Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2014, Article ID 505798, 4 pages http://dx.doi.org/10.1155/2014/505798
Research Article On 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 on the 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 properties; though some are simple and known, others find broad scope in research work. Fibonacci and Lucas numbers cover a wide range of interest in modern mathematics as they appear in the comprehensive works of Koshy [1] and Vajda [2]. The Fibonacci numbers πΉπ are the terms of the sequence {0, 1, 1, 2, 3, 5, 8 β
β
β
} wherein each term is the sum of the two previous terms beginning with the initial values πΉ0 = 0 and πΉ1 = 1. Also the ratio of two consecutive Fibonacci numbers converges to the Golden mean, 0 = (1 + β5)/2. The Fibonacci numbers and Golden mean find numerous applications in modern science and have been extensively used in number theory, applied mathematics, physics, computer science, and biology. The well-known Fibonacci sequence is defined as πΉ0 = 0,
πΉ1 = 1,
πΉπ = πΉπβ1 + πΉπβ2
for π β₯ 2.
for π β₯ 2.
πΉπ,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 as follows. If π = 1, the classical Fibonacci sequence is obtained: πΉ0 = 0,
πΉ1 = 1,
πΉπ+1 = πΉπ + πΉπβ1
for π β₯ 1,
(5)
{πΉπ }πβπ = {0, 1, 1, 2, 3, 5, 8 β
β
β
} . If π = 2, the Pell sequence is obtained: π0 = 0,
πΏ 1 = 1,
πΏ π = πΏ πβ1 + πΏ πβ2
πΉπ,0 = 0,
(1)
In a similar way, Lucas sequence is defined as πΏ 0 = 2,
The π-Fibonacci sequence introduced by FalcΒ΄on and Plaza [3] depends only on one integer parameter π and is defined as follows:
(2)
The second order Fibonacci sequence has been generalized in several ways. Some authors have preserved the recurrence relation and altered the first two terms of the sequence while others have preserved the first two terms of the sequence and altered the recurrence relation slightly.
π = 1,
ππ+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 πLucas numbers have been defined in a similar fashion as πΏ π,0 = 2,
πΏ π,1 = π,
πΏ π,π+1 = ππΏ π,π + πΏ π,πβ1 ,
where π β₯ 1, π β₯ 1.
(7)
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
} .
2. On the Products of π-Fibonacci and π-Lucas Numbers Theorem 1. πΉπ,2π πΏ π,2π = πΉπ,4π , where π β₯ 1. Proof. πΉπ,2π πΏ π,2π = [
(10)
In the 19th century, the French mathematician Binet devised two remarkable analytical formulas for the Fibonacci and Lucas numbers [2]. The same idea has been used to develop Binet formulas for other recursive sequences as well. The wellknown Binetβs formulas for π-Fibonacci numbers and π-Lucas numbers, see [3β5], are given by πΉπ,π =
π1 π β π2 π , π1 β π2
(11)
πΏ π,π = π1 π + π2 π ,
π β β π2 + 4 π2 = . 2
=
1 [π 4π β π2 4π ] π1 β π2 1
Theorem 2. πΉπ,2π πΏ π,2π+1 = πΉπ,4π+1 β 1, where π β₯ 1.
πΉπ,2π πΏ π,2π+1 (12)
(13)
=[ =
π1 2π β π2 2π ] [π1 2π+1 + π2 2π+1 ] π1 βπ2
1 [π 4π+1 + π1 2π π2 2π+1 β π1 2π+1 π2 2π β π2 4π+1 ] π1 β π2 1
(16)
2π
We also note that
=
π1 + π2 = π, π1 π2 = β 1,
1 2π 2π [π 4π + (π1 π2 ) β (π1 π2 ) β π2 4π ] π1 β π2 1 (15)
= πΉπ,4π .
which are given by π + β π2 + 4 , π1 = 2
=
Proof.
where π1 , π2 are roots of characteristic equation π2 β ππ β 1 = 0,
π1 2π β π2 2π ] [π1 2π + π2 2π ] π1 β π2
(14)
π1 β π2 = βπ2 + 4. There are a huge number of simple as well as generalized identities available in the Fibonacci related literature in various forms. Some properties for common factors of Fibonacci and Lucas numbers are studied by Thongmoon [6, 7]. The π-Fibonacci numbers which are of recent origin were found by studying the recursive application of two geometrical transformations used in the well-known fourtriangle longest-edge partition [3], serving as an example between geometry and numbers. Also in [8], authors established some new properties of π-Fibonacci numbers and πLucas numbers in terms of binomial sums. FalcΒ΄on and Plaza [9] studied 3-dimensional π-Fibonacci spirals considering geometric point of view. Some identities for π-Lucas numbers may be found in [9]. In [10] many properties of π-Fibonacci numbers are obtained by easy arguments and related with so-called Pascal triangle. The aim of the present paper is to establish connection formulas between π-Fibonacci and πLucas numbers, thereby deriving some results out of them. In the following section we investigate some products of π-Fibonacci numbers and π-Lucas numbers. Though the results can be established by induction method as well, Binetβs formula is mainly used to prove all of them.
(π π ) 1 [π 4π+1 β π2 4π+1 ] + 1 2 (π β π ) π1 β π2 1 (π1 β π2 ) 2 1
= πΉπ,4π+1 β (β1)2π = πΉπ,4π+1 β 1.
Theorem 3. πΉπ,2π πΏ π,2π+2 = πΉπ,4π+2 β π, where π β₯ 1. Proof. πΉπ,2π πΏ π,2π+2 =[
π1 2π β π2 2π ] [π1 2π+2 + π2 2π+2 ] π1 β π2
=
1 [π 4π+2 + π1 2π π2 2π+2 β π1 2π+2 π2 2π β π2 4π+2 ] π1 β π2 1
=
(π π ) 1 [π 4π+2 β π2 4π+2 ] β 1 2 [π 2 β π2 2 ] π1 β π2 1 (π1 β π2 ) 1
2π
2π
= πΉπ,4π+2 β (π1 π2 ) (π1 + π2 ) = πΉπ,4π+2 β (β1)2π π = πΉπ,4π+2 β π.
(17)
International Journal of Mathematics and Mathematical Sciences
3
Theorem 4. πΉπ,2π πΏ π,2π+3 = πΉπ,4π+3 β (π2 + 1), where π β₯ 1.
Theorem 7. πΉπ,2π+2 πΏ π,2π = πΉπ,4π+2 + π, where π β₯ 1.
Proof.
Theorem 8. πΉπ,2π+2 πΏ π,2π+1 = πΉπ,4π+3 β 1, where π β₯ 1.
πΉπ,2π πΏ π,2π+3 =[
3. Generalized Identities on the Products of π-Fibonacci and π-Lucas Numbers
π1 2π β π2 2π ] [π1 2π+3 + π2 2π+3 ] π1 β π2
Theorem 9. πΉπ,π πΏ π,π = πΉπ,π+π β (β1)π πΉπ,πβπ , for π β₯ π + 1, π β₯ 0.
1 = [π 4π+3 + π1 2π π2 2π+3 β π1 2π+3 π2 2π β π2 4π+3 ] π1 β π2 1 2π
(π π ) 1 = [π 4π+3 β π2 4π+3 ] + 1 2 [π 3 β π1 3 ] π1 β π2 1 (π1 β π2 ) 2
(18)
π βπ = πΉπ,4π+3 β (β1) [ 1 2 ] [π1 2 + π2 2 + π1 π2 ] π1 β π2 2π
Proof. πΉπ,π πΏ π,π =[
= πΉπ,4π+3 β (πΏ π,2 β 1) = πΉπ,4π+3 β (π2 + 1) .
Theorem 5. πΉπ,2πβ1 πΏ π,2π+1 = πΉπ,4π + 1, where π β₯ 1.
π1 π β π2 π ] [π1 π + π2 π ] π1 β π2
=
1 [π π+π + π1 π π2 π β π1 π π2 π β π2 π+π ] π1 β π2 1
=
1 1 [π1 π+π β π2 π+π ] + [π π π π β π1 π π2 π ] π1 β π2 π1 β π2 1 2
= πΉπ,π+π β [
Proof. πΉπ,2πβ1 πΏ π,2π+1 =[
π1
2πβ1
β π2 π1 β π2
2πβ1
π1 π π2 π β π1 π π2 π ] π1 β π2 π
] [π1 2π+1 + π2 2π+1 ]
=
1 [π 4π + π1 2πβ1 π2 2π+1 β π1 2π+1 π2 2πβ1 β π2 4π ] π1 β π2 1 (19)
=
(π π ) π π 1 [π1 4π β π2 4π ] + 1 2 [ 2 β 1 ] π1 β π2 π2 (π1 β π2 ) π1
2π
2πβ1
= πΉπ,4π β (π1 π2 ) = πΉπ,4π + 1.
= πΉπ,π+π β (π1 π2 ) [
π1 πβπ β π2 πβπ ] π1 β π2
= πΉπ,π+π β (β1)π πΉπ,πβπ . (21) For different value of π, we have different results: If π = 0 then πΉπ,0 πΏ π,π = πΉπ,π β πΉπ,π = 0,
πβ₯1
If π = 1 then πΉπ,1 πΏ π,π = πΉπ,π+1 + πΉπ,πβ1 ,
πβ₯2
or πΏ π,π = πΉπ,π+1 + πΉπ,πβ1
Theorem 6. πΉπ,2π+1 πΏ π,2π = πΉπ,4π+1 + 1, where π β₯ 1.
If π = 2 then πΉπ,2 πΏ π,π = πΉπ,π+2 β πΉπ,πβ2 , or πΏ π,π =
Proof.
(22) πβ₯3
πΉπ,π+2 β πΉπ,πβ2 and so on. π
πΉπ,2π+1 πΏ π,2π =[
π1 2πβ1 β π2 2πβ1 ] [π1 2π + π2 2π ] π1 βπ2
=
1 [π 4π+1 + π1 2π+1 π2 2π β π1 2π π2 2π+1 β π2 4π+1 ] π1 β π2 1 (20)
=
(π π ) 1 [π1 4π+1 β π2 4π+1 ] + 1 2 (π β π ) π1 β π2 (π1 β π2 ) 1 2
Theorem 10. πΉπ,π πΏ π,2π+π = πΉπ,3π+π β (β1)π πΉπ,π+π , for π β₯ 1, π β₯ 0. Proof.
2π
= πΉπ,4π+1 + (β1)2π = πΉπ,4π+1 + 1. In the same manner, we obtain the following results.
πΉπ,π πΏ π,2π+π =[ =
π1 π β π2 π ] [π1 2π+π + π2 2π+π ] π1 β π2
1 [π 3π+π + π1 π π2 2π+π β π1 2π+π π2 π β π2 3π+π ] π1 β π2 1
4
International Journal of Mathematics and Mathematical Sciences =
π+π β π1 π+π 1 π π [π1 3π+π β π2 3π+π ] + (π1 π2 ) [ 2 ] π1 β π2 π1 β π2
= πΉπ,3π+π β (β1)π πΉπ,π+π
=
π β π2 π 1 2π π [π1 4π+π β π2 4π+π ] + (π1 π2 ) [ 1 ] π1 β π2 π1 β π2
= πΉπ,4π+π + πΉπ,π . (27)
= πΉπ,3π+π β πΉπ,π+π . (23)
If π = 0 then
For different values of π, we have various results: If π = 0 then πΉπ,π πΏ π,2π = πΉπ,3π β (β1)π πΉπ,π ,
πβ₯1
If π = 1 then πΉπ,π πΏ π,2π+1 = πΉπ,3π+1 β (β1)π πΉπ,π+1 ,
For different values of π, we have various results:
πβ₯1
πΉπ,2π πΏ π,2π = πΉπ,4π ,
πβ₯1
If π = 1 then πΉπ,2π+1 πΏ π,2π = πΉπ,4π+1 + 1,
πβ₯1
If π = 2 then πΉπ,2π+2 πΏ π,2π = πΉπ,4π+2 + π,
πβ₯1
(28)
and so on.
and so on. (24)
Conflict of Interests
Similarly we have the following result. Theorem 11. πΉπ,2π+π πΏ π,π = πΉπ,3π+π + (β1)π πΉπ,π+π , for π β₯ 1, π β₯ 0.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Theorem 12. πΉπ,2π πΏ π,2π+π = πΉπ,4π+π β πΉπ,π , for π β₯ 1, π β₯ 0.
References
Proof.
[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. Β΄ Plaza, βOn the Fibonacci π-numbers,β Chaos, [3] S. FalcΒ΄on and A. Solitons and Fractals, vol. 32, no. 5, pp. 1615β1624, 2007. [4] S. Falcon, βOn the π-Lucas numbers,β International Journal of Contemporary Mathematical Sciences, vol. 6, no. 21, pp. 1039β 1050, 2011. [5] C. Bolat, A. Ipeck, and H. Kose, βOn the sequence related to Lucas numbers and its properties,β Mathematica Aeterna, vol. 2, no. 1, pp. 63β75, 2012. [6] M. Thongmoon, βIdentities for the common factors of Fibonacci and Lucas numbers,β International Mathematical Forum, vol. 4, no. 7, pp. 303β308, 2009. [7] M. Thongmoon, βNew identities for the even and odd Fibonacci and Lucas numbers,β International Journal of Contemporary Mathematical Sciences, vol. 4, no. 14, pp. 671β676, 2009. [8] N. Yilmaz, N. Taskara, K. Uslu, and Y. Yazlik, βOn the binomial sums of π-Fibonacci and π-Lucas sequences,β in Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM β11), pp. 341β344, September 2011. Β΄ Plaza, βOn the 3-dimensional π-Fibonacci [9] S. FalcΒ΄on and A. spirals,β Chaos, Solitons and Fractals, vol. 38, no. 4, pp. 993β1003, 2008. Β΄ Plaza, βThe π-Fibonacci sequence and the [10] S. FalcΒ΄on and A. Pascal 2-triangle,β Chaos, Solitons and Fractals, vol. 33, no. 1, pp. 38β49, 2007.
πΉπ,2π πΏ π,2π+π =[
π1 2π β π2 2π ] [π1 2π+π + π2 2π+π ] π1 β π2
=
1 [π 4π+π + π1 2π π2 2π+π β π1 2π+π π2 2π β π2 4π+π ] π1 β π2 1
=
π β π1 π 1 2π π [π1 4π+π β π2 4π+π ] + (π1 π2 ) [ 2 ] π1 β π2 π1 β π2
= πΉπ,4π+π β πΉπ,π . (25) For different values of π, we have various results: If π = 0 then πΉπ,2π πΏ π,2π = πΉπ,4π ,
πβ₯1
If π = 1 then πΉπ,2π πΏ π,2π+1 = πΉπ,4π+1 β 1,
π β₯ 1 and so on. (26)
Theorem 13. πΉπ,2π+π πΏ π,2π = πΉπ,4π+π + πΉπ,π , for π β₯ 1, π β₯ 0. Proof. πΉπ,2π+π πΏ π,2π =[ =
π1 2π+π β π2 2π+π ] [π1 2π + π2 2π ] π1 β π2
1 [π 4π+π + π1 2π+π π2 2π β π1 2π π2 2π+π β π2 4π+π ] π1 β π2 1
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014