Third-order Jacobsthal $3$-Parameter Generalized Quaternions
Gamaliel Morales
TL;DR
The paper addresses the fusion of third-order Jacobsthal sequences with 3-parameter generalized quaternions by defining TJ3p and MTJ3p generalized quaternions $J\mathcal{G}_n^{(3)}$ and $K\mathcal{G}_n^{(3)}$, built from $J_n^{(3)}$ and $K_n^{(3)}$. It develops generating functions with $\sigma(x)=1-x-x^2-2x^3$, provides Binet-type closed forms using a constant $\Theta$ and auxiliary sequences $X_n$, and establishes a suite of identities including Vajda-type, Cassini, Catalan-type, and summation formulas, plus inter-relations between the two quaternion families. The main contributions are the explicit quaternion-valued recurrences, their closed forms, and a rich set of algebraic identities that extend scalar TJ3p results to quaternion-valued sequences. This framework advances the study of recurrence-based quaternions and offers tools potentially applicable in algebraic combinatorics and number theory, with future work suggested on higher-order generalized tribonacci 3PGQs.
Abstract
The purpose of this article is to bring together the third-order Jacobsthal numbers and 3-parameter generalized quaternions, which are a general form of the quaternion algebra according to 3-parameters. With this purpose, we introduce and examine a new type of quite big special numbers system, which is called third-order Jacobsthal 3-parameter generalized quaternions (shortly, third-order Jacobsthal 3PGQs). Further, we compute both some new equations and classical well-known equations such as: linear recurrence, Binet formulas, generating function, sum formulas, Cassini identity and d'Ocagne identity.