arXiv subjects
Publications and source records attributed to Gamaliel Cerda-Morales.
In \cite{Go}, Gökbaş defined a new type of number sequence called Leonardo-Alwyn sequence. In this paper, we consider the generalized Leonardo-Alwyn hybrid numbers and investigate some of their properties. We also give some applications related to the generalized Leonardo-Alwyn hybrid numbers in matrices.
Spinors are used in physics quite extensively. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined Tribonacci spinors using the generalized Tribonacci quaternions. Later, we have established the structure of algebra for these spinors. Finally, we have proved some important formulas such as Binet and Cassini-like formulas which are given for some series of numbers in mathematics for Tribonacci spinors.
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 polynomials is obtained. This result implies the Catalan, Cassini and d'Ocagne identities. Moreover, generating function and matrix generators for these polynomials are presented.
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 special matrices, we can identify other forms of demonstration and also the description of commutative matrix properties for negative indices. A new generalization of this sequence is used for our purpose.
Modified third-order Jacobsthal sequence is defined in this study. Some properties involving this sequence, including the Binet-style formula and the generating function are also presented.
In \cite{Choi-Jo}, the $am+b$ ($0\leq b<a$) subscripted Tribonacci numbers are studied. This work is devoted to study a new generalization of Fibonacci numbers called Padovan numbers. In particular, the $am+b$ subscripted Padovan numbers will be expressed by three $a$ step apart Padovan numbers for any $0\leq b<a$, where $a\in \mathbb{Z}$.
We study the problem of generalization of Oresme numbers with a new sequence of numbers called Oresme polynomials. Moreover, by using the matrix methods for Oresme polynomials, we obtain the identities including the general bilinear index-reduction formula of these numbers. Further, Oresme polynomials that are natural extensions of the $k$-Oresme numbers are introduced and some relations for the derivatives of these polynomials in the form of convolution are proved.
In \cite{Ka}, the authors obtained a method for deriving special matrices, whose powers are related to Fibonacci and Lucas numbers. In the study, it has been developed a method for deriving special matrices of $3\times 3$ dimensions, whose powers are related to Horadam and generalized Fibonacci numbers, and some special matrices have been found via the method developed.
This paper, in considering aspects of the geometric mean sequence, offers new results connecting generalized Tribonacci and third-order Horadam numbers which are established and then proved independently.
In this paper, we first give new generalizations for third-order Horadam $\{H_{n}^{(3)}\}_{n\in \mathbb{N}}$ and generalized Tribonacci $\{h_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for classic Horadam and generalized Fibonacci numbers. Considering these sequences, we define the matrix sequences which have elements of $\{H_{n}^{(3)}\}_{n\in \mathbb{N}}$ and $\{h_{n}^{(3)}\}_{n\in \mathbb{N}}$. Then we investigate their properties.
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-Cesàro identity is satisfied. Also, we define a family of generalized third-order Jacobsthal sequences $\{\mathbb{J}_{n}^{(3)}\}_{n\geq 0}$ by the recurrence relation $$\mathbb{J}_{n+3}^{(3)}=\mathbb{J}_{n+2}^{(3)}+\mathbb{J}_{n+1}^{(3)}+2\mathbb{J}_{n}^{(3)},\ n\geq0,$$ with initials conditions $\mathbb{J}_{0}^{(3)}=a$, $\mathbb{J}_{1}^{(3)}=b$ and $\mathbb{J}_{2}^{(3)}=c$, where $a$, $b$ and $c$ are non-zero real numbers. Many sequences in the literature are special cases of this sequence. We find the generating function and Binet's formula of the sequence. Then we show that the Cassini and Gelin-Cesàro identities are satisfied by the indices of this generalized sequence.
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 quaternions. Furthermore, the third-order Jacobsthal generalized quaternions are classified by considering the special cases of quaternionic units. We derive the relations between third-order Jacobsthal and third-order Jacobsthal-Lucas generalized quaternions.
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 number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the generalized Tribonacci numbers. We introduce the Tribonacci and generalized Tribonacci octonions (such as Narayana octonion, Padovan octonion and third-order Jacobsthal octonion) and give some of their properties. We derive the relations between generalized Tribonacci numbers and Tribonacci octonions.
The Tribonacci sequence is a well-known example of third order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Tribonacci sequence is introduced and defined by $H_{n+2}=H_{n+1}+H_{n}+H_{n-1}\ \ (n\geq 1)$, where $H_{0}=3$, $H_{1}=0$ and $H_{2}=2$. Also $n$-th power of the generating matrix for this generalized Tribonacci sequence is established and some basic properties of this sequence are obtained by matrix methods. There are many elementary formulae relating the various $H_{n}$, most of which, since the sequence is defined inductively, are themselves usually proved by induction.
In this paper, we give quadratic approximation of generalized Tribonacci sequence $\{V_{n}\}_{n\geq0}$ defined by Eq. (\ref{eq:7}) and use this result to give the matrix form of the $n$-th power of a companion matrix of $\{V_{n}\}_{n\geq0}$. Then we re-prove the cubic identity or Cassini-type formula for $\{V_{n}\}_{n\geq0}$ and the Binet's formula of the generalized Tribonacci quaternions.
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. Considering these sequences, we define the matrix sequences which have elements of $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$. Then we investigate their properties.
In \cite{Oz}, M. Özdemir defined a new non-commutative number system called hybrid numbers. In this paper, we define the hybrid Fibonacci and Lucas numbers. This number system can be accepted as a generalization of the complex ($\textbf{i}^{2}=-1$), hyperbolic ($\textbf{h}^{2}=1$) and dual Fibonacci number ($\varepsilon^{2}=0$) systems. Furthermore, a hybrid Fibonacci number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation $\textbf{ih}=-\textbf{hi}=\varepsilon+\textbf{i}$. Then we used the Binet's formula to show some properties of the hybrid Fibonacci numbers. We get some generalized identities of the hybrid Fibonacci and hybrid Lucas numbers.
Narayana's cows problem is a problem similar to the Fibonacci's rabbit problem. We define the numbers which are the solutions of this problem as Narayana's cows numbers. Narayana's cows sequence satisfies the third order recurrence relation $N_{r}=N_{r-1}+N_{r-3}$ ($r\geq3$) with initial condition $N_{0} =0$, $N_{1} = N_{2}= 1$. In this paper, the $ar+b$ subscripted Narayana numbers will be expressed by three $a$ step apart Narayana numbers for any $1\leq b\leq a$ ($a\in \mathbb{Z}$). Furthermore, we study the sum $S_{N,r}^{(4,b)}=\sum_{k=0}^{r}N_{4k+b}$ of $4$ step apart Narayana numbers for any $1\leq b\leq 4$.