New identities involving generalized Fibonacci and generalized Lucas numbers

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.     

A generalization of generalized Fibonacci and generalized Pell numbers

This paper is concerned with developing a new class of generalized numbers. The main advantage of this class is that it generalizes the two classes of generalized Fibonacci numbers and generalized Pell numbers. Some new identities involving these generalized numbers are obtained. In addition, 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.     

On a new class of hyperbolic functions

This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet's formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into “continuous” theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role played by the hyperbolic functions in geometry and physics, (“Lobatchevski's hyperbolic geometry”, “Four-dimensional Minkowski's world”, etc.), it is possible to expect that the new theory of the hyperbolic functions will bring to new results and interpretations on mathematics, biology, physics, and cosmology. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometry and the hyperbolic symmetrical character of the disintegration of the neural vacuum, as pointed out by El Naschie [Chaos Solitons & Fractals 17 (2003) 631].     

On matrices related with Fibonacci and Lucas numbers

In this paper, we obtain some new results on matrices related with Fibonacci numbers and Lucas numbers. Also, we derive the relation between Pell numbers and its companion sequence by using our representations.     

关于高阶Genocchi数和广义Lucas多项式的恒等式

利用组合数学的方法,得到了一些包含高阶Genocchi数和广义Lucas多项式的恒等式,并且由此建立了Fibonacci数与Riemann Zeta函数的关系式.     

关于一些级数取整值问题的注记

利用Pell方程的理论,讨论了与广义Lucas数有关的一些级数的取整值问题.在一定条件下,解决了一些级数的取整值问题.     

关于涉及广义Fibonacci-Lucas数的一些级数和注记

本文得到涉及广义Fibonacci-Lucas数的幂的一些级数的结果。     

On a new type of distance Fibonacci numbers

In this paper, we define a new kind of Fibonacci numbers generalized in the distance sense. This generalization is related to distance Fibonacci numbers and distance Lucas numbers, introduced quite recently. We also study distinct properties of these numbers for negative integers. Their representations and interpretations in graphs are also studied.     

一些包含Lucas数的恒等式

利用第一类Chebyshev多项式的性质以及其与Lucas数的关系得到了关于Lucas数立方的一些恒等式.     

Introduction to Fibonacci and Lucas hybrinomials

ABSTRACT

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper, we introduce and study the Fibonacci and Lucas hybrinomials, i.e. polynomials, which are a generalization of the Fibonacci hybrid numbers and the Lucas hybrid numbers, respectively.     

Some alternating sums of Lucas numbers

We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers.     

Complex Fibonacci and Lucas Numbers,Continued Fractions,and the Square Root of the Golden Ratio (Condensed Version)

The connections between the Golden Ratio namely (1 + √5)/2, a simple continued fraction, and Fibonacci and Lucas numbers, are familiar. The Fibonacci and Lucas numbers have many fascinating properties. We now point out that the square root of the Golden Ratio is the real part of a simple periodic continued fraction but using (complex) Gaussian integers a + ib instead of the natural integers. This fact provokes a definition and a study of complex Fibonacci and Lucas numbers, and the study again turns out to have a rich theoretic structure. A fuller account will appear in The Fibonacci Quarterly.     

Some new Fibonacci and Lucas identities by matrix methods

The aim of this article is to characterize the 2 × 2 matrices X satisfying X ² = X + I and obtain some new identities concerning with Fibonacci and Lucas numbers.     

Theory of Binet formulas for Fibonacci and Lucas p-numbers

Modern natural science requires the development of new mathematical apparatus. The generalized Fibonacci numbers or Fibonacci p-numbers (p = 0, 1, 2, 3, …), which appear in the “diagonal sums” of Pascal’s triangle and are assigned in the recurrent form, are a new mathematical discovery. The purpose of the present article is to derive analytical formulas for the Fibonacci p-numbers. We show that these formulas are similar to the Binet formulas for the classical Fibonacci numbers. Moreover, in this article, there is derived one more class of the recurrent sequences, which is defined to be a generalization of the Lucas numbers (Lucas p-numbers).     

A class of numbers associated with the Lucas numbers

The main object of this paper is to present a systematic investigation of a new class of numbers associated with the familiar Lucas numbers. The various results obtained here for this class of numbers include explicit hypergeometric representations, generating functions, recurrence relations, and summation formulas.     

Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and meaningful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbers and Lucas numbers are obtained.     

一些关于Chebyshef多项式以及它们应用的恒等式(英文)

本文研究了Chebyshef多项式的一类幂和问题.利用初等方法以及Chebyshef多项式的性质,获得了一些有趣的恒等式,推广了Melham关于Lucas数的奇数次幂和的猜想.     

New asymmetric generalizations of the Filbert and Lilbert matrices

Periodica Mathematica Hungarica - Two new asymmetric generalizations of the Filbert and Lilbert matrices constructed by the products of two Fibonacci and Lucas numbers are considered, with...     

Formula for Fibonacci sequence with arbitrary initial numbers

In this paper the formula for Fibonacci sequences with arbitrary initial numbers has been established by using damped oscillation equation. The formula has an exponential and an oscillatory part, it does not separate the indexes of odd and even members of the series and it is applicable on the continual domain. With elementary conditions the formula is reduced to Lucas series, and the square of Lucas series has a catalytic role in the relation of hyperbolic and trigonometric cosine. A complex function is given and the length of Fibonacci spiral is calculated. Natural phenomena support the validity of the proposed concept.     

A generalization of Lucas polynomial sequence

In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.