On Hyperbolic Polynomials with Four-Term Recurrence and Linear Coefficients

For any real numbers a, b, and c, we form the sequence of polynomials {Pn(z)}∞ n=0 satisfying the four-term recurrence Pn(z) = −azPn−1(z)−bPn−2(z)−czPn−3(z), n ∈N, with the initial conditions P0(z) = 1 and P−n(z) = 0. We find necessary and sufficient conditions on a, b, and c under which the zeros of Pn(z) are real for all n, and provide an explicit real interval on which ∞ [ n=0 Z(Pn) is dense, where Z(Pn) is the set of zeros of Pn(z).