Fox–Wright function

Short description: Generalisation of the generalised hypergeometric function pFq(z)

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):

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

Upon changing the normalisation

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

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):

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

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 $f(x)=\frac{2{\beta }^{\frac{\alpha }{2}}{x}^{\alpha -1}\exp (-\beta x^2+\gamma x)}{\mathrm{\Psi }(\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})}$, where $\mathrm{\Psi }(\alpha ,z)={}_1{\mathrm{\Psi }}_1(\begin{array}{c}(\alpha ,\frac{1}{2})\\ (1,0)\end{array};z)$ 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 ${}_0{\mathrm{\Psi }}_1[\dots ]$ of the Fox–Wright function. Its series representation is

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

This function is used extensively in fractional calculus and the stable count distribution. Recall that $\lim {}_{\lambda \to 0}{W}_{\lambda ,\mu }(z)=e^z/\mathrm{\Gamma }(\mu )$. 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)

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

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

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

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:

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

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). Through the stable count distribution, $\alpha$ is connected to Lévy's stability index $(0<\alpha \le 1)$.

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

References

• Fox, C. (1928). "The asymptotic expansion of integral functions defined by generalized hypergeometric series". Proc. London Math. Soc. 27 (1): 389–400. doi:10.1112/plms/s2-27.1.389. 
• Wright, E. M. (1935). "The asymptotic expansion of the generalized hypergeometric function". J. London Math. Soc. 10 (4): 286–293. doi:10.1112/jlms/s1-10.40.286. 
• Wright, E. M. (1940). "The asymptotic expansion of the generalized hypergeometric function". Proc. London Math. Soc. 46 (2): 389–408. doi:10.1112/plms/s2-46.1.389. 
• Wright, E. M. (1952). "Erratum to "The asymptotic expansion of the generalized hypergeometric function"". J. London Math. Soc. 27: 254. doi:10.1112/plms/s2-54.3.254-s. 
• Srivastava, H.M.; Manocha, H.L. (1984). A treatise on generating functions. E. Horwood. ISBN 0-470-20010-3. 
• Miller, A. R.; Moskowitz, I.S. (1995). "Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters". Computers Math. Applic. 30 (11): 73–82. doi:10.1016/0898-1221(95)00165-u. 
• Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics – Theory and Methods 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20. 

External links

• hypergeom on GitLab