Özkan, Engin; Taştan, Merve; Aktaş, Aysun

We introduce advances in the study of k−Pell and k−Pell-Lucas numbers. Then we give some properties of these families and find the generating functions of the families for some k. In addition, we identify the relationships between the family of k−Pell numbers and the classical Pell numbers as well as the family of k−Pell-Lucas numbers and the classical Pell-Lucas numbers. Further, we define generalized polynomials for these numbers. We also obtain some useful properties of these polynomials. Additionally, we give the relationships between the generalized k−Pell polynomials and the classical Pell polynomials as well as the generalized k−Pell-Lucas polynomials and the classical Pell-Lucas polynomials. These generalizations are given in matrix representation. Results in this paper include Cassini identities for these families and polynomials.

Özkan, E.; Kuloğlu, B.

We introduce a new generalization of the known Jacobstal sequence and we call Jacobstal-Like sequence 〈Jk,n〉 and then we study on Jacobstal-Like sequence〈Jk,n〉 and kJacobsthal-Lucas sequence 〈jk,n〉 side by side by introducing two special matrices for these sequences. After that, by using these matrices we obtain Binet formula for Jacobstal-Like sequence 〈Jk,n〉 Further, we get Cassini identity for Jacobstal-Like sequence 〈Jk,n〉. Also we establish some of the interesting properties of Jacobstal-Like sequence. Then we give a limit for Jacobstal-Like sequence 〈Jk,n〉.