Kampé de Fériet Function

A special function generalizes the generalized hypergeometric function to two variables and includes the Appell hypergeometric function  as a special case. The Kampe de Feriet function can represent derivatives of generalized hypergeometric functions with respect to their parameters, as well as indefinite integrals of two and three Meijer G-functions. Exton and Krupnikov (1998) have derived a large collection of formulas involving this function.

Kampé de Fériet functions are written in the notation

(1)

Special cases include

(2)

(3)

for  and , where  is the incomplete elliptic integral of the second kind and  is the incomplete elliptic integral of the first kind, as well as

(4)

for , where  is the incomplete elliptic integral of the third kind (Exton and Krupnikov 1998, p. 1). Additional identities are given by

(5)

(6)

(7)

(Exton and Krupnikov 1998, p. 3).

REFERENCES:

Appell, P. Sur les fonctions hypergéométriques de plusieurs variables. Paris: Gauthier-Villars, 1925.

Appell, P. and Kampé de Fériet, J. Fonctions hypergéométriques et hypersphériques: polynomes d'Hermite. Paris: Gauthier-Villars, 1926.

Exton, H. "The Kampé de Fériet Function." §1.3.2 in Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, pp. 24-25, 1978.

Exton, H. Multiple Hypergeometric Functions and Applications. Chichester, England: Ellis Horwood, 1976.

Exton, H. and Krupnikov, E. D. A Register of Computer-Oriented Reduction Identities for the Kampé de Fériet Function. Draft manuscript. Novosibirsk, 1998.

Kampé de Fériet, J. La fonction hypergéométrique. Paris: Gauthier-Villars, 1937.

Ragab, F. J. "Expansions of Kampe De Feriet's Double Hypergeometric Function of Higher Order." J. reine angew. Math. 212, 113-119, 1963.

Srivastava, H. M., Karlsson, P. W. Multiple Gaussian Hypergeometric Series. Chichester, England: Ellis Horwood, 1985.