Catalan Identity Research Articles

A New Approach to k-Oresme and k-Oresme-Lucas Sequences

In this study, the k-Oresme and k-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the k-Oresme and k-Oresme-Lucas sequences are presented. In addition, we give these sequences the Binet formulas, generating functions, Cassini identity, Catalan identity etc. Moreover, the k-Oresme and k-Oresme-Lucas sequences are associated with Fibonacci, Pell numbers and Lucas, and Pell- Lucas numbers, respectively. Finally, the Catalan transforms of these sequences are given and Hankel transforms are applied to these Catalan sequences and associated with the terms of the sequence.

On the Leonardo quaternions sequence

In the present work, a new sequence of quaternions related to the Leonardo numbers -- named the Leonardo quaternions sequence -- is defined and studied. Binet's formula and certain sum and binomial-sum identities, some of which derived from the mentioned formula, are established. Tagiuri-Vajda's identity and, as consequences, Catalan's identity, d'Ocagne's identity and Cassini's identity are presented. Furthermore, applying Catalan's identity, and the connection between composition algebras and vector cross product algebras, Gelin-Cesàro's identity is also stated and proved. Finally, the generating function, the exponential generating function and the Poisson generating function are deduced. In addition to the results on Leonardo quaternions, known results on Leonardo numbers and on Fibonacci quaternions are extended.

ON THE GENERALIZED DUAL FIBONACCI AND LUCAS OCTONIONS

Generalized number systems, particularly octonions with their algebraic structure, have drawn much interest in mathematics, physics, and computer technology. Therefore, in this paper, we introduce two new concepts, modified generalized dual Fibonacci and modified generalized dual Lucas octonions, to expand the topic of octonions. Additionally, we explore the well-known Catalan and Cassini identities, shedding light on the characteristics of these new constructs. Also, we give generating functions and the Binet formulas of the modified generalized dual Fibonacci and modified generalized dual Lucas octonions.

Влияние освободительной борьбы на Кубе на формирование каталонского национализма в конце XIX в. и первой трети ХХ в.

The article presents a study of the role of Cuba and its influ-ence on the Catalan national movement in the late 19th to ear-ly 20th centuries. Complex socio-political processes, identity formation, and the development of separatist ideas in the con-text of Catalan-Cuban relations were analyzed. Catalonia sup-ported the Spanish government and Castilian nationalism as long as it held colonies, pursued protectionist policies, and acted in the interest of Catalan capital. The catastrophe of 1898 had a significant impact on Catalan nationalism, prompt-ing the development of the idea of regional independence and its desire for greater autonomy. Special attention is paid to the role of Catalan emigrants in Cuba, their influence on the for-mation of separatist organizations, and financial support for the independence movement. The study also highlights the importance of cultural exchanges between Catalonia and Cuba in preserving Catalan identity. It also underscores the key role of Catalan emigrants in the development of Catalan national-ism and the struggle for autonomy, as well as their contribu-tion to the creation of the 1932 Statut of Catalonia.

On some identities for the DGC Leonardo sequence

In this study, we examine the Leonardo sequence with dual-generalized complex ($\mathcal{DGC}$) coefficients for $\mathfrak{p} \in \mathbb R$. Firstly, we express some summation formulas related to the $\mathcal{DGC}$ Fibonacci, $\mathcal{DGC}$ Lucas, and $\mathcal{DGC}$ Leonardo sequences. Secondly, we present some order-$2$ characteristic relations, involving d’Ocagne's, Catalan's, Cassini's, and Tagiuri's identities. The essential point of the paper is that one can reduce the calculations of the $\mathcal{DGC}$ Leonardo sequence by considering $\mathfrak{p}$. This generalization gives the dual-complex Leonardo sequence for $ \mathfrak p =-1$, hyper-dual Leonardo sequence for $ \mathfrak p =0$, and dual-hyperbolic Leonardo sequence for $ \mathfrak p =1$.

Catalan Identities for Generalized Fibonacci Polynomials

Catalan Identities for Generalized Fibonacci Polynomials

ON GENERALIZED ORESME NUMBERS AND THE FIBONACCI FUNDAMENTAL SYSTEM

In this work we study the generalized Oresme numbers using the algebraic properties of the Fibonacci fundamental system associated with this sequence. We explicitly describe the closed connection between the sequences of the Fibonacci fundamental system and the generalized Oresme numbers. In addition, the matrix approach is considered for studying the combinatorial identities and the generalized Cassini identity for the generalized Oresme numbers. Finally, the analytic representation is established, and the combinatorial and analytical generalized Cassini identity is given. Keywords: Generalized sequence of numbers, Fibonacci fundamental system, combinatorial identities, the Cassini identity

On Higher Order Jacobsthal Hyper Complex Numbers

In this work, we define a new class of hyper complex numbers whose components are higher order Jacobsthal numbers, and call such numbers as the higher order Jacobsthal $ 2^{s} $-ions. We obtain some algebraic properties of the higher order Jacobsthal $ 2^{s} $-ions such as recurrence relation, Binet-like formula, generating function, exponential generating function, Vajda's identity, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. Morever we derive the matrix representation of the higher order Jacobsthal $ 2^{s} $-ions, and so prove Cassini's identity as a further type.

On The Hyperbolic k-Mersenne And k-Mersenne-Lucas Octonions

In this paper, we introduce the hyperbolic k-Mersenne and k-Mersenne-Lucas octonions and investigate their algebraic properties. We give Binet’s formula and present several interrelations and some well-known identities such as Catalan identity, d’Ocagne identity, Vajda identity, generating functions, etc. of these octonions in closed form. Furthermore, we investigate the relations between hyperbolic k-Mersenne octonions and hyperbolic k-Mersenne-Lucas octonions.

On the relations between antitrace, row/column sum of elements and generalized horadam sequence associated with a matrix

Using Cayley-Hamilton theorem for a given matrix A = [aij] ∈ M2(K), (K = ℝ or ℂ), we obtain the recurrence relations for Tra(An), suRi(An) and suCi(An); Tra(An+2) = Tr(A)Tra(An+1) – det(A) Tra(An) suRi(An+2) = Tr(A)suRi(An+1) – det(A)suRi(An) suCi(An+2) = Tr(A)suCi(An+1) – det(A) suCi(An) where Tra(A) = a12 + a21 is the sum of antidiagonal entries of A and suRi(A) or scCu(A) is the ith(i = 1, 2) row or column sum of A respectively. In doing so, the analogy between the above natural recurrences and the Horadam sequence is observed. Motivated by this, we define the Horadam sequence {Wn(a, b : Tr(A), det(A))} and Fibonacci sequence {Fn = Wn (0, 1; Tr(A), det(A))} associated with A. In this paper, Tra(An), {suRi(An)}, {suCi(An)}, {Wn} and {Fn} sequences are defined and their inter-relationships are extensively studied. We also obtain Binet’s formulae, generating functions, Cassini identities, Vajda’s identities and Catalan’s identities for these sequences. Further, identities are illustrated for particular values of trace and determinant of a matrix

On some new families of $k$-Mersenne and generalized $k$-Gaussian Mersenne numbers and their polynomials

In this paper, we define the generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further, we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and their relation with the classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials.

ON DUAL BICOMPLEX BALANCING AND LUCAS-BALANCING NUMBERS

In this paper, dual bicomplex Balancing and Lucas-Balancing numbers are defined, and some identities analogous to the classic properties of the Fibonacci and Lucas sequences are produced. We give the relationship between these numbers and Pell and Pell-Lucas numbers. From these, the basic bicomplex properties for the norm and its conjugate of these numbers are also developed. These in turn lead to the Binet formula, the generating functions and exponential generating functions, which are important concepts for number sequences. The Cassini identity, which is important for number sequences, actually emerged to solve the famous Curry paradox. We calculated the Cassini, Catalan, Vajda and d’Ocagne identities for these numbers.

On Some Properties of Bihyperbolic Numbers of The Lucas Type

To date, many authors in the literature have worked on special arrays in various computational systems. In this article, Lucas type bihyperbolic numbers were defined and their algebraic properties were examined. Bihyperbolic Lucas numbers were studied by Azak in 2021. Therefore, we only examined bihyperbolic Jacobsthal-Lucas and Pell-Lucas numbers. We also gave properties of bihyperbolic Jacobstal-Lucas and bihyperbolic Pell-Lucas numbers such as recursion relation, derivation function, Binet formula, D'Ocagne identity, Cassini identity and Catalan identity.

On Horadam finite operator hybrid numbers

In this paper, we introduce and study a new hybrid number sequence with Horadam numbers and a finite operator, called Horadam finite operator hybrid numbers. We derive recurrence relation, Binet-like formula, ordinary generating function, exponential generating function, Poisson generating function, and summation formula for Horadam finite operator hybrid numbers. Moreover, we give matrix representation and Cassini's identities for these numbers.

ЯЗЫКОВОЙ КАТАЛОНСКИЙ НАЦИОНАЛИЗМ В 1870 – 1930-Е ГГ.: ПОЛИТИЧЕСКИЕ ПОКОЛЕНИЯ И СОЦИАЛЬНАЯ МОДЕРНИЗАЦИЯ

The purpose of this study is to analyze the activities of three generations of Catalan philologists and linguists who have made a significant contribution to the development of linguistic Catalan nationalism. The author analyzes the history, directions, forms and features of the activities of several generations of Catalan intellectuals, whose activity contributed to the normalization and standardization of dialect forms of Romance speech in Catalonia, which led to the emergence of the modern Catalan literary language. The novelty of the study lies in the analysis of the historical, political and cultural contribution of nationalist intellectuals to the development of the Catalan language through the prism of the social consequences of modernization processes. The article analyzes 1) the main stages in the history of various generations of Catalan linguistic nationalists, 2) their participation and role in the processes of standardization and normalization of the Catalan language, 3) the contribution of language activists to the consolidation of the Catalan linguistic identity in particular and the national Catalan identity in general, 4) the role nationalist intellectuals, amateurs and professional linguists, in the development of Catalan nationalism, 5) the contribution of linguistic nationalists to raising the status of the Catalan language in the social, political and cultural spaces of Catalonia. It is assumed that Catalan linguistic nationalism significantly complemented political nationalism, contributing to its strengthening and consolidation of the Catalan national identity in contexts of social transformation of the region's society. The generations of linguistic Catalan nationalism are identical to the stages in the history of political and ethnic nationalisms. The results of the study suggest that in the 19th and 20th centuries, the cultural activity of linguistic nationalists contributed to the processes of modernization of Catalonia, which led to the transformation of the region from a peripheral one to a region with complete social and political structures, the functioning of which was impossible without the Catalan language.

New approaches of (q,k)-Fibonacci–Pell sequences via linear difference equations. Applications

In this paper we establish some explicit formulas of (q,k)-Fibonacci–Pell sequences via linear difference equations of order 2 with variable coefficients, and explore some of their new properties. More precisely, our results are based on two approaches, namely, the determinantal and the nested sums approaches, and their closed relations. As applications, we investigate the q-analogue Cassini identities and examine a pair of Rogers–Ramanujan type identities.

On Quaternions with Gaussian Oresme Coefficients

At this paper, we describe Gaussian Oresme numbers taking into account the Oresme numbers. Furthermore, we investigate their some basic characteristic properties such as Binet formula and Cassini identity, etc. Moreover, we define quaternions with Gaussian Oresme coefficients and obtain their some spectacular properties.

On Hybrid Numbers with Gaussian Leonardo Coefficients

We consider the Gaussian Leonardo numbers and investigate some of their amazing characteristic properties, including their generating function, the associated Binet formula and Cassini identity, and their matrix representation. Then, we define the hybrid Gaussian Leonardo numbers and obtain some of their particular properties. Furthermore, we define nn Hessenberg matrices whose permanents yield the Leonardo and Gaussian Leonardo sequences.

Rebirth and the Eternal Return in Modern and Contemporary Catalan Art and Identity

This article explores how themes of birth, rebirth, genesis and coming into being are present in modern and contemporary Catalan art, focusing on the works of Eugènia Balcells (b. 1942), Xicu Cabanyes (b. 1946), Mari Chordà (b. 1942), Salvador Dalí (1904–1989), and Joan Miró (1893–1983). In particular, the article looks at how these themes emerged for the artists as a way of expressing Catalan identity in the wake of the Spanish Civil War (1936–1939) and the dictatorship of Francisco Franco in Spain (1939–1975), as well as following Catalonia’s broader history as a nation without a state in Europe. In exploring the artists’ lives and works, the article also considers the topics of rebirth and the eternal return as they occur in the philosophy and history of religion of Friedrich Nietzsche (1844–1900), Albert Camus (1913–1960) and Mircea Eliade (1907–1986). Ultimately, the author interprets the artworks of the study as physical representations of rebirth that relate in part to a longstanding Catalan sentiment of an eternal recurrence to life after destruction.

On Harmonic Complex Balancing Numbers

In the present work, we define harmonic complex balancing numbers by considering well-known balancing numbers and inspiring harmonic numbers. Mainly, we investigate some of their basic characteristic properties such as the Binet formula and Cassini identity, etc. In addition, one type of symmetric matrix family whose entries are harmonic complex balancing numbers is constructed. Additionally, some linear algebraic properties are obtained. Furthermore, some inequalities are stated by exploiting the well-known inequalities between various matrix norms. Finally, we illustrate the results with some numerical examples.