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Identities of Common Factors of Generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas Numbers

Yashwant K. Panwar*, Bijendra Singh, V. K. Gupta

Department of Mathematics and MCA, Mandsaur Institute of Technology, Mandsaur, India

Abstract: The Fibonacci sequence is famous for possessing wonderful and amazing properties. In this paper, we present generalized identities involving common factors of generalized Fibonacci, Jacobsthal and jacobsthal-Lucas numbers and related identities consisting even and odd terms. Binet’s formula will employ to obtain the identities.

Keywords: generalized Fibonacci numbers, Jacobsthal and jacobsthal-Lucas numbers, Binet’s formula

Received August 20, 2013; Revised November 11, 2013; Accepted November 15, 2013

Citation: Yashwant K. Panwar, Bijendra Singh, V. K. Gupta, Identities of Common Factors of Generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas Numbers. Applied Mathematics and Physics, 1 (4). 126-128. January 2013. doi:10.12691/amp-1-4-5

1. Introduction

It is well-known that the Fibonacci sequence is most prominent examples of recursive sequence. The Fibonacci sequence is famous for possessing wonderful and amazing properties. Fibonacci numbers are a popular topic for mathematical enrichment and popularization. The Fibonacci appear in numerous mathematical problems. The Fibonacci numbers are terms of the sequence wherein each term is the sum of the two previous terms, beginning with the values and .

There are a lot of identities of Fibonacci and Lucas numbers described in [6]. Thongmoon [10], defined various identities of Fibonacci and Lucas numbers. Singh, Bhadouria and Sikhwal [9], present some generalized identities involving common factors of Fibonacci and Lucas numbers. Gupta and Panwar [1], present identities involving common factors of generalized Fibonacci, Jacobsthal and jacobsthal-Lucas numbers. Panwar, Singh and Gupta [8], present Generalized Identities Involving Common factors of generalized Fibonacci, Jacobsthal and jacobsthal-Lucas numbers. In this paper, we present generalized identities involving common factors of generalized Fibonacci, Jacobsthal and jacobsthal-Lucas numbers.

2. Preliminaries

Before presenting our main theorems, we will need to introduce some known results and notations.

Generalized Fibonacci sequence [2,7] is defined as

[picture] | (2.1)

where are positive integers.

For different values of many sequences can be determined.

If , we get

[picture] | (2.2)

The first few terms of are 2, 0, 4, 4, 12, 20 and so on.

Its Binet forms is defined by

[picture] | (2.3)

The Jacobsthal sequence [3], is defined by the recurrence relation

[picture] | (2.4)

Its Binet’s formula is defined by

[picture] | (2.5)

The Jacobsthal-Lucas sequence [3], is defined by the recurrence relation

[picture] | (2.6)

Its Binet’s formula is defined by

[picture] | (2.7)

where are the roots of the characteristic equation

3. Main Results

Generalized Fibonacci sequence [2,7], similar to the other second order classical sequences. In this section we present generalized identities involving common factors of generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas numbers. We shall use the Binet’s formula for derivation.

Theorem 1: If is the generalized Fibonacci numbers and is Jacobsthal-Lucas numbers, then,

[picture] | (3.1)

.

Proof:

This completes the proof.

Corollary 1.1: For different values of , (3.1) can be expressed for even and odd numbers:

Corollary 1.2:

[picture] | (3.2)

Following theorems can be solved by Binet’s formulae (2.3), (2.5) and (2.7).

Theorem 2:

[picture] | (3.3)

Corollary 2.1: For different values of , (3.3) can be expressed for even and odd numbers:

Corollary 2.2:

[picture] | (3.4)

Theorem 3:

[picture] | (3.5)

Corollary 3.1: For different values of , (3.5) can be expressed for even and odd numbers:

Corollary 3.2:

[picture] | (3.6)

.

Theorem 4:

[picture] | (3.7)

.

Corollary 4.1: For different values of , (3.7) can be expressed for even and odd numbers:

Corollary 4.2:

[picture] | (3.8)

[picture] | (3.9)

Corollary 5.1: For different values of , (3.9) can be expressed for even and odd numbers:

Corollary 5.2:

[picture] | (3.10)

.

Theorem 6:

[picture] | (3.11)

Corollary 6.1: For different values of , (3.11) can be expressed for even and odd numbers:

Corollary 6.2:

[picture] | (3.12)

Theorem 7

[picture] | (3.13)

Corollary 7.1: For different values of , (3.13) can be expressed for even and odd numbers:

Corollary 7.2:

[picture] | (3.14)

4. Conclusion

In this paper we present generalized identities involving common factors of generalized Fibonacci, Jacobsthal and jacobsthal-Lucas numbers and related identities consisting even and odd terms. 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]   | Gupta, V. K. & Panwar, Y. K., Common factors of generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas numbers, International Journal of Applied Mathematical Research, 1(4), 377-382, 2012.
[2]   | Gupta, V. K., Panwar, Y. K. & Sikhwal, O., Generalized Fibonacci sequences, Theoretical Mathematics & Applications, 2(2), 115-124, 2012.
[3]   | Horadam, F., Jacobsthal Representation Numbers, The Fibonacci Quarterly, 34(1), 40-54, 1996.
[4]   | Hoggatt, V.E. Jr., Fibonacci and Lucas numbers. Houghton – Mifflin Co., Boston, 1969.
[5]   | Hoggatt, V.E. Jr., Phillips, J.W. and Leonard, H. Jr., “Twenty-four Master Identities”, The Fibonacci Quarterly, 9(1), 1-17, 1971.
[6]   | Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley, New York, 2001.
[7]   | Panwar, Y. K., Generalized Fibonacci sequences, LAP, Germany, 2012.
[8]   | Panwar, Y. K., Singh, B. & Gupta, V. K., Generalized Identities Involving Common factors of generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas numbers, International Journal of Analysis and Application, 3(1), 53-59, 2013.
[9]   | Singh, B., Bhadouria, P. & Sikhwal, O., Generalized Identities Involving Common Factors of Fibonacci and Lucas Numbers, International Journal of Algebra, 5(13), 637-645, 2011.
[10]   | Thongmoon, M., New Identities for the Even and Odd Fibonacci and Lucas Numbers, Int. J. Contemp. Math. Sciences, 4(7), 303-308, 2009.