DLMF:18.18.E13 (Q5803)

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DLMF:18.18.E13
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    Statements

    Defining formula
    H n ⁡ ( λ ⁢ x ) = λ n ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - n ) 2 ⁢ ℓ ℓ ! ⁢ ( 1 - λ - 2 ) ℓ ⁢ H n - 2 ⁢ ℓ ⁡ ( x ) . Hermite-polynomial-H 𝑛 𝜆 𝑥 superscript 𝜆 𝑛 superscript subscript ℓ 0 𝑛 2 Pochhammer 𝑛 2 ℓ ℓ superscript 1 superscript 𝜆 2 ℓ Hermite-polynomial-H 𝑛 2 ℓ 𝑥 {\displaystyle{\displaystyle H_{n}\left(\lambda x\right)=\lambda^{n}\sum_{\ell% =0}^{\left\lfloor n/2\right\rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}(1-% \lambda^{-2})^{\ell}H_{n-2\ell}\left(x\right).}}
    0 references
    DLMF id
    DLMF:18.18.E13
    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.m2acdec
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    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5afdec
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    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4adec
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    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1cdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ldec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1idec
    0 references
     
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