DLMF:18.27.E23 (Q5974)

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DLMF:18.27.E23
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    h ~ n ⁡ ( x ; q ) = ( q ; q ) n ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ q - 2 ⁢ n ⁢ ℓ ⁢ q ℓ ⁢ ( 2 ⁢ ℓ + 1 ) ⁢ x n - 2 ⁢ ℓ ( q 2 ; q 2 ) ℓ ⁢ ( q ; q ) n - 2 ⁢ ℓ = x n ⁢ ϕ 1 2 ⁡ ( q - n , q - n + 1 0 ; q 2 , - x - 2 ⁢ q 2 ) . discrete-q-Hermite-polynomial-h-II 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ superscript 𝑞 2 𝑛 ℓ superscript 𝑞 ℓ 2 ℓ 1 superscript 𝑥 𝑛 2 ℓ q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 2 ℓ superscript 𝑥 𝑛 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 0 superscript 𝑞 2 superscript 𝑥 2 superscript 𝑞 2 {\displaystyle{\displaystyle\tilde{h}_{n}\left(x;q\right)=\left(q;q\right)_{n}% \sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}q^{-2n\ell}q^{% \ell(2\ell+1)}x^{n-2\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q;q\right)_{n-% 2\ell}}=x^{n}{{}_{2}\phi_{1}}\left({q^{-n},q^{-n+1}\atop 0};q^{2},-x^{-2}q^{2}% \right).}}
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    DLMF id
    DLMF:18.27.E23
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