DLMF:17.6.E14 (Q5381)

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DLMF:17.6.E14
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    Statements

    test
    ∑ n = 0 ∞ ( a ; q ) n ⁢ ( b ; q 2 ) n ⁢ z n ( q ; q ) n ⁢ ( a ⁢ z ⁢ b ; q 2 ) n = ( a ⁢ z , b ⁢ z ; q 2 ) ∞ ( z , a ⁢ z ⁢ b ; q 2 ) ∞ ⁢ ϕ 1 2 ⁡ ( a , b b ⁢ z ; q 2 , z ⁢ q ) . superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑏 superscript 𝑞 2 𝑛 superscript 𝑧 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑧 𝑏 superscript 𝑞 2 𝑛 q-multiple-Pochhammer 𝑎 𝑧 𝑏 𝑧 superscript 𝑞 2 q-multiple-Pochhammer 𝑧 𝑎 𝑧 𝑏 superscript 𝑞 2 q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑏 𝑧 superscript 𝑞 2 𝑧 𝑞 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}\left% (b;q^{2}\right)_{n}z^{n}}{\left(q;q\right)_{n}\left(azb;q^{2}\right)_{n}}=% \frac{\left(az,bz;q^{2}\right)_{\infty}}{\left(z,azb;q^{2}\right)_{\infty}}{{}% _{2}\phi_{1}}\left({a,b\atop bz};q^{2},zq\right).}}
    0 references
    DLMF id
    DLMF:17.6.E14
    0 references
    Symbols used
    Q11298
    test
    ( 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.m2ahdec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    test
    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.m2amdec
    0 references
    Q11713
    test
    ( 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.m6ajdec
    0 references
    Q11710
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1mdec
    0 references
    Q11711
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1fdec
    0 references
     
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