On Hyperbolic Polynomials and Four-term Recurrence with 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 [ 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 $P_n(z)$.