I was wondering whether there is a symbol for a sum like for instance
$$x_1x_2 + x_2x_3 + x_3x_4 + x_4x_1$$
that I'd like to write as $$\sum_{??} x_i x_j$$ for instance, with $??$ being some kind of expression I'd like to find.
The idea being that you iterate through all "shifts" of the indices of $(1,2,3,\ldots,n)$ (or $(1,2,3,4)$) in this case.
Conceptually you could maybe think of this as a conceptually similar thing to a discrete version to a contour integral $\oint$.
Best Answer
These sums are typically called cyclic sums, and a common notation is
$$ \sum_\text{cyc} x_1x_2 = x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n + x_nx_1 $$
if there are $n$ elements in $\{x_i\}$. Frequently, we see that there are variables $a, b, c$ (say), and we write something like
$$ \sum_\text{cyc} ab = ab + bc + ca $$