DLMF:18.10.E1 (Q5625)

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DLMF:18.10.E1
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    Statements

    Defining formula
    P n ( α , α ) ⁡ ( cos ⁡ θ ) P n ( α , α ) ⁡ ( 1 ) = C n ( α + 1 2 ) ⁡ ( cos ⁡ θ ) C n ( α + 1 2 ) ⁡ ( 1 ) = 2 α + 1 2 ⁢ Γ ⁡ ( α + 1 ) π 1 2 ⁢ Γ ⁡ ( α + 1 2 ) ⁢ ( sin ⁡ θ ) - 2 ⁢ α ⁢ ∫ 0 θ cos ⁡ ( ( n + α + 1 2 ) ⁢ ϕ ) ( cos ⁡ ϕ - cos ⁡ θ ) - α + 1 2 ⁢ d ϕ , Jacobi-polynomial-P 𝛼 𝛼 𝑛 𝜃 Jacobi-polynomial-P 𝛼 𝛼 𝑛 1 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 𝑛 𝜃 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 𝑛 1 superscript 2 𝛼 1 2 Euler-Gamma 𝛼 1 superscript 𝜋 1 2 Euler-Gamma 𝛼 1 2 superscript 𝜃 2 𝛼 superscript subscript 0 𝜃 𝑛 𝛼 1 2 italic-ϕ superscript italic-ϕ 𝜃 𝛼 1 2 italic-ϕ {\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta% \right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}% _{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac% {2^{\alpha+\frac{1}{2}}\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma% \left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{% \cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+% \frac{1}{2}}}\mathrm{d}\phi,}}
    0 references
    DLMF id
    DLMF:18.10.E1
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    constraint
    α > - 1 2 𝛼 1 2 {\displaystyle{\displaystyle\alpha>-\tfrac{1}{2}}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    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.m2adec
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