Turán Inequalities and Zeros of Orthogonal Polynomials

Abstract

We use Turán type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence $p_{k+1}=x p_k-c_k p_{k-1},$ with a nondecreasing sequence $\{c_k\}$. As a special case they include a non-asymptotic version of Máté, Nevai and Totik result on the largest zeros of orthogonal polynomials with $c_k=c \; k^{2 \delta} (1+ o(k^{-2/3})).$ Our proof is based on new Turán inequalities which are obtained by analogy with higher order Laguerre inequalities.