DLMF:18.17.E49 (Q5790)

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DLMF:18.17.E49
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    Statements

    Defining formula
    ∫ - ∞ ∞ H ℓ ⁡ ( x ) ⁢ H m ⁡ ( x ) ⁢ H n ⁡ ( x ) ⁢ e - x 2 ⁢ d x = 2 1 2 ⁢ ( ℓ + m + n ) ⁢ ℓ ! ⁢ m ! ⁢ n ! ⁢ π ( 1 2 ⁢ ℓ + 1 2 ⁢ m - 1 2 ⁢ n ) ! ⁢ ( 1 2 ⁢ m + 1 2 ⁢ n - 1 2 ⁢ ℓ ) ! ⁢ ( 1 2 ⁢ n + 1 2 ⁢ ℓ - 1 2 ⁢ m ) ! , superscript subscript Hermite-polynomial-H ℓ 𝑥 Hermite-polynomial-H 𝑚 𝑥 Hermite-polynomial-H 𝑛 𝑥 superscript 𝑒 superscript 𝑥 2 𝑥 superscript 2 1 2 ℓ 𝑚 𝑛 ℓ 𝑚 𝑛 𝜋 1 2 ℓ 1 2 𝑚 1 2 𝑛 1 2 𝑚 1 2 𝑛 1 2 ℓ 1 2 𝑛 1 2 ℓ 1 2 𝑚 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}H_{\ell}\left(x\right)H_{m}% \left(x\right)H_{n}\left(x\right)e^{-x^{2}}\mathrm{d}x=\frac{2^{\frac{1}{2}(% \ell+m+n)}\ell\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-% \tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(% \tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!},}}
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    DLMF id
    DLMF:18.17.E49
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    Symbols used
    Q10922
    Defining formula
    H n ⁡ ( x ) Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r13.m2aedec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2awdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aavdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2avdec
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