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
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
exp ( − β
denotes the Fox–Wright Psi function.
The entire function
λ , μ
is often called the Wright function.
[2] It is the special case of
Its series representation is
λ , μ
( λ n + μ )
This function is used extensively in fractional calculus and the stable count distribution.
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)
{\displaystyle {\begin{aligned}\lambda zW_{\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{aligned}}}
Equation (a) is a recurrence formula.
(b) and (c) provide two paths to reduce a derivative.
A special case of (c) is
λ = − c α , μ = 0
A special case of (a) is
λ = − α , μ = 1
, were used extensively in the literatures:
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
2 π ( 1 − α )