Kampé de Fériet function explained

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, … ,a} _{p\colon} b _{1,b} _{1{}'; … ;b} _{q,b} _{q{}';} \\ c _{1, … ,c} _{r\colon} d _{1,d} _{1{}'; … ;d} _{s,d} _{s{}'; \end{matrix} x,y\right)= \sum} ^{inf ty ((a _{1)} _{m+n} … (a _{p)} _{m+n})/((c _{1)} _{m+n} … (c _{r)} _{m+n})} ^{n =0} ((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 ^{my} ^{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

• • • F. J.. Ragab. Expansions of Kampe de Feriet's double hypergeometric function of higher order. 1963. 212. 113–119. 10.1515/crll.1963.212.113. J. reine angew. Math.. 212. 118329382.

Notes and References

1. http://mathworld.wolfram.com/SexticEquation.html Mathworld - Sextic Equation