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Research ArticleOn the Products of-Fibonacci Numbers and -Lucas Numbers

Bijendra Singh, Kiran Sisodiya, and Farooq Ahmad

School of Studies in Mathematics, Vikram University Ujjain, India

Correspondence should be addressed to Farooq Ahmad; mirfarooq@gmail.com

Received January ; Accepted May ; Published June

Academic Editor: Hernando Quevedo

Copyright Bijendra Singh et al. Tis 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.

In this paper we investigate some products of-Fibonacci and-Lucas numbers. We also present some generalized identities onthe products of-Fibonacci and-Lucas numbers to establish connection formulas between them with the help of Binets formula.

1. Introduction

Fibonacci numbers possess wonderful and amazing proper-ties; though some are simple and known, others nd broadscope in research work. Fibonacci and Lucas numbers cover

a wide range of interest in modern mathematics as theyappear in the comprehensive works of Koshy [] and Vajda[]. Te Fibonacci numbersare the terms of the sequence{0 ,1 ,1 ,2 ,3 ,5 ,8 } wherein each term is the sum of the twoprevious terms beginning with the initial values0= 0and1= 1. Also the ratio of two consecutive Fibonacci numbersconverges to the Golden mean, = (1+5)/2. Te Fibonaccinumbers and Golden mean nd numerous applications inmodern science and have been extensively used in numbertheory, applied mathematics, physics, computer science, andbiology.

Te well-known Fibonacci sequence is dened as

0= 0, 1= 1,= 1+ 2 for 2. ()In a similar way, Lucas sequence is dened as

0= 2, 1= 1,= 1+ 2 for 2. ()

Te second order Fibonacci sequence has been gener-alized in several ways. Some authors have preserved therecurrence relation and altered the rst two terms of thesequence while others have preserved the rst two termsof the sequence and altered the recurrence relation slightly.

Te -Fibonacci sequence introduced by FalconandPlaza[]depends only on one integer parameter and is dened asfollows:

,0

= 0, ,1

= 1,,+1= ,+ ,1, where 1, 1. ()Te rst few terms of this sequence are

0,1,,2 + 1, 2 + 2 . ()Te particular cases of the-Fibonacci sequence are asfollows.

If = 1, the classical Fibonacci sequence is obtained:0= 0, 1= 1,

+1

=

+ 1 for

1,={0 ,1 ,1 ,2 ,3 ,5 ,8 } .()

If = 2, the Pell sequence is obtained:0= 0, = 1, +1= 2+ 1 for 1,

={0,1,2,5,12,29,70 } . ()

Motivated by the study of-Fibonacci numbers in [], the-Lucas numbers have been dened in a similar fashion as

,0= 2, ,1= ,

,+1

= ,

+ ,1

, where

1, 1. ()

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014, Article ID 505798, 4 pageshttp://dx.doi.org/10.1155/2014/505798

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International Journal of Mathematics and Mathematical Sciences

Te rst few terms of this sequence are

2,,2 + 2, 3 + 3 . ()Te particular cases of the-Lucas sequence are as follows.

If = 1, the classical Lucas sequence is obtained:

{2 ,1 ,3 ,4 ,7 ,11,18 } . ()If = 2, the Pell-Lucas sequence is obtained:

{2,2,6,14,34,82 } . ()In the th century, the French mathematician Binet devisedtwo remarkable analytical formulas for the Fibonacci andLucas numbers []. Te same idea has been used to developBinet formulas forotherrecursive sequencesas well. Te well-knownBinets formulas for-Fibonacci numbers and-Lucasnumbers, see [], are given by

,

=1 2

1 2 ,,= 1 + 2,()

where1,2are roots of characteristic equation2 1 = 0, ()

which are given by

1= +2 + 42 , 2= 2 + 4

2 . ()We also note that

1

+ 2

= ,12= 1,1 2=2 + 4.

()

Tere are a huge number of simple as well as general-ized identities available in the Fibonacci related literaturein various forms. Some properties for common factors ofFibonacci and Lucas numbers are studied by Tongmoon[,]. Te-Fibonacci numbers which are of recent originwere found by studying the recursive application of twogeometrical transformations used in the well-known four-triangle longest-edge partition [], serving as an examplebetween geometry and numbers. Also in [], authors estab-

lished some new properties of-Fibonacci numbers and-Lucas numbers in terms of binomial sums. Falcon and Plaza[] studied -dimensional-Fibonacci spirals consideringgeometric point of view. Some identities for-Lucas numbersmay be found in []. In [] many properties of-Fibonaccinumbers are obtained by easy arguments and related withso-called Pascal triangle. Te aim of the present paper is toestablish connection formulas between-Fibonacci and-Lucas numbers, thereby deriving some results out of them.In the following section we investigate some products of-Fibonacci numbers and-Lucas numbers. Tough theresults can be established by induction method as well, Binetsformula is mainly used to prove all of them.

2. On the Products of-Fibonacci and-Lucas NumbersTeorem .,2,2= ,4, where 1.Proof.

,2,2= 12 221 2 12 + 22

= 11 2 14 + 122 122 24

= 11 2 14 24

= ,4.

()

Teorem .

,2

,2+1

= ,4+1

1, where

1.

Proof.

,2,2+1= 12 2212 1

2+1 + 22+1

= 11 2 14+1 + 1222+1 12+122 24+1

= 11 2 14+1 24+1 +12

2

1 22 1

= ,4+1(1)2= ,4+1 1.

()

Teorem .,2,2+2= ,4+2 , where 1.Proof.

,2,2+2= 12 22

1

2

12+2 + 22+2

= 11 2 14+2 + 1222+2 12+222 24+2

= 11 2 14+2 24+2 12

2

1 212 22

= ,4+2 122 1+ 2= ,4+2(1)2= ,4+2 .

()

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Teorem .,2,2+3= ,4+3 (2 + 1), where 1.Proof.

,2,2+3

= 12 221 2

12+3

+ 22+3

= 11 2 1

4+3 + 1222+3 12+322 24+3

= 11 2 14+3 24+3 +12

2

1 223 13

= ,4+3(1)2 1 21 2 12 + 22 + 12

= ,4+3 ,2 1

= ,4+3

2

+ 1 .

()

Teorem .,21,2+1= ,4+ 1, where 1.Proof.

,21,2+1= 121 2211 2 1

2+1 + 22+1= 11 2 1

4 + 12122+1 12+1221 24

= 11 2 14 24 +12

2

1 221

12 = ,4 1221= ,4+ 1.

()

Teorem .,2+1,2= ,4+1+ 1, where 1.Proof.

,2+1,2= 121 22112 1

2 + 22= 11 2 1

4+1 + 12+122 1222+1 24+1

= 11 2 14+1 24+1 +12

2

1 21 2= ,4+1+(1)2= ,4+1+ 1.

()

In the same manner, we obtain the following results.

Teorem .,2+2,2= ,4+2+ , where 1.Teorem .,2+2,2+1= ,4+3 1, where 1.3. Generalized Identities on the Products of

-Fibonacci and

-Lucas Numbers

Teorem .,,= ,+ (1),, for + 1, 0.Proof.

,,= 1 21 2 1

+ 2

= 1

1

2

1+ + 12 12 2+

= 11 2 1+ 2+ + 11 2 1

2 12

= ,+ 12 121 2

= ,+ 12 1 21 2

= ,+(1),.()

For different value of, we have different results:If = 0 then,0,= , ,= 0, 1If = 1 then,1,= ,+1+ ,1, 2

or,= ,+1+ ,1If = 2 then,2,= ,+2 ,2, 3

or,=,+2 ,2 and so on.

()

Teorem .,,2+= ,3+ (1),+, for 1, 0.Proof.

,,2+= 1 21 2 1

2+ + 22+

= 1

1

2

13+ + 122+ 12+2 23+

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International Journal of Mathematics and Mathematical Sciences

= 11 2 13+ 23+ + 12 2

+ 1+1 2 = ,3+(1),+= ,3+ ,+.

()

For different values of, we have various results:If = 0 then,,2= ,3(1),, 1If = 1 then,,2+1= ,3+1(1),+1, 1and so on.

()

Similarly we have the following result.

Teorem .,2+,= ,3++ (1)

,+, for 1, 0.Teorem .,2,2+= ,4+ ,, for 1, 0.Proof.

,2,2+= 12 221 2 1

2+ + 22+

= 1

1

2

14+ + 1222+ 12+22 24+

= 11 2 14+ 24+ + 122 2

11 2 = ,4+ ,.

()

For different values of, we have various results:If = 0 then,2,2= ,4, 1If = 1 then,2,2+1= ,4+1 1, 1 and so on.

()

Teorem .,2+,2= ,4++ ,, for 1, 0.Proof.

,2+,2= 12+ 22+1 2 1

2 + 22

= 1

1

2

14+ + 12+22 1222+ 24+

= 11 2 14+ 24+ + 122 1

21 2 = ,4++ ,.

()

For different values of, we have various results:If = 0 then ,2,2= ,4, 1If = 1 then,2+1,2= ,4+1+ 1, 1If = 2 then,2+2,2= ,4+2+ , 1and so on.

()

Conflict of Interests

Te authors declare that there is no conict of interestsregarding the publication of this paper.

References

[] . Koshy, Fibonacci and Lucas Numbers with Applications,Wiley-Interscience, New York, NY, USA, .

[] S. Vajda,Fibonacci and Lucas Numbers, and the Golden Section,Ellis Horwood, Chichester, UK, .

[] S. Falcon and A. Plaza, On the Fibonacci-numbers,Chaos,Solitons and Fractals, vol. , no. , pp. , .

[] S. Falcon, On the-Lucas numbers,International Journal ofContemporary Mathematical Sciences, vol. , no. , pp. , .

[] C. Bolat, A. Ipeck, and H. Kose, On the sequence related toLucas numbers and its properties,Mathematica Aeterna, vol.,no. , pp. , .

[] M. Tongmoon, Identities for the common factors ofFibonacci and Lucas numbers, International MathematicalForum, vol. , no. , pp. , .

[] M. Tongmoon, New identities for theeven andodd Fibonacciand Lucas numbers, International Journal of ContemporaryMathematical Sciences, vol. , no. , pp. , .

[] N. Yilmaz, N. askara, K. Uslu, and Y. Yazlik, On the binomialsums of-Fibonacci and-Lucas sequences, inProceedings ofthe International Conference on Numerical Analysis and AppliedMathematics (ICNAAM ), pp. , September .

[] S. Falcon and A. Plaza, On the -dimensional-Fibonaccispirals, Chaos, Solitons and Fractals, vol. , no. , pp. ,.

[] S. Falcon and A. Plaza, Te-Fibonacci sequence and thePascal -triangle,Chaos, Solitons and Fractals, vol. , no. , pp., .

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