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
λ , μ
( λ n + μ )
This function is used extensively in fractional calculus and the stable count distribution.
λ , μ
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)
λ , μ − 1
{\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
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 )