Polynomials Defined by 5-Term Recurrence Relations, Banded Toeplitz Matrices, and Reality of Zeros

Abstract

This paper establishes 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

$${P_n}(z) = z{P_{n - 1}}(z) + \alpha {P_{n - 2}}(z) + \beta {P_{n - 3}}(z) + \gamma {P_{n - 4}}(z)$$

with the standard initial conditions

$${P_0}(z) = 1,\,\,\,\,{P_{ - 1}}(z) = {P_{ - 2}}(z) = {P_{ - 3}}(z) = 0,$$

where α, β, γ are real coefficients, γ ≠ 0 and z is a complex variable. We interpret this sequence of polynomials as principal minors of an appropriate banded Toeplitz matrix whose associated Laurent polynomial b(z) is holomorphic in ℂ {0}. We derive a criterion for the preimage b⁻¹ (ℝ) to contain a Jordan curve, which, by a recent result of Shapiro and Stampach, is necessary and sufficient for every polynomial in the sequence $\{P_n(z)\}_{n=1}^{\infty}$ to be hyperbolic.