On the complex k-Fibonacci numbers

Sergio Falcon

doi:10.1080/23311835.2016.1201944

Abstract

We first study the relationship between the k-Fibonacci numbers and the elements of a subset of Q2. Later, and since generally studies that are made on the Fibonacci sequences consider that these numbers are integers, in this article, we study the possibility that the index of the k-Fibonacci number is fractional; concretely, 2n+12. In this way, the k-Fibonacci numbers that we obtain are complex. And in our desire to find integer sequences, we consider the sequences obtained from the moduli of these numbers. In this process, we obtain several integer sequences, some of which are indexed in The Online Enciplopedy of Integer Sequences (OEIS).

Highlights

Classical Fibonacci numbers have been very used in different sciences such biology, demography, or economy (Hoggat, 1969; Koshy, 2001)
Sergio Falcon Santana obtained his PhD in Mathematics from ULPGC
He is a professor in the University of Las Palmas de Gran Canaria in Mathematics, Advanced Calculus, Numerical Calculus, Algebra

Summary

Introduction

Classical Fibonacci numbers have been very used in different sciences such biology, demography, or economy (Hoggat, 1969; Koshy, 2001). As Horadam (1961) and recently by Bolat and Kse (2010), Ramírez (2015), Salas (2011) and the current author Falcon and Plaza (2007a, 2007b, 2009a) In this paper, this last generalization is presented, so called the k-Fibonacci numbers. It follows that, from Equations (1) and (2), we can deduce that (1, 0)−n = (−1)n(−Fk,n−1, Fk,n) This formula allows us to define the k-Fibonacci numbers of negative index (as is known), Fk,−n = (−1)n−1Fk,n (4). The previous definition and the results obtained allow us to find some properties of the k-Fibonacci numbers, previously proven in papers (Falcon & Plaza, 2007a, 2007b, 2009a), in the subsection. This formula is very similar to Formula (4) for the k-Fibonacci numbers of negative integer indices

Binnet identity

On the sequences of k-Fibonacci numbers of half index

Expression of a k-Fibonacci number whose index is a multiple of another index