Hyperbolic polynomials and linear-type generating functions

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