DLMF:18.17.E41 (Q5782)

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DLMF:18.17.E41
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    Statements

    test
    ∫ 0 ∞ e - a ⁢ x ⁢ 𝐻𝑒 n ⁡ ( x ) ⁢ x z - 1 ⁢ d x = Γ ⁡ ( z + n ) ⁢ a - n - 2 ⁢ F 2 2 ⁡ ( - 1 2 ⁢ n , - 1 2 ⁢ n + 1 2 - 1 2 ⁢ z - 1 2 ⁢ n , - 1 2 ⁢ z - 1 2 ⁢ n + 1 2 ; - 1 2 ⁢ a 2 ) , superscript subscript 0 superscript 𝑒 𝑎 𝑥 Hermite-polynomial-He 𝑛 𝑥 superscript 𝑥 𝑧 1 𝑥 Euler-Gamma 𝑧 𝑛 superscript 𝑎 𝑛 2 Gauss-hypergeometric-pFq 2 2 1 2 𝑛 1 2 𝑛 1 2 1 2 𝑧 1 2 𝑛 1 2 𝑧 1 2 𝑛 1 2 1 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\mathit{He}_{n}\left(x% \right)x^{z-1}\mathrm{d}x=\Gamma\left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-% \tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-% \tfrac{1}{2}z-\tfrac{1}{2}n+\tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),}}
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    DLMF id
    DLMF:18.17.E41
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    ℜ ⁡ z > - 1 𝑧 1 {\displaystyle{\displaystyle\Re z>-1}}
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    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    n 𝑛 {\displaystyle{\displaystyle n}}
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    ℜ ⁡ z > - 1 𝑧 1 {\displaystyle{\displaystyle\Re z>-1}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aqdec
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    Q11125
    test
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1adec
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    Q11525
    test
    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.m2ahdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aandec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2atdec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aandec
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    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
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    Q11750
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1hdec
    0 references
     
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