Polynomials with rational generating functions and real zeros

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions gives rise to a sequence of polynomials $\{P_m(z){\}}_{m=0}^{\mathrm{\infty }}$ that is eventually hyperbolic. Moreover, the real zeros of the polynomials $P_m(z)$ form a dense subset of an interval $I\subset {\mathbb{R}}^{+}$, whose length depends on the particular values of the parameters in the generating function.