# Combinatorial sum and generating functions

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!

• something like $$\sum _{i=1}^nf(x_i)\prod _{j\neq i}g(x_j)$$?? – Phicar Sep 13 '17 at 21:23

How about $$\left(\displaystyle \prod_{i=1}^d g(x_i)\right) \left(\displaystyle \sum_{i=1}^d \frac{f(x_i)}{g(x_i)} \right)$$