Third-order $k$-Jacobsthal matrix sequence: Another way of demonstrating their properties

TL;DR

This paper explores properties of third-order Jacobsthal matrix sequences, introducing new demonstrations and generalizations, including commutative matrix properties for negative indices, expanding understanding of these sequences.

Contribution

It presents a new generalization of the third-order Jacobsthal matrix sequence and identifies additional forms of demonstrating their properties, especially for negative indices.

Findings

Identification of new forms of demonstrating sequence properties

Establishment of commutative matrix properties for negative indices

Introduction of a generalized sequence form

Abstract

Recently, Cerda-Morales \cite{Ce6} introduced commutative matrices derived from the third-order Jacobsthal matrix sequence and the third-order Jacobsthal--Lucas matrix sequence. In the present work, through the identification of certain special matrices, we can identify other forms of demonstration and also the description of commutative matrix properties for negative indices. A new generalization of this sequence is used for our purpose.