# Compute a series from a sequence

Let's say I have a sequence $$s_n$$ of numbers, and I want a series $$a_i$$ which computes the sequence; that is

$$\sum_{i=0}^\infty a_i n^i = s_n$$

Clearly $$a_0 = s_0$$, but after that I am stuck. I need $$\sum_{i=0}^\infty a_i = s_1$$, so I'm not sure how I should go about picking the $$a_1$$ -- it seems that I need to choose all the $$a_i$$ "simultaneously".

Are there any methods for doing this? I am most interested in computable methods, though I suspect it may not be a computable problem in general. I am interested in theoretical results about this as well.

• What do you mean by computable? If you truncate the $s_n$ sequence this is just polynomial interpolation. – MR_Q Mar 24 at 4:28
• @MR_Q yeah but I don't want to truncate. I mean can I compute (exactly, or perhaps to arbitrary closeness) each $a_i$ from a finite amount of $s_i$. – luqui Mar 24 at 5:08

## 1 Answer

More generally, for any sequences $$p_j$$ and $$s_j$$ of complex numbers such that $$p_j$$ has no limit points, there exist entire functions $$f$$ such that $$f(p_j) = s_j$$ for all $$j$$. See e.g. Rudin, "Real and Complex Analysis", Theorem 15.15. You then want the Maclaurin series of $$f$$.

EDIT: One way to construct such an $$f$$ in your case is as follows. Let $$g_n(z)$$ be an entire function with $$g_n(n) = (-1)^n s_n/\pi$$ and $$|g_n(z)| \le 1/n$$ for $$|z| \le n/2$$. For example, you might take $$g_0(z) = s_1/\pi$$ and $$g_n(z) = (-1)^n s_n (z/n)^{k_n} \pi$$ for $$n \ge 1$$, where $$k_n > \log_2(n |a_n|/\pi)$$. Let $$g(z) = \sum_{n=0}^\infty \frac{g_n(z)}{z-n}$$. This series converges to a meromorphic function with simple poles at the nonnegative integers, having residue $$(-1)^n s_n/\pi$$ at $$z=n$$. Then $$f(z) = \sin(\pi z) g(z)$$ is entire with $$f(n) = s_n$$ for all nonnegative integers $$n$$.

Note that to compute $$a_j$$ we need only consider the $$g_n$$ where $$k_n \le j$$, and we can arrange it so $$k_n \ge n$$.