Is there an accepted notation for the $n$th sum/integral of a function?

Question

I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the $n$th integral of a function of $n$ variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of $x_1$ from $0$ to $M$, then the sum of $x_1$ from $0$ to $M$, and so on with a total of of $N$ summations.

Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?

edit: specific examples (sorry, didn't know you could use LaTeX here):

2D: $$\sum_{x_1=0}^M \sum_{x_2=0}^M f(x_1,x_2)$$ 3D: $$\sum_{x_1=0}^M \sum_{x_2=0}^M \sum_{x_3=0}^M f(x_1,x_2,x_3)$$ Is there a nicer way to express something like: $$\sum_{x_1=0}^M \sum_{x_2=0}^M \dots \sum_{x_n=0}^M f(x_1,x_2,\dots,x_n)$$

Answer 1

I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $\sum_0^M f$.

Here are two suggestions each using just a single $\sum$.

$$ \sum_{x \in [1, 2, \ldots, M]^n} f(x) $$

$$ \sum_{x_i = 0, \ (i = 1, \ldots n)}^M f(x_i, x_2, \ldots, x_n) $$

Answer 2

How about something like : $$\sum_{x_1=0}^M \sum_{x_2=0}^M \dots \sum_{x_n=0}^M f(x_1,x_2,\dots,x_n) = \large]_{k=x_0}^{k=x_n}\sum_{x_k=0}^{M_n}f(x_1,x_2,\dots,x_n)$$

Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.