Fox–Wright function

<h2 id="wright-function">Wright function</h2>
The entire function 
 
 
 
 
 W
 
 λ
 ,
 μ
 
 
 (
 z
 )
 
 
 {\displaystyle W_{\lambda ,\mu }(z)}
 
 is often called the Wright function.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> It is the special case of

0
 
 
 
 Ψ
 
 1
 
 
 
 [
 …
 ]
 
 
 
 {\displaystyle {}_{0}\Psi _{1}\left[\ldots \right]}
 
 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 <a href="/facts/Fractional_calculus/PxN1pMcj">fractional calculus</a> and the <a href="/facts/Stable_count_distribution/SDhu7Yek">stable count distribution</a>. 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 <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> in such context.
Three properties were stated in Theorem 1 of Wright (1933)<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955)<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> (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)\\[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
 =
 −
 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[W_{-c\alpha ,-1}(-x^{\alpha })+W_{-c\alpha ,0}(-x^{\alpha })\right]\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),\\[1ex]\implies F_{\alpha }(z)&=W_{-\alpha ,0}(-z)=\alpha zM_{\alpha }(z).\end{aligned}}}

<h3>M-Wright function</h3>

 
 
 
 
 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).<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Through the <a href="/facts/Stable_count_distribution/SDhu7Yek">stable count distribution</a>, 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is connected to Lévy's stability index 
 
 
 
 (
 0
 <
 α
 ≤
 1
 )
 
 
 {\displaystyle (0<\alpha \leq 1)}
 
.
Its <a href="/facts/Asymptotic_expansion/4W79BNve">asymptotic expansion</a> 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 }}.}

<h2 id="see-also">See also</h2>
<ul>
<li>Mathematics portal</li></ul>
<ul><li><a href="/facts/Prabhakar_function/mBBnzoIE">Prabhakar function</a></li>
<li><a href="/facts/Hypergeometric_function/yTEC6NBJ">Hypergeometric function</a></li>
<li><a href="/facts/Generalized_hypergeometric_function/fqXXDHdR">Generalized hypergeometric function</a></li>
<li><a href="/facts/Modified_half-normal_distribution/JcmNW2Ix">Modified half-normal distribution</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 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 <a href="/facts/Fox%E2%80%93Wright_Psi_function/U197C9hr">Fox–Wright Psi function</a>.</li></ul>

<ul><li>Fox, C. (1928). "The asymptotic expansion of integral functions defined by generalized hypergeometric series". Proc. London Math. Soc. 27 (1): 389–400. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-27.1.389">10.1112/plms/s2-27.1.389</a>.</li>
<li>Wright, E. M. (1935). "The asymptotic expansion of the generalized hypergeometric function". J. London Math. Soc. 10 (4): 286–293. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fjlms%2Fs1-10.40.286">10.1112/jlms/s1-10.40.286</a>.</li>
<li>Wright, E. M. (1940). "The asymptotic expansion of the generalized hypergeometric function". Proc. London Math. Soc. 46 (2): 389–408. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-46.1.389">10.1112/plms/s2-46.1.389</a>.</li>
<li>Wright, E. M. (1952). <a href="https://doi.org/10.1112%2Fplms%2Fs2-54.3.254-s">"Erratum to "The asymptotic expansion of the generalized hypergeometric function""</a>. J. London Math. Soc. 27: 254. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-54.3.254-s">10.1112/plms/s2-54.3.254-s</a>.</li>
<li>Srivastava, H.M.; Manocha, H.L. (1984). A treatise on generating functions. E. Horwood. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-470-20010-3.</li>
<li>Miller, A. R.; Moskowitz, I.S. (1995). <a href="https://doi.org/10.1016%2F0898-1221%2895%2900165-u">"Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters"</a>. Computers Math. Applic. 30 (11): 73–82. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0898-1221%2895%2900165-u">10.1016/0898-1221(95)00165-u</a>.</li>
<li>Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). <a href="https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20">"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"</a>. Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F03610926.2021.1934700">10.1080/03610926.2021.1934700</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0361-0926">0361-0926</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:237919587">237919587</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://gitlab.com/RZ-FZJ/hypergeom">hypergeom</a> on <a href="/facts/GitLab/3CCW9luS">GitLab</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20" target="_blank">https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Weisstein, Eric W. "Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03. <a href="https://mathworld.wolfram.com/WrightFunction.html" target="_blank">https://mathworld.wolfram.com/WrightFunction.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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. <a href="https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20" target="_blank">https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Fox–Wright function open-in-new

Fox–Wright function