DLMF:18.27.E24 (Q5975)

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DLMF:18.27.E24
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    Statements

    Defining formula
    ∑ ℓ = - ∞ ∞ ( h ~ n ⁡ ( c ⁢ q ℓ ; q ) ⁢ h ~ m ⁡ ( c ⁢ q ℓ ; q ) + h ~ n ⁡ ( - c ⁢ q ℓ ; q ) ⁢ h ~ m ⁡ ( - c ⁢ q ℓ ; q ) ) ⁢ q ℓ ( - c 2 ⁢ q 2 ⁢ ℓ ; q 2 ) ∞ = 2 ⁢ ( q 2 , - c 2 ⁢ q , - c - 2 ⁢ q ; q 2 ) ∞ ( q , - c 2 , - c - 2 ⁢ q 2 ; q 2 ) ∞ ⁢ ( q ; q ) n q n 2 ⁢ δ n , m , superscript subscript ℓ discrete-q-Hermite-polynomial-h-II 𝑛 𝑐 superscript 𝑞 ℓ 𝑞 discrete-q-Hermite-polynomial-h-II 𝑚 𝑐 superscript 𝑞 ℓ 𝑞 discrete-q-Hermite-polynomial-h-II 𝑛 𝑐 superscript 𝑞 ℓ 𝑞 discrete-q-Hermite-polynomial-h-II 𝑚 𝑐 superscript 𝑞 ℓ 𝑞 superscript 𝑞 ℓ q-Pochhammer-symbol superscript 𝑐 2 superscript 𝑞 2 ℓ superscript 𝑞 2 2 q-multiple-Pochhammer superscript 𝑞 2 superscript 𝑐 2 𝑞 superscript 𝑐 2 𝑞 superscript 𝑞 2 q-multiple-Pochhammer 𝑞 superscript 𝑐 2 superscript 𝑐 2 superscript 𝑞 2 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 superscript 𝑛 2 Kronecker 𝑛 𝑚 {\displaystyle{\displaystyle\sum_{\ell=-\infty}^{\infty}\left(\tilde{h}_{n}% \left(cq^{\ell};q\right)\tilde{h}_{m}\left(cq^{\ell};q\right)+\tilde{h}_{n}% \left(-cq^{\ell};q\right)\tilde{h}_{m}\left(-cq^{\ell};q\right)\right)\frac{q^% {\ell}}{\left(-c^{2}q^{2\ell};q^{2}\right)_{\infty}}=2\frac{\left(q^{2},-c^{2}% q,-c^{-2}q;q^{2}\right)_{\infty}}{\left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{% \infty}}\frac{\left(q;q\right)_{n}}{q^{n^{2}}}\delta_{n,m},}}
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    DLMF id
    DLMF:18.27.E24
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    constraint
    c > 0 𝑐 0 {\displaystyle{\displaystyle c>0}}
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    Symbols used
    Q10768
    Defining formula
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5ahdec
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    discrete $$q$$ -Hermite II polynomial
    Defining formula
    h ~ n ⁡ ( x ; q ) discrete-q-Hermite-polynomial-h-II 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle\tilde{h}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}
    xml-id
    C18.S27.E23.m2aadec
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