What is a formal way of notating the expansion of the following product?
$$p_m = \prod^m_{n=1} (1 + a_n).$$
Alternatively, what is the formal way of notating a product operator over combinations of indices?
Context.
Taking $m = 3$, the expansion is
$$\begin{aligned}
p_3 &= (1 + a_1)(1 + a_2)(1+ a_3) \\
&= 1 + a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_1a_3 + a_1a_2a_3
\end{aligned}$$
With a view to finding an expression for $p_m$ I tried using a sum-product to get something a bit like
$$p_3 = 1 + \sum^3_{n=1} \prod^n_{i = ???}(a_i).$$
But I am unsure how to express the fact that I only want a product over combinations of indices. However I can certainly recall seeing probability and statistics papers using a clean notation to express this idea. However, I can't remember what they are and Googling for "product operator over combinations of indices" doesn't yield fruit. I would appreciate some assistance on this.
Best Answer
I would simply write $$ \prod_{n=1}^m (1+a_n) = \sum_{S\subseteq \{1,2,\dots,m\}} \prod_{n\in S} a_n. $$ But if you wanted to collect the terms corresponding to subsets of a particular size, I would write $$ \prod_{n=1}^m (1+a_n) = 1 + \sum_{k=1}^m \sum_{\substack{S\subseteq \{1,2,\dots,m\} \\ \#S = k}} \prod_{n\in S} a_n. $$