Generating Function for Elementary Symmetric Function

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

Define:

Then:

Explanation
Generating function discovery methods can find a formula for $\map G z$.

Let $n=1$.

Then $U$ is a singleton set $\set {x_1}$.

Expand the formal series:

Product of Generating Functions and experience with elementary symmetric functions suggests:


 * $\displaystyle \map G z = \paren {1 + x_1 z} \paren {1 + x_{2} z} \cdots \paren {1 + x_n z}$

Knuth (1997) section 1.2.9 discusses the technique and the issue of a valid proof.