Bounds on Tur{\'a}n determinants

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.