Dual Third-order Jacobsthal Quaternions

TL;DR

This paper introduces dual third-order Jacobsthal quaternions, explores their properties, and establishes relations with third-order Jacobsthal numbers, including formulas and identities, expanding quaternion number theory.

Contribution

The paper defines dual third-order Jacobsthal quaternions and investigates their properties, relations, and identities, which is a novel extension in quaternion and number theory.

Findings

Derived Binet's formulas for these quaternions

Established Cassini-like identities

Presented summation formulas and quadratic properties

Abstract

In 2016, Y\"uce and Torunbalc\i\ Ayd\i n \cite{Yuc-Tor} defined dual Fibonacci quaternions. In this paper, we defined the dual third-order Jacobsthal quaternions and dual third-order Jacobsthal-Lucas quaternions. Also, we investigated the relations between the dual third-order Jacobsthal quaternions and third-order Jacobsthal numbers. Furthermore, we gave some their quadratic properties, the summations, the Binet's formulas and Cassini-like identities for these quaternions.