# How to write an "index set"

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?

• What is $N$ in $\Bbb Z /N$? Do you mean the quotient ring $\Bbb Z/N\Bbb Z$? Dec 20, 2021 at 12:09
• Are you sure about the tag "ring-theory" ? Dec 20, 2021 at 12:10
• Do you mean quotient ring? The title makes no sense so far. Dec 20, 2021 at 12:11
• Don't look for inspiration. Just define the index set with $K = \{ k | ... \}$ so your reader will know what it is without having to parse notation. Dec 20, 2021 at 12:13
• @EthanBolker what shape of $K$ would you suggest? Dec 20, 2021 at 12:59

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.