DLMF:18.17.E47 (Q5788)

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DLMF:18.17.E47
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    Statements

    Defining formula
    ∫ 0 x t α ⁢ L m ( α ) ⁡ ( t ) L m ( α ) ⁡ ( 0 ) ⁢ ( x - t ) β ⁢ L n ( β ) ⁡ ( x - t ) L n ( β ) ⁡ ( 0 ) ⁢ d t = Γ ⁡ ( α + 1 ) ⁢ Γ ⁡ ( β + 1 ) Γ ⁡ ( α + β + 2 ) ⁢ x α + β + 1 ⁢ L m + n ( α + β + 1 ) ⁡ ( x ) L m + n ( α + β + 1 ) ⁡ ( 0 ) . superscript subscript 0 𝑥 superscript 𝑡 𝛼 Laguerre-polynomial-L 𝛼 𝑚 𝑡 Laguerre-polynomial-L 𝛼 𝑚 0 superscript 𝑥 𝑡 𝛽 Laguerre-polynomial-L 𝛽 𝑛 𝑥 𝑡 Laguerre-polynomial-L 𝛽 𝑛 0 𝑡 Euler-Gamma 𝛼 1 Euler-Gamma 𝛽 1 Euler-Gamma 𝛼 𝛽 2 superscript 𝑥 𝛼 𝛽 1 Laguerre-polynomial-L 𝛼 𝛽 1 𝑚 𝑛 𝑥 Laguerre-polynomial-L 𝛼 𝛽 1 𝑚 𝑛 0 {\displaystyle{\displaystyle\int_{0}^{x}t^{\alpha}\frac{L^{(\alpha)}_{m}\left(% t\right)}{L^{(\alpha)}_{m}\left(0\right)}(x-t)^{\beta}\frac{L^{(\beta)}_{n}% \left(x-t\right)}{L^{(\beta)}_{n}\left(0\right)}\mathrm{d}t=\frac{\Gamma\left(% \alpha+1\right)\Gamma\left(\beta+1\right)}{\Gamma\left(\alpha+\beta+2\right)}x% ^{\alpha+\beta+1}\frac{L^{(\alpha+\beta+1)}_{m+n}\left(x\right)}{L^{(\alpha+% \beta+1)}_{m+n}\left(0\right)}.}}
    0 references
    DLMF id
    DLMF:18.17.E47
    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.m2ardec
    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.m2ajdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aatdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aardec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1ddec
    0 references
     
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