DLMF:18.28.E8 (Q6679)

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DLMF:18.28.E8
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    Statements

    DLMF defining formula
    1 2 ⁢ π ⁢ ∫ 0 π Q n ⁡ ( cos ⁡ θ ; a , b | q ) ⁢ Q m ⁡ ( cos ⁡ θ ; a , b | q ) ⁢ | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ; q ) ∞ | 2 ⁢ d θ = δ n , m ( q n + 1 , a ⁢ b ⁢ q n ; q ) ∞ , 1 2 𝜋 superscript subscript 0 𝜋 AlSalam-Chihara-polynomial-Q 𝑛 𝜃 𝑎 𝑏 𝑞 AlSalam-Chihara-polynomial-Q 𝑚 𝜃 𝑎 𝑏 𝑞 q-Pochhammer-symbol superscript 𝑒 2 𝑖 𝜃 𝑞 q-multiple-Pochhammer 𝑎 superscript 𝑒 𝑖 𝜃 𝑏 superscript 𝑒 𝑖 𝜃 𝑞 2 𝜃 Kronecker 𝑛 𝑚 q-multiple-Pochhammer superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{\pi}Q_{n}\left(\cos\theta;% a,b\,|\,q\right)Q_{m}\left(\cos\theta;a,b\,|\,q\right)\*{\left|\frac{\left(e^{% 2i\theta};q\right)_{\infty}}{\left(ae^{i\theta},be^{i\theta};q\right)_{\infty}% }\right|^{2}}\mathrm{d}\theta=\frac{\delta_{n,m}}{\left(q^{n+1},abq^{n};q% \right)_{\infty}},}}
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    constraint
    a , b ∈ ℝ 𝑎 𝑏 {\displaystyle{\displaystyle a,b\in\mathbb{R}}}
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    a = b ¯ 𝑎 𝑏 {\displaystyle{\displaystyle a=\overline{b}}}
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    a = b ¯ 𝑎 ¯ 𝑏 {\displaystyle{\displaystyle a=\overline{b}}}
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    | a ⁢ b | < 1 𝑎 𝑏 1 {\displaystyle{\displaystyle|ab|<1}}
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    | a | , | b | ≤ 1 𝑎 𝑏 1 {\displaystyle{\displaystyle|a|,|b|\leq 1}}
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    Symbols used
    Al-Salam–Chihara polynomial
    DLMF defining formula
    Q n ⁡ ( x ; a , b | q ) AlSalam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle Q_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\,|\,% \NVar{q}\right)}}
    xml-id
    C18.S28.E7.m2aadec
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    Q10742
    DLMF defining formula
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5abdec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    complex conjugate
    DLMF defining formula
    z ¯ 𝑧 {\displaystyle{\displaystyle\overline{\NVar{z}}}}
    xml-id
    C1.S9.E11.m2adec
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    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    Q10758
    DLMF defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q11269
    DLMF 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.m2aedec
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    Q11684
    DLMF 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.m6addec
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    Q10786
    DLMF defining formula
    ℝ {\displaystyle{\displaystyle\mathbb{R}}}
    xml-id
    introduction.Sx4.p2.t1.r14.m2adec
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    real variable
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1edec
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    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1bdec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1edec
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    Identifiers

     
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