The combinatorial power of the companion matrix

Abstract

Using combinatorial methods, we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to Waring's formula on symmetric functions, the general solution to homogeneous linear recurrence relations, the multiplicative inverse of formal power series, the generating function of compositions (of numbers), a unified approach to Chebyshev polynomials including two recently discovered classes that satisfy analogous smallest-norm and orthogonality properties subject to different weight functions, Dickson polynomials of various kinds arising from the theory of finite fields, combinatorial expansions of Toeplitz matrices, and the recent notion of cycle dissections involving a bijective study of Waring's formula.