Bulunan: 42 Adet 0.002 sn
Yazarlar Menken, Hamza
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.

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.

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.

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.

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
170396

ON THE RECURRENCES OF THE JACOBSTHAL SEQUENCE

Dişkaya, Orhan | Menken, Hamza

In the present work, two new recurrences of the Jacobsthal sequence are defined. Some identities of these sequences which we call the Jacobsthal array is examined. Also, the generating and series functions of the Jacobsthal array are obtained.

Makale2023MATHEMATICA 2 | 0 Erişime Açık
170633

ON THE COMPONENTS OF SOME SPECIAL SEQUENCES

Dişkaya, Orhan | Menken, Hamza

The Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences {Fn}, {Ln}, {Pn}, {P Ln}, {Jn} and {JLn} are defined by two order recurrences for n ≥ 0, respectively, Fn+2 = Fn+1 + Fn, Ln+2 = Ln+1 + Ln, Pn+2 = 2Pn+1 + Pn, P Ln+2 = 2P Ln+1 + P Ln, Jn+2 = Jn+1 + 2Jn, JLn+2 = JLn+1 + 2JLn, with the initial conditions, respectively, F0 = 0, and F1 = 1, L0 = 2, and L1 = 1, P0 = 0, and P1 = 1, P L0 = 2, and P L1 = 1, J0 = 0, and J1 = 1, JL0 = 2, and JL1 = 1. In this work we define new component sequences which generalize the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences with different initial conditions. We give some identities of these component sequences. Also, the Binet-like formulas, the generating functions and the exponentia...

170448

Matrix Encryption Standard

Dişkaya, Orhan | Avaroğlu, Erdinç | Menken, Hamza

AES (Advanced Encryption Standard) is a standard for encrypting electronic data. AES operates on a 44 column-major order array of bytes. The operations in the matrix are also performed on the polynomials in a special finite field and using S-box. We firstly recall necessary information about matrix algebra. In the present work, we examine the AES encryption method. We create a new encryption algorithm called matrix encryption standard (MES). MES is performed by similar steps to the AES algorithm over 16x16 matrices with elements {0,1} without using polynomials operations and S- box in the AES algorithm. So, we provide 256-bits plain text to be encrypted by passing it through certain rounds with the 16x16 matrix key. In order to decrypt the cipher text, we take the reverse of t...

170574

A NEW GENERALIZATION OF THE PADOVAN SEQUENCE

Dişkaya, Orhan | Menken, Hamza

Integers number sequences in mathematics are one of the subjects with the most ap- plication area. The Fibonacci number sequence has applications in many branches of science such as nature, anatomy, botany, zoology, art, music, analysis, physics, astronomy, chemistry, biology and computers. Since the positive real root of the Fibonacci number sequence gives the golden ratio, it has many applications. Many scientists deal with Fibonacci sequence and its generalizations in recent years. Some of these generalizations are number sequences such as Lucas, Pell, and Jacobthal [4, 5, 6]. In this study, the Padovan numbers sequence, which has a third-order characteristic equation, and some of its generalizations are examined. The Padovan sequence {Pn} is defined by the third order recurren...

170468

ON THE SEQUENCE OF GELL NUMBERS

Dişkaya, Orhan | Menken, Hamza

In this paper, we consider Pell numbers. We define the gell num- bers which generalize the Pell numbers. Moreover, we derive Binet-like for- mula, generating function and exponential generating function for the gell sequence. Also, we obtain the gell series and some important identities for the gell sequence.

170631

COMPOSITIONS OF POSITIVE INTEGERS AND THE PADOVAN NUMBERS

Dişkaya, Orhan | Menken, Hamza

The Padovan sequence {Pn}n≥0 is defined by the third order recurrence (1) Pn+3 = Pn+1 + Pn with the initial conditions P0 = 1, P1 = 0 and P2 = 1. The Padovan sequence appears as sequence A000931 on the On-Line Encyclopedia of Integer Sequences (OEIS) [1]. For relevance, we consider as P−2 = P−1 = 0. In [2] the Padovan polynomial sequence {Pn(x)}n≥0 is defined by a third order recurrence (2) Pn+3(x) = xPn+1(x) + Pn(x) with the initial conditions P0(x) = 1, P1(x) = 0 and P2(x) = x. For relevance, we consider as P−2(x) = P−1(x) = 0. To simplify notation, take Pn(x) = Pn. A composition of an integer n is a representation of n as a sum of positive integers, for example the eight compositions of 4 are as follows: 4, 3+1, 1+3, 2+2, 2+1+1, 1 + 2 + 1, 1 + 1 + 2, 1 + 1 + 1 + 1. A partition ...

170398

PADOVAN POLYNOMIALS MATRIX

Dişkaya, Orhan | Menken, Hamza

In this paper, we explore the Padovan numbers and polynomials, and define the Padovan polynomials matrix. We obtain its Binet-like formula and a sum formula. Subsequently, we derive the Padovan polynomials matrix series. Additionally, we establish the generating and exponential generating functions for the Padovan polynomials matrix.

170397

ON THE WEIGHTED PADOVAN AND PERRIN SUMS

Dişkaya, Orhan | Menken, Hamza

In this paper, we obtain various weighted sum formulas using several sum formulas of Padovan and Perrin numbers.

170320

On the pulsating Padovan sequence

Dişkaya, Orhan | Menken, Hamza

A novel kind of Padovan sequence is introduced, and precise formulas for the form of its members are given and proven. Furthermore, the pulsating Padovan sequence in its most general form is introduced and the obtained identity is proved.

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