# When is there a closed form for $\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$?

This is a follow up on a previous question of mine.

Out of curiosity, I am wondering more generally when a closed form exists for

$$\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$$

where $P$ and $Q$ are both polynomials. Obviously for convergence $\deg(Q)\ge \deg(P)+2$, but I don't know much from there.

• Have a look at the book $A=B$ math.upenn.edu/~wilf/AeqB.html Such sums tend to be expressed as values of hypergeometric functions, not that it helps too much. – Liviu Nicolaescu Oct 16 '15 at 18:17
• More natural sum $\sum_{n=-\infty}^{\infty} \frac{P(n)}{Q(n)}$ can be expressed via residues of $P/Q$. – Andrew Oct 16 '15 at 18:36

Any function of the form $P(n)/Q(n)$ with $\deg P \leq \deg Q -2$ can be written as a linear combination of $1/(n+1) - 1/(n+\alpha)$ and $1/(n+\alpha)^k$, for various $\alpha$ and various $k>1$. (Use partial fraction decomposition. The bound on $\deg P$ ensures that the coefficients of $1/(n+1)$ will cancel out.)
The sum $\sum \left( \frac{1}{n+1} - \frac{1}{n+\alpha} \right)$ is $\gamma+\psi(\alpha)$, where $\psi$ is the digamma function. The sum $\sum \frac{1}{(n+\alpha)^k}$ is the polygamma function $\psi^k(z)$. In both cases, it is worth noting the recurrence identities and reflection relations in the linked Wikipedia articles, which let you relate values at $\alpha$ to values at $m \pm \alpha$, for $m$ an integer.
Closed form is a subjective term. At any rate, our understanding of these sums is quite limited, e.g. it is not known if $\sum_{n=1}^\infty n^{-5}$ is an irrational number (same for any smaller negative odd exponent).
• "Unknown" meaning unproven, yes, but it's morally certain that $\zeta(5) \notin \bf Q$... – Noam D. Elkies Oct 16 '15 at 18:57