Dalam matematika, **fungsi Fox–Wright** (dikenal sebagai **fungsi Psi Fox–Wright** atau hanya **fungsi Wright**, jangan bingung dengan fungsi Omega Wright) merupakan perumuman dari fungsi hipergeometrik diperumum {\displaystyle \_{p}F\_{q}(z)} yang berasal dari gagasan Charles Fox (1928) dan E, Maitland Wright (1935):



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{\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!}}}

{\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!}}}
Setelah mengubah penormalan




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{\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!}},}

{\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!}},}
fungsi tersebut menjadi {\displaystyle \_{p}F\_{q}(z)} untuk {\displaystyle A\_{1,\dots ,p}=B\_{1,\dots ,q}=1}.

Fungsi Fox–Wright merupakan kasus istimewa dari fungsi-H Fox (Srivastava & Manocha 1984, hlm. 50):




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{\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]}

{\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]}

Referensi
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* Fox. C (1928). "The asymptotic expansion of integral functions defined by generalized hypergeometric series". *Proc. London Math Soc*. **27** (1): 369–400. doi:10.1112/plms/s2-27.1.389.
* Wright, 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.
* Wright, 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.
* Wright, 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.
* Srivastava, H.M.; Manocha, H.L. (1984). *A treatise on generating functions*. ISBN 0-470-20010-3.
* Miller, 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 alt=Dapat diakses gratis.

Pranala luar
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