On Hessenberg and pentadiagonal determinants related with Fibonacci and Fibonacci-like numbers

On Hessenberg and pentadiagonal determinants related

with Fibonacci and Fibonacci-like numbers

Ahmet _Ipek

⇑

, Kamil Arı

Karamanog˘lu Mehmetbey University, Faculty of Kamil Özdag˘ Science, Department of Mathematics, 70100 Karaman, Turkey

a r t i c l e

i n f o

Keywords: Fibonacci numbers Fibonacci-like numbers Hessenberg determinants Pentadiagonal determinants

a b s t r a c t

In this paper, we establish several new connections between the generalizations of Fibo-nacci and Lucas sequences and Hessenberg determinants. We also give an interesting con-jecture related to the determinant of an infinite pentadiagonal matrix with the classical Fibonacci and Gaussian Fibonacci numbers.

Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction

The Fibonacci sequence, say ff gn n2Nis the sequence of positive integers satisfying the recurrence relation f0¼ 0; f1¼ 1 and

fn¼ fn1þ fn2;n P 2. The Lucas sequence, say lf gn n2Nis the sequence of positive integers satisfying the recurrence relation

l0¼ 2; l1¼ 1 and ln¼ ln1þ ln2;n P 2.

In recent years, several connections between the Fibonacci and Lucas sequences with matrices have been given by re-searches. In[3], some classes of identities for some generalizations of Fibonacci numbers have been obtained. The relations between the Bell matrix and the Fibonacci matrix, which provide a unified approach to some lower triangular matrices, such as the Stirling matrices of both kinds, the Lah matrix, and the generalized Pascal matrix, were studied in[23]. In[16], _Ipek computed the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries. In[4], Bozkurt first computed the spectral norms of the matrices related with integer sequences, and then he gave two examples related with Fibonacci, Lucas, Pell and Perrin numbers.

Some of traditional methods for calculation of the determinant of an n  n matrix are based on factorization in a product of certain matrices such as lower, upper, tridiagonal, pentadiagonal and Hessenberg matrices. A brief overview of the theory of determinants can be found, for example, in[14,21].

In some papers related with relationships between the Fibonacci and Lucas sequences with certain matrices, the results on relations between determinants of families of tridiagonal and pentadiagonal matrices with Fibonacci and Lucas numbers have been presented. Cahill and Narayan[7]showed how Fibonacci and Lucas numbers arise as determinants of some tri-diagonal matrices. Strang[20]has introduced real tridiagonal matrices such that their determinants are Fibonacci numbers. Nallı and Civciv[19]gave a generalization of the presented in[7]. Also, Civciv[9]investigated the determinant of a special pentadiagonal matrix with the Fibonacci numbers. In[11], by the determinant of tridiagonal matrix, another proof of the Fibonacci identities is given. In[22], another proof of Pell identities is presented by the determinant of tridiagonal matrix.

In general we use the standard terminology and notation of Hessenberg matrix theory, see[13]. The determinant

Hn¼ aij   

n;

0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.amc.2013.12.071

⇑Corresponding author.

E-mail address:dr.ahmetipek@gmail.com(A. _Ipek). URL:http://drahmetipek.net(A. _Ipek).

Contents lists available atScienceDirect

Applied Mathematics and Computation

where aij¼ 0 when i  j > 1 or when j  i > 1 is known as a Hessenberg determinant or simply Hessenbergian. If aij¼ 0

when i  j > 1, the Hessenbergian takes the from

Hn¼ a11 a12 a13    a1;n1 a1n a21 a22 a23    a2;n1 a2n a32 a33          a43                   an;n1 ann                             n :

If aij¼ 0 when j  i > 1, the triangular array of zero elements appears in top right-hand corner.

It is well known that several famous numbers may be represented as determinants of Hessenberg matrices. In many pa-pers related with relationships between the Fibonacci and Lucas sequences with certain matrices, the results on relations between determinants of families of Hessenberg matrices with Fibonacci and Lucas numbers have been given[5, 6, 8, 20]. In [6], complex Hessenberg matrices such that their determinants are Fibonacci numbers have been introduced. It was showed in[8]that the maximum determinant achieved by n  n Hessenberg 0; 1ð Þ-matrices is the nth Fibonacci number fn. Esmaeili[10]gave several new classes of Fibonacci–Hessenberg matrices whose determinants are in the form tfn1þ fn2

or fn1þ tfn2 for some real or complex number t. In[18], by constructing new Fibonacci–Hessenberg matrices, another

proofs of two results relative to the Pell and Perrin numbers is given.

Besides the usual Fibonacci and Lucas numbers many kinds of generalizations of these numbers have been presented in the literature. For any integer numbers s > 0 and t – 0 with s2_{þ 4t > 0; the nth s; tð Þ-Fibonacci sequence, say F}

nðs; tÞ

f gn2Nis

defined recurrently by

Fnþ1ðs; tÞ ¼ sFnðs; tÞ þ tFn1ðs; tÞ for n P 1; ð1Þ

with F0ðs; tÞ ¼ 0, F1ðs; tÞ ¼ 1.

For any integer numbers s > 0 and t – 0 with s2_{þ 4t > 0; the nth s; tð Þ-Lucas sequence, say L} nðs; tÞ

f gn2Nis defined

recur-rently by

Lnþ1ðs; tÞ ¼ sLnðs; tÞ þ tLn1ðs; tÞ for n P 1; ð2Þ

with L0ðs; tÞ ¼ 2, L1ðs; tÞ ¼ s.

In the rest of the paper, Fnðs; tÞ and Lnðs; tÞ would be written as Fnand Lnrespectively.

The following table summarizes special cases of Fnand Ln:

s; t

ð Þ Fn Ln

1; 1

ð Þ Fibonacci numbers Lucas numbers

2; 1

ð Þ Pell numbers Pell–Lucas numbers

1; 2

ð Þ Jacobsthal numbers Jacobsthal–Lucas numbers

3; 2

ð Þ Mersenne numbers Fermat numbers

Binet’s formula are well known in the Fibonacci numbers theory[17]. Binet’s formula allows us to express the s; tð Þ-Fibo-nacci and Lucas numbers in function of the roots

a

and b of the following characteristic equation, associated to the recur-rence relation(1), or(2):

x2_{¼ sx þ t: ð3Þ}

Theorem 1 (Binet’s formula). The nth s; tð Þ-Fibonacci and Lucas numbers are given by

Fn¼

a

n_{ b}n

a

 b and Ln¼

a

n_{þ b}n

; ð4Þ

where

a

;bare the roots of the characteristic equation(3), and

a

>b(see[17]). Note that, since 0 < s, then

b <0 <

a

and j j <b j j;

a

a

þ b ¼ s and

a

b¼ t;

a

 b ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2_{þ 4t:}

In this paper, we derive some relationships between the s; tð Þ-Fibonacci and Lucas numbers and determinants of some types of Hessenberg matrices, and we give a conjecture on the determinant of an infinite pentadiagonal matrix with the clas-sical Fibonacci and Gaussian Fibonacci numbers.

The main contents of this paper are organized as follows: in Section2, we introduce new classes of Hessenberg matrices whose determinants are the s; tð Þ-Fibonacci and Lucas numbers, where cofactor expansion is used to obtain these determi-nants. We also give the following interesting conjecture on the determinant of an infinite pentadiagonal matrix with the classical Fibonacci and Gaussian Fibonacci numbers in Section3.

2. The determinants of Hessenberg matrices with the s; tð Þ-Fibonacci and Lucas sequences

Theorem 2. For any integer numbers s > 0 and t – 0 with s2_{þ 4t > 0, define the n þ 1ð Þ  n þ 1ð Þ matrix H} nþ1as Hnþ1¼ F2n 2tF2n1 ð Þ2t2F2n2 ð Þ2t3F2n3    ð Þ2tn1Fnþ1 ð Þ2tnFn 1 s s2 _s3 _{     } n 0   s ð Þn 1 2s 3s2 _{     } n 1   s ð Þn1 1 3s       n 2   s ð Þn2             1 sn                                          _nþ1 ; n P 0:

Then, the determinant Hnþ1is given by

Hnþ1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn q F2n; if n is zero or even;  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn1 q L2n; if n odd: 8 > < > :

Proof. If we expand the determinant Hnþ1by the two elements in the last row, and repeat this operation on the

determi-nants of lower order which appear, then we obtain

Hnþ1¼  Xn k¼1 n k   sk_H nþ1kþ 2tð ÞnFn:

From where, the Hnþ1term can be absorbed into the sum, thus we have

1 ð Þnð Þ2tnFn¼ Xn k¼0 n k   sk_H nþ1k: ð5Þ

Since the polynomial in(5)is an Appell polynomial, using inverse relation of Appell polynomial (for more details see[1,2]) we obtain Hnþ1¼ ð1Þn Xn k¼0 n k   2t ð Þnksk_F nþk: ð6Þ

From(4) and (6)we have

Hnþ1¼ 1 ð Þn

a

 b Xn k¼0 n k ! 2t ð Þnksk

_a

nþk_{ b}nþk ¼ð1Þ n

a

 b Xn k¼0 n k ! 2t ð Þnk

a

n_{ð Þ}

_a

_sk  bnð Þbsk h i ð7Þ ¼ð1Þ n

a

 b

a

n_{ð2t þ}

_a

_sÞn  bn_{ð2t þ bsÞ}n  ¼ð1Þ n

a

 b

a

n

_a

2_

_a

b n  bnb2

a

bn h i ¼ð1Þ n

a

 b

a

2n_ð

_a

_{ bÞ}n  1ð Þnb2nð

a

 bÞn  ¼ð

a

 bÞn ð1Þ n

_a

2n_{ b}2n

a

 b ! ; ð8Þ

Hnþ1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn q F2n; if n is zero or even;  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn1 q L2n; if n odd: 8 > < > :

This completes the proof. h

For

u

nðsÞ, we have the following values:

s; t ð Þ H2 H3 H4 H5 1; 1 ð Þ 5F4 5L6 25F8 25L10 1; 2 ð Þ 9F4 9L6 81F8 81L10

Theorem 3. For any integer numbers s > 0 and t – 0 with s2_{þ 4t > 0, define the n þ 1ð Þ  n þ 1ð Þ matrix H} nþ1as H nþ1¼ 1 n! F2n 2tF2n1 ð Þ2t2F2n2 ð Þ2t3F2n3    ð Þ2tn1Fnþ1 ð Þ2tnFn n s n  1 2s n  2 3s             1 ns                                       nþ1 ; n P 0:

Then, the determinant H

nþ1is given by H nþ1¼ 1 n! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn q F2n; if n is zero or even; 1 n! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn1 q L2n; if n odd: 8 > < > :

Proof. Since some of Hnþ1’s elements are functions of n, the minor obtained by removing its last row and column is not

equal to H

n. Hence, this implies that there is no obvious recurrence relation linking H  nþ1;H  n;H  n1, etc. Therefore, by a

series of row operations which reduce some of its elements to zero, the determinant H

nþ1 can be obtained by

transform-ing Hnþ1.

By performing the row operations

R0 i¼ Ri i  1 n þ 1  i   ðsÞRiþ1;

with 2 6 i 6 n, we get the determinant Cnþ1with ðn  1Þ zero elements. Then, again by performing the row operations

R0i¼ Ri i  1 n þ 1  i

 

ðsÞRiþ1;

with 2 6 i 6 n  1, we get the determinant Cnwith ðn  2Þ zero elements. Then, with 2 6 i 6 n  2, etc., and, finally, with

i ¼ 2, we get the determinant H nþ1. h

For Hnþ1, we have the following values:

s; t ð Þ H 2 H3 H4 H5 1; 1 ð Þ 5 2F4 56L6 2524F8 245L10 1; 2 ð Þ 9 2F4 32L6 8124F8 2740L10

Corollary 4. For any integer numbers s > 0 and t – 0 with s2_{þ 4t > 0, define the n þ 1ð Þ  n þ 1ð Þ matrix S}

nþ1 and Tnþ1,

Snþ1¼ n 0   _n 1   _n 2   _n 3      n n  1   _n n   

a

s 2t 

a

s 2t 

a

s 2t             

a

s 2t                                 nþ1 and Tnþ1¼ n 0   _n 1   _n 2   _n 3      n n  1   _n n   bs 2t bs 2t bs 2t             bs 2t                                 nþ1 ; with n P 0. Then, 1 ð Þn

a

 b

a

n_S nþ1 bnTnþ1 ð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn q F2n; if n is zero or even;  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2_{þ 4t} ð Þn1 q L2n; if n odd: 8 > < > : Proof. Since Snþ1¼ Xn k¼0 n k   2t ð Þnkð Þ

a

sk and Tnþ1¼ Xn k¼0 n k   2t ð Þnkð Þbsk;

thus from the Eq.(8)we haveð1Þn

ab

a

n_S nþ1 bnTnþ1 ð Þ ¼ð

a

 bÞn ð1Þn_a2n_b2n ab  

. From the Binet’s formulas of the nth s; tð Þ-Fibo-nacci and Lucas numbers, the result appears. h

Theorem 5. For any integer numbers s > 0 and t – 0 with s2_{þ 4t > 0, define the n þ 1ð Þ  n þ 1ð Þ; n P 0, matrices A}

nþ1and Bnþ1, respectively, as Anþ1¼ 1

a

s ð Þ

a

s2 ð Þ

a

s3    ð Þ

a

sn1 ð Þ

a

sn 1 t 1

a

s ð Þ

a

s 2    ð Þ

a

sn2 ð Þ

a

sn1 1 t 1

a

s    ð Þ

a

s n3

_a

_s ð Þn2 . . . . . .             1 t 1                                 nþ1 and Bnþ1¼ 1 bs ð Þbs2 ð Þbs3    ð Þbsn1 ð Þbsn 1 t 1 bs ð Þbs 2    ð Þbsn2 ð Þbsn1 1 t 1 bs    ð Þbs n3 _{ð Þbs}n2 . . . . . .             1 t 1                                 nþ1 ; with

a

¼sþ ffiffiffiffiffiffiffiffiffi s2_þ4t p 2 and b ¼ spffiffiffiffiffiffiffiffiffis2_þ4t 2 . Then,

Anþ1þ Bnþ1¼ 1 tnL2n:

Proof. If we recall the properties of determinant and use the Binet’s formulas of the nth s; tð Þ-Fibonacci and Lucas numbers, we obtain Anþ1þ Bnþ1¼ tnð

a

s þ tÞ n þ bs þ tð Þn  ¼ tnX n k¼0 n k   tnk _{ð Þ}

_a

_sk þ bsð Þk h i ¼X n k¼0 n k   tk _{ð Þ}

_a

_sk þ bsð Þk h i ¼X n k¼0 n k  

_a

_s t  k þ bs t  k " # ¼

a

s t þ 1  n þ bs t þ 1  n ¼

a

s þ t t  n þ bs þ t t  n ¼1 tn

a

2n_{þ b}2n  ¼1 tnL2n:

Thus, the proof is completed. For n ¼ 0, we have

A1þ B1¼ 2 ¼ 1 t0L0

and for n ¼ 1 we have

A2þ B2¼ 1 t s 2_{þ 2t}  ¼1 tL2: h

3. The determinant of an infinite pentadiagonal matrix with Fibonacci and Gaussian Fibonacci numbers

Gaussian numbers were investigated in 1832 by Gauss[12]. A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z½i. This domain does not have a total ordering that respects arithmetic, since it contains imaginary numbers. Gaussian integers are the set

Z½i ¼ a þ ib : a; b 2 Z and in 2¼ 1o:

Horadam[15] examined Fibonacci numbers on the complex plane and established some interesting properties about them. Gaussian Fibonacci numbers (GFNS) fnðGÞare defined fnðGÞ¼ fn1ðGÞ þ f

ðGÞ n2;n P 2, where f ðGÞ 0 ¼ i; f ðGÞ 1 ¼ 1. The first six GFNs

are 1; 1 þ i; 2 þ i; 3 þ 2i; 5 þ 3i and 8 þ 5i. Therefore, clearly, fðGÞ

n ¼ fnþ ifn1;n P 1. Here, fn is the nth classical Fibonacci

number[17].

In the paper[9], the determinant of a pentadiagonal matrix with Fibonacci numbers such that

Ek¼ 1  fkfk1 fkþ1 fkfk1 fkþ1 1  2fkfk1 . . . . . . fkfk1 fkþ1 . . . . . . . . . . . . . . . . . . fkþ1 fkfk1 . . . . . . 1  2fkfk1 fkþ1 fkfk1 fkþ1 1  fkfk1 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 kk was computed.

In here, we give the following interesting conjecture on the determinant of an infinite pentadiagonal matrix with the clas-sical Fibonacci and Gaussian Fibonacci numbers:

Conjecture 6. f1ðGÞ i ffiffiffiffi f1 p 0 0 0 0 0    ifðGÞ2 ffiffiffiffi f1 p f1ðGÞf ðGÞ 2 if ðGÞ 3 ffiffiffiffi f3 p i2 ffiffiffiffiffiffiffiffif3f5 p 0 0 0    i2 ffiffiffiffiffiffiffiffif1f3 p ifðGÞ 1 ffiffiffiffi f3 p f2ðGÞf3 if2ðGÞ ffiffiffiffi f5 p 0 0 0    0 0 ifðGÞ4 ffiffiffiffi f5 p f3ðGÞf ðGÞ 4 if ðGÞ 5 ffiffiffiffi f7 p i2 ffiffiffiffiffiffiffiffif7f9 p 0    0 0 i2 ffiffiffiffiffiffiffiffif5f7 p if3ðGÞ ffiffiffiffi f7 p f4ðGÞf ðGÞ 5 if ðGÞ 4 ffiffiffiffi f9 p 0    .. . .. . .. . .. . .. . .. . .. . . . .                                     ¼ 0;

where fnand fnðGÞare the nth classical Fibonacci and Gaussian Fibonacci numbers, respectively, i2¼ 1, and fnðGÞis the conjugate of

the nth Gaussian Fibonacci number.

4. Conclusions

We obtain formulas for the determinants of some Hessenberg matrices associated with the s; tð Þ-Fibonacci numbers and the roots of the characteristic equation(3)and they are computational feasible. Also, we give the following interesting con-jecture on the determinant of an infinite pentadiagonal matrix with the classical Fibonacci and Gaussian Fibonacci numbers. Acknowledgements

The authors wish to thank the editor and the referees for their thorough review and very useful suggestions. References

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