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 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 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.

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 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.

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.

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.

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...

Yaklaşım teorisinde q-pozitif lineer operatör dizilerinin genelleştirilmesi

Şimşek, Ersin

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 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.

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 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.

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.