ON A FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-LUCAS SEQUENCE. A. A. Wani, V. H. Badshah, S. Halici, P. Catarino

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1 Acta Universitatis Apulensis ISSN: No. 53/018 pp doi: /j.aua ON A FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-LUCAS SEQUENCE A. A. Wani, V. H. Badshah, S. Halici, P. Catarino Abstract. In the present article we consider a new generalization of classical Fibonacci sequence and we call it as Fibonacci-Like sequence V k,n and then we study Fibonacci-Like sequence V k,n and k-lucas sequence L k,n side by side by introducing two special matrices for these two sequences. After that by using these matrices we obtain Binet formulae for Fibonacci-Like sequence and for k-lucas sequence, we also give Cassini s identity for Fibonacci-Like sequence. 010 Mathematics Subject Classification: 11B37, 11B39. Keywords: Fibonacci Sequence, Generalized Fibonacci Sequence and k-lucas Sequence. 1. Introduction The Fibonacci sequence F n n 0 is the sequence of integers given by F n+ = F n+1 + F n, F 0 = 0 and F 1 = 1 (1.1) Classical Fibonacci numbers have been generalized by lot of the authors in different ways. The two most important generalizations of Fibonacci numbers are k-fibonacci numbers F k,n 1 and k-lucas numbers L k,n. Especially some authors used matrix technique to study these numbers. In 1979, Silvester 3 obtained some properties of classical Fibonacci numbers by matrix methods, particularly here the author used diagonaliztion of matrix to obtain Binet formula for classical Fibonacci numbers. In 4 Kilic introduced Binet formula, sums, combinatorial representations and generating function for the generalized Fibonacci p-numbers by using matrix technique, these numbers are defined by the following recurrence relation: F p (n) = F p (n 1) + F p (n p 1), n > p + 1 (1.) 41

2 With initial conditions F p (1) = F p () = = F p (p) = F p (p + 1) = 1 Jun and Choi 5 presented some basic properties for generalized Fibonacci sequence q n n N. Ying et al. 6 studied the generalized Fibonacci numbers by two different matrix methods via the method of diagonalization and the method of matrix collation. Akyuz and Halici 7 considered Horadam sequence (see 7 ) and showed interest towards two cases of Horadam sequence i.e. the sequences U n and V n, which are delineated by U n = pu n 1 qu n, n and U 0 = 0, U 1 = 1 V n = pv n 1 qv n, n and V 0 =, V 1 = p (1.3) where p and q are integers with p > 0, q 0, in 7 the authors derived some combinatorial identities, the determinant and the n th power of matrix. In 8 the authors found some well-known equalities and Binet formula for Jacobsthal numbers by matrix methods. Some divisible properties have been obtained for the generalized Fibonacci sequence by Yalciner 9. Borges et al. 10 obtained Cassini s identities and Binet formulae for k-fibonacci and k-lucas sequences by using some tools of matrix algebra. In 11, 1 the authors also used matrix terminology to deduce Cassini s identities and Binet formulae for the k-pell and k-pell-lucas Sequences. Srisawat and Sripad 13 applied matrix technique to investigate some generalizations of Pell and Pell-Lucas numbers as (s, t)-pell and (s, t)-pell-lucas numbers, these numbers are defined respectively by P n (s, t) = sp n 1 (s, t) + tp n (s, t) for n Q n (s, t) = sq n 1 (s, t) + tq n (s, t) for n (1.4) with initial conditions P 0 (s, t) = 0, P 1 (s, t) = 1 and Q 0 (s, t) =, P 1 (s, t) = s.. Preliminaries First of all, we consider a sequence G k,n which is defined by the following recurrence relation: G k,n+ = kg k,n+1 + G k,n, n 1 and G k,0 = a, G k,1 = b (.1) where k R + and a, b Z +. For the present study we are interested in the two cases of the sequence G k,n : 4

3 (i) Fibonacci-Like sequence V k,n is defined by the following equation: V k,n+ = kv k,n+1 + V k,n, n 0 and V k,0 = m, V k,1 = p + mk (.) (ii) k-lucas sequence L k,n is defined recurrently by L k,n+ = kl k,n+1 + L k,n, n 0 and L k,0 =, L k,1 = k (.3) Both the recurrence relations (.) and (.3) have same characteristic equation x kx 1 = 0. Let r and s be its roots and are given as r = k + k + 4 and s = k k + 4 We can see easily r and s holds the following properties: (a) rs = 1, r + s = k and r s = k + 4 (.4) (b) r 1 = kr and s 1 = ks (c) r + 1 = kr + = (r s) r and s + 1 = ks + = (r s) s Also we introduce two special matrices V and L which are given by k 1 k + k V = and L = 1 0 k (.5) 3. n th Powers of the Matrices Theorem 1. For n Z +, we have the following result V n = mv k,n+ (p + mk) V k,n+1 m k + 4m p mv k,n+1 (p + mk) V k,n m k + 4m p mv k,n+1 (p + mk) V k,n m k + 4m p mv k,n (p + mk) V k,n 1 m k + 4m p (3.1) Proof. We use principle of mathematical induction on n. Certainly the result is true for n = 1. 43

4 Assume that the result is true for all values j less than or equal n and then mv k,n+ (p + mk) V k,n+1 mv k,n+1 (p + mk) V k,n V n+1 = m k + 4m p m k + 4m p k 1 mv k,n+1 (p + mk) V k,n mv k,n (p + mk) V k,n m k + 4m p m k + 4m p V n+1 = ( m k + 4m p ) a 1 a k 1 a 3 a V n+1 = ( m k + 4m p ) ka 1 + a a 1 ka 3 + a 4 a 3 Here and ka 1 + a = m (kv k,n+ + V k,n+1 ) (p + mk) (kv k,n+1 + V k,n ) = mv k,n+3 (p + mk) V k,n+ ka 3 + a 4 = m (kv k,n+1 + V k,n ) (p + mk) (kv k,n + V k,n 1 ) = mv k,n+ (p + mk) V k,n+1 Hence V n+1 = as required. mv k,n+3 (p + mk) V k,n+ m k + 4m p mv k,n+ (p + mk) V k,n+1 m k + 4m p mv k,n+ (p + mk) V k,n+1 m k + 4m p mv k,n+1 (p + mk) V k,n m k + 4m p V k,n+1 Corollary. For lim n V k,n = r, we have r kr 1 = 0 (3.) Proof. To prove the required result, we should use the concept of limits. Since lim n V n = lim V k,n 1 n mv k,n+ (p + mk) V k,n+1 mv k,n+1 (p + mk) V k,n (m k + 4m p ) V k,n 1 (m k + 4m p ) V k,n 1 mv k,n+1 (p + mk) V k,n mv k,n (p + mk) V k,n 1 (m k + 4m p ) V k,n 1 (m k + 4m p ) V k,n 1 44

5 Now Therefore, lim n mv k,n+ (p + mk) V k,n+1 lim n V k,n 1 mv k,n+1 (p + mk) V k,n lim n V k,n 1 mv k,n (p + mk) V k,n 1 lim n V k,n 1 V n V k,n 1 = ( m k + 4m p ) 1 = r (mr p mk) = r (mr p mk) = mr p mk r (mr p mk) r (mr p mk) r (mr p mk) ( m k + 4m p ) 1 r (mr p mk) r (mr p mk) mr p mk r (mr p mk) mr p mk = ( m k + 4m p ) 1 (kr + 1) (mr p mk) r (mr p mk) r (mr p mk) mr p mk If we equate determinant on both sides, we obtain Hence the result. 0 = kr + 1 r r kr 1 = 0 Theorem 3. For n N, the following result holds where U n = L n = Lk,n+1 L k,n L k,n L k,n 1 { (r s) n 1 U n if n is odd (r s) n V n if n is even (3.3) Proof. To prove the result, we use induction on n. First, we consider odd n. Let n = 1, we have k + k Lk, L k,1 L = = k L k,1 L k,0 45

6 Let us suppose that the result is true for all odd values i less than or equal n and then L n+ = (r s) n 1 U n L = (r s) n+1 U n V = (r s) n+1 U n+ as required. Now we consider even n. Let n =, we have k 4 L + 5k + 4 k 3 + 4k = k 3 + 4k k + 4 k + 1 k = (r s) k 1 = (r s) V Assume that the result is true for all even values j less than or equal n and then as needed. L n+ = (r s) n V n L = (r s) n+ V n V = (r s) n+ V n+ Lemma 4. For n 0, we have Vk,1 V n V k,0 = Vk,n+1 V k,n (3.4) Proof. Here we shall use induction on n. Indeed the result is true for n = 0. Suppose the result is true for all values i less than or equal n and then Vk,1 Vk,1 V n+1 = V V n V k,0 V k,0 k 1 Vk,n+1 = 1 0 V k,n kvk,n+1 + V k,n = 46 V k,n+1

7 = as required. Vk,n+ V k,n+1 4. Binet Formulae In this section we present Binet formulae for k-lucas sequence L k,n and for Fibonacci- Like sequence V k,n. The most worth noticing point is here that we obtain Binet formula for k-lucas in a different way that did the authors in, 10. Theorem 5. For n Z 0, the n th terms for L k,n and V k,n are respectively given by where r and s are determined from equation (.4). L k,n = r n + s n (4.1) V k,n = p rn s n r s + m (rn + s n ) (4.) Proof. To prove the needed result, we diagonalize the matrix L. Sine L is a square matrix, by the Cayley Hamilton theorem the characteristic equation of L is given by det (L xi ) = 0 k + x k = 0 k x x ( k + 4 ) x + ( k + 4 ) = 0 x (r s) x + (r s) = 0 (4.3) This is the characteristic equation of L. Let u and v be the eigen values of equation (4.3) and are given by k (r s) + (r s) u = k + (r s) = (r s) = r (r s) 47

8 and k (r s) + (r s) v = k (r s) = (r s) = s (r s) Now the eigen vector corresponding to eigen value u is given by the following equation: (L ui ) A where A is the column vector of order 1. Then k + u k A1 = 0 k u A ( k + 4 u ) A 1 + ka = 0 ka 1 + ( u) A Consider the system ( k + 4 u ) A 1 + ka = 0 ka 1 + ( u) A = 0 (4.4) We assume that A = l in equation (4.4), we achieve ( (u ) l r 1 ) l A 1 = = = lr k k lr Thus the eigen vectors corresponding to u are of kind. For particular l = 1, the l r s eigen vector assigning to u is. Similarly the eigen vector assigning to v is. 1 1 r s 1 s Let P be matrix of eigen vectors, P = and P 1 = (r s) 1. Now r we consider the diagonal matrix D, in which eigen values of L are on the main diagonal, D =. Then by the principle of matrix diagonalization r (r s) 0 0 s (r s) 14, 15, we have L = P DP 1 48

9 L n = P D n P 1 r s r n = (r s) 1 (r s) n 0 1 s ( 1) n s n (r s) n 1 r r n+1 = (r s) n 1 ( 1) n s n+1 1 s r n ( 1) n s n 1 r r n+1 = (r s) n 1 ( 1) n s n+1 sr n+1 + ( 1) n rs n+1 r n ( 1) n s n sr n + ( 1) n rs n r n+1 = (r s) n 1 ( 1) n s n+1 r n ( 1) n s n r n ( 1) n s n r n 1 ( 1) n s n 1 If n is odd then by equation (3.3), we get r n+1 + s n+1 r n + s n U n = r n + s n r n 1 + s n 1 By equating corresponding terms of the matrices, we have L k,n = r n + s n This is the required Binet s formula for k-lucas sequence. if n is even then again by equation (3.3), we achieve V n = (r s) 1 r n+1 s n+1 r n s n r n s n r n 1 s n 1 (4.5) By using lemma (4), we obtain Vk,n+1 r n+1 = (r s) 1 s n+1 r n s n p + mk V k,n r n s n r n 1 s n 1 m = (r s) 1 b 1 b p + mk r n s n r n 1 s n 1 m where b 1 and b are the corresponding terms of the matrix. Thus Vk,n+1 = (r s) 1 (p + mk) b 1 + mb (p + mk) (r n s n ) + m ( r n 1 s n 1) V k,n 49

10 = (r s) 1 (p + mk) b 1 + mb p (r n s n ) + mkr n + mr n 1 mks n ms n 1 = (r s) 1 (p + mk) b 1 + mb p (r n s n ) + mr n 1 (kr + ) ms n 1 (ks + ) = (r s) 1 (p + mk) b 1 + mb p (r n s n ) + m (r s) r n + m (r s) s n Equating corresponding terms on both sides, we get V k,n = p rn s n r s + m (rn + s n ) This is the Binet s formula for Fibonacci-Like sequence. Now we present a result which establishes a relation between Fibonacci-Like sequence V k,n and k-lucas sequence. Corollary 6. For n Z +, the following result holds mv k,n+ (p + mk) V k,n+1 + mv k,n (p + mk) V k,n 1 (4.6) = ( m k + 4m p ) L k,n Proof. If we equate corresponding terms of matrices in the equation (4.5), we get and Therefore, mv k,n+ (p + mk) V k,n+1 p m + 4m p = rn+1 s n+1 r s mv k,n (p + mk) V k,n 1 p m + 4m p = rn 1 s n 1 r s Thus mv k,n+ (p + mk) V k,n+1 + mv k,n (p + mk) V k,n 1 p m + 4m p = rn 1 ( r + 1 ) s n 1 ( s + 1 ) r s = rn (r s) + s n (r s) r s mv k,n+ (p + mk) V k,n+1 + mv k,n (p + mk) V k,n 1 = ( m k + 4m p ) L k,n Hence the result. 50

11 5. Cassini s Identity for V k,n In this section we obtain Cassinin s identity for Fibonacci-Like sequence by using matrix V. Theorem 7. For n 1, we have V k,n V k,n+1v k,n 1 = ( 1) n ( m k + 4m p ) (5.1) Proof. mv k,n+ (p + mk) V k,n+1 mv k,n+1 (p + mk) V k,n det (V n ) = m k + 4m p m k + 4m p mv k,n+1 (p + mk) V k,n mv k,n (p + mk) V k,n 1 m k + 4m p m k + 4m p det (V n ) = ( m k + 4m p ) { mv k,n+ mvk,n (p + mk) V k,n 1 (p + mk) V k,n+1 mvk,n (p + mk) V k,n 1 mvk,n+1 (p + mk) V k,n } By using expansion of V k,n+ and Vk,n+1, we have det (V n ) = ( m k + 4m p ) 4m kv k,n+1 V k,n mk (p + mk) V k,n+1 V k,n 1 + 4m Vk,n m (p + mk) V k,nv k,n 1 + (p + mk) V k,n+1 V k,n 1 m (p + mk) V k,n+1 V k,n 4m k Vk,n 4m Vk,n 1 8m kv k,n V k,n 1 (p + mk) Vk,n + 4m (p + mk) V k,n+1v k,n det (V n ) = ( m k + 4m p ) 4m kv k,n+1 V k,n m (p + mk) V k,n V k,n 1 + m (p + mk) V k,n+1 V k,n 4m Vk,n 1 8m kv k,n V k,n 1 mk (p + mk) V k,n+1 V k,n 1 + 4m Vk,n + (p + mk) V k,n+1 V k,n 1 4m Vk,n (p + mk) Vk,n = ( m k + 4m p ) ( 6m k + mp ) V k,n+1 V k,n ( mp + m k ) V k,n V k,n 1 4m V k,n 1 8m kv k,n V k,n 1 mk (p + mk) V k,n+1 V k,n 1 + 4m Vk,n + (p + mk) V k,n+1 V k,n 1 4m V (p + mk) Vk,n = ( m k + 4m p ) ( 6m V k,n + mpkv k,n 4m kv k,n+1 V k,n 1 m k V k,n+1 V k,n 1 + 4m V k,n + p V k,n+1 V k,n 1 4m k V k,n 51 k,n

12 p Vk,n m k Vk,n mpkv k,n ) = ( m k + 4m p ) ( m V k,n + 4m V k, n p V k,n m k V k,n+1 V k,n 1 4m V k,n+1 V k,n 1 + p V k,n+1 V k,n 1 ) = ( m k + 4m p ) ( m k + 4m p ) ( Vk,n V ) k,n+1v k,n 1 = ( m k + 4m p ) 1 ( ) V k,n V k,n+1 V k,n 1 Since det (V n ) = ( 1) n, we have V k,n V k,n+1v k,n 1 = ( 1) n ( m k + 4m p ) Hence the result. From the proof of this theorem, we conclude that mvk,n+ (p + mk) V k,n+1 mvk,n (p + mk) V k,n 1 mv k,n+1 (p + mk) V k,n = ( 1) ( m k + 4m p ) (5.) 6. Characteristic Equation of V n In theorem (5) we easily saw the characteristic equation of V. But in this section we obtain the characteristic equation for V n. Theorem 8. For n Z 0, the characteristic equation of V n is given by x L k,n x + ( 1) n = 0 (6.1) Proof. Since V n is a square matrix then by Cayley Hamilton theorem, we have Here det (V n xi ) = 0 det (V n xi ) = mv k,n+ (p + mk) V k,n+1 m k + 4m p mv k,n+1 (p + mk) V k,n m k + 4m p x mv k,n+1 (p + mk) V k,n m k + 4m p mv k,n (p + mk) V k,n 1 m k + 4m p x = ( m k + 4m p ) { mvk,n+ (p + mk) V k,n+1 mvk,n (p + mk) V k,n 1 x ( m k + 4m p ) mv k,n+ (p + mk) V k,n+1 x 5

13 ( m k + 4m p ) mv k,n (p + mk) V k,n 1 + x ( m k + 4m p ) mv k,n+1 (p + mk) V k,n } = ( m k + 4m p ) { x ( m k + 4m p ) x ( m k + 4m p ) mvk,n+ (1 + mk) V k,n+1 + mv k,n (p + mk) V k,n 1 + mv k,n+ (p + mk) V k,n+1 mvk,n (p + mk) V k,n 1 mv k,n+1 (p + mk) V k,n } If we consider corollary (6) and equation (5.), we get det (V n xi ) = ( m k + 4m p ) x ( m k + 4m p ) L k,n x ( m k + 4m p ) + ( 1) n ( m k + 4m p ) = x L k,n x + ( 1) n Hence the characteristic equation of V n is Hence the result. x L k,n x + ( 1) n = 0 References 1 S. Falcon and A. Plaza, On the Fibonacci k-numbers, Chaos Solitons and Fractals, 3 (007), S. Falcon, On the k-lucas Numbers, Int. J. Contemp. Math. Sciences, 6(1) (011), J. R. Silvester, Fibonacci Numbers by Matrix Methods, The Mathematical Gazette, 63(45) (1979), E. Kilic, The Binet Formula, Sums and Representations of Generalized Fibonacci p-numbers, Chaos Solitons and Fractals, 3 (007), S. P. Jun and K. H. Choi, Some Properties of the Generalized Fibonacci Sequence q n by Matrix Methods, Korean J. Math., 4(4) (016), S. J. Ying, H. C. Kit, Ibrahim H. and Ahmad N., Algebraic Properties of Generalized Fibonacci Sequence Via Matrix Methods, Journal of Engineering and Applied Sciences, 11(11) (016), Z. Akyuz and S. Halici, Some Identities Deriving from the nth Power of a Special Matrix, Advances in Difference Equations,

14 8 F. Koken and D. Bozkurt, On the Jacobsthal Numbers by Matrix Methods, Int. J. Contemp. Math. Sciences, 3(13) (008), A. Yalciner, A Matrix Approach for Divisibility Properties of the Generalized Fibonacci Sequence, Discrete Dynamics in Nature and Society, A. Borges, P. Catarino, A. P. Aires, P. Vasco, and H. Campos, Two-by-Two Matrices Involving k-fibonacci and k-lucas Sequences, Applied Mathematical Sciences, 8(34) (014), P. Catarino and P. Vasco, Two-by-Two Matrices Involving k-fibonacci and k- Lucas Sequences, Int. Journal of Math. Analysis, 7(45) (013), P. Catarino and P. Vasco, A Note Involving Two-by-Two Matrices of the k- Pell and k-pell-lucas Sequences, International Mathematical Forum, 8(3) (013), S. Srisawat and W. Sripad, On the (s, t)-pell and (s, t)-pell-lucas Numbers by Matrix Methods, Annales Mathematicae et Informaticae, 8(3) (013), A. A. Wani School of Studies in Mathematics, Vikram University Ujjain, Ujjain, India arfatahmadwani@gmail.com V. H. Badshah School of Studies in Mathematics, Vikram University Ujjain, Ujjain, India vhbadshah@gmail.com S. Halici Department of Mathematics, Faculty of Arts and Sciences, Sakarya University, Sakarya, Turkey shalici@pau.edu.tr Paula Catarino Department of Mathematics, School of Science and Technology, University of Trás-os-Montes e Alto Douro (Vila Real Portugal) pcatarino3@gmail.com 54