Orhan Dişkaya

Mersin University · Department of Mathematics

Galois field, has an important position in cryptology. Advanced Encryption Standard (AES) also used in polynomial operations. In this paper, we consider the polynomial operations on the Galois fields, the Fibonacci polynomial sequences. Using a certain irreducible polynomial, we redefine the elements of Fibonacci polynomial sequences to use in our...

In this study, one-dimensional, two-dimensional, three-dimensional and n−dimensional recurrences of the (p, q)−Fibonacci sequence are examined and their some identities are given.

In the present work, we consider the Padovan numbers. Inspiring of the Hosoya's triangle, we define the Padovan triangle. We give some identities and properties of the Padovan triangle.

In the present work we consider the Jacobsthal and Jacobsthal-Lucas sequences with the Jacobsthal and Jacobsthal-Lucas subscripts sequences; that is, the numbers of the form Xn = J Jn , Zn = J jn , Yn = j jn and Tn = j Jn. We obtain some identities and relations for the Jacobsthal and Jacobsthal-Lucas subscripts sequences. Also, we give recursive d...

In this paper, we consider Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas sequences. We introduce the quadra Fibona-Pell,Fibona-Jacobsthal and Pell-Jacobsthal and the hexa Fibona-Pell-Jacobsthal sequences whose components are the Fibonacci, Pell and Jacobsthal sequences. We derive the Binet-like formulas, the generating functions...

Spinors are components of a complex vector space that can be related to Euclidean space in both geometry and physics. In essence, the forms of usage include quaternions that are equivalent to Pauli spin matrices, which may be produced by thinking of a quaternion matrix as the compound. This study's objective is the spinor structure that forms based...

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.

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.

There are a lot of quaternion numbers that are related to the Fibonacci and Lucas numbers or their generalizations have been described and extensively explored. The coefficients of these quaternions have been chosen from terms of Fibonacci and Lucas numbers. In this study, we define two new quaternions that are pseudo-Fibonacci and pseudo-Lucas qua...

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.

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

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.

This study considers the m−order linear recursive sequences yielding some well-known sequences (such as the Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Perrin sequences). Also, the Binet-like formulas and generating functions of the m−order linear recursive sequences have been derived. Then, we define the m−order linear recursive quaternions a...

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

In this article, we consider compositions of positive integers with 2s and 3s. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling...

In this paper, we introduce 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.

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. MSC 2020. 11B39, 05A15, 11B83.

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-lik...

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-Pado...

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...

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 the exponential generating functions of these sequences. Also, we obtain some binomial identities, series and su...

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,...

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.

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...

In this work, we look at the theory of the harmonic and hyperharmonic Padovan numbers. In addition, we obtain some combinatorial identities such as harmonic and hyperharmonic numbers, as well as some useful formulas for R_n , which is concerned with finite sums of reciprocals of Padovan numbers.

In this article, we construct the plastic number in the three-dimensional space. We examine the nested radicals and continued fraction expansions of the plastic ratio. In addition, we give some properties and geometric interpretations of the plastic constant.

This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio 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.

In this paper, we consider Padovan numbers with different initial values. We define the Gadovan numbers which generalize 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...

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

In this paper we first introduce a class of (s, t)-Padovan and (s, t)-Perrin quaternions which generalizes Padovan and Perrin quaternions, and then we derive new Binet-like formulas, generating functions and certain binomial sums for these quaternions.

In this paper, we consider the generalization of Padovan and Perrin quaternions. We define the split (s, t)-Padovan and (s, t)-Perrin quaternions which generalize Padovan and Perrin quaternions. We derive the Binet-like formulas for the split (s, t)-Padovan and (s, t)-Perrin quaternions. We establish their generating functions. Also, we obtain cert...