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):

[math]\displaystyle{ {}_p\Psi_q \left[\begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} ; z \right] = \sum_{n=0}^\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!}. }[/math]

Upon changing the normalisation

[math]\displaystyle{ {}_p\Psi^*_q \left[\begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} ; z \right] = \frac{ \Gamma(b_1) \cdots \Gamma(b_q) }{ \Gamma(a_1) \cdots \Gamma(a_p) } \sum_{n=0}^\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!} }[/math]

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):

[math]\displaystyle{ {}_p\Psi_q \left[\begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} ; z \right] = H^{1,p}_{p,q+1} \left[ -z \left| \begin{matrix} ( 1-a_1 , A_1 ) & ( 1-a_2 , A_2 ) & \ldots & ( 1-a_p , A_p ) \\ (0,1) & (1- b_1 , B_1 ) & ( 1-b_2 , B_2 ) & \ldots & ( 1-b_q , B_q ) \end{matrix} \right. \right]. }[/math]

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 [math]\displaystyle{ (0, \infty) }[/math] is given as [math]\displaystyle{ f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}} }[/math], where [math]\displaystyle{ \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\(1,0)\end{matrix};z \right) }[/math] denotes the Fox–Wright Psi function.

Wright function

The entire function [math]\displaystyle{ W_{\lambda,\mu}(z) }[/math] is often called the Wright function. It is the special case of [math]\displaystyle{ {}_0\Psi_1 \left[\ldots \right] }[/math] of the Fox–Wright function. Its series representation is

[math]\displaystyle{ W_{\lambda,\mu}(z) = \sum_{n=0}^\infty \frac{z^n}{n!\,\Gamma(\lambda n+\mu)}, \lambda \gt -1. }[/math]

This function is used extensively in fractional calculus and the stable count distribution. Recall that [math]\displaystyle{ \lim\limits_{\lambda \to 0} W_{\lambda,\mu}(z) = e^{z} / \Gamma(\mu) }[/math]. Hence, a non-zero [math]\displaystyle{ \lambda }[/math] with zero [math]\displaystyle{ \mu }[/math] 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)

[math]\displaystyle{ \begin{align} \lambda z W_{\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{align} }[/math]

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 [math]\displaystyle{ \lambda = -c\alpha, \mu = 0 }[/math]. Replacing [math]\displaystyle{ z }[/math] with [math]\displaystyle{ -x^\alpha }[/math], we have

[math]\displaystyle{ \begin{array}{lcl} x {d \over dx} W_{-c\alpha,0 }(-x^\alpha) & = & -\frac{1}{c} \left[ W_{-c\alpha,-1}(-x^\alpha) + W_{-c\alpha,0}(-x^\alpha) \right] \end{array} }[/math]

A special case of (a) is [math]\displaystyle{ \lambda = -\alpha, \mu = 1 }[/math]. Replacing [math]\displaystyle{ z }[/math] with [math]\displaystyle{ -z }[/math], we have [math]\displaystyle{ \alpha z W_{-\alpha,1-\alpha}(-z) = W_{-\alpha,0}(-z) }[/math]

Two notations, [math]\displaystyle{ M_{\alpha}(z) }[/math] and [math]\displaystyle{ F_{\alpha}(z) }[/math], were used extensively in the literatures:

[math]\displaystyle{ \begin{align} M_{\alpha}(z) & = W_{-\alpha,1-\alpha}(-z), \\ [1ex] \implies F_{\alpha}(z) & = W_{-\alpha,0}(-z) = \alpha z M_{\alpha}(z). \end{align} }[/math]

M-Wright function

[math]\displaystyle{ M_\alpha(z) }[/math] 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, [math]\displaystyle{ \alpha }[/math] is connected to Lévy's stability index [math]\displaystyle{ (0 \lt \alpha \leq 1) }[/math].

Its asymptotic expansion of [math]\displaystyle{ M_{\alpha}(z) }[/math] for [math]\displaystyle{ \alpha \gt 0 }[/math] is [math]\displaystyle{ M_\alpha \left ( \frac{r}{\alpha} \right ) = A(\alpha) \, r^{(\alpha -1/2)/(1-\alpha)} \, e^{-B(\alpha) \, r^{1/(1-\alpha)}}, \,\, r\rightarrow \infty, }[/math] where [math]\displaystyle{ A(\alpha) = \frac{1}{\sqrt{2\pi (1-\alpha)}}, }[/math] [math]\displaystyle{ B(\alpha) = \frac{1-\alpha}{\alpha}. }[/math]

See also

• Prabhakar function
• Hypergeometric function
• Generalized hypergeometric function
• Modified half-normal distribution with the pdf on [math]\displaystyle{ (0, \infty) }[/math] is given as [math]\displaystyle{ f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}} }[/math], where [math]\displaystyle{ \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\(1,0)\end{matrix};z \right) }[/math] 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