A Method to Construct Generalized Fibonacci Sequences

Abstract

The main purpose of this paper is to study the convergence properties of Generalized Fibonacci Sequences and the series of partial sums associated with them. When the proper values of an real matrix are real and different, we give a necessary and sufficient condition for the convergence of the matrix sequence to a matrix .

1. Introduction

The Fibonacci Sequence is an interesting numerical sequence that occurs quite frequently in many parts of nature. This sequence has a special feature; every element of this sequence, starting from the third, is the sum of its two predecessors and can be generated recursively by the formula It is clear that we need the first two terms ,   and the recursive formula to define the sequence.

If we want to know the term without constructing the previous terms, we can use the unexplainable formula (see [1]): What do the irrational numbers have to do with the original sequence?

The so-called Golden ratio appears in nature very frequently. It is also considered the most esthetic ratio between the basis and height of a rectangle:If we replace the recursive formula bywe obtain a new sequence and this sequence is no longer divergent; in fact, it converges to .

To define a Generalized Fibonacci Sequence, we fix a natural number and two elements in the Euclidean space . The recursive formula isThe main purpose of this paper is to study the convergence properties of Generalized Fibonacci Sequences and the series of partial sums associated with them. When the proper values of an real matrix are real and different, we give a necessary and sufficient condition for the convergence of the matrix sequence to a matrix : we say if for every ordered pair , where , the sequence of the -entries of converges to the -entry of . As a particular case, we study when do we have the convergence of the powers of a Moebius transformation to a constant function.

Then we would like to list the four published monographs about generalized Fibonacci sequences [1–4] and several more specialized articles [5–8].

2. Main Results

Consider a Generalized Fibonacci Sequence (GFS) with initial terms and recursive formula (7). Define the matrices :The characteristic polynomial of the matrix isSuppose all the roots of are real and pairwise different; that is, , for . Consider Vandermonde's matrix:Since , we deduce that and, hence, is invertible. We need the following matrix relation.

Theorem 1. and are related by the following formula:where is the diagonal matrix:

Proof. Let be the proper vectors of the matrix . We have then for each . If and , we have ,  . Since , we deduce that the vectors are linearly independent and hence they constitute a basis for . Calling , , we have . On the other hand, consider the linear transformation defined by the formula . Clearly, Therefore,

In the next theorem, we relate and with .

Theorem 2. One has the following formulas:(a) and   for every .(b)  for every .

Proof. (a) It is straightforward.
(b) Using (36), we have

Using the formula , we obtain any member of the corresponding Generalized Fibonacci Sequence.

Theorem 3. Consider

Proof. The first row of the matrix is the following: . The first column of the matrix iswhere is the cofactor of the entry of in the position. is the entry in the position of the matrix . Therefore,The expression inside the square brackets coincides withTo see this, develop this determinant by the first row and the last column. The coefficient of is then This completes the proof.

In the particular case , we obtainIf we further assume that and , we obtain

In the original Fibonacci sequence, we have and hence . The roots of this polynomial are and . We justify then the mythical formula .

In the case , we obtainThe roots of are and . Hence, . It is now clear that this last GFS converges to .

We give next a sufficient condition for the convergence of the series of a GFS.

Theorem 4 (main theorem). Suppose the roots of the characteristic polynomial of a GFS are pairwise different and all of them lie in the open interval . Then the series of converges to

Proof. This is a consequence of the identity and the convergence to .

We give now two examples.

Example 5. One has , , , , , and .
The characteristic polynomial of the corresponding GFS is Therefore, ,  ,  , and Hence Clearly

Example 6. One has ,  ,  ,  ,  ,  and . In this case, . Therefore, ,  ,  , and . Hence,In this case, , butGiven different real numbers , we may construct, for every , a GFS; namely, The missing sequence may be obtained from the coefficients of the polynomial: For instance, and .

With the help of this remark, we prove the following.

Theorem 7. Let be an invertible matrix. Suppose the characteristic polynomial of factors into the form:where are pairwise different real numbers. Let be the entry in the position of the matrix . Then

Proof. Let be the GFS determined by (36). It is clear that for . Proceeding by induction, suppose for every , where . We clearly haveBe induction, the terms may be obtained using determinants of type (36). We consider the last columns of determinants (36) and we obtainThereforeThe GFS on the right side of this equation starts with ,   and has the same proper values of the GFS . Hence and the proof is complete.

As an exercise, we calculate the powers of a matrix: In this case we have , , , , , , , . Hence Thereforewhere and . If and , we have This limit matrix has determinant . So in the case of a Moebius transformation we have the following corollary.

Corollary 8. If 1 is a proper value of the matrix and if  , then the powers converge to the constant map .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.