Jacobsthal Quaternions Research Articles

Generalized commutative Jacobsthal quaternions and some matrices

In this paper, some examples of matrix generators for generalized commutative Jacobsthal quaternions were given. The generating matrices are useful tools for the number sequences satisfying a recurrence relation. They can be used for an algebraic representation and for obtaining some identities.

On Quaternions with Higher Order Jacobsthal Numbers Components

In this study, we present higher order Jacobsthal numbers. Then we define higher order Jacobsthal quaternions by using higher order Jacobsthal numbers. We give the concept of the norm and conjugate for these quaternions. We express and prove some propositions related to these quaternions. Also, we find the recurrence relation, the Binet formula and the generating function for these quaternions. Finally, we calculate Cassini, Catalan, Vajda and d’Ocagne identities for higher order Jacobsthal quaternions.

Generalised k- Jacobsthal 2^m Ions (For Fixed m) Quarternions, Sedenions

In this paper we deliberate about Generalised k- Jacobsthal Quaternions, Octonions, and Sedenions. We discuss Binet formula, Generating function, Catalan Identity, Cassini Identity, D’Ocagne’s Identity of them. From that we extent the same results for k- Jacobsthal, Generalised k- Jacobsthal.

On split r-Jacobsthal quaternions

In this paper we introduce a one-parameter generalization of the split Jacobsthal quaternions, namely the split r-Jacobsthal quaternions. We give a generating function, Binet formula for these numbers. Moreover, we obtain some identities, among others Catalan, Cassini identities and convolution identity for the split r-Jacobsthal quaternions.

Dual-complex Jacobsthal Quaternions

In this paper, dual-complex Jacobsthal quaternions are defined. Also, some algebraic properties of dual-complex Jacobsthal quaternionswhich are connected with dual-complex numbers and Lucas numbers are investigated. Furthermore, the Honsberger identity, the d'Ocagne'sidentity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions and their real representations are given.............................................................................................................................................

Dual Jacobsthal Quaternions

In this paper, dual Jacobsthal quaternions were defined. Also, the relations between dual Jacobsthal quaternions which connected with Jacobsthal and Jacobsthal-Lucas numbers were investigated. Furthermore, Binet's formula, Honsberger identity, D'ocagne's identity, Cassini's identity and Catalan's identity for these quaternions were given.

On a new generalization of Jacobsthal quaternions and several identities involving these numbers

In this paper we generalize Jacobsthal quaternions to \((s,p)\)\dywiz Jacobsthal quaternions. We give some of their properties, among others the Binet formula, the generating function and the matrix representation of these quaternions. We will show how a~graph interpretation can be used in proving some identities for quaternions.

On ̄h-Jacobsthal and ̄h-Jacobsthal–Lucas sequences, and related quaternions

Abstract In this paper, inspired by recent articles of A. Szynal-Liana & I. Włoch and F. T. Aydin & S. Yüce (see [26] and [2]), we will introduce the ̄h-Jacobsthal quaternions and the ̄h-Jacobsthal–Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in [26].

Some results on dual third-order Jacobsthal quaternions

Dual Fibonacci and dual Lucas numbers are defined with dual Fibonacci and Lucas quaternions in Nurkan and G?ven [14]. In this study, we define the dual third-order Jacobsthal quaternion and the dual third-order Jacobsthal-Lucas quaternion. We derive the relations between the dual third-order Jacobsthal quaternion and dual third-order Jacobsthal-Lucas quaternion which connected the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers. In addition, we give the generating functions, the Binet and Cassini formulas for these new types of quaternions.

Dual third-order Jacobsthal quaternions

In 2016, Y\"uce and Torunbalc\i\ Ayd\i n \cite{Yuc-Tor} defined dual Fibonacci quaternions. In this paper, we defined the dual third-order Jacobsthal quaternions and dual third-order Jacobsthal-Lucas quaternions. Also, we investigated the relations between the dual third-order Jacobsthal quaternions and third-order Jacobsthal numbers. Furthermore, we gave some their quadratic properties, the summations, the Binet's formulas and Cassini-like identities for these quaternions.

On fourth-order jacobsthal quaternions

In this paper, we present for the first time a sequence of quaternions of order 4 that we will call the fourth-order Jacobsthal and the fourth-order Jacobsthal-Lucas quaternions. In particular, we are interested in the generating function, Binet formula, explicit formula and some interesting results for fourth-order Jacobsthal quaternions and fourth-order Jacobsthal-Lucas quaternions. This generalizes some previous results given by Szynal-Liana and Wloch in [13], Torunbalci Aydin and Yüce in [14] and Cerda-Morales in [2].

A new approach to Jacobsthal quaternions

The Jacobsthal quaternions defined by Szynal-Liana and Wloch [35]. In this paper, we defined some properties of Jacobsthal quaternions. Also, we investigated the relations between the Jacobsthal quaternions which connected with Jacobsthal and Jacobsthal-Lucas numbers. Furthermore, we gave the Binet formulas and Cassini identities for these quaternions.

On (p, q)-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums

Formulas and sums involving many well-known special quaternion sequences (such as the Fibonacci, Pell, Jacobsthal quaternion sequences and so on) play important roles in various branches of science. Binet formulas, generating functions and certain sums of these quaternion sequences based on Fibonacci-like numbers have been investigated by a number of authors in many different ways. In this paper, which is a sequel to earlier works, we first introduce a new class of (p, q)-Fibonacci quaternions which generalizes Fibonacci, Pell, and Jacobsthal quaternions, and then we derive a new Binet formula, generating functions, and certain binomial sums for these quaternions.

Identities for Third Order Jacobsthal Quaternions

In this paper we introduce the third order Jacobsthal quaternions and the third order Jacobsthal–Lucas quaternions and give some of their properties. We derive the relations between third order Jacobsthal numbers and third order Jacobsthal quaternions and we give the matrix representation of these quaternions.

A Note on Jacobsthal Quaternions

In this paper we introduce the Jacobsthal quaternions and the Jacobsthal–Lucas quaternions and give some of their properties. We derive the relations between Jacobsthal quaternions and Jacobsthal–Lucas quaternions and we give the matrix representation of these quaternions.