Polynomials with rational generating functions and real zeros

TL;DR

This paper studies the zeros of polynomials generated by rational functions with binomial denominators, showing they are eventually hyperbolic and have real zeros densely populating an interval, depending on parameters.

Contribution

It introduces a two-parameter family of generating functions producing polynomials with real zeros densely filling an interval, extending understanding of zero distributions.

Findings

Polynomials are eventually hyperbolic.

Real zeros form a dense subset of an interval.

Interval length depends on parameters.

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.