# Zeros of polynomials with four-term recurrence

## TL;DR

This paper characterizes when the zeros of polynomials satisfying a specific four-term recurrence are all real, providing explicit conditions on parameters and describing the density of zeros on a real interval.

## Contribution

It establishes necessary and sufficient conditions for the real zeros of four-term recurrence polynomials and describes the density of their zeros on a specific interval.

## Key findings

- Zeros are real under certain parameter conditions.
- Explicit interval where zeros are dense.
- Conditions depend on parameters b and c.

## Abstract

For any real numbers $b,c\in\mathbb{R}$, we form the sequence of polynomials $\left\{ H_{m}(z)\right\} _{m=0}^{\infty}$ satisfying the four-term recurrence \[ H_{m}(z)+cH_{m-1}(z)+bH_{m-2}(z)+zH_{m-3}(z)=0,\qquad m\ge3, \] with the initial conditions $H_{0}(z)=1$, $H_{1}(z)=$$-c$, and $H_{2}(z)=-b+c^{2}$. We find necessary and sufficient conditions on $b$ and $c$ under which the zeros of $H_{m}(z)$ are real for all $m$, and provide an explicit real interval on which ${\displaystyle \bigcup_{m=0}^{\infty}\mathcal{Z}(H_{m})}$ is dense where $\mathcal{Z}(H_{m})$ is the set of zeros of $H_{m}(z)$.