Is there a closed form of the product $(1+a_1) (1+a_2)\cdots (1+a_N)$?

Question

I have the following product $$ P = \prod_{n=1}^N (1+a_n) $$ where each $a_n$ is different. I know the expanded out form should be something like $$ P = 1+\sum_n^N a_n + \sum_{n < m}^{N-1} a_n a_m + \sum_{n < m < q}^{N-2} a_n a_m a_q + \cdots + a_1 a_2 \cdots a_N . $$ How can I write that in a closed form, i.e. without the $+\cdots + $?

Answer 1

It's not really a closed form, but if you're looking for a compact notation that lists all summands of the expanded product, it'd be this:

$$\sum_{S\subseteq \{1,...,N\}} \prod_{i\in S} a_i$$

(Be aware that the empty product is defined as: $\prod_{i\in \emptyset}a_n$=1)

Answer 2

If $A$ is an NxN matrix with eigenvalues $a_i$ and characteristic polynomial $p_A$ then $$\det(I+A) = p_A(-1)=\prod_{n=1}^N (1+a_n)$$ Also, $\det(I+A)$= sum of all principal minors of $A$