DLMF:18.17.E48 (Q5789)

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DLMF:18.17.E48
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    Statements

    Defining formula
    ∫ - ∞ ∞ H m ⁡ ( y ) ⁢ e - y 2 ⁢ H n ⁡ ( x - y ) ⁢ e - ( x - y ) 2 ⁢ d y = π 1 2 ⁢ 2 - 1 2 ⁢ ( m + n + 1 ) ⁢ H m + n ⁡ ( 2 - 1 2 ⁢ x ) ⁢ e - 1 2 ⁢ x 2 . superscript subscript Hermite-polynomial-H 𝑚 𝑦 superscript 𝑒 superscript 𝑦 2 Hermite-polynomial-H 𝑛 𝑥 𝑦 superscript 𝑒 superscript 𝑥 𝑦 2 𝑦 superscript 𝜋 1 2 superscript 2 1 2 𝑚 𝑛 1 Hermite-polynomial-H 𝑚 𝑛 superscript 2 1 2 𝑥 superscript 𝑒 1 2 superscript 𝑥 2 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}H_{m}\left(y\right)e^{-y^{2% }}H_{n}\left(x-y\right)e^{-(x-y)^{2}}\mathrm{d}y=\pi^{\frac{1}{2}}2^{-\frac{1}% {2}(m+n+1)}H_{m+n}\left(2^{-\frac{1}{2}}x\right)e^{-\frac{1}{2}x^{2}}.}}
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    DLMF id
    DLMF:18.17.E48
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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.m2addec
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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.m2avdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaudec
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