Fungsi Fox–Wright

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 $_{p}F_{q}(z)$ yang berasal dari gagasan Charles Fox (1928) dan E, Maitland Wright (1935):

{}_{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

{}_{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 $_{p}F_{q}(z)$ untuk $A_{1,\dots ,p}=B_{1,\dots ,q}=1$ .

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

{}_{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[sunting | sunting sumber]

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 .