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Fox–Wright function

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(Redirected from Fox–Wright Psi function) 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 pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935):

p 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 ( a 1 + A 1 n ) ⋯ Γ ( a p + A p n ) Γ ( b 1 + B 1 n ) ⋯ Γ ( b q + B q n ) z n n ! . {\displaystyle {}_{p}\Psi _{q}\left=\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 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 ( b 1 ) ⋯ Γ ( b q ) Γ ( a 1 ) ⋯ Γ ( a p ) ∑ n = 0 ∞ Γ ( a 1 + A 1 n ) ⋯ Γ ( a p + A p n ) Γ ( b 1 + B 1 n ) ⋯ Γ ( b q + B q n ) z n n ! {\displaystyle {}_{p}\Psi _{q}^{*}\left={\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 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=H_{p,q+1}^{1,p}\left.}

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 , ∞ ) {\displaystyle (0,\infty )} is given as f ( x ) = 2 β α 2 x α − 1 exp ⁡ ( − β x 2 + γ x ) Ψ ( α 2 , γ 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 ( α , z ) = 1 Ψ 1 ( ( 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

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

W λ , μ ( z ) = ∑ n = 0 ∞ z n n ! Γ ( λ n + μ ) , 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 → 0 W λ , μ ( z ) = e z / Γ ( μ ) {\displaystyle \lim \limits _{\lambda \to 0}W_{\lambda ,\mu }(z)=e^{z}/\Gamma (\mu )} . Hence, a non-zero λ {\displaystyle \lambda } with zero 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) and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955) (p. 212)

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

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

A special case of (a) is λ = − α , μ = 1 {\displaystyle \lambda =-\alpha ,\mu =1} . Replacing z {\displaystyle z} with − z {\displaystyle -z} , we have α z W − α , 1 − α ( − z ) = W − α , 0 ( − z ) {\displaystyle \alpha zW_{-\alpha ,1-\alpha }(-z)=W_{-\alpha ,0}(-z)}

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

M α ( z ) = W − α , 1 − α ( − z ) , ⟹ F α ( z ) = W − α , 0 ( − z ) = α z M α ( z ) . {\displaystyle {\begin{aligned}M_{\alpha }(z)&=W_{-\alpha ,1-\alpha }(-z),\\\implies F_{\alpha }(z)&=W_{-\alpha ,0}(-z)=\alpha zM_{\alpha }(z).\end{aligned}}}

M-Wright function

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

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

See also

  • Prabhakar function
  • Hypergeometric function
  • Generalized hypergeometric function
  • Modified half-normal distribution with the pdf on ( 0 , ∞ ) {\displaystyle (0,\infty )} is given as f ( x ) = 2 β α 2 x α − 1 exp ⁡ ( − β x 2 + γ x ) Ψ ( α 2 , γ 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 ( α , z ) = 1 Ψ 1 ( ( 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.

References

  1. ^ 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.
  2. Weisstein, Eric W. "Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 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. doi:10.1112/JLMS/S1-8.1.71. S2CID 122652898.
  4. Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.
  5. Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey. arXiv:1004.2950.

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