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Jacobi Polynomials are defined for fixed parameters $\alpha,\beta>-1$ by their orthogonality relation $$\int_{-1}^1 (1-x)^\alpha(1+x)^\beta J_n^{(\alpha,\beta)}(x)J_m^{(\alpha,\beta)}(x) dx = c_n^{(\alpha,\beta)}\delta_{n,m}$$ where $c$ is some irrelevant normalization. It can be shown that all $n$ roots of the polynomial $J_n^{(\alpha,\beta)}$ of degree $n$ are distinct, real, and lie inside the interval $[-1,1]$. Is there any good approximation of the location of the roots $x_k$?

Context

For the special case $\alpha=\beta=0$ (Legendre Polynomials) there are very precise formulas like $$x_k = \left(1-\frac{n-1}{8n^3} + O(n^{-4})\right)\cos\left(\frac{4k-1}{4n+2}\pi\right)$$ for $k=1,\dots,n$. And for $\alpha,\beta\in[-\frac{1}{2},\frac{1}{2}]$ there are at least some (non-overlapping) bounds on $x_k$. But what about general $\alpha,\beta>-1$? The purpose of such approximations is to use them as initial value for a Newton-Raphson method in order to find the precise roots of $J_n^{(\alpha,\beta)}$ numerically. I know the roots (which are important for certain quadrature methods) are usually computed as the eigenvalues of certain tridiagonal matrices, but I'm interested in this alternative method.

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A starting approximation I frequently use for the $k$-th root of the Jacobi polynomial $P_n^{(\alpha,\beta)}(x)$ for further polishing with Newton-Raphson or Halley is

$$x_{n,k}\approx\cos\left(\pi\frac{4k-1+2\alpha}{4n+2(\alpha+\beta+1)}\right)$$

This can be derived from the first term of the asymptotic approximation for $P_n^{(\alpha,\beta)}(\cos \theta)$.

There are other references with more precise estimates, but they (often) depend on the zeroes of Bessel functions; see e.g. Gatteschi's paper, Frenzen/Wong's paper, Baratella/Gatteschi's paper, or Szegő's (now classical) book.

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