A generalized Perrin polynomial sequence and its two-dimensional recurrences

Abstract

In this work, the generating matrices for the positive integers of the Perrin polynomial sequence will be investigated, as well as the generalization of their matrix form, extending to the field of non-positive integers. The similarity of Perrin’s sequence with Padovan’s is highlighted, differentiating it only in relation to its initial terms. And yet, there is a discussion about the two-dimensional, or Gaussian, recurring relations of Perrin’s numbers, related to Padovan’s numbers, based on his one-dimensional model. Finally, we introduce new relationships derived from two-dimensional recurrence and some identities resulting from it.