# What is the proper notation for a general number of nested summations?

A sum over one index: $\sum_i f(i)$

A sum over two indices: $\sum_i \sum_j f(i,j)$

A sum over many indices: $\sum_{k_1} \sum_{k_2} \underbrace{\dots}_n \sum_{k_n} f(\mathbf k)$?

• I have seen the use $$\sum_{k_1,k_2,\ldots,k_n}f(\mathbf k)$$optionally with limits $0\leq k_i\leq n_i$ stacked vertically below the summation symbol. – Arthur Jan 21 '15 at 11:48
• What @Arthur said is very common, especially in some combinatorial papers I have read. – Ali Caglayan Jan 21 '15 at 18:22

Probably in most contexts $$\sum_{k_1} \cdots \sum_{k_n} f(k_1, \ldots, k_n)$$ suffices, and sometimes one will just write $$\sum_{k_1, \ldots, k_n} f(k_1, \ldots, k_n),$$ or perhaps slightly more precisely, $$\sum_{(k_1, \ldots, k_n)} f(k_1, \ldots, k_n).$$ As written, this last summation isn't actually a multiple summation, but rather a single summation over all $k$-tuples $(k_1, \ldots, k_n)$ in the Cartesian product $K_1 \times \cdots \times K_n$ of the index sets $K_i$ over which the index variables $k_i$ respectively vary.
Try $$\sum_{x \mathop \in S} f(x)$$ which is the sum of $f(x)$ over all elements $x$ in the set $S$. Your set $S$ could be $\{k_{1},...,k_{n}\}$
• But the OP's summations are equivalent to sum of ordered tuples, not unordered tuples, so in this case $S$ could be $\{(k_1,\ldots,k_n)\}$. – Stan Liou Jan 21 '15 at 15:46