Hyperbolic polynomials and linear-type generating functions

Abstract

In this paper we consider sequences of polynomials {Hm(z)}m=0∞ generated by a relation ∑m=0∞Hm(z)tm=1P(t)+ztrQ(t), where P and Q are real polynomials and r∈N, r≥2. In the main result of the paper (cf. Theorem 1) we give a necessary conditions on P and Q (and their zeros) to ensure that for all sufficiently large m, the zeros of the polynomials Hm(z) are real. We also show that the set of all zeros of the Hm(z)'s for m≫1 is dense in a real ray.