• 検索結果がありません。

On Some Doubly Infinite, Finite and Mixed Sums derived from The N- Fractional Calculus of A Power Function(Sakaguchi Functions in Univalent Function Theory and Its Applications)

N/A
N/A
Protected

Academic year: 2021

シェア "On Some Doubly Infinite, Finite and Mixed Sums derived from The N- Fractional Calculus of A Power Function(Sakaguchi Functions in Univalent Function Theory and Its Applications)"

Copied!
9
0
0

読み込み中.... (全文を見る)

全文

(1)

18

On

Some

Doubly

Infinite,

Finite and Mixed Sums

derived

from The N

-

Fractional

Calculus

of A

Power

Function

Katsuyuki

Nishimoto

Abstract

In

a

previous

paper,

some

doubly infinite,

finite and mixed

sums are

reported

using

the

N-

fractional calculus

$((z-c)^{a\cdot*\beta})_{\gamma}$

by

the author

and his

colleagues.

In

this

article

the

same

doubly infinite

sums

in

a

previous

paper

are

discussed

again

using

$((z-c)^{\beta}\cdot(z -c)^{a})_{\gamma}$

,

the

$\mathrm{N}$

-fractional

calculus of

products

of power

functions.

\S 0.

Introduction

(

Definition

of

Fractional Calculus

)

(I)

Definition.

(by

K.

Nishimoto

)([1 ]

Vol.

1)

Let

$D=\{D_{-}, D_{+}\}$

,

$C=\{C_{-}, C_{+}\}$

,

$C_{-}$

be

a

curve

along

the

cut joining two points

$z$

and

$-\infty+\mathrm{i}{\rm Im}(z)$

,

$C_{+}$

be

a

curve

along

the

cut joining two points

$z$

and

$\infty+\mathrm{i}{\rm Im}(z)$

,

$D_{-}$

be

a

domain surrounded

by

C-,

$D_{+}$

be

a

domain

surrounded

by

$C_{+}$

.

$\langle$

Here

$D$

contains

the

points

over

the

curve

$C$

).

Moreover,

let

$f=f(z)$

be

a

regular

function in

$D(z\in D)$

,

$f_{\backslash },(z)=(J)_{\mathrm{v}}=_{c}(f)_{\mathrm{v}}= \frac{\Gamma(\mathrm{v}+1)}{2\pi i}\int_{c}^{\frac{f(\zeta)}{(\zeta-z)^{\mathrm{t}’+1}}d\zeta}$

.

$(\mathrm{v} \not\in T)$

,

$\langle$

1

$\rangle$

$(f)_{-n1}= \lim_{\mathrm{v}arrow}$

$m$

(

$f]_{\mathrm{v}}$ $(m\in \mathrm{Z}^{*})$

,

(2

$\rangle$

where

$-\pi\leq\arg(\zeta-z)\leq$

$\pi$

for

$C_{-}$

,

$0\leq\arg(\zeta-z)$

$\leq 2\pi$

for

$C_{+}$

,

$\zeta\neq z$

,

$z$

$\in C$

,

$\mathrm{v}$

$\in R$

,

$\Gamma$

; Gamma

function,

then

$(f)_{\mathrm{s}}$

,

is

the fractional differintegration of

arbitrary

order

$\mathrm{v}$

(

derivatives

of

order

$\mathrm{v}$

for

$\mathrm{v}$

$>0$

,

and integrals of order

$-\mathrm{v}$

for

$\mathrm{v}$

$<0$

),

with

respect to

$\mathrm{z}$

,

of

the

function

$f$

,

if

$|(f)_{\mathrm{v}}|<\infty$

.

(2)

$(\mathrm{v} \not\in T)$

,

[

Refer

to

(1)1(3)

Theorem A. Let

fractional

calculus

Nishimoto’s

be

$N^{\mathrm{v}}=( \frac{\Gamma(\mathrm{v}+1)}{2\pi i}\int_{c}\frac{d\zeta}{(\zeta-z)^{\mathrm{v}+1}})$

with

$N^{-\prime\eta}= \lim N^{\mathrm{v}}$

$(m\in Z^{+})$

,

(4)

$\mathrm{v}arrow-ttt$

artd

define

the

binary operation

$\circ$

as

$N^{\beta}\circ N^{\alpha}f=N^{\beta}N^{a}f=N^{\beta}(N^{a}f)$

$(\alpha, \beta\in R)$

,

(5)

then the

set

$\{N^{\mathrm{v}}\}=\{N^{v}|\mathrm{v}\in R\}$

(6)

is

an

Abelian

product

group

(

having corrtin

tous

index

$\mathrm{v}$

)

which has the

inverse

transform

operator

$(N^{\mathrm{v}})^{-1}=N^{-\mathrm{v}}$

to

the

fractional

calculus

operator

$N^{\mathrm{y}}$

.

for

the

function

$f$

such that

$f\in F=\{f$

;

$0\neq|f_{1},|<\infty$

,

$\mathrm{v}$

$\in R\}$

,

where

$f=f(z)$

artd

$z$

$\in C$

.

(vis.

$-\infty$ $<\mathrm{v}$ $<\infty$

).

(For

our

convenience,

we

call

$N^{\beta}\circ N^{a}$

as

product

of

$N^{\beta}$

and

$N^{\mathrm{Q}}$

.

)

Theorem B.

F.O.G.

$\{N^{\mathrm{v}}\}$ ”

is

art

11

Action product

group

which

has

continuous

index

v

for

the

set

of

F

,

(

F.O.G.

; Fractional

calculus

operator

group

)

Theorem C. Let

$S:=\{\pm N^{\mathrm{v}}\}\mathrm{U}\{0\}=\{N^{\mathrm{y}}\}\cup\{-N^{\mathrm{v}}\}\cup\{0\}$ $(\mathrm{v} \in R)$

.

(7

$\rangle$

Then the

set

$S$

is

a

commutative

ring

for

the

function

$f\in F$

,

when the

identity

$N^{\alpha}+N^{\beta}=N^{\gamma}$

$(N^{a}, N^{\beta}, N^{\gamma}\in S)$

$\mathrm{t}$

$8)$

holds.

$\mathrm{E}51$

(III )

Lemma.

We have

[1]

$(i)$

$( (z -c)^{\beta})_{a}=e^{-i\pi a} \frac{\Gamma(\alpha-\beta)}{\Gamma(-\beta)}(\mathrm{z}-c)^{\beta-a}$ $\mathrm{f}$ $| \frac{\Gamma(\alpha-\beta)}{\Gamma(-\beta)}|<\infty \mathrm{I}$

,

$(\mathrm{i}\mathrm{i})$

$(\log (z -c))_{\alpha}=-e^{-t\pi\alpha}\Gamma(\alpha)(z -c)^{-a}$

$\langle|\Gamma(\alpha)1$$<\infty$$)$

,

$(i\mathrm{i}\mathrm{i})$ $((z-c)^{-a})_{-\mathrm{c}x}=-e^{i_{J\mathrm{P}O}} \frac{1}{\Gamma(\alpha)}\log(z -c)$

(I

$\Gamma(\alpha)|<\infty$

),

where

z–

$c\neq 0$

in

(i),

and

$z$

$-c\neq 0$

,

1

in

$(\mathrm{i}\mathrm{i})$

and

$(\mathrm{i}i\mathrm{i})$

.

(

$\Gamma$

; Gamma

function),

(3)

\S 1.

Doubly

Infinite,

Finite and Mixed Infinite Sums

In

the following

$\alpha$

,

$\beta$

,

$\gamma\in R$

.

Theorem

1.

Let

$L( \alpha,\beta, \gamma ; k, m):=\frac{\Gamma(\alpha+1)\Gamma(\gamma+1)\Gamma(k-\alpha+m)\Gamma(\gamma-\beta-m)}{k!\cdot m!\Gamma(\alpha+1-k)\Gamma(\gamma+1-m)\Gamma(k-\alpha)\Gamma(-\beta)}$

.

(1)

(i)

When

$\alpha$

,

$\beta$

,

$\gamma\not\in \mathrm{Z}_{0}^{+}$

we

have the following

doubly

infinite

sums

;

$\Sigma\Sigma L(\alpha,\beta,\gamma;k,m)(\frac{- c}{z})^{\Lambda}($

$\frac{z- c}{z})=Q(\alpha,\beta,\gamma)\frac{\Gamma(\gamma-\alpha-\beta)}{\Gamma(-\alpha-\beta)}m($ $\frac{z- c}{z}\mathrm{I}^{a},(2)$

where

$Q=Q( \alpha, \beta, \gamma):=\frac{\sin\pi\beta\cdot\sin\pi(\gamma-a-\beta)}{\sin\pi(\alpha+\beta)\cdot\sin\pi(\gamma-\beta)}$

$(1 Q|=M<\infty)$

,

$\mathrm{C}$

$3)$

I

$-c/z|<1$

,

$\mathfrak{l}(z-c)/z1$

$<1$

,

and

$(\mathrm{i}\mathrm{i})$

When

$\alpha$

,

$\beta\not\in \mathrm{z}^{+}$

we

have

the

following

mixed

infinite sums

$j$

$\Sigma\Sigma L(\alpha,\beta,s;k,m)(\frac{- c}{Z})\{k$

$\frac{z- c}{z}\mathrm{I}^{tn}=Q\langle\alpha,\beta;s)\frac{\Gamma(s-\alpha-\beta)}{\Gamma(-\alpha-\beta)}($$\frac{\mathrm{z}- c}{\mathrm{z}}),(4)a$

for

$s\in Z^{+}$

where

I

$-c/_{\mathrm{Z}}|<1$

,

I

$(z -c)/_{\mathrm{Z}}|<\infty$

,

and

Proof of

(i).

We

have

$(z -c)^{a}=z^{\alpha}(1- \frac{c}{z})\alpha$

$\langle$

5)

$=z^{a} \sum_{k\neq 0}^{\infty}\frac{(-c)^{k}\Gamma(\alpha+1)}{k!\Gamma(\alpha+1-k)}z^{-k}$

$(|_{\mathrm{Z}}|> \{ -c|)$

(6)

(4)

Next

make

(7) ,

then

operate

to

its both sides,

we

obtain

$((z-c)^{\beta} \cdot(z-c)^{a})_{\gamma}=\sum_{k*0}^{\infty}\frac{(-c)^{k}\Gamma(\alpha+1)}{k!\Gamma(\alpha+1-k)}((z-c)^{\beta}$

.

$z^{a-k})_{\gamma}$

(8)

$= \sum_{\mathrm{A}\Leftrightarrow 0}^{\infty}\frac{(-c)^{L}\Gamma(\alpha+1)}{k!\Gamma(\alpha+1-k)}\sum_{n\iota\Leftarrow 0}^{\infty}\frac{\Gamma(\gamma+1)}{m!\Gamma(\gamma+1-m)}((z-c)^{\beta})_{\gamma-m}(z^{\alpha-\mathrm{A}}.)_{rn}$

.

(9)

Now

we

have

(

$(_{\mathrm{Z}}- \mathrm{C})^{\beta}\mathrm{I}_{\gamma-m}=e^{-j\pi(\gamma-m)}\frac{\Gamma(\gamma-m-\beta)}{\Gamma(-\beta)}(z-c)^{\beta-\gamma+m}$

\langle

10}

$\mathrm{f}$$| \frac{\Gamma(\gamma-m-\beta)}{\Gamma(-\beta)}|<\infty)$

and

$(z^{} )_{m}=e^{-:\pi m} \frac{\Gamma(m+k-\alpha 1}{\Gamma(k-\alpha)}z^{\alpha-\mathrm{A}-n1}$

(11)

$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}$

ctively.

On

the other hand

we

have

$((z-c)^{\beta} \cdot(z-c)^{\alpha})_{\gamma}=\sum_{\mathrm{A}\Leftarrow 0}^{\infty}\frac{\Gamma(\gamma+1)}{k!\Gamma(\gamma+1-k)}((z-c)^{\beta})_{\gamma-k}((z-c)^{a})_{k}$

(12)

$=e^{-i\pi\gamma} \sum_{k\cdot 0}^{\infty}\frac{\Gamma(\gamma+1)\Gamma(\gamma-\beta-k)\Gamma(k-a)}{k!\Gamma(\gamma+1-k)\Gamma(-\beta)\Gamma(-\alpha)}(z-c)^{o+\beta-\gamma}$

(13)

$=e^{-i\eta} \frac{\Gamma(\gamma-\beta)}{\Gamma(-\beta)}(z-c)^{\alpha+\beta-\gamma}\sum_{\mathrm{A}\cdot 0}^{\infty}\frac{[-\alpha]_{\mathrm{A}}[-\gamma]_{k}}{k![1+\beta-\gamma]_{\mathrm{A}}}$

(14)

since

$\backslash \acute{\mathrm{t}}z-c)^{\beta})_{\gamma- \mathrm{A}}/=e^{-i\pi(\gamma-k)}\frac{\Gamma(\gamma-k-\beta)}{\Gamma(-\beta)}(z -c)^{\beta-\gamma+\mathrm{A}}$

(15)

$\mathrm{f}$$| \frac{\Gamma(\gamma-k-\beta)}{\Gamma(-\beta)}|<\infty \mathrm{t}$

,

$((z-c)^{a})_{\mathrm{A}}=e^{-i\pi \mathrm{A}} \frac{\Gamma(k-\alpha)}{\Gamma(-\alpha)}(z -c)^{\alpha- k}$

(16)

and

(5)

where

$[\lambda]_{k}=\lambda(\lambda+1)\cdots(\lambda+k-1)=\Gamma(\lambda+k)/\Gamma(\lambda)$

,

with

$[\lambda]_{0}=1$

(

notation of

Pochhammer

).

Next

we

have the

identity

$\sum_{0}^{\infty}.-,=F_{1}(\mathrm{z}a, b ; c;1)\underline{[a\underline{]}_{\mathrm{L}}\underline{[}\underline{b]}_{4}}k![c]_{\mathrm{A}}$

(18)

$= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$ $\xi_{c\not\in \mathrm{Z}^{\frac{}{0}}}^{{\rm Re}(c-a-b)>[\}})$

.

$\langle$

19)

Therefore,

we

have

$((z-c)^{\beta} \cdot(z -c)^{a})_{\gamma}=e^{-i\pi\gamma}\frac{\Gamma(\gamma-\beta)}{\Gamma(-\beta)}(z -c)^{\alpha+\beta-\gamma}\mathrm{z}F_{1}(-\alpha ,-\gamma;1+\beta-\gamma ; 1)$

(20

$\rangle$

$=e^{-i_{J}\eta} \frac{\Gamma(\gamma-\beta)\Gamma(1+\alpha+\beta)\Gamma(1+\beta-\gamma)}{\Gamma(-\beta)\Gamma(1+\beta)\Gamma(1+a+\beta-\gamma)}(z-c)^{a+\beta-\gamma}$ $\langle$

21)

$\mathrm{f}(1+\beta-\gamma)\not\in Z_{0}^{-)}{\rm Re}(\alpha+\beta+1)>0$

$=e^{-t\pi\gamma} \frac{\sin\pi\beta\cdot\sin\pi(\gamma-\alpha-\beta)}{\sin\pi(\alpha+\beta)\cdot\sin\pi(\gamma-\beta)}\cdot\frac{\Gamma(\gamma-\alpha-\beta)}{\mathrm{R}-\alpha-\beta)}(z-c)^{a+\beta-\gamma}$

(22)

$=e^{-i\pi\gamma}Q( \alpha,\beta,\gamma)\cdot\frac{\Gamma(\gamma-a-\beta)}{\Gamma\langle-\alpha-\beta)}(z-c)^{a+\beta-\gamma}$

(23)

from

(

14

},

because

we

have the

identity

$\Gamma(\lambda)\Gamma(1-\lambda)=\frac{\pi}{\sin\pi\lambda}$ $(\lambda\not\in Z)$

.

(24)

Therefore, substitutimg

(23 ),

(10)

and

(11 )

into

(9)we

obtain

$Q(a , \beta,\gamma)\cdot\frac{\Gamma(\gamma-\alpha-\beta)}{\Gamma(-\alpha-\beta)}(z -c)^{a+\beta-\gamma}$

$= \sum_{k\cdot 0}^{\infty}\frac{(-c)^{k}\Gamma(\alpha+1)}{k!\Gamma\langle\alpha+1-k)}\sum_{m=0}^{\infty}\frac{\Gamma(\gamma+1)\Gamma(\gamma-m-\beta)\Gamma(m+k-\alpha)}{m!\Gamma(\gamma+1-m)\Gamma(-\beta)\Gamma(k-\alpha)}$

$\mathrm{x}(\mathrm{Z} -c)^{\rho_{-,\prime}+nl}z^{a-k-n\iota}$

(25)

we

have

then

(6)

from

(25

),

using the notation

(1)

,under

the conditions.

Proof

of

$\langle$$\mathrm{i}\mathrm{i})$

.

Set

$\gamma=s\in \mathrm{Z}^{+}$

in

(

2

),

we

have then

(4 )

clearly ubder the

condi-ti

$\mathrm{o}\mathrm{n}\mathrm{s}$

.

Corollary

1.

When

$r$

,

$s\in Z^{+}$

we

have the

following doubly

finite

sums;

$\Sigma\Sigma L(r,\beta,s;k,m)(\frac{- c}{z})(\mathrm{A}$

$\frac{z- c}{z}l^{m}=Q(r,\beta,s)\frac{\Gamma(s-r-\beta)}{\Gamma(-r-\beta)}($$\frac{z- c}{z}1^{r}$

,

(26)

where

$|-c/z\mathrm{I}$

,

I

$(z -c)/z$

I

$<\infty$

,

and

Proof. Set

$\alpha=r$

and

$\gamma=s$

in

(2)we

have then this

corollary clearly.

\S

2.

Direct calculation

of the doubly infinite

sums

The direct calculation

$\langle$

without

the

use

of

N-

fractional calculus

)

of

the

doubly

infinite

sum

in

the LH5

of

\S 1. { 2

)

is

shown

as

folJow

s.

Theorem

2.

Let

$L=L(\alpha,\beta, \gamma ; k, m)$

$:= \frac{\Gamma(\alpha+1)\Gamma(\gamma+1)\prod\gamma-\beta-m)\Gamma(k-\alpha+m)}{k!\cdot m!\Gamma(\alpha+1-k)\Gamma(\gamma+1-m)\Gamma(-\beta)\Gamma(k-\alpha)}$

(1)

and

$Q=Q( \alpha, \beta, \gamma):=\frac{\sin\pi\beta\cdot\sin\pi(\gamma-a-\beta)}{\sin\pi(\alpha+\beta)\cdot\sin\pi(\gamma-\beta)}$

$(|Q(\alpha,\beta,\gamma)\mathrm{I}=M<\infty)$

.

$\mathrm{t}$ $2\rangle$

We have

then

$\sum\sum$

$L \cdot(\frac{\overline{\iota}-c}{\sim 7})^{m}($$\frac{-c}{z})^{k}=Q\cdot.\frac{\Gamma(\gamma-a-\beta)}{\Gamma(-\alpha-\beta)}($$\frac{\sim r-C}{Z})a$

(3)

where

I

$c/z$ $|<1$

,

artd

$(\alpha+\beta)$

,

$(\gamma-\beta)$

,

$(\gamma-\alpha-\beta)\not\in Z$

Proof.

Now

we

have

$L$

.

$( \frac{z-c}{z})^{Jl1}($$\frac{-c}{z})^{\Lambda}=\frac{\Gamma(\gamma-\beta)}{\Gamma(-\beta)}\cdot\frac{[-\alpha_{4+m}][-\gamma]_{m}}{\overline{k!\cdot}m\overline{![1+\beta-\gamma]_{m}}}($$\frac{c}{\mathrm{z}})^{k}($

$\langle$

4)

(7)

using

the identity

$\Gamma(\lambda+1-k)=(-1)^{-k}\frac{\Gamma(\lambda+1)\Gamma(-\lambda)}{\Gamma(k-\lambda)}$ $\mathrm{t}5$ $1$

and

$[-\alpha]_{\mathrm{A}+n\iota}=[-a]_{m}[-\alpha+m]_{\mathrm{A}}$

(6)

We have then

$\sum_{\mathrm{A}-0}^{\infty},\sum_{\prime 1\cdot 0}^{\infty}L\cdot(\frac{-c}{z})^{k}($$\frac{z-c}{z})^{m}=\frac{\Gamma(\gamma-\beta)}{\Gamma(-\beta^{\backslash }}$

,

$\mathrm{x}\sum_{m=0}^{\infty}\frac{[-\alpha]_{m}[-\gamma]_{m}}{m![1+\beta-\gamma]_{m}}(\frac{z- c}{z})^{\prime 1l}\sum_{k\Leftarrow 0}^{\infty}\frac{[-\alpha+m]_{k}}{k!}($$\frac{c}{z})k$

(7)

(8)

$= \frac{\Gamma(\gamma-\beta)}{\Gamma(-\beta)}(\frac{z-c}{z})^{a}\sum_{m\neq 0}^{\infty}\frac{[-\alpha]_{l},[-\gamma],]}{m![1+\beta-\gamma]_{m}},$

,

(9

$\rangle$

$= \frac{\Gamma(\gamma-\beta)}{\Gamma(-\beta)}(\frac{z-c}{z})^{a}21F(-a, -\gamma;1+\beta-\gamma ; 1)$

(10

$\rangle$ $=$$\frac{\Gamma(\gamma-\beta)\Gamma(1+\beta-\gamma)\Gamma(1+\alpha+\beta)}{\Gamma(-\beta)\Gamma(1+\beta)\Gamma(1+\alpha+\beta-\gamma)}(\frac{z-c}{z})^{\alpha}$

,

where

$\frac{1-c}{\ulcorner}|z<1,$ $\vdash_{Z}^{z-c}|<1$

,

${\rm Re}(\alpha+\beta)>-1$

Because

we

ahave

$E\infty$

.

$\frac{[-\alpha+m]_{k}}{k!}(\frac{c}{z})^{k}=($ $\ell x- m$ $\frac{z-c}{z})$

(11)

since

$\sum_{k=0}^{\infty}\frac{[\lambda]_{k}}{k!}z^{\mathrm{A}}=(1-z)^{-\lambda}$

(12)

and

$21F(a, b;c;1)= \sum_{m\approx 0}^{\infty}\frac{\underline{[}a_{I4}],[b]_{nf}}{\overline{m![c]_{m}}}=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$ $[_{c\not\in Z_{0}^{-}}^{{\rm Re}(c-a-b)>(?}$

(13)

Moreover

we

have

the identity

$\Gamma(\lambda)\Gamma(1-\lambda)=\frac{\pi}{\sin\pi\lambda}$ $(\lambda\not\in Z)$

,

$\langle$

14)

then applying

(

14

)

to

(10)we

obtain

$\sum_{\mathrm{r}k0m}^{\infty}\sum_{-0}^{\infty}L\cdot(\frac{z- c}{z}1^{m}($

(8)

\S 3.

Commentary

[I1

$\mathrm{h}$

a

previous

PaPer,

the

results

obtained

by

the author

are

derived

by

the

use

of

$((z-c)^{a+\beta})_{\gamma}$

,

how

ever

the

results

shown in

this

article,

the N- fractional

calculus

(

$(z-c)^{\beta}\cdot$ $(z-c)^{\alpha}$

)

is

used

{III

When

$Q=Q(\alpha, \beta, \gamma)=1$

,

\S 1.

(2)

overlaps

Theorem

2

obtained

in

a

previous

paper

$.[11]$

References

[1]

K.

Nishimoto

;Fractional

Calculus,

Vol. 1

(1984),

Vol.

2

\langle 1987},

Vol.

3

(1989),

VoL4

(1991),

Vol.

5,

(1996),

Descartes

Press,

Koriyama, Japan.

[I]

K.

Nishimoto;An Essence

of

Nishimoto’s Fractional Calculus

(Calculus

of the 21st

Century);

Integrals

and Differentiations of

Arbitrary

Order

(1991),

Descartes

Press,

Koriyama,

Japan.

[I]

K.

Nishimoto

j

On

Nishimoto’s

fractional

calculus

operator

$N^{\mathrm{v}}$

(

On

an

action

group

),

J.

Frac.

Calc.

VoL4,

Nov.

(1993),

1-

11.

[4] K.

Nishimoto;

Unification

of

the

integrals

and derivatives

\langle A serendipity in

fractional

cal-su ns

),

J.

Frac.

Calc. Vol.

6,

Nov.

(1994),

1

-

14.

[5]

K.

Nishimoto;

Ring

and Field Produced

from The Set

of

N-

Fractional

Calculus

Operator,

J.

Frac

Calc. Vol.

24,

Nov.

(2003),29

-

36.

[6]

K. Nishimoto,

Ding-

Kuo Chyan, Shy-

Der lin and

Shih-

Tong

Tu;

On

some

infinite

suns

de-rived

by N-

fractional

calculus,

J.

Frac.

Calc.

Vo1.20

(2001),

91

-

97.

[7]

Pin Yu Wang, Tsu-

Chen

Wu

and

Shin-

Tong Tu;

Some

Infinite

Sums via N-

fractional

calculus,

J.

Frac.

Calc.

Vo1.21, May

(2002),

71 -77.

[8]

Shy-

Der

Lin,

Shih-

Tong

Tu

,

Tsai-

Mng

Hsieh

and

H.M. Srivastava

;Some

Finite

and Infinite

Sums

Associated with

the Digamma and

Related

Functions,

J.

Frac.Calc.Vo1.22, Nov.

(2002),

103- 114.

[9]

K.

Nishimoto

;N-

Fractional

Calculus of the

Power

and

Logarithmic

Functions

and

Some

Identities

\langle

Continue

),

J.

Frac.

Calc.

VoL22,

Nov.

(2002),

59

-

65.

[10]

K.

Nishimoto and

Susana S.

de

Romero;

Some Multiple

Infinite

Sums

derived from

The

N-Fractional Calculus

of

some

Power

Functions,

J.

Frac.

Calc.Vo1.24,

Nov.

(2003),

67

-

76.

[1]

K. Nishimoto;

Examinations

for Some

Doubly Infinite, Finite

and Mixed

Sums,

J.

Frac.

Calc.

Vol.

25, May

(2004),

25

-

32.

[12]

Shy-

Der

lin

and H. M.

Srivastava.,

Fractional

Calculus and

Its

Applications Involving

Bila-teral

Expansions

and

Multiple

Infinite

Sums,

J.

Frac.

Calc.

Vol.

25, May

(2004),

47

-

58.

[133

K.

Nishimoto,

Susana

S.

de Romero

and

Ana I. Prieto

,.

Examinations for Some Doubly

Infinite

Sums

derived

by

Means

of

$\mathrm{N}$

-Fractional

Calculus,

J.

Frac.

Calc.Vo1.26,

Nov.

(2004),

1-8.

[14]

K.

Nishimoto.,

Some Multiply

Infinite,

Mixed

and Finite Sums

derived

from

The

N- Fract

0-nal

Calculus of

Some

Power

Functions,

J.

Frac.

Calc.Vol

26,

Nov.

(2004),

9

-

23.

[151

K.

Nishimoto

j

Examinations

for

Some Doubly

Infinite

Sums

derived from

The

N.

Frac-tional

Calculus of A

Logarithmic Function,

J.

Frac.Cak.

Vol. 26, Nov.

(2004),

25

-

34.

[16]

K.

Nishimoto

j

On Some

\langle

$q+1)$

Multiply

Infinite

Sums

(q

$\in Z^{+})$

derived from

the N.

Frac-tional Calculus of Some Power Functions

(Part

I

),

J.

Frac.Calc.VoL

26,

Nov.

(2004),

53

-

60.

[10]

K.

Nishimoto and

Susana S.

de Romero

j

Numerical

Examinations

for

Mixed

and

Double

Finite

Sums

obtained

by Means

of

N-

Fractional

Calculus,

J.

Frac.Calc.VoL

26,

Nov.

(2004),

91

-

99.

[18]

K.S.

Miller and B. Ross

j

An

Introduction

to

$\Pi \mathrm{e}$

Fractional Calculus and Fractional

Diffe-tial

Equations,

John

Wiley

&

Sons,

(1993).

(9)

[20]

R.

Hilfer

(Ed.)

; Applcations

of

Fractional

Calculus

in Physics,

(2000),

World

Scientific,

Singapor,

New

Jersey, London, Hong Kong.

[21]

A.P. Prudnikov, Yu. A Bryckov and O.I.

Marichev

j

Integrals and

Series,

Vol.

I,

Gordon

and

Breach,

New

York,

(1986).

[22]

S. Moriguchi,

K.

Udagawa

and

S. Hitotsumatsu

j

Mathematical

Formulae,

Vo1.2, Iwanami

Z

ensho,

(1957\rangle ,

Iwanami,

Japan.

Katsuyuki

Nishimoto

Institute

of Applied Mathematics

Descartes Press Co.

2-13 -10

Kaguike, Koriyama

参照

関連したドキュメント

We study the finite and infinite irreducible representations of the quantum quadratic algebras though the construction of models in which the symmetries act on spaces of functions of

So far we have shown in this section that the Gross Question (1.1) has actually a negative answer when it is reformulated for general quadratic forms, for totally singular

Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta

In previous work [11], the author shows that in the general case of functions f : G → N between arbitrary finite groups G and N , bundle and graph equivalence have a common source

To complete the “concrete” proof of the “al- gebraic implies automatic” direction of Theorem 4.1.3, we must explain why the field of p-quasi-automatic series is closed

In this paper we consider a class of symbols of infinite order and develop a global calculus for the related pseudodifferential operators in the functional frame of the

Abstract: In this paper, we investigate the uniqueness problems of meromorphic functions that share a small function with its differential polynomials, and give some results which

classes of harmonic functions are introduced and mixed Zaremba’s bound- ary value problem is studed in them, i.e., the problem of constructing a harmonic function when on a part of