SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Fox–Wright function</title>

</head>

Fox–Wright function

</h1>

</header>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with <a href="Wright_Omega_function" class="mw-redirect" title="Wright Omega function">Wright Omega function</a>) is a generalisation of the <a href="Generalized_hypergeometric_function" title="Generalized hypergeometric function">generalised hypergeometric function</a> pFq(z) based on ideas of <a href="Charles_Fox_(mathematician)" title="Charles Fox (mathematician)">Charles Fox</a> (<a href="#CITEREFFox1928">1928</a>) and <a href="E.M._Wright" class="mw-redirect" title="E.M. Wright">E. Maitland Wright</a> (<a href="#CITEREFWright1935">1935</a>):

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{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!}}.}</annotation>

</semantics>

</math>

Upon changing the normalisation

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{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!}}}</annotation>

</semantics>

</math>

it becomes pFq(z) for A1...p = B1...q = 1.

The Fox–Wright function is a special case of the <a href="Fox_H-function" title="Fox H-function">Fox H-function</a> (<a href="#CITEREFSrivastavaManocha1984">Srivastava & Manocha 1984</a>, p. 50):

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{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},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].}</annotation>

</semantics>

</math>

A special case of Fox–Wright function appears as a part of the <a href="Normalizing_constant" title="Normalizing constant">normalizing constant</a> of the <a href="Modified_half-normal_distribution" title="Modified half-normal distribution">modified half-normal distribution</a><a href="#cite_note-Sun,_Kong_and_Pal-1">[1]</a> with the pdf on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,\infty )}</annotation>

</semantics>

</math><img src="./da17102e4ed0886686094ee531df040d2e86352a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (0,\infty )}" loading="lazy"> is given as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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)}}}}">

<mfrac>

<mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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)}}}}</annotation>

</semantics>

</math><img src="./ba362b2bb9616f39c06cb4214bf4d8df1d14dc4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:34.194ex; height:10.509ex;" alt="{\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)}}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}</annotation>

</semantics>

</math><img src="./f5f318f1c6f5b6c50886d35fe09b9205c3e66784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.494ex; height:7.509ex;" alt="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}" loading="lazy"> denotes the <a href="Fox%E2%80%93Wright_Psi_function" class="mw-redirect" title="Fox–Wright Psi function">Fox–Wright Psi function</a>.

<div class="mw-heading mw-heading2"><h2 id="Wright_function">Wright function</h2></div>

The entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\lambda ,\mu }(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{\lambda ,\mu }(z)}</annotation>

</semantics>

</math><img src="./6e4644aeb544d2e20e18ad8896784fccecb378ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.73ex; height:3.009ex;" alt="{\displaystyle W_{\lambda ,\mu }(z)}" loading="lazy"> is often called the Wright function.<a href="#cite_note-2">[2]</a> It is the special case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{0}\Psi _{1}\left[\ldots \right]}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{0}\Psi _{1}\left[\ldots \right]}</annotation>

</semantics>

</math><img src="./3d4d213f41c4cbec3b33408d2c2e338039817259.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.321ex; height:2.843ex;" alt="{\displaystyle {}_{0}\Psi _{1}\left[\ldots \right]}" loading="lazy"> of the Fox–Wright function. Its series representation is

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{\lambda ,\mu }(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!\,\Gamma (\lambda n+\mu )}},\lambda >-1.}</annotation>

</semantics>

</math>

This function is used extensively in <a href="Fractional_calculus" title="Fractional calculus">fractional calculus</a> and the <a href="Stable_count_distribution" title="Stable count distribution">stable count distribution</a>. Recall that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )}</annotation>

</semantics>

</math><img src="./3882b5e5b37df67fba715ee91904af89859ec09c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.551ex; height:4.009ex;" alt="{\displaystyle \lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )}" loading="lazy">. Hence, a non-zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> with zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is the simplest nontrivial extension of the <a href="Exponential_function" title="Exponential function">exponential function</a> in such context.

Three properties were stated in Theorem 1 of Wright (1933)<a href="#cite_note-3">[3]</a> and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955)<a href="#cite_note-4">[4]</a> (p. 212)

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

</semantics>

</math>

Equation (a) is a recurrence formula. (b) and (c) provide two paths to reduce a derivative. And (c) can be derived from (a) and (b).

A special case of (c) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =-c\alpha ,\mu =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda =-c\alpha ,\mu =0}</annotation>

</semantics>

</math><img src="./36a69a9e1caa4ce8d5d5712129b5a1ee9aafa82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.453ex; height:2.676ex;" alt="{\displaystyle \lambda =-c\alpha ,\mu =0}" loading="lazy">. Replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x^{\alpha }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -x^{\alpha }}</annotation>

</semantics>

</math><img src="./fbef7192ee6e8672627412f95e33b4832379a907.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.422ex; height:2.509ex;" alt="{\displaystyle -x^{\alpha }}" loading="lazy">, we have

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcl}x{d \over dx}W_{-c\alpha ,0}(-x^{\alpha })&=&-{\frac {1}{c}}\left[W_{-c\alpha ,-1}(-x^{\alpha })+W_{-c\alpha ,0}(-x^{\alpha })\right]\end{array}}}</annotation>

</semantics>

</math>

A special case of (a) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =-\alpha ,\mu =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda =-\alpha ,\mu =1}</annotation>

</semantics>

</math><img src="./7b0c278f5e9de73b1cdbac26d3108efdff404362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.446ex; height:2.676ex;" alt="{\displaystyle \lambda =-\alpha ,\mu =1}" loading="lazy">. Replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -z}</annotation>

</semantics>

</math><img src="./78c4571a4a3ceb7e7e55712372835ebe65d20f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.896ex; height:2.176ex;" alt="{\displaystyle -z}" loading="lazy">, we have

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha zW_{-\alpha ,1-\alpha }(-z)=W_{-\alpha ,0}(-z)}</annotation>

</semantics>

</math>

Two notations, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\alpha }(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\alpha }(z)}</annotation>

</semantics>

</math><img src="./887cd70b6d5371fb221e776173d0983b96c80c6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.436ex; height:2.843ex;" alt="{\displaystyle M_{\alpha }(z)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\alpha }(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\alpha }(z)}</annotation>

</semantics>

</math><img src="./6238377e847e91e0dc7bf846ed5054ee977a07f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.676ex; height:2.843ex;" alt="{\displaystyle F_{\alpha }(z)}" loading="lazy">, were used extensively in the literatures:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="M-Wright_function">M-Wright function</h3></div>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\alpha }(z)}</annotation>

</semantics>

</math><img src="./887cd70b6d5371fb221e776173d0983b96c80c6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.436ex; height:2.843ex;" alt="{\displaystyle M_{\alpha }(z)}" loading="lazy"> is known as the M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes, generally referred to as time-fractional diffusion processes.

Its properties were surveyed in Mainardi et al (2010).<a href="#cite_note-5">[5]</a>

Through the <a href="Stable_count_distribution" title="Stable count distribution">stable count distribution</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is connected to Lévy's stability index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0<\alpha \leq 1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0<\alpha \leq 1)}</annotation>

</semantics>

</math><img src="./3b6f473a8306b21d43dc999b37ec5948fd498a17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.819ex; height:2.843ex;" alt="{\displaystyle (0<\alpha \leq 1)}" loading="lazy">.

Its <a href="Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\alpha }(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\alpha }(z)}</annotation>

</semantics>

</math><img src="./887cd70b6d5371fb221e776173d0983b96c80c6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.436ex; height:2.843ex;" alt="{\displaystyle M_{\alpha }(z)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation>

</semantics>

</math><img src="./edd4f784b6e8bb68fa774213ceacbab2d97825dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}" loading="lazy"> is

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\alpha }\left({\frac {r}{\alpha }}\right)=A(\alpha )\,r^{(\alpha -1/2)/(1-\alpha )}\,e^{-B(\alpha )\,r^{1/(1-\alpha )}},\,\,r\rightarrow \infty ,}</annotation>

</semantics>

</math>

where

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},}</annotation>

</semantics>

</math><img src="./9b9e063fac756ab1e2dbd60ef9fba8fb2b2b0511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.739ex; height:6.509ex;" alt="{\displaystyle A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},}" loading="lazy">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(\alpha )={\frac {1-\alpha }{\alpha }}.}</annotation>

</semantics>

</math><img src="./df46cb4f424df7f56fc61e1da2dfcee473e51dd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.133ex; height:5.176ex;" alt="{\displaystyle B(\alpha )={\frac {1-\alpha }{\alpha }}.}" loading="lazy">

/* start https://en.wikipedia.org/ */

.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}

/* end https://en.wikipedia.org/ */

</style>

<ul><li><a href="Prabhakar_function" title="Prabhakar function">Prabhakar function</a></li>

<li><a href="Hypergeometric_function" title="Hypergeometric function">Hypergeometric function</a></li>

<li><a href="Generalized_hypergeometric_function" title="Generalized hypergeometric function">Generalized hypergeometric function</a></li>

<li><a href="Modified_half-normal_distribution" title="Modified half-normal distribution">Modified half-normal distribution</a><a href="#cite_note-Sun,_Kong_and_Pal-1">[1]</a> with the pdf on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,\infty )}</annotation>

</semantics>

<mfrac>

<mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</semantics>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFSunKongPal2021" class="citation journal cs1">Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20">"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"</a>. Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610926.2021.1934700">10.1080/03610926.2021.1934700</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0361-0926">0361-0926</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237919587">237919587</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/WrightFunction.html">"Wright Function"</a>. From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFWright1933" class="citation journal cs1">Wright, E. (1933). "On the Coefficients of Power Series Having Exponential Singularities". Journal of the London Mathematical Society. Second Series: 71–79. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2FJLMS%2FS1-8.1.71">10.1112/JLMS/S1-8.1.71</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122652898">122652898</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFErdelyi1955" class="citation book cs1">Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMainardiMuraPagnini2010" class="citation book cs1">Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1004.2950">1004.2950</a>.</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFFox1928" class="citation journal cs1">Fox, C. (1928). "The asymptotic expansion of integral functions defined by generalized hypergeometric series". Proc. London Math. Soc. 27 (1): 389–400. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-27.1.389">10.1112/plms/s2-27.1.389</a>.</cite></li>

<li><cite id="CITEREFWright1935" class="citation journal cs1">Wright, E. M. (1935). "The asymptotic expansion of the generalized hypergeometric function". J. London Math. Soc. 10 (4): 286–293. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjlms%2Fs1-10.40.286">10.1112/jlms/s1-10.40.286</a>.</cite></li>

<li><cite id="CITEREFWright1940" class="citation journal cs1">Wright, E. M. (1940). "The asymptotic expansion of the generalized hypergeometric function". Proc. London Math. Soc. 46 (2): 389–408. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-46.1.389">10.1112/plms/s2-46.1.389</a>.</cite></li>

<li><cite id="CITEREFWright1952" class="citation journal cs1">Wright, E. M. (1952). <a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-54.3.254-s">"Erratum to "The asymptotic expansion of the generalized hypergeometric function""</a>. J. London Math. Soc. 27: 254. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-54.3.254-s">10.1112/plms/s2-54.3.254-s</a>.</cite></li>

<li><cite id="CITEREFSrivastavaManocha1984" class="citation book cs1">Srivastava, H.M.; Manocha, H.L. (1984). A treatise on generating functions. E. Horwood. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-470-20010-3</bdi>.</cite></li>

<li><cite id="CITEREFMillerMoskowitz1995" class="citation journal cs1">Miller, A. R.; Moskowitz, I.S. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0898-1221%2895%2900165-u">"Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters"</a>. Computers Math. Applic. 30 (11): 73–82. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0898-1221%2895%2900165-u">10.1016/0898-1221(95)00165-u</a>.</cite></li>

<li><cite id="CITEREFSunKongPal2021" class="citation journal cs1">Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20">"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"</a>. Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610926.2021.1934700">10.1080/03610926.2021.1934700</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0361-0926">0361-0926</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237919587">237919587</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://gitlab.com/RZ-FZJ/hypergeom">hypergeom</a> on <a href="GitLab" title="GitLab">GitLab</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-02-23" href="https://en.wikipedia.org/wiki/?title=Fox%E2%80%93Wright_function&oldid=1277311308">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>