# Characterization of dimension over $\mathbb{Q}$ of infinite sums of rational functions

Let $$P(n)=(n+r_1)(n+r_2)...(n+r_k)$$ be a polynomial with simple, rational, negative roots (i.e. $$r_i>0$$) and degree $$k\geq 2$$ (I stick with negative roots as I don't have to worry about dividing by $$0$$). My question is: What can we say about the dimension of the set

$$S=\text{span}\left\{\sum_{n=0}^\infty \frac{1}{P(n)},\sum_{n=0}^\infty \frac{n}{P(n)},...,\sum_{n=0}^\infty \frac{n^{k-2}}{P(n)}\right\}$$

over $$\mathbb{Q}$$ in terms of the roots $$r_1,r_2,...,r_k$$? Any characterization past the cases done below would be helpful.

Work so far: For the case $$k=2$$, it is not hard to show that

$$\sum_{n=0}^\infty \frac{1}{(n+a)(n+b)}=\frac{\psi(a)-\psi(b)}{a-b}$$

(where $$\psi(x)$$ is the digamma function). Since this is an increasing function for $$x>0$$ we know

$$\sum_{n=0}^\infty\frac{1}{(n+a)(n+b)}=\frac{\psi(a)-\psi(b)}{a-b}\neq 0$$

and therefore the dimension can only be $$1$$. In fact, we can use this fact to prove the dimension is never $$0$$. Assume by way of contradiction that there exists a polynomial $$P(n)=(n+r_1)(n+r_2)+...+(n+r_k)$$ such that

$$\dim\left(\text{span}\left\{\sum_{n=0}^\infty \frac{1}{P(n)},\sum_{n=0}^\infty \frac{n}{P(n)},...,\sum_{n=0}^\infty \frac{n^{k-2}}{P(n)}\right\}\right)=0$$

(i.e. all of the elements are $$0$$). Then pick rationals $$a_i$$ such that

$$Q(n)=\sum_{i=0}^{k-2}a_i n^i=(n+r_3)(n+r_4)+...+(n+r_k)=\frac{P(n)}{(n+r_1)(n+r_2)}$$

Since the dimension of the space is $$0$$, we know

$$0=\sum_{n=0}^\infty\frac{1}{P(n)}\sum_{i=0}^{k-2}a_in^i=\sum_{n=0}^\infty\frac{Q(n)}{P(n)}=\sum_{n=0}^\infty\frac{1}{(n+r_1)(n+r_2)}\neq 0$$

(which follows from the $$k=2$$ case). And finally, for the $$k=3$$ case, a paper by Saradha and Tijdeman proves the sum is rational if and only if all the roots are integer lengths apart. Using this, it is possible to show that the dimension is $$1$$ if and only if the roots are integers lengths apart (otherwise the dimension will be $$2$$). This pattern extends to higher $$k$$ as well: For $$k\geq 3$$, the dimension of $$S$$ is $$1$$ if and only if all roots of $$P(n)$$ are integer lengths apart (this takes more effort to prove but it is indeed true).

For all other $$k\geq 4$$, the question seems much more difficult.

Motivation: I already submitted this as a question on math.stackexchange (including a bounty) but did not receive any answers/comments pertaining to the dimension question (which is the main focus of the post). A colleague suggested that I might have better luck on mathoverflow as they felt this question was much more research oriented than is generally found on stackexchange.