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

Fibonacci Matrisinin Kökleri Aracılığıyla Fibonacci Ve Lucas Sayılarının Özellikleri Üzerine

Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi

In this study, we exploit general techniques from matrix theory to establish some identities for the complex Fibonacci and Lucas numbers with rational subscripts of the forms 2 n and ns. For this purpose, we establish matrix functions 2 n RR → and ns RR → of the Fibonacci matrix R of order 33  for integer odd n and discuss some relations between two special matrices functions 2 n R and ns R , respectively. Also, some identities related to the complex Fibonacci and Lucas numbers with rational subscripts of the forms 2 n and ns are given for every integer odd n and ( ) gcd , 1, n s s = , respectively.

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.

More identities on Fibonacci and Lucas hybrid numbers

Notes on Number Theory and Discrete Mathematics, 2021

We give several identities about Fibonacci and Lucas hybrid numbers. We introduce the Fibonacci and Lucas hybrid numbers with negative subscripts. We obtain different Cassini identities for the conjugate of the Fibonacci and Lucas hybrid numbers by two different determinant definitions of a hybrid square matrix (whose entries are hybrid numbers).

New identities involving generalized Fibonacci and generalized Lucas numbers

Indian Journal of Pure and Applied Mathematics, 2018

This paper presents two new identities involving generalized Fibonacci and generalized Lucas numbers. One of these identities generalize the two well-known identities of Sury and Marques which are recently developed. Some other interesting identities involving the famous numbers of Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced as special cases of the two derived identities. Performing some mathematical operations on the introduced identities yield some other new identities involving generalized Fibonacci and generalized Lucas numbers.

Fermat kkk-Fibonacci and kkk-Lucas numbers

Mathematica Bohemica, 2018

Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all k-Fibonacci and k-Lucas numbers which are Fermat numbers. Some more general results are given.

On the Binomial Sums of k-Fibonacci and k -Lucas sequences

2011

The main purpose of this paper is to establish some new properties of k-Fibonacci and k-Lucas numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between k-Fibonacci and k-Lucas numbers are revealed to get a more strong result.

One Parameter Generalizations of the Fibonacci and Lucas Numbers

2006

We give one parameter generalizations of the Fibonacci and Lucas numbers denoted by {Fn(θ)} and {Ln(θ)}, respectively. We evaluate the Hankel determinants with entries {1/F j+k+1 (θ) : 0 ≤ i, j ≤ n} and {1/L j+k+1 (θ) : 0 ≤ i, j ≤ n}. We also find the entries in the inverse of {1/F j+k+1 (θ) : 0 ≤ i, j ≤ n} and show that all its entries are integers. Some of the identities satisfied by the Fibonacci and Lucas numbers are extended to more general numbers. All integer solutions to three diophantine equtions related to the Pell equation are also found.

Extending the Theory of k-Fibonacci and k-Lucas Numbers

The a:k:m-Fibonacci sequences are introduced. One might be interested to meet the equally famous Jacobsthal sequence at a=2, k=m=1. Our brief results capture the most important properties relating to the assemblage mechanics of these sequences.

On generalized Fibonacci and Lucas polynomials

Chaos, Solitons & Fractals, 2009

Let hðxÞ be a polynomial with real coefficients. We introduce hðxÞ-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these hðxÞ-Fibonacci polynomials. We also introduce hðxÞ-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ðxÞ that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.

On the sequences related to Fibonacci and Lucas numbers

J. Korean Math. Soc, 2005

In this paper, we obtain some properties of the sequences U q n and V q n introduced in [6]. We find polynomial representations and formulas of them. For q = 5, U 5 n is the Fibonacci sequence Fn and V 5 n is the Lucas sequence Ln.