DLMF:18.18.E8 (Q5798)

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DLMF:18.18.E8
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    Statements

    Defining formula
    C n ( λ ) ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) = ∑ ℓ = 0 n 2 2 ⁢ ℓ ⁢ ( n - ℓ ) ! ⁢ 2 ⁢ λ + 2 ⁢ ℓ - 1 2 ⁢ λ - 1 ⁢ ( ( λ ) ℓ ) 2 ( 2 ⁢ λ ) n + ℓ ⁢ ( sin ⁡ θ 1 ) ℓ ⁢ C n - ℓ ( λ + ℓ ) ⁡ ( cos ⁡ θ 1 ) ⁢ ( sin ⁡ θ 2 ) ℓ ⁢ C n - ℓ ( λ + ℓ ) ⁡ ( cos ⁡ θ 2 ) ⁢ C ℓ ( λ - 1 2 ) ⁡ ( cos ⁡ ϕ ) , ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ superscript subscript ℓ 0 𝑛 superscript 2 2 ℓ 𝑛 ℓ 2 𝜆 2 ℓ 1 2 𝜆 1 superscript Pochhammer 𝜆 ℓ 2 Pochhammer 2 𝜆 𝑛 ℓ superscript subscript 𝜃 1 ℓ ultraspherical-Gegenbauer-polynomial 𝜆 ℓ 𝑛 ℓ subscript 𝜃 1 superscript subscript 𝜃 2 ℓ ultraspherical-Gegenbauer-polynomial 𝜆 ℓ 𝑛 ℓ subscript 𝜃 2 ultraspherical-Gegenbauer-polynomial 𝜆 1 2 ℓ italic-ϕ {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta_{1}\cos\theta_{% 2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\sum_{\ell=0}^{n}2^{2\ell}(n-% \ell)!\frac{2\lambda+2\ell-1}{2\lambda-1}\frac{({\left(\lambda\right)_{\ell}})% ^{2}}{{\left(2\lambda\right)_{n+\ell}}}(\sin\theta_{1})^{\ell}C^{(\lambda+\ell% )}_{n-\ell}\left(\cos\theta_{1}\right)(\sin\theta_{2})^{\ell}C^{(\lambda+\ell)% }_{n-\ell}\left(\cos\theta_{2}\right)C^{(\lambda-\frac{1}{2})}_{\ell}\left(% \cos\phi\right),}}
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    DLMF id
    DLMF:18.18.E8
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    constraint
    λ ≠ 1 2 𝜆 1 2 {\displaystyle{\displaystyle\lambda\neq\frac{1}{2}}}
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    Symbols used
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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