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 $P_n(z)$.
Categories: math.CV math.CA