Related papers: A note on Modified Third-order Jacobsthal numbers
In this study, we introduce a new class of quaternions associated with the well-known modified third-order Jacobsthal numbers. There are many studies about the quaternions with special integer sequences and their generalizations. All of…
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…
In this study, we introduce the generalized Gaussian third-order Jacobsthal numbers with arbitrary initial values and discuss two particular cases, namely, Gaussian third-order Jacobsthal and Gaussian modified third-order Jacobsthal…
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…
In this paper, we first give new generalizations for third-order Jacobsthal $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and third-order Jacobsthal-Lucas $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for Jacobsthal and Jacobsthal-Lucas numbers.…
Dual third order Jacobsthal and dual third order Jacobsthal-Lucas numbers are defined. In this study, we work on these dual numbers and we obtain the properties e.g. some quadratic identities, summation, norm, negadual third order…
In this paper, the third-order Jacobsthal generalized quaternions are introduced. We use the well-known identities related to the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers to obtain the relations regarding these…
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…
Recently, Cerda-Morales \cite{Ce6} introduced commutative matrices derived from the third-order Jacobsthal matrix sequence and the third-order Jacobsthal--Lucas matrix sequence. In the present work, through the identification of certain…
In this paper, we bring into light the matrix representation of bi-periodic Jacobsthal sequence, which we shall call the bi-periodic Jacobsthal Matrix sequence.. We obtained the nth general term of this new matrix sequence. By studying the…
In this paper, we study a generalization of Jacobsthal and Jacobsthal-Lucas numbers, we find their generating function binet formulas, related matrix representation and many other properties
Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many different ways. In addition, formulas and identities involving these…
In this paper, we deal with two families of third-order Jacobsthal sequences. The first family consists of generalizations of the Jacobsthal sequence. We show that the Gelin-Ces\`aro identity is satisfied. Also, we define a family of…
In this article, we introduce and study a new integer sequence referred to as the higher order Mersenne sequence. The proposed sequence is analogous to the higher order Fibonacci numbers and closely associated with the Mersenne numbers.…
In this paper we investigate some divisibility properties of Jacobsthal numbers.
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…
This study applies the binomial, k-binomial, rising k-binomial and falling k-binomial transforms to the modified k-Fibonacci-like sequence. Also, the Binet formulas and generating functions of the above mentioned four transforms are newly…
In this paper, a new generalization of third-order Jacobsthal bihyperbolic polynomials is introduced. Some of the properties of presented polynomials are given. A Vadja formula for the generalized bihyperbolic third-order Jacobsthal…
Binet formulae for three versions of third-order Pell polynomials are derived.
This manuscript introduces $J_3$-numbers, a seemingly missing three-dimensional intermediate between complex numbers related to points in the Cartesian coordinate plane and Hamilton's quaternions in the 4D space. The current development is…