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}\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!}.
Applications
The general sextic equation can be solved in terms of Kampé de Fériet functions.[http://mathworld.wolfram.com/SexticEquation.html Mathworld - Sextic Equation]
See also
- Appell series
- Humbert series
- Lauricella series (three-variable)
References
{{Reflist}}
- {{Citation | last1=Exton | first1=Harold | authorlink=Harold Exton | title=Handbook of hypergeometric integrals | url=https://books.google.com/books?id=fUHvAAAAMAAJ | publisher=Ellis Horwood Ltd. | location=Chichester | series=Mathematics and its Applications | isbn=978-0-85312-122-0 | mr=0474684 | year=1978}}
- {{Citation | last1=Kampé de Fériet | first1=M. J. | title=La fonction hypergéométrique. | url=https://books.google.com/books?id=JObuAAAAMAAJ | publisher=Paris: Gauthier-Villars | language=French | series=Mémorial des sciences mathématiques | jfm=63.0996.03 | year=1937 | volume=85}}
- {{ cite journal
|first1=F. J.
|last1=Ragab
|title = Expansions of Kampe de Feriet's double hypergeometric function of higher order
|year=1963
|issue=212
|pages=113–119
|doi=10.1515/crll.1963.212.113
|journal=J. reine angew. Math.
|volume=212
|s2cid=118329382
}}
External links
- {{MathWorld |title=Kampé de Fériet function |id=KampedeFerietFunction}}
{{DEFAULTSORT:Kampe de Feriet function}}
Category:Hypergeometric functions
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