Matrix methods in horadam sequences  Given the generalized Fibonacci sequence {Wna, b; p, q} we can naturally associate a matrix of order 2, denoted by Wp, q, whose coefficients are integer numbers. In this paper, using this matrix, we find some identities and the Binet formula for the generalized Fibonacci–Lucas numbers.

Tipo de documento: Artículo - Article

Palabras clave: generalized Fibonacci numbers, matrix methods, Binet formula.

Source: http://www.bdigital.unal.edu.co

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19(2), 97–106 (2012) 97 Matrix methods in Horadam sequences Gamaliel Cerda1 Instituto de Matemáticas Universidad Católica de Valparaı́so Valparaı́so Given the generalized Fibonacci sequence {Wn (a, b; p, q)} we can naturally associate a matrix of order 2, denoted by W (p, q), whose coefficients are integer numbers.
In this paper, using this matrix, we find some identities and the Binet formula for the generalized Fibonacci–Lucas numbers. Keywords: generalized Fibonacci numbers, matrix methods, Binet formula. Dada la sucesión generalizada de Fibonacci {Wn (a, b; p, q)} podemos asociar naturalmente una matriz de orden 2, denotada por W (p, q), cuyos coeficientes son números enteros.
En este trabajo, usando esta matriz, encontramos algunas identidades y la fórmula de Binet para los números generalizados de Fibonacci–Lucas. Palabras claves: números generalizados de Fibonacci, métodos matriciales, fórmula de Binet. MSC: 11B39, 11C20, 15A24 Recibido: 12 de abril de 2012 1 gamaliel.cerda.m@mail.pucv.cl Aceptado: 17 de julio de 2012 98 1 Gamaliel Cerda, Matrix methods in Horadam sequences Introduction Let {Wn (a, b; p, q)} be a sequence defined by the recurrence relation  Wn = p Wn−1 − q Wn−2 , (1) for n ≥ 2, with W0 = a, W1 = b, where a, b, p and q are integer numbers with p 0, q 6= 0. We are interested in the following two special cases of {Wn }: (i) {Un } is defined by U0 = 0, U1 = 1; and (ii) {Vn } is defined by V0 = 2, V1 = p. Then {Un } and {Vn } can be expressed in the form αn − β n , α−β = αn β n , Un = Vn √ (2) √ where α = p 2 ∆ , β = p−2 ∆ and the discriminant is denoted by ∆ = p2 − 4q.
If p = 1, q = −1, then {Un } and {Vn } are the usual Fibonacci and Lucas sequences. In this study we define the generalized Fibonacci–Lucas matrix W by W (p, q) =  p −q 1 0  . (3) Then we can write (Un 1 Un )T = W (p, q)(Un Un−1 )T , where {Un } is the n–th gene...