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On Some Identities of K-Jacobsthal-Lucas Numbers 1

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On Some Identities of K-Jacobsthal-Lucas Numbers 1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 10, 489 - 494 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4249 On Some Identities of k-Jacobsthal-Lucas Numbers H. Campos1, P. Catarino1, A. P. Aires2, P. Vasco3 and A. Borges3 Universidade de Tr´as-os-Montes e Alto Douro, UTAD, http://www.utad.pt Quinta de Prados, 5000-801 Vila Real, Portugal Department of Mathematics, School of Science and Technology Copyright c 2014 H. Campos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we present the sequence of the k-Jacobsthal-Lucas numbers that generalizes the Jacobsthal-Lucas sequence introduced by Horadam in 1988. For this new sequence we establish an explicit formula for the term of order n, the well-known Binet’s formula, Catalan’s and d’Ocagne’s Identities and a generating function. Mathematics Subject Classification 2010: 11B37, 11B83 Keywords: Jacobsthal numbers; Jacobsthal-Lucas numbers; Binet’s for- mula 1 Introduction Several recurrence sequences of positive integers have been object of study for many researchers. Examples of these are the Fibonacci, Lucas, Pell, Pell-Lucas, Modified Pell, Jacobhstal, Jacobsthal-Lucas sequences among others(see [8], [10], [12], [13]). About them there is a vast literature studying several prop- erties, ones involving the well-known Binet’s formula, Catalan’s, Cassini’s and d’Ocagne’s identities and there is also a vast literature dedicated to the study of other properties involving each sequence (see [7] and [14]). More recently, some of these sequences were generalized for any positive real number k: the study of the k-Fibonacci sequence, the k-Lucas sequence, the k-Pell sequence, the k-Pell-Lucas sequence,the Modified k-Pell sequence and the k-Jacobhstal sequence appeared (see [1], [11], [2], [4], [5], [6] and [3]). 490 H. Campos, P. Catarino, A. P. Aires, P. Vasco and A. Borges In this paper we generalize the sequence of Jacobsthal-Lucas numbers and study by introducing the sequence of the k-Jacobsthal-Lucas numbers. We give an explicit formula for the term of order n of this sequence, the well-know Binet’s formula, Catalan’s and d’Ocagne’s Identities and a generating function for this recurrence sequence. 2 Identities k {j } Let us define the sequence of the -Jacobsthal-Lucas numbers k,n n∈Æ as follows: jk,n+1 = kjk,n +2jk,n−1 (1) where the initial conditions are: jk,0 =2 (2) jk,1 = k for any positive real number k.Ifk = 1 we get the sequence of Jacobsthal- Lucas numbers defined by Horadam in [9]. The characteristic equation asso- ciated to the recurrence relation (1) is x2 = kx + 2 (3) √ √ r r r k+ k2+8 r k− k2+8 with roots 1 and 2 given by 1 = 2 and 2 √= 2 . 2 Note that r1r2 = −2; r1 + r2 = k and r1 − r2 = k + 8. Associated to (1) the term of order n of the k-Jacobsthal-Lucas sequence, can be written by the n n following identity jk,n = c1r1 + c2r2 for some constants c1,c2. Solving the system of two linear equations corresponding to the initial conditions (2), 2=c1 + c2 (4) k = c1r1 + c2r2, we obtain c1 = c2 = 1. So, we get the next Proposition: Proposition 2.1 (Binet’s Formula): The nth k-Jacobsthal-Lucas number jk,n is given by n n jk,n = r1 + r2 , (5) where r1 and r2 are the roots of the characteristic equation (3) and r1 >r2. On some identities of k-Jacobsthal-Lucas Numbers 491 Proof. We use induction on n. Taking into account the initial conditions (2), we note that the equation (5) is valid for n = 0 and n = 1. Now assume s s that (5) is true for 0 ≤ s ≤ n, that is, jk,s = r1 + r2, for every s ∈{0,... ,n}. Using (1) and taking in account that r1r2 = −2 we have jk,n+1 = kjk,n +2jk,n−1 n n n−1 n−1 = k (r1 + r2 )+2 r1 + r2 n−1 n−1 = r1 (kr1 +2)+r2 (kr2 +2) n−1 n−1 = r1 ((r1 + r2) r1 +2)+r2 ((r1 + r2) r2 +2) n−1 2 n−1 2 = r1 (r1 + r1r2 +2)+r2 (r1r2 + r2 +2) n+1 n+1 = r1 + r2 . Consequently, the Binet’s Formula is true for any positive integer n. The use of the Binet’s Formula (5) and the fact that r1r2 = −2 allows us to obtain Catalan’s Identity. Proposition 2.2 (Catalan’s Identity): 2 n−r 2 r+2 jk,n−rjk,n+r − jk,n =(−2) (jk,r − (−2) ). (6) Proof. We have 2 n−r n−r n+r n+r n n 2 jk,n−rjk,n+r − jk,n = r1 + r2 r1 + r2 − (r1 + r2 ) r r =(−2)n r2 +(−2)n r1 − 2(−2)n r1 r2 rr rr − n 2 1 − =(2) rr + rr 2 1 2 2r 2r r n r2 +r1 −2(r1r2) =(−2) r (r1r2) r2r+r2r−2(r r )r − n 2 1 1 2 =(2) (−2)r − n−r r2r r2r − r r r =(2) ( 2 + 1 2( 1 2) ) n−r r r 2 r =(−2) (r1 + r2) − 4( r1r2) n−r 2 r =(−2) jk,r − 4(−2) , as required. Substituting r = 1 in Catalan’s Identity (6), yields 2 n−1 2 jk,n−1jk,n+1 − jk,n =(−2) jk,1 − 4(−2) and using the initial condition jk,1 = k, we obtain the Cassini’s identity for k-Jacobsthal-Lucas sequence. 492 H. Campos, P. Catarino, A. P. Aires, P. Vasco and A. Borges Proposition 2.3 (Cassini’s Identity): 2 n−1 2 jk,n−1jk,n+1 − jk,n =(−2) k +8 . (7) The d’Ocagne’s identity can also be obtained from the Binet’s Formula (5) and the fact that r1r2 = −2 and m>n. Proposition 2.4 (d’Ocagne’s Identity): For m>n, √ √ m−n n 2 n−m+1 2 jk,mjk,n+1−jk,m+1jk,n =(−2) k +8 jk,m−n − 2 k + k +8 . Proof. For m>n, we have m m n+1 n+1 m+1 m+1 n n jk,mjk,n+1 − jk,m+1jk,n =(r1 + r2 ) r1 + r2 − r1 + r2 (r1 + r2 ) n m−n m−n m−n m−n =(−2) r1 r2 + r1r2 − r1 r1 − r2 r2 n m−n m−n =(−2) r1 (r2− r1)+r2 (r1 − r2) − n r − r rm−n − rm−n =(2) √( 1 2) 2 1 − n k2 rm−n rm−n − rm−n =(2) √ +8 1 + 2 2 1 n 2 m−n =(−2) k +8 jk,m−n − 2r1 √ m−n √ k+ k2+8 n 2 ( ) =(−2) k +8 jk,m−n − 2 m−n 2 √ √ m−n n 2 n−m+1 2 =(−2) k +8 jk,m−n − 2 k + k +8 . as required. The limit property stated in the following Proposition is also deduced using Binet’s Formula (5). Proposition 2.5 For m>n, jk,n limn→∞ = r1. (8) jk,n−1 Proof. We have n n jk,n r1 + r2 limn→∞ = limn→∞ n−1 n−1 . jk,n−1 r1 + r2 n r2 r2 < limn→∞ Since r1 1, then r1 = 0 and therefore n 1+ r2 jk,n r1 limn→∞ = limn→∞ n jk,n−1 1 + r2 1 r1 r1 r2 1 = 1 , r1 and the result follows. On some identities of k-Jacobsthal-Lucas Numbers 493 3 Generating Function In the next Proposition we present a generating function for the sequence of the k-Jacobsthal-Lucas numbers. Proposition 3.1 (Generating function of the k-Jacobsthal-Lucas numbers) 2 − kx jk(x)= 1 − kx − 2x2 Proof. Let us suppose that the k-Jacobsthal-Lucas numbers are the coef- − 1 , 1 ficients of a power series centered at the origin, that is convergent in r1 r1 , taking in account the Proposition (2.5) . To the sum of this power series, jk(x), we call generating function of the k-Jacobsthal-Lucas numbers. So we have 2 n jk(x)=jk,0 + jk,1x + jk,2x + ···+ jk,nx + ··· and then, 2 3 n+1 kxjk(x)=kjk,0x + kjk,1x + kjk,2x + ···+ kjk,nx + ··· 2 2 3 4 n+2 2x jk(x)=2jk,0x +2jk,1x +2jk,2x + ···+2jk,nx + ··· . Since (1) e (2) we obtain 2 jk(x) − kxjk(x) − 2x jk(x)=2− kx and then we conclude that 2 − kx jk(x)= 1 − kx − 2x2 Acknowledgements. This work has been supported by the Portuguese Government through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia - under the project PEst-OE/MAT/ UI4080/2014. References [1] Bolat C., K¨ose H., On the Properties of k-Fibonacci Numbers. Int. J. Contemp. Math. Sciences 2010; 22(5): 1097-1105. [2] Catarino P., On Some Identities and Generating Functions for k-Pell Numbers. Int. Journal of Math. Analysis 2013; 7 (38): 1877-1884. [3] Catarino P., On Some Identities for k-Fibonacci sequence. International Journal of Contemporary mathematical Sciences 2014; 9(1): 37-42. 494 H. Campos, P. Catarino, A. P. Aires, P. Vasco and A. Borges [4] Catarino P., Vasco P., Modified k-Pell Sequence: Some Identities and Ordinary Generating Function. Applied Mathematical Sciences 2013; 7 (121): 6031-6037. [5] Catarino P., Vasco P., On some Identities and Generating Functions for k-Pell-Lucas sequence.
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