Positivity of Tur\'an determinants for orthogonal polynomials

TL;DR

This paper establishes general criteria for when orthogonal polynomials satisfy Turán's inequality, extending known results to new polynomial families like the q-ultraspherical polynomials.

Contribution

It provides new theoretical criteria for Turán's inequality in orthogonal polynomials, including novel results for q-ultraspherical polynomials.

Findings

Criteria for Turán's inequality in orthogonal polynomials

Extension of known results to new polynomial families

Application to q-ultraspherical polynomials

Abstract

The orthogonal polynomials $p_n$ 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 general criteria for orthogonal polynomials to satisfy Tur\'an's inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the $q$--ultraspherical polynomials.