An Explicit Formula for $f(\mathcal{A})$ and the Generating Functions of the Generalized Lucas Polynomials

From $\mathcal{A}^n = \sum_{k = 1}^r {F_{k,n - 1} } ( {I_1 , \cdots ,I_r } )\mathcal{A}^{r - k} $, where $\mathcal{A}$ is a $r \times r$ matrix and $I_1 , \cdots ,I_r $ are the invariants of $\mathcal{A}$ (elementary symmetric functions of the eigenvalues), we first derive a formula for $f(\mathcal{A})$. Then we obtain the generating functions for the $F_{k,n} $ and thence for the generalized Lucas polynomials $F_{1,n} ,n \geqq - 1$.