# DLMF:18.10.E1 (Q5625)

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DLMF:18.10.E1
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## Statements

${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta% \right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}% _{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac% {2^{\alpha+\frac{1}{2}}\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma% \left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{% \cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+% \frac{1}{2}}}\mathrm{d}\phi,}}$
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DLMF:18.10.E1
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${\displaystyle{\displaystyle\alpha>-\tfrac{1}{2}}}$
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${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$
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${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}$
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${\displaystyle{\displaystyle\pi}}$
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${\displaystyle{\displaystyle\cos\NVar{z}}}$
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${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$
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${\displaystyle{\displaystyle\int}}$
${\displaystyle{\displaystyle\sin\NVar{z}}}$
${\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}$
${\displaystyle{\displaystyle n}}$