DLMF:17.13.E4 (Q5469)

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DLMF:17.13.E4
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ t α - 1 ⁢ ( - c ⁢ t ⁢ q α + β ; q ) ∞ ( - c ⁢ t ; q ) ∞ ⁢ d q t = Γ q ⁡ ( α ) ⁢ Γ q ⁡ ( β ) ⁢ ( - c ⁢ q α ; q ) ∞ ⁢ ( - q 1 - α / c ; q ) ∞ Γ q ⁡ ( α + β ) ⁢ ( - c ; q ) ∞ ⁢ ( - q / c ; q ) ∞ . superscript subscript 0 superscript 𝑡 𝛼 1 q-Pochhammer-symbol 𝑐 𝑡 superscript 𝑞 𝛼 𝛽 𝑞 q-Pochhammer-symbol 𝑐 𝑡 𝑞 q-differential 𝑞 𝑡 q-Gamma 𝑞 𝛼 q-Gamma 𝑞 𝛽 q-Pochhammer-symbol 𝑐 superscript 𝑞 𝛼 𝑞 q-Pochhammer-symbol superscript 𝑞 1 𝛼 𝑐 𝑞 q-Gamma 𝑞 𝛼 𝛽 q-Pochhammer-symbol 𝑐 𝑞 q-Pochhammer-symbol 𝑞 𝑐 𝑞 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{% \alpha+\beta};q\right)_{\infty}}{\left(-ct;q\right)_{\infty}}{\mathrm{d}}_{q}t% =\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta\right)\left(-cq^{% \alpha};q\right)_{\infty}\left(-q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}% \left(\alpha+\beta\right)\left(-c;q\right)_{\infty}\left(-q/c;q\right)_{\infty% }}.}}
    0 references
    DLMF id
    DLMF:17.13.E4
    0 references
    Symbols used
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    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.m2abdec
    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.m2acdec
    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.m3abdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1cdec
    0 references
     
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