Powers of a Matrix and the Generalized Lucas Polynomials

The functions $L_{\mathrm{nk}}^{\mathrm{(r)}}{\mathrm{(\Phi }}_1{\mathrm{,\cdots ,\Phi }}_n)$ are defined by $X^r{\mathrm{=\sum }}_{\text{k=1}}^n{L}_{\mathrm{nk}}^{\mathrm{(r)}}{X}^{\mathrm{n-k}}$ , where X is an indeterminate n × n matrix and Φ₁, ⋯, Φ_n are the invariants of X (basic symmetric functions in the eigenvalues of X). In this paper the generalized Lucas polynomial $L_{\text{n1}}^{\mathrm{(r)}}$ is expressed explicitly as a determinant of order r − n + 1 or as a ratio of two determinants of order n.