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Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 26, 2020, No. 2, 177–197 DOI: 10.7546/nntdm.2020.26.2.177-197 On r -circulant matrices with Horadam numbers having arithmetic indices Aldous Cesar F. Bueno Mathematics Unit, Philippine Science High School – Central Luzon Campus Lily Hill, Clark Freeport Zone, Pampanga, Philippines e-mails: [email protected], [email protected] Received: 2 January 2020 Revised: 25 April 2020 Accepted: 27 April 2020 Abstract: We investigate an r-circulant matrix whose entries are Horadam numbers having arithmetic indices. We then solve for the eigenvalues, determinant, Euclidean norm and spectral norm of the matrix. Lastly, we present some special cases and some results on identities and divisibility. Keywords: Horadam Numbers, r-circulant matrix, Eigenvalue, Determinant, Euclidean norm, Spectral norm. 2010 Mathematics Subject Classification: 11B05, 15B36, 11B39. 1 Introduction Given a nonzero complex number r and a finite sequence {s k } n-1 k=0 , a certain r-circulant matrix can be formed. From this type of matrix, it has been a custom to determine and investigate its eigenvalues, determinant, inverse, Euclidean norm and spectral norm. The cases where r = ±1 partnered with a sequence satisfying a recurrence relation are the ones that are usually explored. Some studies that involve such approach are that of Bahsi and Solak [1], Bozkurt [3,5], Bozkurt and Tam [4], Bueno [6–13], Civciv and Turkmen [16], Lind [19], Majumdar [20], Nalli and Sen [21], Shen and Cen [23] and Yalciner [25]. There are also studies that revolve on r-circulant matrices having special sequences. In Radicic [22], r-circulant matrices involving geometric sequences were considered. Bozkurt and Tam [5] worked on the determinants and inverses of r-circulant matrices with Horadam numbers. The bounds of the spectral norms of r-circulant matrices with k-Fibonacci and k-Lucas 177
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Page 1: On r-circulant matrices with Horadam numbers having arithmetic …nntdm.net/papers/nntdm-26/NNTDM-26-2-177-197.pdf · 2020. 7. 3. · Notes on Number Theory and Discrete Mathematics

Notes on Number Theory and Discrete MathematicsPrint ISSN 1310–5132, Online ISSN 2367–8275Vol. 26, 2020, No. 2, 177–197DOI: 10.7546/nntdm.2020.26.2.177-197

On r-circulant matrices with Horadam numbershaving arithmetic indices

Aldous Cesar F. BuenoMathematics Unit, Philippine Science High School – Central Luzon Campus

Lily Hill, Clark Freeport Zone, Pampanga, Philippinese-mails: [email protected], [email protected]

Received: 2 January 2020 Revised: 25 April 2020 Accepted: 27 April 2020

Abstract: We investigate an r-circulant matrix whose entries are Horadam numbers havingarithmetic indices. We then solve for the eigenvalues, determinant, Euclidean norm and spectralnorm of the matrix. Lastly, we present some special cases and some results on identities anddivisibility.Keywords: Horadam Numbers, r-circulant matrix, Eigenvalue, Determinant, Euclidean norm,Spectral norm.2010 Mathematics Subject Classification: 11B05, 15B36, 11B39.

1 Introduction

Given a nonzero complex number r and a finite sequence skn−1k=0 , a certain r-circulant matrixcan be formed. From this type of matrix, it has been a custom to determine and investigate itseigenvalues, determinant, inverse, Euclidean norm and spectral norm.

The cases where r = ±1 partnered with a sequence satisfying a recurrence relation are theones that are usually explored. Some studies that involve such approach are that of Bahsi andSolak [1], Bozkurt [3, 5], Bozkurt and Tam [4], Bueno [6–13], Civciv and Turkmen [16], Lind[19], Majumdar [20], Nalli and Sen [21], Shen and Cen [23] and Yalciner [25].

There are also studies that revolve on r-circulant matrices having special sequences. InRadicic [22], r-circulant matrices involving geometric sequences were considered. Bozkurtand Tam [5] worked on the determinants and inverses of r-circulant matrices with Horadamnumbers. The bounds of the spectral norms of r-circulant matrices with k-Fibonacci and k-Lucas

177

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numbers were derived by Shen and Cen [24]. In Bueno [14], r-circulant matrices with Fibonacciand Lucas numbers having arithmetic indices were introduced and some properties of theeigenvalues, determinants and norms of the matrices were established.

In this research, we will examine the r-circulant matrix given by

M =

Ws Ws+t Ws+2t ... Ws+(n−2)t Ws+(n−1)t

rWs+(n−1)t Ws Ws+t ... Ws+(n−3)t Ws+(n−2)t

rWs+(n−2)t rWs+(n−1)t Ws ... Ws+(n−4)t Ws+(n−3)t...

...... . . . ...

...rWs+2t rWs+3t rWs+4t ... Ws Ws+t

rWs+t rWs+2t rWs+3t ... rWs+(n−1)t Ws

. (1)

In the matrix above, Wjs are Horadam numbers and s, t ∈ Z. Observe that the indices of theHoradam numbers in the matrix form an arithmetic sequence and not only limited to nonnegativevalues. Our objective is to derive for the formulas of the eigenvalues, determinant, Euclideannorm and spectral norm of the matrix. We will also consider some special cases of the Horadamnumbers and establish some identities and results on divisibility.

2 Preliminaries

2.1 The Horadam sequence

We first define the Horadam sequence and enumerate some of its special cases.

Definition 2.1 ( [18]). The sequence given by Wk+∞k=0 that satisfy the following recurrencerelation

Wn (a, b, p, q) = Wn =

a if n = 0

b if n = 1

pWn−1 − qWn−2 if n ≥ 2

, (2)

where a, b, p, q ∈ R, is called the Horadam sequence.

For the n-th Horadam number, we have the Binet’s formula given by

Wn = Aαn +Bβn, (3)

where α =p+

√p2 − 4q

2, β =

p−√p2 − 4q

2, A =

b− aβα− β

and B =aα− bα− β

.

The following well-known sequences are all special cases of the Horadam sequence.

1. Fibonacci sequence: Fk+∞k=0

• Recurrence relation:

Fn = Wn (0, 1, 1,−1) =

0 if n = 0

1 if n = 1

Fn−1 + Fn−2 if n ≥ 2

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• Binet’s formula:

Fn =1√5

[(1 +√

5

2

)n

(1−√

5

2

)n]

2. Lucas sequence: Lk+∞k=0

• Recurrence relation

Ln = Wn (2, 1, 1,−1) =

2 if n = 0

1 if n = 1

Ln−1 + Ln−2 if n ≥ 2

• Binet’s formula:

Ln =

(1 +√

5

2

)n

+

(1−√

5

2

)n

3. Jacobsthal sequence: Jk+∞k=0

• Recurrence relation:

Jn = Wn (0, 1, 1,−2) =

0 if n = 0

1 if n = 1

Jn−1 + 2Jn−2 if n ≥ 2

• Binet’s formula:Jn =

1

3[2n − (−1)n]

4. Jacobsthal–Lucas sequence: Kk+∞k=0

• Recurrence relation:

Kn = Wn (2, 1, 1,−2) =

2 if n = 0

1 if n = 1

Kn−1 + 2Kn−2 if n ≥ 2

• Binet’s formula:Kn = 2n + (−1)n

5. Pell sequence: Pk+∞k=0

• Recurrence relation:

Pn = Wn (0, 1, 2,−1) =

0 if n = 0

1 if n = 1

2Pn−1 + Pn−2 if n ≥ 2

• Binet’s formula:Pn =

1

2√

2

[(1 +

√2)n − (1−

√2)n]

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6. Pell–Lucas sequence: Qk+∞k=0

• Recurrence relation:

Qn = Wn (2, 2, 2,−1) =

2 if n = 0

2 if n = 1

2Qn−1 +Qn−2 if n ≥ 2

• Binet’s formula:Qn = (1 +

√2)n + (1−

√2)n.

2.2 Notations

After the discussion of Horadam numbers, we now introduce some notations. For the rest of thepaper we will use the following notations:

• M(r, ~W ) = M , the r-circulant matrix with Horadam numbers having arithmetic indices

• ∆[r, ~W ] := determinant of M(r, ~W )

• λm[r, ~W ] := an eigenvalue of M(r, ~W ), m = 0, 1, . . . , n− 1

•∥∥∥M(r, ~W )

∥∥∥E

:= Euclidean norm of M(r, ~W )

•∥∥∥M(r, ~W )

∥∥∥2

:= spectral norm of M(r, ~W )

• W−n = Aα−n +Bβ−n

• Vn = αn + βn

• Un = Bαn + Aβn

• ρ = |r|1/n[cos(θ+2πjn

)+ i sin

(θ+2πjn

)], an n-th root of r, j = 0, 1, . . . , n− 1

• ω = e2πi/n = cos(2πjn

)+ i sin

(2πjn

), an n-th root of unity, j = 0, 1, . . . , n− 1

2.3 The r-circulant matrices

We now define the r-circulant and provide some of its properties.

Definition 2.2. An r-circulant matrix is a matrix of the form

M(r,~c) =

c0 c1 c2 ... cn−2 cn−1rcn−1 c0 c1 ... cn−3 cn−2rcn−2 rcn−1 c0 ... cn−4 cn−3

......

... . . . ......

rc2 rc3 rc4 ... c0 c1rc1 rc2 rc3 ... rcn−1 c0

, (4)

where r ∈ C\ 0 and ~c = (c0, c1 , c2, ..., cn−2, cn−1) ∈ Rn. The vector ~c is called the circulantvector and it determines the r-circulant matrix.

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The 1-circulant matrices and −1-circulant matrices are called the right circulant matrices andskew right circulant matrices, respectively. In general, r-circulant matrices are Toeplitz matrices[17]; they are diagonal-constant matrices.

Here are some properties of the r-circulant matrices that we will use.We start with the eigenvalues.

Theorem 2.3 ( [15]). The eigenvalues of M(r,~c) are given by

λm[r,~c] =n−1∑k=0

ck(ρω−m)k, (5)

where m = 0, 1, . . . , n− 1, ρ is an n-th root of r and ω is an n-th root of unity.

Basically, the right-hand side of (5) is the Generalized Discrete Fourier Transform (GDFT) of~c. For r = 1 we have the following result from Gray [17].

Theorem 2.4 ([17]). The eigenvalues of the right circulant matrix M(1,~c) are given by

λm[1,~c] =n−1∑k=0

ckω−mk, (6)

where m = 0, 1, . . . , n− 1.

Note that, this is just the Discrete Fourier Transform (DFT) of ~c. Since right circulant matricesare related to the DFT via their eigenvalues, they appear in signal processing theory and codingtheory.

From the eigenvalues, it follows that the determinant is given by

∆[r,~c] =n−1∏m=0

[n−1∑k=0

ck(ρω−m)k

]. (7)

For the spectral norm, we have

‖M(r,~c)‖2 = max0≤m≤n−1

∣∣∣∣∣n−1∑k=0

ck(ρω−m)k

∣∣∣∣∣. (8)

Finally, for the Euclidean norm, we have the following result.

Lemma 2.5 ([14]). The Euclidean norm of M(r,~c) is given by

∥∥∥M(r, ~W )∥∥∥E

=

√√√√n−1∑k=0

|ck|2 [n− k(1− |r|2)] (9)

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3 Main results

3.1 The eigenvalues of M(r, ~W )

For the eigenvalues M(r, ~W ) we have the following results.

Theorem 3.1. The eigenvalues of M [r, ~W ] are given by

λm[r, ~W ] =

Ws − rWs+nt −(qsUt−s − rqtWs+(n−1)t

)ρω−m

(1− αtρω−m)(1− βtρω−m), s < t

Ws − rW(n+1)s − qs (a− rWns) ρω−m

(1− αsρω−m)(1− βsρω−m), s = t

Ws − rWs+nt − qt(Ws−t − rWs+(n−1)t

)ρω−m

(1− αtρω−m)(1− βtρω−m), s > t

where m = 0, 1, . . . , n− 1.

Proof.

λm[r, ~W ] =n−1∑k=0

Ws+kt(ρω−m)k

=n−1∑k=0

[Aαs+kt +Bβs+kt

](ρω−m)k

= Aαsn−1∑k=0

(αtρω−m

)k+Bβs

n−1∑k=0

(βtρω−m

)k= Aαs

(1− rαnt

1− αtρω−m

)+Bβs

(1− rβnt

1− βtρω−m

)=

(Aαs − rAαs+nt)(1− βtρω−m) + (Bβs − rBβs+nt)(1− αtρω−m)

(1− αtρω−m)(1− βtρω−m)

=[Aαs +Bβs]− r [Aαs+nt +Bβs+nt]

(1− αtρω−m)(1− βtρω−m)−[

(Aαsβt +Bαtβs) + rqt(Aαs+(n−1)t +Bβs+(n−1)t)]ρω−m

(1− αtρω−m)(1− βtρω−m)

=Ws − rWs+nt −

[(Aαsβt +Bαtβs)− rqtWs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m)

Note that

Aαsβt +Bαtβs =

qsUt−s, s < t

aqs, s = t

qtWs−t, s > t

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Using the equation above, we obtain

λm[r, ~W ] =

Ws − rWs+nt −[qsUt−s − rqtWs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s < t

Ws − rW(n+1)s − qs [a− rWns] ρω−m

(1− αsρω−m)(1− βsρω−m), s = t

Ws − rWs+nt − qt[Ws−t − rWs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s > t

.

Here are some special cases of Theorem 3.1.

Corollary 3.2.

λm[r, ~F ] =

Fs − rFs+nt −[(−1)sFt−s − (−1)trFs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s < t

Fs − rF(n+1)s + (−1)srFnsρω−m

(1− αsρω−m)(1− βsρω−m), s = t

Fs − rFs+nt − (−1)t[Fs−t − rFs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s > t

(10)

where α =1 +√5

2and β =

1−√5

2.

Corollary 3.3.

λm[r, ~L] =

Ls − rLs+nt −[(−1)sLt−s − (−1)trLs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s < t

Ls − rL(n+1)s − (−1)s [2− rLns] ρω−m

(1− αsρω−m)(1− βsρω−m), s = t

Ls − rLs+nt − (−1)t[Ls−t − rLs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s > t

(11)

where α =1 +√5

2and β =

1−√5

2.

Corollary 3.4.

λm[r, ~J ] =

Js − rJs+nt +[(−2)sJt−s + (−2)trJs+(n−1)t

]ρω−m

(1− 2tρω−m)(1− (−1)tρω−m), s < t

Js − rJ(n+1)s + (−2)srJnsρω−m

(1− 2sρω−m)(1− (−1)sρω−m), s = t

Js − rJs+nt − (−2)t[Js−t − rJs+(n−1)t

]ρω−m

(1− 2tρω−m)(1− (−1)tρω−m), s > t

(12)

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Corollary 3.5.

λm[r, ~K] =

Ks − rKs+nt −[(−2)sKt−s − (−2)trKs+(n−1)t

]ρω−m

(1− 2tρω−m)(1− (−1)tρω−m), s < t

Ks − rK(n+1)s − (−2)s [2− rKns] ρω−m

(1− 2sρω−m)(1− (−1)sρω−m), s = t

Ks − rKs+nt − (−2)t[Ks−t − rKs+(n−1)t

]ρω−m

(1− 2tρω−m)(1− (−1)tρω−m), s > t

(13)

Corollary 3.6.

λm[r, ~P ] =

Ps − rPs+nt −[(−1)s+1Pt−s − (−1)trPs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s < t

Ps − rP(n+1)s + (−1)srPnsρω−m

(1− αsρω−m)(1− βsρω−m), s = t

Ps − rPs+nt − (−1)t[Ps−t − rPs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s > t

(14)

where α = 1 +√

2 and β = 1−√

2.

Corollary 3.7.

λm[r, ~Q] =

Qs − rQs+nt −[(−1)sQt−s − (−1)trQs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s < t

Qs − rQ(n+1)s − (−1)s [2− rQns] ρω−m

(1− αsρω−m)(1− βsρω−m), s = t

Qs − rQs+nt − (−1)t[Qs−t − rQs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s > t

(15)

where α = 1 +√

2 and β = 1−√

2.

3.2 The determinant of M(r, ~W )

We now discuss the results on the determinant of M(r, ~W ).

Theorem 3.8.

∆[r, ~W ] =

(Ws − rWs+nt)n − r

[qsUt−s − rqtWs+(n−1)t

]n1− rVnt + r2qnt

, s < t

(Ws − rW(n+1)s)n − rqns [a− rWns]

n

1− rVns + r2qns, s = t

(Ws − rWs+nt)n − rqnt

[Ws−t − rWs+(n−1)t

]n1− rVnt + r2qnt

, s > t

(16)

184

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Proof.

∆[r, ~W ] =n−1∏m=0

λm[r, ~W ]

=

∏n−1m=0

Ws − rWs+nt −[qsUt−s − rqtWs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s < t

∏n−1m=0

Ws − rW(n+1)s − qs [a− rWns] ρω−m

(1− αsρω−m)(1− βsρω−m), s = t

∏n−1m=0

Ws − rWs+nt − qt[Ws−t − rWs+(n−1)t

]ρω−m

(1− αtρω−m)(1− βtρω−m), s > t

=

(Ws − rWs+nt)n − r

[qsUt−s − rqtWs+(n−1)t

]n(1− rαnt)(1− rβnt)

, s < t

(Ws − rW(n+1)s)n − rqns [a− rWns]

n

(1− rαns)(1− rβns), s = t

(Ws − rWs+nt)n − rqt

[Ws−t − rWs+(n−1)t

]n(1− rαnt)(1− rβnt)

, s > t

=

(Ws − rWs+nt)n − r

[qsUt−s − rqtWs+(n−1)t

]n1− rVnt + r2qnt

, s < t

(Ws − rW(n+1)s)n − rqns [a− rWns]

n

1− rVns + r2qns, s = t

(Ws − rWs+nt)n − rqnt

[Ws−t − rWs+(n−1)t

]n1− rVnt + r2qnt

, s > t

Here are some special cases of Theorem 3.8.

Corollary 3.9.

∆[r, ~F ] =

(Fs − rFs+nt)n − r[(−1)sFt−s − (−1)trFs+(n−1)t

]n1− rLnt + (−1)ntr2

, s < t

(Fs − rF(n+1)s)n − (−1)nsr (−rFns)n

1− rLns + (−1)nsr2, s = t

(Fs − rFs+nt)n − (−1)ntr(Fs−t − rFs+(n−1)t

)n1− rLnt + (−1)ntr2

, s > t

(17)

Corollary 3.10.

∆[r, ~L] =

(Ls−rLs+nt)n−r[(−1)sLt−s−(−1)trLs+(n−1)t]n

1−rLnt+(−1)ntr2, s < t

(Ls − rL(n+1)s)n − (−1)nsr [2− rLns]n

1− rLns + (−1)nsr2, s = t

(Ls − rLs+nt)n − (−1)ntr[Ls−t − rLs+(n−1)t

]n1− rLnt + (−1)ntr2

, s > t

(18)

185

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Corollary 3.11.

∆[r, ~J ] =

(Js − rJs+nt)n − r[(−2)s+1Jt−s − (−2)trJs+(n−1)t

]n1− rKnt + (−2)ntr2

, s < t

(Js − rJ(n+1)s)n − (−2)nsr (−rJns)n

1− rKns + (−2)nsr2, s = t

(Js − rJs+nt)n − (−2)ntr[Js−t − rJs+(n−1)t

]n1− rKnt + (−2)ntr2

, s > t

(19)

Corollary 3.12.

∆[r, ~K] =

(Ks − rKs+nt)n − r

[(−2)sKt−s − (−2)trKs+(n−1)t

]n1− rKnt + (−2)ntr2

, s < t

(Ks − rK(n+1)s)n − (−2)nsr [2− rKns]

n

1− rKns + (−2)nsr2, s = t

(Ks − rKs+nt)n − (−2)ntr

[Ks−t − rKs+(n−1)t

]n1− rKnt + (−2)ntr2

, s > t

(20)

Corollary 3.13.

∆[r, ~P ] =

(Ps − rPs+nt)n − r[(−1)s+1Pt−s − (−1)trPs+(n−1)t

]n1− rQnt + (−1)ntr2

, s < t

(Ps − rP(n+1)s)n − (−1)nsr (−rPns)n

1− rQns + (−1)nsr2, s = t

(Ps − rPs+nt)n − (−1)ntr[Ps−t − rPs+(n−1)t

]n1− rQnt + (−1)ntr2

, s > t

(21)

Corollary 3.14.

∆[r, ~Q] =

(Qs − rQs+nt)n − r

[(−1)sQt−s − (−1)trQs+(n−1)t

]n1− rQnt + (−1)ntr2

, s < t

(Qs − rQ(n+1)s)n − (−1)nsr [2− rQns]

n

1− rQns + (−1)nsr2, s = t

(Qs − rQs+nt)n − (−1)ntr

[Qs−t − rQs+(n−1)t

]n1− rQnt + (−1)ntr2

, s > t

(22)

3.3 Sum identities

From the eigenvalues of M(r, ~W ), if we choose m = 0, r = 1 and ρ = 1, we will obtain thefollowing sum identities on Horadam numbers and its special cases.

Theorem 3.15.

n−1∑k=0

Ws+kt =

Ws −Ws+nt − qsUt−s + qtWs+(n−1)t

1− Vt + qt, s < t

Ws −W(n+1)s − qs [a−Wns]

1− Vs + qs, s = t

Ws −Ws+nt − qt[Ws−t −Ws+(n−1)t

]1− Vt + qt

, s > t

(23)

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Corollary 3.16.

n−1∑k=0

Fs+kt =

Fs − Fs+nt + (−1)s+1Ft−s + (−1)tFs+(n−1)t

1− Lt + (−1)t, s < t

Fs − F(n+1)s + (−1)sFns1− Ls + (−1)s

, s = t

Fs − Fs+nt − (−1)t[Fs−t − Fs+(n−1)t

]1− Lt + (−1)t

, s > t

(24)

Corollary 3.17.

n−1∑k=0

Ls+kt =

Ls − Ls+nt + (−1)s+1Lt−s + (−1)tLs+(n−1)t

1− Lt + (−1)t, s < t

Ls − L(n+1)s − (−1)s [2− Lns]1− Ls + (−1)s

, s = t

Ls − Ls+nt − (−1)t[Ls−t − Ls+(n−1)t

]1− Lt + (−1)t

, s > t

(25)

Corollary 3.18.

n−1∑k=0

Js+kt =

Js − Js+nt + (−2)sJt−s + (−2)tJs+(n−1)t

1−Kt + (−2)t, s < t

Js − J(n+1)s + (−2)sJns1−Ks + (−2)s

, s = t

Js − Js+nt − (−2)t[Js−t − Js+(n−1)t

]1−Kt + (−2)t

, s > t

(26)

Corollary 3.19.

n−1∑k=0

Ks+kt =

Ks −Ks+nt + (−2)sKt−s + (−2)tKs+(n−1)t

1−Kt + (−2)t, s < t

Ks −K(n+1)s − (−2)s [2−Kns]

1−Ks + (−2)s, s = t

Ks −Ks+nt − (−2)t[Ks−t −Ks+(n−1)t

]1−Kt + (−2)t

, s > t

(27)

Corollary 3.20.

n−1∑k=0

Ps+kt =

Ps − Ps+nt + (−1)sUt−s + (−1)tPs+(n−1)t

1−Qt + (−1)t, s < t

Ps − P(n+1)s + (−1)sPns1−Qs + (−1)s

, s = t

Ps − Ps+nt − (−1)t[Ps−t − Ps+(n−1)t

]1−Qt + (−1)t

, s > t

(28)

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Corollary 3.21.

n−1∑k=0

Qs+kt =

Qs −Qs+nt + (−1)s+1Qt−s + (−1)tQs+(n−1)t

1−Qt + (−1)t, s < t

Qs −Q(n+1)s − (−1)s [2−Qns]

1−Qs + (−1)s, s = t

Qs −Qs+nt − (−1)t[Qs−t −Qs+(n−1)t

]1−Qt + (−1)t

, s > t

(29)

3.4 Results on divisibility

From the results on determinant and sum identities, we now derive results on divisibility.

Theorem 3.22. Let a, b, p, q, r ∈ Z.

1. If s < t, then 1− rVnt + r2qnt divides (Ws − rWs+nt)n − r

[qsUt−s − rqtWs+(n−1)t

]n .

2. If s = t, then 1− rVns + r2qns divides (Ws − rW(n+1)s)n − rqns [a− rWns]

n.

3. If s > t, then 1− rVnt + r2qnt divides (Ws − rWs+nt)n − rqnt

[Ws−t − rWs+(n−1)t

]n.

Proof. Let a, b, p, q, r ∈ Z. This means that M(r, ~W ) is an integer matrix. Hence its determinant∆[r, ~W ] is an integer because integer matrices have integer determinants. From this, Theorem3.22 immediately follows.

Here are the special cases of Theorem 3.22.

Corollary 3.23. Let r ∈ Z.

1. If s < t, then 1−rLnt+(−1)ntr2 divides (Fs−rFs+nt)n−r[(−1)sFt−s − (−1)trFs+(n−1)t

]n.

2. If s = t, then 1− rLns + (−1)nsr2 divides (Fs − rF(n+1)s)n − (−1)nsr (−rFns)n.

3. If s > t, then 1− rLnt + (−1)ntr2 divides (Fs − rFs+nt)n − (−1)ntr(Fs−t − rFs+(n−1)t

)n.

Corollary 3.24. Let r ∈ Z.

1. If s < t, then 1−rLnt+(−1)ntr2 divides (Ls−rLs+nt)n−r[(−1)sLt−s − (−1)trLs+(n−1)t

]n.

2. If s = t, then 1− rLns + (−1)nsr2 divides (Ls − rL(n+1)s)n − (−1)nsr [2− rLns]n.

3. If s > t, then 1− rLnt + (−1)ntr2 divides (Ls − rLs+nt)n − (−1)ntr[Ls−t − rLs+(n−1)t

]n.

Corollary 3.25. Let r ∈ Z.

1. If s < t, then 1−rKnt+(−2)ntr2 divides (Js−rJs+nt)n−r[(−2)s+1Jt−s − (−2)trJs+(n−1)t

]n.

2. If s = t, then 1− rKns + (−2)nsr2 divides (Js − rJ(n+1)s)n − (−2)nsr (−rJns)n.

3. If s > t, then 1− rKnt + (−2)ntr2 divides (Js − rJs+nt)n − (−2)ntr[Js−t − rJs+(n−1)t

]n.

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Corollary 3.26. Let r ∈ Z.

1. If s < t, then 1−rKnt+(−2)ntr2 divides (Ks−rKs+nt)n−r

[(−2)sKt−s − (−2)trKs+(n−1)t

]n.

2. If s = t, then 1− rKns + (−2)nsr2 divides (Ks − rK(n+1)s)n − (−2)nsr [2− rKns]

n.

3. If s > t, then 1− rKnt + (−2)ntr2 divides (Ks− rKs+nt)n− (−2)ntr

[Ks−t − rKs+(n−1)t

]n.

Corollary 3.27. Let r ∈ Z.

1. If s < t, then 1−rQnt+(−1)ntr2 divides (Ps−rPs+nt)n−r[(−1)s+1Pt−s − (−1)trPs+(n−1)t

]n.

2. If s = t, then 1− rQns + (−1)nsr2 divides (Ps − rP(n+1)s)n − (−1)nsr (−rPns)n.

3. If s > t, then 1− rQnt + (−1)ntr2 divides (Ps − rPs+nt)n − (−1)ntr[Ps−t − rPs+(n−1)t

]n.

Corollary 3.28. Let r ∈ Z.

1. If s < t, then 1−rQnt+(−1)ntr2 divides (Qs−rQs+nt)n−r

[(−1)sQt−s − (−1)trQs+(n−1)t

]n.

2. If s = t, then 1− rQns + (−1)nsr2 divides (Qs − rQ(n+1)s)n − (−1)nsr [2− rQns]

n.

3. If s > t, then 1− rQnt + (−1)ntr2 divides (Qs − rQs+nt)n − (−1)ntr

[Qs−t − rQs+(n−1)t

]n.

Theorem 3.29. Let a, b, p, q ∈ Z and r = 1.

1. If s < t, then 1− Vt + qt divides Ws −Ws+nt − qsUt−s + qtWs+(n−1)t.

2. If s = t, then 1− Vs + qs divides Ws −W(n+1)s − qs [a−Wns].

3. If s > t, then 1− Vt + qt divides Ws −Ws+nt − qt[Ws−t −Ws+(n−1)t

].

Proof. If a, b, p, q ∈ Z and r = 1, we obtain the sum identity from Theorem 3.15 where theWs+kt-s are all integers. Hence the sum is an integer, so the theorem immediately follows.

For the special cases, we have the following.

Corollary 3.30. Let r = 1.

1. If s < t, then 1− Lt + (−1)t divides Fs − Fs+nt + (−1)s+1Ft−s + (−1)tFs+(n−1)t.

2. If s = t, then 1− Ls + (−1)s divides Fs − F(n+1)s + (−1)sFns.

3. If s > t, then 1− Lt + (−1)t divides Fs − Fs+nt − (−1)t[Fs−t − Fs+(n−1)t

].

Corollary 3.31. Let r = 1.

1. If s < t, then 1− Lt + (−1)t divides Ls − Ls+nt + (−1)s+1Lt−s + (−1)tLs+(n−1)t.

2. If s = t, then 1− Ls + (−1)s divides Ls − L(n+1)s − (−1)s [2− Lns].

3. If s > t, then 1− Lt + (−1)t divides Ls − Ls+nt − (−1)t[Ls−t − Ls+(n−1)t

].

189

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Corollary 3.32. Let r = 1.

1. If s < t, then 1−Kt + (−2)t divides Js − Js+nt + (−2)sJt−s + (−2)tJs+(n−1)t.

2. If s = t, then 1−Ks + (−2)s divides Js − J(n+1)s + (−2)sJns.

3. If s > t, then 1−Kt + (−2)t divides Js − Js+nt − (−2)t[Js−t − Js+(n−1)t

].

Corollary 3.33. Let r = 1.

1. If s < t, then 1−Kt + (−2)t divides Ks −Ks+nt + (−2)sKt−s + (−2)tKs+(n−1)t.

2. If s = t, then 1−Ks + (−2)s divides Ks −K(n+1)s − (−2)s [2−Kns].

3. If s > t, then 1−Kt + (−2)t divides Ks −Ks+nt − (−2)t[Ks−t −Ks+(n−1)t

].

Corollary 3.34. Let r = 1.

1. If s < t, then 1−Qt + (−1)t divides Ps − Ps+nt + (−1)sUt−s + (−1)tPs+(n−1)t.

2. If s = t, then 1−Qs + (−1)s divides Ps − P(n+1)s + (−1)sPns.

3. If s > t, then 1−Qt + (−1)t divides Ps − Ps+nt − (−1)t[Ps−t − Ps+(n−1)t

].

Corollary 3.35. Let r = 1.

1. If s < t, then 1−Qt + (−1)t divides Qs −Qs+nt + (−1)s+1Qt−s + (−1)tQs+(n−1)t.

2. If s = t, then 1−Qs + (−1)s divides Qs −Q(n+1)s − (−1)s [2−Qns].

3. If s > t, then 1−Qt + (−1)t divides Qs −Qs+nt − (−1)t[Qs−t −Qs+(n−1)t

].

3.5 Norms of M(r, ~W )

For the Euclidean norm, we have the following results.

Theorem 3.36.∥∥∥M(r, ~W )∥∥∥2E

=(A+B)[nΩ1 − (1− |r|2)Ω2

]− AB

[nΩ3 − (1− |r|2)Ω4

]− 2AB

[nΩ5 − (1− |r|2)Ω6

] (30)

where

• Ω1 =

W2s −W2(s+nt) − q2sU2(t−s) + q2tW2(s+(n−1))t

1− V2t + q2t, s < t

W2s −W2(n+1)s − q2s [a−W2ns]

1− V2s + q2s, s = t

W2s −W2(s+nt) − q2t[W2(s−t) −W2(s+(n−1))t

]1− V2t + q2t

, s > t

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• Ω2 =γ1(s, t)− γ2(s, t)(1− V2t + q2t)2

• Ω3 =

V2s − V2(s+nt) − q2sV2(t−s) + q2tV2(s+(n−1))t

1− V2t + q2t, s < t

V2s − V2(n+1)s − q2s [2− V2ns]1− V2s + q2s

, s = t

V2s − V2(s+nt) − q2t[V2(s−t) − V2(s+(n−1))t

]1− V2t + q2t

, s > t

• Ω4 =σ1(s, t)− σ2(s, t)(1− V2t + q2t)2

• Ω5 =qs − qs+nt

1− qt

• Ω6 =qs+t − nqs+nt + (n− 1)qs+(n+1)t

(1− qt)2

• γ1(s, t) =

(W2s −W2(s+nt) − q2sU2(t−s) + q2tW2(s+(n−1)t))(V2t − 2q2t), s < t

(W2s −W2(n+1)s − q2s(a−W2ns))(V2s − 2q2s), s = t

(W2s −W2(s+nt) − q2t(W2(s−t) −W2(s+(n−1)t)))(V2t − 2q2t), s > t

• γ2(s, t) =

(q2sU2(t−s) + nW2(s+nt) − (n+ 1)q2tW2(s+(n−1)t))(1− V2t + q2t), s < t

(aq2s + nW2(n+1)s − (n+ 1)q2sWns)(1− V2s + q2s), s = t

(q2tW2(s−t) + nW2(s+nt) − (n+ 1)q2tW2(s+(n−1)t))(1− V2t + q2t), s > t

• σ1(s, t) =

(V2s − V2(s+nt) − q2sV2(t−s) + q2tV2(s+(n−1)t))(V2t − 2q2t), s < t

(V2s − V2(n+1)s − q2s(2− V2ns))(V2s − 2q2s), s = t

(V2s − V2(s+nt) − q2t(V2(s−t) − V2(s+(n−1)t)))(V2t − 2q2t), s > t

• σ2(s, t) =

(q2sV2(t−s) + nV2(s+nt) − (n+ 1)q2tV2(s+(n−1)t))(1− V2t + q2t), s < t

(2q2s + nV2(n+1)s − (n+ 1)q2sV2ns)(1− V2s + q2s), s = t

(q2tV2(s−t) + nV2(s+nt) − (n+ 1)q2tV2(s+(n−1)t))(1− V2t + q2t), s > t

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Proof.

n−1∑k=0

|ck|2[n− k(1− |r|2)

]=

n−1∑k=0

W 2s+kt

[n− k(1− |r|2)

]=

n−1∑k=0

[Aαs+kt +Bβs+kt

]2 [n− k(1− |r|2)

]=

n−1∑k=0

[A2α2(s+kt) + 2ABqs+kt +B2β2(s+kt)

] [n− k(1− |r|2)

]=

n−1∑k=0

[A2α2(s+kt) + (ABβ2(s+kt) − ABβ2(s+kt))

] [n− k(1− |r|2)

]+

n−1∑k=0

[B2β2(s+kt) + (ABα2(s+kt) − ABα2(s+kt))

] [n− k(1− |r|2)

]+ 2AB

n−1∑k=0

qs+kt[n− k(1− |r|2)

]=

n−1∑k=0

[A(Aα2(s+kt) +Bβ2(s+kt))

] [n− k(1− |r|2)

]+

n−1∑k=0

[B(Aα2(s+kt) +Bβ2(s+kt))

] [n− k(1− |r|2)

]−

n−1∑k=0

[AB(α2(s+kt) + β2(s+kt))

] [n− k(1− |r|2)

]+

n−1∑k=0

[2ABqs+kt

] [n− k(1− |r|2)

]

=n−1∑k=0

[(A+B)W2(s+kt) − ABV2(s+kt) + 2ABqs+kt

] [n− k(1− |r|2)

]= n

n−1∑k=0

[(A+B)W2(s+kt) − ABV2(s+kt) + 2ABqs+kt

]− (1− |r2|)

n−1∑k=0

[(A+B)kW2(s+kt) − ABkV2(s+kt) + 2ABkqs+kt

].

Using the sum identities that we have derived and the sum of finite geometric sequence, weobtain the following expression from the first summation:

n

n−1∑k=0

[(A+B)W2(s+kt) − ABV2(s+kt) + 2ABqs+kt

]= (A+B)nΩ1 − ABnΩ3 − 2ABnΩ5

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For the second summation, we use the following finite generating functions

Fs,t(x) =n−1∑k=0

Ws+ktxk =

Ws − qsUt−sx−Ws+ntxn + qtWs+(n−1)tx

n+1

1− Vtx+ qtx2, s < t

Ws − aqsx−W(n+1)sxn + qsWnsx

n+1

1− Vsx+ qsx2, s = t

Ws − qtWs−tx−Ws+ntxn + qtWs+(n−1)tx

n+1

1− Vtx+ qtx2, s > t

Gs,t(x) =n−1∑k=0

Vs+ktxk =

Vs − qsVt−sx− Vs+ntxn + qtVs+(n−1)txn+1

1− Vtx+ qtx2, s < t

Vs − 2qsx− V(n+1)sxn + qsVnsx

n+1

1− Vsx+ qsx2, s = t

Vs − qtVs−tx− Vs+ntxn + qtVs+(n−1)txn+1

1− Vtx+ qtx2, s > t

Hs,t(x) =n−1∑k=0

qs+ktxk =qs − qs+ntxn

1− qtx.

Consequently, we have

(1− |r2|)n−1∑k=0

[(A+B)kW2(s+kt) − ABkV2(s+kt) + 2ABkqs+kt

]= (1− |r2|)

[(A+B)F ′2s,2t(1)− ABG′2s,2t(1) + 2ABH ′s,t(1)

]= (A+B)(1− |r|2)Ω2 − AB(1− |r|2)Ω4 + 2AB(1− |r|2)Ω6.

Hence we have∥∥∥M(r, ~W )∥∥∥2E

= (A+B)[nΩ1 − (1− |r|2)Ω2

]− AB

[nΩ3 − (1− |r|2)Ω4

]− 2AB

[nΩ5 − (1− |r|2)Ω6

].

For r = ±1 we have the following result.

Corollary 3.37.∥∥∥M(1, ~W )∥∥∥2E

= n [(A+B)Ω1 − ABΩ3 − 2ABΩ5] =∥∥∥M(−1, ~W )

∥∥∥2E. (31)

For the spectral norm, we have the following results.

Theorem 3.38.∥∥∥M(r, ~W )∥∥∥2

= max0≤m≤n−1

√[δ1(r, s, t)]

2 − 2δ1(r, s, t)δ2(r, s, t)|r|1/n cos τm + [δ2(r, s, t)]2 |r|2/n

1− 2Vt|r|1/n cos τm (1 + qt|r|2/n) + |r|2/n (4qt cos2 τm + V2t + q2t|r|2/n)

(32)

where:

193

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• τm =θ + 2(j −m)π

n,

• δ1(r, s, t) =

Ws − rWs+nt, s 6= t

Ws − rW(n+1)s, s = t,

• δ2(r, s, t) =

qsUt−s − rqtWs+(n−1)t, s < t

qs [a− rWns] , s = t

qt[Ws−t − rWs+(n−1)t

], s > t

.

Proof. Let τm =θ + 2(j −m)π

n, δ1(r, s, t) =

Ws − rWs+nt, s 6= t

Ws − rW(n+1)s, s = tand

δ2(r, s, t) =

qsUt−s − rqtWs+(n−1)t, s < t

qs [a− rWns] , s = t

qt[Ws−t − rWs+(n−1)t

], s > t

.

Computing the spectral norm, we have∥∥∥M(r, ~W )∥∥∥2

= max0≤m≤n−1

∣∣∣∣ δ1(r, s, t)− δ2(r, s, t)ρω−m(1− αtρω−m)(1− βtρω−m)

∣∣∣∣ .Note that for any M,N ∈ R, we have the following equation.∣∣M −Nρω−m∣∣ =

∣∣∣∣M −N |r|1/n [cos

(θ + 2π(j −m)

n

)+ i sin

(θ + 2π(j −m)

n

)]∣∣∣∣=

∣∣M −N |r|1/n [cos τm + i sin τm]∣∣

=√M2 − 2MN |r|1/n cos τm +N2|r|2/n.

Applying the norm equation above, results to:∥∥∥M(r, ~W )∥∥∥2

= max0≤m≤n−1

√[δ1(r, s, t)]

2 − 2δ1(r, s, t)δ2(r, s, t)|r|1/n cos τm + [δ2(r, s, t)]2 |r|2/n

1− 2Vt|r|1/n cos τm (1 + qt|r|2/n) + |r|2/n (4qt cos2 τm + V2t + q2t|r|2/n).

For the cases r = ±1, we have the following results.

Corollary 3.39.∥∥∥M(1, ~W )∥∥∥2

= max0≤m≤n−1

√[δ1(1, s, t)]

2 − 2δ1(1, s, t)δ2(1, s, t) cos τm + [δ2(1, s, t)]2

1− 2Vt cos τm (1 + qt) + 4qt cos2 τm + V2t + q2t. (33)

Corollary 3.40.∥∥∥M(−1, ~W )∥∥∥2

= max0≤m≤n−1

√[δ1(−1, s, t)]2 − 2δ1(−1, s, t)δ2(−1, s, t) cos τm + [δ2(−1, s, t)]2

1− 2Vt cos τm (1 + qt) + 4qt cos2 τm + V2t + q2t.

(34)

194

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4 Conclusion and recommendations

We have obtained the explicit forms of the eigenvalues, determinants, Euclidean norms andspectral norms of r-circulant matrices whose entries are Horadam numbers having arithmeticindices. Special cases (Fibonacci, Lucas, Jaconsthal, Jacobsthal–Lucas, Pell and Pell–Lucas)were also obtained for the eigenvalues and determinants. Special cases where r = ±1 werealso obtained for the Euclidean norm and spectral norm. Furthermore, sum identities anddivisibility results were also obtained. Lastly, we have generalized the results that involveeigenvalues, determinants, Euclidean norms and spectral norm in Bozkurt [2–4], Bueno [7–14],Civciv and Turkmen [16], and Lind [19].

For future work, we recommend the following:

• determine the inverse of r-circulant matrices with Horadam numbers having arithmeticindices;

• determine the eigenvalues, determinants, Euclidean norms and spectral norms of the matrixgiven by

M [r, ~G] =

Gs Gs+t Gs+2t · · · Gs+(n−2)t Gs+(n−1)t

rGs+(n−1)t Gs Gs+t · · · Gs+(n−3)t Gs+(n−2)t

rGs+(n−2)t rGs+(n−1)t Gs · · · Gs+(n−4)t Gs+(n−3)t...

...... . . . ...

...rGs+2t rGs+3t rGs+4t · · · Gs Gs+t

rGs+t rGs+2t rGs+3t · · · rGs+(n−1)t Gs

, (35)

where Gj-s are Generalized Fibonacci numbers.

References

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