How to write an "index set"

Question

Imagine I have a finite sequence $\{s_i\}_{1\leq i\leq N}$, for some $N$. Now assume I want to sum all the terms of such sequence apart from the term $s_i$, for some fixed $i$. One way to do it is to simply write $$ S=\sum_{j\neq i}s_j $$ However, I want to explicitly write it by defining a $k$ in some set $K$ such that $$ S=\sum_{k\in K}s_{i+k} $$ Would the set $K=(\mathbb{Z}/N)\setminus\{0\}$ work? Any better way of writing it?

Answer 1

The most obvious (and readable) is to write \begin{align} \sum_{j\neq i}s_j. \end{align} Once in a while, this notation can become ambiguous, because the reader might interpret the sum to be over all pairs of $(i,j)$ such that $j\neq i$ (in this specific case, it's next to impossible to misinterpret though). To be slightly more explicit, you can write \begin{align} \sum_{\substack{1\leq j\leq N\\ j\neq i}}s_j \end{align} If you want to write sets, then I'd suggest defining $J=\{j\in \Bbb{N}\,: 1\leq j\leq N\quad\text{and} \quad j\neq i\}$. Then you can write your sum as \begin{align} \sum_{j\in J}s_j \end{align}

Now, if you want to write it in the form $s_{i+k}$, then you have to define the set $K=\{k\in \Bbb{Z}\,: \text{$1-i\leq k\leq N-i$ and $k\neq 0$}\}$, and then you can write the sum as \begin{align} \sum_{k\in K}s_{i+k}, \end{align} but honestly this is just a confusing way of writing things.

What you could do though is define a new sequence of numbers $\{\sigma_n\}_{n\in\Bbb{Z}}$, such that $\sigma_n=s_n$ if $1\leq n\leq N$, and $\sigma_n=0$ otherwise. Then you can write the sum in question as \begin{align} \sum_{k\in \Bbb{Z}\setminus \{0\}}\sigma_{i+k}. \end{align} This simplifies the index set, but slightly complicates the sequence.