Proyecciones Journal of Mathematics Vol. 31, No 4, pp. 345-354, December 2012. Universidad Cat´olica del Norte Antofagasta - Chile

Matrix representation of the q-Jacobsthal numbers

Gamaliel Cerda-Morales P. Universidad Cat´olicadeValpara´so, Chile Received : September 2012. Accepted : October 2012

Abstract

In this paper, we consider a q-Jacobsthal sequence Jq,n ,with { } initial conditions Jq,0 =0and Jq,1 =1.Thengiveagenerating matrix for the terms of sequence Jq,kn for a positive integer k.With the aid of this matrix, we derive{ some} new identities for the sequence Jq,kn . { } Subjclass : 11B39, 11B37, 15A36.

Keywords : q-Jacobsthal numbers, q-Jacobsthal-Lucas numbers, ma- trix methods. 346 Gamaliel Cerda-Morales

1. Introduction

In [H1], Horadam introduce a sequence Wn(a, b; p, q) ,orbriefly Wn , defined by the recurrence relation { } { }

(1.1) Wn = pWn 1 qWn 2,n 2, − − − ≥

with W0 = a, W1 = b,wherea, b, p and q are integers with p>0, q =0. 6 We are interested in the following two special cases of Wn ,theq- { } Jacohsthal sequence Jq,n defined by { }

(1.2) Jq,n = Jq,n 1 qJq,n 2,Jq,0 =0,Jq,1 =1,n 2, − − − ≥

and the q-Jacobsthal-Lucas sequence jq,n defined by { }

(1.3) jq,n = jq,n 1 qjq,n 2,jq,0 =2,jq,1 =1,n 2. − − − ≥ The above recurrences involve the characteristic equation

(1.4) x2 x + q =0 − 1+√1 4q 1+√1 4q with roots αq = 2 − and βq = 2 − . Explicit closed form expres- sions for Jq,n and jq,n are (n 1) ≥ n n αq βq (1.5) Jq,n = − , αq βq − and n n (1.6) jq,n = αq + βq . Particular cases of the previous definition are:

If q = 1, the classic Fibonacci sequence appears by F0 =0,F1 =1 • − and Fn+1 = Fn + Fn 1 for n 1: Fn n N = 0, 1, 1, 2, 3, 5, 8,... . − ≥ { } ∈ { } If q = 2, the Jacobsthal sequence introduced in [H2], and defined • − by Jn+1 = Jn +2Jn 1 for n 1, J0 =0,J1 =1: Jn n N = 0, 1, 1, 3, 5, 11,... . − ≥ { } ∈ { } We define Jq be the 2 2matrix × 1 q (1.7) Jq = − , " 10# Matrix representation of the q-Jacobsthal numbers 347 then for an integer n with n 1, Jn has the form ≥ q n Jq,n+1 qJq,n (1.8) Jq = − . " Jq,n qJq,n 1 # − − The particular case q = 2, was introduced by Koken¨ and Bozkurt in [KB2, KB3]. Moreover, they− have obtained the Cassini formula for the Jacobsthal numbers. In this paper, we study new relations on q-Jacobsthal sequence, using the matrix Jq defined in (1.7). Initially, the q-Jacobsthal numbers are defined for n 0 but their exis- tence for n<0 is readily extended, yielding ≥ n n Jq, n = q− Jq,n and jq, n = q− jq,n. − − − For n 2andafixed positive integer k, in [KS] the authors study the ≥ sequence Wq,kn(x) and prove the following relation: { } k (1.9) Jq,kn = jq,kJq,k(n 1) q Jq,k(n 2), − − − k (1.10) jq,kn = jq,kjq,k(n 1) q jq,k(n 2), − − − where the initial conditions of the sequences Jq,n and jq,n are 0 and { } { } Jq,k , and 2 and jq,k , respectively. { } { } 2 k If αq,k and βq,k are the roots of equation λ jq,kλ + q = 0, then the − Binet formulas of the sequences Jq,kn and jq,kn are given by { } { } αn βn J = J q,k− q,k and j = αn + βn , q,kn q,k αq,k βq,k q,kn q,k q,k µ − ¶ respectively. It is clear that αq,1 = αq and βq,1 = βq. From the Binet formulas, one can see that Jq,2kn = jq,knJq,kn.

2. Companion matrix for the sequence Jq,kn { } In this section, we define a 2 2matrixAq andthenwegivesomenew × results for the q-Jacobsthal numbers Jq,kn by matrix methods. Define the 2 2matrixAq as follows: × k jq,k q (2.1) Aq = − . " 10# By an inductive argument and using (1.9), we get Proposition 2.1. For any integer n 1 holds: ≥ k n 1 Jq,k(n+1) q Jq,kn (2.2) Aq = k− . Jq,k " Jq,kn q Jq,k(n 1) # − − 348 Gamaliel Cerda-Morales

Proof. (By induction). For n =1: k k 1 jq,k q 1 Jq,2k q Jq,k (2.3) Aq = − = − k 10J Jq,k q Jq,0 " # q,k " − # since Jq,0 =0andJq,2k = jq,kJq,k. Let us suppose that the formula is true for n 1: − k n 1 1 Jq,kn q Jq,k(n 1) (2.4) Aq − = − k − . Jq,k " Jq,k(n 1) q Jq,k(n 2) # − − − k k n n 1 1 1 Jq,kn q Jq,k(n 1) jq,k q Then, Aq = Aq − Aq = J − k − − q,k " Jq,k(n 1) q Jq,k(n 2) #" 10# − − − k k 1 jq,kJq,kn q Jq,k(n 1) q Jq,kn = J − k − k− q,k " jq,kJq,k(n 1) q Jq,k(n 2) q Jq,k(n 1) # − − − − − k 1 Jq,k(n+1) q Jq,kn = J k− 2 q,k " Jq,kn q Jq,k(n 1) # − − Clearly the matrix An satisfies the recurrence relation, for n 1 q ≥ n+1 n k n 1 (2.5) A = jq,kA q A − , q q − q 0 1 where A = I2, A = Aq and I2 is the 2 2 unit matrix. q q × In this study, we define the q-Jacobsthal-Lucas jq-matrix by 2 k k jq,k 2q jq,kq (2.6) jq = − − k . jq,k 2q " − # It is easy to see that

jq,k(n+1) Jq,kn Jq,k(n+1) jq,kn = jq and ∆ = jq " jq,kn # " Jq,k(n 1) # " Jq,kn # " jq,k(n 1) # − − where Jq,kn, jq,kn are as above, and ∆ =1 4q. We obtain Cassini’s formula and properties− of these numbers by a sim- ilar matrix method to the Lucas numbers [KB1].

Proposition 2.2. Let jq be a matrix as in (2.6). Then, for all integers n 1, the following matrix power is held below ≥ k n Jq,k(n+1) q Jq,kn ∆ 2 k− if n even n ⎧ " Jq,kn q Jq,k(n 1) # (2.7) jq = ⎪ − k − ⎪ n 1 jq,k(n+1) q jq,kn ⎪ −2 ⎨ ∆ k− if n odd, " jq,kn q jq,k(n 1) # ⎪ − − ⎪ where Jq,kn and⎩⎪jq,kn are the kn-th q-Jacobsthal and q-Jacobsthal-Lucas numbers, respectively. Matrix representation of the q-Jacobsthal numbers 349

Proof. We use mathematical induction on n. First, we consider odd n. For n =1, k 1 jq,2k q jq,k jq = − k , jq,k q jq,0 " − # 2 k since jq,2k = jq,k 2q and jq,0 = 2. So, (2.7) is indeed true for n =1.Now wesupposeitistruefor− n = t,thatis

t 1 k t − jq,k(t+1) q jq,kt jq = ∆ 2 k− . " jq,kt q jq,k(t 1) # − − 2 Using the induction hypothesis and jq by a direct computation. we can write k t+2 t 2 t+1 jq,k(t+3) q jq,k(t+2) j = j j = ∆ 2 , q q q j −qkj " q,k(t+2) − q,k(t+1) # as desired. Secondly, let us consider even n. For n =2wecanwrite

k 2 Jq,3k q Jq,2k jq = ∆ − k . Jq,2k q Jq,k " − # So, (2.7) is true for n = 2. Now, we suppose it is true for n = t,using properties of the q-Jacobsthal numbers and the induction hypothesis, we can write k t+2 t+2 Jq,k(t+3) q Jq,k(t+2) j = ∆ 2 , q J −qkJ " q,k(t+2) − q,k(t+1) # as desired. Hence, (2.7) holds for all n. 2 n+1 2 n k n 1 If we use the equation (2.5), we can write jq,kA = j A q jq,kA q q,k q − q − = j2 An qk(An + qkAn 2) q,k q − q q − =(j2 qk)An q2kAn 2. q,k − q − q − Comparing the entries in the first row and first column for the above matrix equation, we get

2 k 2k (jq,k q )Jq,k(n+1) q Jq,k(n 1) (2.8) jq,k = − − − . Jq,k(n+2)

n n For n 0, if we consider the fact that det(Aq ) = (det(Aq)) ,thenwe obtain the≥ generalized Cassini identity

2 k(n 1) 2 (2.9) Jq,k(n+1)Jq,k(n 1) Jq,kn = q − Jq,k. − − − 350 Gamaliel Cerda-Morales

2 n 1 For example, for k =1,wegetJq,n+1Jq,n 1 Jq,n = q − , the gener- alized Cassini identity with q-Jacobsthal numbers.− − In this− case, if q = 1, we get Cassini identity on classic Fibonacci sequence. − Now we shall derive some results for Jq,kn by matrix methods. { } Proposition 2.3. For all n, m Z ∈ k (2.10) Jq,kJq,k(n+m) = Jq,kmJq,k(n+1) q Jq,k(m 1)Jq,kn. − −

n+m n m Proof. Since Aq = Aq Aq and after some simplifications, we obtain k k n+m 1 Jq,k(n+1) q Jq,kn Jq,k(m+1) q Jq,km Aq = J2 k− k− q,k " Jq,kn q Jq,k(n 1) #" Jq,km q Jq,k(m 1) # − − − − Jq,km n+1 k Jq,k(m 1) n = Aq q − Aq . Thus we obtain Jq,k − Jq,k

n+m n+1 k n (2.11) Jq,kAq = Jq,kmAq q Jq,k(m 1)Aq , − − which, Comparing the entries in the second row and first column for the matrix equation (2.11), gives the conclusion. 2 When m = n in (2.10), we obtain

k (2.12) Jq,kJq,2kn = Jq,kn(Jq,k(n+1) q Jq,k(n 1)), − −

and the follow equality for Jq,kn(x) =0 6 k (2.13) Jq,kjq,kn = Jq,k(n+1) q Jq,k(n 1). − − Comparing the entries in the first row and first column of the equality (2.11) and by taking m = n,weobtain

2 k 2 (2.14) Jq,kJ = J q J , q,k(2n+1) q,k(n+1) − q,kn a particular case of the next equality

2n n+1 k n (2.15) Jq,kAq = Jq,knAq q Jq,k(n 1)Aq . − − Corollary 2.4. For k 1 and n Z, ≥ ∈

1 2 k 2 2 (2.16)Jq,k(2n+1) + Jq,k(2n 1) = Jq,k(n+1) +(1 q )Jq,kn Jq,k(n+1) . − Jq,k − − ³ ´ Matrix representation of the q-Jacobsthal numbers 351

2n Proof. Considering the first row and first column of the matrix Aq = n 2 2 k 2 (Aq ) ,wegetJq,kJq,k(2n+1) = Jq,k(n+1) q Jq,kn, and the second row and 2n − 2 k 2 second column of the matrix Aq , Jq,kJq,k(2n 1) = Jq,kn q Jq,k(n 1).By adding side by side in the above equations, we− have the conclusion.− − 2 2n n+m n m For any integer m,wehaveAq = Aq Aq − .Hereifweconsiderthe n+m n m entries in the second row and first column in the product Aq Aq − and 2n the matrix Aq ,weget

k (2.17) Jq,kJq,2kn = Jq,k(n+m)Jq,k(n m+1) q Jq,k(n+m 1)Jq,k(n m). − − − − 3. sums with q-Jacobsthal numbers

We define the q-Jacobsthal Gq-matrix by

10 0 k (3.1) Gq = ⎡ 1 jq,k q ⎤ . 01− 0 ⎢ ⎥ ⎣ ⎦ By an inductive argument and using (1.9), we get

Proposition 3.1. For any integer n 1 holds: ≥ 100 1 n Js J qkJ (3.2) Gq = ⎡ q,kn q,k(n+1) q,kn ⎤ , Jq,k s k− Jq,k(n 1) Jq,kn q Jq,k(n 1) ⎢ − − ⎥ ⎣ − ⎦ s s n where Jq,kn is defined such that Jq,kn = i=1 Jq,ki. P Proof. (By induction). For n =1:

10 0 10 0 1 k 1 s k (3.3) Gq = ⎡ 1 jq,k q ⎤ = ⎡ Jq,k Jq,2k q Jq,k ⎤ − Jq,k s − k 01 0 J Jq,k q Jq,0 ⎢ ⎥ ⎢ q,0 − ⎥ ⎣ ⎦ ⎣ ⎦ since Jq,0 =0andJq,2k = jq,kJq,k. Let us suppose that the formula is true for n 1: − 100 1 n 1 Js J qkJ (3.4) Gq − = ⎡ q,k(n 1) q,kn q,k(n 1) ⎤ Jq,k s − − k − Jq,k(n 2) Jq,k(n 1) q Jq,k(n 2) ⎢ − − − ⎥ ⎣ − ⎦ 352 Gamaliel Cerda-Morales

Then, 10010 0 n n 1 1 1 s k k G = G G = J Jq,kn q Jq,k(n 1) 1 jq,k q q q − q Jq,k ⎡ q,k(n 1) ⎤ ⎡ ⎤ s − − k − − Jq,k(n 2) Jq,k(n 1) q Jq,k(n 2) 01 0 ⎢ − − − − ⎥ ⎢ ⎥ 100⎣ ⎦ ⎣ ⎦ 1 s k k = J + Jq,kn jq,kJq,kn q Jq,k(n 1) q Jq,kn Jq,k ⎡ q,k(n 1) ⎤ s − − k − k− Jq,k(n 2) + Jq,k(n 1) jq,kJq,k(n 1) q Jq,k(n 2) q Jq,k(n 1) ⎢ − − − − − − − ⎥ ⎣ 100 ⎦ 1 s k = J Jq,k(n+1) q Jq,kn . 2 Jq,k ⎡ q,kn ⎤ s k− Jq,k(n 1) Jq,kn q Jq,k(n 1) ⎢ − − ⎥ ⎣ − ⎦

Corollary 3.2. If Jq,kn is the kn-th q-Jacobsthal number, then

s s s k Jq,kJq,k(n+m) = Jq,kn + Jq,k(m 1)(Jq,k(n+1) q Jq,kn)+Jq,k(n+1)Jq,km. − − (3.5)

n+m Proof. The second row and firstcolumninthematrixGq is equal s n m to Jq,k(n+m). Considering the (2,1)-entries of the matrix equation Gq Gq , s s s k s Jq,kJq,k(n+m) = Jq,kn + Jq,k(n+1)Jq,km q Jq,knJq,k(m 1) s s − k s − = Jq,kn + Jq,k(n+1)(Jq,k(m 1) + Jq,km) q Jq,knJq,k(m 1) s s − k − − = Jq,kn + Jq,k(m 1)(Jq,k(n+1) q Jq,kn)+Jq,k(n+1)Jq,km. Thus, the proof is completed. 2 − − If q = 2andk = 1, the classic Jacobsthal sequence satisfy − s s s (3.6) Jn+m = Jn + Jm 1Jn+2 + Jn+1Jq,m. −

4. conclusion

In this paper, we consider the amazing relationships between the q-Jacobsthal numbers and matrices. Sum formula involving the terms of q-Jacobsthal numbers is one of the most important results obtained in this study. Note that the q-Jacobsthal numbers represent a generalization of the classical Fibonacci sequence. In particular, we have obtained new results by matrix method in these numbers. Matrix representation of the q-Jacobsthal numbers 353

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Gamaliel Cerda-Morales Instituto de Matem´aticas P. Universidad Cat´olica de Valpara´iso Blanco Viel 596 Cerro Bar´on Valpara´iso Chile e-mail : [email protected]