Generalization of the k-Leonardo sequence and their hyperbolic quaternions

Abstract

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.