Root geometry of polynomial sequences I: Type $(0,1)$

TL;DR

This paper investigates the roots of a specific polynomial sequence defined by a second-order linear recursion, proving real-rootedness, establishing bounds for roots, and identifying limit points, with applications in combinatorics and graph theory.

Contribution

It provides a comprehensive analysis of the root distribution of polynomial sequences of type (0,1), including real-rootedness proofs and zero-set bounds, which were previously unestablished.

Findings

All polynomials in the sequence are real-rooted.

The best bounds for the zero-set are derived.

Three precise limit points of the zero-set are identified.

Abstract

This paper is concerned with the distribution in the complex plane of the roots of a polynomial sequence $\{W_n(x)\}_{n\ge0}$ given by a recursion $W_n(x)=aW_{n-1}(x)+(bx+c)W_{n-2}(x)$, with $W_0(x)=1$ and $W_1(x)=t(x-r)$, where $a>0$, $b>0$, and $c,t,r\in\mathbb{R}$. Our results include proof of the distinct-real-rootedness of every such polynomial $W_n(x)$, derivation of the best bound for the zero-set $\{x\mid W_n(x)=0\ \text{for some $n\ge1$}\}$, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.