Newton's identities | Newton's relations

The identities between the symmetric polynomials Π_k(x_1, ..., x_n) and the sums of kth powers of their variables S_k(x_1, ..., x_n) = sum_(j = 1)^n x_j^k. The identities are given by (-1)^m m Π_m(x_1, ..., x_n) + sum_(k = 1)^m (-1)^(k + m) S_k(x_1, ..., x_n) Π_(m - k)(x_1, ..., x_n) = 0 for each 1<=m<=n and for an arbitrary number of variables n. The first few identities are S_1 - Π_1 | = | 0 S_2 - S_1 Π_1 + 2Π_2 | = | 0 S_3 - S_2 Π_1 + S_1 Π_2 - 3Π_3 | = | 0 S_4 - S_3 Π_1 + S_2 Π_2 - S_1 Π_3 + 4Π_4 | = | 0.

power sum | symmetric polynomial