Kampé de Fériet function

Template:Short description 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

^{p + q} F_{r + s} (\begin{matrix} a_{1}, \dots, a_{p} : b_{1}, b_{1}^{'}; \dots; b_{q}, b_{q}^{'}; \\ c_{1}, \dots, c_{r} : d_{1}, d_{1}^{'}; \dots; d_{s}, d_{s}^{'}; \end{matrix} x, y) = \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \frac{(a_{1})_{m + n} \dots (a_{p})_{m + n}}{(c_{1})_{m + n} \dots (c_{r})_{m + n}} \frac{(b_{1})_{m} (b_{1}^{'})_{n} \dots (b_{q})_{m} (b_{q}^{'})_{n}}{(d_{1})_{m} (d_{1}^{'})_{n} \dots (d_{s})_{m} (d_{s}^{'})_{n}} \cdot \frac{x^{m} y^{n}}{m! n!} .

Applications

The general sextic equation can be solved in terms of Kampé de Fériet functions.^[1]