Is there a general formula for infinite series of a rational function

Question

Is there some sort of formula to calculate $$\sum_x \frac{P_1(x)}{P_2(x)}$$ In particular, what is $$\sum_x \frac{1}{ax^2+bx+c}$$ And what is $$\sum_x \frac{1}{x^3+1}$$

Answer 1

$$S_p=\sum_{x=0}^p \frac{1}{ax^2+bx+c}=\frac 1a\sum_{x=0}^p \frac{1}{(x-r)(x-s)}=\frac 1{a(r-s)}\sum_{x=0}^p \left(\frac 1{x-r}- \frac 1{x-s}\right)$$ Assuming that $r$ and $s$ are not integers, use $$\sum_{x=0}^p\frac 1{x-t}=\psi (p-t+1)-\psi (-t)$$ $$S_p=\frac{\psi (p-r+1)-\psi(p-s+1)-\psi (-r)+\psi (-s)}{a(r-s)}$$ Now, for large values of $p$, using asymptotics $$S_p=\frac{\psi (-s)-\psi (-r)}{a (r-s)}-\frac{1}{a p}+\frac{1-r-s}{2 a p^2}+O\left(\frac{1}{p^3}\right)$$

For the other cases, it is the same story : use partial fraction decomposition.