DLMF:18.18.E17 (Q5807)

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DLMF:18.18.E17
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    Statements

    Defining formula
    ( 2 ⁢ x ) n = n ! ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ λ + n - 2 ⁢ ℓ λ ⁢ 1 ( λ + 1 ) n - ℓ ⁢ ℓ ! ⁢ C n - 2 ⁢ ℓ ( λ ) ⁡ ( x ) . superscript 2 𝑥 𝑛 𝑛 superscript subscript ℓ 0 𝑛 2 𝜆 𝑛 2 ℓ 𝜆 1 Pochhammer 𝜆 1 𝑛 ℓ ℓ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 2 ℓ 𝑥 {\displaystyle{\displaystyle(2x)^{n}=n!\sum_{\ell=0}^{\left\lfloor n/2\right% \rfloor}\frac{\lambda+n-2\ell}{\lambda}\frac{1}{{\left(\lambda+1\right)_{n-% \ell}}\,\ell!}C^{(\lambda)}_{n-2\ell}\left(x\right).}}
    0 references
    DLMF id
    DLMF:18.18.E17
    0 references
    Symbols used
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aedec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aidec
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