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

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Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3addec
Property / Symbols used
 
Property / Symbols used: Q11298 / rank
 
Normal rank
Property / Symbols used: Q11298 / qualifier
 
Defining formula:

( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}

\qPochhammer{\NVar{a}}{\NVar{q}}{\NVar{n}}
Property / Symbols used: Q11298 / qualifier
 
xml-id: C17.S2.SS1.p1.m2aldec
Property / Symbols used
 
Property / Symbols used: Q11728 / rank
 
Normal rank
Property / Symbols used: Q11728 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11728 / qualifier
 
xml-id: C18.S1.XMD3.m1ndec
Property / Symbols used
 
Property / Symbols used: Q11730 / rank
 
Normal rank
Property / Symbols used: Q11730 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11730 / qualifier
 
xml-id: C18.S1.XMD5.m1edec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1ldec

Latest revision as of 13:24, 2 January 2020

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DLMF:18.28.E17
No description defined

    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
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aidec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    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.m2aldec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1ndec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1edec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ldec
    0 references
     
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