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title: "Fox–Wright function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["factorial-and-binomial-topics", "hypergeometric-functions", "series-expansions"] description: "Generalisation of the generalised hypergeometric function pFq(z)" topic_path: "arts/film" source: "https://en.wikipedia.org/wiki/Fox–Wright_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Generalisation of the generalised hypergeometric function pFq(z) ::

In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of and :

{}_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!}.

Upon changing the normalisation

{}_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!}

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

The Fox–Wright function is a special case of the Fox H-function :

{}_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^{1,p}_{p,q+1} \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].

A special case of Fox–Wright function appears as a part of the normalizing constant of the modified half-normal distribution with the pdf on (0, \infty) is given as 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)}}, where \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\(1,0)\end{matrix};z \right) denotes the Fox–Wright Psi function.

Wright function

The entire function W_{\lambda,\mu}(z) is often called the Wright function. It is the special case of {}_0\Psi_1 \left[\ldots \right] of the Fox–Wright function. Its series representation is

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

This function is used extensively in fractional calculus. Recall that \lim\limits_{\lambda \to 0} W_{\lambda,\mu}(z) = e^{z} / \Gamma(\mu). Hence, a non-zero \lambda with zero \mu is the simplest nontrivial extension of the exponential function in such context.

Three properties were stated in Theorem 1 of Wright (1933) and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955) (p. 212)

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

\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 \lambda = -\alpha, \mu = 1. Replacing z with -z, we have \alpha z W_{-\alpha,1-\alpha}(-z) = W_{-\alpha,0}(-z)

Two notations, M_{\alpha}(z) and F_{\alpha}(z), were used extensively in the literatures:

\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-Wright function

M_\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).

Its asymptotic expansion of M_{\alpha}(z) for \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(\alpha) = \frac{1}{\sqrt{2\pi (1-\alpha)}}, B(\alpha) = \frac{1-\alpha}{\alpha}.

References

References

1. (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics – Theory and Methods.
2. Weisstein, Eric W.. "Wright Function".
3. Wright, E.. (1933). "On the Coefficients of Power Series Having Exponential Singularities". Journal of the London Mathematical Society.
4. Erdelyi, A. (1955). "The Bateman Project, Volume 3". California Institute of Technology.
5. (2010-02-11). "The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey". International Journal of Differential Equations.

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