DLMF:18.27.E16 (Q5967)

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DLMF:18.27.E16
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ L n ( α ) ⁡ ( x ; q ) ⁢ L m ( α ) ⁡ ( x ; q ) ⁢ x α ( - x ; q ) ∞ ⁢ d x = ( q α + 1 ; q ) n ( q ; q ) n ⁢ q n ⁢ h 0 ( 1 ) ⁢ δ n , m , superscript subscript 0 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 q-Laguerre-polynomial-L 𝛼 𝑚 𝑥 𝑞 superscript 𝑥 𝛼 q-Pochhammer-symbol 𝑥 𝑞 𝑥 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 𝑛 superscript subscript ℎ 0 1 Kronecker 𝑛 𝑚 {\displaystyle{\displaystyle\int_{0}^{\infty}L^{(\alpha)}_{n}\left(x;q\right)L% ^{(\alpha)}_{m}\left(x;q\right)\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}% \mathrm{d}x=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}h% _{0}^{(1)}\delta_{n,m},}}
    0 references
    DLMF id
    DLMF:18.27.E16
    0 references
    constraint
    α > - 1 𝛼 1 {\displaystyle{\displaystyle\alpha>-1}}
    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.m5acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    $$q$$ -Laguerre polynomial
    Defining formula
    L n ( α ) ⁡ ( x ; q ) q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};% \NVar{q}\right)}}
    xml-id
    C18.S27.E15.m2aadec
    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.m2abdec
    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.m1cdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1kdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1kdec
    0 references
    DLMF:18.2.E5
    Defining formula
    h n subscript ℎ 𝑛 {\displaystyle{\displaystyle h_{n}}}
    xml-id
    C18.S2.XMD4.m1cdec
    0 references
     
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