DLMF:18.18.E9 (Q5799)

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DLMF:18.18.E9
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    Statements

    Defining formula
    P n ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) = P n ⁡ ( cos ⁡ θ 1 ) ⁢ P n ⁡ ( cos ⁡ θ 2 ) + 2 ⁢ ∑ ℓ = 1 n ( n - ℓ ) ! ⁢ ( n + ℓ ) ! 2 2 ⁢ ℓ ⁢ ( n ! ) 2 ⁢ ( sin ⁡ θ 1 ) ℓ ⁢ P n - ℓ ( ℓ , ℓ ) ⁡ ( cos ⁡ θ 1 ) ⁢ ( sin ⁡ θ 2 ) ℓ ⁢ P n - ℓ ( ℓ , ℓ ) ⁡ ( cos ⁡ θ 2 ) ⁢ cos ⁡ ( ℓ ⁢ ϕ ) . Legendre-spherical-polynomial 𝑛 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ Legendre-spherical-polynomial 𝑛 subscript 𝜃 1 Legendre-spherical-polynomial 𝑛 subscript 𝜃 2 2 superscript subscript ℓ 1 𝑛 𝑛 ℓ 𝑛 ℓ superscript 2 2 ℓ superscript 𝑛 2 superscript subscript 𝜃 1 ℓ Jacobi-polynomial-P ℓ ℓ 𝑛 ℓ subscript 𝜃 1 superscript subscript 𝜃 2 ℓ Jacobi-polynomial-P ℓ ℓ 𝑛 ℓ subscript 𝜃 2 ℓ italic-ϕ {\displaystyle{\displaystyle P_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin% \theta_{1}\sin\theta_{2}\cos\phi\right)={P_{n}\left(\cos\theta_{1}\right)P_{n}% \left(\cos\theta_{2}\right)+2\sum_{\ell=1}^{n}\frac{(n-\ell)!\;(n+\ell)!}{2^{2% \ell}(n!)^{2}}(\sin\theta_{1})^{\ell}P^{(\ell,\ell)}_{n-\ell}\left(\cos\theta_% {1}\right)(\sin\theta_{2})^{\ell}P^{(\ell,\ell)}_{n-\ell}\left(\cos\theta_{2}% \right)\cos\left(\ell\phi\right)}.}}
    0 references
    DLMF id
    DLMF:18.18.E9
    0 references
    Symbols used
    Q11660
    Defining formula
    P n ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r2.m2abdec
    0 references
    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1hdec
    0 references
     
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