Notation for multiple summation for two functions

Question

Suppose I have two functions, namely $f_1(x_i)$ and $f_2(x_i)$. In addition, I also have $n$ numbers of $x$ (i.e., $x_1, x_2, ..., x_n$). Is there an alternate way to represent the multiple summations given below?

$$ \sum_{i_n=1}^{2}...\sum_{i_2=1}^{2}\sum_{i_1=1}^{2}{f_{i_1}(x_1)\cdot f_{i_2}(x_2) ... f_{i_n}(x_n)} $$

Thanks!

Answer 1

You can sum over $n$-uples of $1$'s and $2$'s:

$$ \sum_{i \in \{1,2\}^n} f_{i_1}(x_1) \cdot \ldots \cdot f_{i_n}(x_n) $$

Answer 2

Yes, you can further shorten this by writing $$ \sum_{i \in \{1,2\}^n} \prod_{j=1}^{n} f_{i_j} (x_j) $$ but maybe this may be hard to read for some.