Fox–Wright function

In mathematics, the Fox–Wright function is a generalisation of the generalised hypergeometric function _pF_q based on ideas of and :
Upon changing the normalisation
it becomes _pF_q for A_1...p = B_1...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. 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 and 18.1 of Erdelyi, Bateman Project, Vol 3
Equation is a recurrence formula. and provide two paths to reduce a derivative. And can be derived from and.
A special case of is. Replacing with, we have
A special case of 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.
Its asymptotic expansion of for is
where