On the difference property of higher orders for differentiable functions

Abstract

In this paper, inspired by some results concerning the double difference property, we show that the class $C^p(\mathbb{R},\mathbb{R})$ of $p$-times continuously differentiable functions has the difference property of $p$-th order, i.e. if a function $f\colon \mathbb{R}\to \mathbb{R}$ is such that $\Delta^p_h f\in C^p(\mathbb{R}\times \mathbb{R},\mathbb{R})$, where $\Delta^p_h f$ is the $p$-th iterate of the well-known difference operator $\Delta_h f(x):=f(x+h)-f(x)$, then there exists a polynomial function $\Gamma\!_{p-1}\colon \mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order such that $f-\Gamma\!_{p-1}\in C^p(\mathbb{R},\mathbb{R})$. Moreover, some new equalities connected with the difference operator are also presented.