# Some Properties of Convolvedk-Fibonacci Numbers

Authors: José L. Ramírez
Journal: ISRN Combinatorics
 (2013)

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## Abstract

We define the convolvedk-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.


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## Body

## 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 Fn are the terms of the sequence 0,1,1,2,3,5,…, wherein each term is the sum of the two previous terms, beginning with the values F0=0 and F1=1.Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature. In particular, a generalization is thek-Fibonacci numbers.For any positive real numberk, the k-Fibonacci sequence, say {Fk,n}n∈ℕ, is defined recurrently by
(1)Fk,0=0,Fk,1=1,Fk,n+1=kFk,n+Fk,n-1,n≥1.In [2], k-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 numbersFj(r) are defined by
(2)(1-x-x2)-r=∑j=0∞‍Fj+1(r)xj,r∈ℤ+.
If r=1 we have classical Fibonacci numbers. Note that
(3)Fm+1(r)=∑j1+j2+⋯+jr=m‍Fj1+1Fj2+1⋯Fjr+1.
Moreover, using a result of Gould [8, page 699] on Humbert polynomials (with n=j,m=2,x=1/2,y=-1,p=-r, and C=1), we have
(4)Fj+1(r)=∑k=0⌊j/2⌋‍(j+r-k-1j-k)(j-kk).
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 k-Fibonacci numbers.


## 2. Some Properties ofk-Fibonacci Numbers and k-Lucas Numbers

The characteristic equation associated with the recurrence relation (1) is x2=kx+1. The roots of this equation are
(5)r1=k+k2+42,r2=k-k2+42.
Then we have the following basic identities:
(6)r1+r2=k,r1-r2=k2+4,r1r2=-1.
Some of the properties that the k-Fibonacci numbers verify are summarized below (see [2, 6] for the proofs).(i)
Binet formula:Fk,n=(r1n-r2n)/(r1-r2).(ii)
Combinatorial  formula:Fk,n=∑i=0⌊(n-1)/2⌋‍(n-1-ii)×kn-1-2i.(iii)
Generating function:fk(x)=x/(1-kx-x2).Definition 1.
For any positive real numberk, the k-Lucas sequence, say {Lk,n}n∈ℕ, is defined recurrently by
(7)Lk,0=2,Lk,1=k,Lk,n+1=kLk,n+Lk,n-1.
If  k=1 we have the classical Lucas numbers. Some properties that the k-Lucas numbers verify are summarized below (see [13] for the proofs).(i)
Binet formula:Lk,n=r1n+r2n.(ii)
Relation withk-Fibonacci numbers: Lk,n=Fk,n-1+Fk,n+1,n≥1.


## 3. Convolvedk-Fibonacci Numbers

Definition 2.
The convolvedk-Fibonacci numbers  Fk,j(r)  are defined by
(8)fk(r)(x)=(1-kx-x2)-r=∑j=0∞‍Fk,j+1(r)xj,r∈ℤ+.Note that(9)Fk,m+1(r)=∑j1+j2+⋯+jr=m‍Fk,j1+1Fk,j2+1⋯Fk,jr+1.
Moreover, from Humbert polynomials (with n=j,m=2,x=k/2,y=-1,p=-r, and C=1), we have
(10)Fk,j+1(r)=∑l=0⌊j/2⌋‍(j+r-l-1j-l)(j-ll)kj-2l.
If r=1 we obtain the combinatorial formula of k-Fibonacci numbers. In Tables 1, 2, and 3 some values of convolved k-Fibonacci numbers are provided. The purpose of this paper is to investigate the properties of these numbers.Table 1
F
1
,
n
(
r
), with r=1,2,3,4,5,6,7,8….
n
F
1
,
n
(
1
)
F
1
,
n
(
2
)
F
1
,
n
(
3
)
F
1
,
n
(
4
)
F
1
,
n
(
5
)
F
1
,
n
(
6
)
F
1
,
n
(
7
)
F
1
,
n
(
8
)
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
F
2
,
n
(
r
), with r=1,2,3,4,5,6,7,8,….
n
F
2
,
n
(
1
)
F
2
,
n
(
2
)
F
2
,
n
(
3
)
F
2
,
n
(
4
)
F
2
,
n
(
5
)
F
2
,
n
(
6
)
F
2
,
n
(
7
)
F
2
,
n
(
8
)
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
F
3
,
n
(
r
), with r=1,2,3,4,5,6,7,8….
n
F
3
,
n
(
1
)
F
3
,
n
(
2
)
F
3
,
n
(
3
)
F
3
,
n
(
4
)
F
3
,
n
(
5
)
F
3
,
n
(
6
)
F
3
,
n
(
7
)
F
3
,
n
(
8
)
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)
Fk,2(r)=kr,(2)
Fk,n(r)=Fk,n(r-1)+kFk,n-1(r)+Fk,n-2(r),n≥2,(3)
nFk,n+1(r)=r(kFk,n(r+1)+2Fk,n-1(r+1)),n≥1.Proof.
(
1
) Taking j=1 in (10), we obtain
(11)Fk,2(r)=(r1)(10)k=rk.
(
2
) This identity is obtained from observing that
(12)∑j=0∞‍Fk,j+1(r)xj=(kx+x2)∑j=0∞‍Fk,j+1(r)xj+∑j=0∞‍Fk,j+1(r-1)xj.
(
3
)  Taking the first derivative of fk(r)(x)=(1-kx-x2)-r, we obtain
(13)(fk(r)(x))′=∑j=1∞‍Fk,j+1(r)jxj-1=r(11-kx-x2)r-1(k+2x(1-kx-x2)2)=r(k+2x)fk(r+1)(x).
Therefore the identity is clear.In the next theorem we show that the convolvedk-Fibonacci numbers can be expressed in terms of k-Fibonacci and k-Lucas numbers. This theorem generalizes Theorem 4 of  [11].Theorem 4.
Letj≥0 and r≥1. We have
(14)Fk,j+1(r)=∑l=0r+l≅0mod2r-1‍(r+l-1l)(r-l+j-1j)×××××1(k2+4)(r+l)/2Lk,r+j-l(r)××××+∑l=0r+l≅1mod2r-1‍(r+l-1l)(r-l+j-1j)×××××××××××1(k2+4)(r+l-1)/2Fk,r+j-l(r).Proof.
Givenα,β∈ℂ, such that αβ≠0 and α≠β. Then we have the following partial fraction decomposition:
(15)(1-αz)-r(1-βz)-r=∑l=0r-1‍(-rl)αrβl(α-β)r+l(1-αz)l-r+∑l=0r-1‍(-rl)βrαl(β-α)r+l(1-βz)l-r,
where (t0)=1 and (tl)=(t(t-1)⋯(t-l+1))/l! with t∈ℝ. Using the Taylor expansion
(16)(1-z)t=∑j=0∞‍(-1)j(tj)zj.
Then (1-αz)-r(1-βz)-r=∑j=0∞‍γ(j)zj, where
(17)γ(j)=∑l=0r-1‍(-rl)αrβl(α-β)r+l(-1)j(l-rj)αj+∑l=0r-1‍(-rl)βrαl(β-α)r+l(-1)j(l-rj)βj.
Note that 1-kz-z2=(1-r1z)(1-r2z). On substituting these values of α=r1 and β=r2 and using the identities (6), we obtain
(18)Fk,j+1(r)=∑l=0r-1‍(-rl)(-1)lαr-l(k2+4)(r+l)/2(-1)j(l-rj)αj+∑l=0r-1‍(-rl)(-1)lβr-l(k2-4)(r+l)/2(-1)j(l-rj)βj=∑l=0r-1‍(-1)l(-rl)(-1)j(l-rj)×××1(k2+4)(r+l)/2(αr+j-l+(-1)r+lβr+j-l).
Since that (-1)l(-rl)=(r+l-1l)  and  (-1)j(l-rj)=(r-l+j-1j),  then
(19)Fk,j+1(r)=(r+l-1l)(r-l+j-1j)×1(k2+4)(r+l)/2(αr+j-l+(-1)r+lβr+j-l).
From the above equality and Binet formula, we obtain (14).


## 4. Hessenberg Matrices and Convolvedk-Fibonacci Numbers

An upper Hessenberg matrix,An, is an n×n matrix, where ai,j=0 whenever i>j+1 and aj+1,j≠0 for some j. That is, all entries below the superdiagonal are 0 but the matrix is not upper triangular:
(20)An=(a1,1a1,2a1,3⋯a1,n-1a1,na2,1a2,2a2,3⋯a2,n-1a2,n0a3,2a3,3⋯a3,n-1a3,n⋮⋮⋮⋯⋮⋮000⋯an-1,n-1an-1,n000⋯an,n-1an,n).
We consider a type of upper Hessenberg matrix whose determinants are k-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.
Leta1, pi,j(i≤j)  be arbitrary elements of a commutative ring R, and let the sequence a1,a2,… be defined by
(21)an+1=∑i=1n‍pi,nai,(n=1,2,…).
If
(22)An=(p1,1p1,2p1,3⋯p1,n-1p1,n-1p2,2p2,3⋯p2,n-1p2,n0-1p3,3⋯p3,n-1p3,n⋮⋮⋮⋯⋮⋮000⋯an-1,n-1an-1,n000⋯-1an,n),
then
(23)an+1=a1detAn.In particular, if(24)Fn(k)=(k10⋯00-1k1⋯000-1k⋯00⋮⋮⋮⋯⋮⋮000⋯k1000⋯-1k),
then from Theorem 5 we have that
(25)detFn(k)=Fk,n+1,(n=1,2,…).
It is clear that the principal minor M(k)(i) of Fn(k) is equal to Fk,iFk,n-i+1. It follows that the principal minor M(k)(i1,i2,…,il) of the matrix Fn(k) is obtained by deleting rows and columns with indices 1≤i1<i2<⋯<il≤n:
(26)M(k)(i1,i2,…,il)=Fk,i1Fk,i2-i1⋯Fk,il-il-1Fk,n-il+1.
Then we have the following theorem.Theorem 6.
LetSn-l(k),(l=0,1,2,…,n-1) be the sum of all principal minors of  Fn(k) or order n-l. Then
(27)Sn-l(k)=∑j1+j2+⋯+jl+1=n-l‍Fk,j1+1Fk,j2+1⋯Fk,jl+1+1=Fk,n-l+1(l+1).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 convolvedk-Fibonacci number Fk,n-l+1(l+1) is equal, up to the sign, to the coefficient of tl in the characteristic polynomial pn(t) of Fn(k).Corollary 8.
The following identity holds:(28)Fk,n-l+1(l+1)=∑i=0⌊(n-l)/2⌋‍(n-ii)(n-2il)kn-2i-l.Proof.
The characteristic matrix ofFn(k) has the form
(29)(t-k10⋯00-1t-k1⋯000-1t-k⋯00⋮⋮⋮⋯⋮⋮000⋯t-k1000⋯-1t-k).
Then pn(t)=fn+1(t-k), where fn+1(t) is a Fibonacci polynomial. Then from Corollary 8 and the following identity for Fibonacci polynomial [5]:
(30)fn+1(x)=∑i=0⌊n/2⌋‍(n-ii)xn-2i,
we obtain that
(31)fn+1(t-k)=∑i=0⌊n/2⌋‍(n-ii)∑l=0n-2i‍(n-2il)×××(-1)n-lkn-2i-ltl.
Therefore the corollary is obtained.



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*Source: 759641-2013-06-12.xml*