Generating Function for Elementary Symmetric Function/Proof 1
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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 $U$:
\(\ds \map {e_m} U\) | \(=\) | \(\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n} \paren {\prod_{i \mathop = 1}^m x_{j_i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n} x_{j_1} x_{j_2} \cdots x_{j_m}\) |
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:
- $\ds \map G z = \prod_{k \mathop = 1}^n \paren {1 + x_k z}$
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 $x = -1 / z$:
\(\ds \prod_{k \mathop = 1}^n \paren {-\frac 1 z - x_k}\) | \(=\) | \(\ds \paren {-z}^{-n} + \sum_{m \mathop = 0}^{n - 1} \paren {-1}^{n - m} \map {e_{n - m} } U \, \paren {-z}^{-m}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \prod_{k \mathop = 1}^n \paren {1 + x_k z}\) | \(=\) | \(\ds z^n + \sum_{m \mathop = 0}^{n - 1} \map {e_{n - m} } U \, \paren z^{n - m}\) | all signs cancel | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \prod_{k \mathop = 1}^n \paren {1 + x_k z}\) | \(=\) | \(\ds \sum_{m \mathop = 0}^n \map {e_m} U \, z^m\) |
$\blacksquare$