DLMF:18.28.E16 (Q5991)

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DLMF:18.28.E16
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    Statements

    Defining formula
    H n ⁡ ( cos ⁡ θ | q ) = ∑ ℓ = 0 n ( q ; q ) n ⁢ e i ⁢ ( n - 2 ⁢ ℓ ) ⁢ θ ( q ; q ) ℓ ⁢ ( q ; q ) n - ℓ = e i ⁢ n ⁢ θ ⁢ ϕ 0 2 ⁡ ( q - n , 0 - ; q , q n ⁢ e - 2 ⁢ i ⁢ θ ) . continuous-q-Hermite-polynomial-H 𝑛 𝜃 𝑞 superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑒 imaginary-unit 𝑛 2 ℓ 𝜃 q-Pochhammer-symbol 𝑞 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 ℓ superscript 𝑒 imaginary-unit 𝑛 𝜃 q-hypergeometric-rphis 2 0 superscript 𝑞 𝑛 0 𝑞 superscript 𝑞 𝑛 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle H_{n}\left(\cos\theta\,|\,q\right)=\sum_{\ell=0}^% {n}\frac{\left(q;q\right)_{n}e^{\mathrm{i}(n-2\ell)\theta}}{\left(q;q\right)_{% \ell}\left(q;q\right)_{n-\ell}}=e^{\mathrm{i}n\theta}{{}_{2}\phi_{0}}\left({q^% {-n},0\atop-};q,q^{n}e^{-2\mathrm{i}\theta}\right).}}
    0 references
    DLMF id
    DLMF:18.28.E16
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2agdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
    0 references
    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.m2akdec
    0 references
    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.m2aedec
    0 references
     
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