: Mathématiques: Kampé de Fériet function

mathematical physics > special function > hypergeometric function > Kampé de Fériet function

mathematical analysis > functional analysis > special function > hypergeometric function > Kampé de Fériet function

mathematical analysis > real analysis > real-valued function > function of several real variables > Kampé de Fériet function

Kampé de Fériet function

In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet.
The Kampé de Fériet function is given by

${\\displaystyle {}^{p+q}F_{r+s}\\left({\egin{matrix}a_{1},\\cdots ,a_{p}\\colon b_{1},b_{1}{}';\\cdots ;b_{q},b_{q}{}';\\\\c_{1},\\cdots ,c_{r}\\colon d_{1},d_{1}{}';\\cdots ;d_{s},d_{s}{}';\\end{matrix}}x,y\ ight)=\\sum _{m=0}^{\\infty }\\sum _{n=0}^{\\infty }{\ rac {(a_{1})_{m+n}\\cdots (a_{p})_{m+n}}{(c_{1})_{m+n}\\cdots (c_{r})_{m+n}}}{\ rac {(b_{1})_{m}(b_{1}{}')_{n}\\cdots (b_{q})_{m}(b_{q}{}')_{n}}{(d_{1})_{m}(d_{1}{}')_{n}\\cdots (d_{s})_{m}(d_{s}{}')_{n}}}\\cdot {\ rac {x^{m}y^{n}}{m!n!}}.}$

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