There's a generalization of the mean value theorem that can be
applied here, namely that if $f$ is smooth enough then for each $x$
there is $y$ such that
$$\Delta^kf(x)=f^{(k)}(y)$$
and $x < y < x+k$.

**Added** see
https://mathoverflow.net/questions/18054/is-it-true-that-all-the-irrational-power-functions-are-almost-polynomial/18056#18056 .