DLMF:17.6.E11 (Q5378)

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DLMF:17.6.E11
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    Statements

    Defining formula
    1 - z 1 - b ⁢ ϕ 1 2 ⁡ ( q , a ⁢ q b ⁢ q ; q , z ) = ∑ n = 0 ∞ ( a ⁢ q ; q ) n ⁢ ( a ⁢ z ⁢ q / b ; q ) 2 ⁢ n ⁢ b n ( z ⁢ q , a ⁢ q / b ; q ) n - a ⁢ q ⁢ ∑ n = 0 ∞ ( a ⁢ q ; q ) n ⁢ ( a ⁢ z ⁢ q / b ; q ) 2 ⁢ n + 1 ⁢ ( b ⁢ q ) n ( z ⁢ q ; q ) n ⁢ ( a ⁢ q / b ; q ) n + 1 , 1 𝑧 1 𝑏 q-hypergeometric-rphis 2 1 𝑞 𝑎 𝑞 𝑏 𝑞 𝑞 𝑧 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑧 𝑞 𝑏 𝑞 2 𝑛 superscript 𝑏 𝑛 q-multiple-Pochhammer 𝑧 𝑞 𝑎 𝑞 𝑏 𝑞 𝑛 𝑎 𝑞 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑧 𝑞 𝑏 𝑞 2 𝑛 1 superscript 𝑏 𝑞 𝑛 q-Pochhammer-symbol 𝑧 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑏 𝑞 𝑛 1 {\displaystyle{\displaystyle\frac{1-z}{1-b}{{}_{2}\phi_{1}}\left({q,aq\atop bq% };q,z\right)=\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q\right% )_{2n}b^{n}}{\left(zq,aq/b;q\right)_{n}}-aq\sum_{n=0}^{\infty}\frac{\left(aq;q% \right)_{n}\left(azq/b;q\right)_{2n+1}(bq)^{n}}{\left(zq;q\right)_{n}\left(aq/% b;q\right)_{n+1}},}}
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    DLMF id
    DLMF:17.6.E11
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