Determinants and permanents of Hessenberg matrices and generalized Lucas polynomials

In this paper, we give some determinantal and permanental representations of generalized Lucas polynomials, which are a general form of generalized bivariate Lucas p polynomials, ordinary Lucas and Perrin sequences etc., by using various Hessenberg matrices. In addition, we show that determinant and permanent of these Hessenberg matrices can be obtained by using combinations. Then we show, the conditionsunderwhich the determinants of the Hessenberg matrix become its permanents.