DLMF:17.9.E20 (Q5446)

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DLMF:17.9.E20
No description defined

    Statements

    Defining formula
    ∑ n = 0 ∞ ( a ; q k ) n ⁢ ( b ; q ) k ⁢ n ⁢ z n ( q k ; q k ) n ⁢ ( c ; q ) k ⁢ n = ( b ; q ) ∞ ⁢ ( a ⁢ z ; q k ) ∞ ( c ; q ) ∞ ⁢ ( z ; q k ) ∞ ⁢ ∑ n = 0 ∞ ( c / b ; q ) n ⁢ ( z ; q k ) n ⁢ b n ( q ; q ) n ⁢ ( a ⁢ z ; q k ) n , superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑘 𝑛 q-Pochhammer-symbol 𝑏 𝑞 𝑘 𝑛 superscript 𝑧 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑘 superscript 𝑞 𝑘 𝑛 q-Pochhammer-symbol 𝑐 𝑞 𝑘 𝑛 q-Pochhammer-symbol 𝑏 𝑞 q-Pochhammer-symbol 𝑎 𝑧 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑐 𝑞 q-Pochhammer-symbol 𝑧 superscript 𝑞 𝑘 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑐 𝑏 𝑞 𝑛 q-Pochhammer-symbol 𝑧 superscript 𝑞 𝑘 𝑛 superscript 𝑏 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑧 superscript 𝑞 𝑘 𝑛 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{\left(a;q^{k}\right)_{n}% \left(b;q\right)_{kn}z^{n}}{\left(q^{k};q^{k}\right)_{n}\left(c;q\right)_{kn}}% =\frac{\left(b;q\right)_{\infty}\left(az;q^{k}\right)_{\infty}}{\left(c;q% \right)_{\infty}\left(z;q^{k}\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c% /b;q\right)_{n}\left(z;q^{k}\right)_{n}b^{n}}{\left(q;q\right)_{n}\left(az;q^{% k}\right)_{n}},}}
    0 references
    DLMF id
    DLMF:17.9.E20
    0 references
    constraint
    k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle k=1,2,3,\dots}}
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    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.m2ajdec
    0 references
    Q11715
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C17.S1.XMD1.m1dec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1sdec
    0 references
     
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