Fox–Wright function explained

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 and :

Upon changing the normalisation

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 :

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

( 0 , inf ty) is given as f(x) = (2 \beta ^{(\alpha)/(2)} x ^{\alpha-1} \exp( - \beta x ^{2+} \gamma x ))/(\Psi{\left( (\alpha)/(, (\gamma)/(\sqrt{\beta) 2)) \right)}} , where \Psi(\alpha,z) = {} _{1\Psi} ^{1 \left(\begin{matrix}\left(\alpha, (1)/(2)} \right)\\( 1 , 0 )\end{matrix};z \right) denotes the Fox–Wright Psi function.

Wright function

The entire function

W _{λ,\mu} (z) is often called the Wright function.^[2] It is the special case of {} _{0\Psi} _{1} \left[\ldots \right] of the Fox–Wright function. Its series representation is

$$W\_(z)\; =\; \backslash sum\_^\backslash infty\backslash frac,\; \backslash lambda\; >\; -1.$$

This function is used extensively in fractional calculus and the stable count distribution. Recall that

\lim\limits _{λ} W _{λ,\mu} (z) = e ^{z} / \Gamma(\mu) . Hence, a non-zero λ 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)^[3] and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955)^[4] (p. 212)

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

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

A special case of (a) is

λ = - \alpha, \mu = 1 . Replacing z with - z , we have$$\backslash alpha\; z\; W\_(-z)\; =\; W\_(-z)$$

Two notations,

M _{\alpha} (z) and F _{\alpha} (z) , were used extensively in the literatures:

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).^[5] Through the stable count distribution,

\alpha is connected to Lévy's stability index ( 0 < \alpha \leq 1 ) .

Its asymptotic expansion of

M _{\alpha} (z) for \alpha > 0 is$$M\_\backslash alpha\; \backslash left\; (\backslash frac\; \backslash right)\; =A(\backslash alpha)\; \backslash ,\; r^\backslash ,\; e^,\; \backslash ,\backslash ,\; r\backslash rightarrow\; \backslash infty,$$whereA(\alpha) = (1)/(\sqrt{ 2 \pi ( 1- \alpha)) }, B(\alpha) = (1- \alpha)/(\alpha) .

See also

• Prabhakar function
• Hypergeometric function
• Generalized hypergeometric function
• Modified half-normal distribution with the pdf on

( 0 , inf ty) is given as f(x) = (2 \beta ^{(\alpha)/(2)} x ^{\alpha-1} \exp( - \beta x ^{2+} \gamma x ))/(\Psi{\left( (\alpha)/(, (\gamma)/(\sqrt{\beta) 2)) \right)}} , where \Psi(\alpha,z) = {} _{1\Psi} ^{1 \left(\begin{matrix}\left(\alpha, (1)/(2)} \right)\\( 1 , 0 )\end{matrix};z \right) denotes the Fox–Wright Psi function.

References

• Fox . C. . The asymptotic expansion of integral functions defined by generalized hypergeometric series . Proc. London Math. Soc. . 1928 . 27 . 1 . 389–400 . 10.1112/plms/s2-27.1.389 .
• Wright . E. M. . The asymptotic expansion of the generalized hypergeometric function . J. London Math. Soc. . 1935 . 10 . 4 . 286–293 . 10.1112/jlms/s1-10.40.286 .
• Wright . E. M. . The asymptotic expansion of the generalized hypergeometric function . Proc. London Math. Soc. . 1940 . 46 . 2 . 389–408 . 10.1112/plms/s2-46.1.389.
• Wright . E. M. . Erratum to "The asymptotic expansion of the generalized hypergeometric function" . J. London Math. Soc. . 1952 . 27 . 254 . 10.1112/plms/s2-54.3.254-s . free .
• Book: Srivastava . H.M. . Manocha . H.L. . A treatise on generating functions . 1984 . E. Horwood . 0-470-20010-3 .
• Miller . A. R. . Moskowitz . I.S. . Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters . Computers Math. Applic. . 1995 . 30 . 11 . 73–82 . 10.1016/0898-1221(95)00165-u. free .
• Sun . Jingchao . Kong . Maiying . Pal . Subhadip . The Modified-Half-Normal distribution: Properties and an efficient sampling scheme . Communications in Statistics – Theory and Methods . 22 June 2021 . 52 . 5 . 1591–1613 . 10.1080/03610926.2021.1934700 . 237919587 . 0361-0926.

External links

• hypergeom on GitLab

Notes and References

1. Sun . Jingchao . Kong . Maiying . Pal . Subhadip . The Modified-Half-Normal distribution: Properties and an efficient sampling scheme . Communications in Statistics – Theory and Methods . 22 June 2021 . 52 . 5 . 1591–1613 . 10.1080/03610926.2021.1934700 . 237919587 . 0361-0926.
2. Web site: Wright Function . Weisstein . Eric W. . From MathWorld--A Wolfram Web Resource . 2022-12-03.
3. Wright . E. . 1933 . On the Coefficients of Power Series Having Exponential Singularities . Journal of the London Mathematical Society . Second Series . 71–79 . 10.1112/JLMS/S1-8.1.71 . 122652898 . en.
4. Book: Erdelyi, A . The Bateman Project, Volume 3 . California Institute of Technology . 1955.
5. Book: Mainardi . Francesco . Mura . Antonio . Pagnini . Gianni . 2010-04-17 . The M-Wright function in time-fractional diffusion processes: a tutorial survey . 1004.2950.