DLMF:17.5.E1 (Q5363)

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DLMF:17.5.E1
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    Statements

    Defining formula
    ϕ 0 0 ⁡ ( - ; - ; q , z ) = ∑ n = 0 ∞ ( - 1 ) n ⁢ q ( n 2 ) ⁢ z n ( q ; q ) n = ( z ; q ) ∞ , q-hypergeometric-rphis 0 0 𝑞 𝑧 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 binomial 𝑛 2 superscript 𝑧 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑧 𝑞 {\displaystyle{\displaystyle{{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{% \infty}\frac{(-1)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_% {n}}=\left(z;q\right)_{\infty},}}
    0 references
    DLMF id
    DLMF:17.5.E1
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    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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    Symbols used
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
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    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.m2adec
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    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.m2adec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1dec
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    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1dec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1dec
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