Positivity of Tur\'an determinants for orthogonal polynomials II

TL;DR

This paper provides a general criterion for orthogonal polynomials to satisfy Turán's inequality, extending previous results and applicable to many classes via their recurrence relations.

Contribution

It introduces a new, broad criterion for Turán's inequality in orthogonal polynomials, expanding the scope of previous findings.

Findings

Provides a unified criterion for Turán's inequality

Extends previous results to more polynomial classes

Applicable to various orthogonal polynomial families

Abstract

The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n(1)=1,$ satisfy Tur\'an's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Tur\'an's inequality. This extends essentially the results of \cite{szw}. In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.