DLMF:18.18.E28 (Q5818)

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DLMF:18.18.E28
No description defined

    Statements

    Defining formula
    ∑ n = 0 ∞ H n ⁡ ( x ) ⁢ H n ⁡ ( y ) 2 n ⁢ n ! ⁢ z n = ( 1 - z 2 ) - 1 2 ⁢ exp ⁡ ( 2 ⁢ x ⁢ y ⁢ z - ( x 2 + y 2 ) ⁢ z 2 1 - z 2 ) , superscript subscript 𝑛 0 Hermite-polynomial-H 𝑛 𝑥 Hermite-polynomial-H 𝑛 𝑦 superscript 2 𝑛 𝑛 superscript 𝑧 𝑛 superscript 1 superscript 𝑧 2 1 2 2 𝑥 𝑦 𝑧 superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 1 superscript 𝑧 2 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)H_{n}% \left(y\right)}{2^{n}n!}z^{n}=(1-z^{2})^{-\frac{1}{2}}\exp\left(\frac{2xyz-(x^% {2}+y^{2})z^{2}}{1-z^{2}}\right),}}
    0 references
    DLMF id
    DLMF:18.18.E28
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    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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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.m2afdec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aadec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aodec
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    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1bdec
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    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1bdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1aadec
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    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1wdec
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