Analytical continuations of partial sums

Question

Take a function $f(x) \in C^k(\mathbb{C})$ and consider the partial sum, $$F(n) = \sum_{i=0}^n f(i), \quad n \in \mathbb{N}$$ Is there a general method on how $F(n)$ can be extended to the real or complex plane? At least, is there a sufficient and/or necessary condition on $f$ that ensures that resulting extension well-defined? For example, I would guess a necessary condition to be $k = \infty$? As a simple (naive?) example, consider, $$F(n) = \sum_{i=1}^n i = \frac{n (n+1)}{2}.$$ Since we here have an explicit expression for the summation, it is quite straightforward to simply declare, $$F(z) = \frac{z (z+1)}{2}, \qquad z \in \mathbb{C}.$$

Answer 1

This is a good question, I would say that for starters you want some decay assumptions on $f$ for large n otherwise you'll have convergence issues. But suppose $\lim_{n\to \infty} F(n)$ exits then you have a function defined on $\mathbb{N}$. You certainly can extend it to a continuous function on $\mathbb{R}$. I'm actually not sure you need any assumption for $f$ being differentiable to extend it to a continuous function on $\mathbb{R}$. Now if we want to extend it to a smooth function on $\mathbb{R}$ I also don't see anything wrong with it. You could take a sum of bump functions of height $f(n)$ with support $[n-\frac{1}{2},n-\frac{1}{2}]$.

As far as extending it to a complex analytic function, I also think you can do it but I can't think of a proof right now.

Hope this helps