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Date: 19-8-2019
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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.
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