폭스-라이트 함수

수학에서 폭스-라이트 함수(Fox-Wright Psi 함수라고도 함)는 찰스 폭스(1928)와 E의 아이디어를 바탕으로 일반화된 초기하 함수 F_q(z)를 일반화한 것입니다. 메이틀랜드 라이트 (1935):

{}_{p}\Psi_{q}\left[{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix}};z\right]=\sum_{n=0}^{\infty}{\frac {\Gamma(a_{1}+)A_{1}n)\cdots \Gamma (a_{p}+A_{p}n}}{\Gamma(b_{1}+B_{1}n)\cdots \Gamma(b_{q}+B_{q}n}}\,{\frac {z^{n}}}{n!}}.

정규화 변경 시

{}_{p}\Psi _{q}^{*}\left[{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix};z\right]={\frac {\Gamma (b_{1})\cdots \Gamma (b_{q})}{\Gamma (a_{1})}\cdots \Gamma (a_{p}}}}\sum _{n=0}^{\infty}{\frac {\Gamma (a_{1}+)}A_{1}n)\cdots \Gamma (a_{p}+A_{p}n}}{\Gamma(b_{1}+B_{1}n)\cdots \Gamma(b_{q}+B_{q}n}}\,{\frac {z^{n}}}{n!}}

A = B = 1에 대하여 F(z)가 됩니다.

Fox-Wright 함수는 Fox H 함수의 특수한 경우입니다(Srivastava & Manocha 1984, 페이지 50).

{}_{p}\Psi_{q}\left[{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix}};z\right]=H_{p,q+1}^{1,p}\left[-z\left {\begin{matrix}(1-a_{1},A_{1})&(1-a_{2})&\ldots &(1-a_{p}),A_{p})\\(0,1)&(1-b_{1},B_{1})&(1-b_{2},B_{2})&\ldots &(1-b_{q},B_{q})\end{matrix}}\right.\right.

A special case of Fox–Wright function appears as a part of the normalizing constant of the modified half-normal distribution^[1] with the pdf on $(0,\infty )$ is given as ${\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma$ $x) }{\Psi {\left}({\frac {\alpha}{2}},{\frac {\gamma}{\sqrt {\beta}}}\right)$ $}}}}$ 여기서 $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ $ψ$ $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ (α, z ) $=$ 1 $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ $ψ$ 1 $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ ( $α,$ 12 ) (1, 0 ) ; z ) {\displaystyle \Psi (\alpha,z) $=$ {}_{1}\Psi _{1}\left $({$ \ $begin$ { $matrix$ }\left(\alpha,{\frac {1}{2}}\right)\(1,0)\end{ $matrix$ };z\right)}는 Fox-Wright Psi 함수를 나타냅니다.

라이트 함수

전체 함수 $W_{\lambda ,\mu }(z)$ λ $W_{\lambda ,\mu }(z)$ μ ( z) {\ $displaystyle W_{\lambda,\$ mu }(z)}을(를) 라이트 함수라고 합니다.Fox-Wright 함수의 0 ${}_{0}\Psi _{1}\left[\ldots \right]$ ψ 1 ${}_{0}\Psi _{1}\left[\ldots \right]$ [ … ] ${\displaystyle$ {}_ ${0}\Psi$ _{1}\ $left[\ldots \$ right]}의 특수한 경우입니다.그 시리즈는 다음과.

W_{\lambda,\mu }(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!\,\Gamma(\lambda n+\mu )}},\lambda >-1.

이 함수는 분수 미적분학과 안정적인 카운트 분포에서 광범위하게 사용됩니다. $\lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )$ λ → $\lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )$ 0 $W$ λ, μ ( z ) = ez / γ lambda ( μ ) {\displaystyle \lim \limits _{\lambda \to 0}W_{\lambda,\mu }(z) = e^{z}/\Gamma(\mu )}. 따라서 0μ {\displaystyle \mu }인 0이 아닌 λ {\displaystyle \lambda }은 이러한 맥락에서 지수 함수의 가장 단순한 비사소 확장입니다.

라이트의 정리 1(1933)^[3]과 18.1(30-32), Bateman Project, Vol 3(1955)(^[4]p.212)에서 세 가지 성질이 언급되었습니다.

{\begin{aligned}\lambda zW_{\lambda,\mu +\lambda}(z)&=W_{\lambda,\mu -1}(z)+(1-\mu)W_{\lambda,\mu }(z)&(a)\[6pt]{d \over dz}W_{\lambda,\mu }(z)&=W_{\lambda,\mu +\lambda }(z)&(b)\[6pt]\lambda z{d \over dz}W_{\lambda,\mu }(z)&=W_{\lambda,\mu -1}(z)+(1-\mu)W_{\lambda,\mu }(z)&(c)\end{aligned}

식 (a)는 반복 공식이며, (b)와 (c)는 도함수를 줄이기 위한 두 가지 경로를 제공합니다.그리고 (c)는 (a) 및 (b)로부터 도출될 수 있습니다.

(c)의 특수한 경우는 λ = - $\lambda =-c\alpha ,\mu =0$ c $\lambda =-c\alpha ,\mu =0$ α, $\lambda =-c\alpha ,\mu =0$ = 0 ${\displaystyle$ \lambda = $-$ c\alpha,\mu =0}입니다. z {\displaystyle z}를 - x α {\displaystyle -x^{\alpha}}로 대체하면 다음과 같습니다.

{\begin{array}{lcl}x{d \overdx}W_{-c\alpha}(-x^{\alpha})&=&-{\frac {1}{c}}\left[W_{-c\alpha,-1}(-x^{\alpha})+W_{-c\alpha,0}(-x^{\alpha})\right]\end{array}

(a)의 특수한 경우는 λ = - $,$ μ = 1 {\ $displaystyle$ \lambda =-\alpha,\mu =1}입니다. z {\displaystyle z}를 -z {\displaystyle -z}로 대체하면 다음과 같습니다.

\alpha zW_{-\alpha,1-\alpha}(-z)=W_{-\alpha,0}(-z)

M $M_{\alpha }(z)$ ( $M_{\alpha }(z)$ $M_{\alpha}(z)$ 와 $M_{\alpha }(z)$ $F_{\alpha }(z)$ α $F_{\alpha }(z)$ $F_{\alpha}(z)$ 의 두 표기법이 문헌에서 광범위하게 사용되었습니다 $F_{\alpha }(z)$

{\begin{aligned}M_{\alpha }(z)&=W_{-\alpha,1-\alpha }(-z),\[1ex]\implies F_{\alpha }(z)&=W_{-\alpha,0}(-z)=\alpha zM_{\alpha }(z).\end{aligned}

엠라이트 함수

$M_{\alpha }(z)$ ( $M_{\alpha }(z)$ $M_{\alpha }(z)$ 는 M-Wright 함수로 $M_{\alpha }(z)$ 알려져 있으며, 일반적으로 시간-분절 확산 프로세스라고 불리는 관련 클래스의 자기 유사 확률 프로세스에서 확률 밀도로 들어갑니다.

Mainardi et al(2010)에서 그 속성을 조사했습니다.^[5]안정 카운트 분포를 통해 $\alpha$ $\alpha$ 는 레비의 안정 지수 $(0<\alpha \leq 1)$ < $(0<\alpha \leq 1)$ $(0<\alpha \leq 1)$ $(0<\alpha \leq 1)$ $(0<\alpha \leq 1)$ 와 연결됩니다 $\alpha$ $(0<\alpha \leq 1)$

$\alpha >0$ > $\alpha >0$ $\alpha >0$ 에 대한 $M_{\alpha }(z)$ M $M_{\alpha }(z)$ ( $M_{\alpha }(z)$ ${\displaystyle M_{\alpha}(z)$ 의 점근적 확장은 $\alpha >0$

M_{\alpha}}\left ({\frac {r}{\alpha}}\right)=A(\alpha )\,r^{(\alpha -1/2)/(1-\alpha )}\,e^{-B(\alpha )}\,r^{1/(1-\alpha )}},\,\,r\직화살 \infty,

여기서

A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},

(

A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},

)

A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},

=

A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},

B(\alpha )={\frac {1-\alpha }{\alpha }}.

π

(

1 - α ), {\

displaystyle

A (\alpha ) = {\frac {1}{\sqrt {2\pi (1 -\alpha )}},} B (α ) = 1 - α . {\displaystyle B (\alpha ) = {\frac {1 -\alpha }}.

참고 항목

초기하함수
일반화된 초기하함수
Modified half-normal distribution^[1] with the pdf on $(0,\infty )$ is given as ${\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)$ $}}}}$ 여기서 $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ $ψ$ $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ (α, z ) $=$ 1 $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ $ψ$ 1 $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ ( $α,$ 12 ) (1, 0 ) ; z ) {\displaystyle \Psi (\alpha,z) $=$ {}_{1}\Psi _{1}\left $({$ \ $begin$ { $matrix$ }\left(\alpha,{\frac {1}{2}}\right)\(1,0)\end{ $matrix$ };z\right)}는 Fox-Wright Psi 함수를 나타냅니다.

참고문헌

^ ^a ^b Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.
^ Weisstein, Eric W. "Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03.
^ Wright, E. (1933). "On the Coefficients of Power Series Having Exponential Singularities". Journal of the London Mathematical Society. Second Series: 71–79. doi:10.1112/JLMS/S1-8.1.71. S2CID 122652898.
^ Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.
^ Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey. arXiv:1004.2950.

Fox, C. (1928). "The asymptotic expansion of integral functions defined by generalized hypergeometric series". Proc. London Math. Soc. 27 (1): 389–400. doi:10.1112/plms/s2-27.1.389.
Wright, E. M. (1935). "The asymptotic expansion of the generalized hypergeometric function". J. London Math. Soc. 10 (4): 286–293. doi:10.1112/jlms/s1-10.40.286.
Wright, E. M. (1940). "The asymptotic expansion of the generalized hypergeometric function". Proc. London Math. Soc. 46 (2): 389–408. doi:10.1112/plms/s2-46.1.389.
Wright, E. M. (1952). "Erratum to "The asymptotic expansion of the generalized hypergeometric function"". J. London Math. Soc. 27: 254. doi:10.1112/plms/s2-54.3.254-s.
Srivastava, H.M.; Manocha, H.L. (1984). A treatise on generating functions. ISBN 0-470-20010-3.
Miller, A. R.; Moskowitz, I.S. (1995). "Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters". Computers Math. Applic. 30 (11): 73–82. doi:10.1016/0898-1221(95)00165-u.
Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.

외부 링크

GitLab에서 하이퍼검

[Sun,_Kong_and_Pal-1] Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.

[2] Weisstein, Eric W. "Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03.

[3] Wright, E. (1933). "On the Coefficients of Power Series Having Exponential Singularities". Journal of the London Mathematical Society. Second Series: 71–79. doi:10.1112/JLMS/S1-8.1.71. S2CID 122652898.

[4] Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.

[5] Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey. arXiv:1004.2950.

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