Excluding a specific index when writing out a sum

Question

Given the sum: $$ \sum_{i=1}^{m}a_i\prod_{j\neq i}(n_j+x)=0 $$ If I were to write it without the sum symbol how would I go about excluding $j\neq i$ when writing it in the form $$ a_1\prod_{j=2}^{m}(n_j+x)(where j\neq 1)+a_2\prod_{j=1}^{m}(n_j+x)(where j\neq 2)+\cdots+a_m\prod_{j=1}^{m-1}(n_j+x)(where j\neq m) $$ It seems clear for the first and last summand but how would one go about excluding specific summands inbetween? My idea was to write it like so: $$ a_1\prod_{j=2}^{m}(n_j+x)+a_2\prod_{j=1\setminus \{2\}}^{m}(n_j+x)+\cdots+a_{m-1}\prod_{j=1\setminus\{m-1\}}^{m}(n_j+x)+a_m\prod_{j=1}^{m-1}(n_j+x) $$ Another approach I thought about is this one: $$ a_1\prod_{j=2}^{m}(n_j+x)+a_2\prod_{j=1}^{m}(n_j+x)(j\neq 2)+\cdots+a_{m-1}\prod_{j=1}^{m}(n_j+x)(j\neq m-1)+a_m\prod_{j=1}^{m-1}(n_j+x) $$ Are these approaches correct and are there more elegant approaches?

Answer 1

$$ a_1\prod_{j=1,\\j\ne 1}^{m}(n_j+x)+a_2\prod_{j=1,\\j\ne 2}^{m}(n_j+x)+\cdots+a_m\prod_{j=1,\\j\ne m}^{m}(n_j+x). $$

You can "specialize" the extreme terms, but I don't think it makes it more readable.

Answer 2

This is a common way to do it: $$ \prod_{1 \leq j \leq m, \, j \neq i} c_j $$ where $c_j$ can be, for instance, $(n_j + x)$, as in your case.

EDIT: Now, using the above notation, you can write the sum as follows: $$ a_1 \prod_{2 \leq j \leq m} (n_j + x) + a_2 \prod_{1 \leq j \leq m, \, j \neq 2} (n_j + x) \, + \, ... \, + \, a_m \, \prod_{1 \leq j \leq m - 1} (n_j + x) $$

Answer 3

You will often encounter these very common notations

$$ \prod_{\substack{j=1 \\ j \neq i}}^m c_j \qquad \prod_{\substack{j \in [\![ 1,m]\!] \\ j \neq i}} c_j \qquad \prod_{\substack{1 \leq j \leq m \\ j \neq i}} c_j $$

which all mean the same thing.

Rk: to get this in Latex you type '\prod_{\substack{j=1 \ j \neq i}}^m'