[Paper Review] Bicomplex Third-order Jacobsthal Quaternions

Gamaliel Cerda|arXiv (Cornell University)|Sep 19, 2018

Advanced Mathematical Theories and Applications3 citations

TL;DR

This paper introduces bicomplex third-order Jacobsthal quaternions, a novel sequence defined by a third-order linear recurrence over bicomplex numbers. It derives Binet-style formulas, generating functions, Cassini’s and d’Ocagne’s identities, and presents a determinant-based method using four-diagonal matrices to compute the nth term, offering a new computational approach for this hypercomplex sequence.

ABSTRACT

The aim of this work is to consider the bicomplex third-order Jacobsthal quaternions and to present some properties involving this sequence, including the Binet-style formulae and the generating functions. Furthermore, Cassini's identity and d'Ocagne's identity for this type of bicomplex quaternions are given, and a different way to find the $n$-th term of this sequence is stated using the determinant of a four-diagonal matrix whose entries are bicomplex third-order quaternions.

Motivation & Objective

• To define and study bicomplex third-order Jacobsthal quaternions as a generalization of third-order Jacobsthal numbers in the bicomplex algebra.
• To establish Binet-style formulae and generating functions for the sequence.
• To prove Cassini’s and d’Ocagne’s identities for the bicomplex quaternion sequence.
• To present an alternative method for computing the nth term using determinants of four-diagonal matrices with bicomplex quaternion entries.

Proposed method

• The bicomplex third-order Jacobsthal quaternions are defined via a recurrence relation: $\mathrm{BC}_{J,n}^{(3)} = \mathrm{BC}_{J,n-1}^{(3)} + \mathrm{BC}_{J,n-2}^{(3)} + 2\mathrm{BC}_{J,n-3}^{(3)}$ with initial values $\mathrm{BC}_{J,0}^{(3)} = i + j + 2ij$, $\mathrm{BC}_{J,1}^{(3)} = 1 + i + 2j + 5ij$, $\mathrm{BC}_{J,2}^{(3)} = 1 + 2i + 5j + 9ij$.
• The Binet formula is derived using the roots $\omega_1, \omega_2$ of the characteristic equation $x^3 - x^2 - x - 2 = 0$, expressing $\mathrm{BC}_{J,n}^{(3)}$ as a linear combination of $\widehat{2} \cdot 2^n$, $\widehat{\omega_1} \cdot \omega_1^n$, and $\widehat{\omega_2} \cdot \omega_2^n$.
• Generating functions are constructed based on the recurrence and initial conditions, providing a formal power series representation of the sequence.
• Cassini’s and d’Ocagne’s identities are proven using algebraic manipulation of the Binet formula and properties of the bicomplex quaternions.
• A novel method for computing the nth term is introduced using the determinant of an $(n+1) \times (n+1)$ four-diagonal matrix with entries derived from the initial terms and recurrence coefficients.
• The matrix formulation uses coefficients $r=1$, $s=1$, $t=2$, and incorporates inverses and scaled terms to maintain consistency with the recurrence.

Experimental results

Research questions

• RQ1 How can the bicomplex third-order Jacobsthal quaternion sequence be formally defined and characterized algebraically within the bicomplex number system?
• RQ2 What is the closed-form Binet-style formula for the nth term of this sequence?
• RQ3 What are the generating functions and identities (e.g., Cassini’s, d’Ocagne’s) satisfied by this sequence?
• RQ4 Can the nth term be computed via a determinant of a structured matrix, and if so, what is the form of this matrix?
• RQ5 How do the algebraic properties of bicomplex numbers influence the behavior and structure of this quaternion sequence?

Key findings

• The Binet formula for the bicomplex third-order Jacobsthal quaternions is given by $\mathrm{BC}_{J,n}^{(3)} = \frac{1}{7}\left(\widehat{2} \cdot 2^{n+1} - \frac{\widehat{\omega_1} \omega_1^n - \widehat{\omega_2} \omega_2^n}{\omega_1 - \omega_2}\right)$, where $\widehat{2}, \widehat{\omega_1}, \widehat{\omega_2}$ are bicomplex quaternions derived from the roots of the characteristic equation.
• The generating function for the sequence is derived as a formal power series based on the recurrence and initial conditions, enabling analytical manipulation of the sequence.
• Cassini’s identity and d’Ocagne’s identity are established for the sequence, extending classical identities to the bicomplex quaternion setting.
• A new method to compute the nth term is provided via the determinant of an $(n+1) \times (n+1)$ four-diagonal matrix with entries involving $\mathrm{BC}_{J,0}^{(3)}$, $\mathrm{BC}_{J,1}^{(3)}$, $\mathrm{BC}_{J,2}^{(3)}$, and scaled coefficients from the recurrence.
• The identity $\left(\mathrm{BC}_{J,n}^{(3)}\right)^2 + \left(\mathrm{BC}_{J,n+1}^{(3)}\right)^2 + \left(\mathrm{BC}_{J,n+2}^{(3)}\right)^2 = \frac{1}{7}\left(3\cdot\widehat{2}^2 \cdot 2^{2n+2} - \widehat{2} \cdot 2^{n+2} \cdot \mathrm{BC}_{U,n}^{(3)} + 2ij\right)$ is proven, linking the sum of squares to a linear combination of bicomplex quaternions.
• The sequence satisfies $V_n^{(3)} = V_{n+3}^{(3)}$ and $V_n^{(3)} = 2$ if $n \equiv 0 \pmod{3}$, $-3$ if $n \equiv 1 \pmod{3}$, and $1$ if $n \equiv 2 \pmod{3}$, which plays a key role in the Binet formula.