DLMF:16.11.E9 (Q5243)

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DLMF:16.11.E9
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    Statements

    Defining formula
    ( ∏ ℓ = 1 p Γ ⁡ ( a ℓ ) / ∏ ℓ = 1 q Γ ⁡ ( b ℓ ) ) ⁢ F q p ⁡ ( a 1 , … , a p b 1 , … , b q ; - z ) ∼ E p , q ⁢ ( z ⁢ e - π ⁢ i ) + E p , q ⁢ ( z ⁢ e π ⁢ i ) , asymptotic-expansion superscript subscript product ℓ 1 𝑝 Euler-Gamma subscript 𝑎 ℓ superscript subscript product ℓ 1 𝑞 Euler-Gamma subscript 𝑏 ℓ Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 subscript 𝐸 𝑝 𝑞 𝑧 superscript 𝑒 𝜋 imaginary-unit subscript 𝐸 𝑝 𝑞 𝑧 superscript 𝑒 𝜋 imaginary-unit {\displaystyle{\displaystyle\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}% \Gamma\left(a_{\ell}\right)}{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}% \right)}}\right){{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% -z\right)\sim E_{p,q}(ze^{-\pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),}}
    0 references
    DLMF id
    DLMF:16.11.E9
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1acdec
    0 references
     
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