Determinants and permanents of Hessenberg matrices and generalized Lucas polynomials (Q469104)

Let \(k\) be a positive integer. \textit{T. MacHenry} [Rocky Mt. J. Math. 29, No. 3, 1055--1065 (1999; Zbl 0989.11002)] defined the generalized Lucas polynomials \(G_{k,n}(t)\), where \(n\) is an integer and \(t=(t_1,\dots,t_k)\), inductively by \(G_{k,n}(t)=0\) for \(n<0\), \(G_{k,0}(t)=k\), \(G_{k,n}(t)=G_{k-1,n}(t)\) for \(1\leq n<k\), and \(G_{k,n}(t)=t_1G_{k,n-1}(t)+\dots+t_kG_{k,n-k}(t)\) for \(n\geq k\). The present authors express these and some related polynomials as determinants and permanents of certain Hessenberg matrices.