DLMF:18.28.E15 (Q5990)

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DLMF:18.28.E15
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ 0 π C n ⁡ ( cos ⁡ θ ; β | q ) ⁢ C m ⁡ ( cos ⁡ θ ; β | q ) ⁢ | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 ⁢ d θ = ( β , β ⁢ q ; q ) ∞ ( β 2 , q ; q ) ∞ ⁢ ( 1 - β ) ⁢ ( β 2 ; q ) n ( 1 - β ⁢ q n ) ⁢ ( q ; q ) n ⁢ δ n , m , 1 2 𝜋 superscript subscript 0 𝜋 q-ultraspherical-polynomial 𝑛 𝜃 𝛽 𝑞 q-ultraspherical-polynomial 𝑚 𝜃 𝛽 𝑞 q-Pochhammer-symbol superscript 𝑒 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝛽 superscript 𝑒 2 imaginary-unit 𝜃 𝑞 2 𝜃 q-multiple-Pochhammer 𝛽 𝛽 𝑞 𝑞 q-multiple-Pochhammer superscript 𝛽 2 𝑞 𝑞 1 𝛽 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 1 𝛽 superscript 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 Kronecker 𝑛 𝑚 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{\pi}C_{n}\left(\cos\theta;% \beta\,|\,q\right)C_{m}\left(\cos\theta;\beta\,|\,q\right)\*{\left|\frac{\left% (e^{2\mathrm{i}\theta};q\right)_{\infty}}{\left(\beta e^{2\mathrm{i}\theta};q% \right)_{\infty}}\right|^{2}}\mathrm{d}\theta=\frac{\left(\beta,\beta q;q% \right)_{\infty}}{\left(\beta^{2},q;q\right)_{\infty}}\frac{(1-\beta)\left(% \beta^{2};q\right)_{n}}{(1-\beta q^{n})\left(q;q\right)_{n}}\delta_{n,m},}}
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    DLMF id
    DLMF:18.28.E15
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    constraint
    - 1 < β < 1 1 𝛽 1 {\displaystyle{\displaystyle-1<\beta<1}}
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    Symbols used
    Q10768
    Defining formula
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5addec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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    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.m2ajdec
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    Q11713
    Defining formula
    ( a 1 , a 2 , … , a r ; q ) n q-multiple-Pochhammer subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}% \right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m6afdec
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    continuous $$q$$ -ultraspherical polynomial
    Defining formula
    C n ⁡ ( x ; β | q ) q-ultraspherical-polynomial 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle C_{\NVar{n}}\left(\NVar{x};\NVar{\beta}\,|\,\NVar% {q}\right)}}
    xml-id
    C18.S28.E13.m2abdec
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    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1ldec
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    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1ddec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1jdec
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