DLMF:17.6.E12 (Q5379)

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DLMF:17.6.E12
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    Statements

    Defining formula
    ( 1 - z ) ⁢ ϕ 1 2 ⁡ ( q , a ⁢ q b ⁢ q ; q , z ) = ∑ n = 0 ∞ ( a ⁢ q , a ⁢ z ⁢ q / b ; q ) n ( b ⁢ q , z ⁢ q ; q ) n ⁢ ( 1 - a ⁢ z ⁢ q 2 ⁢ n + 1 ) ⁢ ( b ⁢ z ) n ⁢ q n 2 , 1 𝑧 q-hypergeometric-rphis 2 1 𝑞 𝑎 𝑞 𝑏 𝑞 𝑞 𝑧 superscript subscript 𝑛 0 q-multiple-Pochhammer 𝑎 𝑞 𝑎 𝑧 𝑞 𝑏 𝑞 𝑛 q-multiple-Pochhammer 𝑏 𝑞 𝑧 𝑞 𝑞 𝑛 1 𝑎 𝑧 superscript 𝑞 2 𝑛 1 superscript 𝑏 𝑧 𝑛 superscript 𝑞 superscript 𝑛 2 {\displaystyle{\displaystyle(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z% \right)=\sum_{n=0}^{\infty}\frac{\left(aq,azq/b;q\right)_{n}}{\left(bq,zq;q% \right)_{n}}(1-azq^{2n+1})(bz)^{n}q^{n^{2}},}}
    0 references
    DLMF id
    DLMF:17.6.E12
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    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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    Symbols used
    basic hypergeometric (or $$q$$ -hypergeometric) function
    Defining formula
    ϕ s r + 1 ⁡ ( a 0 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rphis 𝑟 1 𝑠 subscript 𝑎 0 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E1.m2akdec
    0 references
    Q11713
    Defining formula
    ( a 1 , a 2 , … , a r ; q ) n q-multiple-Pochhammer subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}% \right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m6ahdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1kdec
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    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1edec
    0 references
     
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