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Araştırmacılar Orhan Dişkaya
Doç.Dr. Orhan DişkayaFEN FAKÜLTESİ MATEMATİK BÖLÜMÜ CEBİR VE SAYILAR TEORİSİ ANABİLİM DALI
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
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.

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
169326

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.

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...

170573

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.

Bildiri20242024 2 | 0 Erişime Açık
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.

169327

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 coeffici...

Makale2025The Ramanujan Journal 9 | 0 Erişime Açık
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 ...

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