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

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.

24 pages, 1 figure