana-2799	311	6	𝑉	𝑉	PROPN
cana-2799	311	7	𝑐	𝑐	PROPN
cana-2799	311	8	δ𝑒1((𝑠𝜅)𝑠	δ𝑒1((𝑠𝜅)𝑠	NOUN
cana-2799	311	9	)	)	PUNCT
cana-2799	311	10	=	=	PUNCT
cana-2799	311	11	𝑉	𝑉	PROPN
cana-2799	311	12	𝑐	𝑐	PROPN
cana-2799	311	13	𝑒1((𝑠𝜅)(𝑠))(𝑒1((1)(𝑠	𝑒1((𝑠𝜅)(𝑠))(𝑒1((1)(𝑠	NOUN
cana-2799	311	14	)	)	PUNCT
cana-2799	311	15	)	)	PUNCT
cana-2799	312	1	−	−	ADP
cana-2799	312	2	1	1	NUM
cana-2799	312	3	)	)	PUNCT
cana-2799	312	4	.	.	PUNCT
cana-2799	313	1	substituting	substitute	VERB
cana-2799	313	2	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	313	3	)	)	PUNCT
cana-2799	313	4	and	and	CCONJ
cana-2799	313	5	δ𝐼(𝜅	δ𝐼(𝜅	NUM
cana-2799	313	6	)	)	PUNCT
cana-2799	313	7	in	in	ADP
cana-2799	313	8	the	the	DET
cana-2799	313	9	above	above	ADJ
cana-2799	313	10	equation	equation	NOUN
cana-2799	313	11	,	,	PUNCT
cana-2799	313	12	we	we	PRON
cana-2799	313	13	arrive	arrive	VERB
cana-2799	313	14	𝐼(𝜅)𝑅	𝐼(𝜅)𝑅	NOUN
cana-2799	313	15	+	+	X
cana-2799	313	16	𝐿δ𝐼(𝜅	𝐿δ𝐼(𝜅	NUM
cana-2799	313	17	)	)	PUNCT
cana-2799	313	18	=	=	SYM
cana-2799	313	19	𝑅	𝑅	PROPN
cana-2799	313	20	𝑉	𝑉	PROPN
cana-2799	313	21	𝑐	𝑐	PROPN
cana-2799	313	22	𝑒1((𝑠𝜅)𝑠	𝑒1((𝑠𝜅)𝑠	NOUN
cana-2799	313	23	)	)	PUNCT
cana-2799	314	1	+	+	CCONJ
cana-2799	314	2	𝐿	𝐿	PROPN
cana-2799	314	3	[	[	PUNCT
cana-2799	314	4	𝑉	𝑉	PROPN
cana-2799	314	5	𝑐	𝑐	PROPN
cana-2799	314	6	𝑒(𝑠𝜅)(𝑒1(1(𝑠	𝑒(𝑠𝜅)(𝑒1(1(𝑠	PROPN
cana-2799	314	7	)	)	PUNCT
cana-2799	314	8	)	)	PUNCT
cana-2799	315	1	−	−	ADP
cana-2799	316	1	1	1	NUM
cana-2799	316	2	)	)	PUNCT
cana-2799	316	3	]	]	PUNCT
cana-2799	316	4	,	,	PUNCT
cana-2799	316	5	which	which	PRON
cana-2799	316	6	yields	yield	VERB
cana-2799	316	7	[	[	PUNCT
cana-2799	316	8	𝑅	𝑅	PROPN
cana-2799	316	9	+	+	PROPN
cana-2799	316	10	𝐿δ]𝐼	𝐿δ]𝐼	PROPN
cana-2799	316	11	=	=	SYM
cana-2799	316	12	𝑉	𝑉	PROPN
cana-2799	316	13	𝑐	𝑐	PROPN
cana-2799	316	14	[	[	X
cana-2799	316	15	𝐿(𝑒1(ℓ(𝑠	𝐿(𝑒1(ℓ(𝑠	NUM
cana-2799	316	16	)	)	PUNCT
cana-2799	316	17	)	)	PUNCT
cana-2799	317	1	−	−	ADP
cana-2799	317	2	1	1	NUM
cana-2799	317	3	+	+	NUM
cana-2799	317	4	𝑅)]𝑒1((𝑠𝜅)𝑠	𝑅)]𝑒1((𝑠𝜅)𝑠	ADJ
cana-2799	317	5	)	)	PUNCT
cana-2799	317	6	.	.	PUNCT
cana-2799	318	1	hence	hence	ADV
cana-2799	318	2	taking	take	VERB
cana-2799	318	3	𝑐	𝑐	NOUN
cana-2799	318	4	=	=	PUNCT
cana-2799	318	5	𝐿(𝑒1(1(𝑠	𝐿(𝑒1(1(𝑠	PROPN
cana-2799	318	6	)	)	PUNCT
cana-2799	318	7	)	)	PUNCT
cana-2799	319	1	−	−	ADP
cana-2799	319	2	1	1	NUM
cana-2799	319	3	+	+	CCONJ
cana-2799	319	4	𝑅	𝑅	NOUN
cana-2799	319	5	)	)	PUNCT
cana-2799	319	6	,	,	PUNCT
cana-2799	319	7	we	we	PRON
cana-2799	319	8	find	find	VERB
cana-2799	319	9	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	319	10	)	)	PUNCT
cana-2799	319	11	=	=	SYM
cana-2799	319	12	𝑉	𝑉	PROPN
cana-2799	319	13	𝐿(𝑒1(1(𝑠))−1+𝑅	𝐿(𝑒1(1(𝑠))−1+𝑅	PROPN
cana-2799	319	14	)	)	PUNCT
cana-2799	319	15	𝑒𝑠𝜅	𝑒𝑠𝜅	NOUN
cana-2799	319	16	is	be	AUX
cana-2799	319	17	a	a	DET
cana-2799	319	18	particular	particular	ADJ
cana-2799	319	19	solution	solution	NOUN
cana-2799	319	20	of	of	ADP
cana-2799	319	21	equation	equation	NOUN
cana-2799	319	22	when	when	SCONJ
cana-2799	319	23	𝜈	𝜈	X
cana-2799	319	24	=	=	SYM
cana-2799	319	25	1	1	NUM
cana-2799	319	26	and	and	CCONJ
cana-2799	319	27	(	(	PUNCT
cana-2799	319	28	24	24	NUM
cana-2799	319	29	)	)	PUNCT
cana-2799	319	30	follows	follow	VERB
cana-2799	319	31	.	.	PUNCT
cana-2799	320	1	theorem	theorem	VERB
cana-2799	320	2	5.3	5.3	NUM
cana-2799	320	3	.	.	PUNCT
cana-2799	321	1	for	for	ADP
cana-2799	321	2	for	for	ADP
cana-2799	321	3	0	0	NUM
cana-2799	321	4	<	<	X
cana-2799	321	5	𝜈	𝜈	X
cana-2799	321	6	<	<	X
cana-2799	321	7	1	1	NUM
cana-2799	321	8	,	,	PUNCT
cana-2799	321	9	the	the	DET
cana-2799	321	10	energizing	energize	VERB
cana-2799	321	11	fractional	fractional	ADJ
cana-2799	321	12	difference	difference	NOUN
cana-2799	321	13	equation	equation	NOUN
cana-2799	321	14	𝑉𝑒1((𝑠𝜅)(𝑠	𝑉𝑒1((𝑠𝜅)(𝑠	NOUN
cana-2799	321	15	)	)	PUNCT
cana-2799	321	16	)	)	PUNCT
cana-2799	321	17	=	=	PUNCT
cana-2799	321	18	𝐼(𝜅)𝑅	𝐼(𝜅)𝑅	X
cana-2799	321	19	+	+	CCONJ
cana-2799	321	20	δ	δ	PROPN
cana-2799	321	21	𝜈𝐼(𝜅	𝜈𝐼(𝜅	NOUN
cana-2799	321	22	)	)	PUNCT
cana-2799	321	23	,	,	PUNCT
cana-2799	321	24	(	(	PUNCT
cana-2799	321	25	25	25	NUM
cana-2799	321	26	)	)	PUNCT
cana-2799	321	27	has	have	VERB
cana-2799	321	28	an	an	DET
cana-2799	321	29	extorial	extorial	ADJ
cana-2799	321	30	solution	solution	NOUN
cana-2799	321	31	of	of	ADP
cana-2799	321	32	the	the	DET
cana-2799	321	33	form	form	NOUN
cana-2799	321	34	𝑉𝑒1((𝑠𝜅)(𝑠	𝑉𝑒1((𝑠𝜅)(𝑠	NOUN
cana-2799	321	35	)	)	PUNCT
cana-2799	321	36	)	)	PUNCT
cana-2799	322	1	𝐿(𝑒1(ℓ(𝑠))−1)𝜈+𝑅	𝐿(𝑒1(ℓ(𝑠))−1)𝜈+𝑅	PROPN
cana-2799	322	2	+	+	CCONJ
cana-2799	322	3	𝐼0𝑒1	𝐼0𝑒1	PROPN
cana-2799	322	4	(	(	PUNCT
cana-2799	322	5	(	(	PUNCT
cana-2799	322	6	−𝑅	−𝑅	PROPN
cana-2799	322	7	𝐿	𝐿	PROPN
cana-2799	322	8	)	)	PUNCT
cana-2799	322	9	1	1	NUM
cana-2799	322	10	𝜈𝜅	𝜈𝜅	INTJ
cana-2799	322	11	(	(	PUNCT
cana-2799	322	12	−𝑅	−𝑅	PROPN
cana-2799	322	13	𝐿	𝐿	PROPN
cana-2799	322	14	)	)	PUNCT
cana-2799	322	15	1	1	NUM
cana-2799	322	16	𝜈	𝜈	NOUN
cana-2799	322	17	)	)	PUNCT
cana-2799	322	18	.	.	PUNCT
cana-2799	323	1	(	(	PUNCT
cana-2799	323	2	26	26	NUM
cana-2799	323	3	)	)	PUNCT
cana-2799	323	4	proof	proof	NOUN
cana-2799	323	5	.	.	PUNCT
cana-2799	324	1	we	we	PRON
cana-2799	324	2	try	try	VERB
cana-2799	324	3	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	324	4	)	)	PUNCT
cana-2799	324	5	=	=	SYM
cana-2799	324	6	𝑉𝑐𝑒1((𝑠𝜅)(𝑠	𝑉𝑐𝑒1((𝑠𝜅)(𝑠	PROPN
cana-2799	324	7	)	)	PUNCT
cana-2799	324	8	)	)	PUNCT
cana-2799	324	9	as	as	ADP
cana-2799	324	10	a	a	DET
cana-2799	324	11	solution	solution	NOUN
cana-2799	324	12	of	of	ADP
cana-2799	324	13	equation	equation	NOUN
cana-2799	324	14	(	(	PUNCT
cana-2799	324	15	25	25	NUM
cana-2799	324	16	)	)	PUNCT
cana-2799	324	17	,	,	PUNCT
cana-2799	324	18	where	where	SCONJ
cana-2799	324	19	c	c	NOUN
cana-2799	324	20	is	be	AUX
cana-2799	324	21	to	to	PART
cana-2799	324	22	be	be	AUX
cana-2799	324	23	determined	determine	VERB
cana-2799	324	24	.	.	PUNCT
cana-2799	325	1	δ𝐼(𝜅	δ𝐼(𝜅	NUM
cana-2799	325	2	)	)	PUNCT
cana-2799	325	3	=	=	SYM
cana-2799	325	4	𝑉𝑐(𝑒1(1(𝑠	𝑉𝑐(𝑒1(1(𝑠	PROPN
cana-2799	325	5	)	)	PUNCT
cana-2799	325	6	)	)	PUNCT
cana-2799	326	1	−	−	ADP
cana-2799	326	2	1)𝑒1((𝑠𝜅)(𝑠	1)𝑒1((𝑠𝜅)(𝑠	NUM
cana-2799	326	3	)	)	PUNCT
cana-2799	326	4	)	)	PUNCT
cana-2799	326	5	,	,	PUNCT
cana-2799	326	6	δ	δ	PROPN
cana-2799	326	7	2𝐼(𝜅	2𝐼(𝜅	NUM
cana-2799	326	8	)	)	PUNCT
cana-2799	326	9	=	=	SYM
cana-2799	326	10	𝑉𝑐(𝑒1(1(𝑠	𝑉𝑐(𝑒1(1(𝑠	PROPN
cana-2799	326	11	)	)	PUNCT
cana-2799	326	12	)	)	PUNCT
cana-2799	327	1	−	−	ADP
cana-2799	327	2	1)2𝑒1((𝑠𝜅)(𝑠	1)2𝑒1((𝑠𝜅)(𝑠	NUM
cana-2799	327	3	)	)	PUNCT
cana-2799	327	4	)	)	PUNCT
cana-2799	328	1	⋯	⋯	PROPN
cana-2799	328	2	,	,	PUNCT
cana-2799	328	3	δ	δ	PROPN
cana-2799	328	4	𝜈𝐼(𝜅	𝜈𝐼(𝜅	PROPN
cana-2799	328	5	)	)	PUNCT
cana-2799	328	6	=	=	SYM
cana-2799	328	7	𝑉𝑐(𝑒1(1(𝑠	𝑉𝑐(𝑒1(1(𝑠	PROPN
cana-2799	328	8	)	)	PUNCT
cana-2799	328	9	)	)	PUNCT
cana-2799	329	1	−	−	ADP
cana-2799	330	1	1)𝜈𝑒1((𝑠𝜅)(𝑠	1)𝜈𝑒1((𝑠𝜅)(𝑠	NUM
cana-2799	330	2	)	)	PUNCT
cana-2799	330	3	)	)	PUNCT
cana-2799	330	4	is	be	AUX
cana-2799	330	5	obtained	obtain	VERB
cana-2799	330	6	from	from	ADP
cana-2799	330	7	δ𝐼(𝜅	δ𝐼(𝜅	NUM
cana-2799	330	8	)	)	PUNCT
cana-2799	331	1	=	=	PUNCT
cana-2799	331	2	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	331	3	+	+	NOUN
cana-2799	331	4	1	1	NUM
cana-2799	331	5	)	)	PUNCT
cana-2799	331	6	−	−	PROPN
cana-2799	331	7	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	331	8	)	)	PUNCT
cana-2799	331	9	.	.	PUNCT
cana-2799	332	1	substituting	substitute	VERB
cana-2799	332	2	𝐼	𝐼	PROPN
cana-2799	332	3	and	and	CCONJ
cana-2799	332	4	δ	δ	PROPN
cana-2799	332	5	𝜈𝐼	𝜈𝐼	PROPN
cana-2799	332	6	in	in	ADP
cana-2799	332	7	(	(	PUNCT
cana-2799	332	8	25	25	NUM
cana-2799	332	9	)	)	PUNCT
cana-2799	332	10	,	,	PUNCT
cana-2799	332	11	we	we	PRON
cana-2799	332	12	find	find	VERB
cana-2799	332	13	communications	communication	NOUN
cana-2799	332	14	on	on	ADP
cana-2799	332	15	applied	apply	VERB
cana-2799	332	16	nonlinear	nonlinear	ADJ
cana-2799	332	17	analysis	analysis	NOUN
cana-2799	332	18	issn	issn	NOUN
cana-2799	332	19	:	:	PUNCT
cana-2799	332	20	1074	1074	NUM
cana-2799	332	21	-	-	PUNCT
cana-2799	332	22	133x	133x	NUM
cana-2799	332	23	vol	vol	NOUN
cana-2799	332	24	32	32	NUM
cana-2799	332	25	no	no	NOUN
cana-2799	332	26	.	.	PUNCT
cana-2799	333	1	4s	4s	NUM
cana-2799	333	2	(	(	PUNCT
cana-2799	333	3	2025	2025	NUM
cana-2799	333	4	)	)	PUNCT
cana-2799	333	5	267	267	NUM
cana-2799	333	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	333	7	𝐼(𝜅)𝑅	𝐼(𝜅)𝑅	NOUN
cana-2799	333	8	+	+	CCONJ
cana-2799	333	9	𝐿	𝐿	PROPN
cana-2799	333	10	ℓ	ℓ	PROPN
cana-2799	333	11	δ	δ	PROPN
cana-2799	333	12	𝜈𝐼(𝜅	𝜈𝐼(𝜅	PROPN
cana-2799	333	13	)	)	PUNCT
cana-2799	333	14	=	=	SYM
cana-2799	333	15	𝑅𝑉𝑐𝑒1((𝑠𝜅)(𝑠	𝑅𝑉𝑐𝑒1((𝑠𝜅)(𝑠	X
cana-2799	333	16	)	)	PUNCT
cana-2799	333	17	)	)	PUNCT
cana-2799	334	1	+	+	CCONJ
cana-2799	334	2	𝐿[𝑉𝑐(𝑒1(1(𝑠	𝐿[𝑉𝑐(𝑒1(1(𝑠	PROPN
cana-2799	334	3	)	)	PUNCT
cana-2799	334	4	)	)	PUNCT
cana-2799	335	1	−	−	ADP
cana-2799	336	1	1)𝜈𝑒1((𝑠𝜅)(𝑠	1)𝜈𝑒1((𝑠𝜅)(𝑠	NUM
cana-2799	336	2	)	)	PUNCT
cana-2799	336	3	)	)	PUNCT
cana-2799	336	4	]	]	PUNCT
cana-2799	337	1	=	=	PUNCT
cana-2799	337	2	𝑉𝑐	𝑉𝑐	PROPN
cana-2799	337	3	[	[	PUNCT
cana-2799	337	4	𝐿	𝐿	PROPN
cana-2799	337	5	ℓ	ℓ	PROPN
cana-2799	337	6	(	(	PUNCT
cana-2799	337	7	𝑒1(1(𝑠	𝑒1(1(𝑠	PROPN
cana-2799	337	8	)	)	PUNCT
cana-2799	337	9	)	)	PUNCT
cana-2799	338	1	−	−	PROPN
cana-2799	338	2	1)𝜈	1)𝜈	NUM
cana-2799	338	3	+	+	CCONJ
cana-2799	338	4	𝑅]𝑒1((𝑠𝜅)(𝑠	𝑅]𝑒1((𝑠𝜅)(𝑠	NOUN
cana-2799	338	5	)	)	PUNCT
cana-2799	338	6	)	)	PUNCT
cana-2799	338	7	.	.	PUNCT
cana-2799	339	1	hence	hence	ADV
cana-2799	339	2	𝐼(𝜅	𝐼(𝜅	NOUN
cana-2799	339	3	)	)	PUNCT
cana-2799	339	4	=	=	SYM
cana-2799	339	5	𝑉	𝑉	PROPN
cana-2799	339	6	𝐿(𝑒1(1(𝑠))−1)𝜈+𝑅	𝐿(𝑒1(1(𝑠))−1)𝜈+𝑅	NOUN
cana-2799	339	7	𝑒1((𝑠𝜅)(𝑠	𝑒1((𝑠𝜅)(𝑠	NOUN
cana-2799	339	8	)	)	PUNCT
cana-2799	339	9	)	)	PUNCT
cana-2799	339	10	is	be	AUX
cana-2799	339	11	a	a	DET
cana-2799	339	12	particular	particular	ADJ
cana-2799	339	13	solution	solution	NOUN
cana-2799	339	14	of	of	ADP
cana-2799	339	15	(	(	PUNCT
cana-2799	339	16	26	26	NUM
cana-2799	339	17	)	)	PUNCT
cana-2799	339	18	.	.	PUNCT
cana-2799	340	1	thus	thus	ADV
cana-2799	340	2	extorial	extorial	ADJ
cana-2799	340	3	function	function	NOUN
cana-2799	340	4	is	be	AUX
cana-2799	340	5	used	use	VERB
cana-2799	340	6	to	to	PART
cana-2799	340	7	obtain	obtain	VERB
cana-2799	340	8	the	the	DET
cana-2799	340	9	solution	solution	NOUN
cana-2799	340	10	to	to	ADP
cana-2799	340	11	rl	rl	NOUN
cana-2799	340	12	-	-	PUNCT
cana-2799	340	13	circuit	circuit	NOUN
cana-2799	340	14	difference	difference	NOUN
cana-2799	340	15	equation	equation	NOUN
cana-2799	340	16	.	.	PUNCT
cana-2799	341	1	also	also	ADV
cana-2799	341	2	we	we	PRON
cana-2799	341	3	have	have	AUX
cana-2799	341	4	obtained	obtain	VERB
cana-2799	341	5	solution	solution	NOUN
cana-2799	341	6	of	of	ADP
cana-2799	341	7	rl	rl	NOUN
cana-2799	341	8	circuit	circuit	NOUN
cana-2799	341	9	of	of	ADP
cana-2799	341	10	chaos	chaos	NOUN
cana-2799	341	11	situation	situation	NOUN
cana-2799	341	12	represented	represent	VERB
cana-2799	341	13	by	by	ADP
cana-2799	341	14	fractional	fractional	ADJ
cana-2799	341	15	order	order	NOUN
cana-2799	341	16	difference	difference	NOUN
cana-2799	341	17	equation	equation	NOUN
cana-2799	341	18	.	.	PUNCT
cana-2799	342	1	6	6	NUM
cana-2799	342	2	.	.	PUNCT
cana-2799	342	3	fractional	fractional	ADJ
cana-2799	342	4	difference	difference	NOUN
cana-2799	342	5	heat	heat	NOUN
cana-2799	342	6	equation	equation	NOUN
cana-2799	342	7	model	model	NOUN
cana-2799	342	8	in	in	ADP
cana-2799	342	9	this	this	DET
cana-2799	342	10	section	section	NOUN
cana-2799	342	11	,	,	PUNCT
cana-2799	342	12	we	we	PRON
cana-2799	342	13	apply	apply	VERB
cana-2799	342	14	the	the	DET
cana-2799	342	15	alpha	alpha	NOUN
cana-2799	342	16	and	and	CCONJ
cana-2799	342	17	fibonacci	fibonacci	NOUN
cana-2799	342	18	difference	difference	NOUN
cana-2799	342	19	operators	operator	NOUN
cana-2799	342	20	and	and	CCONJ
cana-2799	342	21	obtain	obtain	VERB
cana-2799	342	22	new	new	ADJ
cana-2799	342	23	model	model	NOUN
cana-2799	342	24	of	of	ADP
cana-2799	342	25	heat	heat	NOUN
cana-2799	342	26	equations	equation	NOUN
cana-2799	342	27	.	.	PUNCT
cana-2799	343	1	the	the	DET
cana-2799	343	2	solution	solution	NOUN
cana-2799	343	3	of	of	ADP
cana-2799	343	4	these	these	DET
cana-2799	343	5	equations	equation	NOUN
cana-2799	343	6	are	be	AUX
cana-2799	343	7	expressed	express	VERB
cana-2799	343	8	in	in	ADP
cana-2799	343	9	terms	term	NOUN
cana-2799	343	10	of	of	ADP
cana-2799	343	11	extorial	extorial	ADJ
cana-2799	343	12	functions	function	NOUN
cana-2799	343	13	.	.	PUNCT
cana-2799	344	1	the	the	DET
cana-2799	344	2	materials	material	NOUN
cana-2799	344	3	up	up	ADP
cana-2799	344	4	to	to	PART
cana-2799	344	5	three	three	NUM
cana-2799	344	6	dimensions	dimension	NOUN
cana-2799	344	7	i.e.	i.e.	X
cana-2799	344	8	,	,	PUNCT
cana-2799	344	9	rod	rod	NOUN
cana-2799	344	10	,	,	PUNCT
cana-2799	344	11	thin	thin	ADJ
cana-2799	344	12	plate	plate	NOUN
cana-2799	344	13	and	and	CCONJ
cana-2799	344	14	medium	medium	NOUN
cana-2799	344	15	are	be	AUX
cana-2799	344	16	taken	take	VERB
cana-2799	344	17	for	for	ADP
cana-2799	344	18	study	study	NOUN
cana-2799	344	19	and	and	CCONJ
cana-2799	344	20	the	the	DET
cana-2799	344	21	transfer	transfer	NOUN
cana-2799	344	22	of	of	ADP
cana-2799	344	23	heat	heat	NOUN
cana-2799	344	24	is	be	AUX
cana-2799	344	25	examined	examine	VERB
cana-2799	344	26	.	.	PUNCT
cana-2799	345	1	the	the	DET
cana-2799	345	2	two	two	NUM
cana-2799	345	3	operators	operator	NOUN
cana-2799	345	4	(	(	PUNCT
cana-2799	345	5	alpha	alpha	NOUN
cana-2799	345	6	and	and	CCONJ
cana-2799	345	7	fibonacci	fibonacci	NOUN
cana-2799	345	8	)	)	PUNCT
cana-2799	345	9	are	be	AUX
cana-2799	345	10	used	use	VERB
cana-2799	345	11	for	for	ADP
cana-2799	345	12	the	the	DET
cana-2799	345	13	study	study	NOUN
cana-2799	345	14	of	of	ADP
cana-2799	345	15	transfer	transfer	NOUN
cana-2799	345	16	of	of	ADP
cana-2799	345	17	heat	heat	NOUN
cana-2799	345	18	and	and	CCONJ
cana-2799	345	19	are	be	AUX
cana-2799	345	20	defined	define	VERB
cana-2799	345	21	accordingly	accordingly	ADV
cana-2799	345	22	.	.	PUNCT
cana-2799	346	1	let	let	VERB
cana-2799	346	2	𝛼	𝛼	VERB
cana-2799	346	3	≠	≠	PROPN
cana-2799	346	4	0	0	NUM
cana-2799	346	5	,	,	PUNCT
cana-2799	346	6	𝑙	𝑙	NOUN
cana-2799	346	7	=	=	PUNCT
cana-2799	346	8	(	(	PUNCT
cana-2799	346	9	1,1,1	1,1,1	NUM
cana-2799	346	10	,	,	PUNCT
cana-2799	346	11	.	.	PUNCT
cana-2799	346	12	.	.	PUNCT
cana-2799	347	1	.	.	PUNCT
cana-2799	348	1	,	,	PUNCT
cana-2799	348	2	1	1	NUM
cana-2799	348	3	)	)	PUNCT
cana-2799	348	4	,	,	PUNCT
cana-2799	348	5	𝜅	𝜅	X
cana-2799	348	6	=	=	PUNCT
cana-2799	348	7	(	(	PUNCT
cana-2799	348	8	𝜅1	𝜅1	PROPN
cana-2799	348	9	,	,	PUNCT
cana-2799	348	10	𝜅2	𝜅2	PROPN
cana-2799	348	11	,	,	PUNCT
cana-2799	348	12	⋯	⋯	PROPN
cana-2799	348	13	,	,	PUNCT
cana-2799	348	14	𝜅𝑛	𝜅𝑛	NOUN
cana-2799	348	15	)	)	PUNCT
cana-2799	348	16	∈	∈	PROPN
cana-2799	349	1	ℝ𝑛	ℝ𝑛	PROPN
cana-2799	349	2	and	and	CCONJ
cana-2799	349	3	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	349	4	)	)	PUNCT
cana-2799	349	5	be	be	AUX
cana-2799	349	6	a	a	DET
cana-2799	349	7	real	real	ADV
cana-2799	349	8	valued	value	VERB
cana-2799	349	9	n	n	CCONJ
cana-2799	349	10	-	-	PUNCT
cana-2799	349	11	variable	variable	ADJ
cana-2799	349	12	function	function	NOUN
cana-2799	349	13	defined	define	VERB
cana-2799	349	14	on	on	ADP
cana-2799	349	15	ℝ𝑛.	ℝ𝑛.	PROPN
cana-2799	349	16	the	the	DET
cana-2799	349	17	n	n	CCONJ
cana-2799	349	18	-	-	PUNCT
cana-2799	349	19	variable	variable	NOUN
cana-2799	349	20	𝛼-difference	𝛼-difference	NOUN
cana-2799	349	21	operator	operator	NOUN
cana-2799	349	22	,	,	PUNCT
cana-2799	349	23	denoted	denote	VERB
cana-2799	349	24	as	as	ADP
cana-2799	349	25	δ𝛼	δ𝛼	ADV
cana-2799	349	26	,	,	PUNCT
cana-2799	349	27	on	on	ADP
cana-2799	349	28	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	349	29	)	)	PUNCT
cana-2799	349	30	is	be	AUX
cana-2799	349	31	defined	define	VERB
cana-2799	349	32	by	by	ADP
cana-2799	349	33	δ	δ	PROPN
cana-2799	349	34	𝛼	𝛼	PROPN
cana-2799	349	35	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	349	36	)	)	PUNCT
cana-2799	349	37	=	=	PUNCT
cana-2799	349	38	𝑣(𝜅1	𝑣(𝜅1	ADJ
cana-2799	349	39	+	+	CCONJ
cana-2799	349	40	1	1	NUM
cana-2799	349	41	,	,	PUNCT
cana-2799	349	42	𝜅2	𝜅2	VERB
cana-2799	349	43	+	+	NOUN
cana-2799	349	44	1	1	NUM
cana-2799	349	45	,	,	PUNCT
cana-2799	349	46	.	.	PUNCT
cana-2799	349	47	.	.	PUNCT
cana-2799	350	1	.	.	PUNCT
cana-2799	351	1	,	,	PUNCT
cana-2799	351	2	𝜅𝑛	𝜅𝑛	ADP
cana-2799	351	3	+	+	ADJ
cana-2799	351	4	1	1	X
cana-2799	351	5	)	)	PUNCT
cana-2799	351	6	−	−	PROPN
cana-2799	351	7	𝛼𝑣(𝜅1	𝛼𝑣(𝜅1	NOUN
cana-2799	351	8	,	,	PUNCT
cana-2799	351	9	𝜅2	𝜅2	PROPN
cana-2799	351	10	,	,	PUNCT
cana-2799	351	11	.	.	PUNCT
cana-2799	351	12	.	.	PUNCT
cana-2799	352	1	.	.	PUNCT
cana-2799	353	1	,	,	PUNCT
cana-2799	353	2	𝜅𝑛	𝜅𝑛	PROPN
cana-2799	353	3	)	)	PUNCT
cana-2799	353	4	.	.	PUNCT
cana-2799	354	1	(	(	PUNCT
cana-2799	354	2	27	27	NUM
cana-2799	354	3	)	)	PUNCT
cana-2799	354	4	this	this	DET
cana-2799	354	5	operator	operator	NOUN
cana-2799	354	6	becomes	become	VERB
cana-2799	354	7	partial	partial	ADJ
cana-2799	354	8	𝛼-difference	𝛼-difference	NOUN
cana-2799	354	9	operator	operator	NOUN
cana-2799	354	10	if	if	SCONJ
cana-2799	354	11	we	we	PRON
cana-2799	354	12	replace	replace	VERB
cana-2799	354	13	by	by	ADP
cana-2799	354	14	𝜅𝑖	𝜅𝑖	NOUN
cana-2799	354	15	+	+	CCONJ
cana-2799	354	16	1	1	NUM
cana-2799	354	17	in	in	ADP
cana-2799	354	18	centain	centain	PROPN
cana-2799	354	19	component	component	NOUN
cana-2799	354	20	i.	i.	NOUN
cana-2799	354	21	thus	thus	ADV
cana-2799	354	22	the	the	DET
cana-2799	354	23	above	above	ADJ
cana-2799	354	24	definition	definition	NOUN
cana-2799	354	25	of	of	ADP
cana-2799	354	26	the	the	DET
cana-2799	354	27	alpha	alpha	NOUN
cana-2799	354	28	and	and	CCONJ
cana-2799	354	29	fibonacci	fibonacci	NOUN
cana-2799	354	30	difference	difference	NOUN
cana-2799	354	31	operators	operator	NOUN
cana-2799	354	32	and	and	CCONJ
cana-2799	354	33	its	its	PRON
cana-2799	354	34	equations	equation	NOUN
cana-2799	354	35	are	be	AUX
cana-2799	354	36	employed	employ	VERB
cana-2799	354	37	in	in	ADP
cana-2799	354	38	the	the	DET
cana-2799	354	39	forthcoming	forthcoming	ADJ
cana-2799	354	40	sections	section	NOUN
cana-2799	354	41	and	and	CCONJ
cana-2799	354	42	solutions	solution	NOUN
cana-2799	354	43	are	be	AUX
cana-2799	354	44	derived	derive	VERB
cana-2799	354	45	for	for	ADP
cana-2799	354	46	heat	heat	NOUN
cana-2799	354	47	equations	equation	NOUN
cana-2799	354	48	.	.	PUNCT
cana-2799	355	1	also	also	ADV
cana-2799	355	2	we	we	PRON
cana-2799	355	3	present	present	VERB
cana-2799	355	4	solutions	solution	NOUN
cana-2799	355	5	of	of	ADP
cana-2799	355	6	partial	partial	ADJ
cana-2799	355	7	fractional	fractional	ADJ
cana-2799	355	8	alpha	alpha	NOUN
cana-2799	355	9	difference	difference	NOUN
cana-2799	355	10	equation	equation	NOUN
cana-2799	355	11	with	with	ADP
cana-2799	355	12	polynomial	polynomial	ADJ
cana-2799	355	13	factorial	factorial	ADJ
cana-2799	355	14	and	and	CCONJ
cana-2799	355	15	extorial	extorial	ADJ
cana-2799	355	16	functions	function	NOUN
cana-2799	355	17	.	.	PUNCT
cana-2799	356	1	we	we	PRON
cana-2799	356	2	also	also	ADV
cana-2799	356	3	apply	apply	VERB
cana-2799	356	4	these	these	DET
cana-2799	356	5	type	type	NOUN
cana-2799	356	6	of	of	ADP
cana-2799	356	7	solutions	solution	NOUN
cana-2799	356	8	to	to	PART
cana-2799	356	9	heat	heat	VERB
cana-2799	356	10	flows	flow	NOUN
cana-2799	356	11	.	.	PUNCT
cana-2799	357	1	in	in	ADP
cana-2799	357	2	the	the	DET
cana-2799	357	3	following	follow	VERB
cana-2799	357	4	lemma	lemma	PROPN
cana-2799	357	5	,	,	PUNCT
cana-2799	357	6	some	some	DET
cana-2799	357	7	identities	identity	NOUN
cana-2799	357	8	related	relate	VERB
cana-2799	357	9	to	to	ADP
cana-2799	357	10	alpha	alpha	NOUN
cana-2799	357	11	difference	difference	NOUN
cana-2799	357	12	operator	operator	NOUN
cana-2799	357	13	on	on	ADP
cana-2799	357	14	extorial	extorial	ADJ
cana-2799	357	15	function	function	NOUN
cana-2799	357	16	are	be	AUX
cana-2799	357	17	given	give	VERB
cana-2799	357	18	.	.	PUNCT
cana-2799	358	1	lemma	lemma	PROPN
cana-2799	358	2	6.1	6.1	NUM
cana-2799	358	3	.	.	PUNCT
cana-2799	359	1	let	let	VERB
cana-2799	359	2	𝜅(𝑟𝑛	𝜅(𝑟𝑛	NUM
cana-2799	359	3	)	)	PUNCT
cana-2799	359	4	≠	≠	PROPN
cana-2799	359	5	0	0	NUM
cana-2799	359	6	,	,	PUNCT
cana-2799	359	7	𝑛	𝑛	DET
cana-2799	359	8	∈	∈	NOUN
cana-2799	359	9	𝑁.	𝑁.	PROPN
cana-2799	359	10	then	then	ADV
cana-2799	359	11	we	we	PRON
cana-2799	359	12	have	have	VERB
cana-2799	359	13	the	the	DET
cana-2799	359	14	following	follow	VERB
cana-2799	359	15	identities	identity	NOUN
cana-2799	359	16	with	with	ADP
cana-2799	359	17	extorial	extorial	ADJ
cana-2799	359	18	function	function	NOUN
cana-2799	359	19	:	:	PUNCT
cana-2799	359	20	(	(	PUNCT
cana-2799	359	21	i	i	NOUN
cana-2799	359	22	)	)	PUNCT
cana-2799	359	23	.	.	PUNCT
cana-2799	360	1	δ𝛼𝑒1(𝜅	δ𝛼𝑒1(𝜅	NUM
cana-2799	360	2	)	)	PUNCT
cana-2799	361	1	=	=	SYM
cana-2799	361	2	𝑒1(𝜅)[1	𝑒1(𝜅)[1	PROPN
cana-2799	362	1	+	+	CCONJ
cana-2799	362	2	1	1	NUM
cana-2799	362	3	−	−	NOUN
cana-2799	362	4	𝛼	𝛼	NOUN
cana-2799	362	5	]	]	X
cana-2799	362	6	,	,	PUNCT
cana-2799	362	7	(	(	PUNCT
cana-2799	362	8	ii	ii	NOUN
cana-2799	362	9	)	)	PUNCT
cana-2799	362	10	.	.	PUNCT
cana-2799	363	1	δ𝛼𝑒(−1)(𝜅	δ𝛼𝑒(−1)(𝜅	NOUN
cana-2799	363	2	)	)	PUNCT
cana-2799	363	3	=	=	SYM
cana-2799	363	4	𝑒(−1)(𝜅)[𝑒(−1	𝑒(−1)(𝜅)[𝑒(−1	NOUN
cana-2799	363	5	)	)	PUNCT
cana-2799	363	6	−	−	PROPN
cana-2799	363	7	𝛼](−1	𝛼](−1	NUM
cana-2799	363	8	)	)	PUNCT
cana-2799	363	9	,	,	PUNCT
cana-2799	363	10	(	(	PUNCT
cana-2799	363	11	iii	iii	NOUN
cana-2799	363	12	)	)	PUNCT
cana-2799	363	13	.	.	PUNCT
cana-2799	364	1	δ𝛼𝑒1((−𝜅	δ𝛼𝑒1((−𝜅	NOUN
cana-2799	364	2	)	)	PUNCT
cana-2799	364	3	)	)	PUNCT
cana-2799	365	1	=	=	SYM
cana-2799	366	1	𝑒1((−𝜅))[1	𝑒1((−𝜅))[1	PROPN
cana-2799	366	2	+	+	CCONJ
cana-2799	366	3	1	1	NUM
cana-2799	366	4	−	−	NOUN
cana-2799	366	5	𝛼	𝛼	NOUN
cana-2799	366	6	]	]	X
cana-2799	366	7	,	,	PUNCT
cana-2799	366	8	𝜅	𝜅	X
cana-2799	366	9	>	>	X
cana-2799	366	10	0	0	X
cana-2799	366	11	.	.	PUNCT
cana-2799	366	12	proof	proof	NOUN
cana-2799	366	13	.	.	PUNCT
cana-2799	367	1	(	(	PUNCT
cana-2799	367	2	i	i	NOUN
cana-2799	367	3	)	)	PUNCT
cana-2799	367	4	.	.	PUNCT
cana-2799	368	1	by	by	ADP
cana-2799	368	2	(	(	PUNCT
cana-2799	368	3	27	27	NUM
cana-2799	368	4	)	)	PUNCT
cana-2799	368	5	,	,	PUNCT
cana-2799	368	6	and	and	CCONJ
cana-2799	368	7	applying	apply	VERB
cana-2799	368	8	δ𝛼	δ𝛼	ADP
cana-2799	368	9	on	on	ADP
cana-2799	368	10	𝑒1(𝜅	𝑒1(𝜅	NOUN
cana-2799	368	11	)	)	PUNCT
cana-2799	368	12	,	,	PUNCT
cana-2799	368	13	we	we	PRON
cana-2799	368	14	arrive	arrive	VERB
cana-2799	368	15	δ𝛼𝑒1(𝜅	δ𝛼𝑒1(𝜅	NUM
cana-2799	368	16	)	)	PUNCT
cana-2799	368	17	=	=	SYM
cana-2799	369	1	𝑒1((𝜅	𝑒1((𝜅	INTJ
cana-2799	369	2	+	+	NOUN
cana-2799	369	3	1	1	NUM
cana-2799	369	4	)	)	PUNCT
cana-2799	369	5	)	)	PUNCT
cana-2799	370	1	−	−	ADP
cana-2799	370	2	𝛼𝑒1(𝜅	𝛼𝑒1(𝜅	NUM
cana-2799	370	3	)	)	PUNCT
cana-2799	370	4	=	=	SYM
cana-2799	370	5	𝑒1(𝜅	𝑒1(𝜅	PROPN
cana-2799	370	6	)	)	PUNCT
cana-2799	370	7	.	.	PUNCT
cana-2799	371	1	𝑒1	𝑒1	NOUN
cana-2799	371	2	−	−	NOUN
cana-2799	371	3	𝛼𝑒1(𝜅	𝛼𝑒1(𝜅	NOUN
cana-2799	371	4	)	)	PUNCT
cana-2799	371	5	=	=	SYM
cana-2799	371	6	𝑒1(𝜅)[𝑒1(1	𝑒1(𝜅)[𝑒1(1	NOUN
cana-2799	371	7	)	)	PUNCT
cana-2799	371	8	−	−	NOUN
cana-2799	372	1	𝛼	𝛼	X
cana-2799	372	2	]	]	X
cana-2799	372	3	=	=	PUNCT
cana-2799	372	4	𝑒1(𝜅)[1	𝑒1(𝜅)[1	PROPN
cana-2799	373	1	+	+	CCONJ
cana-2799	373	2	1	1	NUM
cana-2799	373	3	1	1	NUM
cana-2799	373	4	!	!	PUNCT
cana-2799	374	1	+	+	CCONJ
cana-2799	374	2	1	1	NUM
cana-2799	374	3	2	2	NUM
cana-2799	374	4	!	!	PUNCT
cana-2799	375	1	+	+	CCONJ
cana-2799	375	2	⋯	⋯	ADP
cana-2799	375	3	−	−	NOUN
cana-2799	375	4	𝛼	𝛼	NOUN
cana-2799	375	5	]	]	X
cana-2799	375	6	=	=	PUNCT
cana-2799	375	7	𝑒1(𝜅)[1	𝑒1(𝜅)[1	PROPN
cana-2799	376	1	+	+	CCONJ
cana-2799	376	2	1	1	NUM
cana-2799	376	3	−	−	NOUN
cana-2799	376	4	𝛼	𝛼	NOUN
cana-2799	376	5	]	]	X
cana-2799	376	6	.	.	PUNCT
cana-2799	377	1	(	(	PUNCT
cana-2799	377	2	ii	ii	NOUN
cana-2799	377	3	)	)	PUNCT
cana-2799	377	4	.	.	PUNCT
cana-2799	378	1	by	by	ADP
cana-2799	378	2	(	(	PUNCT
cana-2799	378	3	27	27	NUM
cana-2799	378	4	)	)	PUNCT
cana-2799	378	5	,	,	PUNCT
cana-2799	378	6	and	and	CCONJ
cana-2799	378	7	applying	apply	VERB
cana-2799	378	8	δ𝛼	δ𝛼	ADP
cana-2799	378	9	on	on	ADP
cana-2799	378	10	𝑒(𝜅(−1	𝑒(𝜅(−1	NOUN
cana-2799	378	11	)	)	PUNCT
cana-2799	378	12	)	)	PUNCT
cana-2799	378	13	,	,	PUNCT
cana-2799	378	14	we	we	PRON
cana-2799	378	15	arrive	arrive	VERB
cana-2799	378	16	δ𝛼𝑒(−1)(𝜅	δ𝛼𝑒(−1)(𝜅	ADV
cana-2799	378	17	)	)	PUNCT
cana-2799	378	18	=	=	PUNCT
cana-2799	379	1	𝑒(−1)((𝜅	𝑒(−1)((𝜅	ADV
cana-2799	379	2	+	+	NUM
cana-2799	379	3	1	1	NUM
cana-2799	379	4	)	)	PUNCT
cana-2799	379	5	)	)	PUNCT
cana-2799	380	1	−	−	ADP
cana-2799	380	2	𝛼𝑒(−1)(𝜅	𝛼𝑒(−1)(𝜅	NUM
cana-2799	380	3	)	)	PUNCT
cana-2799	380	4	=	=	SYM
cana-2799	380	5	𝑒(−1)(𝜅	𝑒(−1)(𝜅	NOUN
cana-2799	380	6	)	)	PUNCT
cana-2799	380	7	.	.	PUNCT
cana-2799	381	1	𝑒(−1	𝑒(−1	NOUN
cana-2799	381	2	)	)	PUNCT
cana-2799	382	1	−	−	PROPN
cana-2799	382	2	𝛼𝑒(−1)(𝜅	𝛼𝑒(−1)(𝜅	NUM
cana-2799	382	3	)	)	PUNCT
cana-2799	382	4	.	.	PUNCT
cana-2799	383	1	=	=	SYM
cana-2799	383	2	𝑒(−1)(𝜅)[𝑒(−1	𝑒(−1)(𝜅)[𝑒(−1	NOUN
cana-2799	383	3	)	)	PUNCT
cana-2799	383	4	−	−	PROPN
cana-2799	383	5	𝛼](−1	𝛼](−1	NUM
cana-2799	383	6	)	)	PUNCT
cana-2799	383	7	.	.	PUNCT
cana-2799	384	1	(	(	PUNCT
cana-2799	384	2	iii	iii	NOUN
cana-2799	384	3	)	)	PUNCT
cana-2799	384	4	.	.	PUNCT
cana-2799	385	1	follows	follow	VERB
cana-2799	385	2	from	from	ADP
cana-2799	385	3	(	(	PUNCT
cana-2799	385	4	ii	ii	NOUN
cana-2799	385	5	)	)	PUNCT
cana-2799	385	6	by	by	ADP
cana-2799	385	7	replacing	replace	VERB
cana-2799	385	8	𝜅	𝜅	PRON
cana-2799	385	9	as	as	SCONJ
cana-2799	385	10	−𝜅.	−𝜅.	PROPN
cana-2799	385	11	communications	communication	NOUN
cana-2799	385	12	on	on	ADP
cana-2799	385	13	applied	apply	VERB
cana-2799	385	14	nonlinear	nonlinear	ADJ
cana-2799	385	15	analysis	analysis	NOUN
cana-2799	385	16	issn	issn	NOUN
cana-2799	385	17	:	:	PUNCT
cana-2799	385	18	1074	1074	NUM
cana-2799	385	19	-	-	PUNCT
cana-2799	385	20	133x	133x	NUM
cana-2799	385	21	vol	vol	NOUN
cana-2799	385	22	32	32	NUM
cana-2799	385	23	no	no	NOUN
cana-2799	385	24	.	.	PUNCT
cana-2799	386	1	4s	4s	NUM
cana-2799	386	2	(	(	PUNCT
cana-2799	386	3	2025	2025	NUM
cana-2799	386	4	)	)	PUNCT
cana-2799	386	5	268	268	NUM
cana-2799	386	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	386	7	theorem	theorem	VERB
cana-2799	386	8	6.2	6.2	NUM
cana-2799	386	9	.	.	PUNCT
cana-2799	387	1	if	if	SCONJ
cana-2799	387	2	𝑣(𝜅1	𝑣(𝜅1	VERB
cana-2799	387	3	,	,	PUNCT
cana-2799	387	4	𝜅2	𝜅2	ADJ
cana-2799	387	5	)	)	PUNCT
cana-2799	387	6	=	=	SYM
cana-2799	387	7	𝑒1((𝜅1	𝑒1((𝜅1	NUM
cana-2799	387	8	)	)	PUNCT
cana-2799	387	9	)	)	PUNCT
cana-2799	387	10	.	.	PUNCT
cana-2799	388	1	𝑒1((𝜅2	𝑒1((𝜅2	X
cana-2799	388	2	)	)	PUNCT
cana-2799	388	3	)	)	PUNCT
cana-2799	389	1	then	then	ADV
cana-2799	389	2	we	we	PRON
cana-2799	389	3	have	have	VERB
cana-2799	389	4	the	the	DET
cana-2799	389	5	identities	identity	NOUN
cana-2799	389	6	:	:	PUNCT
cana-2799	389	7	(	(	PUNCT
cana-2799	389	8	𝑖)δ𝛼𝑣(𝜅1	𝑖)δ𝛼𝑣(𝜅1	PROPN
cana-2799	389	9	,	,	PUNCT
cana-2799	389	10	𝜅2	𝜅2	ADJ
cana-2799	389	11	)	)	PUNCT
cana-2799	389	12	=	=	SYM
cana-2799	389	13	𝑒1((𝜅1	𝑒1((𝜅1	NUM
cana-2799	389	14	)	)	PUNCT
cana-2799	389	15	)	)	PUNCT
cana-2799	389	16	.	.	PUNCT
cana-2799	390	1	𝑒1((𝜅2))[𝑒1(1	𝑒1((𝜅2))[𝑒1(1	ADV
cana-2799	390	2	)	)	PUNCT
cana-2799	390	3	−	−	ADP
cana-2799	391	1	𝛼	𝛼	X
cana-2799	391	2	]	]	X
cana-2799	391	3	,	,	PUNCT
cana-2799	391	4	(	(	PUNCT
cana-2799	391	5	𝑖𝑖)δ𝛼𝑣(𝜅1	𝑖𝑖)δ𝛼𝑣(𝜅1	ADJ
cana-2799	391	6	,	,	PUNCT
cana-2799	391	7	𝜅2	𝜅2	ADJ
cana-2799	391	8	)	)	PUNCT
cana-2799	391	9	=	=	SYM
cana-2799	392	1	𝑒1((𝜅2	𝑒1((𝜅2	PROPN
cana-2799	392	2	)	)	PUNCT
cana-2799	392	3	)	)	PUNCT
cana-2799	392	4	.	.	PUNCT
cana-2799	393	1	𝑒1((𝜅1))[𝑒1((1	𝑒1((𝜅1))[𝑒1((1	PROPN
cana-2799	393	2	)	)	PUNCT
cana-2799	393	3	)	)	PUNCT
cana-2799	394	1	−	−	PUNCT
cana-2799	394	2	𝛼	𝛼	X
cana-2799	394	3	]	]	PUNCT
cana-2799	394	4	.	.	PUNCT
cana-2799	395	1	proof	proof	NOUN
cana-2799	395	2	.	.	PUNCT
cana-2799	396	1	(	(	PUNCT
cana-2799	396	2	𝑖)δ𝛼𝑣(𝜅1	𝑖)δ𝛼𝑣(𝜅1	PROPN
cana-2799	396	3	,	,	PUNCT
cana-2799	396	4	𝜅2	𝜅2	ADJ
cana-2799	396	5	)	)	PUNCT
cana-2799	396	6	=	=	SYM
cana-2799	397	1	𝑒1((𝜅1))[δ𝛼𝑒1((𝜅2	𝑒1((𝜅1))[δ𝛼𝑒1((𝜅2	NOUN
cana-2799	397	2	)	)	PUNCT
cana-2799	397	3	)	)	PUNCT
cana-2799	397	4	]	]	PUNCT
cana-2799	398	1	=	=	PUNCT
cana-2799	398	2	𝑒1((𝜅1))[𝑒1((𝜅2	𝑒1((𝜅1))[𝑒1((𝜅2	X
cana-2799	398	3	+	+	CCONJ
cana-2799	398	4	1	1	NUM
cana-2799	398	5	)	)	PUNCT
cana-2799	398	6	)	)	PUNCT
cana-2799	399	1	−	−	PROPN
cana-2799	399	2	𝛼𝑒1((𝜅2	𝛼𝑒1((𝜅2	PROPN
cana-2799	399	3	)	)	PUNCT
cana-2799	399	4	)	)	PUNCT
cana-2799	399	5	]	]	PUNCT
cana-2799	400	1	=	=	PUNCT
cana-2799	400	2	𝑒1((𝜅1))𝑒1((𝜅2))[𝑒1(1	𝑒1((𝜅1))𝑒1((𝜅2))[𝑒1(1	PROPN
cana-2799	400	3	)	)	PUNCT
cana-2799	400	4	−	−	NOUN
cana-2799	400	5	𝛼	𝛼	NOUN
cana-2799	400	6	]	]	X
cana-2799	400	7	.	.	PUNCT
cana-2799	401	1	in	in	ADP
cana-2799	401	2	the	the	DET
cana-2799	401	3	similar	similar	ADJ
cana-2799	401	4	way	way	NOUN
cana-2799	401	5	,	,	PUNCT
cana-2799	401	6	the	the	DET
cana-2799	401	7	proof	proof	NOUN
cana-2799	401	8	of	of	ADP
cana-2799	401	9	(	(	PUNCT
cana-2799	401	10	ii	ii	NOUN
cana-2799	401	11	)	)	PUNCT
cana-2799	401	12	follows	follow	VERB
cana-2799	401	13	.	.	PUNCT
cana-2799	402	1	assume	assume	VERB
cana-2799	402	2	that	that	SCONJ
cana-2799	402	3	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	402	4	,	,	PUNCT
cana-2799	402	5	𝜅2	𝜅2	PRON
cana-2799	402	6	)	)	PUNCT
cana-2799	402	7	be	be	VERB
cana-2799	402	8	the	the	DET
cana-2799	402	9	temperature	temperature	NOUN
cana-2799	402	10	of	of	ADP
cana-2799	402	11	a	a	DET
cana-2799	402	12	rod	rod	NOUN
cana-2799	402	13	at	at	ADP
cana-2799	402	14	position	position	NOUN
cana-2799	402	15	𝜅1	𝜅1	NOUN
cana-2799	402	16	at	at	ADP
cana-2799	402	17	time	time	NOUN
cana-2799	402	18	𝜅2	𝜅2	PROPN
cana-2799	402	19	,	,	PUNCT
cana-2799	402	20	ℓ1	ℓ1	NOUN
cana-2799	402	21	and	and	CCONJ
cana-2799	402	22	ℓ2	ℓ2	NOUN
cana-2799	402	23	be	be	AUX
cana-2799	402	24	shift	shift	NOUN
cana-2799	402	25	values	value	NOUN
cana-2799	402	26	of	of	ADP
cana-2799	402	27	𝜅1	𝜅1	NOUN
cana-2799	402	28	and	and	CCONJ
cana-2799	402	29	𝜅2	𝜅2	VERB
cana-2799	402	30	respectively	respectively	ADV
cana-2799	402	31	and	and	CCONJ
cana-2799	402	32	𝛾	𝛾	PART
cana-2799	402	33	be	be	AUX
cana-2799	402	34	the	the	DET
cana-2799	402	35	rate	rate	NOUN
cana-2799	402	36	of	of	ADP
cana-2799	402	37	conductivity	conductivity	NOUN
cana-2799	402	38	of	of	ADP
cana-2799	402	39	rod	rod	NOUN
cana-2799	402	40	.	.	PUNCT
cana-2799	403	1	when	when	SCONJ
cana-2799	403	2	considering	consider	VERB
cana-2799	403	3	impact	impact	NOUN
cana-2799	403	4	of	of	ADP
cana-2799	403	5	external	external	ADJ
cana-2799	403	6	climate	climate	NOUN
cana-2799	403	7	change	change	NOUN
cana-2799	403	8	on	on	ADP
cana-2799	403	9	the	the	DET
cana-2799	403	10	rod	rod	NOUN
cana-2799	403	11	,	,	PUNCT
cana-2799	403	12	the	the	DET
cana-2799	403	13	partial	partial	ADJ
cana-2799	403	14	𝛼	𝛼	NOUN
cana-2799	403	15	difference	difference	NOUN
cana-2799	403	16	equation	equation	NOUN
cana-2799	403	17	of	of	ADP
cana-2799	403	18	heat	heat	NOUN
cana-2799	403	19	flow	flow	NOUN
cana-2799	403	20	in	in	ADP
cana-2799	403	21	the	the	DET
cana-2799	403	22	rod	rod	NOUN
cana-2799	403	23	becomes	become	VERB
cana-2799	403	24	fractional	fractional	ADJ
cana-2799	403	25	𝛼difference	𝛼difference	NOUN
cana-2799	403	26	equation	equation	NOUN
cana-2799	403	27	δ	δ	NOUN
cana-2799	403	28	𝜈	𝜈	X
cana-2799	403	29	𝛼	𝛼	X
cana-2799	403	30	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	403	31	,	,	PUNCT
cana-2799	403	32	𝜅2	𝜅2	ADJ
cana-2799	403	33	)	)	PUNCT
cana-2799	403	34	=	=	SYM
cana-2799	403	35	𝛾[δ𝜈	𝛾[δ𝜈	PROPN
cana-2799	403	36	𝛼	𝛼	NOUN
cana-2799	403	37	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	403	38	,	,	PUNCT
cana-2799	403	39	𝜅2	𝜅2	ADJ
cana-2799	403	40	)	)	PUNCT
cana-2799	403	41	+	+	CCONJ
cana-2799	403	42	δ	δ	X
cana-2799	403	43	𝜈	𝜈	X
cana-2799	403	44	𝛼	𝛼	X
cana-2799	403	45	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	403	46	,	,	PUNCT
cana-2799	403	47	𝜅2	𝜅2	ADJ
cana-2799	403	48	)	)	PUNCT
cana-2799	403	49	]	]	PUNCT
cana-2799	403	50	.	.	PUNCT
cana-2799	404	1	(	(	PUNCT
cana-2799	404	2	28	28	NUM
cana-2799	404	3	)	)	PUNCT
cana-2799	404	4	theorem	theorem	VERB
cana-2799	404	5	6.3	6.3	NUM
cana-2799	404	6	.	.	PUNCT
cana-2799	405	1	if	if	SCONJ
cana-2799	405	2	𝛾	𝛾	PROPN
cana-2799	405	3	=	=	PUNCT
cana-2799	406	1	[	[	X
cana-2799	406	2	𝑒1(1	𝑒1(1	ADJ
cana-2799	406	3	)	)	PUNCT
cana-2799	406	4	−	−	PROPN
cana-2799	406	5	𝛼/𝑒1(±(1	𝛼/𝑒1(±(1	NOUN
cana-2799	406	6	)	)	PUNCT
cana-2799	406	7	)	)	PUNCT
cana-2799	407	1	−	−	ADP
cana-2799	408	1	𝛼	𝛼	X
cana-2799	408	2	]	]	X
cana-2799	408	3	,	,	PUNCT
cana-2799	408	4	then	then	ADV
cana-2799	408	5	the	the	DET
cana-2799	408	6	function	function	NOUN
cana-2799	408	7	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	408	8	,	,	PUNCT
cana-2799	408	9	𝜅2	𝜅2	ADJ
cana-2799	408	10	)	)	PUNCT
cana-2799	408	11	=	=	SYM
cana-2799	408	12	𝑒1((𝜅1	𝑒1((𝜅1	NUM
cana-2799	408	13	)	)	PUNCT
cana-2799	408	14	)	)	PUNCT
cana-2799	408	15	.	.	PUNCT
cana-2799	409	1	𝑒1((𝜅2	𝑒1((𝜅2	X
cana-2799	409	2	)	)	PUNCT
cana-2799	409	3	)	)	PUNCT
cana-2799	410	1	is	be	AUX
cana-2799	410	2	the	the	DET
cana-2799	410	3	exact	exact	ADJ
cana-2799	410	4	solution	solution	NOUN
cana-2799	410	5	of	of	ADP
cana-2799	410	6	the	the	DET
cana-2799	410	7	𝛼difference	𝛼difference	NOUN
cana-2799	410	8	equation	equation	NOUN
cana-2799	410	9	(	(	PUNCT
cana-2799	410	10	28	28	NUM
cana-2799	410	11	)	)	PUNCT
cana-2799	410	12	.	.	PUNCT
cana-2799	411	1	proof	proof	NOUN
cana-2799	411	2	.	.	PUNCT
cana-2799	412	1	by	by	ADP
cana-2799	412	2	applying	apply	VERB
cana-2799	412	3	the	the	DET
cana-2799	412	4	theorem	theorem	NOUN
cana-2799	412	5	6	6	NUM
cana-2799	412	6	,	,	PUNCT
cana-2799	412	7	we	we	PRON
cana-2799	412	8	get	get	VERB
cana-2799	412	9	the	the	DET
cana-2799	412	10	proof	proof	NOUN
cana-2799	412	11	.	.	PUNCT
cana-2799	413	1	corollary	corollary	ADJ
cana-2799	413	2	6.4	6.4	NUM
cana-2799	413	3	.	.	PUNCT
cana-2799	414	1	the	the	DET
cana-2799	414	2	fractional	fractional	ADJ
cana-2799	414	3	partial	partial	ADJ
cana-2799	414	4	𝛼-difference	𝛼-difference	NOUN
cana-2799	414	5	heat	heat	NOUN
cana-2799	414	6	equation	equation	NOUN
cana-2799	414	7	(	(	PUNCT
cana-2799	414	8	28	28	NUM
cana-2799	414	9	)	)	PUNCT
cana-2799	414	10	has	have	VERB
cana-2799	414	11	a	a	DET
cana-2799	414	12	solution	solution	NOUN
cana-2799	414	13	of	of	ADP
cana-2799	414	14	the	the	DET
cana-2799	414	15	form	form	NOUN
cana-2799	414	16	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	414	17	,	,	PUNCT
cana-2799	414	18	𝜅2	𝜅2	ADJ
cana-2799	414	19	)	)	PUNCT
cana-2799	414	20	=	=	SYM
cana-2799	414	21	𝑒1((𝜅1	𝑒1((𝜅1	NUM
cana-2799	414	22	)	)	PUNCT
cana-2799	414	23	)	)	PUNCT
cana-2799	414	24	.	.	PUNCT
cana-2799	415	1	𝑒1((𝜅2	𝑒1((𝜅2	X
cana-2799	415	2	)	)	PUNCT
cana-2799	415	3	)	)	PUNCT
cana-2799	416	1	if	if	SCONJ
cana-2799	416	2	𝛾	𝛾	NOUN
cana-2799	416	3	=	=	SYM
cana-2799	417	1	[	[	X
cana-2799	417	2	(	(	PUNCT
cana-2799	417	3	𝑒1(1	𝑒1(1	ADJ
cana-2799	417	4	)	)	PUNCT
cana-2799	417	5	−	−	PRON
cana-2799	417	6	𝛼)𝜈/(𝑒1(±(1	𝛼)𝜈/(𝑒1(±(1	NOUN
cana-2799	417	7	)	)	PUNCT
cana-2799	417	8	)	)	PUNCT
cana-2799	418	1	−	−	ADP
cana-2799	418	2	𝛼)𝜈	𝛼)𝜈	NOUN
cana-2799	418	3	]	]	PUNCT
cana-2799	418	4	.	.	PUNCT
cana-2799	419	1	assume	assume	VERB
cana-2799	419	2	that	that	SCONJ
cana-2799	419	3	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	419	4	,	,	PUNCT
cana-2799	419	5	𝜅2	𝜅2	ADJ
cana-2799	419	6	,	,	PUNCT
cana-2799	419	7	𝜅3	𝜅3	ADJ
cana-2799	419	8	)	)	PUNCT
cana-2799	419	9	be	be	VERB
cana-2799	419	10	the	the	DET
cana-2799	419	11	temperature	temperature	NOUN
cana-2799	419	12	of	of	ADP
cana-2799	419	13	a	a	DET
cana-2799	419	14	thin	thin	ADJ
cana-2799	419	15	plate	plate	NOUN
cana-2799	419	16	at	at	ADP
cana-2799	419	17	position	position	NOUN
cana-2799	419	18	(	(	PUNCT
cana-2799	419	19	𝜅1	𝜅1	NOUN
cana-2799	419	20	,	,	PUNCT
cana-2799	419	21	𝜅2	𝜅2	PROPN
cana-2799	419	22	)	)	PUNCT
cana-2799	419	23	at	at	ADP
cana-2799	419	24	time	time	NOUN
cana-2799	419	25	𝜅3	𝜅3	PROPN
cana-2799	419	26	.	.	PUNCT
cana-2799	420	1	let	let	VERB
cana-2799	420	2	(	(	PUNCT
cana-2799	420	3	1,1,1	1,1,1	NUM
cana-2799	420	4	)	)	PUNCT
cana-2799	420	5	be	be	AUX
cana-2799	420	6	the	the	DET
cana-2799	420	7	shift	shift	NOUN
cana-2799	420	8	values	value	NOUN
cana-2799	420	9	of	of	ADP
cana-2799	420	10	(	(	PUNCT
cana-2799	420	11	𝜅1	𝜅1	ADJ
cana-2799	420	12	,	,	PUNCT
cana-2799	420	13	𝜅2	𝜅2	PROPN
cana-2799	420	14	)	)	PUNCT
cana-2799	420	15	and	and	CCONJ
cana-2799	420	16	𝜅3	𝜅3	ADJ
cana-2799	420	17	and	and	CCONJ
cana-2799	420	18	𝛾	𝛾	PART
cana-2799	420	19	be	be	AUX
cana-2799	420	20	the	the	DET
cana-2799	420	21	rate	rate	NOUN
cana-2799	420	22	of	of	ADP
cana-2799	420	23	conductivity	conductivity	NOUN
cana-2799	420	24	of	of	ADP
cana-2799	420	25	thin	thin	ADJ
cana-2799	420	26	plate	plate	NOUN
cana-2799	420	27	.	.	PUNCT
cana-2799	421	1	the	the	DET
cana-2799	421	2	fractional	fractional	ADJ
cana-2799	421	3	partial	partial	ADJ
cana-2799	421	4	𝛼-difference	𝛼-difference	NOUN
cana-2799	421	5	heat	heat	NOUN
cana-2799	421	6	equation	equation	NOUN
cana-2799	421	7	of	of	ADP
cana-2799	421	8	thin	thin	ADJ
cana-2799	421	9	plate	plate	NOUN
cana-2799	421	10	is	be	AUX
cana-2799	421	11	given	give	VERB
cana-2799	421	12	by	by	ADP
cana-2799	421	13	δ	δ	PROPN
cana-2799	421	14	𝜈	𝜈	X
cana-2799	421	15	𝛼	𝛼	X
cana-2799	421	16	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	421	17	,	,	PUNCT
cana-2799	421	18	𝜅2	𝜅2	ADJ
cana-2799	421	19	,	,	PUNCT
cana-2799	421	20	𝜅3	𝜅3	ADJ
cana-2799	421	21	)	)	PUNCT
cana-2799	421	22	=	=	SYM
cana-2799	421	23	𝛾	𝛾	AUX
cana-2799	421	24	{	{	PUNCT
cana-2799	421	25	δ	δ	X
cana-2799	421	26	𝜈	𝜈	X
cana-2799	421	27	𝛼	𝛼	X
cana-2799	421	28	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	421	29	,	,	PUNCT
cana-2799	421	30	𝜅2	𝜅2	ADJ
cana-2799	421	31	,	,	PUNCT
cana-2799	421	32	𝜅3	𝜅3	ADJ
cana-2799	421	33	)	)	PUNCT
cana-2799	421	34	+	+	NUM
cana-2799	421	35	δ	δ	X
cana-2799	421	36	𝜈	𝜈	X
cana-2799	421	37	𝛼	𝛼	X
cana-2799	421	38	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	421	39	,	,	PUNCT
cana-2799	421	40	𝜅2	𝜅2	ADJ
cana-2799	421	41	,	,	PUNCT
cana-2799	421	42	𝜅3	𝜅3	ADJ
cana-2799	421	43	)	)	PUNCT
cana-2799	421	44	}	}	PUNCT
cana-2799	421	45	.	.	PUNCT
cana-2799	422	1	(	(	PUNCT
cana-2799	422	2	29	29	NUM
cana-2799	422	3	)	)	PUNCT
cana-2799	422	4	corollary	corollary	NOUN
cana-2799	422	5	6.5	6.5	NUM
cana-2799	422	6	.	.	PUNCT
cana-2799	423	1	if	if	SCONJ
cana-2799	423	2	𝛾	𝛾	NOUN
cana-2799	423	3	=	=	PUNCT
cana-2799	424	1	[	[	X
cana-2799	424	2	1	1	NUM
cana-2799	424	3	+	+	SYM
cana-2799	424	4	1	1	NUM
cana-2799	424	5	−	−	NUM
cana-2799	424	6	𝛼)𝜈/(𝑒1(±(1)1	𝛼)𝜈/(𝑒1(±(1)1	NUM
cana-2799	424	7	)	)	PUNCT
cana-2799	424	8	−	−	ADP
cana-2799	424	9	𝛼)𝜈	𝛼)𝜈	NOUN
cana-2799	424	10	]	]	PUNCT
cana-2799	424	11	,	,	PUNCT
cana-2799	424	12	then	then	ADV
cana-2799	424	13	the	the	DET
cana-2799	424	14	function𝑣(𝜅	function𝑣(𝜅	NOUN
cana-2799	424	15	)	)	PUNCT
cana-2799	424	16	=	=	VERB
cana-2799	424	17	∏3	∏3	NOUN
cana-2799	424	18	𝑖=1	𝑖=1	PROPN
cana-2799	424	19	𝑒1(𝜅𝑖(1𝑖	𝑒1(𝜅𝑖(1𝑖	NOUN
cana-2799	424	20	)	)	PUNCT
cana-2799	424	21	)	)	PUNCT
cana-2799	424	22	is	be	AUX
cana-2799	424	23	an	an	DET
cana-2799	424	24	exact	exact	ADJ
cana-2799	424	25	solution	solution	NOUN
cana-2799	424	26	of	of	ADP
cana-2799	424	27	the	the	DET
cana-2799	424	28	fractional	fractional	ADJ
cana-2799	424	29	partial	partial	ADJ
cana-2799	424	30	heat	heat	NOUN
cana-2799	424	31	equation	equation	NOUN
cana-2799	424	32	(	(	PUNCT
cana-2799	424	33	29	29	NUM
cana-2799	424	34	)	)	PUNCT
cana-2799	424	35	.	.	PUNCT
cana-2799	425	1	assume	assume	VERB
cana-2799	425	2	that	that	SCONJ
cana-2799	425	3	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	425	4	,	,	PUNCT
cana-2799	425	5	𝜅2	𝜅2	ADJ
cana-2799	425	6	,	,	PUNCT
cana-2799	425	7	𝜅3	𝜅3	ADJ
cana-2799	425	8	,	,	PUNCT
cana-2799	425	9	𝜅4	𝜅4	NOUN
cana-2799	425	10	,	,	PUNCT
cana-2799	425	11	𝜅5	𝜅5	NOUN
cana-2799	425	12	)	)	PUNCT
cana-2799	425	13	be	be	VERB
cana-2799	425	14	the	the	DET
cana-2799	425	15	temperature	temperature	NOUN
cana-2799	425	16	of	of	ADP
cana-2799	425	17	a	a	DET
cana-2799	425	18	medium	medium	NOUN
cana-2799	425	19	at	at	ADP
cana-2799	425	20	position	position	NOUN
cana-2799	425	21	(	(	PUNCT
cana-2799	425	22	𝜅1	𝜅1	NOUN
cana-2799	425	23	,	,	PUNCT
cana-2799	425	24	𝜅2	𝜅2	PROPN
cana-2799	425	25	,	,	PUNCT
cana-2799	425	26	𝜅3	𝜅3	ADJ
cana-2799	425	27	)	)	PUNCT
cana-2799	425	28	at	at	ADP
cana-2799	425	29	time	time	NOUN
cana-2799	425	30	𝜅4	𝜅4	VERB
cana-2799	425	31	and	and	CCONJ
cana-2799	425	32	at	at	ADP
cana-2799	425	33	density	density	NOUN
cana-2799	425	34	𝜅5	𝜅5	PROPN
cana-2799	425	35	.	.	PUNCT
cana-2799	426	1	let	let	VERB
cana-2799	426	2	(	(	PUNCT
cana-2799	426	3	1,1,1,1,1	1,1,1,1,1	NUM
cana-2799	426	4	)	)	PUNCT
cana-2799	426	5	be	be	AUX
cana-2799	426	6	the	the	DET
cana-2799	426	7	shift	shift	NOUN
cana-2799	426	8	values	value	NOUN
cana-2799	426	9	of	of	ADP
cana-2799	426	10	(	(	PUNCT
cana-2799	426	11	𝜅1	𝜅1	PROPN
cana-2799	426	12	,	,	PUNCT
cana-2799	426	13	𝜅2	𝜅2	PROPN
cana-2799	426	14	,	,	PUNCT
cana-2799	426	15	𝜅3	𝜅3	ADJ
cana-2799	426	16	)	)	PUNCT
cana-2799	426	17	,	,	PUNCT
cana-2799	426	18	𝜅4	𝜅4	NOUN
cana-2799	426	19	and	and	CCONJ
cana-2799	426	20	𝜅5	𝜅5	NOUN
cana-2799	426	21	and	and	CCONJ
cana-2799	426	22	𝛾	𝛾	AUX
cana-2799	426	23	be	be	AUX
cana-2799	426	24	the	the	DET
cana-2799	426	25	rate	rate	NOUN
cana-2799	426	26	of	of	ADP
cana-2799	426	27	conductivity	conductivity	NOUN
cana-2799	426	28	of	of	ADP
cana-2799	426	29	medium	medium	NOUN
cana-2799	426	30	.	.	PUNCT
cana-2799	427	1	the	the	DET
cana-2799	427	2	fractional	fractional	ADJ
cana-2799	427	3	partial	partial	ADJ
cana-2799	427	4	𝛼-difference	𝛼-difference	NOUN
cana-2799	427	5	equation	equation	NOUN
cana-2799	427	6	of	of	ADP
cana-2799	427	7	heat	heat	NOUN
cana-2799	427	8	flow	flow	NOUN
cana-2799	427	9	in	in	ADP
cana-2799	427	10	medium	medium	NOUN
cana-2799	427	11	is	be	AUX
cana-2799	427	12	δ	δ	X
cana-2799	427	13	𝜈	𝜈	X
cana-2799	427	14	𝛼	𝛼	X
cana-2799	427	15	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	427	16	,	,	PUNCT
cana-2799	427	17	𝜅2	𝜅2	ADJ
cana-2799	427	18	,	,	PUNCT
cana-2799	427	19	𝜅3	𝜅3	ADJ
cana-2799	427	20	,	,	PUNCT
cana-2799	427	21	𝜅4	𝜅4	NOUN
cana-2799	427	22	,	,	PUNCT
cana-2799	427	23	𝜅5	𝜅5	NOUN
cana-2799	427	24	)	)	PUNCT
cana-2799	427	25	=	=	SYM
cana-2799	428	1	𝛾	𝛾	PROPN
cana-2799	428	2	{	{	PUNCT
cana-2799	428	3	δ	δ	PROPN
cana-2799	428	4	𝜈	𝜈	X
cana-2799	428	5	𝛼(±1	𝛼(±1	PROPN
cana-2799	428	6	)	)	PUNCT
cana-2799	428	7	𝑣(𝜅1	𝑣(𝜅1	NOUN
cana-2799	428	8	,	,	PUNCT
cana-2799	428	9	𝜅2	𝜅2	NOUN
cana-2799	428	10	,	,	PUNCT
cana-2799	428	11	𝜅3	𝜅3	ADJ
cana-2799	428	12	,	,	PUNCT
cana-2799	428	13	𝜅4	𝜅4	NOUN
cana-2799	428	14	,	,	PUNCT
cana-2799	428	15	𝜅5	𝜅5	NOUN
cana-2799	428	16	)	)	PUNCT
cana-2799	428	17	}	}	PUNCT
cana-2799	428	18	.	.	PUNCT
cana-2799	429	1	(	(	PUNCT
cana-2799	429	2	30	30	NUM
cana-2799	429	3	)	)	PUNCT
cana-2799	429	4	corollary	corollary	NOUN
cana-2799	429	5	6.6	6.6	NUM
cana-2799	429	6	.	.	PUNCT
cana-2799	430	1	if	if	SCONJ
cana-2799	430	2	𝛾	𝛾	NOUN
cana-2799	430	3	=	=	PUNCT
cana-2799	431	1	[	[	X
cana-2799	431	2	1	1	NUM
cana-2799	431	3	+	+	SYM
cana-2799	431	4	1	1	NUM
cana-2799	431	5	−	−	NOUN
cana-2799	431	6	𝛼)𝜈	𝛼)𝜈	NOUN
cana-2799	431	7	+	+	CCONJ
cana-2799	431	8	𝑒1(1	𝑒1(1	ADJ
cana-2799	431	9	+	+	X
cana-2799	431	10	1	1	NUM
cana-2799	431	11	−	−	NUM
cana-2799	431	12	𝛼)𝜈/(𝑒1(±(1)1	𝛼)𝜈/(𝑒1(±(1)1	NUM
cana-2799	431	13	)	)	PUNCT
cana-2799	431	14	−	−	ADP
cana-2799	431	15	𝛼)𝜈	𝛼)𝜈	NOUN
cana-2799	431	16	]	]	PUNCT
cana-2799	431	17	,	,	PUNCT
cana-2799	431	18	then	then	ADV
cana-2799	431	19	𝑣(𝜅	𝑣(𝜅	PROPN
cana-2799	431	20	)	)	PUNCT
cana-2799	431	21	=	=	PUNCT
cana-2799	431	22	∏5	∏5	NUM
cana-2799	431	23	𝑖=1	𝑖=1	PROPN
cana-2799	431	24	𝑒1(𝜅𝑖(1𝑖	𝑒1(𝜅𝑖(1𝑖	NOUN
cana-2799	431	25	)	)	PUNCT
cana-2799	431	26	)	)	PUNCT
cana-2799	431	27	is	be	AUX
cana-2799	431	28	a	a	DET
cana-2799	431	29	closed	closed	ADJ
cana-2799	431	30	form	form	NOUN
cana-2799	431	31	solution	solution	NOUN
cana-2799	431	32	of	of	ADP
cana-2799	431	33	the	the	DET
cana-2799	431	34	fractional	fractional	ADJ
cana-2799	431	35	partial	partial	ADJ
cana-2799	431	36	𝛼-difference	𝛼-difference	NOUN
cana-2799	431	37	equation	equation	NOUN
cana-2799	431	38	(	(	PUNCT
cana-2799	431	39	30	30	NUM
cana-2799	431	40	)	)	PUNCT
cana-2799	431	41	.	.	PUNCT
cana-2799	432	1	7	7	X
cana-2799	432	2	.	.	X
cana-2799	432	3	conclusion	conclusion	NOUN
cana-2799	432	4	in	in	ADP
cana-2799	432	5	conclusion	conclusion	NOUN
cana-2799	432	6	,	,	PUNCT
cana-2799	432	7	the	the	DET
cana-2799	432	8	resistor	resistor	NOUN
cana-2799	432	9	and	and	CCONJ
cana-2799	432	10	inductor	inductor	NOUN
cana-2799	432	11	,	,	PUNCT
cana-2799	432	12	as	as	ADP
cana-2799	432	13	fundamental	fundamental	ADJ
cana-2799	432	14	linear	linear	ADJ
cana-2799	432	15	and	and	CCONJ
cana-2799	432	16	passive	passive	ADJ
cana-2799	432	17	circuit	circuit	NOUN
cana-2799	432	18	elements	element	NOUN
cana-2799	432	19	,	,	PUNCT
cana-2799	432	20	form	form	VERB
cana-2799	432	21	the	the	DET
cana-2799	432	22	basis	basis	NOUN
cana-2799	432	23	of	of	ADP
cana-2799	432	24	rl	rl	NOUN
cana-2799	432	25	circuits	circuit	NOUN
cana-2799	432	26	,	,	PUNCT
cana-2799	432	27	which	which	PRON
cana-2799	432	28	can	can	AUX
cana-2799	432	29	be	be	AUX
cana-2799	432	30	configured	configure	VERB
cana-2799	432	31	in	in	ADP
cana-2799	432	32	series	series	NOUN
cana-2799	432	33	or	or	CCONJ
cana-2799	432	34	parallel	parallel	NOUN
cana-2799	432	35	.	.	PUNCT
cana-2799	433	1	the	the	DET
cana-2799	433	2	mathematical	mathematical	ADJ
cana-2799	433	3	analysis	analysis	NOUN
cana-2799	433	4	of	of	ADP
cana-2799	433	5	rl	rl	ADP
cana-2799	433	6	communications	communication	NOUN
cana-2799	433	7	on	on	ADP
cana-2799	433	8	applied	apply	VERB
cana-2799	433	9	nonlinear	nonlinear	ADJ
cana-2799	433	10	analysis	analysis	NOUN
cana-2799	433	11	issn	issn	NOUN
cana-2799	433	12	:	:	PUNCT
cana-2799	433	13	1074	1074	NUM
cana-2799	433	14	-	-	PUNCT
cana-2799	433	15	133x	133x	NUM
cana-2799	433	16	vol	vol	NOUN
cana-2799	433	17	32	32	NUM
cana-2799	433	18	no	no	NOUN
cana-2799	433	19	.	.	PUNCT
cana-2799	434	1	4s	4s	NUM
cana-2799	434	2	(	(	PUNCT
cana-2799	434	3	2025	2025	NUM
cana-2799	434	4	)	)	PUNCT
cana-2799	434	5	269	269	NUM
cana-2799	434	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2799	434	7	circuits	circuit	NOUN
cana-2799	434	8	involves	involve	VERB
cana-2799	434	9	differential	differential	ADJ
cana-2799	434	10	equations	equation	NOUN
cana-2799	434	11	,	,	PUNCT
cana-2799	434	12	with	with	SCONJ
cana-2799	434	13	solutions	solution	NOUN
cana-2799	434	14	often	often	ADV
cana-2799	434	15	expressed	express	VERB
cana-2799	434	16	in	in	ADP
cana-2799	434	17	terms	term	NOUN
cana-2799	434	18	of	of	ADP
cana-2799	434	19	extorial	extorial	ADJ
cana-2799	434	20	functions	function	NOUN
cana-2799	434	21	.	.	PUNCT
cana-2799	435	1	by	by	ADP
cana-2799	435	2	developing	develop	VERB
cana-2799	435	3	the	the	DET
cana-2799	435	4	theory	theory	NOUN
cana-2799	435	5	of	of	ADP
cana-2799	435	6	extorial	extorial	ADJ
cana-2799	435	7	functions	function	NOUN
cana-2799	435	8	,	,	PUNCT
cana-2799	435	9	we	we	PRON
cana-2799	435	10	successfully	successfully	ADV
cana-2799	435	11	applied	apply	VERB
cana-2799	435	12	it	it	PRON
cana-2799	435	13	to	to	PART
cana-2799	435	14	derive	derive	VERB
cana-2799	435	15	solutions	solution	NOUN
cana-2799	435	16	for	for	ADP
cana-2799	435	17	rl	rl	NOUN
cana-2799	435	18	circuits	circuit	NOUN
cana-2799	435	19	and	and	CCONJ
cana-2799	435	20	extended	extend	VERB
cana-2799	435	21	its	its	PRON
cana-2799	435	22	utility	utility	NOUN
cana-2799	435	23	to	to	ADP
cana-2799	435	24	solving	solve	VERB
cana-2799	435	25	problems	problem	NOUN
cana-2799	435	26	in	in	ADP
cana-2799	435	27	wave	wave	NOUN
cana-2799	435	28	motion	motion	NOUN
cana-2799	435	29	,	,	PUNCT
cana-2799	435	30	demonstrating	demonstrate	VERB
cana-2799	435	31	its	its	PRON
cana-2799	435	32	versatility	versatility	NOUN
cana-2799	435	33	and	and	CCONJ
cana-2799	435	34	practical	practical	ADJ
cana-2799	435	35	significance	significance	NOUN
cana-2799	435	36	.	.	PUNCT
cana-2799	436	1	references	reference	NOUN
cana-2799	436	2	[	[	X
cana-2799	436	3	1	1	NUM
cana-2799	436	4	]	]	PUNCT
cana-2799	436	5	britto	britto	PROPN
cana-2799	436	6	antony	antony	PROPN
cana-2799	436	7	xavier.g	xavier.g	PROPN
cana-2799	436	8	,	,	PUNCT
cana-2799	436	9	gerly.t.g	gerly.t.g	NOUN
cana-2799	436	10	and	and	CCONJ
cana-2799	436	11	nasira	nasira	ADP
cana-2799	436	12	begum.h	begum.h	PROPN
cana-2799	436	13	,	,	PUNCT
cana-2799	436	14	finite	finite	PROPN
cana-2799	436	15	series	series	NOUN
cana-2799	436	16	of	of	ADP
cana-2799	436	17	polynomials	polynomial	NOUN
cana-2799	436	18	and	and	CCONJ
cana-2799	436	19	polynomial	polynomial	ADJ
cana-2799	436	20	factorials	factorial	NOUN
cana-2799	436	21	arising	arise	VERB
cana-2799	436	22	from	from	ADP
cana-2799	436	23	generalized	generalized	ADJ
cana-2799	436	24	q	q	ADJ
cana-2799	436	25	-	-	PUNCT
cana-2799	436	26	difference	difference	NOUN
cana-2799	436	27	operator	operator	NOUN
cana-2799	436	28	,	,	PUNCT
cana-2799	436	29	far	far	PROPN
cana-2799	436	30	east	east	PROPN
cana-2799	436	31	journal	journal	PROPN
cana-2799	436	32	of	of	ADP
cana-2799	436	33	mathematical	mathematical	ADJ
cana-2799	436	34	sciences,94(1)(2014	sciences,94(1)(2014	PROPN
cana-2799	436	35	)	)	PUNCT
cana-2799	436	36	,	,	PUNCT
cana-2799	436	37	47	47	NUM
cana-2799	436	38	-	-	SYM
cana-2799	436	39	63	63	NUM
cana-2799	436	40	.	.	PUNCT
cana-2799	437	1	[	[	X
cana-2799	437	2	2	2	NUM
cana-2799	437	3	]	]	PUNCT
cana-2799	437	4	britto	britto	PROPN
cana-2799	437	5	antony	antony	PROPN
cana-2799	437	6	xavier.g	xavier.g	PROPN
cana-2799	437	7	,	,	PUNCT
cana-2799	437	8	john	john	PROPN
cana-2799	437	9	borg	borg	PROPN
cana-2799	437	10	.	.	PUNCT
cana-2799	438	1	s	s	X
cana-2799	438	2	,	,	PUNCT
cana-2799	438	3	meganathan	meganathan	PROPN
cana-2799	438	4	.	.	PUNCT
cana-2799	439	1	m	m	PROPN
cana-2799	439	2	,	,	PUNCT
cana-2799	439	3	discrete	discrete	ADJ
cana-2799	439	4	heat	heat	NOUN
cana-2799	439	5	equation	equation	NOUN
cana-2799	439	6	model	model	NOUN
cana-2799	439	7	with	with	ADP
cana-2799	439	8	shift	shift	NOUN
cana-2799	439	9	values	value	NOUN
cana-2799	439	10	,	,	PUNCT
cana-2799	439	11	applied	apply	VERB
cana-2799	439	12	mathematics	mathematic	NOUN
cana-2799	439	13	,	,	PUNCT
cana-2799	439	14	2017	2017	NUM
cana-2799	439	15	,	,	PUNCT
cana-2799	439	16	8	8	NUM
cana-2799	439	17	,	,	PUNCT
cana-2799	439	18	1343	1343	NUM
cana-2799	439	19	-	-	SYM
cana-2799	439	20	1350	1350	NUM
cana-2799	439	21	.	.	PUNCT
cana-2799	440	1	[	[	X
cana-2799	440	2	3	3	NUM
cana-2799	440	3	]	]	PUNCT
cana-2799	440	4	weisstein	weisstein	ADV
cana-2799	440	5	,	,	PUNCT
cana-2799	440	6	eric	eric	PROPN
cana-2799	440	7	.	.	PUNCT
cana-2799	441	1	harmonic	harmonic	PROPN
cana-2799	441	2	series	series	PROPN
cana-2799	441	3	mathworld	mathworld	PROPN
cana-2799	441	4	.	.	PUNCT
cana-2799	441	5	web	web	PROPN
cana-2799	441	6	.	.	PROPN
cana-2799	442	1	24	24	NUM
cana-2799	442	2	august	august	PROPN
cana-2799	442	3	2014	2014	NUM
cana-2799	442	4	.	.	PUNCT
cana-2799	443	1	[	[	X
cana-2799	443	2	4	4	NUM
cana-2799	443	3	]	]	X
cana-2799	443	4	jerzy	jerzy	X
cana-2799	443	5	popenda	popenda	NOUN
cana-2799	443	6	and	and	CCONJ
cana-2799	443	7	blazej	blazej	NOUN
cana-2799	443	8	szmanda	szmanda	ADV
cana-2799	443	9	,	,	PUNCT
cana-2799	443	10	on	on	ADP
cana-2799	443	11	the	the	DET
cana-2799	443	12	oscillation	oscillation	NOUN
cana-2799	443	13	of	of	ADP
cana-2799	443	14	solutions	solution	NOUN
cana-2799	443	15	of	of	ADP
cana-2799	443	16	certain	certain	ADJ
cana-2799	443	17	difference	difference	NOUN
cana-2799	443	18	equations	equation	NOUN
cana-2799	443	19	,	,	PUNCT
cana-2799	443	20	demonstratio	demonstratio	PROPN
cana-2799	443	21	mathematica	mathematica	PROPN
cana-2799	443	22	,	,	PUNCT
cana-2799	443	23	xvii(1	xvii(1	PROPN
cana-2799	443	24	)	)	PUNCT
cana-2799	443	25	,	,	PUNCT
cana-2799	443	26	(	(	PUNCT
cana-2799	443	27	1984	1984	NUM
cana-2799	443	28	)	)	PUNCT
cana-2799	443	29	,	,	PUNCT
cana-2799	443	30	153	153	NUM
cana-2799	443	31	164	164	NUM
cana-2799	443	32	.	.	PUNCT
cana-2799	444	1	[	[	X
cana-2799	444	2	5	5	X
cana-2799	444	3	]	]	PUNCT
cana-2799	444	4	maria	maria	PROPN
cana-2799	444	5	susai	susai	PROPN
cana-2799	444	6	manuel.m	manuel.m	PROPN
cana-2799	444	7	,	,	PUNCT
cana-2799	444	8	chandrasekar.v	chandrasekar.v	PUNCT
cana-2799	444	9	and	and	CCONJ
cana-2799	444	10	britto	britto	PROPN
cana-2799	444	11	antony	antony	PROPN
cana-2799	444	12	xavier.g	xavier.g	PROPN
cana-2799	444	13	,	,	PUNCT
cana-2799	444	14	solutions	solution	NOUN
cana-2799	444	15	and	and	CCONJ
cana-2799	444	16	applications	application	NOUN
cana-2799	444	17	of	of	ADP
cana-2799	444	18	certain	certain	ADJ
cana-2799	444	19	class	class	NOUN
cana-2799	444	20	of	of	ADP
cana-2799	444	21	𝛼-difference	𝛼-difference	NOUN
cana-2799	444	22	equations	equation	NOUN
cana-2799	444	23	,	,	PUNCT
cana-2799	444	24	international	international	ADJ
cana-2799	444	25	journal	journal	NOUN
cana-2799	444	26	of	of	ADP
cana-2799	444	27	applied	apply	VERB
cana-2799	444	28	mathematics	mathematic	NOUN
cana-2799	444	29	,	,	PUNCT
cana-2799	444	30	24(6	24(6	NUM
cana-2799	444	31	)	)	PUNCT
cana-2799	444	32	(	(	PUNCT
cana-2799	444	33	2011	2011	NUM
cana-2799	444	34	)	)	PUNCT
cana-2799	444	35	,	,	PUNCT
cana-2799	444	36	943	943	NUM
cana-2799	444	37	-	-	SYM
cana-2799	444	38	954	954	NUM
cana-2799	444	39	.	.	PUNCT
cana-2799	445	1	[	[	X
cana-2799	445	2	6	6	NUM
cana-2799	445	3	]	]	PUNCT
cana-2799	445	4	m.maria	m.maria	NOUN
cana-2799	445	5	susai	susai	PROPN
cana-2799	445	6	manuel	manuel	NOUN
cana-2799	445	7	,	,	PUNCT
cana-2799	445	8	g.britto	g.britto	PROPN
cana-2799	445	9	antony	antony	PROPN
cana-2799	445	10	xavier	xavier	PROPN
cana-2799	445	11	and	and	CCONJ
cana-2799	445	12	v.chandrasekar	v.chandrasekar	NOUN
cana-2799	445	13	,	,	PUNCT
cana-2799	445	14	theory	theory	NOUN
cana-2799	445	15	of	of	ADP
cana-2799	445	16	generalized	generalized	ADJ
cana-2799	445	17	difference	difference	NOUN
cana-2799	445	18	operator	operator	NOUN
cana-2799	445	19	of	of	ADP
cana-2799	445	20	the	the	DET
cana-2799	445	21	𝑛𝑡ℎ	𝑛𝑡ℎ	PRON
cana-2799	445	22	kind	kind	VERB
cana-2799	445	23	and	and	CCONJ
cana-2799	445	24	its	its	PRON
cana-2799	445	25	application	application	NOUN
cana-2799	445	26	to	to	ADP
cana-2799	445	27	number	number	NOUN
cana-2799	445	28	theory	theory	NOUN
cana-2799	445	29	(	(	PUNCT
cana-2799	445	30	part	part	NOUN
cana-2799	445	31	i	i	NOUN
cana-2799	445	32	)	)	PUNCT
cana-2799	445	33	,	,	PUNCT
cana-2799	445	34	international	international	ADJ
cana-2799	445	35	journal	journal	NOUN
cana-2799	445	36	of	of	ADP
cana-2799	445	37	pure	pure	ADJ
cana-2799	445	38	and	and	CCONJ
cana-2799	445	39	applied	applied	ADJ
cana-2799	445	40	mathematics	mathematic	NOUN
cana-2799	445	41	[	[	X
cana-2799	445	42	7	7	X
cana-2799	445	43	]	]	X
cana-2799	445	44	r.p	r.p	PROPN
cana-2799	445	45	agarwal	agarwal	PROPN
cana-2799	445	46	difference	difference	NOUN
cana-2799	445	47	equations	equation	NOUN
cana-2799	445	48	and	and	CCONJ
cana-2799	445	49	inequalities	inequality	NOUN
cana-2799	445	50	,	,	PUNCT
cana-2799	445	51	marceldekker	marceldekker	NOUN
cana-2799	445	52	,	,	PUNCT
cana-2799	445	53	new	new	PROPN
cana-2799	445	54	york	york	PROPN
cana-2799	445	55	,	,	PUNCT
cana-2799	445	56	2000	2000	NUM
cana-2799	445	57	.	.	PUNCT
cana-2799	446	1	[	[	X
cana-2799	446	2	8	8	NUM
cana-2799	446	3	]	]	PUNCT
cana-2799	446	4	m.maria	m.maria	NOUN
cana-2799	446	5	susai	susai	PROPN
cana-2799	446	6	manuel	manuel	NOUN
cana-2799	446	7	,	,	PUNCT
cana-2799	446	8	g.britto	g.britto	PROPN
cana-2799	446	9	antony	antony	PROPN
cana-2799	446	10	xavier	xavier	PROPN
cana-2799	446	11	and	and	CCONJ
cana-2799	446	12	e.thandapani	e.thandapani	PROPN
cana-2799	446	13	,	,	PUNCT
cana-2799	446	14	theory	theory	NOUN
cana-2799	446	15	of	of	ADP
cana-2799	446	16	generalised	generalise	VERB
cana-2799	446	17	difference	difference	NOUN
cana-2799	446	18	operator	operator	NOUN
cana-2799	446	19	and	and	CCONJ
cana-2799	446	20	its	its	PRON
cana-2799	446	21	applications	application	NOUN
cana-2799	446	22	,	,	PUNCT
cana-2799	446	23	far	far	ADV
cana-2799	446	24	east	east	PROPN
cana-2799	446	25	journal	journal	PROPN
cana-2799	446	26	of	of	ADP
cana-2799	446	27	mathematical	mathematical	ADJ
cana-2799	446	28	sciences	science	NOUN
cana-2799	446	29	,	,	PUNCT
cana-2799	446	30	vol.20,no.2(2006	vol.20,no.2(2006	ADJ
cana-2799	446	31	)	)	PUNCT
cana-2799	446	32	,	,	PUNCT
cana-2799	446	33	pp	pp	ADP
cana-2799	446	34	163	163	NUM
cana-2799	446	35	171	171	NUM
cana-2799	446	36	.	.	PUNCT
cana-2799	447	1	[	[	X
cana-2799	447	2	9	9	NUM
cana-2799	447	3	]	]	X
cana-2799	447	4	ronald	ronald	NOUN
cana-2799	447	5	e.mickens	e.micken	NOUN
cana-2799	447	6	,	,	PUNCT
cana-2799	447	7	difference	difference	NOUN
cana-2799	447	8	equations	equation	NOUN
cana-2799	447	9	,	,	PUNCT
cana-2799	447	10	van	van	PROPN
cana-2799	447	11	nostrand	nostrand	PROPN
cana-2799	447	12	reinhold	reinhold	PROPN
cana-2799	447	13	company	company	PROPN
cana-2799	447	14	,	,	PUNCT
cana-2799	447	15	new	new	PROPN
cana-2799	447	16	york	york	PROPN
cana-2799	447	17	,	,	PUNCT
cana-2799	447	18	1990	1990	NUM
cana-2799	447	19	.	.	PUNCT
cana-2799	448	1	[	[	X
cana-2799	448	2	10	10	NUM
cana-2799	448	3	]	]	PUNCT
cana-2799	448	4	ferreira.r	ferreira.r	PROPN
cana-2799	448	5	.	.	PUNCT
cana-2799	449	1	a.	a.	PROPN
cana-2799	449	2	c	c	PROPN
cana-2799	449	3	and	and	CCONJ
cana-2799	449	4	torres.d	torres.d	PROPN
cana-2799	449	5	.	.	PUNCT
cana-2799	450	1	f.	f.	PROPN
cana-2799	450	2	m	m	PROPN
cana-2799	450	3	,	,	PUNCT
cana-2799	450	4	fractional	fractional	ADJ
cana-2799	450	5	h	h	NOUN
cana-2799	450	6	-	-	PUNCT
cana-2799	450	7	difference	difference	NOUN
cana-2799	450	8	equations	equation	NOUN
cana-2799	450	9	arising	arise	VERB
cana-2799	450	10	from	from	ADP
cana-2799	450	11	the	the	DET
cana-2799	450	12	calculus	calculus	NOUN
cana-2799	450	13	of	of	ADP
cana-2799	450	14	variations	variation	NOUN
cana-2799	450	15	,	,	PUNCT
cana-2799	450	16	applicable	applicable	ADJ
cana-2799	450	17	analysis	analysis	NOUN
cana-2799	450	18	and	and	CCONJ
cana-2799	450	19	discrete	discrete	ADJ
cana-2799	450	20	mathematics	mathematic	NOUN
cana-2799	450	21	,	,	PUNCT
cana-2799	450	22	5(1	5(1	NUM
cana-2799	450	23	)	)	PUNCT
cana-2799	450	24	(	(	PUNCT
cana-2799	450	25	2011	2011	NUM
cana-2799	450	26	)	)	PUNCT
cana-2799	450	27	,	,	PUNCT
cana-2799	450	28	110	110	NUM
cana-2799	450	29	-	-	SYM
cana-2799	450	30	121	121	NUM
cana-2799	450	31	.	.	PUNCT
cana-2799	451	1	[	[	X
cana-2799	451	2	11	11	NUM
cana-2799	451	3	]	]	X
cana-2799	451	4	g.britto	g.britto	PROPN
cana-2799	451	5	antony	antony	PROPN
cana-2799	451	6	xavier	xavier	PROPN
cana-2799	451	7	,	,	PUNCT
cana-2799	451	8	s.u.vasantha	s.u.vasantha	PROPN
cana-2799	451	9	kumar	kumar	PROPN
cana-2799	451	10	,	,	PUNCT
cana-2799	451	11	and	and	CCONJ
cana-2799	451	12	b.mohan	b.mohan	ADV
cana-2799	451	13	,	,	PUNCT
cana-2799	451	14	sum	sum	NOUN
cana-2799	451	15	of	of	ADP
cana-2799	451	16	two	two	NUM
cana-2799	451	17	dimensional	dimensional	ADJ
cana-2799	451	18	fibonacci	fibonacci	NOUN
cana-2799	451	19	sequence	sequence	NOUN
cana-2799	451	20	by	by	ADP
cana-2799	451	21	solutions	solution	NOUN
cana-2799	451	22	of	of	ADP
cana-2799	451	23	higher	high	ADJ
cana-2799	451	24	order	order	NOUN
cana-2799	451	25	difference	difference	NOUN
cana-2799	451	26	equations	equation	NOUN
cana-2799	451	27	,	,	PUNCT
cana-2799	451	28	ser.a	ser.a	ADJ
cana-2799	451	29	:	:	PUNCT
cana-2799	451	30	appl.math.inform.and	appl.math.inform.and	PRON
cana-2799	451	31	mech.vol.8,2	mech.vol.8,2	NOUN
cana-2799	451	32	(	(	PUNCT
cana-2799	451	33	2016	2016	NUM
cana-2799	451	34	)	)	PUNCT
cana-2799	451	35	,	,	PUNCT
cana-2799	451	36	131	131	NUM
cana-2799	451	37	-	-	SYM
cana-2799	451	38	138	138	NUM
cana-2799	451	39	.	.	PUNCT
cana-2799	452	1	[	[	X
cana-2799	452	2	12	12	NUM
cana-2799	452	3	]	]	X
cana-2799	452	4	g.	g.	PROPN
cana-2799	452	5	britto	britto	PROPN
cana-2799	452	6	antony	antony	PROPN
cana-2799	452	7	xavier	xavier	PROPN
cana-2799	452	8	,	,	PUNCT
cana-2799	452	9	s.john	s.john	PROPN
cana-2799	452	10	borg	borg	PROPN
cana-2799	452	11	,	,	PUNCT
cana-2799	452	12	s.jaraldpushparaj	s.jaraldpushparaj	ADJ
cana-2799	452	13	,	,	PUNCT
cana-2799	452	14	extorial	extorial	ADJ
cana-2799	452	15	solutions	solution	NOUN
cana-2799	452	16	for	for	ADP
cana-2799	452	17	fractional	fractional	ADJ
cana-2799	452	18	and	and	CCONJ
cana-2799	452	19	partial	partial	ADJ
cana-2799	452	20	difference	difference	NOUN
cana-2799	452	21	equations	equation	NOUN
cana-2799	452	22	with	with	ADP
cana-2799	452	23	application	application	NOUN
cana-2799	452	24	,	,	PUNCT
cana-2799	452	25	aip	aip	PROPN
cana-2799	452	26	conference	conference	NOUN
cana-2799	452	27	proceedings	proceeding	NOUN
cana-2799	452	28	2095	2095	NUM
cana-2799	452	29	,	,	PUNCT
cana-2799	452	30	030004(2019	030004(2019	NUM
cana-2799	452	31	)	)	PUNCT
cana-2799	452	32	https://doi.org/10.1063/1.5097515	https://doi.org/10.1063/1.5097515	PROPN
cana-2799	452	33	.	.	PUNCT
cana-2799	453	1	[	[	X
cana-2799	453	2	13	13	NUM
cana-2799	453	3	]	]	PUNCT
cana-2799	453	4	britto	britto	PROPN
cana-2799	453	5	antony	antony	PROPN
cana-2799	453	6	xavier.g	xavier.g	PROPN
cana-2799	453	7	,	,	PUNCT
cana-2799	453	8	m.	m.	NOUN
cana-2799	453	9	meganathan	meganathan	PROPN
cana-2799	453	10	,	,	PUNCT
cana-2799	453	11	fractional	fractional	ADJ
cana-2799	453	12	order	order	NOUN
cana-2799	453	13	alpha	alpha	NOUN
cana-2799	453	14	laplace	laplace	NOUN
cana-2799	453	15	and	and	CCONJ
cana-2799	453	16	extorial	extorial	ADJ
cana-2799	453	17	transform	transform	NOUN
cana-2799	453	18	by	by	ADP
cana-2799	453	19	inverse	inverse	ADJ
cana-2799	453	20	difference	difference	NOUN
cana-2799	453	21	operator	operator	NOUN
cana-2799	453	22	,	,	PUNCT
cana-2799	453	23	aip	aip	PROPN
cana-2799	453	24	conference	conference	NOUN
cana-2799	453	25	proceedings	proceeding	NOUN
cana-2799	453	26	2095	2095	NUM
cana-2799	453	27	,	,	PUNCT
cana-2799	453	28	030016(2019	030016(2019	NUM
cana-2799	453	29	)	)	PUNCT
cana-2799	453	30	https://doi.org/10.1063/1.5097527	https://doi.org/10.1063/1.5097527	ADJ
cana-2799	453	31	.	.	PUNCT
cana-2799	454	1	[	[	X
cana-2799	454	2	14	14	NUM
cana-2799	454	3	]	]	X
cana-2799	454	4	sathinathan	sathinathan	PROPN
cana-2799	454	5	,	,	PUNCT
cana-2799	454	6	t.	t.	PROPN
cana-2799	454	7	,	,	PUNCT
cana-2799	454	8	sherine	sherine	PROPN
cana-2799	454	9	,	,	PUNCT
cana-2799	454	10	v.r	v.r	PROPN
cana-2799	454	11	.	.	PROPN
cana-2799	454	12	and	and	CCONJ
cana-2799	454	13	xavier	xavier	PROPN
cana-2799	454	14	,	,	PUNCT
cana-2799	454	15	g.b.a	g.b.a	PROPN
cana-2799	454	16	.	.	PUNCT
cana-2799	454	17	,	,	PUNCT
cana-2799	454	18	fractional	fractional	ADJ
cana-2799	454	19	order	order	NOUN
cana-2799	454	20	of	of	ADP
cana-2799	454	21	alpha	alpha	NOUN
cana-2799	454	22	-	-	PUNCT
cana-2799	454	23	delta	delta	NOUN
cana-2799	454	24	and	and	CCONJ
cana-2799	454	25	its	its	PRON
cana-2799	454	26	sums	sum	NOUN
cana-2799	454	27	on	on	ADP
cana-2799	454	28	extorial	extorial	ADJ
cana-2799	454	29	functions	function	NOUN
cana-2799	454	30	.	.	PUNCT
cana-2799	455	1	advances	advance	NOUN
cana-2799	455	2	in	in	ADP
cana-2799	455	3	mathematics	mathematic	NOUN
cana-2799	455	4	:	:	PUNCT
cana-2799	455	5	scientific	scientific	ADJ
cana-2799	455	6	journal	journal	NOUN
cana-2799	455	7	9	9	NUM
cana-2799	455	8	(	(	PUNCT
cana-2799	455	9	2020	2020	NUM
cana-2799	455	10	)	)	PUNCT
cana-2799	455	11	,	,	PUNCT
cana-2799	455	12	no.8	no.8	PROPN
cana-2799	455	13	,	,	PUNCT
cana-2799	455	14	6097–6106	6097–6106	NOUN
cana-2799	455	15	.	.	PUNCT
cana-2799	456	1	[	[	X
cana-2799	456	2	15	15	NUM
cana-2799	456	3	]	]	X
cana-2799	456	4	h.	h.	PROPN
cana-2799	456	5	byeon	byeon	PROPN
cana-2799	456	6	,	,	PUNCT
cana-2799	456	7	m.	m.	NOUN
cana-2799	456	8	abisha	abisha	PROPN
cana-2799	456	9	,	,	PUNCT
cana-2799	456	10	v.	v.	PROPN
cana-2799	456	11	r.	r.	PROPN
cana-2799	456	12	sherine	sherine	PROPN
cana-2799	456	13	,	,	PUNCT
cana-2799	456	14	g.	g.	PROPN
cana-2799	456	15	b.	b.	PROPN
cana-2799	456	16	a.	a.	PROPN
cana-2799	456	17	xavier	xavier	PROPN
cana-2799	456	18	,	,	PUNCT
cana-2799	456	19	s.	s.	PROPN
cana-2799	456	20	prema	prema	PROPN
cana-2799	456	21	,	,	PUNCT
cana-2799	456	22	v.	v.	ADP
cana-2799	456	23	govindan	govindan	PROPN
cana-2799	456	24	,	,	PUNCT
cana-2799	456	25	h.	h.	PROPN
cana-2799	456	26	ahmad	ahmad	PROPN
cana-2799	456	27	,	,	PUNCT
cana-2799	456	28	d.	d.	PROPN
cana-2799	456	29	piriadarshani	piriadarshani	PROPN
cana-2799	456	30	,	,	PUNCT
cana-2799	456	31	s.	s.	PROPN
cana-2799	456	32	elmorsy	elmorsy	PROPN
cana-2799	456	33	,	,	PUNCT
cana-2799	456	34	journal	journal	NOUN
cana-2799	456	35	of	of	ADP
cana-2799	456	36	mathematics	mathematic	NOUN
cana-2799	456	37	and	and	CCONJ
cana-2799	456	38	computer	computer	NOUN
cana-2799	456	39	science	science	NOUN
cana-2799	456	40	,	,	PUNCT
cana-2799	456	41	34(4	34(4	NUM
cana-2799	456	42	)	)	PUNCT
cana-2799	456	43	,	,	PUNCT
cana-2799	456	44	381	381	NUM
cana-2799	456	45	-	-	SYM
cana-2799	456	46	393	393	NUM
cana-2799	456	47	.	.	PUNCT
cana-2799	457	1	[	[	X
cana-2799	457	2	16	16	NUM
cana-2799	457	3	]	]	X
cana-2799	457	4	sherine	sherine	PROPN
cana-2799	457	5	,	,	PUNCT
cana-2799	457	6	v.r	v.r	PROPN
cana-2799	457	7	.	.	PROPN
cana-2799	457	8	,	,	PUNCT
cana-2799	457	9	gerly	gerly	ADV
cana-2799	457	10	,	,	PUNCT
cana-2799	457	11	t.	t.	PROPN
cana-2799	457	12	and	and	CCONJ
cana-2799	457	13	xavier	xavier	PROPN
cana-2799	457	14	,	,	PUNCT
cana-2799	457	15	g.b.a	g.b.a	PROPN
cana-2799	457	16	.	.	PUNCT
cana-2799	457	17	,	,	PUNCT
cana-2799	457	18	infinite	infinite	ADJ
cana-2799	457	19	series	series	NOUN
cana-2799	457	20	of	of	ADP
cana-2799	457	21	fractional	fractional	ADJ
cana-2799	457	22	order	order	NOUN
cana-2799	457	23	of	of	ADP
cana-2799	457	24	fibonacci	fibonacci	PROPN
cana-2799	457	25	delta	delta	NOUN
cana-2799	457	26	operator	operator	NOUN
cana-2799	457	27	and	and	CCONJ
cana-2799	457	28	its	its	PRON
cana-2799	457	29	sum	sum	NOUN
cana-2799	457	30	.	.	PUNCT
cana-2799	458	1	advances	advance	NOUN
cana-2799	458	2	in	in	ADP
cana-2799	458	3	mathematics	mathematic	NOUN
cana-2799	458	4	:	:	PUNCT
cana-2799	458	5	scientific	scientific	ADJ
cana-2799	458	6	journal	journal	NOUN
cana-2799	458	7	9	9	NUM
cana-2799	458	8	(	(	PUNCT
cana-2799	458	9	2020	2020	NUM
cana-2799	458	10	)	)	PUNCT
cana-2799	458	11	,	,	PUNCT
cana-2799	458	12	no.8	no.8	PROPN
cana-2799	458	13	,	,	PUNCT
cana-2799	458	14	5891–5900	5891–5900	NUM
cana-2799	458	15	.	.	PUNCT