Waring’s formula

Let $x_1,\mathrm{\dots },x_n$ be $n$ indeterminates. For $k\ge 1$, let${\sigma }_k$ be the $k$th elementary symmetric polynomials in $x_1,\mathrm{\dots },x_n$, 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 relationbetween ${\sigma }_k$ and $S_k$:

$$S_k=\sum {(-1)}^{(i_2+i_4+i_6+\mathrm{\dots })}\frac{(i_1+i_2+\mathrm{\dots }+i_n-1)!k}{i_1!i_2!\mathrm{\cdots }{i}_n!}{\sigma }_1^{i_1}{\sigma }_2^{i_2}\mathrm{\cdots }{\sigma }_n^{i_n},$$

where the summation is over all $n$-tuples $(i_1,\mathrm{\dots },i_n)\in {\mathbb{Z}}^n$ with non-negative components, such that

$$i_1+2i_2+\mathrm{\dots }+n{i}_n=k.$$

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

$$x_1^k+x_2^k=\sum _{i=0}^{\lfloor k/2\rfloor }{(-1)}^i\frac{k}{k-i}\left(\genfrac{}{}{0pt}{}{k-i}{i}\right){(x_1+x_2)}^{k-2i}{(x_1{x}_2)}^i.$$