DLMF:17.6.E29 (Q5396)

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DLMF:17.6.E29
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    Statements

    Defining formula
    ϕ 1 2 ⁡ ( a , b c ; q , z ) = ( - 1 2 ⁢ π ⁢ i ) ⁢ ( a , b ; q ) ∞ ( q , c ; q ) ∞ ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ ( q 1 + ζ , c ⁢ q ζ ; q ) ∞ ( a ⁢ q ζ , b ⁢ q ζ ; q ) ∞ ⁢ π ⁢ ( - z ) ζ sin ⁡ ( π ⁢ ζ ) ⁢ d ζ , q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 𝑞 𝑧 1 2 𝜋 𝑖 q-multiple-Pochhammer 𝑎 𝑏 𝑞 q-multiple-Pochhammer 𝑞 𝑐 𝑞 superscript subscript 𝑖 𝑖 q-multiple-Pochhammer superscript 𝑞 1 𝜁 𝑐 superscript 𝑞 𝜁 𝑞 q-multiple-Pochhammer 𝑎 superscript 𝑞 𝜁 𝑏 superscript 𝑞 𝜁 𝑞 𝜋 superscript 𝑧 𝜁 𝜋 𝜁 𝜁 {\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=% \left(\frac{-1}{2\pi i}\right)\frac{\left(a,b;q\right)_{\infty}}{\left(q,c;q% \right)_{\infty}}\int_{-i\infty}^{i\infty}\frac{\left(q^{1+\zeta},cq^{\zeta};q% \right)_{\infty}}{\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}}\frac{\pi(-z)^% {\zeta}}{\sin\left(\pi\zeta\right)}\mathrm{d}\zeta,}}
    0 references
    DLMF id
    DLMF:17.6.E29
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    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.m2aabdec
    0 references
    Q11713
    Defining formula
    ( a 1 , a 2 , … , a r ; q ) n q-multiple-Pochhammer subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}% \right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m6aodec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1abdec
    0 references
     
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