On the third-order Jacobsthal and third-order Jacobsthal-Lucas sequences and their matrix representations

TL;DR

This paper introduces new generalizations of third-order Jacobsthal and Jacobsthal-Lucas sequences, explores their matrix representations, and investigates their mathematical properties and behaviors.

Contribution

It provides novel generalizations of third-order Jacobsthal sequences and develops their matrix representations, analyzing their properties for the first time.

Findings

New generalizations of third-order Jacobsthal and Lucas sequences

Matrix representations of these sequences

Properties and behaviors of the matrix sequences

Abstract

In this paper, we first give new generalizations for third-order Jacobsthal $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and third-order Jacobsthal-Lucas $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for Jacobsthal and Jacobsthal-Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$. Then we investigate their properties.