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Research ArticleOn the Products of π-Fibonacci Numbers and π-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; [email protected]
Received 3 January 2014; Accepted 22 May 2014; Published 12 June 2014
Academic Editor: Hernando Quevedo
Copyright Β© 2014 Bijendra Singh 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.
In this paper we investigate some products of π-Fibonacci and π-Lucas numbers. We also present some generalized identities onthe products of π-Fibonacci and π-Lucas numbers to establish connection formulas between them with the help of Binetβs formula.
1. Introduction
Fibonacci numbers possess wonderful and amazing proper-ties; though some are simple and known, others find broadscope in research work. Fibonacci and Lucas numbers covera wide range of interest in modern mathematics as theyappear in the comprehensive works of Koshy [1] and Vajda[2]. The Fibonacci numbers πΉ
πare the terms of the sequence
{0, 1, 1, 2, 3, 5, 8 β β β } wherein each term is the sum of the twoprevious terms beginning with the initial values πΉ
0= 0 and
πΉ1= 1. Also the ratio of two consecutive Fibonacci numbers
converges to the Goldenmean, 0 = (1+β5)/2.The Fibonaccinumbers and Golden mean find numerous applications inmodern science and have been extensively used in numbertheory, applied mathematics, physics, computer science, andbiology.
The well-known Fibonacci sequence is defined as
πΉ0= 0, πΉ
1= 1,
πΉπ= πΉπβ1+ πΉπβ2
for π β₯ 2.(1)
In a similar way, Lucas sequence is defined as
πΏ0= 2, πΏ
1= 1,
πΏπ= πΏπβ1+ πΏπβ2
for π β₯ 2.(2)
The second order Fibonacci sequence has been gener-alized in several ways. Some authors have preserved therecurrence relation and altered the first two terms of thesequence while others have preserved the first two termsof the sequence and altered the recurrence relation slightly.
The π-Fibonacci sequence introduced by Falcon and Plaza [3]depends only on one integer parameter π and is defined asfollows:
πΉπ,0= 0, πΉ
π,1= 1,
πΉπ,π+1= ππΉπ,π+ πΉπ,πβ1, where π β₯ 1, π β₯ 1.
(3)
The first few terms of this sequence are
{0, 1, π, π2
+ 1, π2
+ 2 β β β } . (4)
The particular cases of the π-Fibonacci sequence are asfollows.
If π = 1, the classical Fibonacci sequence is obtained:
πΉ0= 0, πΉ
1= 1,
πΉπ+1= πΉπ+ πΉπβ1
for π β₯ 1,
{πΉπ}πβπ= {0, 1, 1, 2, 3, 5, 8 β β β } .
(5)
If π = 2, the Pell sequence is obtained:
π0= 0, π = 1, π
π+1= 2ππ+ ππβ1
for π β₯ 1,
{ππ}πβπ= {0, 1, 2, 5, 12, 29, 70 β β β } .
(6)
Motivated by the study of π-Fibonacci numbers in [4], the π-Lucas numbers have been defined in a similar fashion as
πΏπ,0= 2, πΏ
π,1= π,
πΏπ,π+1= ππΏπ,π+ πΏπ,πβ1, where π β₯ 1, π β₯ 1.
(7)
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014, Article ID 505798, 4 pageshttp://dx.doi.org/10.1155/2014/505798
2 International Journal of Mathematics and Mathematical Sciences
The first few terms of this sequence are
{2, π, π2
+ 2, π3
+ 3 β β β } . (8)
The particular cases of the π-Lucas sequence are as follows.If π = 1, the classical Lucas sequence is obtained:
{2, 1, 3, 4, 7, 11, 18 β β β } . (9)
If π = 2, the Pell-Lucas sequence is obtained:
{2, 2, 6, 14, 34, 82 β β β } . (10)
In the 19th century, the French mathematician Binet devisedtwo remarkable analytical formulas for the Fibonacci andLucas numbers [2]. The same idea has been used to developBinet formulas for other recursive sequences aswell.Thewell-knownBinetβs formulas for π-Fibonacci numbers and π-Lucasnumbers, see [3β5], are given by
πΉπ,π=π1
π
β π2
π
π1β π2
,
πΏπ,π= π1
π
+ π2
π
,
(11)
where π1, π2are roots of characteristic equation
π2
β ππ β 1 = 0, (12)
which are given by
π1=π + βπ2 + 4
2, π
2=π β βπ2 + 4
2. (13)
We also note thatπ1+ π2= π,
π1π2= β 1,
π1β π2= βπ2 + 4.
(14)
There 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 Thongmoon[6, 7]. The π-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 [3], serving as an examplebetween geometry and numbers. Also in [8], authors estab-lished some new properties of π-Fibonacci numbers and π-Lucas numbers in terms of binomial sums. Falcon and Plaza[9] studied 3-dimensional π-Fibonacci spirals consideringgeometric point of view. Some identities for π-Lucas numbersmay be found in [9]. In [10] many properties of π-Fibonaccinumbers are obtained by easy arguments and related withso-called Pascal triangle. The aim of the present paper is toestablish connection formulas between π-Fibonacci and π-Lucas numbers, thereby deriving some results out of them.In the following section we investigate some products ofπ-Fibonacci numbers and π-Lucas numbers. Though theresults can be established by inductionmethod as well, Binetβsformula is mainly used to prove all of them.
2. On the Products of π-Fibonacci andπ-Lucas Numbers
Theorem 1. πΉπ,2ππΏπ,2π= πΉπ,4π
, where π β₯ 1.
Proof.
πΉπ,2ππΏπ,2π= [π1
2π
β π2
2π
π1β π2
] [π1
2π
+ π2
2π
]
=1
π1β π2
[π1
4π
+ (π1π2)2π
β (π1π2)2π
β π2
4π
]
=1
π1β π2
[π1
4π
β π2
4π
]
= πΉπ,4π.
(15)
Theorem 2. πΉπ,2ππΏπ,2π+1= πΉπ,4π+1β 1, where π β₯ 1.
Proof.
πΉπ,2ππΏπ,2π+1
= [π1
2π
β π2
2π
π1βπ2
] [π1
2π+1
+ π2
2π+1
]
=1
π1β π2
[π1
4π+1
+ π1
2π
π2
2π+1
β π1
2π+1
π2
2π
β π2
4π+1
]
=1
π1β π2
[π1
4π+1
β π2
4π+1
] +(π1π2)2π
(π1β π2)(π2β π1)
= πΉπ,4π+1β (β1)
2π
= πΉπ,4π+1β 1.
(16)
Theorem 3. πΉπ,2ππΏπ,2π+2= πΉπ,4π+2β π, where π β₯ 1.
Proof.
πΉπ,2ππΏπ,2π+2
= [π1
2π
β π2
2π
π1β π2
] [π1
2π+2
+ π2
2π+2
]
=1
π1β π2
[π1
4π+2
+ π1
2π
π2
2π+2
β π1
2π+2
π2
2π
β π2
4π+2
]
=1
π1β π2
[π1
4π+2
β π2
4π+2
] β(π1π2)2π
(π1β π2)[π1
2
β π2
2
]
= πΉπ,4π+2β (π1π2)2π
(π1+ π2)
= πΉπ,4π+2β (β1)
2π
π
= πΉπ,4π+2β π.
(17)
International Journal of Mathematics and Mathematical Sciences 3
Theorem 4. πΉπ,2ππΏπ,2π+3= πΉπ,4π+3β (π2
+ 1), where π β₯ 1.
Proof.
πΉπ,2ππΏπ,2π+3
= [π1
2π
β π2
2π
π1β π2
] [π1
2π+3
+ π2
2π+3
]
=1
π1β π2
[π1
4π+3
+ π1
2π
π2
2π+3
β π1
2π+3
π2
2π
β π2
4π+3
]
=1
π1β π2
[π1
4π+3
β π2
4π+3
] +(π1π2)2π
(π1β π2)[π2
3
β π1
3
]
= πΉπ,4π+3β (β1)
2π
[π1β π2
π1β π2
] [π1
2
+ π2
2
+ π1π2]
= πΉπ,4π+3β (πΏπ,2β 1)
= πΉπ,4π+3β (π2
+ 1) .
(18)
Theorem 5. πΉπ,2πβ1πΏπ,2π+1= πΉπ,4π+ 1, where π β₯ 1.
Proof.
πΉπ,2πβ1πΏπ,2π+1
= [π1
2πβ1
β π2
2πβ1
π1β π2
] [π1
2π+1
+ π2
2π+1
]
=1
π1β π2
[π1
4π
+ π1
2πβ1
π2
2π+1
β π1
2π+1
π2
2πβ1
β π2
4π
]
=1
π1β π2
[π1
4π
β π2
4π
] +(π1π2)2π
(π1β π2)[π2
π1
βπ1
π2
]
= πΉπ,4πβ (π1π2)2πβ1
= πΉπ,4π+ 1.
(19)
Theorem 6. πΉπ,2π+1πΏπ,2π= πΉπ,4π+1+ 1, where π β₯ 1.
Proof.
πΉπ,2π+1πΏπ,2π
= [π1
2πβ1
β π2
2πβ1
π1βπ2
] [π1
2π
+ π2
2π
]
=1
π1β π2
[π1
4π+1
+ π1
2π+1
π2
2π
β π1
2π
π2
2π+1
β π2
4π+1
]
=1
π1β π2
[π1
4π+1
β π2
4π+1
] +(π1π2)2π
(π1β π2)(π1β π2)
= πΉπ,4π+1+ (β1)
2π
= πΉπ,4π+1+ 1.
(20)
In the same manner, we obtain the following results.
Theorem 7. πΉπ,2π+2πΏπ,2π= πΉπ,4π+2+ π, where π β₯ 1.
Theorem 8. πΉπ,2π+2πΏπ,2π+1= πΉπ,4π+3β 1, where π β₯ 1.
3. Generalized Identities on the Products ofπ-Fibonacci and π-Lucas Numbers
Theorem 9. πΉπ,ππΏπ,π= πΉπ,π+πβ (β1)
π
πΉπ,πβπ
, for π β₯ π + 1,π β₯ 0.
Proof.
πΉπ,ππΏπ,π
= [π1
π
β π2
π
π1β π2
] [π1
π
+ π2
π
]
=1
π1β π2
[π1
π+π
+ π1
π
π2
π
β π1
π
π2
π
β π2
π+π
]
=1
π1β π2
[π1
π+π
β π2
π+π
] +1
π1β π2
[π1
π
π2
π
β π1
π
π2
π
]
= πΉπ,π+πβ [π1
π
π2
π
β π1
π
π2
π
π1β π2
]
= πΉπ,π+πβ (π1π2)π
[π1
πβπ
β π2
πβπ
π1β π2
]
= πΉπ,π+πβ (β1)
π
πΉπ,πβπ.
(21)
For different value ofπ, we have different results:
If π = 0 then πΉπ,0πΏπ,π= πΉπ,πβ πΉπ,π= 0, π β₯ 1
If π = 1 then πΉπ,1πΏπ,π= πΉπ,π+1+ πΉπ,πβ1, π β₯ 2
or πΏπ,π= πΉπ,π+1+ πΉπ,πβ1
If π = 2 then πΉπ,2πΏπ,π= πΉπ,π+2β πΉπ,πβ2, π β₯ 3
or πΏπ,π=πΉπ,π+2β πΉπ,πβ2
πand so on.
(22)
Theorem 10. πΉπ,ππΏπ,2π+π= πΉπ,3π+πβ (β1)
π
πΉπ,π+π
, for π β₯ 1,π β₯ 0.
Proof.
πΉπ,ππΏπ,2π+π
= [π1
π
β π2
π
π1β π2
] [π1
2π+π
+ π2
2π+π
]
=1
π1β π2
[π1
3π+π
+ π1
π
π2
2π+π
β π1
2π+π
π2
π
β π2
3π+π
]
4 International Journal of Mathematics and Mathematical Sciences
=1
π1β π2
[π1
3π+π
β π2
3π+π
] + (π1π2)π
[π2
π+π
β π1
π+π
π1β π2
]
= πΉπ,3π+πβ (β1)
π
πΉπ,π+π
= πΉπ,3π+πβ πΉπ,π+π.
(23)
For different values ofπ, we have various results:
If π = 0 then πΉπ,ππΏπ,2π= πΉπ,3πβ (β1)
π
πΉπ,π, π β₯ 1
If π = 1 then πΉπ,ππΏπ,2π+1= πΉπ,3π+1β (β1)
π
πΉπ,π+1, π β₯ 1
and so on.(24)
Similarly we have the following result.
Theorem 11. πΉπ,2π+ππΏπ,π= πΉπ,3π+π+ (β1)
π
πΉπ,π+π
, for π β₯ 1,π β₯ 0.
Theorem 12. πΉπ,2ππΏπ,2π+π= πΉπ,4π+πβ πΉπ,π
, for π β₯ 1,π β₯ 0.
Proof.
πΉπ,2ππΏπ,2π+π
= [π1
2π
β π2
2π
π1β π2
] [π1
2π+π
+ π2
2π+π
]
=1
π1β π2
[π1
4π+π
+ π1
2π
π2
2π+π
β π1
2π+π
π2
2π
β π2
4π+π
]
=1
π1β π2
[π1
4π+π
β π2
4π+π
] + (π1π2)2π
[π2
π
β π1
π
π1β π2
]
= πΉπ,4π+πβ πΉπ,π.
(25)
For different values ofπ, we have various results:
If π = 0 then πΉπ,2ππΏπ,2π= πΉπ,4π, π β₯ 1
If π = 1 then πΉπ,2ππΏπ,2π+1= πΉπ,4π+1β 1, π β₯ 1 and so on.
(26)
Theorem 13. πΉπ,2π+ππΏπ,2π= πΉπ,4π+π+ πΉπ,π
, for π β₯ 1,π β₯ 0.
Proof.
πΉπ,2π+ππΏπ,2π
= [π1
2π+π
β π2
2π+π
π1β π2
] [π1
2π
+ π2
2π
]
=1
π1β π2
[π1
4π+π
+ π1
2π+π
π2
2π
β π1
2π
π2
2π+π
β π2
4π+π
]
=1
π1β π2
[π1
4π+π
β π2
4π+π
] + (π1π2)2π
[π1
π
β π2
π
π1β π2
]
= πΉπ,4π+π+ πΉπ,π.
(27)
For different values ofπ, we have various results:
If π = 0 then πΉπ,2ππΏπ,2π= πΉπ,4π, π β₯ 1
If π = 1 then πΉπ,2π+1πΏπ,2π= πΉπ,4π+1+ 1, π β₯ 1
If π = 2 then πΉπ,2π+2πΏπ,2π= πΉπ,4π+2+ π, π β₯ 1
and so on.
(28)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
References
[1] T. Koshy, Fibonacci and Lucas Numbers with Applications,Wiley-Interscience, New York, NY, USA, 2001.
[2] S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section,Ellis Horwood, Chichester, UK, 1989.
[3] S. Falcon and A. Plaza, βOn the Fibonacci π-numbers,β Chaos,Solitons and Fractals, vol. 32, no. 5, pp. 1615β1624, 2007.
[4] S. Falcon, βOn the π-Lucas numbers,β International Journal ofContemporary Mathematical Sciences, vol. 6, no. 21, pp. 1039β1050, 2011.
[5] C. Bolat, A. Ipeck, and H. Kose, βOn the sequence related toLucas numbers and its properties,βMathematica Aeterna, vol. 2,no. 1, pp. 63β75, 2012.
[6] M. Thongmoon, βIdentities for the common factors ofFibonacci and Lucas numbers,β International MathematicalForum, vol. 4, no. 7, pp. 303β308, 2009.
[7] M.Thongmoon, βNew identities for the even and odd Fibonacciand Lucas numbers,β International Journal of ContemporaryMathematical Sciences, vol. 4, no. 14, pp. 671β676, 2009.
[8] N. Yilmaz, N. Taskara, K. Uslu, and Y. Yazlik, βOn the binomialsums of π-Fibonacci and π-Lucas sequences,β in Proceedings ofthe International Conference on Numerical Analysis and AppliedMathematics (ICNAAM β11), pp. 341β344, September 2011.
[9] S. Falcon and A. Plaza, βOn the 3-dimensional π-Fibonaccispirals,βChaos, Solitons and Fractals, vol. 38, no. 4, pp. 993β1003,2008.
[10] S. Falcon and A. Plaza, βThe π-Fibonacci sequence and thePascal 2-triangle,β Chaos, Solitons and Fractals, vol. 33, no. 1, pp.38β49, 2007.
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