Tur\'an Inequalities for Three Term Recurrences with Monotonic Coefficients

TL;DR

This paper derives new Turán inequalities for orthogonal polynomials with monotonic recurrence coefficients and uses these to establish asymptotic bounds on their extreme zeros.

Contribution

It introduces novel Turán inequalities for a class of orthogonal polynomials and applies them to analyze zero distribution asymptotics.

Findings

New Turán inequalities for orthogonal polynomials with monotonic coefficients

Asymptotic bounds on the extreme zeros of these polynomials

Results applicable to polynomials with polynomially growing recurrence coefficients

Abstract

We establish some new Tur\'an's type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. As a corollary we deduce asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing coefficients of the three-term recurrence.