What's a good symbol to represent many summations? Can a tensor product glyph be used?

Question

What's a good symbol for many repeated summations? I vaguely remember seeing something like

$$ \otimes_{j=1}^N \sum_{n_j=-\infty}^\infty f(\vec{n})= \sum_{n_1=-\infty}^\infty\sum_{n_2=-\infty}^\infty\cdots\sum_{n_N=-\infty}^\infty f(\vec{n}).$$

Is this used in practice? If not, is there a commonly accepted shorthand?

Answer 1

You essentially want to sum over all vectors of length $N$ with integer entries. If we say $\mathcal{Z} = \otimes^N_{j=1} \mathbb{Z}$, then you can write the above sum compactly as: $$\sum_{\vec{z} \in \mathcal{Z}} f(\vec{z})$$

Answer 2

Here we are summing over all $N$-tuples of $\mathbb{Z}^N$. If the order of summation does not matter, it is common to write \begin{align*} \sum_{\vec{n}\in\mathbb{Z}^N} f(\vec{n}) \end{align*}

If the notation $(n_1,n_2,\ldots,n_N)=\vec{n}$ has already been specified, it is also common to write just one sigma-symbols instead of $N$ as in \begin{align*} \sum_{n_1,\ldots,n_N=-\infty}^{\infty}f(\vec{n}) \end{align*}