Related papers: A method for obtaining Fibonacci identities
We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions.…
Many kinds of convolution identities have been considered about several numbers, including Bernoulli, Euler, Genocchi, Cauchy, Stirling, and Fibonacci numbers. The well-known basic result about Bernoulli numbers is due to Euler. The…
In this paper we present two families of Fibonacci-Lucas identities, with the Sury's identity being the best known representative of one of the families. While these results can be proved by means of the basic identity relating Fibonacci…
In this paper, firstly, we introduce the Q_{l}-Generating matrix for the bi-periodic Lucas numbers. Then, by taking into account this matrix representation, we obtain some properties for the bi-periodic Fibonacci and Lucas numbers.
In this study, we define a new type of Fibonacci and Lucas num- bers which are called bicomplex Fibonacci and bicomplex Lucas numbers. We obtain the well-known properties e.g. Docagnes, Cassini, Catalan for these new types. We also give the…
Based on well-known properties of Fibonacci and Lucas numbers and polynomials we give a self-contained approach to some bivariate analogs.
We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
In this paper, by presenting bi-periodic Lucas numbers as a binomial sum, we introduce the bi-periodic incomplete Lucas numbers. After that, by using the bi-periodic incomplete Lucas numbers, we derive the recurrence relation and the…
In this paper, we define Tribonacci and Tribonacci-Lucas matrix sequences and investigate their properties.
We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas…
We use probability urn models to discover some known and unknown series identities involving Fibonacci numbers.
Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and…
This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion…
In 2016, Edgar and, independently of him, Bhatnagar sta\-ted a nice polynomial identity that connects Fibonacci and Lucas numbers. Shortly after their publications, this identity has been generalized in two different ways: Dafnis, Phillipou…
In this article we provide with combinatorial proofs of some recent identities due to Sury and McLaughlin. We show that, the solution of a general linear recurrence with constant coefficients can be interpreted as a determinant of a matrix.…
The goal of this paper is twofold: (1) extend theory on certain statistics in the Fibonacci and Lucas sequences modulo $m$ to the Lucas sequences $U := \left(U_n(p,q)\right)_{n \geq 0}$ and $V := \left(V_n(p,q)\right)_{n \geq 0}$, and (2)…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
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…
Horadam introduced a new generalized sequence of numbers, describing its key features and the special sub-sequences that are obtained depending on the choices of initial parameters. This sequence and its sub-sequences are known as the…