Orhan Dişkaya
FEN FAKÜLTESİ MATEMATİK BÖLÜMÜ CEBİR VE SAYILAR TEORİSİ ANABİLİM DALI
Matematik Bölümü
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Varyant-
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E-Postaorhandiskaya@mersin.edu.tr
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Web Sitesi
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Durum-
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ÜnvanDoç.Dr.
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YÖK Araştırmacı No359013
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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.
Degenerate Bernoulli–Fibonacci and Euler–Fibonacci polynomials
Dişkaya, Orhan
In this article, we first introduce the degenerate Bernoulli–Fibonacci numbers and degenerate Euler–Fibonacci numbers. Using these definitions, we then define the degenerate Bernoulli–Fibonacci polynomials and degenerate Euler–Fibonacci polynomials and examine their graphs for several initial values of λ. Subsequently, we define the degenerate Bernoulli and Euler F-polynomials and derive new exponential generating functions for these polynomials. Additionally, we investigate various identities associated with these polynomials.
Fibonacci-based generalizations of degenerate Stirling numbers
Dişkaya, Orhan
This paper introduces and systematically investigates Fibonacci-based analogues and generalizations of degenerate Stirling and Lah numbers. We begin by recalling the classical definitions and key properties of Stirling numbers of both kinds, Lah numbers, Fibonacci numbers, and Fibonomial coefficients, along with F-falling and F-rising factorials. The foundational concept of degenerate numbers and their associated degenerate factorials, as initiated by Carlitz, is also reviewed. Our primary contribution is the definition of four new families of numbers: the degenerate F-Stirling numbers of the first kind and S1,λF(n,k), the degenerate F-Stirling numbers of the second kind S2,λF(n,k), and the degenerate F-Lah number LλF(n,k). These numbers are precisely characterized as connection coefficients between the standard power basis {xn:n≥0} and the F-falling and F-rising factorial bases, and their degenerate counterparts. We derive fundamental recurrence relations for each class of these new numbers, providing a structural foundation for their analysis. Furthermore, we establish their generating functions and prove several combinatorial identities, including inverse relations. Our methodology primarily utilizes generating function techniques and properties of degenerate Fibonacci-based...
ON THE QUATERNION PADOVAN NUMBERS
Dişkaya, Orhan | Menken, Hamza
In this paper, we define the quaternion Padovan sequence. We obtain some identities for this quaternion sequence. Also, we derive a plastic-like ratio.
Nötrosofik Kompleks Fibonacci Sayıları
Dişkaya, Orhan
Bu çalışmada, nötrosofik kompleks sayılar (NKS) ile Fibonacci ve Lucas sayıları arasında yeni bir ilişki tanımlanmaktadır. Öncelikle, Fibonacci dizisinin klasik özdeşlikleri olan Binet, Honsberger, d’Ocagne, Catalan ve Cassini eşitlikleri nötrosofik kompleks sayılar bağlamında incelenmiştir. Ayrıca, binom açılımı, kısmi toplam formülleri ve üreteç fonksiyonları NKS çerçevesinde ele alınmıştır. Elde edilen sonuçlar, nötrosofik kompleks sayılar bağlamında Fibonacci dizilerinin daha geniş bir perspektifte anlaşılmasına katkı sağlamakta ve hem teorik hem de uygulamalı alanlarda yeni araştırma olanakları sunmaktadır.
Padovan ve Perrin Spinorları Hakkında
Dişkaya, Orhan | Menken, Hamza
Spinorlar, hem geometride hem de fizikte Öklit uzayıyla ilişkilendirilebilen karmaşık bir vektör uzayının bileşenleridir. Özünde, kullanım biçimleri, bir kuaterniyon matrisini bileşik olarak düşünerek üretilebilen Pauli spin matrislerine eşdeğer olan kuaterniyonları içerir. Bu çalışmanın amacı, kuaterniyon cebirine dayalı olarak oluşan spinor yapısıdır. Bu çalışmada, öncelikle spinorlar matematiksel olarak sunulmuştur. Daha sonra, Padovan ve Perrin spinorları Padovan ve Perrin kuaterniyonları kullanılarak tanımlanmıştır. Ayrıca, bu spinorlar için cebirsel yapı oluşturulmuştur. Son olarak, Padovan ve Perrin spinörleri için Binet benzeri formüller ve üreteç fonksiyonları gibi belirli özdeşlikler elde edilmiştir.
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.
On the Pseudo-Fibonacci and Pseudo-Lucas Quaternions
Dişkaya, Orhan | Menken, Hamza
There are a lot of quaternion numbers that are related to theFibonacci 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 quaternions. Then, we give their Binet-like formulas, generating functions, certain binomial sums and Honsberg-like, d’Ocagne-like, Catalan-like and Cassini-like identities.
On the Non-Newtonian Padovan and Non-Newtonian Perrin Numbers
Dişkaya, Orhan
In this work, we introduce a novel version of Padovan and Perrin numbers which we refer to as non-Newtonian Padovan and non-Newtonian Perrin numbers. Furthermore, we examine about a number of their properties. Additionally, we provide a variety of identities and formulas involving these new kinds, including the Binet-like formulas, the generating functions, the partial sum formulas, and the binomial sum formulas.
On the F-Bernstein polynomials
Erdem, Alper | Dişkaya, Orhan | Menken, Hamza
In this present work, we construct a new Bernstein operator which is called the F -Bernstein operator obtained by using F - factorial (Fibonacci factorial) and Fibonomial (Fibonacci binomial). Then we examine the F-Bernstein basis polynomials and some properties. Moreover, we acquire some connection between the F -Bernstein polynomials and the Fibonacci numbers.
ON A GENERALIZATION OF THE HOSOYA TRIANGLE
Menken, Hamza | Dişkaya, Orhan
In this talk, we introduce the Fibonacci polynomial triangle, inspired by the struc- ture of the Hosoya triangle, utilizing Fibonacci polynomials. Additionally, we present and prove several identities and properties associated with the Fibonacci polynomial triangle.
On the Quaternion Padovan Numbers
Dişkaya, Orhan | Menken, Hamza
In the present work, we explore the concept of quaternion Padovan sequences, extending the classical Padovan sequence into the realm of quaternions. Quaternions, introduced by Hamilton in 1843, are a number system that generalizes complex numbers into four dimensions, with applications in various fields such as quantum physics, computer graphics, and three-dimensional rotations. The study begins by reviewing the definition and properties of quaternions, including their non- commutative multiplication rules and the formulation of quaternion numbers. The Padovan sequence is defined by the recurrence relation Pn+3 = Pn+1 + Pn with initial conditions P0 = P1 = P2 =1. This sequence is generalized to quaternions by defining quaternion Padovan numbers, denoted QP (a,b, c, d) , which follow a similar recurrence relation for each coordinate of the quaternion. The study provides detailed recurrence relations and initial conditions for these quaternion Padovan numbers and derives several new identities and plastic-like ratio. Furthermore, the study introduces the concept of quaternion Padovan sequences in four dimensions and investigates their algebraic properties, including the calculation of the norm and conjugate of these quaternion sequences.
On Iterative Relations in Pell Subscripts
Dişkaya, Orhan
In this study, we define iterative relations in Pell subscripts using Pell numbers, and examine the divisibility properties of these iterative relations. Furthermore, despite the computational difficulty of determining the divisibility of very large Pell numbers, the study possible to obtain results through theoretical methods. Thus, we have successfully generalized Desmond's work on the Fibonacci sequence by using Pell numbers to develop a dissimilar approach
On the Lagrange Interpolations of the Jacobsthal and Jacobsthal-Lucas Sequences
Dişkaya, Orhan
This study explores the formation of polynomials of at most degree nusing the first n+1 terms of the Jacobsthal and Jacobsthal-Lucas sequences through Lagrange interpolation. The paper provides a detailed examination of the recurrence relations and various identities associated with the Jacobsthal and Jacobsthal-Lucas Lagrange Interpolation Polynomials.
PADOVAN AND PERRIN SPINORS
Dişkaya, Orhan | Menken, Hamza
Spinors are components of a complex vector space that can be re- lated 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 on the quaternion al- gebra. In this work, first, spinors have been mathematically presented. Then, Padovan and Perrin spinors have been defined using the Padovan and Perrin quaternions. Later, we defined the algebraic structure for these spinors. Fi- nally, we have established certain identities such as the Binet formulas and generating functions for Padovan and Perrin spinors.
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.
SOME IDENTITIES FOR SEQUENCES OF BINOMIAL SUMS OF GENERALIZED PADOVAN IDENTITIES INCLUDING POWERS AND BINOMIAL COEFFICIENTS
Dişkaya, Orhan
In this work, we investigate several identities of the generalized Padovan sequence, including binomial coefficients, using the method of ordinary power series generat- ing functions. We explore the characteristics and identities of generalized Padovan sequences, examining binomial coefficients and other extended identities through this method. Nevertheless, specific cases of third-order sequences have been ex- plored and applied in this context. By employing various techniques, we aim to derive new results concerning third-order recurrence relations, enhancing our com- prehension of these complex sequences.
On the Incomplete (p,q)−Fibonacci and (p,q)−Lucas Numbers
Dişkaya, Orhan | Menken, Hamza
In this present work, the incomplete (p,q)−Fibonacci and (p,q)−Lucas numbers are defined. We examine their recurrence relations as well as some of their properties. We derive their generating functions.
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.
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.
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.
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
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.
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.
Compositions of positive integers with 2s and 3s
Dişkaya, Orhan | Menken, Hamza
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 composi- tions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2s and 3s related to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2s and 3s.
The m−Order Linear Recursive Quaternions
Dişkaya, Orhan | Menken, Hamza
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, and give the Binet-like formulas and generating functions for them.
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.
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.
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.
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.
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.
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 recurrence Pn+3 = Pn+1 + Pn with the initial conditions P0 = 1, P1 = 0 and P2 = 1. Feinberg [2] defined Tribonacci numbers. Hoggatt and Bicknell [3] gave Tribonacci poliynomials. Vieira and Alves [9] expressed Sequences of Tridovan and their iden- tities. The Tridovan sequence {T Pn} is defined by the forth order recurrence T Pn+4 = T Pn+2 + T Pn+1 + T Pn with the initial conditions T P−1 = 0, T P0 = 1, T P1 = 0 and T P2 = 1. G. P. Dresden and Z. Du have defined of the k-bonacci sequence in [1]...
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.
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 changed intensity (UACI). In addition, the root mean squared error (RMSE) suggests that the decoded images are consistent with the original images. It can therefore be summarized that the proposed encryption algorithm is suitable for image encryption.
ON THE (p, q)− FIBONACCI N−DIMENSIONAL RECURRENCES
Dişkaya, Orhan | Menken, Hamza
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.
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 of n is a representation of n as a sum of positive integers where the order of the summands is considered irrelevant[5]. A composition of a positive integer n is an ordered sum of 1s and 2s [3, 4]. For example, 2 has two distinct such compositions, and 3 has three, and 4 has five. In the present work, we study compositions of positive integers with 2s and 3s. We prove that these compositions can be given in terms of Padovan numbers. We create some flooring models of these compositions rela...
SOME PROPERTIES OF THE PLASTIC CONSTANT
Dişkaya, Orhan | Menken, Hamza
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.
SOME NOTES ON THE PLASTIC CONSTANT
Dişkaya, Orhan | Menken, Hamza
The Padovan sequence is named after Richard Padovan who attributed its dis- covery to Dutch architect Hans van der Laan in his 1994 essay Dom. The Padovan sequence {Pn}n≥0 is defined by (1) P0 = P1 = P2 = 1 and Pn+3 = Pn+1 + Pn for all n ≥ 0. Here, Pn is the nth Padovan number. First few terms of this sequence are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21. The ratio of successive the Padovan number converges to the plastic constant. As n gets larger, it appers that PPn+1 n approaches a limit, namely, 1, 32471795724474602596 . . . The plastic number p (also known as the plastic constant, the plastic ratio, the platin number and the minimal Pisot number) is a mathematical constant which is the unique real solution of the cubic equation x3 − x − 1 = 0. It has the exact value p = s3 9 + √ 69 18 + s3 9 − √ 69 18 that was firstly defined in 1924 by Gerard Cordonnier. He described applications to architecture and illustrated the use of the plastic constant in many buildings (for the details see [1, 2, 3]). Its decimal expansion begins with α ≈ 1.3247 = p = P lastic ratio. In the present work we construct the plastic number in three-dimensional space. We examine the nested radicals and continued fraction expansions of the plastic ratio. Also, ...
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.
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.
THE CLASSICAL AES-LIKE CRYPTOLOGY VIA THE FIBONACCI POLYNOMIAL MATRIX
Dişkaya, Orhan | Menken, Hamza
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 cryptology algorithm. So, we find the classical AES-like cryptology via the Fibonacci polynomial matrix. Successful results were achieved with the method used.
On the Padovan Triangle
Dişkaya, Orhan | Menken, Hamza
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.
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.
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 exponential generating functions are obtained.
ON THE (s, t)-PADOVAN AND (s, t)-PERRIN QUATERNIONS
Dişkaya, Orhan | Menken, Hamza
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.
On the Split (s, t)-Padovan and (s, t)-Perrin Quaternions
Dişkaya, Orhan | Menken, Hamza
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 certain binomial sums regarding the split (s, t)- Padovan and (s, t)-Perrin quaternions.
ON THE JACOBSTHAL AND JACOBSTHAL-LUCAS SUBSCRIPTS
Dişkaya, Orhan | Menken, Hamza
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 = JJn, Zn = Jjn, Yn = jjn and Tn = jJn. We obtain some identities and relations for the Jacobsthal and Jacobsthal- Lucas subscripts sequences. Also, we give recursive definitions for the sequences Xn, Zn, Yn and Tn.
On the Quadra Fibona-Pell and Hexa Fibona-Pell-Jacobsthal Sequences
Dişkaya, Orhan | Menken, Hamza
In this paper, we consider the 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 compounds are the Fibonacci, Pell and Jacobsthal sequences. We derive the Binet-like formulas, the generating functions and the exponential generating functions of these sequences. Also, we obtain some binomial identities for them.
ON THE SEQUENCE OF GELL NUMBERS
Dişkaya, Orhan | Menken, Hamza
The Pell sequence {Pn}n≥0 is defined by the initial values P0 = 0 and P1 = 1 and the recurrence relation Pn+2 = 2Pn+1 + Pn, n ≥ 0. In this work we introduce a new class of Gell numbers which generalizes Pell num- bers. We derive Binet-like formula, generating function and some identities of Gell numbers.