On the difference property of higher orders for differentiable functions

In this paper, inspired by some results concerning the double difference property, we show that the class C_p(R,R) of p-times continuously differentiable functions has the difference property of p-th order, i.e. if a function f: R → R is such that Δ^p _hf ∈ C^p(R × R,R), where Δ^p _hf is the p-th iterate of the well-known difference operator Δ_hf(x) := f(x + h) - f(x), then there exists a polynomial function Γ_p-1: R → R of (p - 1)-th order such that f - Γ_p-1 ∈ C^p(R,R). Moreover, some new equalities connected with the difference operator are also presented.