research article on the products of -fibonacci numbers and...

Research ArticleOn the Products of 𝑘-Fibonacci Numbers and 𝑘-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 © 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 𝑘-Fibonacci and 𝑘-Lucas numbers. We also present some generalized identities onthe products of 𝑘-Fibonacci and 𝑘-Lucas numbers to establish connection formulas between them with the help of Binet’s 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 𝐹

𝑛are 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 values 𝐹

0= 0 and

𝐹1= 1. Also the ratio of two consecutive Fibonacci numbers

converges to the Goldenmean, 0 = (1+√5)/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

𝐹0= 0, 𝐹

1= 1,

𝐹𝑛= 𝐹𝑛−1+ 𝐹𝑛−2

for 𝑛 ≥ 2.(1)

In a similar way, Lucas sequence is defined as

𝐿0= 2, 𝐿

1= 1,

𝐿𝑛= 𝐿𝑛−1+ 𝐿𝑛−2

for 𝑛 ≥ 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 𝑘-Fibonacci sequence introduced by Falcon and Plaza [3]depends only on one integer parameter 𝑘 and is defined asfollows:

𝐹𝑘,0= 0, 𝐹

𝑘,1= 1,

𝐹𝑘,𝑛+1= 𝑘𝐹𝑘,𝑛+ 𝐹𝑘,𝑛−1, where 𝑛 ≥ 1, 𝑘 ≥ 1.

(3)

The first few terms of this sequence are

{0, 1, 𝑘, 𝑘2

+ 1, 𝑘2

+ 2 ⋅ ⋅ ⋅ } . (4)

The particular cases of the 𝑘-Fibonacci sequence are asfollows.

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

𝐹0= 0, 𝐹

1= 1,

𝐹𝑛+1= 𝐹𝑛+ 𝐹𝑛−1

for 𝑛 ≥ 1,

{𝐹𝑛}𝑛∈𝑁= {0, 1, 1, 2, 3, 5, 8 ⋅ ⋅ ⋅ } .

(5)

If 𝑘 = 2, the Pell sequence is obtained:

𝑃0= 0, 𝑃 = 1, 𝑃

𝑛+1= 2𝑃𝑛+ 𝑃𝑛−1

for 𝑛 ≥ 1,

{𝑃𝑛}𝑛∈𝑁= {0, 1, 2, 5, 12, 29, 70 ⋅ ⋅ ⋅ } .

(6)

Motivated by the study of 𝑘-Fibonacci numbers in [4], the 𝑘-Lucas numbers have been defined in a similar fashion as

𝐿𝑘,0= 2, 𝐿

𝑘,1= 𝑘,

𝐿𝑘,𝑛+1= 𝑘𝐿𝑘,𝑛+ 𝐿𝑘,𝑛−1, where 𝑛 ≥ 1, 𝑘 ≥ 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, 𝑘, 𝑘2

+ 2, 𝑘3

+ 3 ⋅ ⋅ ⋅ } . (8)

The particular cases of the 𝑘-Lucas sequence are as follows.If 𝑘 = 1, the classical Lucas sequence is obtained:

{2, 1, 3, 4, 7, 11, 18 ⋅ ⋅ ⋅ } . (9)

If 𝑘 = 2, the Pell-Lucas sequence is obtained:

{2, 2, 6, 14, 34, 82 ⋅ ⋅ ⋅ } . (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-knownBinet’s formulas for 𝑘-Fibonacci numbers and 𝑘-Lucasnumbers, see [3–5], are given by

𝐹𝑘,𝑛=𝑟1

𝑛

− 𝑟2

𝑛

𝑟1− 𝑟2

,

𝐿𝑘,𝑛= 𝑟1

𝑛

+ 𝑟2

𝑛

,

(11)

where 𝑟1, 𝑟2are roots of characteristic equation

𝑟2

− 𝑘𝑟 − 1 = 0, (12)

which are given by

𝑟1=𝑘 + √𝑘2 + 4

2, 𝑟

2=𝑘 − √𝑘2 + 4

2. (13)

We also note that𝑟1+ 𝑟2= 𝑘,

𝑟1𝑟2= − 1,

𝑟1− 𝑟2= √𝑘2 + 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 𝑘-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 𝑘-Fibonacci numbers and 𝑘-Lucas numbers in terms of binomial sums. Falcon and Plaza[9] studied 3-dimensional 𝑘-Fibonacci spirals consideringgeometric point of view. Some identities for 𝑘-Lucas numbersmay be found in [9]. In [10] many properties of 𝑘-Fibonaccinumbers are obtained by easy arguments and related withso-called Pascal triangle. The aim of the present paper is toestablish connection formulas between 𝑘-Fibonacci and 𝑘-Lucas numbers, thereby deriving some results out of them.In the following section we investigate some products of𝑘-Fibonacci numbers and 𝑘-Lucas numbers. Though theresults can be established by inductionmethod as well, Binet’sformula is mainly used to prove all of them.

2. On the Products of 𝑘-Fibonacci and𝑘-Lucas Numbers

Theorem 1. 𝐹𝑘,2𝑛𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛

, where 𝑛 ≥ 1.

Proof.

𝐹𝑘,2𝑛𝐿𝑘,2𝑛= [𝑟1

2𝑛

− 𝑟2

2𝑛

𝑟1− 𝑟2

] [𝑟1

2𝑛

+ 𝑟2

2𝑛

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛

+ (𝑟1𝑟2)2𝑛

− (𝑟1𝑟2)2𝑛

− 𝑟2

4𝑛

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛

− 𝑟2

4𝑛

]

= 𝐹𝑘,4𝑛.

(15)

Theorem 2. 𝐹𝑘,2𝑛𝐿𝑘,2𝑛+1= 𝐹𝑘,4𝑛+1− 1, where 𝑛 ≥ 1.

Proof.

𝐹𝑘,2𝑛𝐿𝑘,2𝑛+1

= [𝑟1

2𝑛

− 𝑟2

2𝑛

𝑟1−𝑟2

] [𝑟1

2𝑛+1

+ 𝑟2

2𝑛+1

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+1

+ 𝑟1

2𝑛

𝑟2

2𝑛+1

− 𝑟1

2𝑛+1

𝑟2

2𝑛

− 𝑟2

4𝑛+1

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+1

− 𝑟2

4𝑛+1

] +(𝑟1𝑟2)2𝑛

(𝑟1− 𝑟2)(𝑟2− 𝑟1)

= 𝐹𝑘,4𝑛+1− (−1)

2𝑛

= 𝐹𝑘,4𝑛+1− 1.

(16)

Theorem 3. 𝐹𝑘,2𝑛𝐿𝑘,2𝑛+2= 𝐹𝑘,4𝑛+2− 𝑘, where 𝑛 ≥ 1.

Proof.


= [𝑟1

2𝑛

− 𝑟2

2𝑛

𝑟1− 𝑟2

] [𝑟1

2𝑛+2

+ 𝑟2

2𝑛+2

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+2

+ 𝑟1

2𝑛

𝑟2

2𝑛+2

− 𝑟1

2𝑛+2

𝑟2

2𝑛

− 𝑟2

4𝑛+2

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+2

− 𝑟2

4𝑛+2

] −(𝑟1𝑟2)2𝑛

(𝑟1− 𝑟2)[𝑟1

2

− 𝑟2

2

]

= 𝐹𝑘,4𝑛+2− (𝑟1𝑟2)2𝑛

(𝑟1+ 𝑟2)

= 𝐹𝑘,4𝑛+2− (−1)

2𝑛

𝑘

= 𝐹𝑘,4𝑛+2− 𝑘.

(17)

International Journal of Mathematics and Mathematical Sciences 3

Theorem 4. 𝐹𝑘,2𝑛𝐿𝑘,2𝑛+3= 𝐹𝑘,4𝑛+3− (𝑘2

+ 1), where 𝑛 ≥ 1.

Proof.


= [𝑟1

2𝑛

− 𝑟2

2𝑛

𝑟1− 𝑟2

] [𝑟1

2𝑛+3

+ 𝑟2

2𝑛+3

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+3

+ 𝑟1

2𝑛

𝑟2

2𝑛+3

− 𝑟1

2𝑛+3

𝑟2

2𝑛

− 𝑟2

4𝑛+3

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+3

− 𝑟2

4𝑛+3

] +(𝑟1𝑟2)2𝑛

(𝑟1− 𝑟2)[𝑟2

3

− 𝑟1

3

]

= 𝐹𝑘,4𝑛+3− (−1)

2𝑛

[𝑟1− 𝑟2

𝑟1− 𝑟2

] [𝑟1

2

+ 𝑟2

2

+ 𝑟1𝑟2]

= 𝐹𝑘,4𝑛+3− (𝐿𝑘,2− 1)

= 𝐹𝑘,4𝑛+3− (𝑘2

+ 1) .

(18)

Theorem 5. 𝐹𝑘,2𝑛−1𝐿𝑘,2𝑛+1= 𝐹𝑘,4𝑛+ 1, where 𝑛 ≥ 1.

Proof.

𝐹𝑘,2𝑛−1𝐿𝑘,2𝑛+1

= [𝑟1

2𝑛−1

− 𝑟2

2𝑛−1

𝑟1− 𝑟2

] [𝑟1

2𝑛+1

+ 𝑟2

2𝑛+1

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛

+ 𝑟1

2𝑛−1

𝑟2

2𝑛+1

− 𝑟1

2𝑛+1

𝑟2

2𝑛−1

− 𝑟2

4𝑛

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛

− 𝑟2

4𝑛

] +(𝑟1𝑟2)2𝑛

(𝑟1− 𝑟2)[𝑟2

𝑟1

−𝑟1

𝑟2

]

= 𝐹𝑘,4𝑛− (𝑟1𝑟2)2𝑛−1

= 𝐹𝑘,4𝑛+ 1.

(19)

Theorem 6. 𝐹𝑘,2𝑛+1𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛+1+ 1, where 𝑛 ≥ 1.

Proof.

𝐹𝑘,2𝑛+1𝐿𝑘,2𝑛

= [𝑟1

2𝑛−1

− 𝑟2

2𝑛−1

𝑟1−𝑟2

] [𝑟1

2𝑛

+ 𝑟2

2𝑛

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+1

+ 𝑟1

2𝑛+1

𝑟2

2𝑛

− 𝑟1

2𝑛

𝑟2

2𝑛+1

− 𝑟2

4𝑛+1

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+1

− 𝑟2

4𝑛+1

] +(𝑟1𝑟2)2𝑛

(𝑟1− 𝑟2)(𝑟1− 𝑟2)

= 𝐹𝑘,4𝑛+1+ (−1)

2𝑛

= 𝐹𝑘,4𝑛+1+ 1.

(20)

In the same manner, we obtain the following results.

Theorem 7. 𝐹𝑘,2𝑛+2𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛+2+ 𝑘, where 𝑛 ≥ 1.

Theorem 8. 𝐹𝑘,2𝑛+2𝐿𝑘,2𝑛+1= 𝐹𝑘,4𝑛+3− 1, where 𝑛 ≥ 1.

3. Generalized Identities on the Products of𝑘-Fibonacci and 𝑘-Lucas Numbers

Theorem 9. 𝐹𝑘,𝑚𝐿𝑘,𝑛= 𝐹𝑘,𝑚+𝑛− (−1)

𝑚

𝐹𝑘,𝑛−𝑚

, for 𝑛 ≥ 𝑚 + 1,𝑚 ≥ 0.

Proof.

𝐹𝑘,𝑚𝐿𝑘,𝑛

= [𝑟1

𝑚

− 𝑟2

𝑚

𝑟1− 𝑟2

] [𝑟1

𝑛

+ 𝑟2

𝑛

]

=1

𝑟1− 𝑟2

[𝑟1

𝑚+𝑛

+ 𝑟1

𝑚

𝑟2

𝑛

− 𝑟1

𝑛

𝑟2

𝑚

− 𝑟2

𝑚+𝑛

]

=1

𝑟1− 𝑟2

[𝑟1

𝑚+𝑛

− 𝑟2

𝑚+𝑛

] +1

𝑟1− 𝑟2

[𝑟1

𝑚

𝑟2

𝑛

− 𝑟1

𝑛

𝑟2

𝑚

]

= 𝐹𝑘,𝑚+𝑛− [𝑟1

𝑛

𝑟2

𝑚

− 𝑟1

𝑚

𝑟2

𝑛

𝑟1− 𝑟2

]

= 𝐹𝑘,𝑚+𝑛− (𝑟1𝑟2)𝑚

[𝑟1

𝑛−𝑚

− 𝑟2

𝑛−𝑚

𝑟1− 𝑟2

]

= 𝐹𝑘,𝑚+𝑛− (−1)

𝑚

𝐹𝑘,𝑛−𝑚.

(21)

For different value of𝑚, we have different results:

If 𝑚 = 0 then 𝐹𝑘,0𝐿𝑘,𝑛= 𝐹𝑘,𝑛− 𝐹𝑘,𝑛= 0, 𝑛 ≥ 1

If 𝑚 = 1 then 𝐹𝑘,1𝐿𝑘,𝑛= 𝐹𝑘,𝑛+1+ 𝐹𝑘,𝑛−1, 𝑛 ≥ 2

or 𝐿𝑘,𝑛= 𝐹𝑘,𝑛+1+ 𝐹𝑘,𝑛−1

If 𝑚 = 2 then 𝐹𝑘,2𝐿𝑘,𝑛= 𝐹𝑘,𝑛+2− 𝐹𝑘,𝑛−2, 𝑛 ≥ 3

or 𝐿𝑘,𝑛=𝐹𝑘,𝑛+2− 𝐹𝑘,𝑛−2

𝑘and so on.

(22)

Theorem 10. 𝐹𝑘,𝑛𝐿𝑘,2𝑛+𝑚= 𝐹𝑘,3𝑛+𝑚− (−1)

𝑛

𝐹𝑘,𝑛+𝑚

, for 𝑛 ≥ 1,𝑚 ≥ 0.

Proof.

𝐹𝑘,𝑛𝐿𝑘,2𝑛+𝑚

= [𝑟1

𝑛

− 𝑟2

𝑛

𝑟1− 𝑟2

] [𝑟1

2𝑛+𝑚

+ 𝑟2

2𝑛+𝑚

]

=1

𝑟1− 𝑟2

[𝑟1

3𝑛+𝑚

+ 𝑟1

𝑛

𝑟2

2𝑛+𝑚

− 𝑟1

2𝑛+𝑚

𝑟2

𝑛

− 𝑟2

3𝑛+𝑚

]

4 International Journal of Mathematics and Mathematical Sciences

=1

𝑟1− 𝑟2

[𝑟1

3𝑛+𝑚

− 𝑟2

3𝑛+𝑚

] + (𝑟1𝑟2)𝑛

[𝑟2

𝑛+𝑚

− 𝑟1

𝑛+𝑚

𝑟1− 𝑟2

]

= 𝐹𝑘,3𝑛+𝑚− (−1)

𝑛

𝐹𝑘,𝑛+𝑚

= 𝐹𝑘,3𝑛+𝑚− 𝐹𝑘,𝑛+𝑚.

(23)

For different values of𝑚, we have various results:

If 𝑚 = 0 then 𝐹𝑘,𝑛𝐿𝑘,2𝑛= 𝐹𝑘,3𝑛− (−1)

𝑛

𝐹𝑘,𝑛, 𝑛 ≥ 1

If 𝑚 = 1 then 𝐹𝑘,𝑛𝐿𝑘,2𝑛+1= 𝐹𝑘,3𝑛+1− (−1)

𝑛

𝐹𝑘,𝑛+1, 𝑛 ≥ 1

and so on.(24)

Similarly we have the following result.

Theorem 11. 𝐹𝑘,2𝑛+𝑚𝐿𝑘,𝑛= 𝐹𝑘,3𝑛+𝑚+ (−1)

𝑛

𝐹𝑘,𝑛+𝑚

, for 𝑛 ≥ 1,𝑚 ≥ 0.

Theorem 12. 𝐹𝑘,2𝑛𝐿𝑘,2𝑛+𝑚= 𝐹𝑘,4𝑛+𝑚− 𝐹𝑘,𝑚

, for 𝑛 ≥ 1,𝑚 ≥ 0.

Proof.

𝐹𝑘,2𝑛𝐿𝑘,2𝑛+𝑚

= [𝑟1

2𝑛

− 𝑟2

2𝑛

𝑟1− 𝑟2

] [𝑟1

2𝑛+𝑚

+ 𝑟2

2𝑛+𝑚

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+𝑚

+ 𝑟1

2𝑛

𝑟2

2𝑛+𝑚

− 𝑟1

2𝑛+𝑚

𝑟2

2𝑛

− 𝑟2

4𝑛+𝑚

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+𝑚

− 𝑟2

4𝑛+𝑚

] + (𝑟1𝑟2)2𝑛

[𝑟2

𝑚

− 𝑟1

𝑚

𝑟1− 𝑟2

]

= 𝐹𝑘,4𝑛+𝑚− 𝐹𝑘,𝑚.

(25)


If 𝑚 = 0 then 𝐹𝑘,2𝑛𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛, 𝑛 ≥ 1

If 𝑚 = 1 then 𝐹𝑘,2𝑛𝐿𝑘,2𝑛+1= 𝐹𝑘,4𝑛+1− 1, 𝑛 ≥ 1 and so on.

(26)

Theorem 13. 𝐹𝑘,2𝑛+𝑚𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛+𝑚+ 𝐹𝑘,𝑚

, for 𝑛 ≥ 1,𝑚 ≥ 0.

Proof.

𝐹𝑘,2𝑛+𝑚𝐿𝑘,2𝑛

= [𝑟1

2𝑛+𝑚

− 𝑟2

2𝑛+𝑚

𝑟1− 𝑟2

] [𝑟1

2𝑛

+ 𝑟2

2𝑛

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+𝑚

+ 𝑟1

2𝑛+𝑚

𝑟2

2𝑛

− 𝑟1

2𝑛

𝑟2

2𝑛+𝑚

− 𝑟2

4𝑛+𝑚

]

=1

𝑟1− 𝑟2

[𝑟1

4𝑛+𝑚

− 𝑟2

4𝑛+𝑚

] + (𝑟1𝑟2)2𝑛

[𝑟1

𝑚

− 𝑟2

𝑚

𝑟1− 𝑟2

]

= 𝐹𝑘,4𝑛+𝑚+ 𝐹𝑘,𝑚.

(27)


If 𝑚 = 0 then 𝐹𝑘,2𝑛𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛, 𝑛 ≥ 1

If 𝑚 = 1 then 𝐹𝑘,2𝑛+1𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛+1+ 1, 𝑛 ≥ 1

If 𝑚 = 2 then 𝐹𝑘,2𝑛+2𝐿𝑘,2𝑛= 𝐹𝑘,4𝑛+2+ 𝑘, 𝑛 ≥ 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, “On the Fibonacci 𝑘-numbers,” Chaos,Solitons and Fractals, vol. 32, no. 5, pp. 1615–1624, 2007.

[4] S. Falcon, “On the 𝑘-Lucas numbers,” International Journal ofContemporary Mathematical Sciences, vol. 6, no. 21, pp. 1039–1050, 2011.

[5] C. Bolat, A. Ipeck, and H. Kose, “On the sequence related toLucas numbers and its properties,”Mathematica Aeterna, vol. 2,no. 1, pp. 63–75, 2012.

[6] M. Thongmoon, “Identities for the common factors ofFibonacci and Lucas numbers,” International MathematicalForum, vol. 4, no. 7, pp. 303–308, 2009.

[7] M.Thongmoon, “New identities for the even and odd Fibonacciand Lucas numbers,” International Journal of ContemporaryMathematical Sciences, vol. 4, no. 14, pp. 671–676, 2009.

[8] N. Yilmaz, N. Taskara, K. Uslu, and Y. Yazlik, “On the binomialsums of 𝑘-Fibonacci and 𝑘-Lucas sequences,” in Proceedings ofthe International Conference on Numerical Analysis and AppliedMathematics (ICNAAM ’11), pp. 341–344, September 2011.

[9] S. Falcon and A. Plaza, “On the 3-dimensional 𝑘-Fibonaccispirals,”Chaos, Solitons and Fractals, vol. 38, no. 4, pp. 993–1003,2008.

[10] S. Falcon and A. Plaza, “The 𝑘-Fibonacci sequence and thePascal 2-triangle,” Chaos, Solitons and Fractals, vol. 33, no. 1, pp.38–49, 2007.

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