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Koleksiyon Matematik Bölümü
170350

Bernoulli-Padovan polynomials and Pado-Bernoulli matrices

Dişkaya, Orhan | Menken, Hamza

In the present work we introduce Bernoulli-Padovan numbers and polynomials. We give their generating functions of the Bernoulli-Padovan numbers and polynomials. We establish various relations involving the Bernoulli-Padovan numbers and polynomials by considering the Pado-derivative. We describe Pado-Bernoulli matrices in terms of the Bernoulli-Padovan numbers and polynomials. We establish a factorisation of the Pado- Bernoulli matrix by using a generalised Pado-Pascal matrix, and obtain the inverse of the Pado-Bernoulli matrix. Also, we give a relationship between the Pado-Bernoulli matrix and the Pado-Pascal matrix.

170354

ON THE BICOMPLEX PADOVAN AND BICOMPLEX PERRIN NUMBERS

Dişkaya, Orhan | Menken, Hamza

In this paper we first introduce the bicomplex Padovan and bicom- plex Perrin numbers which generalize Padovan and Perrin numbers, and then we derive the Binet-like formulas, the generating functions and the exponential gen- erating functions, series, sums of these sequences. Also, we obtain some binomial identities for them.

170384

On the bivariate Padovan polynomials matrix

Dişkaya, Orhan | Menken, Hamza

In this paper, we intruduce the bivariate Padovan sequence we examine its various identities. We define the bivariate Padovan polynomials matrix. Then, we find the Binet formula, generating function and exponential generating function of the bivariate Padovan polynomials matrix. Also, we obtain a sum formula and its series representation.

170385

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

Dişkaya, Orhan | Menken, Hamza

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’ proper- ties 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.

Makale2023Journal of New Theory 1 | 0 Erişime Açık
170392

ON THE NOVEL GENERALIZATIONS OF THE PADOVAN SEQUENCE

Dişkaya, Orhan | Menken, Hamza

In the present work, we consider the Padovan sequence and define a sequence (called Quadrovan) that is a new generalization. In addition, we give the previously defined Tridovan sequence as a generalization of the Padovan sequence. We derive the Binet-like formulas, the generating functions and the exponential generating functions for the Tridovan and Quadrovan sequences. Also, we establish their series and matrices.

170393

On the Quinary Fibonacci-Padovan Sequences

Dişkaya, Orhan | Menken, Hamza

In this paper, we consider the Fibonacci and Padovan sequences. We introduce the quinary Fibonacci-Padovan sequences whose compounds are the Fibonacci and Padovan sequences. We derive the Binet-like formulas, the generating functions and exponential generating functions of these sequences. Also, we obtain some binomial identities, series and sums for them.

Makale2023CREAT. MATH. INFORM. 1 | 0 Erişime Açık
170402

ON THE PADOVAN ARRAYS

Dişkaya, Orhan | Menken, Hamza

In the present work, two new recurrences of the Padovan sequence given with delayed initial conditions are defined. Some identities of these sequences which we call the Padovan arrays were examined. Also, generating and series functions of the Padovan arrays are examined.

170404

On the Richard and Raoul numbers

Dişkaya, Orhan | Menken, Hamza

In this study, we define and examine the Richard and Raoul sequences and we deal with, in detail, two special cases, namely, Richard and Raoul sequences. We indicate that there are close relations between Richard and Raoul numbers and Padovan and Perrin numbers. Moreover, we present the Binet-like formulas, generating functions, summation formulas, and some identities for these sequences.

170409

On the Fibonacci quaternion sequence with quadruple-produce components

Dişkaya, Orhan | Menken, Hamza

This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ra- tio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.

170470

SOME IDENTITIES OF GADOVAN NUMBERS

Dişkaya, Orhan | Menken, Hamza

In this parer, we consider Padovan numbers with different initial values. We define the Gadovan numbers which generalizes a new class of Padovan numbers, and we derive Binet-like formulas, generating functions, exponential generating functions for the Gadovan numbers. Also, we obtain binomial sums, some identities and a matrix of the Gadovan numbers.

170391

On the incomplete narayana numbers

Dişkaya, Orhan | Menken, Hamza

In this paper, we first express with sums of binomial coefficients of the Narayana sequence. Moreover, we define the incomplete Narayana numbers and examine their recurrence relations, some properties of these numbers, and the generating function of the incomplete Narayana numbers.

170356

On the bi-periodic Padovan sequences

Dişkaya, Orhan | Menken, Hamza

In this study, we define a new generalization of the Padovan numbers, which shall also be called the bi-periodic Padovan sequence. Also, we consider a generalized bi-periodic Padovan matrix sequence. Finally, we investigate the Binet formulas, generating functions, series and partial sum formulas for these sequences

Makale2023Mathematica Moravica 4 | 0 Erişime Açık
170400

A New Encryption Algorithm Based on Fibonacci Polynomials and Matrices

Dişkaya, Orhan | Avaroğlu, Erdinç | Menken, Hamza

Confusion and diffusion features are two fundamental needs of encoded text or images. These features have been used in various encryption algorithms such as Advanced Encryption Standard (AES) and Data Encryption Standard (DES). The AES adopts the S- box table formed with irreducible polynomials, while the DES employs the Feistel and S- box structures. This study proposes a new encryption algorithm based on Fibonacci polynomials and matrices, which meets the fundamental needs of image encryption and provides an alternative to other encryption algorithms. The success of the proposed method was tested on three different images, as evidenced by the histogram analysis results of the sample images, together with the number of changing pixel rate (NPCR) and the unified averaged ch...

Makale2022Traitement du Signal 2 | 0 Erişime Açık
169215

On Uniform Exhuastiveness and Korovkin-type Theorem

Erdem, Alper | Tunç, Tuncay | Arpacıoğlu, Ali

This study considers a uniform version of exhaustiveness for function sequences and examined its connections with alpha convergence, uniform alpha convergence, and classical uniform convergence. A Korovkin-type approximation theorem is also established, showing that the classical conditions can be loosened under this new concept.

170291

ARCTANGENT IDENTITIES INVOLVING THE JACOBSTHAL AND JACOBSTHAL-LUCAS NUMBERS

Dişkaya, Orhan

This study presents novel arctangent identities that establish connections between the Jacobsthal and Jacobsthal-Lucas numbers. These findings contribute to the understanding of the interplay between trigonometric functions and number theory, particularly in relation to well-known mathematical sequences and constants.

169405

ANALYSIS OF CONICS UNDER KANTOROVICH OPERATORS OF TWO VARIABLES

Elkahwa, Esraa Dakak | Tunç, Tuncay

This study examines some shape-preserving properties of two-variable Kantorovich polynomials. We examine which types of conic equations transform into conic equations under two types of two-variable Kantorovich polynomials, single-index and double-index, and if so, which conic equations they transform into. While it is observed that conic equations transform into the same type of conic equations under the single-index two-variable Kantorovich polynomial, they are shown to transform into different types under the double-index two-variable Kantorovich polynomial, for example, a circle can transform into an ellipse or parabola under certain conditions. Furthermore, all the  findings are supported by numerous graphical examples.

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