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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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