A Matrix Approach for General Higher Order Linear Recurrences

Abstract

We consider k sequences of generalized order-k linear recurrences with arbitrary initial conditions and coefficients, and we give their generalized Binet formulas and generating functions. We also obtain a new matrix method to derive explicit formulas for the sums of terms of the k sequences. Further, some relationships between determinants of certain Hessenberg matrices and the terms of these sequences are obtained.