On the Lucas-Leonardo Numbers in Complex and Dual-Complex Number Systems

Abstract

This study aimed to introduce the Lucas-Leonardo numbers in 2-dimensional real algebra and 4-dimensional real Clifford algebra, namely, complex and dual-complex Lucas-Leonardo numbers, respectively. In this sense, basic algebraic properties of these numbers were presented as well as some Fibonacci-type identities such as Cassini, Catalan, and d'Ocagne. The generating function and Binet formula were constructed for the complex and dual-complex forms of Lucas-Leonardo numbers. Some relations between these numbers and other well-known integer sequences were proven. Moreover, some formulas related to the sums of the terms of these sequences were established.