Piecewise interlacing zeros of polynomials

TL;DR

The paper introduces piecewise interlacing zeros to analyze root distributions of polynomials, confirming real-rootedness in polynomial sequences with linear recurrence coefficients, extending previous work.

Contribution

It presents a novel concept of piecewise interlacing zeros for studying polynomial root distributions and extends prior results to polynomials with linear polynomial coefficients.

Findings

Confirmed real-rootedness of certain polynomials with linear recurrence coefficients

Developed a method to construct disjoint intervals for interlacing zeros

Extended previous work on polynomial root analysis

Abstract

We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly speaking, one constructs a collection of disjoint intervals such that one may show by induction that consecutive polynomials have interlacing zeros over each of the intervals. We confirm the real-rootedness of some polynomials satisfying a recurrence with linear polynomial coefficients. This extends Gross et al.'s work where one of the polynomial coefficients is a constant.