I would like to show that for $s \in \mathbb{R}$ and a nonnegative integer $k$
$$
\triangle^k ((1+n)^s) \lesssim (1+|n|)^{s-k}
$$


where $\triangle$ is the discrete derivative, i.e. $\triangle^1 ((1+n)^s) = (2+n)^s - (1+n)^s$. 




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This is easy when $s \in \mathbb{Z}$, and in the continuous analogue because 
$$
\partial\_x^k (1+x)^s = (s)(s-1)\cdots (s-k+1) (1+x)^{s-k}
$$




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I think that you can use the generalized binomial theorem to prove this, but I was wondering if there was anything more straightforward, e.g. some kind of convexity argument to use the continuous case.




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Note: I wasn't sure about the tags, feel free to re-tag as appropriate. 



Elementary proof of bounds on discrete derivative applied to $(1+n)^s$
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
[Is it true that all the "irrational power" functions are almost polynomial ?](https://mathoverflow.net/questions/18054/is-it-true-that-all-the-irrational-power-functions-are-almost-polynomial/18056#18056) .


