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.     

Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials

In this article, we find elements of the Lucas polynomials by using two matrices. We extend the study to the n-step Lucas polynomials. Then the Lucas polynomials and their relationship are generalized in the paper. Furthermore, we give relationships between the Fibonacci polynomials and the Lucas polynomials.     

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.     

The Lucas p-matrix

In this note, we study the Fibonacci and Lucas p-numbers. We introduce the Lucas p-matrix and companion matrices for the sums of the Fibonacci and Lucas p-numbers to derive some interesting identities of the Fibonacci and Lucas p-numbers.     

Continued fractions,the group GL(2, ℤ), and Pisot numbers

The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.     

A generalization of Fibonacci and Lucas matrices

We define the matrix of type s, whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix , whose nonzero elements are generalized Fibonacci numbers. We observe two regular cases of these matrices (s=0 and s=1). Generalized Fibonacci matrices in certain cases give the usual Fibonacci matrix and the Lucas matrix. Inverse of the matrix is derived. In partial case we get the inverse of the generalized Fibonacci matrix and later known results from [Gwang-Yeon Lee, Jin-Soo Kim, Sang-Gu Lee, Factorizations and eigenvalues of Fibonaci and symmetric Fibonaci matrices, Fibonacci Quart. 40 (2002) 203–211; P. Staˇnicaˇ, Cholesky factorizations of matrices associated with r-order recurrent sequences, Electron. J. Combin. Number Theory 5 (2) (2005) #A16] and [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)]. Correlations between the matrices , and the generalized Pascal matrices are considered. In the case a=0,b=1 we get known result for Fibonacci matrices [Gwang-Yeon Lee, Jin-Soo Kim, Seong-Hoon Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math. 130 (2003) 527–534]. Analogous result for Lucas matrices, originated in [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)], can be derived in the partial case a=2,b=1. Some combinatorial identities involving generalized Fibonacci numbers are derived.     

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.     

关于与广义Fibonacci,Lucas数有关的一些级数的注记

求出了一些与广义Fibonacci,Lucas数有关的一些倒数级数的值。     

涉及广义Fibonacci-Lucas数的幂的一些级数的近似值

计算出涉及广义Fibonacci-Lucas数的幂的一些级数的近似值。     

Fibonacci identities via the determinant of tridiagonal matrix

In this paper, using the method of Laplace expansion to evaluate the determinant tridiagonal matrices, we construct a kind of determinants to give new proof of the Fibonacci identities.     

探索以(Fn-k,Fn,Fn)为边长的Fibonacci三角形

Fibonacci三角形是边长为Fibonacci数、面积为整数的三角形.存在以(F<,n-k>,F<,n>.F<,n>)为边长的Fibonacci三角形的情形可以被划分为三类(k时,不存在边长为(F<,n-k>,F<,n>.F<,n>)的Fibonacci三角形.     

广义Fibonacci数的一些求和公式

This note provides the some sum formulas for generalized Fibonacci numbers. The results are proved using clever rearrangements, rather than using induction.     

k=2t·3时不存在以(Fn-k,Fn,Fn)为边长的Fibonacci三角形

Fibonacci三角形是边长为Fibonacci数、面积为整数的三角形.利用平方剩余的方法得到:当k=2'·3时,不存在边长为(Fn-k,Fn,Fn)的Fibonacci三角形(k＜2).     

Reflecting the pascal matrix about its main antidiagonal

Let B denote either of two varieties of order n Pascal matrix, i.e., one whose entries are the binomial coefficients. Let B_R denote the reflection of B about its main antidiagonal. The matrix B is always invertible modulo n; our main result asserts that B^-1 ≡ B^R mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of the difficulties with the matrix exponential under modular arithmetic. We then use our main result to extend the "Fibonacci diagonal" property of Pascal matrices.     

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.     

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 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.     

Some combinatorial identities via Fibonacci numbers

The Pascal matrix and the Stirling matrices of the first kind and the second kind obtained from the Fibonacci matrix are studied, respectively. Also, we obtain combinatorial identities from the matrix representation of the Pascal matrix, the Stirling matrices of the first kind and the second kind and the Fibonacci matrix.     

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.     

Identities for generalized Fibonacci numbers: a combinatorial approach

This note shows a combinatorial approach to some identities for generalized Fibonacci numbers. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem.