Kampé de Fériet function

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

The Kampé de Fériet function is given by


{}^{p+q}f_{r+s}\left(
\begin{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\right)=
\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{(a_1)_{m+n}\cdots(a_p)_{m+n}}{(c_1)_{m+n}\cdots(c_r)_{m+n}}\frac{(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\frac{x^my^n}{m!n!}.

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