Polynomials with rational generating functions and real zeros (1601.02582v1)

Abstract: 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 ${P_{m}(z)}{m=0}^{\infty}$ 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.