The Humbert series Φ1 can also be written as a one-dimensional Euler-type integral:
This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.
Similarly, the function Φ2 is defined for all x, y by the series:
the function Φ3 for all x, y by the series:
the function Ψ1 for |x| < 1 by the series:
the function Ψ2 for all x, y by the series:
the function Ξ1 for |x| < 1 by the series:
and the function Ξ2 for |x| < 1 by the series:
Related seriesedit
There are four related series of two variables, F1, F2, F3, and F4, which generalize Gauss's hypergeometric series2F1 of one variable in a similar manner and which were introduced by Paul Émile Appell in 1880.
Referencesedit
Appell, Paul; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13. (see p. 126)
Bateman, H.; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I(PDF). New York: McGraw–Hill. (see p. 225)
Humbert, Pierre (1920). "Sur les fonctions hypercylindriques". Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French). 171: 490–492. JFM 47.0348.01.