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

TL;DR

This paper studies the zeros of polynomials generated by rational functions with hyperbolic polynomial denominators, showing they are eventually hyperbolic and their zeros densely fill an interval on the positive real line.

Contribution

It establishes conditions under which the generated polynomial sequences are hyperbolic and characterizes the distribution of their zeros.

Findings

Polynomials are eventually hyperbolic for the given generating functions.

Zeros of the polynomials densely fill an interval on the positive real axis.

The interval length depends on parameters of the generating function.

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 $G(z,t)=P(t)+zt^{r}$, 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 $P(0)>0$ 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.