Keywords: Fibonacci number, Fibonacci sequence, k Fibonacci number and k Lucas number
American Journal of Mathematical Analysis, 2014 2 (3),
pp 3335.
DOI: 10.12691/ajma231
Received April 14, 2014; Revised May 27, 2014; Accepted June 16, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
The Fibonacci sequence and the Lucas sequence are the two shining stars in the vast array of integer sequences. They have fascinated both amateurs and professional mathematicians for centuries, and they continue to charm us with their beauty, abundant applications, and ubiquitous habit of occurring in totally surprising and unrelated places.
The Fibonacci numbers are terms of the sequence {0, 1, 1, 2, 3, 5, 8, 13, 21,. ..}, where each term is the sum of the two previous terms, beginning with the values. The Lucas numbers are terms of the sequence {2, 1, 3, 4, 7, 11, 18, 29, 47,...}, where each term is the sum of the two previous terms, beginning with the values. Many kinds of generalizations of Fibonacci numbers and Lucas numbers have been presented in the literature ^{[6, 7, 9]}. One example of this recent generalization is kFibonacci numbers ^{[2]} and kLucas numbers ^{[4]}, which are defined by the recurrence relation, depend on one real parameter k. Falcon and Plaza ^{[3]} defined k Fibonacci hyperbolic functions and deduced some properties of k Fibonacci hyperbolic functions related with the analogous identities for the k Fibonacci numbers. Falcon ^{[4]} studied the k Lucas numbers and proved various properties related with the k Fibonacci numbers. Thongmoon ^{[11]} and Hoggatt ^{[5]} defined various identities for the Fibonacci and Lucas numbers. Gupta and Panwar ^{[10]}, present generalized identities involving common factors of generalized Fibonacci, Jacobsthal and jacobsthalLucas numbers and Binet’s formula will employ to obtain the identities. There are a lot of identities of Fibonacci and Lucas numbers described in ^{[1, 8]}. In this paper, we present generalized identities involving common factors of k Fibonacci, and kLucas numbers.
2. The KFibonacci and the KLucas Numbers
Before presenting our main results, we will need to introduce some known results and notations.
The kFibonacci numbers have been defined in ^{[2]} for any real k as follows:
For any positive real number k, the kFibonacci sequence say is define recurrently by
 (1) 
With initial condition
 (2) 
Particular cases of the kFibonacci sequence are
If k = 1, the classical Fibonacci sequence is obtained
If k = 2, the classical Pell sequence is obtained
If k = 3, the following sequence is obtained
Binet formula for the n^{th} kFibonacci numbers is given by
 (3) 
where , are the roots of the characteristic equation and ;, which gives
 (4) 
Similarly for any positive real number k, the kLucas sequence ^{[4]} say is define recurrently by
 (5) 
with initial condition
 (6) 
Particular cases of the kLucas sequence are
If k = 1, the classical Lucas sequence is obtained
If k = 2, the classical Pell Lucas sequence is obtained
If k = 3, the following sequence is obtained
Binet formula for the n^{th} kLucas numbers is given by
 (7) 
where , are the roots of the characteristic equation which are given in equation (4).
There are a lot of identities for the common factor of Fibonacci and the Lucas numbers define in ^{[12]}.
3. Identities for The Common Factors of KFibonacci and KLucas Number
In this section we present generalized identities involving common factors of kFibonacci and kLucas numbers. We shall use the Binet’s formula for the kFibonacci numbers and kLucas numbers for derivation.
3.1. PropositionProof
 (8) 
3.2. PropositionProof
 (9) 
3.3. PropositionProof
 (10) 
3.4. PropositionProof
 (11) 
3.5. PropositionProof
 (12) 
3.6. PropositionProof
 (13) 
3.7. PropositionProof
 (14) 
In ^{[13]}, if we take in Lemma 1(i), then we have the following proposition.
3.8. PropositionProof
 (15) 
3.9. PropositionProof
 (16) 
In ^{[13]}, if we take in Lemma 1(ii), then we have the following proposition.
3.10. PropositionProof
 (17) 
For different values of p, Proposition (3.1) to Proposition (3.10) can be expressed for even and odd kFibonacci and kLucas numbers.
Identities (8) to (17) give a relation between kFibonacci and kLucas numbers by using their subscripts.
4. Conclusion
In this paper we present generalized identities involving common factors of k Fibonacci and kLucas numbers. Mainly Binet’s formula employ for the identities. The concept can be executed for generalized second order recursive sequences as well as polynomials.
References
[1]  Benjamin, A. T. and Quinn, J. J., “Recounting Fibonacci and Lucas identies”, The College Mathematics Journal, 30 (5) (1999), 359366. 
 In article  CrossRef 

[2]  Falcon, S. & Plaza, A., “On the Fibonacci knumbers”, Chaos, Solitons & Fractals, 32 (5), (2007), 161524. 
 In article  CrossRef 

[3]  Falcon S. and Plaza A., The kFibonacci hyperbolic functions, Chaos Solitons and Fractals, 38 (2) (2008), 409420. 
 In article  CrossRef 

[4]  Falcon, S., “On the kLucas Numbers”, Int. J. Contemp. Math. Sciences, 6 (21), (2011), 10391050. 
 In article  

[5]  Hoggatt, V.E. Jr., “Fibonacci and Lucas Numbers”, Houghton – Mifflin Co., Boston (1969). 
 In article  

[6]  Horadam A. F., A generalized Fibonacci sequences, Mathematical Magazine. 68 (1961), 455459. 
 In article  

[7]  Horadam A. F. and Shanon A. G.., Generalized Fibonacci triples, American Mathmatical Monthly, 80 (1973), 187190. 
 In article  CrossRef 

[8]  Koshy, T., “Fibonacci and Lucas Numbers with Applications”, John Wiley, New York (2001). 
 In article  CrossRef 

[9]  Kilic E., The Binet formula, sums and representation of generalized Fibonacci pnumbers, Eur. J. Combin, 29 (3) (2008), 701711. 
 In article  CrossRef 

[10]  Panwar, Y. K., B. Singh and Gupta, V. K., “identities of Common Factors of generalized Fibonacci, Jacobsthal and JacobsthalLucas numbers”, Applied Mathematics and Physics, 1 (4), (2013), 126128. 
 In article  

[11]  Thongmoon, M., “New Identities for the Even and Odd Fibonacci and Lucas Numbers”, Int. J. Contemp. Math. Sciences, 4 (14), (2009), 671676. 
 In article  

[12]  Thongmoon, M., “Identities for the common factors of Fibonacci and Lucas numbers”. International Mathematical Forum, 4 (7), (2009), 303308. 
 In article  

[13]  Yilmaz, N., Taskara, N., Uslu,K., Yazlik, Y., On the binomial sums of kFibonacci and kLucas Sequences, American institue of Physics (AIP) Conf. Proc. 
 In article  
