On a Generalization for Tribonacci Quaternions

TL;DR

This paper introduces a new quaternion sequence generalizing Tribonacci numbers, providing formulas for its Binet expression, summation, and norm, and unifying several known quaternion sequences.

Contribution

The paper presents a novel quaternion sequence that encompasses Tribonacci, Padovan, Narayana, and Jacobsthal quaternions, with explicit formulas derived.

Findings

Derived Binet formula for the new quaternion sequence.

Established summation and norm formulas.

Unified several known quaternion sequences under a new general framework.

Abstract

Let $V_{n}$ denote the third order linear recursive sequence defined by the initial values $V_{0}$, $V_{1}$ and $V_{2}$ and the recursion $V_{n}=rV_{n-1}+sV_{n-2}+tV_{n-3}$ if $n\geq 3$, where $r$, $s$, and $t$ are real constants. The $\{V_{n}\}_{n\geq0}$ are generalized Tribonacci numbers and reduce to the usual Tribonacci numbers when $r=s=t=1$ and to the $3$-bonacci numbers when $r=s=1$ and $t=0$. In this study, we introduced a quaternion sequence which has not been introduced before. We show that the new quaternion sequence that we introduced includes the previously introduced Tribonacci, Padovan, Narayana and Third order Jacobsthal quaternion sequences. We obtained the Binet formula, summation formula and the norm value for this new quaternion sequence.