Zeros of polynomials with four-term recurrence and linear coefficients

Abstract

This paper investigates the zero distribution of a sequence of polynomials ${\left\{P_m(z)\right\}}_{m=0}^{\infty }$ generated by the reciprocal of $1+ct+B(z)t^2+A(z)t^3$ where $c\in \mathbb{R}$ and $A(z)$, $B(z)$ are real linear polynomials. We study necessary and sufficient conditions for the reality of the zeros of $P_m(z)$. Under these conditions, we find an explicit interval containing these zeros, whose union forms a dense subset of this interval.