Some Identities for the Fibonacci and Lucas Sequences with Rational Subscript via Matrix Methods (original) (raw)

On Some identities for Generalized Fibonacci and Lucas Sequences with Rational Subscript

Journal of Informatics and Mathematical Sciences, 2019

In this paper, we exploit general techniques from matrix theory to establish some identities for generalized Fibonacci and Lucas sequences with rationalsubscripts of the forms \(\frac{n}{2}\) and \(\frac{r}{s}\). For this purpose, we consider matrix functions \(X\mapsto X^{n/2}\) (resp. \(X\mapsto X^{r/s}\)) of two special matrices, and discuss whether the \(\frac{n}{2}\) (resp. \(\frac{r}{s}\)) are integers or irreducible fractions.

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.

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 .

Generalized Fibonacci – Lucas sequence its Properties

Global Journal of Mathematical Analysis, 2014

Sequences have been fascinating topic for mathematicians for centuries. The Fibonacci sequences are a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci number, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula F F F n 2 n n-1 n-2 ,    and F 0,F 1 01  , where F n are an n th number of sequences. The Lucas Sequence is defined by the recurrence formula L L L n 2 n n-1 n-2 ,    and L =2, L =1 01 , where L n an nth number of sequences are. In this paper, we present generalized Fibonacci-Lucas sequence that is defined by the recurrence relation 12 B B B n n n   , 2 n  with B 0 = 2s, B 1 = s. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.

Generalized Fibonacci-Lucas Sequence

Turkish Journal of Analysis and Number Theory, 2014

The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula − − = + , 2 n ≥ with B 0 = 2b, B 1 = s, where b and s are integers. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.

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

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.

Identities for Fibonacci and Lucas Numbers

In this paper several new identities are given for the Fibonacci and Lucas numbers. This is accomplished by by solving a class of non-homogeneous, linear recurrence relations.

On generalized Fibonacci and Lucas polynomials

Chaos, Solitons & Fractals, 2009

Let hðxÞ be a polynomial with real coefficients. We introduce hðxÞ-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these hðxÞ-Fibonacci polynomials. We also introduce hðxÞ-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ðxÞ that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.

Identities of Generalized Fibonacci-Like Sequence

Turkish Journal of Analysis and Number Theory, 2014

The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula n