On hyperbolic polynomials with four-term recurrence and linear coefficients

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

$$\begin{aligned} P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in {\mathbb {N}}, \end{aligned}$$

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 $\bigcup \nolimits _{n=0}^\infty {\mathcal {Z}}(P_n)$ is dense, where ${\mathcal {Z}}(P_n)$ is the set of zeros of $P_n(z)$.