DLMF:18.17.E49 (Q5790)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:18.17.E49
No description defined

    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\,)\,!},}}
    0 references
    DLMF id
    DLMF:18.17.E49
    0 references
    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
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2awdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aavdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2avdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5andec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aatdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1dec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1fdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1avdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1atdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5790&oldid=30695"