DLMF:18.10.E1

MaRDI QIDQ6322

• Label: DLMF:18.10.E1

Digital Library of Mathematical Functions ID 18.10.E1

${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\mathrm{d}\phi,}}$

Constraint(s)

${\displaystyle{\displaystyle\alpha>-\tfrac{1}{2}}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Jacobi polynomial
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)}}$: ultraspherical (or Gegenbauer) polynomial
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

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