I'm looking for a formula expressing the difference between the Riemann sum and the Riemann integral. If $S=\sum_{n=-\infty}^{+\infty} f(an)$, $I=\int_{-\infty}^{+\infty} dx f(x)$ and $a>0$, then I'm looking for an expression for the coefficients $c_i$ in the following formula:
$$S=I+a c_1+....+a^n c_n + O(a^{n+1})$$
where the Big $O$ notation refers to the limit $a\rightarrow 0$.
In other words, $I$ is the first approximation of $S$, while the $c_i$ give the remainder (I let the reader give hypothesis on the function $f$)