DLMF:18.28.E17 (Q5992): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1addec

Revision as of 13:24, 2 January 2020

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DLMF:18.28.E17
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ 0 π H n ⁡ ( cos ⁡ θ | q ) ⁢ H m ⁡ ( cos ⁡ θ | q ) ⁢ | ( e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 ⁢ d θ = δ n , m ( q n + 1 ; q ) ∞ . 1 2 𝜋 superscript subscript 0 𝜋 continuous-q-Hermite-polynomial-H 𝑛 𝜃 𝑞 continuous-q-Hermite-polynomial-H 𝑚 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑒 2 imaginary-unit 𝜃 𝑞 2 𝜃 Kronecker 𝑛 𝑚 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑞 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{\pi}H_{n}\left(\cos\theta% \,|\,q\right)H_{m}\left(\cos\theta\,|\,q\right){\left|\left(e^{2\mathrm{i}% \theta};q\right)_{\infty}\right|^{2}}\mathrm{d}\theta=\frac{\delta_{n,m}}{% \left(q^{n+1};q\right)_{\infty}}.}}
    0 references
    DLMF id
    DLMF:18.28.E17
    0 references
    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.m5aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    continuous $$q$$ -Hermite polynomial
    Defining formula
    H n ⁡ ( x | q ) continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\,|\,\NVar{q}\right)}}
    xml-id
    C18.S28.E16.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
     
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