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

Abstract

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form $G(z,t)=P(t)+z{t}^r$, 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 $P(0)>0$ the real zeros of the polynomials $H_m(z)$ 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.

References in corpus (1)

• On rational approximation of algebraic functions