DLMF:18.28.E18 (Q5993)

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DLMF:18.28.E18
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    Statements

    test
    h n ⁡ ( sinh ⁡ t | q ) = ∑ ℓ = 0 n q 1 2 ⁢ ℓ ⁢ ( ℓ + 1 ) ⁢ ( q - n ; q ) ℓ ( q ; q ) ℓ ⁢ e ( n - 2 ⁢ ℓ ) ⁢ t = e n ⁢ t ⁢ ϕ 1 1 ⁡ ( q - n 0 ; q , - q ⁢ e - 2 ⁢ t ) = i - n ⁢ H n ⁡ ( i ⁢ sinh ⁡ t | q - 1 ) . continuous-q-inverse-Hermite-polynomial-h 𝑛 𝑡 𝑞 superscript subscript ℓ 0 𝑛 superscript 𝑞 1 2 ℓ ℓ 1 q-Pochhammer-symbol superscript 𝑞 𝑛 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ superscript 𝑒 𝑛 2 ℓ 𝑡 superscript 𝑒 𝑛 𝑡 q-hypergeometric-rphis 1 1 superscript 𝑞 𝑛 0 𝑞 𝑞 superscript 𝑒 2 𝑡 imaginary-unit 𝑛 continuous-q-Hermite-polynomial-H 𝑛 imaginary-unit 𝑡 superscript 𝑞 1 {\displaystyle{\displaystyle h_{n}\left(\sinh t\,|\,q\right)=\sum_{\ell=0}^{n}% q^{\frac{1}{2}\ell(\ell+1)}\frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right% )_{\ell}}e^{(n-2\ell)t}=e^{nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-qe^{-2t% }\right)={\mathrm{i}^{-n}}H_{n}\left(\mathrm{i}\sinh t\,|\,q^{-1}\right).}}
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    DLMF id
    DLMF:18.28.E18
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    Symbols used
    continuous $$q$$ -Hermite polynomial
    test
    H n ⁡ ( x | q ) continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\,|\,\NVar{q}\right)}}
    xml-id
    C18.S28.E16.m2abdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
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    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ajdec
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    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.m2amdec
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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.m2afdec
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    Q11728
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1odec
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    Q11744
    test
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1edec
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    Q11726
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1mdec
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    Q11727
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1gdec
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