Geometry of limits of zeros of polynomial sequences of type $(1,1)$

TL;DR

This paper investigates the geometric distribution of zeros of polynomial sequences of type (1,1), characterizing their limit sets and establishing conditions for real-rootedness and bounds on zeros.

Contribution

It provides a comprehensive geometric description of zero limits and a criterion for real-rootedness of such polynomial sequences.

Findings

Limit sets of zeros are arcs, circles, lollipops, or intervals.

Established necessary and sufficient conditions for real-rootedness.

Derived sharp bounds for zeros when they are real.

Abstract

In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, or a "lollipop", or an interval. As an application, we discover a sufficient and necessary condition for the universal real-rootedness of the polynomials, subject to certain sign condition on the coefficients of the recurrence. Moreover, we obtain the sharp bound for all the zeros when they are real.