On generalized Lucas sequences

We introduce the notions of unsigned and signed generalized Lucas sequences and prove certain polynomial recurrence relations on their characteristic polynomials. We also characterize when these characteristic polynomials are irreducible polynomials over a finite field. Moreover, we obtain the explicit expressions of the remainders of Dickson polynomials of the first kind divided by the characteristic polynomial of generalized Lucas sequences. Using these remainders, we show an application of generalized Lucas sequences in the characterization of a class of permutation polynomials and their compositional inverses.