Akkus, Hakan; Ozkan, Engin

In this study, we define the k-Copper Fibonacci and k-Copper Lucas sequences, and the first few terms of these sequences are given. Then we obtain properties such as special generating functions, Binet formulas, etc. Also, we present special relations between these sequences and the roots of their characteristic equations. We obtain the relationship between the ratio of the biggest to the smallest of two consecutive terms of these sequences. In addition, we give important relations among the positive and negative index terms of these sequences, and the sum of the squares of two consecutive terms is related to these sequences. We calculate the significant identities of these sequences. Besides, we apply Catalan transformations of these sequences and obtain the first few terms. We present some features of these Catalan transformations, such as generating functions. Furthermore, we apply the Hankel transform to the Catalan transform of these sequences.

Akkus, Hakan; Özkan, Engin

In this study, we examine the-Edouard,-Edouard-Lucas, and Modified-Edouard sequences, and some terms of these sequences, are given. Then, we give 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 among the terms of the-Edouard,-Edouard-Lucas, Modified-Edouard, Edouard, Edouard-Lucas, Modified Edouard, Balancing, Balancing-Lucas, and Modified Balancing sequences. Finally, we associate the terms of these sequences with matrices.

Kuloglu, Bahar; Özkan, Engin

In this study, a different perspective was brought to Narayana sequences and one-, two-, three- and n -dimensional recurrence relations of these sequences were created. Then, some identities ranging from one to n -dimensions of these recurrences were created.

Uysal, Mine; Özkan, Engin; Shannon, Anthony G.

In this paper, dual bicomplex Balancing and Lucas -Balancing numbers are defined, and some identities analogous to the classic properties of the Fibonacci and Lucas sequences are produced. We give the relationship between these numbers and Pell and Pell -Lucas numbers. From these, the basic bicomplex properties for the norm and its conjugate of these numbers are also developed. These in turn lead to the Binet formula, the generating functions and exponential generating functions, which are important concepts for number sequences. The Cassini identity, which is important for number sequences, actually emerged to solve the famous Curry paradox. We calculated the Cassini, Catalan, Vajda and d' Ocagne identities for these numbers.

Kuloglu, Bahar; Özkan, Engin

In this paper, inspiring Hosoya's triangle, we define a new Narayana triangle. Then, we represent this Narayana triangle geometrically on the plane. In addition, we give some identities and properties of the new Narayana triangle.

Uysal, Mine; Özkan, Engin

We define Padovan hybrid quaternions by using Padovan hybrid numbers and Padovan quaternion. We give the basic operation properties of Padovan hybrid quaternion numbers. We give some properties and identities such as the Binet formula, sum formula, the matrix representation, characteristic equation, norm, characteristic and generating function for these quaternions.

Altunbas, Murat

The aim of this paper is to give some characterizations for N-Legendre and N-slant curves in the unit tangent bundles of surfaces endowed with natural diagonal lifted structures.

Kuloglu, Bahar; Özkan, Engin

We define the generalized (k, r) - Gauss Pell numbers by using the definition of a distance between numbers. Then we examine their properties and give some important identities for these numbers. In addition, we present the generating functions for these numbers and the sum of the terms of the generalized (k, r) - Gauss Pell numbers.

Özkan, Engin; Tastan, Merve

In this paper, we define the new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal-Lucas numbers. We obtain some exciting properties of the families. We give the relationships between the family of the Gauss k-Jacobsthal numbers and the known Gauss Jacobsthal numbers, the family of the Gauss k-Jacobsthal-Lucas numbers and the known Gauss Jacobsthal-Lucas numbers. We also define the generalized polynomials for these numbers. Further, we obtain some interesting properties of the polynomials. In addition, we give the relationships between the generalized Gauss k-Jacobsthal polynomials and the known Gauss Jacobsthal polynomials, the generalized Gauss k-Jacobsthal-Lucas polynomials and the known Gauss Jacobsthal-Lucas polynomials. Furthermore, we find the new generalizations of these families and the polynomials in matrix representation. Then we prove Cassini's Identities for the families and their polynomials.