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16 Generalized Hypergeometric Functions & Meijer G-FunctionTwo-Variable Hypergeometric Functions

§16.15 Integral Representations and Integrals

16.15.1 F1(α;β,β;γ;x,y)=Γ(γ)Γ(α)Γ(γ-α)01uα-1(1-u)γ-α-1(1-ux)β(1-uy)βdu,
α>0, (γ-α)>0,
16.15.2 F2(α;β,β;γ,γ;x,y)=Γ(γ)Γ(γ)Γ(β)Γ(β)Γ(γ-β)Γ(γ-β)×0101uβ-1vβ-1(1-u)γ-β-1(1-v)γ-β-1(1-ux-vy)αdudv,
γ>β>0, γ>β>0,
16.15.3 F3(α,α;β,β;γ;x,y)=Γ(γ)Γ(β)Γ(β)Γ(γ-β-β)Δuβ-1vβ-1(1-u-v)γ-β-β-1(1-ux)α(1-vy)αdudv,
(γ-β-β)>0, β>0, β>0,

where Δ is the triangle defined by u0, v0, u+v1.

16.15.4 F4(α;β;γ,γ;x(1-y),y(1-x))=Γ(γ)Γ(γ)Γ(α)Γ(β)Γ(γ-α)Γ(γ-β)×0101uα-1vβ-1(1-u)γ-α-1(1-v)γ-β-1(1-ux)γ+γ-α-1(1-vy)γ+γ-β-1(1-ux-vy)α+β-γ-γ+1dudv,
γ>α>0, γ>β>0.

For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large x, large y, or both. For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).