DLMF:18.18.E20 (Q5810)

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DLMF:18.18.E20
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    Statements

    Defining formula
    ( 2 ⁢ x ) n = ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - n ) 2 ⁢ ℓ ℓ ! ⁢ H n - 2 ⁢ ℓ ⁡ ( x ) . superscript 2 𝑥 𝑛 superscript subscript ℓ 0 𝑛 2 Pochhammer 𝑛 2 ℓ ℓ Hermite-polynomial-H 𝑛 2 ℓ 𝑥 {\displaystyle{\displaystyle(2x)^{n}=\sum_{\ell=0}^{\left\lfloor n/2\right% \rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}H_{n-2\ell}\left(x\right).}}
    0 references
    DLMF id
    DLMF:18.18.E20
    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.m2addec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1ahdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5akdec
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    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4acdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1jdec
    0 references
     
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