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

p Ψ Ψ q [ ( a 1 , A 1 ) ( a 2 , A 2 ) … … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … … ( b q , B q ) ; z ] = ∑ ∑ n = 0 ∞ ∞ Γ Γ ( a 1 + A 1 n ) ⋯ ⋯ Γ Γ ( a p + A p n ) Γ Γ ( b 1 + B 1 n ) ⋯ ⋯ Γ Γ ( b q + B q n ) z n n ! . {\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

p Ψ Ψ q ∗ ∗ [ ( a 1 , A 1 ) ( a 2 , A 2 ) … … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … … ( b q , B q ) ; z ] = Γ Γ ( b 1 ) ⋯ ⋯ Γ Γ ( b q ) Γ Γ ( a 1 ) ⋯ ⋯ Γ Γ ( a p ) ∑ ∑ n = 0 ∞ ∞ Γ Γ ( a 1 + A 1 n ) ⋯ ⋯ Γ Γ ( a p + A p n ) Γ Γ ( b 1 + B 1 n ) ⋯ ⋯ Γ Γ ( b q + B q n ) z n n ! {\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 pFq(z) for A1...p = B1...q = 1.

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

p Ψ Ψ q [ ( a 1 , A 1 ) ( a 2 , A 2 ) … … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … … ( b q , B q ) ; z ] = H p , q + 1 1 , p [ − − z | ( 1 − − a 1 , A 1 ) ( 1 − − a 2 , A 2 ) … … ( 1 − − a p , A p ) ( 0 , 1 ) ( 1 − − b 1 , B 1 ) ( 1 − − b 2 , B 2 ) … … ( 1 − − b q , B q ) ] . {\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 distributioncite-ref-sun-kong-and-pal-1-0[1] with the pdf on ( 0 , ∞ ∞ ) {\displaystyle (0,\infty )} is given as f ( x ) = 2 β β α α 2 x α α − − 1 exp ⁡ ⁡ ( − − β β x 2 + γ γ x ) Ψ Ψ ( α α 2 , γ γ β β ) {\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)}}}} , where Ψ Ψ ( α α , z ) = 1 Ψ Ψ 1 ( ( α α , 1 2 ) ( 1 , 0 ) ; z ) {\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)} denotes the Fox–Wright Psi function.

• Wright function

• M-Wright function

• See also

• References

• External links

The entire function W λ λ , μ μ ( z ) {\displaystyle W_{\lambda ,\mu }(z)} is often called the Wright function.cite-ref-2[2] It is the special case of 0 Ψ Ψ 1 [ … … ] {\displaystyle {}_{0}\Psi _{1}\left[\ldots \right]} of the Fox–Wright function. Its series representation is

W λ λ , μ μ ( z ) = ∑ ∑ n = 0 ∞ ∞ z n n ! Γ Γ ( λ λ n + μ μ ) , λ λ > − − 1. {\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 lim λ λ → → 0 W λ λ , μ μ ( z ) = e z / Γ Γ ( μ μ ) {\displaystyle \lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )} . Hence, a non-zero λ λ {\displaystyle \lambda } with zero μ μ {\displaystyle \mu } is the simplest nontrivial extension of the exponential function in such context.

Three properties were stated in Theorem 1 of Wright (1933)cite-ref-3[3] and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955)cite-ref-4[4] (p. 212)

λ λ z W λ λ , μ μ + λ λ ( z ) = W λ λ , μ μ − − 1 ( z ) + ( 1 − − μ μ ) W λ λ , μ μ ( z ) ( a ) d d z W λ λ , μ μ ( z ) = W λ λ , μ μ + λ λ ( z ) ( b ) λ λ z d d z W λ λ , μ μ ( z ) = W λ λ , μ μ − − 1 ( z ) + ( 1 − − μ μ ) W λ λ , μ μ ( z ) ( c ) {\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 λ λ = − − c α α , μ μ = 0 {\displaystyle \lambda =-c\alpha ,\mu =0} . Replacing z {\displaystyle z} with − − x α α {\displaystyle -x^{\alpha }} , we have

x d d x W − − c α α , 0 ( − − x α α ) = − − 1 c [ W − − c α α , − − 1 ( − − x α α ) + W − − c α α , 0 ( − − x α α ) ] {\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 λ λ = − − α α , μ μ = 1 {\displaystyle \lambda =-\alpha ,\mu =1} . Replacing z {\displaystyle z} with − − z {\displaystyle -z} , we have α α z W − − α α , 1 − − α α ( − − z ) = W − − α α , 0 ( − − z ) {\displaystyle \alpha zW_{-\alpha ,1-\alpha }(-z)=W_{-\alpha ,0}(-z)}

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

M α α ( z ) = W − − α α , 1 − − α α ( − − z ) , ⟹ ⟹ F α α ( z ) = W − − α α , 0 ( − − z ) = α α z M α α ( z ) . {\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}}}

M α α ( z ) {\displaystyle 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).cite-ref-5[5] Through the stable count distribution, α α {\displaystyle \alpha } is connected to Lévy's stability index ( 0 < α α ≤ ≤ 1 ) {\displaystyle (0<\alpha \leq 1)} .

Its asymptotic expansion of M α α ( z ) {\displaystyle M_{\alpha }(z)} for α α > 0 {\displaystyle \alpha >0} is M α α ( r α α ) = A ( α α ) r ( α α − − 1 / 2 ) / ( 1 − − α α ) e − − B ( α α ) r 1 / ( 1 − − α α ) , r → → ∞ ∞ , {\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 ,} where A ( α α ) = 1 2 π π ( 1 − − α α ) , {\displaystyle A(\alpha )={\frac {1}{\sqrt {2\pi (1-\alpha )}}},} B ( α α ) = 1 − − α α α α . {\displaystyle B(\alpha )={\frac {1-\alpha }{\alpha }}.}

• Prabhakar function

• Hypergeometric function

• Generalized hypergeometric function

• Modified half-normal distributioncite-ref-sun-kong-and-pal-1-1[1] with the pdf on ( 0 , ∞ ∞ ) {\displaystyle (0,\infty )} is given as f ( x ) = 2 β β α α 2 x α α − − 1 exp ⁡ ⁡ ( − − β β x 2 + γ γ x ) Ψ Ψ ( α α 2 , γ γ β β ) {\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)}}}} , where Ψ Ψ ( α α , z ) = 1 Ψ Ψ 1 ( ( α α , 1 2 ) ( 1 , 0 ) ; z ) {\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)} denotes the Fox–Wright Psi function.

cite-note-sun-kong-and-pal-11. ↑ citerefsunkongpal2021Sun, 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.

cite-note-22. ↑ citerefweissteinWeisstein, Eric W. "Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03.

cite-note-33. ↑ citerefwright1933Wright, 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.

cite-note-44. ↑ citereferdelyi1955Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.

cite-note-55. ↑ citerefmainardimurapagnini2010Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey. arXiv:1004.2950.

• citereffox1928Fox, 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.

• citerefwright1935Wright, 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.

• citerefwright1940Wright, 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.

• citerefwright1952Wright, 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.

• citerefsrivastavamanocha1984Srivastava, H.M.; Manocha, H.L. (1984). A treatise on generating functions. E. Horwood. ISBN 0-470-20010-3.

• citerefmillermoskowitz1995Miller, 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.

• citerefsunkongpal2021Sun, 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.

• hypergeom on GitLab