Turan inequalities and zeros of orthogonal polynomials

TL;DR

The paper addresses non-asymptotic, uniform bounds on the extreme zeros of orthogonal polynomials defined by a three-term recurrence, with a focus on the symmetric case. It develops second-order zero bounds by exploiting Turán inequalities and higher-order TI, deriving explicit bounds such as $x_{kk}^2<4c_k\left(1-\frac{d_{k+1}^{2/3}}{(2^{1/3}+d_{k+1}^{1/3})^2}\right)$ under suitable monotonicity conditions on $c_k$, and extends these results to power-growth sequences and Hermite polynomials via refined quadratic and higher-order forms. The work introduces two new TI families and a Delta-operator approach to generate TI-based bounds for a wide class of symmetric recurrences, providing practical criteria in terms of $d_k=(c_k-c_{k-1})/c_k$. These results yield non-asymptotic, parameter-uniform estimates for extreme zeros, connecting to and generalizing Mate–Nevai–Totik-type asymptotics.

Abstract

We use Turan 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 Mate, Nevai and Totik result on the largest zeros of orthogonal polynomials with $c_k=k^δ (1+ o(k^{-2/3})).$

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• Theorem 6
• proof
• Lemma 7
• proof