Algebraic properties of bi$-$periodic dual Fibonacci quaternions

TL;DR

This paper introduces bi-periodic dual Fibonacci quaternions, a new generalized quaternion representation, and explores their algebraic properties, including generating functions, Binet formulas, and Catalan's identities.

Contribution

It presents a novel class of dual quaternions called bi-periodic dual Fibonacci quaternions, extending existing quaternion types with new algebraic properties and formulas.

Findings

Defined bi-periodic dual Fibonacci quaternions

Derived generating functions and Binet formulas

Established Catalan's identity for these quaternions

Abstract

The purpose of the paper is to construct a new representation of dual quaternions called bi$-$periodic dual Fibonacci quaternions. These quaternions are originated as a generalization of the known quaternions in literature such as dual Fibonacci quaternions, dual Pell quaternions and dual $k-$Fibonacci quaternions. Furthermore, some of them have not been introduced until this time. Then, we give generating function, Binet formula and Catalan's identity in terms of these quaternions.