Let $$\{x_1,x_2,x_3,...,x_d\}$$ be an array of $`d'$ non-negative real numbers and $`f'$ and $`g'$ be some real valued functions. If I wanted to express the following sum;
$$\{f\left(x_1\right)\times g\left(x_2\right)\times g\left(x_3\right)...\times g\left(x_d\right)\,+\\
\,g\left(x_1\right)\times f\left(x_2\right)\times g\left(x_3\right)...\times g\left(x_d\right)\,\,\,+\\
\,g\left(x_1\right)\times g\left(x_2\right)\times f\left(x_3\right)...\times g\left(x_d\right)\,\,\,+\\
...\\...\\...\\
\,g\left(x_1\right)\times g\left(x_2\right)\times g\left(x_3\right)...\times f\left(x_d\right)\,\,\,\}\\
$$
in a compact form (may be in terms of generating functions?!), is there any way to do so?
(Since $f(x_i)\ne f(x_j)$ for general $i$ and $j$, I cannot take an usual combinatorial sum)
Thank you very much in advance!
How about $$\left(\displaystyle \prod_{i=1}^d g(x_i)\right) \left(\displaystyle \sum_{i=1}^d \frac{f(x_i)}{g(x_i)} \right)$$