Explicit Newton's formula for \({\mathfrak gl}_n\) (Q1273402)

Let \(Z_n\) be the center of the universal enveloping algebra \(U({\mathfrak {gl}}_n)\). There are two classical sets of generators known for \(Z_n\). The first one is the set of Capelli elements (\(\{C_i\}\)), and the second one is the set of Gelfand's trace generators (\(\{P_i\}\)). From the viewpoint of symmetric polynomials, these sets correspond to the elementary symmetric polynomials and the power sums. Based on the classical Newton formula for symmetric polynomials, the author obtains two analogues of it connecting \(\{C_i\}\) and \(\{P_i\}\). These are the special explicit relations between \(\{C_i\}\) and \(\{P_i\}\) in \(U({\mathfrak {gl}}_n)\).