Fox–Wright function

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 _pF_q(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935):

$${}_p{\Psi }_q[\begin{array}{cccc}(a_1,A_1)& (a_2,A_2)& \dots & (a_p,A_p)\\ (b_1,B_1)& (b_2,B_2)& \dots & (b_q,B_q)\end{array};z]=\sum _{n=0}^{\mathrm{\infty }}\frac{\Gamma (a_1+A_{1}n)\cdots \Gamma (a_p+A_{p}n)}{\Gamma (b_1+B_{1}n)\cdots \Gamma (b_q+B_{q}n)}\frac{z^n}{n!}.$$

Upon changing the normalisation

$${}_p{\Psi }_q^{*}[\begin{array}{cccc}(a_1,A_1)& (a_2,A_2)& \dots & (a_p,A_p)\\ (b_1,B_1)& (b_2,B_2)& \dots & (b_q,B_q)\end{array};z]=\frac{\Gamma (b_1)\cdots \Gamma (b_q)}{\Gamma (a_1)\cdots \Gamma (a_p)}\sum _{n=0}^{\mathrm{\infty }}\frac{\Gamma (a_1+A_{1}n)\cdots \Gamma (a_p+A_{p}n)}{\Gamma (b_1+B_{1}n)\cdots \Gamma (b_q+B_{q}n)}\frac{z^n}{n!}$$

it becomes _pF_q(z) for A_1...p = B_1...q = 1.

The Fox–Wright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50):

$${}_p{\Psi }_q[\begin{array}{cccc}(a_1,A_1)& (a_2,A_2)& \dots & (a_p,A_p)\\ (b_1,B_1)& (b_2,B_2)& \dots & (b_q,B_q)\end{array};z]=H_{p,q+1}^{1,p}[-z|\begin{array}{cccc}(1-a_1,A_1)& (1-a_2,A_2)& \dots & (1-a_p,A_p)\\ (0,1)& (1-b_1,B_1)& (1-b_2,B_2)& \dots & (1-b_q,B_q)\end{array}].$$

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 $(0,\mathrm{\infty })$ is given as $f(x)=\frac{2{\beta }^{\frac{\alpha }{2}}{x}^{\alpha -1}\exp (-\beta x^2+\gamma x)}{\Psi (\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})}$, where $\Psi (\alpha ,z)={}_1{\Psi }_1(\begin{array}{c}(\alpha ,\frac{1}{2})\\ (1,0)\end{array};z)$ denotes the Fox–Wright Psi function.

Wright function

The entire function $W_{\lambda ,\mu }(z)$ is often called the Wright function. It is the special case of ${}_0{\Psi }_1[\dots ]$ of the Fox–Wright function. Its series representation is

$$W_{\lambda ,\mu }(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!\Gamma (\lambda n+\mu )},\lambda >-1.$$

This function is used extensively in fractional calculus. Recall that $\lim {\mathrm{\backslash limits}}_{\lambda \to 0}{W}_{\lambda ,\mu }(z)=e^z/\Gamma (\mu )$. Hence, a non-zero $\lambda$ with zero $\mu$ 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)

$$\begin{array}{ccc}\lambda z{W}_{\lambda ,\mu +\lambda }(z)& =W_{\lambda ,\mu -1}(z)+(1-\mu )W_{\lambda ,\mu }(z)& (a)\\ \frac{d}{dz}{W}_{\lambda ,\mu }(z)& =W_{\lambda ,\mu +\lambda }(z)& (b)\\ \lambda z\frac{d}{dz}{W}_{\lambda ,\mu }(z)& =W_{\lambda ,\mu -1}(z)+(1-\mu )W_{\lambda ,\mu }(z)& (c)\end{array}$$

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 $\lambda =-c\alpha ,\mu =0$. Replacing $z$ with $-x^{\alpha }$, we have

$$\begin{array}{ccc}x\frac{d}{dx}{W}_{-c\alpha ,0}(-x^{\alpha })& =& -\frac{1}{c}[W_{-c\alpha ,-1}(-x^{\alpha })+W_{-c\alpha ,0}(-x^{\alpha })]\end{array}$$

A special case of (a) is $\lambda =-\alpha ,\mu =1$. Replacing $z$ with $-z$, we have $$\alpha z{W}_{-\alpha ,1-\alpha }(-z)=W_{-\alpha ,0}(-z)$$

Two notations, $M_{\alpha }(z)$ and $F_{\alpha }(z)$, were used extensively in the literatures:

$$\begin{array}{cc}{M}_{\alpha }(z)& =W_{-\alpha ,1-\alpha }(-z),\\ [1ex]⟹F_{\alpha }(z)& =W_{-\alpha ,0}(-z)=\alpha z{M}_{\alpha }(z).\end{array}$$

M-Wright function

$M_{\alpha }(z)$ 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 $M_{\alpha }(z)$ for $\alpha >0$ is $$M_{\alpha }\left(\frac{r}{\alpha }\right)=A(\alpha )r^{(\alpha -1/2)/(1-\alpha )}{e}^{-B(\alpha )r^{1/(1-\alpha )}},r\to \mathrm{\infty },$$ where $A(\alpha )=\frac{1}{\sqrt{2\pi (1-\alpha )}},$ $B(\alpha )=\frac{1-\alpha }{\alpha }.$

See also

• Prabhakar function
• Hypergeometric function
• Generalized hypergeometric function
• Modified half-normal distribution with the pdf on $(0,\mathrm{\infty })$ is given as $f(x)=\frac{2{\beta }^{\frac{\alpha }{2}}{x}^{\alpha -1}\exp (-\beta x^2+\gamma x)}{\Psi (\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})}$, where $\Psi (\alpha ,z)={}_1{\Psi }_1(\begin{array}{c}(\alpha ,\frac{1}{2})\\ (1,0)\end{array};z)$ denotes the Fox–Wright Psi function.

External links

• hypergeom on GitLab