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 p F q z ) based on ideas of Charles Fox (1928 ) and E. Maitland Wright (1935 ):
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{\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!}}.}
Upon changing the normalisation
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{\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!}}}
it becomes p F 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):
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{\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_{p,q+1}^{1,p}\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].}
A special case of Fox–Wright function appears as a part of the normalizing constant of the modified half-normal distribution [ 1]
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{\displaystyle (0,\infty )}
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{\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)}}}}
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{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}
Fox–Wright Psi function .
The entire function
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{\displaystyle W_{\lambda ,\mu }(z)}
the Wright function .[ 2]
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{\displaystyle {}_{0}\Psi _{1}\left[\ldots \right]}
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{\displaystyle W_{\lambda ,\mu }(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!\,\Gamma (\lambda n+\mu )}},\lambda >-1.}
This function is used extensively in fractional calculus and the stable count distribution . Recall that
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{\displaystyle \lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )}
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Three properties were stated in Theorem 1 of Wright (1933)[ 3] [ 4]
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{\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. And (c) can be derived from (a) and (b).
A special case of (c) is
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{\displaystyle \lambda =-c\alpha ,\mu =0}
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{\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}}}
A special case of (a) is
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{\displaystyle \lambda =-\alpha ,\mu =1}
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{\displaystyle \alpha zW_{-\alpha ,1-\alpha }(-z)=W_{-\alpha ,0}(-z)}
Two notations,
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{\displaystyle M_{\alpha }(z)}
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{\displaystyle F_{\alpha }(z)}
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{\displaystyle {\begin{aligned}M_{\alpha }(z)&=W_{-\alpha ,1-\alpha }(-z),\\[1ex]\implies F_{\alpha }(z)&=W_{-\alpha ,0}(-z)=\alpha zM_{\alpha }(z).\end{aligned}}}
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{\displaystyle M_{\alpha }(z)}
Its properties were surveyed in Mainardi et al (2010).[ 5] stable count distribution ,
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{\displaystyle (0<\alpha \leq 1)}
Its asymptotic expansion of
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{\displaystyle M_{\alpha }(z)}
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{\displaystyle \alpha >0}
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{\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 ,}
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{\displaystyle A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},}
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{\displaystyle B(\alpha )={\frac {1-\alpha }{\alpha }}.}
Prabhakar function Hypergeometric function Generalized hypergeometric function Modified half-normal distribution [ 1]
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{\displaystyle (0,\infty )}
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{\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)}}}}
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{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}
Fox–Wright Psi function .
^ a b 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 . S2CID 237919587 . ^ Weisstein, Eric W. "Wright Function" . From MathWorld--A Wolfram Web Resource . Retrieved 2022-12-03 . ^ Wright, E. (1933). "On the Coefficients of Power Series Having Exponential Singularities". Journal of the London Mathematical Society . Second Series: 71–79. doi :10.1112/JLMS/S1-8.1.71 . S2CID 122652898 . ^ Erdelyi, A (1955). The Bateman Project, Volume 3 . California Institute of Technology. ^ Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey . arXiv :1004.2950
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 . S2CID 237919587 . 
![{\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!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab45ae4b1283978ae4ccb07a7a46a36957d9fb4)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c530d8bd1ce434141c9d2b83eb3c98d9b22fa0)
![{\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_{p,q+1}^{1,p}\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe83e472f89f4a998dc750566a980c28def03b3)
![{\displaystyle {}_{0}\Psi _{1}\left[\ldots \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4d213f41c4cbec3b33408d2c2e338039817259)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a07b7f942cb218ae53fc4b1889e67096fc264977)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe07892a2fb601ec2f29988305f4a2beda9a29a)
![{\displaystyle {\begin{aligned}M_{\alpha }(z)&=W_{-\alpha ,1-\alpha }(-z),\\[1ex]\implies F_{\alpha }(z)&=W_{-\alpha ,0}(-z)=\alpha zM_{\alpha }(z).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97ecbfca463abbfd3d327dfa481c9acc621f7f7d)
