﻿ On the k–Lucas Numbers of Arithmetic Indexes

Applied Mathematics
Vol.3 No.10(2012), Article ID:23366,5 pages DOI:10.4236/am.2012.310175

On the k-Lucas Numbers of Arithmetic Indexes

Sergio Falcon

Department of Mathematics and Institute for Applied Microelectronics (IUMA), University of Las Palmas de Gran Canaria, Las Palmas, Spain

Email: sfalcon@dma.ulpgc.es

Received January 23, 2012; revised September 5, 2012; accepted June 12, 2012

Keywords: k-Fibonacci Numbers; k-Lucas Numbers; Generating Function

ABSTRACT

In this paper, we study the k-Lucas numbers of arithmetic indexes of the form an + r, where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k-Fibonacci numbers of indexes of the form 2rn and the k-Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k-Lucas numbers.

1. Introduction

Let us remember the k-Lucas numbers Lk,n are deﬁned  by the recurrence relation with the initial conditions Among other properties, the Binnet Identity establishes being and the characteristic roots of the recurrence equation .

Evidently, .

Moreover, it is veriﬁed [1, Theorem 2.4] that .

If we apply iteratively the equation then we will find a formula that relates the k–Lucas numbers to the k–Fibonacci numbers: (1.1)

This formula is similar to the Convolution formula for the k–Fibonacci numbers [2,3].

Moreover, we deﬁne . Then, if we do p = −n in Formula (1.1) obtain .

2. On the k-Lucas Numbers of Arithmetic Index

We begin this section with a formula that relates each other some k-Lucas numbers.

2.1. Theorem 1 (The k-Lucas Numbers of Arithmetic Index)

If a is a nonnull natural number and r = 0, 1, 2, ... a − 1, then (2.1)

Proof. In  it is proved .

Then If r = 0, then In this case, if a = 2p + 1, then an odd k-Lucas number can be expressed in the form Applying iteratively Formula (2.1), the general term, for , can be written like a non-linear combination of the form In particular, if m = n, then 2.2. Generating Function of the Sequence {Lk, an + r}

Let be the generating function of the sequence . That is, Then, and from where no more to take into account Formula (2.1). So, the generating function of the sequence is .

As particular case, if a = 1, then r = 0 and the generating function of the k-Lucas sequence is , that, for the classical Lucas sequence is If we want to take out the two bisection sequences of the classical Lucas sequence (k = 1), the respective generating functions are a = 2 and r = 0: that generates the sequence a = 2 and r = 1: that generates the sequence .

2.3. Theorem 2 (Sum of the k-Lucas Numbers of Arithmetic Index)

If a is a nonnull natural number and r = 0, 1, 2, ... a − 1, then (2.2)

Proof. because and after applying the formula for the sum of a geometric progression.

2.4. Corollary 1 (Sum of Consecutive Odd k-Lucas Numbers)

If r = 0 and a = 2p + 1, Equation (2.2) is In this case, the sum of the ﬁrst k-Lucas numbers is (for p = 0), (2.3)

that for the classical Lucas numbers is 2.5. Corollary 2 (Sum of Consecutive Even k-Lucas Numbers)

If r = 0 and a = 2p, then Equation (2.2) is (2.4)

In this case, if p = 1 we obtain the formula for the sum of the first even k-Lucas numbers , and for the classical Lucas numbers is 2.6. Theorem 3 (Sum of Alternated k-Lucas Numbers of Arithmetic Index)

For a > 0 and r = 0, 1, 2, ...a − 1, the sum of alternated k-Lucas numbers is Proof. As in the previous theorem, 2.7. Corollary 3 (Sum of Consecutive Alternated Odd k-Lucas Numbers)

As particular case, if a = 2p + 1 and r = 0, Then, for p = 0 we obtain the sum of the first alternated k-Lucas numbers , that for the classical Lucas numbers is .

2.8. Corollary 4 (Sum of Consecutive Alternated Even k-Lucas Numbers)

If r = 0 and a = 2p + 1, then And for the first consecutive alternated even k-Lucas numbers that for the classical Lucas numbers is .

3. On the k-Fibonacci Numbers of Indexes n and the k-Lucas Numbers

In this section we will study a relation between the numbers and .

3.1. Theorem 4 (A Relation between Some k-Fibonacci and the k-Lucas Numbers)

For r ≥ 1, it is (3.1)

Proof. In particular, if r = 1, it is Taking into account , if we expand Formula (3.1), we find that this formula can be expressed as or, that is the same, Then, applying Formula (2.2) to the second hand right of this equation with , a = 4n, and r = 3n for the first term and r = n for the second, (3.2)

We tray to simplify the second hand right of this equation. For that, we will prove the following Lemma.

3.2. Lemma 1 (3)

Proof. We will apply the following formulas: (relation) (negative) (convolution) (definition)

Then: (by relation) (by convolution) (by negative) (by definition)

And applying this Lemma to Equation (3.2), we will have: that is from where If in Equation (3.3) it is a = 0, then it is , and applying the Formulas (2.5) and (2.4), That is In particular, for the classical Lucas numbers (k = 1), it is .

4. Acknowledgements

This work has been supported in part by CICYT Project number MTM200805866-C03-02 from Ministerio de Educación y Ciencia of Spain.

REFERENCES

1. S. Falcon, “On the k-Lucas Numbers,” International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 21, 2011, pp. 1039-1050
2. S. Falcon and A. Plaza, “On the Fibonacci k-Numbers,” Chaos, Solitons & Fractals, Vol. 32, No. 5, 2007, pp. 1615-1624. doi:10.1016/j.chaos.2006.09.022
3. S. Falcon and A. Plaza, “The k-Fibonacci Sequence and the Pascal 2-Triangle,” Chaos, Solitons & Fractals, Vol. 33, No. 1, 2007, pp. 38-49. doi:10.1016/j.chaos.2006.10.022
4. S. Falcon and A. Plaza, “On k-Fibonacci Numbers of Arithmetic Indexes,” Applied Mathematics and Computation, Vol. 208, 2009, pp. 180-185 doi:10.1016/j.amc.2008.11.031