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.     

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.     

Hessenberg matrices and the Pell and Perrin numbers

In this paper, we investigate the Pell sequence and the Perrin sequence and we derive some relationships between these sequences and permanents and determinants of one type of Hessenberg matrices.     

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.     

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.     

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...     

一种三对角矩阵的特征值及其应用

给出了一种三对角矩阵的特征值和特征向量的算法,利用矩阵方法和对称多项式证明了一些与Lucas数以及第一类Chebyshev多项式有关的三角恒等式.     

Hankel matrices for the period-doubling sequence

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.     

递推数列的非线性表达式的构造

A type of nonlinear expressions of Lucas sequences are established inspired by Hsu [A nonlinear expression for Fibonacci numbers and its consequences.J.Math.Res.Appl.,2012,32(6):654–658].Using the relationships between the Lucas sequence and other linear recurring sequences satisfying the same recurrence relation of order 2,i.e.,the Horadam sequences,we may transfer the identities of Lucas sequences to the latter.     

On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices

In this paper we consider certain generalizations of the well-known Fibonacci and Lucas numbers, the generalized Fibonacci and Lucas p-numbers. We give relationships between the generalized Fibonacci p-numbers, F_p(n), and their sums, , and the 1-factors of a class of bipartite graphs. Further we determine certain matrices whose permanents generate the Lucas p-numbers and their sums.     

On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

The aim of this paper is to give new results about factorizations of the Fibonacci numbers F _n and the Lucas numbers L _n . These numbers are defined by the second order recurrence relation a _n+2 = a _n+1+a _n with the initial terms F ₀ = 0, F ₁ = 1 and L ₀ = 2, L ₁ = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in [CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].     

An Analog of the Poincarée Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem

We establish an analog of the Cauchy–Poincarée separation theorem for normal matrices in terms of majorization. A solution to the inverse spectral problem (Borg type result) is also presented. Using this result, we generalize and extend the Gauss–Lucas theorem about the location of roots of a complex polynomial and of its derivative. The generalization is applied to prove old conjectures due to de Bruijn–Springer and Schoenberg.     

Perfect fibonacci and lucas numbers

In this note, we show that the classical Fibonacci and Lucas sequence do not contain any perfect number.     

Perfect fibonacci and lucas numbers

In this note, we show that the classical Fibonacci and Lucas sequence do not contain any perfect number.     

On the matrix algebra of elliptic biquaternions

In this study, we introduce the concept of elliptic biquaternion matrices. Firstly, we obtain elliptic matrix representations of elliptic biquaternion matrices and establish a universal similarity factorization equality for elliptic biquaternion matrices. Afterwards, with the aid of these representations and this equality, we obtain various results on some basic topics such as generalized inverses, eigenvalues and eigenvectors, determinants, and similarity of elliptic biquaternion matrices. These valuable results may be useful for developing a perfect theory on matrix analysis over elliptic biquaternion algebra in the future.     

Products of members of Lucas sequences with indices in an interval being a power

In this paper, we show that the product of sufficiently many distinct members of a Lucas sequence with indices in an interval of a fixed length cannot be a perfect power of exponent larger than 1.     

关于Lucas数列同余性质的研究

将二项式系数的性质应用到Lucas数列的研究中,并结合Fibonacci数列与Lucas数列的恒等式得到几个有趣的Lucas数列的同余式.     

Extreme determinants on the convex hull of matrices with prescribed singular values

In a recent paper[1] the author carried through a comprehesive analysis of the diagonal elements of matrices having prescribed singular values, and as an application of this analysis a characterization was obtained of matrices with prescribed singular values. In the present note we obtain the greatest and least values for the determinants of the matrices in such convex hulls.     

正定Hermite矩阵加权幂平均的行列式不等式

本文给出了m个正定Hermite矩阵加权幂平均的行列式的一个不等式，它是m个正数的加权益平均不等式的自然推广，也是正定Hermite矩阵行列式的凸性不等式的推广．