Kampé de Fériet function

Short description: Special function in mathematics

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]

Exton, Harold (1978), Handbook of hypergeometric integrals, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-122-0, https://books.google.com/books?id=fUHvAAAAMAAJ
Kampé de Fériet, M. J. (1937) (in French), La fonction hypergéométrique., Mémorial des sciences mathématiques, 85, Paris: Gauthier-Villars, https://books.google.com/books?id=JObuAAAAMAAJ
Ragab, F. J. (1963). "Expansions of Kampe de Feriet's double hypergeometric function of higher order". J. reine angew. Math. 212 (212): 113–119. doi:10.1515/crll.1963.212.113.

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