Newton-Girard formula for symmetric polynomials

Let $E_{k}$ be the elementary symmetric polynomials in $n$ variables and $S_{k}$ be defined by

S_{k}(x_{1},...,x_{n})=\sum_{i=1}^{n}{x_{i}^{k}}.

Then the $S_{k}$ and $E_{k}$ are related as follows:

	$\displaystyle S_{1}$	$\displaystyle=E_{1}$
	$\displaystyle S_{2}$	$\displaystyle=S_{1}E_{1}-2E_{2}$
	$\displaystyle S_{3}$	$\displaystyle=S_{2}E_{1}-S_{1}E_{2}+3E_{3}$
		$\displaystyle\vdots$
	$\displaystyle S_{k}$	$\displaystyle=-\left(\sum_{j=1}^{k-1}{(-1)^{j}S_{k-j}E_{j}}\right)-(-1)^{k}kE_% {k}$

By applying these formulas recursively, $S_{k}$ can be expressed solely in terms of the $E_{k}$ , which is often desirable. For example, since $S_{1}=E_{1}$ , $S_{2}=E_{1}^{2}-2E_{2}$ , and then $S_{3}=(E_{1}^{2}-2E_{2})E_{1}-E_{1}E_{2}+3E_{3}=E_{1}^{3}-3E_{1}E_{2}+3E_{3}$ , and so on.

Note that $E_{0}=1$ and $E_{k}=0$ for $k>n$ .

Title	Newton-Girard formula for symmetric polynomials
Canonical name	NewtonGirardFormulaForSymmetricPolynomials
Date of creation	2013-03-22 15:32:40
Last modified on	2013-03-22 15:32:40
Owner	kschalm (9486)
Last modified by	kschalm (9486)
Numerical id	5
Author	kschalm (9486)
Entry type	Theorem
Classification	msc 11C08
Related topic	WaringsFormula
Related topic	ElementarySymmetricPolynomialInTermsOfPowerSums