On some identities for k-Fibonacci sequence

2012

To facilitate rapid numerical calculations of identities pertaining to Fibonacci numbers, we present each identity as a binomial sum. Mathematics Subject Classification: 05A10,11B39

Turkish Journal of Analysis and Number Theory, 2014

The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences. 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 n

Cornell University - arXiv, 2022

We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci-Lucas summation identities. 1 Introduction Our goal is to derive, from elementary identities, some presumably new Fibonacci and Lucas identities including binomial coefficients. The research is a continuation of the recent works by Adegoke [1, 2] and Adegoke et al. [3]. The results are similar to those found in the classical articles of Carlitz and Ferns [5], Hoggatt et al. [6], Layman [9], Carlitz [4] and Long [10]. Recall that the Fibonacci numbers F j and the Lucas numbers L j are defined, for j ∈ Z, through the recurrence relations

Mathematica Montisnigri, 2021

In the present paper, we propose some properties of the new family 𝑘-generalized Fibonacci numbers which related to generalized Fibonacci numbers. Moreover, we give some identities involving binomial coefficients for 𝑘-generalized Fibonacci numbers.

International Journal of Nonlinear Sciences and Numerical Simulation, 2009

In this paper, we apply the binomial, k-binomial, rising, and falling transforms to the k-Fibonacci sequence. Many formulas relating the so obtained new sequences are presented and proved. Finally, we define and find the inverse transforms of the sequences previously obtained.

International Journal of Mathematical Analysis, 2014

The purpose of this paper is to obtain some identities for the s-Generalized Fibonacci sequence as a consequence of its Binet's formula. We also give the generating function for this sequence, as well as another expression for its general term using the ordinary generating function. Moreover, we find the connection between s-Generalized Fibonacci numbers and Pythagorean numbers triples.

Global Journal of Mathematical Analysis, 2014

The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Like Polynomials are introduced and defined by 1 2 () () (), n 2. n n n m x xm x m x      with 0 () 2 m x s  and 1 () 1 m x s , where s is integer. Further, some basic identities are generated and derived by standard methods.

In this paper, we define a new family of k-generalized Fibonacci numbers. Furthermore, we give sums and recurrence relations of this numbers and obtain generating functions of this numbers for k = 2.

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.