DLMF:18.17.E26 (Q5767)

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DLMF:18.17.E26
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    Statements

    Defining formula
    ∫ 0 ∞ e - 1 2 ⁢ x 2 ⁢ 𝐻𝑒 n ⁡ ( x ) ⁢ 𝐻𝑒 n + 2 ⁢ m + 1 ⁡ ( x ) ⁢ sin ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) m ⁢ 1 2 ⁢ π ⁢ n ! ⁢ y 2 ⁢ m + 1 ⁢ e - 1 2 ⁢ y 2 ⁢ L n ( 2 ⁢ m + 1 ) ⁡ ( y 2 ) . superscript subscript 0 superscript 𝑒 1 2 superscript 𝑥 2 Hermite-polynomial-He 𝑛 𝑥 Hermite-polynomial-He 𝑛 2 𝑚 1 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑚 1 2 𝜋 𝑛 superscript 𝑦 2 𝑚 1 superscript 𝑒 1 2 superscript 𝑦 2 Laguerre-polynomial-L 2 𝑚 1 𝑛 superscript 𝑦 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_% {n}\left(x\right)\mathit{He}_{n+2m+1}\left(x\right)\sin\left(xy\right)\mathrm{% d}x=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}n!\,y^{2m+1}e^{-\frac{1}{2}y^{2}}L^{(2m+1)}_% {n}\left(y^{2}\right).}}
    0 references
    DLMF id
    DLMF:18.17.E26
    0 references
    Symbols used
    Q11525
    Defining formula
    𝐻𝑒 n ⁡ ( x ) Hermite-polynomial-He 𝑛 𝑥 {\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r14.m2addec
    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.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aydec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    0 references
     
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