2

Is there any standard notation, other than an ellipsis, for a chain of nested sigma summations?

For instance, I have: $$ \sum_{b_0=0}^{L} \sum_{b_1=0}^{L-b_0} \sum_{b_2=0}^{L-b_0-b_1} \cdots \sum_{b_N}^{L-b_0-b_1- \cdots - b_{N-1}} f\left(b_0, b_1, b_2, \ldots, b_N \right) $$

It seems like there should be some better way to write this. For instance, something like:

$$ \left( \bigodot\limits_{i=0}^{N} \sum_{b_i=0}^{L-\sum_{k=0}^{i-1} b_i} \right) \left( f\left(b_0, b_1, b_2, \ldots, b_N \right) \right) $$

brianmearns
  • 837

1 Answers1

2

It is the sum on $\sum_{b_1\geq 0,\cdots, b_N\geq 0, b_1+\cdots+b_n\leq L}$, I think.

More generally, if you have a nested summation, you can always write the set of the $(b_1,\cdots,b_n)$ which are in the sum and write $\sum_{(b_1,\cdots,b_n)\in I}$.

In your example, $I=\{(b_1,\cdots,b_N)\in\mathbb{N}^N, b_1+\cdots+b_n\leq L\}$

D.L.
  • 1,306
  • $\sum_{0 \leq b_1 + \ldots + b_n \leq L}$ – user66081 Feb 21 '14 at 19:06
  • No, for example, $(1,-1,0,...)$ must not be in the sum, I think... – D.L. Feb 21 '14 at 19:07
  • Correct; I'm so used to this notation with $b_i$ being nonnegative integers, that I didn't notice this. – user66081 Feb 21 '14 at 19:10
  • Me too, my first idea to simplify was the same. – D.L. Feb 21 '14 at 19:10
  • Oh, that's right, I forgot you can iterate over a set of tuples. But it doesn't really resolve the question of how to neatly define the set from which they draw, which is kind of the big problem in the originally posted example. – brianmearns Feb 21 '14 at 19:11
  • In a very general case, you won't have necessarily a simple description of the set, but in your example, you have one. – D.L. Feb 21 '14 at 19:13