There is an "iterated" version of the mean value theorem that states that for a smooth enough function $f$ on an interval that $\Delta^k f(x)=k! f^{(k)}(\xi)$ where $\xi$ is between $x$ and $x+k$. This should be enough for your purposes.

There is an "iterated" version of the mean value theorem which states that for a smooth enough function $f$ on an interval that $\Delta^k f(x)=f^{(k)}(\xi)$ where $\xi$ is between $x$ and $x+k$. This should be enough for your purposes.