DLMF:17.6.E28 (Q5395)

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DLMF:17.6.E28
No description defined

    Statements

    Defining formula
    ϕ 1 2 ⁡ ( q α , q β q γ ; q , z ) = Γ q ⁡ ( γ ) Γ q ⁡ ( β ) ⁢ Γ q ⁡ ( γ - β ) ⁢ ∫ 0 1 t β - 1 ⁢ ( t ⁢ q ; q ) γ - β - 1 ( x ⁢ t ; q ) α ⁢ d q t . q-hypergeometric-rphis 2 1 superscript 𝑞 𝛼 superscript 𝑞 𝛽 superscript 𝑞 𝛾 𝑞 𝑧 q-Gamma 𝑞 𝛾 q-Gamma 𝑞 𝛽 q-Gamma 𝑞 𝛾 𝛽 superscript subscript 0 1 superscript 𝑡 𝛽 1 q-Pochhammer-symbol 𝑡 𝑞 𝑞 𝛾 𝛽 1 q-Pochhammer-symbol 𝑥 𝑡 𝑞 𝛼 q-differential 𝑞 𝑡 {\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({q^{\alpha},q^{\beta}\atop q% ^{\gamma}};q,z\right)=\frac{\Gamma_{q}\left(\gamma\right)}{\Gamma_{q}\left(% \beta\right)\Gamma_{q}\left(\gamma-\beta\right)}\int_{0}^{1}\frac{t^{\beta-1}% \left(tq;q\right)_{\gamma-\beta-1}}{\left(xt;q\right)_{\alpha}}{\mathrm{d}}_{q% }t.}}
    0 references
    DLMF id
    DLMF:17.6.E28
    0 references
    Symbols used
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    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.m2akdec
    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.m3adec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    Defining formula
    ϕ s r + 1 ⁡ ( a 0 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rphis 𝑟 1 𝑠 subscript 𝑎 0 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E1.m2aaadec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1aadec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1udec
    0 references
    Q11719
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C17.S1.XMD8.m1dec
    0 references
     
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