Unrestricted modified third-order Jacobsthal quaternions

TL;DR

This paper introduces a new class of quaternions based on modified third-order Jacobsthal numbers, extending previous work by using arbitrary sequence elements and exploring their properties, including generating functions and Binet-like formulas.

Contribution

It extends the definition of Jacobsthal quaternions to a broader structure using arbitrary sequence elements and introduces complex variants with new properties.

Findings

Derived generating functions for the new quaternions.

Established Binet-like formulas for the complex modified third-order Jacobsthal numbers.

Analyzed properties and potential applications of these quaternions.

Abstract

In this study, we introduce a new class of quaternions associated with the well-known modified third-order Jacobsthal numbers. There are many studies about the quaternions with special integer sequences and their generalizations. All of these studies used consecutive elements of the considered sequences. Here, we extend the usual definitions into a wider structure by using arbitrary modified third-order Jacobsthal numbers. Moreover, we present unrestricted complex modified third-order Jacobsthal numbers. In addition, we give some properties of this type of quaternions and complex modified third-order Jacobsthal numbers, including generating function and Binet-like formula.