DLMF:17.14.E5 (Q5474)

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DLMF:17.14.E5
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    Statements

    Defining formula
    ∑ n = 0 ∞ q n 2 + 2 ⁢ n ( q 2 ; q 2 ) n ⁢ ( q ; q 2 ) n + 1 =  coeff. of  ⁢ z 0 ⁢  in  ⁢ ( - z ⁢ q ; q 2 ) ∞ ⁢ ( - z - 1 ⁢ q ; q 2 ) ∞ ⁢ ( q 2 ; q 2 ) ∞ ( - q 2 ⁢ z - 1 ; q 2 ) ∞ ⁢ ( q ; q 2 ) ∞ ⁢ ( z - 1 ⁢ q 2 ; q 2 ) ∞ = 1 ( q ; q 2 ) ∞ ⁢  coeff. of  ⁢ z 0 ⁢  in  ⁢ ( - z ⁢ q ; q 2 ) ∞ ⁢ ( - z - 1 ⁢ q ; q 2 ) ∞ ⁢ ( q 2 ; q 2 ) ∞ ( q 4 ⁢ z - 2 ; q 4 ) ∞ = H ⁢ ( q 4 ) ( q ; q 2 ) ∞ . superscript subscript 𝑛 0 superscript 𝑞 superscript 𝑛 2 2 𝑛 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 q-Pochhammer-symbol 𝑞 superscript 𝑞 2 𝑛 1  coeff. of  superscript 𝑧 0  in  q-Pochhammer-symbol 𝑧 𝑞 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑧 1 𝑞 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑧 1 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑧 1 superscript 𝑞 2 superscript 𝑞 2 1 q-Pochhammer-symbol 𝑞 superscript 𝑞 2  coeff. of  superscript 𝑧 0  in  q-Pochhammer-symbol 𝑧 𝑞 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑧 1 𝑞 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑞 4 superscript 𝑧 2 superscript 𝑞 4 𝐻 superscript 𝑞 4 q-Pochhammer-symbol 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{q^{n^{2}+2n}}{\left(q^{2}% ;q^{2}\right)_{n}\left(q;q^{2}\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in % }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}% \left(q^{2};q^{2}\right)_{\infty}}{\left(-q^{2}z^{-1};q^{2}\right)_{\infty}% \left(q;q^{2}\right)_{\infty}\left(z^{-1}q^{2};q^{2}\right)_{\infty}}=\frac{1}% {\left(q;q^{2}\right)_{\infty}}\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-% zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}% \right)_{\infty}}{\left(q^{4}z^{-2};q^{4}\right)_{\infty}}=\frac{H(q^{4})}{% \left(q;q^{2}\right)_{\infty}}.}}
    0 references
    DLMF id
    DLMF:17.14.E5
    0 references
    Symbols used
    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
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1ddec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1ddec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1cdec
    0 references
     
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