Newton's Identities

Theorem
Let $X$ be a set of $n$ numbers $\set {x_1, x_2, \ldots, x_n}$.

Define:

Then Newton's Identities are:

Historical Note
Isaac Newton published in Arithmetica universalis (1707) a generalization of the $ n \le 4$ formulas of A. Girard (1629), without proof. Formulas (1)-(2) make it possible to recursively solve for the symmetric functions $\set {e_k} $ in terms of power sums $\set {p_k}$ (Knuth 1997, p 94) and conversely. A history of Girard's work is in Funkhouser (1930). Various proofs of (1)-(2) exist: Baker (1959), Eidswick (1968), Mead (1992), Kalman (2000), Tignol (2001). Boklan (2018) reported calculus differentiation recursions to directly generate Girard's identities.

Also named as
Newton-Girard Identities, Newton-Girard Formulas.