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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(γα)01uα1(1u)γα1(1ux)β(1uy)βdu,
α>0, (γα)>0,
16.15.2 F2(α;β,β;γ,γ;x,y)=Γ(γ)Γ(γ)Γ(β)Γ(β)Γ(γβ)Γ(γβ)×0101uβ1vβ1(1u)γβ1(1v)γβ1(1uxvy)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(γββ)Δuβ1vβ1(1uv)γββ1(1ux)α(1vy)αdudv,
(γββ)>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(1y),y(1x))=Γ(γ)Γ(γ)Γ(α)Γ(β)Γ(γα)Γ(γβ)×0101uα1vβ1(1u)γα1(1v)γβ1(1ux)γ+γα1(1vy)γ+γβ1(1uxvy)α+βγγ+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).