Distances of roots of classical orthogonal polynomials

TL;DR

This paper provides lower bounds on the minimal distances between roots of classical orthogonal polynomials and their boundary distances, using eigenvalue results from random matrix theory.

Contribution

It introduces new estimates for root distances of Hermite, Laguerre, and Jacobi polynomials based on eigenvalue analysis from random matrix ensembles.

Findings

Lower bounds for root spacing of classical orthogonal polynomials.

Estimates for root distances from orthogonality interval boundaries.

Application of random matrix eigenvalue results to polynomial root analysis.

Abstract

Let $(P_N)_{N\ge0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots, z_{N,N}$ 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.