Newton-Girard formula for symmetric polynomials

Let Ek be the elementary symmetric polynomials in n variables and Sk be defined by


Then the Sk and Ek are related as follows:

S1 =E1
S2 =S1E1-2E2
S3 =S2E1-S1E2+3E3
Sk =-(j=1k-1(-1)jSk-jEj)-(-1)kkEk

By applying these formulas recursively, Sk can be expressed solely in terms of the Ek, which is often desirable. For example, since S1=E1, S2=E12-2E2, and then S3=(E12-2E2)E1-E1E2+3E3=E13-3E1E2+3E3, and so on.

Note that E0=1 and Ek=0 for k>n.

Title Newton-Girard formula for symmetric polynomialsMathworldPlanetmath
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