On Factorization of the Fibonacci and Lucas Numbers Using Tridiagonal Determinants
نویسندگان
چکیده
The aim of this paper is to give new results about factorizations of the Fibonacci numbers Fn and the Lucas numbers Ln. These numbers are defined by the second order recurrence relation an+2 = an+1+an with the initial terms F0 = 0, F1 = 1 and L0 = 2, L1 = 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]. c ©2012 Mathematical Institute Slovak Academy of Sciences
منابع مشابه
Tridiagonal matrices related to subsequences of balancing and Lucas-balancing numbers
It is well known that balancing and Lucas-balancing numbers are expressed as determinants of suitable tridiagonal matrices. The aim of this paper is to express certain subsequences of balancing and Lucas-balancing numbers in terms of determinants of tridiagonal matrices. Using these tridiagonal matrices, a factorization of the balancing numbers is also derived.
متن کاملThe sum and product of Fibonacci numbers and Lucas numbers, Pell numbers and Pell-Lucas numbers representation by matrix method
Denote by {Fn} and {Ln} the Fibonacci numbers and Lucas numbers, respectively. Let Fn = Fn × Ln and Ln = Fn + Ln. Denote by {Pn} and {Qn} the Pell numbers and Pell-Lucas numbers, respectively. Let Pn = Pn × Qn and Qn = Pn + Qn. In this paper, we give some determinants and permanent representations of Pn, Qn, Fn and Ln. Also, complex factorization formulas for those numbers are presented. Key–Wo...
متن کاملNegativity Subscripted Fibonacci And Lucas Numbers And Their Complex Factorizations
In this paper, we nd families of (0; 1; 1) tridiagonal matrices whose determinants and permanents equal to the negatively subscripted Fibonacci and Lucas numbers. Also we give complex factorizations of these numbers by the rst and second kinds of Chebyshev polynomials. 1. Introduction The well-known Fibonacci sequence, fFng ; is de ned by the recurrence relation, for n 2 Fn+1 = Fn + Fn 1 (1.1...
متن کاملOn determinants of tridiagonal matrices with (−1, 1)-diagonal or superdiagonal in relation to Fibonacci numbers
The aim of the paper is to find some new determinants connected with Fibonacci numbers. We generalize the result provided in Strang’s book because we derive that two sequences of similar tridiagonal matrices are connected with Fibonacci numbers. AMS subject classification: Primary 15A15, 11B39; Secondary 11B37, 11B83.
متن کاملOn The Second Order Linear Recurrences By Generalized Doubly Stochastic Matrices
In this paper, we consider the relationships between the second order linear recurrences, and the generalized doubly stochastic permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Lucas Sequence, fLng ; is de ned by the recurrence relation, for n 1 Ln+1 = Ln + Ln 1 (1.2) where ...
متن کامل