Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 829535, 4 pages

http://dx.doi.org/10.1155/2013/829535

Research Article

A Matrix Approach for Divisibility Properties of the Generalized Fibonacci Sequence

Aynur Yalçiner

Department of Mathematics, Science Faculty, Selcuk University, 42075 Konya, Turkey

Received 14 March 2013; Accepted 9 May 2013

Academic Editor: Gerald Teschl

Copyright © 2013 Aynur Yalçiner. 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.

Abstract.

We give divisibility properties of the generalized Fibonacci sequence by matrix methods. We also present new recursive identities for the generalized Fibonacci and Lucas sequences.

1. Introduction

The generalized Fibonacci sequence and the generalized Lucas sequence are defined for , by,

where , and , respectively.

Let and be the roots of the equation . Then the Binet formulas of the sequences and are given by

If , then (th Fibonacci number) and (th Lucas number).

It is a well-known fact that

It is also known that is a multiple of , for all integers and . In [1], the author showed that, for , the Fibonacci number is a multiplication of if and only if is multiplication of (for more details see [2]). Also, in [3], the author obtained the following divisibility properties:

(i) is divisible by ;(ii) is divisible by ,

where . Kiliç [4] generalized these results for a general second-order linear recursion as follows:

In this paper, we investigate divisibility properties of the generalized Fibonacci numbers by , where . For , we show that

We use matrix methods to prove the claim. We recall that matrix methods are useful tools for deriving some properties of linear recurrences (see [4–9]). We consider the quotient

for all positive integers and . We define a generating matrix for this quotient for fixed and increasing values of . Then we give an explicit statement for the quotient. Also, by considering this explicit statement, we find new recursive identities for the general second-order linear recurrences. Finally, we give divisibility properties of the generalized Fibonacci numbers in the case . Thus we obtain a generalization of the results given in [4].

2. Main Results

We denote the quotient by .

Define a second-order linear sequence , for ,

with initial conditions and .

By the definitions of and , we have

Define a matrix by

where

We next define a matrix of order 4 as follows:

and are given by

where

Thus we give our first main result.

Theorem 1. For ,

Proof. We will use induction on . The result is clear for . Now assume that . Then, by the definitions of , and , we have

Thus the proof is complete.

As a consequence of this theorem, we can see that matrix generates . Since the elements of are integers, the quotient are integers for all positive integers and .

Lemma 2. For , the eigenvalues of are , , , and .

Proof. The characteristic polynomial of is

and it is factorized as

which completes the proof.

As another main result, we have the following theorem.

Theorem 3. For ,

where is defined as shown previously.

Proof. Since the eigenvalues of are distinct, is diagonalizable as

where

and . Therefore, we obtain . By Theorem 1, we write . Then we have the following linear equation system:

The solution of the above linear equation system gives the claimed result.

By considering the definition of , we have the following consequence of Theorem 3.

Corollary 4. For ,

The next results generalize the result given by Corollary 4.

Theorem 5. For all integers ,

Proof. The proof can be seen by the Binet formulas of the sequences and .

For , we give the general case of divisibility properties in the following result.

Corollary 6. For all integers ,

is divisible by .

3. Generalization of the Divisibility Properties

In this section, for a positive integer , we generalize divisibility properties. For this purpose we introduce some new notations.

Let be an integer for . Let

We denote the above product by for .

Corollary 7. (a) For an even positive integer ,

is divisible by .

(b) For an odd positive integer ,

is divisible by .

As an example, if we take , , , , and , then

Acknowledgments

This work is supported by Tubitak and the Scientific Research Projects Office (BAP) of Selcuk University.

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