DLMF:18.17.E14 (Q5755)

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DLMF:18.17.E14
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    Statements

    Defining formula
    x α + μ ⁢ L n ( α + μ ) ⁡ ( x ) Γ ⁡ ( α + μ + n + 1 ) = ∫ 0 x y α ⁢ L n ( α ) ⁡ ( y ) Γ ⁡ ( α + n + 1 ) ⁢ ( x - y ) μ - 1 Γ ⁡ ( μ ) ⁢ d y , superscript 𝑥 𝛼 𝜇 Laguerre-polynomial-L 𝛼 𝜇 𝑛 𝑥 Euler-Gamma 𝛼 𝜇 𝑛 1 superscript subscript 0 𝑥 superscript 𝑦 𝛼 Laguerre-polynomial-L 𝛼 𝑛 𝑦 Euler-Gamma 𝛼 𝑛 1 superscript 𝑥 𝑦 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{x^{\alpha+\mu}L^{(\alpha+\mu)}_{n}\left(x% \right)}{\Gamma\left(\alpha+\mu+n+1\right)}=\int_{0}^{x}\frac{y^{\alpha}L^{(% \alpha)}_{n}\left(y\right)}{\Gamma\left(\alpha+n+1\right)}\frac{(x-y)^{\mu-1}}% {\Gamma\left(\mu\right)}\mathrm{d}y,}}
    0 references
    DLMF id
    DLMF:18.17.E14
    0 references
    constraint
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    Q11333
    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.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1idec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1mdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1kdec
    0 references
     
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