Hyperbolic polynomials and linear-type generating functions

We prove that the polynomials generated by the relation $\displaystyle{\sum_{m=0}^{\infty} H_m(z)t^m=\frac{1}{P(t)+z t^r Q(t)}}$ 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.