DLMF:18.28.E13 (Q5988)

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DLMF:18.28.E13
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    Statements

    test
    C n ⁡ ( cos ⁡ θ ; β | q ) = ∑ ℓ = 0 n ( β ; q ) ℓ ⁢ ( β ; q ) n - ℓ ( q ; q ) ℓ ⁢ ( q ; q ) n - ℓ ⁢ e i ⁢ ( n - 2 ⁢ ℓ ) ⁢ θ = ( β ; q ) n ( q ; q ) n ⁢ e i ⁢ n ⁢ θ ⁢ ϕ 1 2 ⁡ ( q - n , β β - 1 ⁢ q 1 - n ; q , β - 1 ⁢ q ⁢ e - 2 ⁢ i ⁢ θ ) . q-ultraspherical-polynomial 𝑛 𝜃 𝛽 𝑞 superscript subscript ℓ 0 𝑛 q-Pochhammer-symbol 𝛽 𝑞 ℓ q-Pochhammer-symbol 𝛽 𝑞 𝑛 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 𝑛 ℓ superscript 𝑒 imaginary-unit 𝑛 2 ℓ 𝜃 q-Pochhammer-symbol 𝛽 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑒 imaginary-unit 𝑛 𝜃 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 𝛽 superscript 𝛽 1 superscript 𝑞 1 𝑛 𝑞 superscript 𝛽 1 𝑞 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)=\sum_{% \ell=0}^{n}\frac{\left(\beta;q\right)_{\ell}\left(\beta;q\right)_{n-\ell}}{% \left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}e^{\mathrm{i}(n-2\ell)\theta}% =\frac{\left(\beta;q\right)_{n}}{\left(q;q\right)_{n}}e^{\mathrm{i}n\theta}{{}% _{2}\phi_{1}}\left({q^{-n},\beta\atop\beta^{-1}q^{1-n}};q,\beta^{-1}qe^{-2% \mathrm{i}\theta}\right).}}
    0 references
    DLMF id
    DLMF:18.28.E13
    0 references
    Symbols used
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
    0 references
     
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