DLMF:18.18.E40 (Q5830)

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DLMF:18.18.E40
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    Statements

    Defining formula
    ∑ ℓ = 0 n ( n ℓ ) ⁢ H 2 ⁢ ℓ ⁡ ( x ) ⁢ H 2 ⁢ n - 2 ⁢ ℓ ⁡ ( y ) = ( - 1 ) n ⁢ 2 2 ⁢ n ⁢ n ! ⁢ L n ⁡ ( x 2 + y 2 ) . superscript subscript ℓ 0 𝑛 binomial 𝑛 ℓ Hermite-polynomial-H 2 ℓ 𝑥 Hermite-polynomial-H 2 𝑛 2 ℓ 𝑦 superscript 1 𝑛 superscript 2 2 𝑛 𝑛 shorthand-Laguerre-polynomial-L 𝑛 superscript 𝑥 2 superscript 𝑦 2 {\displaystyle{\displaystyle\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}H% _{2\ell}\left(x\right)H_{2n-2\ell}\left(y\right)=(-1)^{n}2^{2n}n!L_{n}\left(x^% {2}+y^{2}\right).}}
    0 references
    DLMF id
    DLMF:18.18.E40
    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.m2ahdec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1addec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5apdec
    0 references
    Q10921
    Defining formula
    L n ⁡ ( x ) = L n ( 0 ) ⁡ ( x ) shorthand-Laguerre-polynomial-L 𝑛 𝑥 Laguerre-polynomial-L 0 𝑛 𝑥 {\displaystyle{\displaystyle L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}% \left(x\right)}}
    xml-id
    C18.S1.I1.ix7.p1.m3adec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1edec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1aadec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1amdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1aidec
    0 references
     
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