Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 759641, 5 pages
http://dx.doi.org/10.1155/2013/759641
Research Article

Some Properties of Convolved -Fibonacci Numbers

Instituto de Matemáticas y sus Aplicaciones, Universidad Sergio Arboleda, Bogotá, Colombia

Received 30 April 2013; Accepted 28 May 2013

Academic Editors: S. Falcon and Y. Zhang

Copyright © 2013 José L. Ramírez. 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 define the convolved k-Fibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the k-Fibonacci and k-Lucas numbers. Moreover we obtain the convolved k-Fibonacci numbers from a family of Hessenberg matrices.

1. Introduction

Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art (e.g., see [1]). The Fibonacci numbers are the terms of the sequence , wherein each term is the sum of the two previous terms, beginning with the values and .

Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature. In particular, a generalization is the -Fibonacci numbers.

For any positive real number , the -Fibonacci sequence, say , is defined recurrently by

In [2], -Fibonacci numbers were found by studying the recursive application of two geometrical transformations used in the four-triangle longest-edge (4TLE) partition. These numbers have been studied in several papers; see [27].

The convolved Fibonacci numbers   are defined by If we have classical Fibonacci numbers. Note that Moreover, using a result of Gould [8, page 699] on Humbert polynomials (with , and ), we have It seems that convolved Fibonacci numbers first appeared in the classical Riordan's book [9]. These numbers have been studied in several papers; see [1012]. In this paper, we obtain new identities for convolved -Fibonacci numbers.

2. Some Properties of -Fibonacci Numbers and -Lucas Numbers

The characteristic equation associated with the recurrence relation (1) is . The roots of this equation are Then we have the following basic identities: Some of the properties that the -Fibonacci numbers verify are summarized below (see [2, 6] for the proofs).(i)Binet formula: . (ii)Combinatorial  formula: . (iii)Generating function: .

Definition 1. For any positive real number , the -Lucas sequence, say , is defined recurrently by If   we have the classical Lucas numbers. Some properties that the -Lucas numbers verify are summarized below (see [13] for the proofs).(i)Binet formula: . (ii)Relation with -Fibonacci numbers: .

3. Convolved -Fibonacci Numbers

Definition 2. The convolved -Fibonacci numbers    are defined by

Note that Moreover, from Humbert polynomials (with , and ), we have If we obtain the combinatorial formula of -Fibonacci numbers. In Tables 1, 2, and 3 some values of convolved -Fibonacci numbers are provided. The purpose of this paper is to investigate the properties of these numbers.

tab1
Table 1: , with .
tab2
Table 2: , with .
tab3
Table 3: , with .

Theorem 3. The following identities hold:(1), (2), (3).

Proof. Taking in (10), we obtain
This identity is obtained from observing that
  Taking the first derivative of , we obtain
Therefore the identity is clear.

In the next theorem we show that the convolved -Fibonacci numbers can be expressed in terms of -Fibonacci and -Lucas numbers. This theorem generalizes Theorem 4 of  [11].

Theorem 4. Let and . We have

Proof. Given , such that and . Then we have the following partial fraction decomposition: where and with . Using the Taylor expansion Then , where Note that . On substituting these values of and and using the identities (6), we obtain Since that   and  ,  then From the above equality and Binet formula, we obtain (14).

4. Hessenberg Matrices and Convolved -Fibonacci Numbers

An upper Hessenberg matrix, , is an matrix, where whenever and for some . That is, all entries below the superdiagonal are 0 but the matrix is not upper triangular: We consider a type of upper Hessenberg matrix whose determinants are -Fibonacci numbers. Some results about Fibonacci numbers and Hessenberg can be found in [14]. The following known result about upper Hessenberg matrices will be used.

Theorem 5. Let ,   be arbitrary elements of a commutative ring , and let the sequence be defined by If then

In particular, if then from Theorem 5 we have that It is clear that the principal minor of is equal to . It follows that the principal minor of the matrix is obtained by deleting rows and columns with indices : Then we have the following theorem.

Theorem 6. Let be the sum of all principal minors of   or order . Then

Since the coefficients of the characteristic polynomial of a matrix are, up to the sign, sums of principal minors of the matrix, then we have the following.

Corollary 7. The convolved -Fibonacci number is equal, up to the sign, to the coefficient of in the characteristic polynomial of .

Corollary 8. The following identity holds:

Proof. The characteristic matrix of has the form Then , where is a Fibonacci polynomial. Then from Corollary 8 and the following identity for Fibonacci polynomial [5]: we obtain that Therefore the corollary is obtained.

Acknowledgments

The author would like to thank the anonymous referees for their helpful comments. The author was partially supported by Universidad Sergio Arboleda under Grant no. USA-II-2012-14.

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