DLMF:18.5.E11 (Q5519)

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DLMF:18.5.E11
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    Statements

    Defining formula
    C n ( λ ) ⁡ ( cos ⁡ θ ) = ∑ ℓ = 0 n ( λ ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ cos ⁡ ( ( n - 2 ⁢ ℓ ) ⁢ θ ) = e i ⁢ n ⁢ θ ⁢ ( λ ) n n ! ⁢ F 1 2 ⁡ ( - n , λ 1 - λ - n ; e - 2 ⁢ i ⁢ θ ) . ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 ℓ 𝑛 2 ℓ 𝜃 superscript 𝑒 imaginary-unit 𝑛 𝜃 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝜆 1 𝜆 𝑛 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{% \ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}% {\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)=e^{\mathrm{i}n\theta}\frac{% {\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n% };e^{-2\mathrm{i}\theta}\right).}}
    0 references
    DLMF id
    DLMF:18.5.E11
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5addec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aedec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5afdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11475
    Defining formula
    C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r3.m2acdec
    0 references
    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.m1kdec
    0 references
     
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