Root geometry of polynomial sequences III: Type $(1,1)$ with positive coefficients

TL;DR

This paper investigates the root distribution of certain polynomial sequences defined by a second-order recurrence with positive linear coefficients, establishing conditions for real-rootedness and describing the zero limits.

Contribution

It provides a necessary and sufficient condition for the real-rootedness of these polynomials and characterizes the limit set of their zeros.

Findings

Characterization of when all polynomials are real-rooted

Description of the zero limit set as a union of an interval and isolated points

Sufficient conditions for large polynomials to have real zeros

Abstract

In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the overall real-rootedness of all the polynomials, in terms of the polynomial coefficients of the recurrence. Moreover, in the real-rooted case, we find the set of limits of zeros, which turns out to be the union of a closed interval and one or two isolated points; when non-real-rooted polynomial exists, we present a sufficient condition under which every polynomial with $n$ large has a real zero.