DLMF:5.18.E12 (Q2199)

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DLMF:5.18.E12
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    Statements

    Defining formula
    B q ⁡ ( a , b ) = ∫ 0 1 t a - 1 ⁢ ( t ⁢ q ; q ) ∞ ( t ⁢ q b ; q ) ∞ ⁢ d q t , q-Beta 𝑞 𝑎 𝑏 superscript subscript 0 1 superscript 𝑡 𝑎 1 q-Pochhammer-symbol 𝑡 𝑞 𝑞 q-Pochhammer-symbol 𝑡 superscript 𝑞 𝑏 𝑞 q-differential 𝑞 𝑡 {\displaystyle{\displaystyle\mathrm{B}_{q}\left(a,b\right)=\int_{0}^{1}\frac{t% ^{a-1}\left(tq;q\right)_{\infty}}{\left(tq^{b};q\right)_{\infty}}{\mathrm{d}}_% {q}t,}}
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    DLMF id
    DLMF:5.18.E12
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    0 < q < 1 0 𝑞 1 {\displaystyle{\displaystyle 0<q<1}}
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    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    Symbols used
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    $$q$$ -beta function
    Defining formula
    B q ⁡ ( a , b ) q-Beta 𝑞 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}_{\NVar{q}}\left(\NVar{a},\NVar{b}\right% )}}
    xml-id
    C5.S18.E11.m2aadec
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    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.m2addec
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    Q11299
    Defining formula
    d q x q-differential 𝑞 𝑥 {\displaystyle{\displaystyle{\mathrm{d}}_{\NVar{q}}\NVar{x}}}
    xml-id
    C17.S2.SS5.p1.m3adec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
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    Q11280
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C5.S1.XMD10.m1kdec
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