N-Fractional Calculus of Some Irrationa1 Functions.

Katsuyuki Nishimoto

Institute for Applied Mathematics, Descartes Press Co.

2-13-10 Kaguike, Koriyama, 963-8833, JAPAN.

Keywords; Fractional Calculus, N-Fractional Calculus (NFC),

NFC of Products, NFC of Composite Functions

AbstractIn this amcle N- fractional calculus of the fmctions (which have multiple root

signs)

$f(z)=\sqrt{\sqrt{z-b}-c-d}$

are discussed.

Theorem 1. We have

(i) $(f( z))_{\gamma}-e^{-igt\gamma}(\overline{<}-b)^{(1/8)-\gamma}\sum_{m.k-0}^{\infty}\frac{[-12]_{m\triangleleft}[\frac{m}{2!}.\perp A\perp n284}{mk!\Gamma(-+)}$

$x\backslash (\frac{c}{\sqrt{z-b}})^{k}(\frac{d}{\sqrt[4]{z-b}})^{m}$ , $(| \frac{\Gamma(\underline{k1a}-++\gamma)}{\Gamma(\frac{k}{2}-+)}|<\infty)$

and

$(ii)$ $(f(z))_{n}=(-1)^{n}(z-b)^{(1/8)-n} \sum_{m,k-0}^{\infty}\frac{[-\perp]_{m}[-]_{k}[\cdot-+]_{n}}{m!\cdot k!}$

$x(\frac{c}{\sqrt{z-b}})^{k}(\frac{d}{\sqrt[4]{z-b}}\backslash ^{m})$

’$(n\in Z_{0}^{*})$

$v\^{r}here$

$| \frac{c}{\sqrt{z-b}}|<1$ , $| \frac{d}{\sqrt[4]{z-b}}|<1$ .

数理解析研究所講究録第 1626巻 2009年 56-67 56

Katsuyuki Nishimoto

\S $0$ . Introduction (Definition of Fractional Calculus)

(I) Defimition. (by K. NI$s$himoto) ([1] Vol. 1)

Let $D=\{D_{-}, D_{+}\},$ $C\simeq\{C_{-}, C_{+}\}$ ,$C_{-}$ be a curve along the cutjoming two points $z$ and $-\infty+i{\rm Im}(z)$ ,$C_{+}$ be a curve along the cutjoining two points $z$ and $\infty+i{\rm Im}(z)$ .$D_{-}$ be a domain $su\Pi ounded$ by $C_{-}$ , $D_{+}$ be a domain surrounded by $C_{+}$ .

(Here $D$ contains the points over the curve $C$ ).Moreover, let $f\sim f(z)$ be a regular function in $D(z\in D)$ ,

$f_{v} \mapsto(f)_{vC}-(f)_{v}\infty\frac{\Gamma(v+1)}{2\pi i}\int_{c}\frac{f(\zeta)}{(\zeta-z)^{\nu+1}}d\zeta$ $(v\not\in\tau)$ , (1)

$(f)_{-m}- \lim_{\nuarrow-m}(J)_{v}$ $(m\in Z^{+})$ , (2)

where $-\pi\leq\arg(\zeta-z)\leq\pi$ for $C_{\sim}$ , $0\leq\arg(\zeta-z)\leq 2\pi$ for $C_{+}$ ,

$\zeta\neq z$ , $z\in C$ , $v\in R$ , $\Gamma$ ; Ganuna function,then $(f)_{v}$ is the fractIonal $diffe\dot{n}ntegra\mathfrak{U}on$ of arbitrary order $v$ (derivatives of

order $v$ for $v>0$ , and integrals of order $-v$ for $v<0$ ), with respect to $z$ , ofthe function $f$ , if $|(f)_{\tau},|<\infty$ .(I I) On the fractional calculus operator $N^{v}[3]$

Theorem A. Let fractional calculus operator (Nishimoto‘s Operator) $N^{V}$ be

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

with $N^{-m}- \lim_{varrow- m}$ ハズ

and deflne the $bina\varphi$ operation $\circ$ as

$(v\not\in Z^{-})$ , [Refer to (1)] (3)

$(m\in Z^{+})$ , (4)

$N^{\beta}\circ N^{\alpha}f-N^{\beta}N^{a}f-N^{\beta}(N^{a}f)$ $(\alpha, \beta\in R)$ , (5)then the set

$\{N^{v}\}arrow\{N^{v}|v\in R\}$ (6)

is an Abelian product group (having continuous index $v$ ) which has the inverse

transfom operator $(N^{v})^{-1}-N^{-v}$ to the $fra\alpha ional$ calculus operator $N^{v}$ , for the

funct ion $f$ such that $f\in F-\{f;0\neq|f_{v}|<\infty,$ $v\in R\}$ , where $farrow f(z)$ and $z\in C$ .$($ vis. $-$ oo $<v<\infty)$ .

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N- Fractional Calculus of Some Functions Wluich Have Multiple Root Signs

(For our convenience, we call $N^{\beta}\circ N^{\alpha}$ as product of $N^{\beta}$ and $N^{\alpha}$ . )

Theorem B. “ F.0.G. $\{N^{\nu}\}$“ is $an$

“ Action product group which has continuous

index $v”$ for the set of F. (F.O. $G$ ; Fractional calculus operator group) [3]

(III) Lemma. We have [1]

(i) $((z-c)^{b})_{\alpha} arrow e^{-i_{J}r\alpha}\frac{\Gamma(\alpha-b)}{\Gamma(-b)}(z-c)^{b-a}$ $(| \frac{\Gamma(\alpha-b)}{\Gamma(-b)}|<\infty)$ (7)

$(ii)$ $(\log(z-c))_{a}--e^{-i\pi a}\Gamma(\alpha)(z-c)^{-\alpha}$ $(|\Gamma(\alpha)|<\infty)$ , (8)

$(iiI)$ $((z-c)^{-\alpha})_{-\alpha}--e^{\dot{\iota}\pi\sigma} \frac{1}{\Gamma(\alpha)}\log(z-c)$ $(|\Gamma(\alpha)|<\infty)$ , (9)

where $z-c\neq 0$ for (i) and $z-cx0,1$ for (I i), $(iiI)$ ,

\S 1. Preliminary[1] The theorem below is reported by K. Nishlmoto already (cf. J. Frac. Calc.

Vol. 29, May (2006), p.37). [12]

Theorem D. We have

(i) $(((z-b)^{\beta}-c)^{\alpha})_{\gamma}\simeq e^{-i\pi\gamma}(z-b)^{a\beta-\gamma}$

$x\sum_{k\triangleleft}^{\infty}\frac{[-a]_{k}\Gamma(\beta k-\alpha\beta+\gamma)}{k!\Gamma(\beta k-\alpha\beta)}(\frac{c}{(z-b)^{\beta}})^{k}$ $(| \frac{\Gamma(\beta k-\alpha\beta+\gamma)}{\Gamma(\beta k-\alpha\beta)}|<\infty)$ (1)

and

(Ii) $(((z-b)^{\beta}-c)^{\alpha})_{n}=(-1)^{n}(z-b)^{\alpha\beta-n}$

$x\sum_{0}^{\infty}\frac{[-\alpha]_{k}[\beta k-\phi]_{n}}{k!}(\frac{c}{(z-b)^{\beta}})^{k}$ $(n\in Z_{0}^{+})$ (2)

whereI $c/(z-b)^{\beta}1<1$ ,

and$[\lambda]_{k}-\lambda(\lambda+1)\cdots(\lambda+k-1)=\Gamma(\lambda+k)/\Gamma(\lambda)$ with $[\lambda]_{0}=1$ .(NotatIon of Pochhanuner).

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KatsuyukI Nishlmoto

[I I] The theorem below Is reported by K. NIshimoto already (cf. J. Frac. Calc.Vol. 31, May (2007), p.13). [13]

Theorem E. We have(i) $((((z-b)^{\beta}-c)^{a}-d)^{\delta})_{\gamma}=e^{-i\pi\gamma}(z-b)^{a\beta t-\gamma}$

$x\sum_{m,karrow 0}^{\infty}\frac{[-\delta L[-\alpha(\delta-m)]_{k}\Gamma(\beta k-\phi_{\frac{(}{m}}\delta-m)+\gamma)}{m!\cdot k!\Gamma(\beta k-\alpha\beta(\delta-))}(\frac{c}{(z-b)^{\beta}})^{k}(\frac{d}{(z-b)^{\phi}})^{m}$

(3)

$(| \frac{\Gamma(\beta k-a\beta(\delta-m)+\gamma)}{\Gamma(\beta k-\phi(\delta-m))}|<\infty)$

and

(Ii) $((((z-b)^{\beta}-c)^{\alpha}-d)^{\delta})_{n}<(-1)^{n}(z-b)^{\alpha\beta\delta-n}$

$x_{m}\sum_{-0}\frac{[-\delta 1_{n}[-\alpha(\delta-m)]_{k}[\beta k-a\beta(\delta-m)]_{n}}{m!\cdot k!}(\frac{c}{(z-b)^{\beta}})^{k}(\frac{d}{(z-b)^{\alpha\beta}})^{m}\infty$

(4)$(n\in Z_{0}^{+})$

where

$((z-b)^{\beta}-c)^{a}-d\neq 0$ , $|c/(z-b)^{\beta}1<1$ , $|d/(z-b)^{\alpha\beta}|<1$ .\S 2. N- Fractional Calculus of Functions

$\sqrt{\sqrt{z-b}-c-d}$

Theorem 1. We have

(i) $(\sqrt{\frac{-b}{z}-c-d})_{\gamma}\approx e^{-in\gamma}(z-b)^{(1/8)-\gamma}$

$x_{m}\sum^{\infty}\cdot 0\frac{[-2]_{m2}[\ovalbox{\tt\small REJECT}_{-}\frac{1}{!4}]_{k}\Gamma(\frac{k}{2}-+}{m!\cdot k\Gamma(-+)}(\frac{C}{\sqrt{z-b}})^{k}(\frac{d}{\sqrt[4]{z-b}})^{m}$ (1)

and

$(| \frac{\Gamma(\cdot-++\gamma)}{\Gamma(-+)}|<\infty)$

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N- Fractional Calculus of Some Functions Whuich Have Multiple Root Signs

$(ii)$ $(\sqrt{\sqrt{z-b}-c-d})_{n}=(-1)^{n}(z-b)^{(118)-n}$

$x\sum_{m,k-0}^{\infty}\frac{[-\perp 224}{m!\cdot k!}(\frac{}{\frac{c-b}{z}})^{k}(\frac{d}{\sqrt[4]{z-b}}I^{m}$ (2)

$(n\in Z_{0}^{+})$

where

$| \frac{}{\frac{c-b}{z}}|<1$ , $| \frac{d}{\sqrt[4]{z-b}}|<1$ .

Proof of (i). We have

$\sqrt{\frac{-b}{z}-c-d}\infty(((z-b)^{1/2}\sim c)^{1/2}-d)^{1/2}$ , (3)

hence, operating $N^{\gamma}$ to the both sides of (3), we obtain

$(\sqrt{\sqrt{z-b}-c-d})_{\gamma}\infty((((z-b)^{1/2}-c)^{1/2}-d)^{1/2})_{\gamma}$ (4)

$=e^{-i\pi\gamma}(z-b)^{(1/8)-\gamma}$

$x\sum_{m.k-0}^{\infty}\frac{[-]_{m}[-]_{\iota}\Gamma(-++\gamma)}{m!k!\Gamma(-+)}(\frac{c}{\sqrt{z-b}})^{\iota}-(\frac{d}{\sqrt[4]{z-b}})^{m}$ (5)

setting $\alpha-\betaarrow\delta-1/2$ in Theorem E. (i) in prehminary, under the conditionsstated befor.Proof of $(ii)$ . Set $\gamma\approx n$ in (i).

Corollary 1. We have

(i) $(\sqrt{z-b-c})_{\gamma}\approx e^{-i\eta}(z-b)^{(1/8)-\gamma}$

$xP_{0}\infty\frac{[-\perp 4]_{k}\Gamma(-+\gamma)}{k!\Gamma(-)}(\frac{c}{\sqrt{z-b}})^{\iota}$ (6)

$($ $|$ $rightarrow^{\Gamma(\underline\Gamma(\frac-\frac)k_{-}\iota_{+\gamma)}k128}$ $|$ $<$ $\infty)$

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Katsuyuki Nishimoto

and

(II) $(\sqrt{z-b-c})_{n}\Leftarrow(-1)^{n}(z-b)^{(1/8)- n}$

$x\sum_{k1-0}^{\infty}\frac{[-4]_{k}[-1]_{n}}{k!}(\frac{c}{\sqrt{z-b}})^{k}$ (7)

$(n\in Z_{0}^{+})$

where

$| \frac{c}{\sqrt{z-b}}|<1$ .

Proof of (i). Set $d\approx 0$ in Theorem 1 (i).

We have (6) from Theorem $D,$ $(i)$, setting $\beta\approx 1/2$ and $\alpha=1/4$ .Proof of $(ii)$. Set $\gamma=n$ in (i).

Corollary 2. We have

(i) $( \sqrt{z-b})_{\gamma}-e^{-\dot{l}n\gamma}\frac{\Gamma(-81+\gamma)}{\Gamma(-J_{)}8}(z-b)^{(1/8)-\gamma}$ (8)

$(|\Gamma(-1_{+\gamma)1<\infty)}8$

and

$(ii)$ $(\sqrt{z-b})_{n}\approx(-1)^{n}[-\S]_{n}(z-b)^{(1/8)-n}$ (9)

$(n\in Z_{0}^{+})$

where$z-b\neq 0$ .

Proof of (i). Set $c-0$ in Corollary 1 (I).

Or set $c-d\approx 0$ in Theorem 1. (I).

Proof of $(iI)$ . Set $\gammarightarrow n$ in (1).

Note. We have

$( \sqrt{z-b})_{\gamma}=((z-b)^{1/8})_{\gamma}-e^{-i\pi\gamma}\frac{\Gamma(-8^{+}\perp\gamma)}{\Gamma(-\Delta_{)}8}(z-b)^{(1/8)-\gamma}$ (10)

from Lemma (I), directly.

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N-Fractional Calculus of Some Function$s$ Which Have Multiple Root Signs

\S 3. Semi Derivatives and Integrals

Theorem 2. We have

(i) $(\sqrt{\sqrt{z-b}-c-d})_{1/2}--i(z-b)^{-3/8}$ ,

$x\sum_{m,k-0}^{\infty}\frac{[-]_{m}[-]_{k}\Gamma(++)}{m!\cdot k!\Gamma(-+)}(\frac{c}{\sqrt{z-b}})^{k}(\frac{d}{\sqrt[4]{z-b}})^{m}$ (1)

(semi derivauves)

and

$(iI)$ $(\sqrt{\sqrt{z-b}-c-d})_{-1/2}-i(z-b)^{5/8}$

$x_{m}\sum_{-0}^{\infty}\frac{[-]_{m}[-]_{i}\Gamma(-+)}{}(\frac{c}{\sqrt{z-b}})^{k}(\frac{d}{\sqrt[4]{z-b}}I^{m}$ (2)$m! \cdot k!\Gamma(\frac{k}{2}-+)$

(semi integrals)where

$| \frac{c}{\sqrt{z-b}}|<1$ , $| \frac{d}{\sqrt[4]{z-b}}|<1$ .

Proof. Set $\gammaarrow 1/2$ and $-1/2$ in Theorem 1 (i) we have then (i) and (Ii)

respectively, clearly.

Corollary 3. We have

(i) $( \sqrt{z-b-c})_{1/2}\approx-i(z-b)^{-3/8}\sum_{k\triangleleft}^{\infty}\frac{[-\perp 4]_{k}\Gamma(+)}{k!\Gamma(-)}(\frac{c}{\sqrt{z-b}}I$

た

(3)

where

$| \frac{c}{\sqrt{z-b}}|<1$

and(semi derivatives)

(Ii) $( \sqrt{z-b-c})_{-1/2}-i(z-b)^{s/8}\sum_{k-0}^{\infty}\frac{[-4]_{k}\Gamma(28}{k!\Gamma(-)}(\frac{c}{\sqrt{z-b}})^{k}$ (4)

(semi integrals)

62

Katsuyub Nishimoto

Proof. Set $d\approx 0$ in Theorem 2.Corollary 4. We have

(i) $( \sqrt{z-b-d})_{1/2}\Leftarrow-i(z-b)^{-3/8}\sum_{m-0}^{\infty}\frac{[-]\Gamma(+)}{m!!\Gamma(-+)}(\frac{d}{\sqrt[4]{z-b}}I^{m}$ (5)

(semi derivatives)

and

(Ii) $( \sqrt{z-b-d})_{-1/2}\approx i(z-b)^{5/8}\sum_{m,\text{た}-0}^{\infty}\frac{[-2]_{m}\Gamma(-+)}{m!\Gamma(-+)}(\frac{d}{\sqrt[4]{z-b}})^{m}$ (2)

(semi integrals)where

$| \frac{d}{\sqrt[4]{z-b}}|<1$ .

Proof. Set $c-0$ in Theorem 2.

\S 4. Some Special Cases[I] When $n-0$ , we have

$x\sum_{m,k-0}^{\infty}rightarrow^{[-\underline 1_{m!.k!^{-]_{l}}}]_{m}[\underline m\perp}(\frac{c}{\sqrt{z-b}})^{k}(\frac{d}{\sqrt[4]{z-b}}I^{m}$ (1)

from \S 2. (2).

Now we have

RHS of (1) $-(z-b)^{1/8} \sum_{m\triangleleft}^{\infty}\frac{[-12]_{m}}{m!}T^{m}\geq_{0}\infty A^{\perp}[\ovalbox{\tt\small REJECT}_{k!}]_{k}\oint$ (2)

$(S- \frac{c}{\sqrt{z-b}}$ , $T- \frac{d}{\sqrt[4]{z-b}})$

$-(z-b)^{1/8} \sum_{m\cdot 0}^{\infty}arrow[-1]_{m}T^{m}(1-S)^{(\iota/4)-(m/2)}m!$ $t3)$

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N-Fractional Calculus of Some Functions Whuich Have Multiple Root Signs

$-(z-b)^{1/8}(1-S)^{1/4} \sum_{m\cdot 0}^{\infty}\frac{[-\perp 2]_{m}T^{m}}{m!(1-S)^{m/2}}$ (4)

$=(z-b)^{1/8}(1-S \gamma^{1/4}\sum_{m-0}^{\infty}\frac{[-\perp 2]_{m}}{m!}(\frac{d}{\sqrt{z-b-c}}I^{m}$ $t5)$

$-(z-b)^{\iota/8}(1- \frac{c}{\sqrt{z-b}})^{1/4}(1-\frac{d}{\sqrt{z-b-c}})^{1/2}$ (6)

$-\sqrt{\sqrt{z-b}-c-d}$

[I I] When $nrightarrow 1$ , we have

(lHS of (1)). (7)

$( \sqrt{\frac{-b}{z}-c-d})_{1}\approx-(z-b)^{-7/8}m\sum_{\triangleleft}\frac{[-2]_{m}[-]_{k}[\cdot+-]_{1}}{m!k!}\oint\infty T^{m}$

(8)

$<-(z-b)^{-7/8} \sum_{m,k-0}^{\infty}\frac{[-2]_{m}[-]_{k}(+-)}{m!k!}$ $ff$ $T^{m}$ (9)

from \S 2. (2).

Now we have the identities;

$\geq_{-0}\infty\frac{[\lambda]_{k}}{k!}z^{k}-(1-z)^{-\lambda}$ , (10)

$[\lambda]_{0}arrow 1$ , $[\lambda]_{1}-\lambda$ , (11)

$[\lambda]_{\text{た}+1}-\lambda[\lambda+1]_{k}$ , (12)

hence

$?_{0} \infty\frac{[\lambda]_{k}k}{k!}z^{k}-\geq_{-1}\infty\frac{[\lambda]_{k}}{(k-1)!}z^{\text{た}}arrow z\sum^{\infty}0^{A}[\lambda]k!z^{k}$

$\infty\lambda z(1-z)^{-\lambda-1}$ . (13)

Therefore, we have

$m2_{-0} \infty\frac{[-\iota]_{m}[-]_{\text{た}}()}{m!k!}s^{k}T^{m}-\frac{1}{2}\sum_{m-0}^{\infty}rightarrow[-\perp]_{m}T^{m}m!\sum^{\infty}0\frac{[_{24}^{n_{-}\perp}]_{k}k}{k!}\#$ (14)

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Katsuyuki Nishimoto

$= \frac{1}{2}\sum_{m\mapsto 0}^{\infty}\frac{[-J2]_{m}}{m!}\tau_{\text{�}}^{m}\frac{m}{2}-\frac{1}{4})S\sum_{k-0}^{\infty}\frac{[_{24}^{\alpha_{+}2}]_{k}}{k!}S^{k}$ (15)

$- \frac{1}{2}S(1-S)^{-3/4}\sum_{m-0}^{\infty}\frac{[-2\iota\ovalbox{\tt\small REJECT} 124}{m!}U^{n}$ $(U \propto\frac{d}{\sqrt{z-b-c}})$ (16)

$- \frac{1}{2}S(1-S)^{-3/4}\{-\frac{1}{4}U(1-U)^{-1/2}-\frac{1}{4}(1-U)^{1/2}\}$ (17)

$– \frac{1}{8}S(1-S)^{-3/4}(1-U)^{-1/2}$ , (18)

$mz_{-0} \infty\frac{[-]_{m}[-]_{k}()}{m!k!}S^{k}T^{m}-\frac{1}{4}\sum_{m\triangleleft}^{\infty}\frac{[-12]_{m}m}{m!}T^{m}z_{-0}\infty\mapsto[\alpha_{k!}l]_{k}\oint$ (19)

$\approx\frac{1}{4}(1-S)^{1/4}\sum_{m-1}^{\infty}\frac{[-\iota_{2}]_{n}}{(m-1)!}U^{m}$ (20)

$= \frac{1}{4}(1-S)^{1/4}U(-\frac{1}{2})\sum_{m-0}^{\infty}-[1\angle]m^{\frac{m}{!}U^{m}}$ (21)

$– \frac{1}{8}(1-5)^{1l4}U(1-U)^{-1/2}$ . (22)

and

$m2_{0} \infty.\frac{[-]_{m}[-]_{\text{た}}(-\downarrow)}{m!\cdot k!}S^{\text{た}}T^{m}--\frac{1}{8}\sum_{m\cdot 0}^{\infty}rightarrow[-\underline{1}]_{m}T^{m}m!\sum^{\infty}0\mapsto[^{\ovalbox{\tt\small REJECT}_{k!}1}]_{k}S^{\text{た}}$ (23)

$– \frac{1}{8}(1-S)^{1/4}\sum_{m-0}^{\infty}arrow[-11_{U^{m}}m!^{n}$ (24)

1$–\overline{8}(1-S)^{1/4}(1-U)^{1/2}$ . (25)

Then applying (18), (22) and (25) to (9) we obtain

$( \sqrt{\sqrt{z-b}-c-d})_{1}-\frac{1}{8}(z-b)^{-7/8}(1-S)^{1/4}(1-U)^{-1/2}\{S(1-S)^{-1}+1\}$ (26)

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N- Fractional Calculus of Some Functions $Wl\dot{u}ch$ Have Multiple Root Signs

$- \frac{1}{8}(z-b)^{-7/8}(1-S)^{1/4}(1-U_{J^{- 1/2}}^{\neg}(1-S)^{-1}$ (27)

$= \frac{1}{8}(z-b)^{-7/8}(1-S)^{-3/4}(1-U)^{-1/2}$ (28)

$\approx\frac{1}{8}(z-b)^{-1/2}(\sqrt{z-b}-c)^{-1/2}(\sqrt{z-b-c}-d)^{-1/2}$ (29)

This result (29) coIncides with the one obtained by classIcal calculus.

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Vol. 4 (1991), Vol. 5, (1996), Descartes Press, Koriyama, Japan.[2] K. Nishimoto ; An Essence of NishImoto’s Fractional Calculus (Calculus of the

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Katsuyuki Nishimoto

[16] T. Miyakoda ; N- Fractional Calculus of Certain Algebraic Functions, J. Frac.Calc.Vol. 31, May (2007), 63 -76.

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Iwanami Zensho, (1957), Iwanami, Japan.

Katsuyuki NIshimotoInstitute for Applied MathematicsDescartes Press Co.2-13-10 Kaguike, Koriyama963-8833 Japan

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