DOI: 0.545/mjis.06.4009 Fibonacci and k Lucas Sequences as Series of Fractions A. D. GODASE AND M. B. DHAKNE V. P. College, Vaijapur, Maharashtra, India Dr. B. A. M. University, Aurangabad, Maharashtra, India E-mail: mathematicsdept@vpcollege.net Received: July 9, 05 Revised: September 30, 05 Accepted: October 3, 05 Published online: March 30, 06 The Author(s) 06. This article is published with open access at www.chitkara.edu.in/publications Abstract In this paper, we defined new relationship between k Fibonacci and k Lucas sequences using continued fractions and series of fractions, this approach is different and never tried in k Fibonacci sequence literature. Keywords: k-fibonacci sequence, k-lucas sequence, Recurrence relation Mathematics Subject Classification: B39, B83 INTRODUCTION The Fibonacci sequence is a source of many nice and interesting identities. Many identities have been documented in [], [3], [0], [], [], [3], [7]. A similar interpretation exists for k Fibonacci and k Lucas numbers. Many of these identities have been documented in the work of Falcon and Plaza[], [4], [5], [7], [8], [9], where they are proved by algebraic means. In this paper, we obtained some new properties for k Fibonacci and k Lucas sequences using series of fraction. PRELIMINARY Definition.. The k Fibonacci sequence {F k,n } n= is defined as, F k,n+ = k F k,n + F k,n, with F k,0 = 0, F k, =, for n Mathematical Journal of Interdisciplinary Sciences Volume 4, No., March 06 pp. 07 9 07
Godase, AD, Dhakne, MB Definition.. The k Lucas sequence {L k,n } n= is defined as, L k,n+ = k L k,n + L k,n, with L k,0 =, L k, = k, for n Characteristic equation of the initial recurrence relation is, Characteristic roots are r k r = 0 () r = k + k + 4 () and r = k k + 4 (3) Characteristic roots verify the properties r r = k + 4 = = δ (4) r + r = k (5) Binet forms for F k,n and L k,n are r. r = (6) and F k,n = rn rn r r (7) L k,n = r n + rn (8) 08
. The first few members of this k Fibonacci family are, k, Fibonacci and k Lucas Sequences as Series of Fractions + k, k + k 3, + 3k + k 4, 3k + 4k 3 + k 5, + 6k + 5k 4 + k 6, 4k + 0k 3 + 6k 5 + k 7, + 0k + 5k 4 + 7k 6 + k 8, 5k + 0k 3 + k 5 + 8k 7 + k 9, + 5k + 35k 4 + 8k 6 + 9k 8 + k 0, 6k + 35k 3 + 56k 5 + 36k 7 + 0k 9 + k, + k + 70k 4 + 84k 6 + 45k 8 + k 0 + k, 7k + 56k 3 + 6k 5 + 0k 7 + 55k 9 + k + k 3, + 8k + 6k 4 + 0k 6 + 65k 8 + 66k 0 + 3k + k 4, 8k + 84k 3 + 5k 5 + 330k 7 + 0k 9 + 78k + 4k 3 + k 5, + 36k + 0k 4 + 46k 6 + 495k 8 + 86k 0 + 9k + 5k 4 + k 6, 9k + 0k 3 + 46k 5 + 79k 7 + 75k 9 + 364k + 05k 3 + 6k 5 + k 7, +45k +330k 4 +94k 6 +87k 8 +00k 0 +455k +0k 4 +7k 6 +k 8, 0k+65k 3 +79k 5 +76k 7 +00k 9 +365k +560k 3 +36k 5 +8k 7 +k 9 09
Godase, AD, Dhakne, MB. k Fibonacci sequences in Encyclopaedia of Integer Sequences F k,n F,n F,n F 3,n F 4,n F 5,n F 6,n F 7,n F 8,n F 9,n F 0,n F,n Classification A000045 A0009 A00690 A00076 A0598 A005668 A05443 A0405 A09937 A0404 A049666 3 RELATIONSHIP OF THE SEQUENCES F K,N AND L K,N AS CONTINUED FRACTIONS: In general, a (simple) continued fraction is an expression of the form [a 0, a,...,a n ] = a 0 + a + a + a 3 + a 4 + a 5 + a 6 +...... (9) The letters a, a,... denote positive integers. The letter a 0 denotes an integer. The expansion F k,n+ F k,n in continued fraction is written as F k,n+ F k,n = k + k + k+ k+ k+ k+ k+...... (0) Here n denotes the number of quantities equal to k. We knew that F k,n F k,n F k,n+ = ( ) n () 0
Moreover, in general we have F k,n+ r r ) n+ ( = r F k,n ( r r ) n Fibonacci and k Lucas Sequences as Series of Fractions Let, r denote the larger of the root, we have F k,n+ lim = r n F k,n More generaly formula (9) is written as F k,(n+)t F k,nt = ( ) t ( ) t ( ) t ( ) t ( ) t ( )t...... () Here, n denotes the number of L k,t s When n increases indefinitely, we have F k,(n+)t lim n F k,nt = (r ) t As equation (9), we have relation for L k,n L k,(n)t F k,(n )t = ( ) t ( ) t ( ) t ( ) t ( ) t ( ) t ( )t ( L k,t ) (3) Here n denotes the number of quantities equal to L k,t. We knew that L k,n L k,n L k,n+ = ( ) n (4) More generaly, equations (0) and (3) are modified as F k,nt F k,(n )tf k,(n+)t = ( ) (n )t (F k,t ) (5)
Godase, AD, Dhakne, MB L k,nt L k,(n )tl k,(n+)t = ( ) (n )t (F k,t ) (6) Moreover using (7) and (8), gives Fk,nt = rn+t + r n+t ( ) n+t (7) Again by subtracting (6) and (7), gives and L k,n = rn + rn ( )n (8) (F k,n+t ( )t F k,n ) = (rn+t r n+t )(r t + rt ) (9) F k,n+t ( )t F k,n = F k,tf k,n+t (0) Similarly, we obtain L k,n+t ( )t L k,n = F k,tf k,n+t () 4 SEQUENCES F K,N AND L K,N AS A SERIES OF FRACTIONS: Theorem 4.. For n, k > 0,. F k,n+ F k,n = F k, F k, ( ) F k, F k, ( ) F k, F k,3 ( )3 F k,3 F k,4... ( )n F k,n F k,n. L k,n+ L k,n = L k, L k, ( ) L k, L k, ( )3 L k, L k,3 ( )4 L k,3 L k,4... ( )n L k,n L k,n
Proof. We can write expressions of F k,n+ F k,n and L k,n+ L k,n in series as F k,n+ = F ( k, Fk,3 + F ) ( k, Fk,4 + F ) k,3 +... F k,n F k, F k, F k, F k,3 F k, ( Fk,n+ + F ) k,n F k,n F k,n Fibonacci and k Lucas Sequences as Series of Fractions And = F k, (F k, F k,3f k, ) (F k,3 F k,f k,4 )... F k, F k, F k, F k, F k,3 (F k,n F k,n+f k,n ) F k,n F k,n L k,n+ = L ( k, Lk,3 + L ) ( k, Lk,4 + L ) k,3 +... L k,n L k, L k, L k, L k,3 L k, ( Lk,n+ + L ) k,n L k,n L k,n = L k, (L k, L k,3l k, ) (L k,3 L k,l k,4 )... L k, L k, L k, L k, L k,3 (L k,n L k,n+l k,n ) L k,n L k,n Using the equations () and (4) Gives F k,n F k,n F k,n+ = ( ) n L k,n L k,n L k,n+ = ( ) n F k,n+ F k,n = F k, F k, ( ) F k, F k, ( ) F k, F k,3 ( )3 F k,3 F k,4... ( )n F k,n F k,n 3
Godase, AD, Dhakne, MB And L k,n+ L k,n = L k, L k, ( ) L k, L k, ( )3 L k, L k,3 ( )4 L k,3 L k,4... ( )n L k,n L k,n Taking limit as lim n, gives r = + k + 4 = k +. k k.(k + 4) +... For Fibonacci series + 5 = +.. +. 3 3. 5 + 5. 8 8. 3 +... Now, we obtain more general relation for F k,n and L k,n as a series of fractions:- Theorem 4.. For n, k > 0,. F k,(n+)t = F k,t ( )t Fk,t ( )t Fk,t F k,nt F k,t F k,t F k,t F k,t F k,3t ( )3t Fk,t... F k,3t F k,4t. ( )(n )t F k,t F k,(n )t F k,nt L k,(n+)t = L k,t + F k,t + ( )t Fk,t + ( )t Fk,t +... L k,nt L k,0 L k,0 L k,t L k,t L k,t L k,t L k,3t ( )(n )t F k,t L k,(n )t L k,nt 4
Proof. We can write expressions of F k,(n+)t F k,nt F k,(n+)t F k,nt and L k,(n+)t L k,nt in series as = F ( k,t Fk,3t + F ) ( k,t Fk,4t + F ) k,3t +... F k,t F k,t F k,t F k,3t F k,t ( Fk,(n+)t + F ) k,nt F k,nt F k,(n )t Fibonacci and k Lucas Sequences as Series of Fractions = F k,t (F k,t F k,3tf k,t ) (F k,3t F k,tf k,4t )... F k,t F k,t F k,t F k,t F k,3t (F k,nt F k,(n+)tf k,(n )t ) F k,(n )t F k,nt And L k,(n+)t L k,nt = L ( k,t Lk,t + L ) ( k,t Lk,3t + L ) k,t +... L k,0 L k,t L k,0 L k,t L k,t ( Lk,(n+)t + L ) k,nt L k,nt L k,(n )t = L k,t ( L k,0l k,t ) ( L k,tl k,3t )... L k,0 L k,0 L k,t L k,t L k,t (L k,nt L k,(n+)tl k,(n )t ) L k,(n )t L k,nt Using the equations (5) and (6) F k,nt F k,(n )tf k,(n+)t = ( ) (n )t (F k,t ) () L k,nt L k,(n )tl k,(n+)t = ( ) (n )t (F k,t ) (3) 5
Godase, AD, Dhakne, MB Gives F k,(n+)t = F k,t ( )t Fk,t ( )t Fk,t F k,nt F k,t F k,t F k,t F k,t F k,3t ( )3t Fk,t... F k,3t F k,4t ( )(n )t F k,t F k,(n )t F k,nt And L k,(n+)t L k,nt = L k,t L k,0 + F k,t + ( )t Fk,t L k,0 L k,t L k,t L k,t + ( )t Fk,t +... ( )(n )t Fk,t L k,t L k,3t L k,(n )t L k,nt Theorem 4.3. For n, m, k > 0,. F k,n+mt = F [ k,n + ( ) n F k,t + ( )t + ( )t L k,n L k,n L k,n+t l k,n+t L k,n+t L k,n+t L k,n+3t +...+ ( ] )(m )t L k,n+(m )t. F k,n+mt = F [ k,n + ( ) n F k,t + ( )t + ( )t L k,n L k,n L k,n+t l k,n+t L k,n+t L k,n+t L k,n+3t +...+ ( ] )(m )t L k,n+(m )t 6
Proof. We can write expressions of F k,n+mt F k,n+mt and in series as = F ( k,n Fk,n+t + F ) ( k,n Fk,n+t + F ) k,n+t +... L k,n L k,n+t L k,n L k,n+t L k,n+t ( Fk,n+mt + F ) k,n+(m )t L k,n+(m )t = F k,n L k,n + (F k,n+tl k,n F k,n L k,n+t ) L k,n L k,n+t + (F k,n+tl k,n+t F k,n+t L k,n+t ) L k,n+t L k,n+t Fibonacci and k Lucas Sequences as Series of Fractions +...+ (F k,n+mtl k,n+(m )t F k,n+(m )t ) L k,nt L k,n+(m )t And F k,n+mt = L ( k,n Lk,n+t + L ) ( k,n Lk,n+t + L ) k,n+t F k,n F k,n+t F k,n F k,n+t F k,n+t ( Lk,n+mt +...+ L ) k,n+(m )t F k,n+mt F k,n+(m )t = L k,n F k,n (F k,n+tl k,n F k,n L k,n+t ) L k,n L k,n+t (F k,n+tl k,n+t F k,n+t L k,n+t ) L k,n+t L k,n+t +... (F k,n+mtl k,n+(m )t F k,n+(m )t ) L k,nt L k,n+(m )t Using the equations (5) and (6) F k,nt F k,(n )tf k,(n+)t = ( ) (n )t (F k,t ) L k,nt L k,(n )tl k,(n+)t = ( ) (n )t (F k,t ) Gives F k,n+mt = F [ k,n + ( ) n F k,t + ( )t + ( )t L k,n L k,n L k,n+t l k,n+t L k,n+t L k,n+t L k,n+3t +...+ ( ] )(m )t L k,n+(m )t 7
Godase, AD, Dhakne, MB And F k,n+mt = F [ k,n + ( ) n F k,t + ( )t + ( )t L k,n L k,n L k,n+t l k,n+t L k,n+t L k,n+t L k,n+3t +...+ ( ] )(m )t L k,n+(m )t 5 CONCLUSIONS Some new relationship between k Fibonacci and k Lucas sequences using continued fractions and series of fractions are derived, this approach is different and never tried in k Fibonacci sequence literature. REFERENCES. A. D. Godase, M. B. Dhakne, On the properties of k Fibonacci and k Lucas numbers, International Journal of Advances in Applied Mathematics and Mechanics, 0 (04), 00-06.. A. D. Godase, M. B. Dhakne, Fundamental Properties of Multiplicative Coupled Fibonacci Sequences of Fourth Order Under Two Specific Schemes, International Journal of Mathematical Archive, 04, No. 6 (03), 74-8. 3. A. D. Godase, M. B. Dhakne, Recurrent Formulas of The Generalized Fibonacci Sequences of Fifth Order, "International Journal of Mathematical Archive", 04, No. 6 (03), 6-67. 4. A. D. Godase, M. B. Dhakne, Summation Identities for k Fibonacci and k Lucas Numbers using Matrix Methods, "International Journal of Mathematics and Scientific Computing", 05, No. (05), 75-78. 5. A. D. Godase, M. B. Dhakne, Determinantal Identities for k Lucas Sequence, "Journal of New Theory", (05), 0-07. 6. A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly, 03, No. 3 (965), 6-76. 7. S. Falcon, A. Plaza, On the k Fibonacci numbers, Chaos, Solitons and Fractals, 05, No. 3 (007), 65-64. 8. S. Falcon, A. Plaza, The k Fibonacci sequence and the Pascal triangle, Chaos, Solitons and Fractals, 33, No. (007), 38-49. 9. S. Falcon, A. Plaza, The k Fibonacci hyperbolic functions, Chaos, Solitons and Fractals Vol. 38, No. (008), 409-0. 0. P. Filipponi, A. F. Horadam, A Matrix Approach to Certain Identities, The Fibonacci Quarterly, 6, No. (988), 5-6. 8
. H. W. Gould, A History of the Fibonacci q - Matrix and a Higher - Dimensional Problem, The Fibonacci Quarterly, 9, No. 3 (98), 50-57.. A. F. Horadam, Jacobstal Representation of Polynomials, The Fibonacci Quarterly, 35, No. (997) 37-48. 3. T. Koshy, Fibonacci and Lucas numbers with applications, Wiley - Intersection Publication (00). 4. N. J. Sloane, The on-line encyclopedia of integer sequences 006. 5. G. Strang, Introduction to Linear Algebra, Wellesley - Cambridge:Wellesley, M. A. Publication (003). 6. S. Vajda, Fibonacci and Lucas numbers and the Golden Section: Theory and applications, Chichester: Ellis Horwood, (989). 7. M. E. Waddill, Matrices and Generalized Fibonacci Sequences, The Fibonacci Quarterly, 55, No. (974) 38-386. Fibonacci and k Lucas Sequences as Series of Fractions 9