# Hermite-Poulain theorems for linear finite difference operators

## TL;DR

This paper extends Hermite-Poulain theorems to finite difference operators, analyzing root locations of polynomials and entire functions, and establishing properties like root simplicity and mesh preservation.

## Contribution

It introduces new analogues of Hermite-Poulain theorems for finite difference operators and studies root behavior and mesh preservation in polynomials.

## Key findings

- Roots of difference operators are simple under certain conditions.
- The operator does not decrease the mesh of polynomials with roots on a line.
- Asymptotic root behavior as |h| approaches infinity is characterized.

## Abstract

We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form $$ \Delta_{\theta, h}(f)(z)=e^{i\theta}f(z+ih)-e^{-i\theta}f(z-ih), \quad\theta\in[0,\pi),\ \ h\in\mathbb{C}\setminus\{0\}, $$ where $f$ is a polynomial or an entire function of a certain kind, and prove that the roots of $\Delta_{\theta, h}(f)$ are simple under some conditions. Moreover, we prove that the operator $\Delta_{\theta, h}$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of $\Delta_{\theta, h}(p)$ as $|h|\to\infty$ is found for any complex polynomial $p$. Some other interesting roots preserving properties of the operator $\Delta_{\theta, h}$ are also studied, and a few examples are presented.