Abstract
In this paper, we find some formulas for finding some special sums of the k-Fibonacci or the k-Lucas numbers. We find also some formulas that relate the k-Fibonacci or the k--Lucas numbers to some sums of these numbers..
Highlights
There exist generalizations of the classical Fibonacci numbers given by many researchers as Horadam [4] and recently by Falcon and Plaza[3]
In similar formFalcon [2], the k–Lucas numbers are defined as Lk,n 1 k Lk,n Lk,n 1 with initial conditions
Some properties of 1 and 2 that we will use in this paper are the following:
Summary
Keywords k-Fibonacci numbers, k-Lucas numbers, Binomial transform, Geometric sum. There exist generalizations of the classical Fibonacci numbers given by many researchers as Horadam [4] and recently by Falcon and Plaza[3]. For any positive real number k, the k–Fibonacci sequence, say For k = 1, classical Fibonacci sequence is obtained and for k = 2, Pell sequence appears. We define the negative k–Fibonacci numbers as Fk, n ( 1)n 1 Fk,n .
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