Faber polynomials, Cayley-Hamilton equation and Newton symmetric functions

We give an explicit formula for the Faber polynomials and for generalized Faber polynomials introduced by H. Airault and J. Ren in H. Airault, J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math. 126 (5) (2002) 343-367]. We introduce a new family of polynomials related to the Faber polynomials of the second kind. This allows us to give a generalized Cayley-Hamilton equation.