Zeros of polynomials with four-term recurrence and linear coefficients

Abstract

This paper investigates the zero distribution of a sequence of polynomials $$\left\{ P_{m}(z)\right\} _{m=0}^{\infty }$$ which satisfy a four-term recurrence whose coefficients are linear polynomials in z. In particular, 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.