DLMF:17.14.E2 (Q5471): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11724 / rank
 
Normal rank
Property / Symbols used: Q11724 / qualifier
 
Defining formula:

H ⁢ ( q ) 𝐻 𝑞 {\displaystyle{\displaystyle H(q)}}

H(q)
Property / Symbols used: Q11724 / qualifier
 
xml-id: C17.S14.XMD2.m1dec

Latest revision as of 15:11, 2 January 2020

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DLMF:17.14.E2
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    Statements

    Defining formula
    ∑ n = 0 ∞ q n ⁢ ( n + 1 ) ( 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 ) ∞ ( z - 1 ⁢ q 2 ; q 2 ) ∞ ⁢ ( - q ; q 2 ) ∞ ⁢ ( z - 1 ⁢ q ; q 2 ) ∞ = 1 ( - q ; q 2 ) ∞ ⁢  coeff. of  ⁢ z 0 ⁢  in  ⁢ ( - z ⁢ q ; q 2 ) ∞ ⁢ ( - z - 1 ⁢ q ; q 2 ) ∞ ⁢ ( q 2 ; q 2 ) ∞ ( z - 1 ⁢ q ; q ) ∞ = H ⁢ ( q ) ( - q ; q 2 ) ∞ , superscript subscript 𝑛 0 superscript 𝑞 𝑛 𝑛 1 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 𝑧 1 superscript 𝑞 2 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑧 1 𝑞 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 𝑧 1 𝑞 𝑞 𝐻 𝑞 q-Pochhammer-symbol 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\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(z^{-1}q^{2};q^{2}\right)_{\infty}\left(-q;% q^{2}\right)_{\infty}\left(z^{-1}q;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(z^{-1}q;q\right)_{\infty}}=\frac{H(q)}{\left(-q;q^{2}\right)_{% \infty}},}}
    0 references
    DLMF id
    DLMF:17.14.E2
    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.m2aadec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1adec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1adec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1dec
    0 references
    Q11724
    Defining formula
    H ⁢ ( q ) 𝐻 𝑞 {\displaystyle{\displaystyle H(q)}}
    xml-id
    C17.S14.XMD2.m1dec
    0 references
     
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