Properties of Discrete Analogue of the Differential Operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$

TL;DR

This paper investigates the properties of a discrete analogue of a specific differential operator, demonstrating that it shares similar zero properties with its continuous counterpart.

Contribution

It introduces and analyzes the discrete analogue of the differential operator, showing it retains key properties of the continuous operator, including its zeros.

Findings

Discrete analogue $D_m(heta)$ has similar zero properties to the differential operator.

The zeros of the discrete analogue include functions $e^x$, $e^{-x}$, and polynomial $P_{2m-3}(x)$.

The study confirms the discrete analogue preserves essential characteristics of the continuous operator.

Abstract

In the paper properties of the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are studied. It is known, that zeros of differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are functions $e^x$, $e^{-x}$ and $P_{2m-3}(x)$. It is proved that discrete analogue $D_m(h\beta)$ of this differential operator also have similar properties.