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Kampé de Fériet function

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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.

See also

References

  1. Mathworld - Sextic Equation

External links


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