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 :

Upon changing the normalisation

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

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

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  is given as , where  denotes the Fox–Wright Psi function.

Wright function 

The entire function  is often called the Wright function. It is the special case of  of the Fox–Wright function. Its series representation is

This function is used extensively in fractional calculus and the stable count distribution. Recall that . Hence, a non-zero  with zero  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)

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 . Replacing  with , we have

A special case of (a) is . Replacing  with , we have

Two notations,  and , were used extensively in the literatures:

M-Wright function 

 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,  is connected to Lévy's stability index .

Its asymptotic expansion of  for  is

where

See also 

 Prabhakar function
 Hypergeometric function
 Generalized hypergeometric function
 Modified half-normal distribution with the pdf on  is given as , where  denotes the Fox–Wright Psi function.

References

External links
 hypergeom on GitLab

Factorial and binomial topics
Hypergeometric functions
Series expansions