Hyperbolic polynomials and linear-type generating functions

Abstract

We prove that the polynomials generated by the relation $\sum _{m=0}^{\infty }{H}_m(z)t^m=\frac{1}{P(t)+z{t}^{r}Q(t)}$ are hyperbolic for $m\gg 1$ given that the zeros of the real polynomials $P$ and $Q$ are real and sufficiently separated. The paper also contains a result on a certain family of exponential polynomials, which are demonstrated to have infinitely many real zeros.