Hyperbolic k-Fibonacci Quaternions

TL;DR

This paper introduces hyperbolic k-Fibonacci quaternions, explores their algebraic properties, and derives several classical identities, expanding the mathematical understanding of these generalized quaternion structures.

Contribution

It defines hyperbolic k-Fibonacci quaternions and investigates their properties, including identities like Binet's and Cassini's, connecting hyperbolic numbers and k-Fibonacci numbers.

Findings

Derived algebraic properties of hyperbolic k-Fibonacci quaternions

Established identities such as D'Ocagne's, Honsberger, Binet's, Cassini's, and Catalan's for these quaternions

Connected hyperbolic numbers with k-Fibonacci numbers in quaternion context

Abstract

In this paper, hyperbolic k-Fibonacci quaternions are defined. Also, some algebraic properties of hyperbolic k-Fibonacci quaternions which are connected with hyperbolic numbers and k-Fibonacci numbers are investigated. Furthermore, D'Ocagne's identity, the Honsberger identity, Binet's formula, Cassini's identity and Catalan's identity for these quaternions are given.