Appell Hypergeometric Function

A formal extension of the Hypergeometric Function to two variables, resulting in four kinds of functions (Appell 1925),

$$F_1(\alpha;\beta,\beta';\gamma;x,y) = \sum_{m=0}^\infty\sum_{n=0}^\infty {(\alpha)_{m+n}(\beta)_m(\beta')_n\over m!n!(\gamma)_{m+n}} x^my^n$$

$$F_2(\alpha;\beta,\beta';\gamma,\gamma';x,y) = \sum_{m=0}^\infty\sum_{n=0}^\infty {(\alpha)_{m+n}(\beta)_m(\beta')_n\over m!n!(\gamma)_m(\gamma')_n} x^my^n$$

$$F_3(\alpha,\alpha';\beta,\beta';\gamma;x,y) = \sum_{m=0}^\infty\sum_{n=0}^\infty {(\alpha)_m(\alpha')_n(\beta)_m(\beta')_n\over m!n!(\gamma)_{m+n}} x^my^n$$

$$F_4(\alpha;\beta;\gamma,\gamma';x,y) = \sum_{m=0}^\infty\sum_{n=0}^\infty {(\alpha)_{m+n}(\beta)_{m+n}\over m!n!(\gamma)_m(\gamma')_n} x^my^n.$$

Appell defined the functions in 1880, and Picard showed in 1881 that they may all be expressed by Integrals of the form

$$\int_0^1 u^\alpha(1-u)^\beta(1-xu)^\gamma(1-yu)^\delta\,du.$$

References

Appell, P. ``Sur les fonctions hypergéométriques de plusieurs variables.'' In Mémoir. Sci. Math. Paris: Gauthier-Villars, 1925.

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 73, 1935.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1461, 1980.