New identities involving generalized Fibonacci and generalized Lucas numbers (original) (raw)

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.

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 identities of generalized Fibonacci and generalized Pell numbers via a new type of numbers

arXiv: Number Theory, 2015

This paper is concerned with developing some new identities of generalized Fibonacci numbers and generalized Pell numbers. A new class of generalized numbers is introduced for this purpose. The two well-known identities of Sury and Marques which are recently developed are deduced as special cases. Moreover, some other interesting identities involving the celebrated Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced

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).

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.

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.

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 Fibonacci-Lucas Sequence

Turkish Journal of Analysis and Number Theory, 2014

The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula − − = + , 2 n ≥ with B 0 = 2b, B 1 = s, where b and s are integers. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.