ISRN Combinatorics

Volume 2013 (2013), Article ID 759641, 5 pages

http://dx.doi.org/10.1155/2013/759641

Research Article

Some Properties of Convolved -Fibonacci Numbers

José L. Ramírez

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 [2–7].

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 [10–12]. 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.

Table 1: , with .

0
0
0
0
0
0
0
0
0

1
1
1
1
1
1
1
1
1

2
1
2
3
4
5
6
7
8

3
2
5
9
14
20
27
35
44

4
3
10
22
40
65
98
140
192

5
5
20
51
105
190
315
490
726

6
8
38
111
256
511
924
1554
2472

7
13
71
233
594
1295
2534
4578
7776

Table 2: , with .

0
0
0
0
0
0
0
0
0

1
1
1
1
1
1
1
1
1

2
2
4
6
8
10
12
14
16

3
5
14
27
44
65
90
119
152

4
12
44
104
200
340
532
784
1104

5
29
131
366
810
1555
2709
4396
6756

6
70
376
1212
3032
6482
12432
22008
36624

7
169
1052
3842
10716
25235
52808
101220
181224

Table 3: , with .

0
0
0
0
0
0
0
0
0

1
1
1
1
1
1
1
1
1

2
3
6
9
12
15
18
21
24

3
10
29
57
94
140
195
259
332

4
33
126
306
600
1035
1638
2436
3456

5
109
516
1491
3385
6630
11739
19306
30006

6
360
2034
6813
17568
38493
75348
135702
229176

7
1189
7807
29737
85826
207935
445718
872130
1589664

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.

References

T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience,, New York, NY, USA, 2001.

S. Falcón and Á. Plaza, “On the Fibonacci k-numbers,” Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1615–1624, 2007.

C. Bolat and H. Köse, “On the properties of k-Fibonacci numbers,” International Journal of Contemporary Mathematical, vol. 5, no. 22, pp. 1097–1105, 2010.

S. Falcon, “The k-Fibonacci matrix and the Pascal matrix,” Central European Journal of Mathematics, vol. 9, no. 6, pp. 1403–1410, 2011.

S. Falcón and Á. Plaza, “On k-Fibonacci sequences and polynomials and their derivatives,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1005–1019, 2009.

S. Falcón and Á. Plaza, “The k-Fibonacci sequence and the Pascal 2-triangle,” Chaos, Solitons and Fractals, vol. 33, no. 1, pp. 38–49, 2007.

A. Salas, “About k-Fibonacci numbers and their associated numbers,” International Mathematical Forum, vol. 50, no. 6, pp. 2473–2479, 2011.

H. W. Gould, “Inverse series relations and other expansions involving Humbert polynomials,” Duke Mathematical Journal, vol. 32, no. 4, pp. 697–711, 1965.

J. Riordan, Combinatorial Identities, Wiley, New York, NY, USA, 1968.

V. E. Hoggatt Jr. and M. Bicknell-Johnson, “Fibonacci convolution sequences,” Fibonacci Quarterly, vol. 15, no. 2, pp. 117–122, 1977.

P. Moree, “Convoluted convolved Fibonacci numbers,” Journal of Integer Sequences, vol. 7, article 04.2.2, 2004.

G. Liu, “Formulas for convolution Fibonacci numbers and polynomials,” Fibonacci Quarterly, vol. 40, no. 4, pp. 352–357, 2002.

S. Falcon, “On the k-Lucas numbers,” International Journal of Contemporary Mathematical, vol. 6, no. 21, pp. 1039–1050, 2011.

M. Janjić, “Hessenberg matrices and integer sequences,” Journal of Integer Sequences, vol. 13, no. 7, pp. 1–10, 2010.