DLMF:18.17.E40 (Q5781)

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DLMF:18.17.E40
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    Statements

    Defining formula
    ∫ 0 ∞ e - a ⁢ x ⁢ L n ( α ) ⁡ ( b ⁢ x ) ⁢ x z - 1 ⁢ d x = Γ ⁡ ( z + n ) n ! ⁢ ( a - b ) n ⁢ a - n - z ⁢ F 1 2 ⁡ ( - n , 1 + α - z 1 - n - z ; a a - b ) , superscript subscript 0 superscript 𝑒 𝑎 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑏 𝑥 superscript 𝑥 𝑧 1 𝑥 Euler-Gamma 𝑧 𝑛 𝑛 superscript 𝑎 𝑏 𝑛 superscript 𝑎 𝑛 𝑧 Gauss-hypergeometric-F-as-2F1 𝑛 1 𝛼 𝑧 1 𝑛 𝑧 𝑎 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}L^{(\alpha)}_{n}\left(bx% \right)x^{z-1}\mathrm{d}x=\frac{\Gamma\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{-n-% z}\*{{}_{2}F_{1}}\left({-n,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right),}}
    0 references
    DLMF id
    DLMF:18.17.E40
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2apdec
    0 references
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5adec
    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.m2aidec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aamdec
    0 references
     
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