Kampé de Fériet function

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 ( a 1 , ⋯ , a p : b 1 , b 1 ′ ; ⋯ ; b q , b q ′ ; c 1 , ⋯ , c r : d 1 , d 1 ′ ; ⋯ ; d s , d s ′ ; x , y ) = ∑ m = 0 ∞ ∑ n = 0 ∞ ( a 1 ) m + n ⋯ ( a p ) m + n ( c 1 ) m + n ⋯ ( c r ) m + n ( b 1 ) m ( b 1 ′ ) n ⋯ ( b q ) m ( b q ′ ) n ( d 1 ) m ( d 1 ′ ) n ⋯ ( d s ) m ( d s ′ ) n ⋅ x m y n m ! n ! . {\displaystyle {}^{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^{m}y^{n}}{m!n!}}.}

Applications

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

See also

  • Appell series
  • Humbert series
  • Lauricella series (three-variable)

References

  1. ^ Mathworld - Sextic Equation
  • Exton, Harold (1978), Handbook of hypergeometric integrals, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-122-0, MR 0474684
  • Kampé de Fériet, M. J. (1937), La fonction hypergéométrique., Mémorial des sciences mathématiques (in French), vol. 85, Paris: Gauthier-Villars, JFM 63.0996.03
  • 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. S2CID 118329382.
  • Weisstein, Eric W. "Kampé de Fériet function". MathWorld.


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