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 + $?
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$\begingroup$ Notice that if you define $g(x)=\prod_{n=1}^N (x+a_n)$, then $P=g(1)$,I'm not sure what's the motivation for getting a closed form representation for the value of a polynomial at some point $\endgroup$– A S DCommented Jul 19, 2021 at 10:22
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3$\begingroup$ You can write it is $ 1+\sum_{i=1}^N \sigma_i $, where the $\sigma_i$s denote the elementary symmetric functions of $a_1,\dots, a_N$. $\endgroup$– BernardCommented Jul 19, 2021 at 10:28
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$\begingroup$ You've a representation of form $P=1+\sum_{j=1}^{n}\sigma_{j} (x)$ where $\sigma_j$ represent elementary symmetric polynomials in one variable, but I'm not sure, whether, this is of any help $\endgroup$– A S DCommented Jul 19, 2021 at 10:28
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2$\begingroup$ How about $$\sum_{k=0}^N \sum_{1\le i_1<i_2<\cdots<i_k\le N} \prod_{j=1}^k a_{i_j}?$$ $\endgroup$– ChristophCommented Jul 19, 2021 at 10:59
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2 Answers
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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)
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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 $A$
