Hyperbolic polynomials and linear-type generating functions

TL;DR

This paper proves that certain polynomials generated by a specific rational function are hyperbolic for large degrees, and also shows that a family of exponential polynomials have infinitely many real zeros, advancing understanding of polynomial zero distributions.

Contribution

It establishes conditions under which polynomials generated by a rational function are hyperbolic and analyzes the real zeros of a family of exponential polynomials, providing new insights into their zero distributions.

Findings

Generated polynomials are hyperbolic for large m under certain conditions.

A family of exponential polynomials has infinitely many real zeros.

Zeros of P and Q are real and sufficiently separated, ensuring hyperbolicity.

Abstract

We prove that the polynomials generated by the relation $\displaystyle{\sum_{m=0}^{\infty} H_m(z)t^m=\frac{1}{P(t)+z t^r Q(t)}}$ are hyperbolic for $m \gg 1$ given that the zeros of the real polynomials $P$ and $Q$ are real and sufficiently separated. The paper also contains a result on a certain family of exponential polynomials, which are demonstrated to have infinitely many real zeros.