# A characterisation of functions of $\mathbb{N} \rightarrow \mathbb{R}$ of the form $f(n) = \sum_k a_k k^n$

I'm looking for a characterisation of functions $f : \mathbb{N} \rightarrow \mathbb{R}$ for which there exists $(a_k) \in \mathbb{R}^\mathbb{N}$ such as $f(n) = \sum_k a_k k^n$ for all $n \in \mathbb{N}$.

I don't think that every functions of $\mathbb{N} \rightarrow \mathbb{R}$ can be written like this. An easy calculus (using Vandermonde matrix) show that for $N \in \mathbb{N}$, one can find a function $f_N$ such as $f_N(N) = 1$ and $f_N(n) = 0$ if $n < N$. So for a given function $f$, we can find a function $g$ of the form we want such as $f(n) = g(n)$ for $n < N$, with of course no information after $N$ on $g$.

I'm looking for a condition on $\lim f(n)$ as $n$ go to infinite for $f$ to be of the given form (by the result before, it's the only kind of condition we can put on $f$).

• I'm certain this is a duplicate but I can't find it in my answers. – mercio Nov 17 '15 at 16:51
• @Normal Human see math.stackexchange.com/questions/175810/do-such-sequences-exist – mercio Nov 17 '15 at 17:00
• @mercio Thanks. I see that the question there is actually different, but your answer precisely addresses the question here. I don't know if this qualifies this question as a duplicate of the other one. Maybe you could post a short answer here pointing out that the Vandermonde matrix approach can give the full result, and referring to your other answer? – user147263 Nov 17 '15 at 17:20

The method used in this answer show that for every function $f : \Bbb N \to \Bbb R$ there is a sequence $(a_k)$ such that forall $n$, $\sum_{k \ge 0} a_k k^n$ converges absolutely to $f(n)$.