Generating Function for Elementary Symmetric Function

Theorem
Let $U$ be a set of $n$ numbers $\set {x_1, x_2, \ldots, x_n}$.

Let $\map {e_m} U$ be the elementary symmetric function of degree $m$ on $U$:

Let $a_m := \map {e_m} U$ for $m = 0, 1, 2, \ldots$

Let $\map G z$ be a generating function for the sequence $\sequence {a_m}$:


 * $\ds \map G z = \sum_{m \mathop = 0}^\infty a_m z^m$

Then:


 * $\ds \map G z = \prod_{k \mathop = 1}^n \paren {1 + x_k z}$