Hello all, the $\Delta$ operator on functions $\mathcal{N} \to \mathbb{R}$ (where $\mathcal N$ denotes $\lbrace 1,2, \ldots , \rbrace$ )defined by $\Delta(f)(n)=f(n+1)-f(n)$ is well-known and it is not very hard to show by induction that $f$ is a polynomial of degree $\leq k$ iff $\Delta^{k+1}(f)$ is identically zero, where $\Delta^{k+1}$ denotes $\Delta$ iterated $k+1$ times. Now I say that a function $f : \mathcal{N} \to \mathbb{R}$ is "almost polynomial" iff $\Delta^{k}(f)$ is a bounded function for some $k$.

My question is : let $\lambda >0$ be a non-integer, and let $f(n)=n^{\lambda}$. Is it true that $f$ is almost polynomial ?