sequence of the hyperbolic k-Padovan quaternions (original) (raw)
The sequence of the hyperbolic k-Padovan quaternions
2023
This work introduces the hyperbolic k-Padovan quaternion sequence, performing the process of complexification of linear and recurrent sequences, more specifically of the generalized Padovan sequence. In this sense, there is the study of some properties around this sequence, deepening the investigative mathematical study of these numbers.
The Sequences of the Hyperbolic k-Perrin and k-Leonardo Quaternions
In this article, hyperbolic k-Perrin and k-Leonardo quaternions are defined. In this sense, bulletin, are evaluated as sequences of k-Perrin and k-Leonardo, their respective quaternions and thus the definition of hyperbolic quaternions. Thus, there are some algebraic properties around numbers, generating function, Binet's formula and properties inherent to these numbers.
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 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.
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.
Hyperbolic k-Fibonacci Quaternions
Journal of Computer Science & Computational Mathematics, 2019
In this paper, hyperbolic k-Fibonacci quaternions are defined. Also, some algebraic properties of hyperbolic k-Fibonacci quaternions which are connected with hyperbolic numbers and k-Fibonacci numbers are investigated. Furthermore, d'Ocagne's identity, the Honsberger identity, Binet's formula, Cassini's identity and Catalan's identity for these quaternions are given.
On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions
DergiPark (Istanbul University), 2022
In this paper, we present a new definition, referred to as the Francois sequence, related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic Leonardo and hyperbolic Francois quaternions. Afterward, we derive the Binet-like formulas and their generating functions. Moreover, we provide some binomial sums, Honsberger-like, d'Ocagne-like, Catalan-like, and Cassini-like identities of the hyperbolic Leonardo quaternions and hyperbolic Francois quaternions that allow an understanding of the quaternions' properties and their relation to the Francois sequence and Leonardo sequence. Finally, considering the results presented in this study, we discuss the need for further research in this field.
Circular-hyperbolic Fibonacci quaternions
Notes on Number Theory and Discrete Mathematics, 2020
In this paper, circular-hyperbolic Fibonacci numbers and quaternions are defined. Also, some algebraic properties of circular-hyperbolic Fibonacci numbers and quaternions which are connected with circular-hyperbolic numbers and Fibonacci numbers are investigated.
On a Generalization for Tribonacci Quaternions
Mediterranean Journal of Mathematics
Let Vn denote the third order linear recursive sequence defined by the initial values V 0 , V 1 and V 2 and the recursion Vn = rV n−1 +sV n−2 +tV n−3 if n ≥ 3, where r, s, and t are real constants. The {Vn} n≥0 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.
On a quaternionic sequence with Vietoris’ numbers
Filomat, 2021
Special integers sequences have been the center of attention for many researchers, as well as the sequences of quaternions where its components are the elements of these sequences. Motivated by a rational sequence, we consider the quaternions with components Vietoris' numbers and investigate some of its properties. For this sequence a two and three term recurrence relation is established, as well as a Binet's type formula. Moreover the generating function for this sequence is introduced and also the determinant of some tridiagonal matrices are used in order to find elements of this sequence.
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 Hyperbolic Padovan and Hyperbolic Pell-Padovan Sequences
DergiPark (Istanbul University), 2022
In this article, we extend Padovan and Pell-Padovan numbers to Hyperbolic Padovan and Hyperbolic Pell-Padovan numbers, respectively. Moreover, we obtain Binet-like formulas, generating functions and some identities related to Hyperbolic Padovan and Hyperbolic Pell-Padovan numbers.
Perfect Sequences and Arrays of Unbounded Lengths and Sizes over the Basic Quaternions
2013
The aim of this Thesis is to provide new understanding of the existence of perfect sequences and arrays over the alphabets of quaternions and complex numbers, and multi-dimensional arrays with recursive autocorrelation. Perfect sequences over the quaternion algebra H were first introduced by O. Kuznetsov in 2009. The quaternion algebra is a non-commutative ring, and for this reason, the concepts of right and left autocorrelation and right and left perfection were introduced. Kuznetsov showed that the concepts of right and left perfection are equivalent. One year later, O. Kuznetsov and T. Hall showed a construction of a perfect sequence of length 5,354,228,880 over a quaternion alphabet with 24 elements, namely the double-trahedron group H24. The authors made the following conjecture: there are perfect sequences of unbounded lengths over the double tetrahedron group H_24. We worked on this conjecture and found a family of perfect sequences of unbounded lengths over H8 = {±1, ±i, ±j,...
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.
Gaussian Mersenne numbers and generalized Mersenne quaternions
Notes on Number Theory and Discrete Mathematics
In this study, we introduce a new class of quaternions associated with the well-known Mersenne numbers. There are many studies about the quaternions with special integer sequences and their generalizations. All of these studies used consecutive elements of the considered sequences. Here, we extend the usual definitions into a wider structure by using arbitrary Mersenne numbers. Moreover, we present Gaussian Mersenne numbers. In addition, we give some properties of this type of quaternions and Gaussian Mersenne numbers, including generating function and Binet-like formula.
Journal of Science and Arts, 2021
In this paper, we define and study another interesting generalization of the Fibonacci quaternions is called k-order Fibonacci quaternions. Then we obtain for Fibonacci quaternions, for Tribonacci quaternions and for Tetranacci quaternions. We give generating function, the summation formula and some properties about k-order Fibonacci quaternions. Also, we identify and prove the matrix representation for k-order Fibonacci quaternions. The matrix given for k-order Fibonacci numbers is defined for k-order Fibonacci quaternions and after the matrices with the k-order Fibonacci quaternions is obtained with help of auxiliary matrices, important relationships and identities are established.
The k-Fibonacci dual quaternions
International Journal of Mathematical Analysis, 2018
In this paper, k-Fibonacci dual quaternions are defined. Also, some algebraic properties of k-Fibonacci dual quaternions which are connected with k-Fibonacci numbers and Fibonacci numbers are investigated. Furthermore, d'Ocagne's identity, the Honsberger identity, Binet's formula, Cassini's identity and Catalan's identity for these quaternions are given.
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.