Matrix methods in Horadam Sequences 1 Introduction

Matrix methods in Horadam Sequences Gamaliel Cerda-Morales Instituto de Matem´aticas, Pontificia Universidad Cat´olica de Valpara´ıso Blanco Viel 596, Cerro Bar´on, Valpara´ıso, Chile [email protected]

Abstract Given the generalized Fibonacci sequence Wn (a, b; p, q) by notation of Horadam [1], we can naturally associate an matrix of order 2, denoted by W (p, q), whose coefficients are integers numbers. In this paper, using this matrix, we find some identities and the Binet-like formula for the Generalized Fibonacci-Lucas numbers.

Subject Classification: 11B39, 11C20, 15A24 Keywords: Generalized Fibonacci Numbers, Matrix method, Binet formula.

1

Introduction

By notation of Horadam [1], consider a sequence {Wn (a, b; p, q)} defined by the recurrence relation Wn = pWn−1 − qWn−2 , n ≥ 2, (1) where W0 = a, W1 = b, with a, b, p and q are integers with p > 0, q ̸= 0. We are interested in the following two special cases of {Wn }: {Un } is defined by U0 = 0, U1 = 1, and {Vn } is defined by V0 = 2, V1 = p. It is well known that {Un } and {Vn } can be expressed in the form Un = √

αn − β n , Vn = α n + β n , α−β

(2)

√

where α = p+2 ∆ , β = p−2 ∆ and the discriminant is denoted by ∆ = p2 − 4q. Especially, if p = −q = 1, {Un } and {Vn } are the usual Fibonacci and Lucas sequences. In this study, we have defined the generalized Fibonacci-Lucas W -matrix by [ ] p −q W (p, q) = . (3) 1 0 It is easy to see that we can write (Un+1 Un )T = W (p, q)(Un Un−1 )T , where {Un } is the n-th generalized Fibonacci sequence and v T is the transpose of the vector v. Similarly, the n-th generalized Fibonacci-Lucas sequence (Vn+1 Vn )T is W (p, q)(Vn Vn−1 )T . Using this representations, we obtain the determinants and elements of W n (p, q), and we get Cassini-like formula for the generalized Fibonacci-Lucas numbers. In [3], the authors define the Jacobsthal F -matrix similar to Fibonacci Q-matrix.

2

2

Gamaliel Cerda-Morales

Generalized Fibonacci-Lucas W (p, q)-matrix

We calculated the generalized characteristic roots and Binet formula of the matrix W n (p, q), with n ≥ 1. Theorem 2.1 Let W (p, q) be a matrix as in (3). Then [ ] Un+1 −qUn W n (p, q) = , Un −qUn−1

(4)

where n is a positive integer. Proof. We will use induction (PMI). When [ the principle ] [of mathematical ] p −q U2 −qU1 n = 1, W (p, q) = = , so the result is true. We 1 0 U1 −qU0 assume it is true for any positive integer n = k. Now, we show that it is true for the n = k + 1, then we can write: [ ][ ] Uk+1 −qUk U2 −qU1 k+1 k W (p, q) = W (p, q)W (p, q) = , Uk −qUk−1 U1 −qU0 and the result follows.  Corollary 2.2 For every positive integer n, it holds (i) det(W n (p, q)) = q n and (ii) Un+1 Un−1 − Un2 = −q n−1 (Cassini-like formula). Proof. It is easy to see that det(W (p, q)) = q, then we can write det(W n (p, q)) as the product of n times det(W (p, q)) equal to q n . The det(W n (p, q)) in (2.1), it follows (ii).  Theorem 2.3 Let n a positive integer. The well-known Binet-like formula of the generalized Fibonacci numbers is Un = where α =

√ p+ ∆ 2

and β =

αn − β n , α−β

(5)

√ p− ∆ . 2

Proof. Let the matrix W (p, q) be as in (4). If we calculate√ the eigenvalues √ and eigenvectors of W -matrix are α = p+2 ∆ and β = p−2 ∆ , roots of the characteristic polynomial x2 − px + q, and v1 = (α, 1),v2 = (β, 1) respectively. Then, we can diagonalize of the W -matrix by D = P −1 W (p, q)P , where P is (v1T , v2T ) and D = diag(α, β). From the properties of the similar matrices, we can write Dn = P −1 W n (p, q)P , where n is any integer. Furthermore, we can write W n (p, q) = P Dn P −1 , where [ n+1 ] 1 α − β n+1 −q(αn − β n ) n W (p, q) = . (6) −q(αn−1 − β n−1 ) α − β αn − β n Thus, the proof is completed.  Consequently, limiting ratio of the successive generalized Fibonacci numbers is αn+1 − β n+1 Un+1 = lim = α, lim n→∞ n→∞ Un αn − β n with α =

√ p+ ∆ 2

for W (p, q).

3

Matrix methods in Horadam Sequences

Theorem 2.4 The characteristic roots of W n (p, q) are λ1,2 = where λ1 = αn and λ2 = β n .Then, Vn = αn + β n .

Vn ±(α−β)Un , 2

Proof. If we write the characteristic polynomial of W n (p, q), we get determinant of (W n (p, q) − λI2 ) as λ2 − (Un+1 − qUn−1 )λ − q(Un+1 Un−1 − Un2 ) equal to λ2 − Vn λ + q n , by identities Un+1 − qUn−1 = Vn and Un+1 Un−1 − Un2 = −q n−1 . Thus, the characteristic equation of W n (p, q) is λ2 − Vn λ + q n = 0 and we get the generalized characteristic roots as following √ Vn ± Vn2 − 4q n . λ1,2 = 2 n . Consequently, Since Vn2 − 4q n = (α − β)2 Un2 , we can write λ1,2 = Vn ±(α−β)U 2 V +(α−β)U V −(α−β)U n n n n n αn = and β = .  2 2 We know the W -matrix in (4), then we can write [ U ] n+1 n −q UUn−1 W n (p, q) Un−1 Rn = = , Un −q Un−1 Un−1

since lim

n→∞

Un+1 = α, it follows that Un

[

Rn (α) = lim Rn = n→∞

pα − q α

−qα −q

] .

(7)

Theorem 2.5 Let Rn (α) be an 2 × 2 matrix as in (7). If α = p, then [ ] U2n+1 −qU2n Rn (p) = , U2n −qU2n−1 for any n ≥ 1. Proof. It can be show easily by PMI.  Corollary 2.6 For every positive integer n, it holds (i) det(Rn (p)) = q 2n 2 = −q 2n−1 . and (ii) U2n+1 U2n−1 − U2n Proof. Proofs are easily seen similar to corollary (2.2). 

3

Sums of Generalized Fibonacci numbers

When n = 1, the equation λ2 − Vn λ + q n = 0 becomes λ2 − pλ + q = 0, which is the characteristic equation of generalized Fibonacci W (p, q)-matrix. Notice that W 2 (p, q) − pW (p, q) + qI = 0 (from the Cayley-Hamilton theorem), with I the identity matrix of order 2. Now, we have following equation (I + G + G2 + ...... + Gn )(G − I) = Gn+1 − I.

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Since W 2 (p, q) − pW (p, q) = −qI, we can write W (p, q)(W (p, q) − pI) = −qI. 1 Thus, W −1 (p, q) = −p q ( p W (p, q) − I). If we multiply both sides of equation (8) by inverse of (G − I), with G = p1 W (p, q), we get I + G + ... + Gn by (Gn+1 − I) −p q W (p, q), and n ∑ W k (p, q) k=0

pk

=

−1 n+2 p W (p, q) + W (p, q). n p q q

Equating the (2, 1)-entry of both side, we prove:

(9)

4

Gamaliel Cerda-Morales

Theorem 3.1 For any integer n ≥ 1,

n ∑ Uk k=0

pk

=

p Un+2 − n . q p q

A particular case of the when p = 1 and q = −2, known as Jacobsthal ∑above, n succession, we can write k=0 Uk = 12∑ (Un+2 − 1), and if p = −q = 1, the case n of the Fibonacci sequence, we obtain k=0 Uk = Un+2 − 1. Theorem 3.2 Let n and m positive integers. Then following relation between the generalized Fibonacci and generalized Fibonacci-Lucas numbers Vn+m = Um+1 Vn − qUm Vn−1 ,

(10)

is valid. Proof. From the definition of the generalized Fibonacci and Fibonacci-Lucas numbers, (Vn+1 Vn )T = W (p, q)(Vn Vn−1 )T can be written easily. If we multiply both sides by W n (p, q), we get W n (p, q)(Vn+1 Vn )T = W n+1 (p, q)(Vn Vn−1 )T . Using (4) obtain [ ] [ ] Vn+m+1 Um+2 Vn − qUm+1 Vn−1 = . (11) Vn+m Um+1 Vn − qUm Vn−1 Thus, the proof is completed.  Let n and m are positive integers. Since W n+m = W n W m , we can write [ ] [ ][ ] U(n+m)+1 −qUn+m Un+1 −qUn Um+1 −qUm = , Un+m −qU(n+m)−1 Un −qUn−1 Um −qUm−1 then, Un+m = Un Um+1 − qUn−1 Um . In particular, if n = m we get U2n = Un (Un+1 − qUn−1 ), 2 2 − qUm if n = m + 1. If we ie., U2n = Un Vn . Furthermore, U2m+1 = Um+1 −n calculate W , we get [ ] 1 −qUn−1 qUn W −n (p, q) = n . −Un Un+1 q

Since W n−m = W n W −m , we can write: [ ] [ 1 U(n−m)+1 −qUn−m Un+1 = m Un−m −qU(n−m)−1 Un q

−qUn −qUn−1

][

−qUm−1 −Um

qUm Um+1

] .

Definition 3.3 In this study, we have defined the generalized FibonacciLucas S-matrix by [ 2 ] p − 2q −pq S = S(p, q) = . (12) p −2q It is feasible that, it can be written (Vn+2 Vn+1 )T = S(Un+1 Un )T , where Un and Vn are the nth generalized Fibonacci and Fibonacci-Lucas numbers, respectively. Furthermore, by (10) we get Vn+1 = V1 Un+1 − qV0 Un , for all n ≥ 1. Then [ ] V2 −qV1 S(p, q) = V1 −qV0 in function of the succession {Vn }.

5

Matrix methods in Horadam Sequences

4

The S-matrix representation

In this section, we will get some properties of the generalized Fibonacci-Lucas S-matrix. Moreover, using this matrix, we have obtained the Cassini and Binetlike formulas for the generalized Fibonacci and Fibonacci-Lucas numbers. Theorem 4.1 Let S(p, q) be a matrix as in (12). Then, for all integers n, the following matrix power is held below  [ ] √ −qUn   ( ∆)n Un+1 if n even  −qUn−1 ] [Un S n = S n (p, q) = (13) √ Vn+1 −qVn   if n odd,  ( ∆)n−1 Vn −qVn−1 where ∆ = p2 − 4q. Proof. We will use the PMI on odd and even n, separately. First n = 1, we get [ 2 ] [ ] p − 2q −pq V2 −qV1 S 1 (p, q) = = p −2q V1 −qV0 is true. We assume that it is correct for odd n = k. Now show that it is correct for n = k + 2. We can write S k+2 = S k S 2 , where [ ] [ ] √ k−1 Vk+1 √ 2 p2 − q −pq −qVk k 2 S = ( ∆) , S = ( ∆) . Vk −qVk−1 p −q By multiplying the two results, we obtain [ √ Vk+3 S k+2 = ( ∆)k+1 Vk+2

−qVk+2 −qVk+1

] .

(14)

When n = 2 and using the above equality, S 2 (p, q) is correct. We assume it is correct for even n = k. Finally, we show that it is correct for n = k + 2. We calculate [ ][ 2 ] [ ] S k+2 Uk+1 −qUk p − q −pq Uk+3 −qUk+2 √ = = , Uk −qUk−1 p −q Uk+2 −qUk+1 ( ∆)k+2 and the proof is completed.  Let S n (p, q) be as in (13). For all positive integers n, the determinant of n S is (−q∆)n , given that det(S(p, q)) = −q∆. Furthermore, Un+1 Un−1 − Un2 is (−1)n (−q)n−1 . The well-known identity 2 Un+1 − qUn2 = U2n+1

(15)

2 Vn+1 − qVn2 = ∆U2n+1 .

(16)

has as its Lucas counterpart

Indeed, since Vn+1 = Un+2 − qUn = pUn+1 − 2qUn and Vn = 2Un+1 − pUn , the equation (16) follows from (15). We define R(p, q) be the 2 × 2 matrix [ ] 1 p ∆ R(p, q) = , (17) 2 1 p

6

Gamaliel Cerda-Morales

then for an integer n, Rn (p, q) has the form [ ] 1 Vn ∆Un Rn (p, q) = . Vn 2 Un

(18)

Theorem 4.2 Vn2 − ∆Un2 = 4q n , for all n ∈ Z. Proof. Since det(R(p, q)) = q, det(Rn (p, q)) = (det(R(p, q)))n = q n . Moreover, since (18), we get det(Rn (p, q)) = 14 (Vn2 − ∆Un2 ). The proof is completed. 

References [1] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quarterly, 3, 161-176, 1965. [2] A. F. Horadam, Jacobsthal Representation Numbers, Fibonacci Quarterly, 34, 40-54, 1996. [3] F. Koken and D. Bozkurt, On The Jacobsthal Numbers by Matrix Methods, Int. J. Contemp. Math. Sciences, Vol.3, no.13, 605-614, 2008. [4] E. Karaduman, On determinants of matrices with general Fibonacci numbers entries, Appl. Math. and Comp., 167, 670-676, 2005. [5] T. Koshy, Fibonacci and Lucas Numbers with Applications, WileyInterscience, 2001. [6] A. Stakhov, Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos, Solitons and Fractals, 30, 56-66, 2006.