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 Charles Fox (1928) and E. Maitland Wright (1935):
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 (Srivastava & Manocha 1984, p. 50):
A special case of Fox–Wright function appears as a part of the normalizing constant of the modified half-normal distribution[1] with the pdf on is given as , where denotes the Fox–Wright Psi function.
The entire function is often called the Wright function.[2] 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)[3] and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955)[4] (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).[5]
Through the stable count distribution, is connected to Lévy's stability index .
Its asymptotic expansion of for is
where