Gaussian Mersenne numbers and generalized Mersenne quaternions (original) (raw)

On a Generalization for Quaternion Sequences

arXiv: Rings and Algebras, 2016

In this study, we introduce a new classes of quaternion numbers. We show that this new classes quaternion numbers include all of quaternion numbers such as Fibonacci, Lucas, Pell, Jacobsthal, Pell-Lucas, Jacobsthal-Lucas quaternions have been studied by many authors. Moreover, for this newly defined quaternion numbers we give the generating function, norm value, Cassini identity, summation formula and their some properties.

A note on Gaussian and Quaternion Repunit Numbers

RMAT, 2024

This work introduces two new sequences: the gaussian repunit numbers and the quaternion repunit numbers. Weestablish some properties of these sequences, as well as, recurrence relations, the Binet formula, and Catalan’s,Cassini’s, and d’Ocganes identities.

On the Mersenne and Mersenne-Lucas hybrinomial quaternions

Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science

In this paper, we introduce Mersenne and Mersenne-Lucas hybrinomial quaternions and present some of their properties. Some identities are derived for these polynomials. Furthermore, we give the Binet formulas, Catalan, Cassini, d’Ocagne identity and generating and exponential generating function of these hybrinomial quaternions.

Oresme Hybrid Quaternion Numbers

In literature until today, many authors have studied special sequences in different number systems. In this paper, we have introduced the Oresme hybrid quaternion numbers. We give some properties and identities such as Binet’s formula, generating function, norm and characteristic equation for these quaternions. Furthermore, matrix and determinant forms for these quaternion numbers are given.

On Quaternion-Gaussian Fibonacci Numbers and Their Properties

2021

We study properties of Gaussian Fibonacci numbers. We start with some basic identities. Thereafter, we focus on properties of the quaternions that accept gaussian Fibonacci numbers as coefficients. Using the Binet form we prove fundamental relations between these numbers. Moreover, we investigate whether the quaternions newly defined provide existing some important identities such as Cassini’s identity for quaternions.

On a generalization for fibonacci quaternions

Chaos, Solitons & Fractals, 2017

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 Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas quaternion sequences. We obtained the Binet formula and calculated the Cassini identity, summation formula and the norm value for this new quaternion sequence.

On a Generalization of Fibonacci and Lucas Quaternions

In this study, we define a type of generalized quaternions whose coefficients are a generalization of Fibonacci and Lucas quaternions. We give Binet-like formulas and generating functions for these kind of quaternions. By using Binet-like formulas, we obtain generalizations of some well-known identities such as, Vajda's, Catalan's, Cassini's and d'Ocagne's identities.

On hyperbolic k-Pell quaternions sequences

Annales Mathematicae et Informaticae

In this paper we introduce the hyperbolic k-Pell functions and new classes of quaternions associated with this type of functions are presented. In addition, the Binet formulas, generating functions and some properties of these functions and quaternions sequences are studied.

On Quaternion Gaussian Bronze Fibonacci Numbers

Annales Mathematicae Silesianae

In the present work, a new sequence of quaternions related to the Gaussian Bronze numbers is defined and studied. Binet’s formula, generating function and certain properties and identities are provided. Tridiagonal matrices are considered to determine the general term of this sequence.

On a Special Quaternionic Sequence

Universal Journal of Applied Mathematics, 2018

In this study, we investigate Fibonacci quaternions and their some important properties. Then, we define a special sequence using the elements of the Fibonacci quaternion sequence. Furthermore, we calculate the autocorrelation, right and left periodic autocorrelation values by using the elements of the newly defined sequence.