# Waring’s formula

Let $x_{1},\ldots,x_{n}$ be $n$ indeterminates. For $k\geq 1$, let $\sigma_{k}$ be the $k$th elementary symmetric polynomials in $x_{1},\ldots,x_{n}$, and $S_{k}$ be the $k$th power sum defined as

 $S_{k}=\sum_{i=1}^{n}x_{i}^{k}.$

Like the Newton’s formula, the Waring formula is a relation between $\sigma_{k}$ and $S_{k}$:

 $S_{k}=\sum(-1)^{(i_{2}+i_{4}+i_{6}+\ldots)}\frac{(i_{1}+i_{2}+\ldots+i_{n}-1)!% k}{i_{1}!i_{2}!\cdots i_{n}!}\sigma_{1}^{i_{1}}\sigma_{2}^{i_{2}}\cdots\sigma_% {n}^{i_{n}},$

where the summation is over all $n$-tuples $(i_{1},\ldots,i_{n})\in\mathbb{Z}^{n}$ with non-negative components, such that

 $i_{1}+2i_{2}+\ldots+ni_{n}=k.$

In particular, when there are two indeterminates, i.e. $n=2$, the Waring formula reads

 $x_{1}^{k}+x_{2}^{k}=\sum_{i=0}^{\lfloor k/2\rfloor}(-1)^{i}\frac{k}{k-i}\binom% {k-i}{i}(x_{1}+x_{2})^{k-2i}(x_{1}x_{2})^{i}.$
Title Waring’s formula WaringsFormula 2013-03-22 15:34:26 2013-03-22 15:34:26 alozano (2414) alozano (2414) 9 alozano (2414) Theorem msc 11C08 Waring formula NewtonGirardFormulaSymmetricPolynomials