Abbreviating lots of consecutive indexed summations

Question

I have lots of equations of the following form $$\sum_{r_0}\sum_{r_1}\sum_{r_2}\cdots\sum_{r_N} x_{r_0} x_{r_1} x_{r_2}\cdots x_{r_N} $$ I can use the following notation for the product of $x$s $$x_{r_0} x_{r_1} x_{r_2}\cdots x_{r_N}=\prod_{j=0}^{N} x_{r_j} $$ Is it also possible to shorten the summation notation $\sum_{r_0}\sum_{r_1}\sum_{r_2}\cdots\sum_{r_N}$?

Answer 1

You could do something like that. For each $n+1$-tuple $r = (r_0, r_1, \ldots, r_n)$, let $$x_r = \prod_{0 \leqslant i \leqslant n} x_{r_i} \text{ and }S = \sum_r x_r$$