How to expand/rewrite the product $\prod\limits_{i=1}^n (x-a_i)$ into a sigma notation $\sum\limits_{j=0}^n b_jx^j$?
I am trying to extract all coefficients ($b_j$ in the above expression). In my attempts I see that $b_j$ should contain $n-j$ of the $a_i$ s and is sort of the sum over all products of them somehow.
Intuitively and after working with smaller examples, I know how it should look like but cannot give a closed form expression for $b_j$.
$$ \prod_{i=1}^n (x-a_i)= \sum_{j=0}^n (-1)^j \left( \sum_{1 \le k_1 \lt \cdots \lt k_j \le n} \ \prod_{m=1}^j a_{k_m} \right) x^j $$
The enclosed parenthesis requested terms the Elementary Symmetric Polynomial terms $b_j=(-1)^j e_j(a_1,...a_n)$, closely related to the Vieta's Formulas, as very well indicated in the question comments.