Compact notation for this "sums of tuples"

Question

I would like to know if you could suggest a compact notation for the following sequence of functions (explanation below).

$ f(a_1) = g(a_1) - g(-a_1) $

$ f(a_1,a_2) = g(a_1,a_2) - g(-a_1,a_2) -g(a_1,-a_2) + g(-a_1,-a_2) $

$ f(a_1,a_2,a_3) = g(a_1,a_2,a_3) - g(-a_1,a_2,a_3) - g(a_1,-a_2,a_3) - g(a_1,a_2,-a_3) + g(-a_1,-a_2,a_3) + g(-a_1,a_2,-a_3) + g(a_1,-a_2,-a_3) - g(-a_1,-a_2,-a_3) $

$ f(a_1,...,a_n) = \sum ... ?$

Note that the function $f$ takes $n$ arguments, and is written as $2^n$ functions $g$. Furthermore, the arguments inside the functions $g$ have all possible tuples of signs $\pm$, and the sign in front of the $g$ depends itself on the parity of the signs of the arguments.

Answer 1

You seem to allow the dots in $f(a_1, \ldots, a_n)$, so I shall do the same. How about

$$f(a_1, \ldots, a_n) = \sum_{s_i\in\{-1,1\}\text{ for }1\le i\le n}\left(\prod_{i=1}^n s_i \right)g(s_1\cdot a_1,\ldots,s_n\cdot a_n)$$

If it is not clear, each $s_i$ is either $-1$ or $1$, the summation is taken over all possible combinations of n of them (so $2^n$ terms), and the product is done for just one such combination at a time. Another way to write the limitations on the $s$'s under the summation would be

$$(s_1,\ldots,s_n)\in\{-1,1\}^n$$

but that presupposes knowledge of the function notation from one set to another (from $\{1,\ldots,n\}$ to $\{-1,1\}$).