# 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 NewtonGirardFormulaForSymmetricPolynomials 2013-03-22 15:32:40 2013-03-22 15:32:40 kschalm (9486) kschalm (9486) 5 kschalm (9486) Theorem msc 11C08 WaringsFormula ElementarySymmetricPolynomialInTermsOfPowerSums