On Hyperbolic Polynomials and Four-term Recurrence with Linear Coefficients

TL;DR

This paper characterizes conditions on parameters for which a sequence of polynomials defined by a four-term recurrence has all real zeros and describes the density of their zeros on a specific interval.

Contribution

It provides necessary and sufficient conditions for real zeros of polynomials from a four-term recurrence and explicitly describes the interval where their zeros are dense.

Findings

Zeros are real under specific parameter conditions.

Explicit interval where zeros are dense.

Conditions are both necessary and sufficient.

Abstract

For any real numbers $a,\ b$, and $c$, we form the sequence of polynomials $\{P_n(z)\}_{n=0}^\infty$ satisfying the four-term recurrence \[ P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in\mathbb{N}, \] with the initial conditions $P_0(z)=1$ and $P_{-n}(z)=0$. We find necessary and sufficient conditions on $a,\ b$, and $c$ under which the zeros of $P_n(z)$ are real for all $n$, and provide an explicit real interval on which $\displaystyle\bigcup_{n=0}^\infty\mathcal{Z}(P_n)$ is dense, where $\mathcal{Z}(P_n)$ is the set of zeros of $P_n(z)$.