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DLMF:18.18.E5 - MaRDI portal

DLMF:18.18.E5 (Q6492)

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DLMF:18.18.E5
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    Statements

    DLMF defining formula
    b n = n ! Γ ⁡ ( n + α + 1 ) ⁢ ∫ 0 ∞ f ⁢ ( x ) ⁢ L n ( α ) ⁡ ( x ) ⁢ e - x ⁢ x α ⁢ d x . subscript 𝑏 𝑛 𝑛 Euler-Gamma 𝑛 𝛼 1 superscript subscript 0 𝑓 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 𝑒 𝑥 superscript 𝑥 𝛼 𝑥 {\displaystyle{\displaystyle b_{n}=\frac{n!}{\Gamma\left(n+\alpha+1\right)}% \int_{0}^{\infty}f(x)L^{(\alpha)}_{n}\left(x\right)e^{-x}x^{\alpha}\mathrm{d}x% .}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    Laguerre (or generalized Laguerre) polynomial
    DLMF defining formula
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r12.m2aadec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ddec
    0 references
    analytic function
    DLMF defining formula
    f ⁢ ( z ) 𝑓 𝑧 {\displaystyle{\displaystyle f(z)}}
    xml-id
    C18.S18.XMD1.m1ddec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1cdec
    0 references

    Identifiers

     
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