Fox–Wright function

The entire function $W\_\{\backslash lambda,\backslash mu\}(z)$ is often called the Wright function.{{cite web |url=https://mathworld.wolfram.com/WrightFunction.html |last=Weisstein |first=Eric W. |title=Wright Function |website=From MathWorld--A Wolfram Web Resource |access-date=2022-12-03}} It is the special case of $\{\}\_0\backslash Psi\_1\; \backslash left[\backslash ldots\; \backslash right]$ of the Fox–Wright function. Its series representation is

$$W\_\{\backslash lambda,\backslash mu\}(z)\; =\; \backslash sum\_\{n=0\}^\backslash infty$$

\frac{z^n}{n!\,\Gamma(\lambda n+\mu)}, \lambda > -1.

This function is used extensively in fractional calculus and the stable count distribution. Recall that $\backslash lim\backslash limits\_\{\backslash lambda\; \backslash to\; 0\}\; W\_\{\backslash lambda,\backslash mu\}(z)\; =\; e^\{z\}\; /\; \backslash Gamma(\backslash mu)$. Hence, a non-zero $\backslash lambda$ with zero $\backslash mu$ is the simplest nontrivial extension of the exponential function in such context.

Three properties were stated in Theorem 1 of Wright (1933){{cite journal |last=Wright |first=E. |date=1933 |title=On the Coefficients of Power Series Having Exponential Singularities |journal=Journal of the London Mathematical Society |series=Second Series |pages=71–79 |doi=10.1112/JLMS/S1-8.1.71 |s2cid=122652898 |language=en}} and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955){{Cite book |last=Erdelyi |first=A |title=The Bateman Project, Volume 3 |publisher=California Institute of Technology |year=1955}} (p. 212)

$$\backslash begin\{align\}$$

\lambda z W_{\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{align}

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 $\backslash lambda\; =\; -c\backslash alpha,\; \backslash mu\; =\; 0$. Replacing $z$ with $-x^\backslash alpha$, we have

$$\backslash begin\{array\}\{lcl\}$$

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}

A special case of (a) is $\backslash lambda\; =\; -\backslash alpha,\; \backslash mu\; =\; 1$. Replacing $z$ with $-z$, we have

$$\backslash alpha\; z\; W\_\{-\backslash alpha,1-\backslash alpha\}(-z)\; =\; W\_\{-\backslash alpha,0\}(-z)$$

Two notations, $M\_\{\backslash alpha\}(z)$ and $F\_\{\backslash alpha\}(z)$, were used extensively in the literatures:

$$\backslash begin\{align\}$$

M_{\alpha}(z) & = W_{-\alpha,1-\alpha}(-z), \\ [1ex]

\implies

F_{\alpha}(z) & = W_{-\alpha,0}(-z) = \alpha z M_{\alpha}(z).

\end{align}

$M\_\backslash alpha(z)$ 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).{{cite book |last1=Mainardi |first1=Francesco |last2=Mura |first2=Antonio |last3=Pagnini |first3=Gianni |date=2010-04-17 |title=The M-Wright function in time-fractional diffusion processes: a tutorial survey |arxiv=1004.2950}}

Through the stable count distribution, $\backslash alpha$ is connected to Lévy's stability index .

Its asymptotic expansion of $M\_\{\backslash alpha\}(z)$ for $\backslash alpha\; >\; 0$ is

$$$$

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,

where

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

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

{{Portal|Mathematics}}

• Prabhakar function
• Hypergeometric function
• Generalized hypergeometric function
• Modified half-normal distribution with the pdf on $(0,\; \backslash infty)$ is given as $f(x)=\; \backslash frac\{2\backslash beta^\{\backslash frac\{\backslash alpha\}\{2\}\}\; x^\{\backslash alpha-1\}\; \backslash exp(-\backslash beta\; x^2+\; \backslash gamma\; x\; )\}\{\backslash Psi\{\backslash left(\backslash frac\{\backslash alpha\}\{2\},\; \backslash frac\{\; \backslash gamma\}\{\backslash sqrt\{\backslash beta\}\}\backslash right)\}\}$, where $\backslash Psi(\backslash alpha,z)=\{\}\_1\backslash Psi\_1\backslash left(\backslash begin\{matrix\}\backslash left(\backslash alpha,\backslash frac\{1\}\{2\}\backslash right)\backslash \backslash (1,0)\backslash end\{matrix\};z\; \backslash right)$ denotes the Fox–Wright Psi function.

{{reflist}}

• {{cite journal | last= Fox | first= C. | title= The asymptotic expansion of integral functions defined by generalized hypergeometric series | journal= Proc. London Math. Soc. | year= 1928 | volume= 27 | issue= 1 | pages= 389–400 | doi=10.1112/plms/s2-27.1.389 }}
• {{cite journal | last= Wright | first= E. M. | title= The asymptotic expansion of the generalized hypergeometric function | journal= J. London Math. Soc. | year= 1935 | volume= 10 | issue= 4 | pages= 286–293 | doi=10.1112/jlms/s1-10.40.286 }}
• {{cite journal | last= Wright | first= E. M. | title= The asymptotic expansion of the generalized hypergeometric function | journal= Proc. London Math. Soc. | year= 1940 | volume= 46 | issue= 2 | pages= 389–408 | doi=10.1112/plms/s2-46.1.389}}
• {{cite journal | last= Wright | first= E. M. | title= Erratum to "The asymptotic expansion of the generalized hypergeometric function" | journal= J. London Math. Soc. | year= 1952 | volume= 27 | pages= 254 | doi= 10.1112/plms/s2-54.3.254-s | doi-access= free }}
• {{cite book | last1= Srivastava | first1= H.M. | last2=Manocha | first2= H.L. | title= A treatise on generating functions | year= 1984 | publisher= E. Horwood | isbn= 0-470-20010-3 }}
• {{cite journal | last1= Miller | first1= A. R. | last2= Moskowitz | first2= I.S. | title= Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters | journal= Computers Math. Applic. | year= 1995 | volume= 30 | issue= 11 | pages= 73–82 | doi=10.1016/0898-1221(95)00165-u| doi-access= free }}
• {{cite journal |last1=Sun |first1=Jingchao |last2=Kong |first2=Maiying |last3=Pal |first3=Subhadip |title=The Modified-Half-Normal distribution: Properties and an efficient sampling scheme |journal=Communications in Statistics – Theory and Methods |date=22 June 2021 |volume=52 |issue=5 |pages=1591–1613 |doi=10.1080/03610926.2021.1934700 |s2cid=237919587 |url=https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20 |issn=0361-0926}}

• [https://gitlab.com/RZ-FZJ/hypergeom hypergeom] on GitLab

{{DEFAULTSORT:Fox-Wright function}}

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