Distances of roots of classical orthogonal polynomials

Abstract

Let $(P_N{)}_{N\ge 0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\dots ,z_{N,N}$ of $P_N$ we derive lower estimates for ${\min }_{i\ne j}|z_{i,N}-z_{j,N}|$ and the distances from the boundary of the orthogonality intervals. The proofs are based on recent results on the eigenvalues of the covariance matrices in central limit theorems for associated $\beta$-random matrix ensembles where these entities appear as entries, and where the eigenvalues of these matrices are known.

In the revised version some estimates were improved, and the comments on the existing literature were extended

References in corpus (2)

• Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices
• Limit theorems for Jacobi ensembles with large parameters