DLMF:18.15.E1

MaRDI QIDQ6388

• Label: DLMF:18.15.E1

Digital Library of Mathematical Functions ID 18.15.E1

${\displaystyle{\displaystyle\left(\sin\tfrac{1}{2}\theta\right)^{\alpha+\frac{1}{2}}\left(\cos\tfrac{1}{2}\theta\right)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)=\pi^{-1}2^{2n+\alpha+\beta+1}\mathrm{B}\left(n+\alpha+1,n+\beta+1\right)\*\left(\sum_{m=0}^{M-1}\frac{f_{m}(\theta)}{2^{m}{\left(2n+\alpha+\beta+2\right)_{m}}}+O\left(n^{-M}\right)\right),}}$

Symbols List

• ${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$: order not exceeding
• ${\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\right)}}$: beta function
• ${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Jacobi polynomial
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle f_{m}(\theta)}}$: 18.15.E2
 

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