DLMF:17.13.E3 (Q5468)

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DLMF:17.13.E3
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    Statements

    Defining formula
    ∫ 0 ∞ t α - 1 ⁢ ( - t ⁢ q α + β ; q ) ∞ ( - t ; q ) ∞ ⁢ d t = Γ ⁡ ( α ) ⁢ Γ ⁡ ( 1 - α ) ⁢ Γ q ⁡ ( β ) Γ q ⁡ ( 1 - α ) ⁢ Γ q ⁡ ( α + β ) , superscript subscript 0 superscript 𝑡 𝛼 1 q-Pochhammer-symbol 𝑡 superscript 𝑞 𝛼 𝛽 𝑞 q-Pochhammer-symbol 𝑡 𝑞 𝑡 Euler-Gamma 𝛼 Euler-Gamma 1 𝛼 q-Gamma 𝑞 𝛽 q-Gamma 𝑞 1 𝛼 q-Gamma 𝑞 𝛼 𝛽 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{% \alpha+\beta};q\right)_{\infty}}{\left(-t;q\right)_{\infty}}\mathrm{d}t=\frac{% \Gamma\left(\alpha\right)\Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(1-\alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},}}
    0 references
    DLMF id
    DLMF:17.13.E3
    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.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    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.m2aadec
    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.m2abdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1bdec
    0 references
     
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