On the properties of k-Fibonacci and k-Lucas numbers (original) (raw)

Identities on generalized Fibonacci and Lucas numbers

Notes on Number Theory and Discrete Mathematics

In this article, the concepts of Fibonacci, Tribonacci, Lucas and Tetranacci numbers are generalized as continued sum. The generalized Fibonacci identity is proved by using induction and the binomial theorem. Further, it is proved that the generalized Fibonacci and Lucas sequences are logarithmically convex (concave) and some special identities are obtained.

Some new formulas on the K-Fibonacci numbers

JOURNAL OF ADVANCES IN MATHEMATICS

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..

An identity relating Fibonacci and Lucas numbers of order k

Electronic Notes in Discrete Mathematics, 2018

The following relation between Fibonacci and Lucas numbers of order k, n ∑ i=0 m i L (k) i + (m − 2)F (k) i+1 − k ∑ j=3 (j − 2)F (k) i− j+1 = m n+1 F (k) n+1 + k − 2, is derived by means of colored tiling. This relation generalizes the well-known Fibonacci-Lucas identities, ∑ n i=0 2 i L i = 2 n+1 F n+1 , ∑ n i=0 3 i (L i + F i+1) = 3 n+1 F n+1 and ∑ n i=0 m i (L i + (m − 2)F i+1) = m n+1 F n+1 of A.

Generalized Lucas Numbers and Relations with Generalized Fibonacci Numbers

2011

In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas numbers and generalized order-k Fibonacci numbers. In addition, we obtain Binet formula and combinatorial representation for generalized order-k Lucas numbers by using properties of generalized Fibonacci numbers.

Identities for Fibonacci and Lucas Numbers

In this paper several new identities are given for the Fibonacci and Lucas numbers. This is accomplished by by solving a class of non-homogeneous, linear recurrence relations.

Generalized Fibonacci – Lucas sequence its Properties

Global Journal of Mathematical Analysis, 2014

Sequences have been fascinating topic for mathematicians for centuries. The Fibonacci sequences are a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci number, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula F F F n 2 n n-1 n-2 ,    and F 0,F 1 01  , where F n are an n th number of sequences. The Lucas Sequence is defined by the recurrence formula L L L n 2 n n-1 n-2 ,    and L =2, L =1 01 , where L n an nth number of sequences are. In this paper, we present generalized Fibonacci-Lucas sequence that is defined by the recurrence relation 12 B B B n n n   , 2 n  with B 0 = 2s, B 1 = s. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.

Some Properties of the Product of (P,Q) – Fibonacci and (P,Q) - Lucas Number

International Journal of GEOMATE, 2017

Some mathematicians study the basic concept of the generalized Fibonacci sequence and Lucas sequence which are the (p,q)-Fibonacci sequence and the (p,q)-Lucas sequence. For example, Singh, Sisodiya and Ahmad studied the product of the k-Fibonacci and k-Lucas numbers. Moreover, Suvarnamani and Tatong showed some results of the (p, q)-Fibonacci number. They found some properties of the (p,q)-Fibonacci number and the (p,q)-Lucas number. There are a lot of open problem about them. Moreover, the example for the application of the Fibonacci number to the generalized function was showed by Djordjevicand Srivastava. In this paper, we consider the (p,q)-Fibonacci sequence and the (p,q)-Lucas sequence. We used the Binet's formulas to show that some properties of the product of the (p,q)-Fibonacci number and the (p,q)-Lucas number. We get some generalized properties on the product of the (p,q)-Fibonacci number and the (p,q)-Lucas number.

A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients

Applied Mathematics and Computation, 2013

In this study, Fibonacci and Lucas numbers have been obtained by using generalized Fibonacci numbers. In addition, some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated to write generalized Fibonacci sequences in a new direct way. Furthermore, it has been given a new formula for some Lucas numbers.