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 :
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 (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).
Its asymptotic expansion of for is
where
See also
Prabhakar function Hypergeometric function Generalized hypergeometric function
References
External links hypergeom on GitLab