DLMF:16.15.E4 (Q5266)

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DLMF:16.15.E4
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    Defining formula
    F 4 ⁡ ( α , β ; γ , γ ′ ; x ⁢ ( 1 - y ) , y ⁢ ( 1 - x ) ) = Γ ⁡ ( γ ) ⁢ Γ ⁡ ( γ ′ ) Γ ⁡ ( α ) ⁢ Γ ⁡ ( β ) ⁢ Γ ⁡ ( γ - α ) ⁢ Γ ⁡ ( γ ′ - β ) ⁢ ∫ 0 1 ∫ 0 1 u α - 1 ⁢ v β - 1 ⁢ ( 1 - u ) γ - α - 1 ⁢ ( 1 - v ) γ ′ - β - 1 ( 1 - u ⁢ x ) γ + γ ′ - α - 1 ⁢ ( 1 - v ⁢ y ) γ + γ ′ - β - 1 ⁢ ( 1 - u ⁢ x - v ⁢ y ) α + β - γ - γ ′ + 1 ⁢ d u ⁢ d v , Appell-F-4 𝛼 𝛽 𝛾 superscript 𝛾 ′ 𝑥 1 𝑦 𝑦 1 𝑥 Euler-Gamma 𝛾 Euler-Gamma superscript 𝛾 ′ Euler-Gamma 𝛼 Euler-Gamma 𝛽 Euler-Gamma 𝛾 𝛼 Euler-Gamma superscript 𝛾 ′ 𝛽 superscript subscript 0 1 superscript subscript 0 1 superscript 𝑢 𝛼 1 superscript 𝑣 𝛽 1 superscript 1 𝑢 𝛾 𝛼 1 superscript 1 𝑣 superscript 𝛾 ′ 𝛽 1 superscript 1 𝑢 𝑥 𝛾 superscript 𝛾 ′ 𝛼 1 superscript 1 𝑣 𝑦 𝛾 superscript 𝛾 ′ 𝛽 1 superscript 1 𝑢 𝑥 𝑣 𝑦 𝛼 𝛽 𝛾 superscript 𝛾 ′ 1 𝑢 𝑣 {\displaystyle{\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x% (1-y),y(1-x)\right)=\frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}% \right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-% \alpha\right)\Gamma\left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{% 0}^{1}\frac{u^{\alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{% \prime}-\beta-1}}{(1-ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+% \gamma^{\prime}-\beta-1}(1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}% \mathrm{d}u\mathrm{d}v,}}
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    DLMF id
    DLMF:16.15.E4
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