DLMF:17.9.E20 (Q5446): Difference between revisions

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Property / constraint
 

k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle k=1,2,3,\dots}}

k=1,2,3,\dots
Property / constraint: k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle k=1,2,3,\dots}} / rank
 
Normal rank

Revision as of 17:59, 30 December 2019

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DLMF:17.9.E20
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    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}},}}
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    DLMF id
    DLMF:17.9.E20
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    constraint
    k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle k=1,2,3,\dots}}
    0 references
     
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