Hyperbolic polynomials and linear-type generating functions

In this paper we consider sequences of polynomials {H-m(z)}(m)(infinity)=0 generated by a relation Sigma H-infinity(m=0)m(z)t(m) =1/p(t)+zt(r)Q(t), where P and Q are real polynomials and r is an element of N , r >= 2. In the main result of the paper (cf. Theorem I) 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 H-m(z) are real. We also show that the set of all zeros of the H-m(z)'s for m >> 1 is dense in a real ray. (C) 2020 Elsevier Inc. All rights reserved.