On the zeros of polynomials generated by rational functions with a hyperbolic polynomial type denominator

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form $G(z,t)=P(t)+zt^{r}$, where the zeros of $P$ are positive and real. We show that every member of a family of such generating functions - parametrized by the degree of $P$ and $r$ - gives rise to a sequence of polynomials ${H_{m}(z)}_{m=0}^{infty}$ that is eventually hyperbolic. Moreover, when $P(0)u003e0$ the real zeros of the polynomials $H_{m}(z)$ form a dense subset of an interval $Isubsetmathbb{R}^{+}$, whose length depends on the particular values of the parameters in the generating function.