Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros

Below we establish the conditions guaranteeing the reality of all the zeros of polynomials $P_n(z)$ in the polynomial sequence $\{P_n(z)\}_{n=1}^{\infty}$ satisfying a five-term recurrence relation $$P_{n}(z)= zP_{n-1}(z) + \alpha P_{n-2}(z)+\beta P_{n-3}(z)+\gamma P_{n-4}(z),$$ with the standard initial conditions $$P_0(z) = 1, P_{-1}(z) = P_{-2}(z) =P_{-3}(z) = 0,$$ where $\alpha, \beta, \gamma$ are real coefficients, $\gamma\neq 0$ and $z$ is a complex variable. We interprete this sequence of polynomials as principal minors of an appropriate banded Teoplitz matrix whose associated Laurent polynomial $b(z)$ is holomorphic in $\mathbb{C}\setminus \{0\}$. We show that when either the critical points of $b(z)$ are all real; or when they are two real and one pair of complex conjugate critical points with some extra conditions on the parameters, the set $b^{-1}(\mathbb{R})$ contains a Jordan curve with $0$ in its interior and in some cases a non-simple curve enclosing $0$. The presence of the said curves is necessary and sufficient for every polynomial in the sequence $\{P_n(z)\}_{n=1}^{\infty}$ to be hyperbolic (real-rooted).