The Appell hypergeometric functions are a formal extension of the hypergeometric function to two variables, resulting in four kinds of functions (Appell 1925; Picard 1880ab, 1881; Goursat 1882, Ex. 22, p. 300), F_1(α;β, β' ;γ;x, y) | = | sum_(m = 0)^∞ sum_(n = 0)^∞ ((α)_(m + n) (β)_m (β')_n)/(m!n!(γ)_(m + n)) x^m y^n F_2(α;β, β' ;γ, γ' ;x, y) | = | sum_(m = 0)^∞ sum_(n = 0)^∞ ((α)_(m + n) (β)_m (β')_n)/(m!n!(γ)_m (γ')_n) x^m y^n F_3(α, α' ;β, β' ;γ;x, y) | = | sum_(m = 0)^∞ sum_(n = 0)^∞ ((α)_m (α')_n (β)_m (β')_n)/(m!n!(γ)_(m + n)) x^m y^n F_4(α;β;γ, γ' ;x, y) | = | sum_(m = 0)^∞ sum_(n = 0)^∞ ((α)_(m + n) (β)_(m + n))/(m!n!(γ)_m (γ')_n) x^m y^n.
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