3x3 Dimensional Special Matrices Associated with Fibonacci and Lucas Numbers (original) (raw)
A generalization of Fibonacci and Lucas matrices
Discrete Applied Mathematics, 2008
We define the matrix U (a,b,s) n of type s, whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix F (a,b,s) n , 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 U (a,b,s) n is derived. In partial case we get the inverse of the generalized Fibonacci matrix F
The Representations of the Fibonacci and Lucas Matrices
Iranian Journal of Science and Technology, Transactions A: Science, 2019
In this study, a matrix R L is defined by the properties associated with the Pascal matrix, and two closed-form expressions of the matrix function f (R L) = R n L are determined by methods in matrix theory. These expressions satisfy a connection between the integer sequences of the first-second kinds and the Pascal matrices. The matrix R n L is the Fibonacci Lucas matrix, whose entries are the Fibonacci and Lucas numbers. Also, the representations of the Lucas matrix are derived by the matrix function f (R L − 5I) , and various forms of the matrix (R L − 5I) n in terms of a binomial coefficient are studied by methods in number theory. These representations give varied ways to obtain the new Fibonacci-and Lucas-type identities via several properties of the matrices R n L and (R L − 5I) n .
Fibonacci Matrisinin Kökleri Aracılığıyla Fibonacci Ve Lucas Sayılarının Özellikleri Üzerine
Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi
In this study, we exploit general techniques from matrix theory to establish some identities for the complex Fibonacci and Lucas numbers with rational subscripts of the forms 2 n and ns. For this purpose, we establish matrix functions 2 n RR → and ns RR → of the Fibonacci matrix R of order 33 for integer odd n and discuss some relations between two special matrices functions 2 n R and ns R , respectively. Also, some identities related to the complex Fibonacci and Lucas numbers with rational subscripts of the forms 2 n and ns are given for every integer odd n and ( ) gcd , 1, n s s = , respectively.
A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences
Hacettepe Journal of Mathematics and Statistics
In this study, a matrix R v is defined, and two closed form expressions of the matrix R n v , for an integer n ≥ 1, are evaluated by the matrix functions in matrix theory. These expressions satisfy a connection between the generalized Fibonacci and Lucas numbers with the Pascal matrices. Thus, two representations of the matrix R n v and various forms of matrix (R v +q I) n are studied in terms of the generalized Fibonacci and Lucas numbers and binomial coefficients. By modifying results of 2 × 2 matrix representations given in the references of our study, we give various 3 × 3 matrix representations of the generalized Fibonacci and Lucas sequences. Many combinatorial identities are derived as applications.
On the generating matrices of the Κ-Fibonacci numbers
Proyecciones (Antofagasta), 2013
In this paper we define some tridiagonal matrices depending of a parameter from which we will find the k-Fibonacci numbers. And from the cofactor matrix of one of these matrices we will prove some formulas for the k-Fibonacci numbers differently to the traditional form. Finally, we will study the eigenvalues of these tridiagonal matrices.
Circulant Type Matrices with the Sum and Product of Fibonacci and Lucas Numbers
Abstract and Applied Analysis, 2014
Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant and -circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the left circulant and -circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and -circulant matrices by utilizing the relation between left circulant, and -circulant matrices and circulant matrix, respectively.
Singular case of generalized Fibonacci and Lucas matrices
The notion of the generalized Fibonacci matrix F (a,b,s) n of type s, whose nonzero elements are generalized Fibonacci numbers, is introduced in the paper [23]. Regular case s = 0 is investigated in [23]. In the present article we consider singular case s = −1. Pseudoinverse of the generalized Fibonacci matrix F (a,b,−1) n is derived. Correlations between the matrix F (a,b,−1) n and the Pascal matrices are considered. Some combinatorial identities involving generalized Fibonacci numbers are derived. A class of test matrices for computing the Moore-Penrose inverse is presented in the last section.
A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers
Chaos, Solitons & Fractals, 2009
In this paper, we construct the symmetric tridiagonal family of matrices M Àa;Àb ðkÞ; k ¼ 1; 2;. .. whose determinants form any linear subsequence of the Fibonacci numbers. Furthermore, we construct the symmetric tridiagonal family of matrices T Àa;Àb ðkÞ; k ¼ 1; 2;. .. whose determinants form any linear subsequence of the Lucas numbers. Thus we give a generalization of the presented in Cahill and Narayan (2004) [Cahill ND, Narayan DA. Fibonacci and Lucas numbers as tridiagonal matrix determinants. Fibonacci Quart 2004;42(3):216-21].
Generalized Lucas Numbers and Relations with Generalized Fibonacci Numbers
2011
In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas numbers and generalized order-k Fibonacci numbers. In addition, we obtain Binet formula and combinatorial representation for generalized order-k Lucas numbers by using properties of generalized Fibonacci numbers.
On Lucas Numbers By The Matrix Method
In this study we define the Lucas QL-matrix similar to the Fibonacci Q-matrix. The Lucas QL-matrix is different from the Fibonacci Q- matrix, but is related to it. Using this matrix representation, we have found some well-known equalities and a Binet-like formula for the Lucas numbers.
On the Relations between Lucas Sequence and Fibonacci-like Sequence by Matrix Methods
International Journal of Mathematical Sciences and Computing, 2017
In the present paper first and foremost we introduce a generalization of a classical Fibonacci sequence which is known as Fibonacci-Like sequence and at hindmost we obtain some relationships between Lucas sequence and Fibonacci-Like sequence by using two cross two matrix representation to the Fibonacci-Like sequence. The most worth noticing cause of this article is our proof method, since all the identities are proved by using matrix methods.
On Lucas Numbers by the Matrix Method ABSTRACT | FULL TEXT
Hacettepe Journal of Mathematics and Statistics, 2010
In this study we define the Lucas QL-matrix similar to the Fibonacci Q-matrix. The Lucas QL-matrix is different from the Fibonacci Qmatrix, but is related to it. Using this matrix representation, we have found some well-known equalities and a Binet-like formula for the Lucas numbers.
A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients
Applied Mathematics and Computation, 2013
In this study, Fibonacci and Lucas numbers have been obtained by using generalized Fibonacci numbers. In addition, some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated to write generalized Fibonacci sequences in a new direct way. Furthermore, it has been given a new formula for some Lucas numbers.