Fox–Wright function

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 :

$${}_p\Psi_q \left[\begin{matrix} ( a_1, A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1, B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} = \sum_{n=0}^\infty \frac{\Gamma( a_1 + A_1 n )\cdots\Gamma( a_p + A_p n )}{\Gamma( b_1 + B_1 n )\cdots\Gamma( b_q + B_q n )} \, \frac {z^n} {n!}. $$
 * z \right]

Upon changing the normalisation

$${}_p\Psi^*_q \left[\begin{matrix} ( a_1, A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1, B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} = \frac{ \Gamma(b_1) \cdots \Gamma(b_q) }{ \Gamma(a_1) \cdots \Gamma(a_p) } \sum_{n=0}^\infty \frac{\Gamma( a_1 + A_1 n )\cdots\Gamma( a_p + A_p n )}{\Gamma( b_1 + B_1 n )\cdots\Gamma( b_q + B_q n )} \, \frac {z^n} {n!} $$
 * z \right]

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

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

$${}_p\Psi_q \left[\begin{matrix} ( a_1, A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1, B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} = H^{1,p}_{p,q+1} \left[ -z \left| \begin{matrix} ( 1-a_1, A_1 ) & ( 1-a_2 , A_2 ) & \ldots & ( 1-a_p , A_p ) \\ (0,1) & (1- b_1, B_1 ) & ( 1-b_2 , B_2 ) & \ldots & ( 1-b_q , B_q ) \end{matrix} \right. \right]. $$
 * z \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 and the stable count distribution. 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). Through the stable count distribution, $$\alpha$$ is connected to Lévy's stability index $$(0 < \alpha \leq 1)$$.

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