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Fox–Wright function
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with <a href="/facts/Wright_Omega_function/z12Wtabg">Wright Omega function</a>) is a generalisation of the <a href="/facts/Generalized_hypergeometric_function/fqXXDHdR">generalised hypergeometric function</a> <i>p</i><i>F</i><i>q</i>(<i>z</i>) based on ideas of <a href="/facts/Charles_Fox_(mathematician)/otjpuC2T">Charles Fox</a> (1928) and <a href="/facts/E.M._Wright/SS01ywc2">E. Maitland Wright</a> (1935): </p><p> 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[{\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!}}.} </p><p>Upon changing the normalisation </p><p> 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[{\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!}}} </p><p>it becomes <i>p</i><i>F</i><i>q</i>(<i>z</i>) for <i>A</i>1...<i>p</i> = <i>B</i>1...<i>q</i> = 1. </p><p>The Fox–Wright function is a special case of the <a href="/facts/Fox_H-function/QwfvM8IN">Fox H-function</a> (Srivastava & Manocha 1984, p. 50): </p><p> 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[{\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].} </p><p>A special case of Fox–Wright function appears as a part of the <a href="/facts/Normalizing_constant/KKTfybqr">normalizing constant</a> of the <a href="/facts/Modified_half-normal_distribution/JcmNW2Ix">modified half-normal distribution</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>. </p>