Jacobi Polynomials - Asymptotics of Jacobi Polynomials

Asymptotics of Jacobi Polynomials

For x in the interior of, the asymptotics of Pn(α,β) for large n is given by the Darboux formula

where

\begin{align} k(\theta) &= \pi^{-1/2} \sin^{-\alpha-1/2} \frac{\theta}{2} \cos^{-\beta-1/2} \frac{\theta}{2}~,\\ N &= n + \frac{\alpha+\beta+1}{2}~,\\ \gamma &= - (\alpha + \frac{1}{2}) \frac{\pi}{2}~, \end{align}

and the "O" term is uniform on the interval for every ε>0.

The asymptotics of the Jacobi polynomials near the points ±1 is given by the Mehler–Heine formula

\begin{align} \lim_{n \to \infty} n^{-\alpha}P_n^{(\alpha,\beta)}\left(\cos \frac{z}{n}\right) &= \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z)~,\\ \lim_{n \to \infty} n^{-\beta}P_n^{(\alpha,\beta)}\left(\cos \left \right) &= \left(\frac{z}{2}\right)^{-\beta} J_\beta(z)~, \end{align}

where the limits are uniform for z in a bounded domain.

The asymptotics outside is less explicit.

Read more about this topic:  Jacobi Polynomials

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