Bounds for extreme zeros of some classical orthogonal polynomials

TL;DR

This paper establishes improved bounds for the extreme zeros of Laguerre, Jacobi, and Gegenbauer polynomials using recurrence relations and zero interlacing properties, enhancing existing bounds for these classical orthogonal polynomials.

Contribution

It introduces a novel method leveraging mixed three-term recurrence relations to derive tighter bounds for the extreme zeros of these polynomials.

Findings

Bounds are tighter than previous estimates for Jacobi and Gegenbauer polynomials.

Numerical evidence shows improved bounds for the largest and smallest zeros.

Interlacing properties restrict the possible locations of common zeros.

Abstract

We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different parameter(s) within the same classical family. We prove that interlacing properties of the zeros impose restrictions on the possible location of common zeros of the polynomials involved and deduce strict bounds for the extreme zeros of polynomials belonging to each of these three classical families. We show numerically that the bounds generated by our method improve known lower (upper) bounds for the largest (smallest) zeros of polynomials in these families, notably in the case of Jacobi and Gegenbauer polynomials.