DLMF:18.28.E14 (Q5989)

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DLMF:18.28.E14
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    Statements

    Defining formula
    C n ⁡ ( cos ⁡ θ ; β | q ) = ( β 2 ; q ) n ( q ; q ) n ⁢ β 1 2 ⁢ n ⁢ ϕ 3 4 ⁡ ( q - n , β 2 ⁢ q n , β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ e - i ⁢ θ β ⁢ q 1 2 , - β , - β ⁢ q 1 2 ; q , q ) . q-ultraspherical-polynomial 𝑛 𝜃 𝛽 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝛽 1 2 𝑛 q-hypergeometric-rphis 4 3 superscript 𝑞 𝑛 superscript 𝛽 2 superscript 𝑞 𝑛 superscript 𝛽 1 2 superscript 𝑒 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑒 imaginary-unit 𝜃 𝛽 superscript 𝑞 1 2 𝛽 𝛽 superscript 𝑞 1 2 𝑞 𝑞 {\displaystyle{\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)=\frac{% \left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}\beta^{\frac{1}{2}n}}{{}_{4}% \phi_{3}}\left({q^{-n},\beta^{2}q^{n},\beta^{\frac{1}{2}}e^{\mathrm{i}\theta},% \beta^{\frac{1}{2}}e^{-\mathrm{i}\theta}\atop\beta q^{\frac{1}{2}},-\beta,-% \beta q^{\frac{1}{2}}};q,q\right).}}
    0 references
    DLMF id
    DLMF:18.28.E14
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    0 references
     
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