Category:Generating Function for Elementary Symmetric Function

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

$\ds \map {e_m} U$

$=$

$\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n} \paren {\prod_{i \mathop = 1}^m x_{j_i} }$

$\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:

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

Pages in category "Generating Function for Elementary Symmetric Function"

The following 5 pages are in this category, out of 5 total.