Altiparmak, İkramiye; Akkuş, Hakan; Özkan, Engin

Summary. In this study, we introduce a new recurrence relation of the perfect square sequence. We establish the relationship of perfect square sequence concerned with Fibonacci and Lucas sequences. We compute some important identities such as Catalan, Cassini, and special summation formula for this sequence. We associate the perfect square sequence with Lucas sequence and we call it the perfect square Lucas sequence. In addition, the Binet formula, generating function and summation formula of this sequence is obtained, as well as some properties are satisfied. Furthermore, we present the relationship of perfect square Lucas sequence with Fibonacci and Lucas sequences. Also, we obtain the relationship of perfect square Lucas sequence with perfect square sequence and we present matrix representation of this sequence. Besides, we described polynomials of perfect square and perfect square Lucas sequences. We get Binet formulas, generating functions, and Simpson formula for these polynomials. Eventually, satisfied some intriguing relations between these two polynomials, as well as we give the matrix representation of them.

Kuloğlu, Bahar; Peters, James F.; Uysal, Mine; Özkan, Engin

This paper gives results for higher-order Horadam numbers, which are akin to higher-order Fibonacci sequence, it weaves intricate connections with the classic Horadam, Pell, Jacobsthal, Mersenne, Balancing numbers. Also, all number sequences used in higher-order Horadam numbers, from Fibonacci to Lucas Mersenne to Fermat numbers, exhibit self-similarity at certain points, and so self-similarity has been exploited in these areas to simplify the analysis of complex systems, improve predictive models, and create efficient designs by exploiting the inherent reproducible and scalable nature of self-similar structures. Methods: We introduce the basic properties and results for Horadam numbers, leading to the Catalan transform of higher order Horadam numbers. Results: We give results for Cassini identity, d’Ocagne identity, Vajda identity, Honsberger identity, Melham’s identity. Moreover, this study examines the "Catalan transforms" of these higher-order Horadam numbers, aiming to solidify their comprehension through matrix representations. These powerful tools, along with other equations, act as keys that unlock the heart of these numbers, paving the way for future exploration and deeper insights.

Akkuş, Hakan; Özkan, Engin

In this study, we define the k-Olivier, k-Olivier-Lucas, and Modified k-Olivier sequences and give some terms of these sequences. Then, we obtain the generating functions, summation formulas, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence's generating functions, and the third is with the help of the matrices. In addition, the relations between the terms of the k-Olivier, k-Olivier-Lucas, Modified k-Olivier, third-order Fibonacci-Pell, Adjusted Pell-Padovan, Olivier, Olivier-Lucas, Modified Olivier, third-order Lucas-Pell, Pell-Padovan, and Pell-Perrin sequences are presented. Finally, we associate the terms of these sequences with matrices.

Akkuş, Hakan; Özkan, Engin

In this study, we define the k-Leonardo, k-Leonardo-Lucas, and Modified k-Leonardo sequences, and some terms of these sequences are given. Then, we obtain the generating functions, summation formulas, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence's generating functions, and the third is with the help of the matrices. In addition, we examine the relations between the terms of the k-Leonardo, k-Leonardo-Lucas, Modified k-Leonardo, Leonardo, Leonardo-Lucas, Modified Leonardo, Francois, Fibonacci, and Lucas sequences. Moreover, we associate the terms of these sequences with matrices. Furthermore, we present on the application of these sequences to hyperbolic quaternions. For these quaternions, we give many properties such as Binet formulas. Finally, the terms of the k-Leonardo, k-Leonardo-Lucas, and Modified k-Leonardo sequences are associated with their hyperbolic quaternion values.

Kuloğlu, Bahar; Özkan, Engin

In this article, we have developed a new method for coding\decoding the Jacobsthal and Jacobsthal-Lucas sequences via matrix representations. In this method, coding\decoding is done by transforming the messages into a square matrix. This process aims to not only increase the reliability of information security technology but also to provide the ability to verify information at a high rate.

Özkan, E.; Alp, T.

In this paper, we define biGaussian Pell and Pell-Lucas Polynomials. We give Binet‘s formulas, generating functions, Catalan’s identities, Cassini’s identities for these polynomials. Matrix presentations of biGaussian Pell and Pell-Lucas polynomials are found. Also, NegabiGaussian Pell and Pell-Lucas Polynomials are defined. Finally, we give some properties for these polynomials.

Yilmaz, N.; Aydoğdu, A.; Özkan, E.

In the present paper, we propose some properties of the new family K-generalized Fibonacci numbers which related to generalized Fibonacci numbers. Moreover, we give some identities involving binomial coefficients for K-generalized Fibonacci numbers.

Özkan, Engin; Uysal, Mine

We introduce Mersenne-Lucas hybrid numbers. We give the Binet formula, the generating function, the sum, the character, the norm and the vector representation of these numbers. We find some relations among Mersenne-Lucas hybrid numbers, Jacopsthal hybrid numbers, Jacopsthal-Lucas hybrid numbers and Mersenne hybrid numbers. Then we present some important identities such as Cassini identities for Mersenne-Lucas hybrid numbers.