DLMF:17.13.E2 (Q5467)

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DLMF:17.13.E2
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    Statements

    Defining formula
    ∫ - c d ( - q ⁢ x / c ; q ) ∞ ⁢ ( q ⁢ x / d ; q ) ∞ ( - x ⁢ q α / c ; q ) ∞ ⁢ ( x ⁢ q β / d ; q ) ∞ ⁢ d q x = Γ q ⁡ ( α ) ⁢ Γ q ⁡ ( β ) Γ q ⁡ ( α + β ) ⁢ c ⁢ d c + d ⁢ ( - c / d ; q ) ∞ ⁢ ( - d / c ; q ) ∞ ( - q β ⁢ c / d ; q ) ∞ ⁢ ( - q α ⁢ d / c ; q ) ∞ . superscript subscript 𝑐 𝑑 q-Pochhammer-symbol 𝑞 𝑥 𝑐 𝑞 q-Pochhammer-symbol 𝑞 𝑥 𝑑 𝑞 q-Pochhammer-symbol 𝑥 superscript 𝑞 𝛼 𝑐 𝑞 q-Pochhammer-symbol 𝑥 superscript 𝑞 𝛽 𝑑 𝑞 q-differential 𝑞 𝑥 q-Gamma 𝑞 𝛼 q-Gamma 𝑞 𝛽 q-Gamma 𝑞 𝛼 𝛽 𝑐 𝑑 𝑐 𝑑 q-Pochhammer-symbol 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑑 𝑐 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛽 𝑐 𝑑 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 𝑑 𝑐 𝑞 {\displaystyle{\displaystyle\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}% \left(qx/d;q\right)_{\infty}}{\left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{% \beta}/d;q\right)_{\infty}}{\mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac% {cd}{c+d}\frac{\left(-c/d;q\right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left% (-q^{\beta}c/d;q\right)_{\infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.}}
    0 references
    DLMF id
    DLMF:17.13.E2
    0 references
    Symbols used
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    $$q$$ -gamma function
    Defining formula
    Γ q ⁡ ( z ) q-Gamma 𝑞 𝑧 {\displaystyle{\displaystyle\Gamma_{\NVar{q}}\left(\NVar{z}\right)}}
    xml-id
    C5.S18.E4.m2adec
    0 references
    Q11298
    Defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2aadec
    0 references
    Q11299
    Defining formula
    d q x q-differential 𝑞 𝑥 {\displaystyle{\displaystyle{\mathrm{d}}_{\NVar{q}}\NVar{x}}}
    xml-id
    C17.S2.SS5.p1.m3aadec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1adec
    0 references
    Q11719
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C17.S1.XMD8.m1adec
    0 references
     
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