# proof of Newton-Girard formula for symmetric polynomials

The following is a proof of Newton-Girard formula using formal power series. Let $z$ be an indeterminate and $f(z)$ be the polynomial

 $1-E_{1}z+\ldots+(-1)^{n}E_{n}z^{n}.$

Take log and differentiate both sides of the equation

 $f(z)=\prod_{i=1}^{n}(1-x_{i}z).$

We obtain

 $f^{\prime}(z)/f(z)=\sum_{i=1}^{n}\frac{-x_{i}}{1-x_{i}z},$ (1)

where $f^{\prime}(z)$ is the derivative of $f(z)$

 $f^{\prime}(z)=-E_{1}+2E_{2}z-\ldots+(-1)^{n}nE_{n}z^{n-1}.$

The right hand side of (1) is equal to

 $-\sum_{i=1}^{n}\sum_{k=0}^{\infty}x_{i}^{k+1}z^{k}=-\sum_{k=0}^{\infty}S_{k+1}% z^{k}.$

By equating coefficients of

 $f^{\prime}(z)=-f(z)(S_{1}+S_{2}z+S_{3}z^{2}+\ldots)$

we get the Newton-Girard formula.

Title proof of Newton-Girard formula for symmetric polynomials ProofOfNewtonGirardFormulaForSymmetricPolynomials 2013-03-22 15:34:37 2013-03-22 15:34:37 kshum (5987) kshum (5987) 4 kshum (5987) Proof msc 11C08
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