Properties of Discrete Analogue of the Differential Operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$
Provides a theoretical extension of known properties from continuous to discrete operators, but the result is incremental for mathematicians working on discrete analogues.
The paper studies the discrete analogue of a differential operator and proves that its zeros have similar properties to the continuous case, including functions e^x, e^{-x}, and a polynomial of degree 2m-3.
In the paper properties of the discrete analogue $D_m(hβ)$ 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 $D_m(hβ)$ of this differential operator also have similar properties.