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 $$ x_1^k + x_2^k = \sum_{i=0}^{\lfloor k/2 \rfloor} (-1)^i\frac{k}{k-i}\binom{k-i}{i}(x_1+x_2)^{k-2i}(x_1x_2)^i. $$