Generating Function for Elementary Symmetric Function/Proof 1

Proof
The summation for $\map G z$ is a finite sum $m = 0, 1, \ldots, n$, which settles convergence issues.

Begin with Viète's Formulas:


 * $\ds \prod_{k \mathop = a}^b \paren {x - x_k} = x^n + \sum_{m \mathop = 0}^{n - 1} \paren {-1}^{n - m} \map {e_{n - m} } U \, x^m$

Change variables $x = -1 / z$: