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DLMF:18.18.E11 - MaRDI portal

DLMF:18.18.E11 (Q6498)

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DLMF:18.18.E11
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    Statements

    DLMF defining formula
    ( a 1 2 + … + a r 2 ) 1 2 ⁢ n n ! ⁢ H n ⁡ ( a 1 ⁢ x 1 + ⋯ + a r ⁢ x r ( a 1 2 + ⋯ + a r 2 ) 1 2 ) = ∑ m 1 + ⋯ + m r = n a 1 m 1 ⁢ ⋯ ⁢ a r m r m 1 ! ⁢ ⋯ ⁢ m r ! ⁢ H m 1 ⁡ ( x 1 ) ⁢ ⋯ ⁢ H m r ⁡ ( x r ) . superscript superscript subscript 𝑎 1 2 … superscript subscript 𝑎 𝑟 2 1 2 𝑛 𝑛 Hermite-polynomial-H 𝑛 subscript 𝑎 1 subscript 𝑥 1 ⋯ subscript 𝑎 𝑟 subscript 𝑥 𝑟 superscript superscript subscript 𝑎 1 2 ⋯ superscript subscript 𝑎 𝑟 2 1 2 subscript subscript 𝑚 1 ⋯ subscript 𝑚 𝑟 𝑛 superscript subscript 𝑎 1 subscript 𝑚 1 ⋯ superscript subscript 𝑎 𝑟 subscript 𝑚 𝑟 subscript 𝑚 1 ⋯ subscript 𝑚 𝑟 Hermite-polynomial-H subscript 𝑚 1 subscript 𝑥 1 ⋯ Hermite-polynomial-H subscript 𝑚 𝑟 subscript 𝑥 𝑟 {\displaystyle{\displaystyle\frac{(a_{1}^{2}+\dots+a_{r}^{2})^{\frac{1}{2}n}}{% n!}H_{n}\left(\frac{a_{1}x_{1}+\cdots+a_{r}x_{r}}{(a_{1}^{2}+\cdots+a_{r}^{2})% ^{\frac{1}{2}}}\right)=\sum_{m_{1}+\cdots+m_{r}=n}\frac{a_{1}^{m_{1}}\cdots a_% {r}^{m_{r}}}{m_{1}!\cdots m_{r}!}H_{m_{1}}\left(x_{1}\right)\cdots H_{m_{r}}% \left(x_{r}\right).}}
    0 references
    Symbols used
    Hermite polynomial
    DLMF 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.m2abdec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aedec
    0 references
    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1adec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1jdec
    0 references
    18.18.E1
    DLMF defining formula
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C18.S18.XMD3.m1cdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1gdec
    0 references

    Identifiers

     
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