Nishimoto, Katsuyuki

Kyoto University Research Information Repository

TitleN-Fractional Calculus of Some Multi-Powers Functions(Studyon Geometric Univalent Function Theory)Author(s)Nishimoto, KatsuyukiCitation数理解析研究所講究録 (2008), 1579: 79-87Issue Date2008-02URL http://hdl.handle.net/2433/81390RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityN- Fractional Calculus of SomeMulti- Powers FunctionsKatsuyuki NishimotoInstitute for Applied Mathematics, Descartes Press Co.2-13-10 Kaguike, Koriyama, 963-8833. JAPANFax: +81-24-922-7596Abstractin a previous article of the author, N- fractional calculus$(((z-b)^{\beta}-c)^{a})_{\gamma}$ , $((z-b)^{\beta}-c\neq 0)$are discussed. In this article that of more extended forms$((((z-b)^{\beta}-c)^{\alpha}-d)^{\delta})_{\gamma}$ , $(((z-b)^{\beta}-c)^{\alpha}-d\neq 0)$are discussed. Moreover their specIal cases$((((z-b)^{\beta}-c)^{\alpha}-d)^{l})_{n}$ , $(n\in Z_{0}^{*}, ((z-b)^{\beta}-c)^{a}-dz0)$are presented.\S $0$ . Introduction (Definition of Fractional Calculus)(I) Definition. (by K. Nishimoto) ([1] Vol. 1)Let $Drightarrow\{D_{-}.D_{*}\}$ . $C-\{C_{-}, C_{*}\}$ ,$C_{-}$ be a curve along the cut joining two points $z\bm{t}d-\infty+i$ Im(z),$C_{*}$ be a curve along the cut joining two points $z$ and $\infty+i{\rm Im}(z)$ ,$D_{-}$ be a domain surrounded by $C_{-}$ , $D_{*}$ be a domain surrounded by $C_{*}$ .(Here $D$ contains the points over the curve $C$ ).Moreover, let $f-f(z)$ be a regular funcdon in $D(z\in D)$ ,$f_{\nu}-(f)_{\vee^{-}C}(f)_{V}- \frac{\Gamma(v+1)}{2\pi i}\int_{c^{\frac{f(\zeta)}{(\zeta-z)^{\nu*1}}d\zeta}}$ $(v\not\in T)$ , (1)$(f)_{-m}- \lim_{\nu\wedge-m}(J)_{v}$ $(m\in Y)$ , (2)where $-\pi\leq\arg(\zeta-z)\leq\pi$ for $C_{-}$ . $0\leq\arg(\zeta-z)\leq 2\pi$ for $C_{*}$ ,$\zeta*z$ . $z\in C$ , $v\in R_{*}$ $\Gamma$ ; Gamma function,then $(f)_{v}$ is the fractional differtntegration of arbitrary order $v$ (derivatives oforder $v$ for $v>0$ , and integrals of $order-v$ for $v<0$ ), with respect to $Z$ , ofthe functton $f$ . if $|(f)_{V}|<\infty$ .1579 2008 79-87 79(I I) On the fractional calculus operator $N^{v}[3]$Theorem A. Let fractional calculus operator (Nishimoro’s Operator) $N^{v}$ be$N^{v} \approx(\frac{\Gamma(v+1)}{2\pi i}\int_{C^{\frac{d\zeta}{(\zeta-z)^{v*1}}}})$ $(v\not\in T)$ , [Refer to (1)] (3)with $N^{-m}= \lim_{varrow-,n}N^{\nu}$ $(m\in Z^{*})$ , (4)and defi ne the $bina\eta$ operation $\circ$ as$N^{\beta}\circ N^{a}f-N^{\beta}N^{\alpha}f-N^{\beta}(N^{a}f)$ $(\alpha, \beta\in R)$ , (5)then the set$\{N^{v}\}-\{N^{v}|v\in R\}$ (6)is an Abelian Product group (having continuous index $v$ ) which has the inversetransfom operaror $(N^{v})^{-1}=W^{v}$ to the fractional calculus operator $N^{v}$ , for theftznction $f$ such that $f\in F=\{f;0\neq|f_{\vee}|<\infty,$ $v\in R\}$ . where $f-f(z)$ and $z\in C$ .(vis. $-\infty<v<\infty$ ).(For our convenience, we call $N^{\beta}\circ N^{a}$ as product of $N^{\beta}$ and $N^{a}$ . )Theorem B. “ F.O.G. $\langle N^{\nu}$ } “ is an * Action product group which has continuousindex $v$ “ for the set of F. (F.O.G. ; Fractional calculus operator group) 131(III) Lemma. We have [1](i) $((z-c)^{b})$ $-e^{-ir}$ $\frac{\Gamma(\alpha-b)}{\Gamma(-b)}(z-c)^{b-\alpha}$ $(| \frac{\Gamma(\alpha-b)}{\Gamma(-b)}|<\infty)$ , (7)(I i) $(\log(z-c))_{a}--e^{-ira}\Gamma(\alpha)(z-c)^{-a}$ $(|\Gamma(\alpha)|<\infty)$ , (8)$(iiI)$ $((z-c)^{-\alpha})_{-\alpha}=-e^{i\pi a} \frac{1}{\Gamma(\alpha)}\log(z-c)$ $(|\Gamma(\alpha)|<\infty)$ , $t9$ )where $z-cx0$ for (i) and $z-c*O,$ $1$ for (i1), ( $i$ il),(1 v) $(u\cdot v)_{\alpha}$ $:- \sum_{k\cdot 0}^{\infty}\frac{\Gamma(\alpha+1)}{k!\Gamma(\alpha+1-k)}u_{\alpha-k}v_{k}$ $(\begin{array}{l}u-u(z)arrow vv(z)\end{array})$ . (10)\S 1. PreliminaryThe Teorem below is reported by the author already (cf. J. Frac. Calc. Vol. 29,May $(2006),pp.35- 44.)$ . $[12]$Theorem D. We have(1) $(((z-b)^{\beta}-c)^{a})_{\gamma}-e^{-ix\gamma}(z-b)^{\alpha\beta-\gamma}$80$x\sum_{k*0}^{\infty}\frac{[-\alpha]_{k}\Gamma(\beta k-\alpha\beta+\gamma)}{k!\Gamma(\beta k\sim\alpha\beta)}(\frac{c}{(z-b)^{\beta}}I^{\text{ }}$ (1)and$(ii)$ $(((z-b)^{\beta}-c)^{a})_{\iota}-(-1)^{n}(z-b)^{\alpha\beta-n}$$x\sum_{k\cdot 0}^{\infty}\frac{[-a]_{k}[\beta k-\alpha\beta]_{n}}{k!}(\frac{c}{(z-b)^{\beta}})^{k}$ $(n\in Z_{0}^{*})$ (2)where$| \frac{c}{(z-b)^{\beta}}|<1$ ,and$[\lambda]_{k}\approx\lambda(\lambda+1)\cdots(\lambda+k-1)\approx\Gamma(\lambda+k)/\Gamma(\lambda)$ with $[\lambda]_{0}arrow 1$ ,(Notation of Pochhammer).\S 2. N- Fractional Calculus of Functions$(((z-b)^{\beta}-c)^{a}-d)^{\delta}$Theorem 1. We have(i) $((((z-b)^{\beta}-c)^{\alpha}-d)^{\delta})_{\gamma}-e^{-i\kappa\gamma}(z-b)^{a\beta\delta-\gamma}$$x\sum_{k-0}^{\infty}\frac{[-\delta]_{m}[-\alpha(\delta-m)]_{k}\Gamma(\beta k-\alpha\beta(\delta-m)+\gamma)}{m!\cdot k!\Gamma(\beta k-\alpha\beta(\delta-m))}(\frac{c}{(z-b)^{\beta}})^{k}(\frac{d}{(z-b)^{\alpha\beta}}l^{m}$ (1)and(Ii) $(((z-b)^{\beta}-c)^{\alpha}-d)^{\delta})_{n}\approx(-1)^{n}(z-b)^{\alpha\beta\delta-n}$$xm\sum_{-0^{m!k!}}^{\ovalbox{\tt\small REJECT}^{-}(\frac{c}{(z-b)^{\beta}})^{k}}^{\infty}[-\delta][-a(\delta m)].[\beta k-\alpha\beta(\delta-m)](\frac{d}{(z-b)^{\alpha\beta}})^{m}$ . (2)81where$((z-b)^{\beta}-c)^{\alpha}-d\neq 0$ , $| \frac{c}{(z-b)^{\beta}}|<1$ , $| \frac{d}{(z-b)^{\alpha\beta}}|<1$ , $n\in Z_{0}^{*}$ .Proof of (i). We have$(((z-b)^{\beta}-c)^{a}-d)^{\delta}\approx X^{\delta}(1-\cdot 4X)^{\delta}$ $(X\approx((z-b)^{\beta}-c)^{a})$ (3)$arrow X^{s}\sum_{m\triangleleft}^{\infty}\frac{[-\delta 1_{n}}{m!}(\frac{d}{X})^{m}$ (4)$- \sum_{m\cdot 0}^{\infty}\frac{[-\delta]_{m}(d^{m})}{m!}X^{\delta\sim m}$ (5)(6)hence, operating N- fractional calculus operator $N^{\gamma}$ to the $bo$th sIdes of (6), weobtain$((((z-b)^{\beta}-c)^{a}-d)^{\delta})_{\gamma}$$\approx\sum_{m-0}^{\infty}\frac{[-\delta]_{m}(d^{m})}{m!}(((z-b)^{\beta}-c)^{\alpha(\delta-\iota n)})_{\gamma}$ (7)$- \sum_{m-0}^{\infty}\frac{[-\delta]_{m}(d^{m})}{m!}[e^{-\dot{l}*\gamma}(z-b)^{\alpha\beta(\delta-m)-\gamma}$$xz_{-0^{k!\Gamma(\beta k-\alpha\beta(\delta-m))}}^{\ovalbox{\tt\small REJECT}}\infty[-a(\delta-m)]\Gamma(\beta k-\alpha\beta(\delta-m)+\gamma)(\frac{c}{(z-b)^{\beta}}I^{k}$ (8)applying Theorem D. (1), under the conditions stated before.We have then (1) from (8) clearly.Proof of (1i). Set $y-n$ in (1).Note 1. We use the notatlons $\sum_{m.k-0}^{\infty}\cdots\approx\sum_{m-0}^{\infty}\sum_{k-0}^{\infty}\cdots$ for our convenience.Note.2. When $d-0$ and $\delta\simeq 1$ , Theorem 1 is reduced to Theorem D., clearly.82Corollary 1. We have(i) $((((z^{\beta}-c)^{\alpha}-d)^{\delta})_{\gamma}=e^{-i\pi\gamma}z^{\alpha\beta\delta-\gamma}$$xm\sum_{-0}^{\ovalbox{\tt\small REJECT}^{k}}[-\delta][-m\alpha!(\delta k-!m)]\Gamma(\beta k-\alpha\beta(\delta-m)+\gamma)(\frac{c}{z^{\beta}}I^{k}(\frac{d}{z^{\alpha\beta}}I^{m}\infty\cdot$ (9)and$(ii)$ $(((z^{\beta}-c)^{\alpha}-d)^{\delta})_{n}\approx(-1)^{n}z^{\alpha\beta l-n}$$x\sum_{m.k-0}^{\infty}\frac{[-\delta]_{m}[-\alpha(\delta-m)]_{k}[\beta k-\alpha\beta(\delta-m)]_{*}}{m!\cdot k!}(\frac{c}{z^{\beta}})^{k}(\frac{d}{z^{\alpha\beta}})^{m}$ , (10)where$(z^{\beta}-c)^{\alpha}-.d\prime 0$ , $| \frac{c}{z^{\beta}}|<1$ , $| \frac{d}{z^{\alpha\beta}}|<1$ , $n\in Z_{0}^{*}$ .Proof. Set $b-0$ in Theorem 1.\S 3. Some Special Cases[I] When $\beta\approx\alpha-1$ , we obtain$((((z-b-c-\phi^{\delta})_{\gamma}\infty e^{-i\kappa\gamma}(z-b)^{\delta-\gamma}$$x\sum_{mk-01}^{\infty}\frac{[-\delta]_{m}[m.-\delta]_{k}I\chi k+m-\delta+y)}{m!k!\Gamma(k+m-\delta)}(\frac{c}{z-b})^{k}(\frac{d}{z-b})^{m}$ (1)$(\vdash_{z-}c_{b}|$ , $| \frac{d}{z-b}|<1$ , $| \frac{\Gamma(k+m-\delta+\gamma)}{r\chi k+m-\delta)}|<\infty)$from Theorem 1. (i).Now we have the identities$\Gamma(m-\delta)-\Gamma(-\delta)[-\delta]_{m}$ , (2)$\Gamma(k+m-\delta)\approx\Gamma(m-\delta)[m-\delta\iota$ , (3)$\Gamma(k+m-\delta+\gamma)-\Gamma(m-\delta\cdot+y)[m-\delta+\gamma]_{k}$ , (4)83then applying (2) $\sim(4)$ into the RHS of (1) , we obtainRHS of (1) $=e^{-in\gamma}(z-b)^{\delta-\gamma}$$x\sum_{m-0}^{\infty}\frac{\Gamma(m-\delta+y)}{m!\Gamma(-\delta)}(\frac{d}{z-b})^{m}\sum_{k-0}^{\infty}\frac{[m-\delta+\gamma]_{k}}{k!}(\frac{c}{z-b})^{k}$ (5)$arrow e^{-i\pi\gamma}(z-b)^{\delta-\gamma}$ $\sum_{m-0}^{\infty}\frac{\Gamma(m-\delta+\gamma)}{m!\Gamma(-\delta)}(\frac{d}{z-b})^{m}(1-\frac{c}{z-b})^{\delta-\gamma-m}$ (6)$\approx e^{-i\eta}(z-b-c)^{\delta-\gamma}$ $\frac{\Gamma(-\delta+y)}{\Gamma(-\delta)}\sum_{m-0}^{\infty}\frac{[-\delta+\gamma]_{m}}{m!}(\frac{d}{z-b-c})^{m}$ (7)$=e^{-in\gamma} \frac{\Gamma(-\delta+y)}{\Gamma(-\delta)}(z-b-c-d\gamma^{\delta-\gamma}$ . $(| \frac{\Gamma(-\delta+y)}{\Gamma(-\delta)}|<\infty)$ (8)$us\dot{i}g$ the relationship$\sum_{-0}^{\infty}\frac{[\lambda]_{k}}{k!}z^{k}=(1-z)^{-\lambda}$ . (9)The result (8) ls same as the one obtained by Lemma (i).[II] When $d-0$ and $\delta\approx 1$ , we obtaIn$(((z-b)^{\beta}-c)^{a})_{\gamma}-e^{-l\kappa\gamma}(z-b)^{a\beta-\gamma}$$xm2_{-0^{m!\cdot k!\Gamma(\beta k-\alpha\beta(1-m))}}^{\ovalbox{\tt\small REJECT}(\frac{c}{(z-b)^{\beta}})^{k}(\frac{0}{(z-b)^{\phi}}I^{m}}\infty[-1][-\alpha(1-m)]\Gamma(\beta k-\alpha\beta(1-m)+\gamma)$ (10)$\approx e^{-i\eta}(z-b)^{a\beta-\gamma}’\geq\triangleleft\frac{[-\alpha]_{k}\Gamma(\beta k-\alpha\beta+y)}{k!\Gamma(\beta k-\alpha\beta)}\infty(\frac{c}{(z-b)^{\beta}}I^{k}$ (11)from Theorem 1. (i).That is, in this case we have Theorem D. (1)lt self, clearly.[III I When $n-0$ , we have$(((z-b)^{\beta}-c)^{a}-d)^{\delta})_{0}-(z-b)^{\alpha\beta\delta}$$xmz_{-0}’\frac{[-\delta]_{m}[-\alpha(\delta-m)]_{k}}{m!\cdot k!}(\frac{c}{(z-b)^{\beta}})^{k}(\frac{d}{(z-b)^{\alpha\beta}})^{m}$ . (12)84$((z-b)^{\beta}-c)^{\alpha}-d\neq 0$ , $| \frac{c}{(z-b)^{\beta}}|<1$ , $| \frac{d}{(z-b)^{\alpha\beta}}|<1$ , .from Theorem 1. $(ii)$.Indeed we obtainRHS of (12) . $(z-b)^{a\beta\delta}$$x\sum_{m-0}^{\infty}\frac{[-\delta]_{m}}{m!}(\frac{d}{(z-b)^{\alpha\beta}}I^{m}z_{\triangleleft}\infty\frac{[-\alpha(\delta-m)]_{k}}{k!}(\frac{c}{(z-b)^{\beta}})^{k}$ (13)$\approx(z-b)^{\alpha\mu}$ $\sum_{m-0}^{\infty}\frac{[-\delta]_{m}}{m!}(\frac{d}{(z-b)^{\alpha\beta}}\}^{m}(1$ $\frac{c}{(z-b)^{\beta}}1^{\alpha\delta}$ (14)$arrow((z-b)^{\beta}-c)^{a\delta}\sum_{\dot{m}-0}^{\infty}\frac{[-\delta]_{m}}{m!}(\frac{d}{((z-b)^{\beta}-c)^{\alpha}})^{m}$ (15)$-((z-b)^{\beta}-c)^{\alpha\delta}$ $(1- \frac{d}{((z-b)^{\beta}-c)^{\alpha}})^{\delta}$ (16)$-(((z-b)^{\beta}-c)^{\alpha}-d)^{\delta}$ , (17)dearly.[IV] When $n-1$ , we have$(((z-b)^{\beta}-c)^{\alpha}-d)^{\delta})_{1}--(z-b)^{a\beta\delta-1}$$x\sum_{m.k-0}^{\infty}\frac{[-\delta]_{m}[-\alpha(\delta-m)]_{k}\cdot(\beta k-\alpha\beta\delta+a\beta m)}{m!\cdot k!}(\frac{c}{(z-b)^{\beta}})^{k}(\frac{d}{(z-b)^{a\beta}}I^{n}$ . (18)from Theorem 1. $(ii)$.Then lettig$R:-( \frac{u}{(z-b)^{\beta}}I^{\alpha}$ $(u-(z-b)^{\beta}-c)$ (19)and$S$ : ., $(1- \frac{d}{u^{\alpha}})^{t}$ (20)85we have$\sum_{m.k0}^{\infty}.\frac{[-\delta]_{m}[-\alpha(\delta-m)]_{k}\cdot\beta k}{m!\cdot k!}(\frac{c}{(z-b)^{\beta}}I^{k}(\frac{d}{(z-b)^{\alpha\beta}}I^{m}$$- \frac{-c\alpha\beta\delta}{u}R^{\delta}S-\frac{cd\alpha\beta\delta}{u(u^{a}-d)}R^{\delta}S$ (21)$- \alpha\beta\delta m\sum_{\triangleleft}^{\infty}\frac{[-\delta L[-\alpha.(\delta-m)]_{k}}{m!k!}(\frac{c}{(z-b)^{\beta}}I^{\iota}(\frac{d}{(z-b)^{a\beta}})^{m}$$–a\beta\delta R^{\delta}S$ , (22)and$\alpha\beta\sum_{m,k0}^{\infty}.\frac{[-\delta]_{m}\cdot m[-.\alpha(\delta-m)]_{k}}{m!k!}(\frac{c}{(z-b)^{\beta}})^{k}(\frac{d}{(z-b)^{\alpha\beta}}I^{m}$$– \frac{da\beta\delta}{a}R^{\delta}S$ (23)$(u-d)$Therefore, applying (23), (22) and (21) into (18), we obtain$(((z-b)^{\beta}-c)^{a}-d)^{\delta})_{1}-(z-b)^{\alpha\beta\delta-1}R^{\delta}S$$x[\frac{ca\beta\delta}{u}+\frac{cda\beta\delta}{u(u^{a}-d)}+\alpha\beta\delta+\frac{da\beta\delta}{u^{a}-d}]$ (24)$\approx a\beta\delta(z-b)^{\alpha\beta\delta-1}(\frac{c}{u}+1)(\frac{d}{u^{\alpha}-d}+\iota)$ (25)(26)$\simeq\alpha\beta\delta(z-b)^{-1}(u^{\alpha}-d)^{\delta}(\frac{(z-b)^{\beta}}{u}I\{\frac{u^{\alpha}}{u^{a}-d}I$$-a\beta\delta(u^{\alpha}-d)^{\delta-1}(u^{\alpha-1})(z-b)^{\beta-1}$ (27)$-\alpha\beta\delta(((z-b)^{\beta}-c)^{\alpha}-d)^{\delta-1}((z-b)^{\beta}-c)^{a-1}(z-b)^{\beta-1}$ (28)clearly.86References[1] K. 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KIryakova; Generalized fractional calculus and applications, Mtman Rese-arch Notes, No. 301, (1994), Longman.[20] Igor $p_{0}nubny$ ; Fractional Differendal Equations (1999), Academic Press.[21] R. Hilfer (Ed.); Applications of Fractional Calculus in PhysIcs, (2000), WorldScientific, Singapor, New Jersey, London, Hong Kong.[22] A.P. PIudnikov, Yu. A. Bryckov and 0.I. Marichev; Integrals and Series, Vol.I, Gordon and Breach, New York, (1986).[23] S. Morlguchi, K. Udagawa and S. Hitotsumatsu $j$ Mathematical Formulae, Vol.2, Iwanami Zensho, (1957), Iwanami, Japan.Katsuyuki NishimotoIn$s$titute for Appled MathematicsDescartes Press Co.2-13-10 Kaguike, KorIyama963-8833 Japan87

N-Fractional Calculus of Some Multi-Powers Functions(Study on Geometric Univalent Function Theory)

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N-Fractional Calculus of Some Multi-Powers Functions(Study on Geometric Univalent Function Theory)

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