Bounds on Tur{\'a}n determinants

TL;DR

This paper establishes bounds on the normalized Turán determinants for orthogonal polynomials associated with symmetric measures on [-1,1], revealing their structure as Turán determinants of lower order.

Contribution

It proves that the normalized Turán determinant is itself a Turán determinant of lower order for orthogonal polynomials, providing new bounds within the interval.

Findings

Normalized Turán determinants are Turán determinants of lower order.

Derived explicit lower and upper bounds for these determinants.

Results apply to orthogonal polynomials with respect to symmetric measures.

Abstract

Let \mu denote a symmetric probability measure on [-1,1] and let (p_n) be the corresponding orthogonal polynomials normalized such that p_n(1)=1. We prove that the normalized Tur{\'a}n determinant \Delta_n(x)/(1-x^2), where \Delta_n=p_n^2-p_{n-1}p_{n+1}, is a Tur{\'a}n determinant of order n-1 for orthogonal polynomials with respect to (1-x^2)d\mu(x). We use this to prove lower and upper bounds for the normalized Tur{\'a}n determinant in the interval -1<x<1.