DLMF:18.33.E15 (Q6025)

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DLMF:18.33.E15
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    Statements

    test
    ϕ n ⁢ ( z ) = ∑ ℓ = 0 n ( a ⁢ q 2 ; q 2 ) ℓ ⁢ ( a ; q 2 ) n - ℓ ( q 2 ; q 2 ) ℓ ⁢ ( q 2 ; q 2 ) n - ℓ ⁢ ( q - 1 ⁢ z ) ℓ = ( a ; q 2 ) n ( q 2 ; q 2 ) n ⁢ ϕ 1 2 ⁡ ( a ⁢ q 2 , q - 2 ⁢ n a - 1 ⁢ q 2 - 2 ⁢ n ; q 2 , q ⁢ z a ) , subscript italic-ϕ 𝑛 𝑧 superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol 𝑎 superscript 𝑞 2 𝑛 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 ℓ superscript superscript 𝑞 1 𝑧 ℓ q-Pochhammer-symbol 𝑎 superscript 𝑞 2 𝑛 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 q-hypergeometric-rphis 2 1 𝑎 superscript 𝑞 2 superscript 𝑞 2 𝑛 superscript 𝑎 1 superscript 𝑞 2 2 𝑛 superscript 𝑞 2 𝑞 𝑧 𝑎 {\displaystyle{\displaystyle\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{\left(aq^{2};q^% {2}\right)_{\ell}\left(a;q^{2}\right)_{n-\ell}}{\left(q^{2};q^{2}\right)_{\ell% }\left(q^{2};q^{2}\right)_{n-\ell}}(q^{-1}z)^{\ell}=\frac{\left(a;q^{2}\right)% _{n}}{\left(q^{2};q^{2}\right)_{n}}{{}_{2}\phi_{1}}\left({aq^{2},q^{-2n}\atop a% ^{-1}q^{2-2n}};q^{2},\frac{qz}{a}\right),}}
    0 references
    DLMF id
    DLMF:18.33.E15
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    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.m2adec
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    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.m2adec
    0 references
    Q11750
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1mdec
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    Q11728
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1dec
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    Q11744
    test
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1cdec
    0 references
    Q11726
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ldec
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    Q11791
    test
    ϕ n ⁢ ( z ) subscript italic-ϕ 𝑛 𝑧 {\displaystyle{\displaystyle\phi_{n}(z)}}
    xml-id
    C18.S33.XMD1.m1ldec
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