Related papers: Some notes on a Fibonacci-Lucas identity
This note gives an elementary exposition of a variant of the spread polynomials in terms of Fibonacci and Lucas polynomials.
We give alternative proof of Melham's and Howard's identities on generalised Fibonacci numbers
We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…
Following Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence…
Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas…
We prove new theorems for the polynomial expansions of $x^n \pm y^n$ in terms of the binary quadratic forms $\alpha x^2 + \beta xy + \alpha y^2 $ and $a x^2 + bxy + a y^2 $. The paper gives new arithmetic differential approach to compute…
In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.
This contribution presents all possible solutions to the Diophantine equations $F_k=L_mL_n$ and $L_k=F_mF_n$. To be clear, Fibonacci numbers that are the product of two arbitrary Lucas numbers and Lucas numbers that are the product of two…
We dedicate this paper to investigate the most generalized form of Fibonacci Sequence, one of the most studied sections of the mathematical literature. One can notice that, we have discussed even a more general form of the conventional one.…
The Lucas polynomials, $\{n\}$, are polynomials in $s$ and $t$ given by $\{ n \} = s \{ n-1 \} + t \{ n-2 \}$ for $n \geq 2$ with $ \{ 0 \} = 0$ and $\{ 1 \} = 1$. The lucanomial coefficients, an analogue of the binomial coefficients, are…
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical…
A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse, generalizing a classic problem about circles. We give a brief history of the circle problem, an account of…
Unlike in the case of Fibonacci and Lucas numbers, there is a paucity of literature dealing with summation identities involving the Padovan and Perrin numbers. In this paper, we derive various summation identities for these numbers,…
A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.
Motivated by observations of Guillera we generalise the so-called Ramanujan-type supercongruences to a further level in which the sequences of Fibonacci, Lucas, Ap\'ery numbers and their friends all receive a natural appearance.
This paper analyzes the concept of orthogonality in second-order polynomial sequences that have Binet formula similar to that of the Fibonacci and Lucas numbers, referred to as Generalized Fibonacci Polynomials (GFP). We give a technique to…
The equation commonly known as Sury's identity is a deceptively simple summation formula that connects the Lucas numbers, Fibonacci numbers, and powers of two. Many authors have given extensions and generalizations over the years; in this…
This paper introduces a variation on an identity by Bruckman and Good. Using this identity, we are able to derive various well-known sums involving reciprocals of Fibonacci and Lucas numbers, including the case when the indices form an…
In this paper we extend the notion of Melham sum to the Pell and Pell-Lucas sequences. While the proofs of general statements rely on the binomial theorem, we prove some spacial cases by the known Pell identities. We also give extensions of…
Let $(U_{n}(P,Q) $ and $(V_{n}(P,Q) $ denote the generalized Fibonacci and Lucas sequence, respectively. In this study, we assume that $Q=1.$ We determine all indices $n$ such that $U_{n}=5\square $ and $U_{n}=5U_{m}\square $ under some…