Some Properties of Hyperbolic Generalized Tribonacci Quaternions

Abstract

In this independent study, we introduce the concept of hyperbolic generalized tribonacci quaternions, a novel extension within the realm of hyperbolic quaternion sequences, denoted by HW_n, defined by

HW_n=W_n+(W_{n+1}){j_1}+(W_{n+2}){j_2}+(W_{n+3}){j_3}, n≥0,

where W_n is the tribonacci number and {j_1}, {j_2}, {j_3} satisfy equalities {j_1}^2}={j_2}^2}={j_3}^2}={j_1j_2j_3}=1, {j_1j_2}={j_3}=-{j_2j_1}, {j_2j_3}={j_1}=-{j_3j_2}, {j_3j_1}={j_2}=-{j_1j_3}.

Through rigorous analysis, we derive the generating function for hyperbolic generalized tribonacci quaternions and establish Binet's formula, providing a closed-form expression for the terms of the sequence. Furthermore, we present a comprehensive summation formula, enhancing the utility and application of these quaternions in mathematical studies. This expanded framework offers deeper insights and broader applicability, contributing to the advancement of research in hyperbolic quaternion sequences and their related fields.