Generalized k-order Fibonacci and Lucas hybrid numbers

Abstract

In this study, we describe the generalized k-order Fibonacci and Lucas numbers and give some important results with specific choices. The main purpose of this study is to define the generalized k-order Fibonacci hybrid and Lucas hybrid numbers and give some of their properties. With this definition, we obtain some important number sequences by making special choices. For example, we give the Horadam hybrid number for k=2 and the other series of hybrid numbers, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal end Jacobsthal-Lucas hybrid numbers. We give the generating functions and some properties about the generalized k-order Fibonacci hybrid and Lucas hybrid numbers. Also, we identify and prove the matrix representation for the generalized k-order Fibonacci hybrid and Lucas hybrid numbers. The Qk matrix given for k-order Fibonacci numbers are defined for the generalized k-order Fibonacci hybrid and Lucas hybrid numbers and after the matrices with the generalized k-order Fibonacci hybrid and Lucas hybrid numbers are obtained with help of auxiliary matrices, important relationships and identities are established.