Kuloǧlu, Bahar; Özkan, Engin

In this paper, an expansion of the classical hyperbolic functions is presented and studied. Also, many features of the k-Jacobsthal hyperbolic functions are given. Finally, we introduced some graph and curved surfaces related to the k-Jacobsthal hyperbolic functions.

Uysal, Mine; Inam, Ilker; Özkan, Engin

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. First, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.

Özkan, Engi˙n; Uysal, Mi˙ne; Kuloǧlu, Bahar

We introduce the Catalan transform of the Incomplete Jacobsthal numbers. We apply the Hankel transform to the Catalan transforms of these numbers. We calculate determinants of matrixes formed with CJk,nlby using Hankel transform. Then we define the incomplete h(x)-Jacobsthal polynomials. Then we examine the recurrence relation and some properties of these polynomials.

Taştan, Merve; Yllmaz, Nazmiye; Özkan, Engin

We present a new family of Gauss (k,t)-Horadam numbers and obtain Binet formula of this family. We give the relationship between this family and the known Gauss t-Horadam number. Then we prove the Cassini and Catalan identities for this family. Furthermore, we investigate the sums, the recurrence relations and generating functions of this family.

Özkan, Engi˙n; Taştan, Merve

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

Özkan, Engi˙n; Kuloǧlu, Bahar

We give a new definition of Narayana polynomials and show that there is a relationship between the coefficient of the new Narayana polynomials and Pascal's triangle. We define the Gauss Narayana numbers and their polynomials. Then we show that there is a relationship between the Gauss Narayana polynomials and the new Narayana polynomials. Also, we show that there is a relationship between the derivatives of the new Narayana polynomials and Pascal's triangle. We also explain the relationship between the new Narayana polynomials and the known Pell numbers. Finally, we give the Hankel transform of the new Narayana polynomials.