DLMF:18.10.E9 (Q5633)

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DLMF:18.10.E9
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    Statements

    test
    L n ( α ) ⁡ ( x ) = e x ⁢ x - 1 2 ⁢ α n ! ⁢ ∫ 0 ∞ e - t ⁢ t n + 1 2 ⁢ α ⁢ J α ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t , Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 𝑒 𝑥 superscript 𝑥 1 2 𝛼 𝑛 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝑛 1 2 𝛼 Bessel-J 𝛼 2 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\frac{e^{x}x^{-% \frac{1}{2}\alpha}}{n!}\int_{0}^{\infty}e^{-t}t^{n+\frac{1}{2}\alpha}J_{\alpha% }\left(2\sqrt{xt}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:18.10.E9
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    constraint
    α > - 1 𝛼 1 {\displaystyle{\displaystyle\alpha>-1}}
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    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
     
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