New results on generalized quaternion algebra involving

generalized Pell-Pell Lucas quaternions

RACHID CHAKER, ABDELKARIM BOUA

Polydisciplinary Faculty, B. P 1223, Taza

Department of Mathematics, LSI

Sidi Mohammed Ben Abdellah University, Fez,

MOROCCO

Abstract: This work presents a new sequence, generalized Pell-Pell-Lucas quaternions, we prove that the

set of these elements forms an order of generalized quaternions with 3-parameters kλ1,λ2,λ3as deﬁned by

ring theory. In addition, some properties of these elements are presented. The properties in this article

refer to kλ1,λ2,λ3algebras and sometimes to the 2-parameter algebra H(α, β).

Key-Words: Quaternion algebra, Generalized quaternion algebra, Pell-Lucas quaternions, Generalized

Pell-Pell-Lucas quaternions, Order, Centralizer.

Received: October 4, 2023. Revised: May 3, 2024. Accepted: June 26, 2024. Published: July 19, 2024.

1 Introduction

In 1830, the Irish mathematician Sir William

Rowan Hamilton began an exploration of com-

plex numbers with the intention of generaliza-

tion. After years of contemplation and extensive

research, he ﬁnally introduced real quaternions in

1843 as a solution to this long-standing problem,

[1], [2].

Then the researchers L. E. Dickson and L. W.

Griﬃths wrote two seminal articles on the subject

of generalized quaternions, [3], [4]. Recently, the

most general form of the quaternion algebra de-

pending on 3-parameters (3PGQ) was introduced

by [5], which prompted us to look for some prop-

erties associated with this algebra, which is called

kλ1,λ2,λ3in this article. For more information on

the properties of this algebra (see, [6], [7]).

Nowadays, quaternions hold signiﬁcant impor-

tance in various domains including computer sci-

ence, quantum physics, and signal and color im-

age processing, as evidenced by [8]. Furthermore,

numerous researchers have explored various types

of quaternion sequences such as Pseudo-Lucas

Quaternions, Balancing Split Quaternions, and

Pell-Lucas numbers, as documented in studies

referenced by [9], [10]. The authors in [11], stud-

ied the quaternions whose coeﬃcients were from

the generalized Fibonacci and Lucas sequences.

The authors in [12], studied the quaternions

whose coeﬃcients are Pell and PellLucas num-

bers. Liano and Wolch worked on Pell and Jacob-

sthal quaternions, [13]. Moreover, Catarino stud-

ied the Fibonacci quaternion polynomials and the

modiﬁed Pell quaternions, [14], and obtained the

norm values, generating functions, Binet formulas

and identities (similar to Cassini) of these poly-

nomials.

The Fibonacci, Lucas, Pell, and Pell-Lucas se-

quences are among the most popular and widely

used sequences in the mathematical community

because of their fascination and possible applica-

tions in other ﬁelds. Pell and Pell-Lucas num-

bers weave a common thread that spans analysis,

geometry, trigonometry, and various areas of dis-

crete mathematics, including disciplines such as

number theory and linear algebra.

The study, [15], was able to introduce and gener-

alize the Pell numbers and some interesting spe-

cial results related to them. The study, [16], in-

troduced split Pell and split Pell-Lucas quater-

nions and also considered some properties of these

sequences, including the Catalan identity, the

Cassini identity, and the Ducani identity. In this

work, our main goal is to generate new quater-

nion sequences from two important families: Pell

numbers and Pell-Lucas numbers, and to present

some of their properties, and we have also en-

riched the repertoire of kλ1,λ2,λ3quaternion alge-

bras with new properties.

The following is the organization of this paper:

Section 2 contains preliminary results for the al-

gebra of quaternions, the Pell numbers, and the

Pell-Lucas numbers. In Section 3, we detail the

results obtained for the properties of the Pell-Pell-

Lucas quaternion sequence generalized in the al-

gebra kλ1,λ2,λ3, and we strengthen these results

by giving an order and a center of the algebra of

quaternions Finally, Section 4 presents a conclu-

sion and perspectives of research of this work in

the practical area.

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2 Deﬁnitions and Notations

In this section we state the deﬁnitions and

the main results concerning the quaternions

algebra has been introduce depending on 3-

parameters (3PGQs), Generalized Quaternions

(2-Parameter), the Pell and Pell-Lucas Numbers.

2.1 3-Parameter Generalized

Quaternions

In the following, we’ll go over some key concepts

and notations that will help us understand and

expand on this topic. From [5], we deﬁne the

set of generalized quaternions with 3-parameters

(3PGQ) as follows:

Deﬁnition 2.1 The set kλ1,λ2,λ3deﬁned by:

{a+be1+ce2+de3|a, b, c, d, λ1, λ2, λ3∈R,}

where e2

1=−λ1λ2,e2

2=−λ1λ3,e2

3=−λ2λ3,

e1e2e3=−λ1λ2λ3is called the set of generalized

quaternions with 3-parameters (3PGQ).

Each element p=x0+x1e1+x2e2+x3e3of the

set kλ1,λ2,λ3is called a 3-parameter generalized

quaternion (3PGQ). The numbers x0, x1, x2, x3

are called components of p. The basis vectors

e0, e1, e2, e3of the kλ1,λ2,λ3satisfy the following

multiplication table:

Table 1. Multiplication Table

. e0e1e2e3

e01e1e2e3

e1e1−λ1λ2λ1e3−λ2e2

e2e2−λ1e3−λ1λ3λ3e1

e3e3λ2e2−λ3e1−λ2λ3

Special cases:

(i) If λ1= 1, λ2=α,λ3=β, then we get the

algebra of 2PGQs.

(ii) If λ1= 1, λ2= 1, λ3= 1,then we get the

algebra of Hamilton quaternions.

(iii) If λ1= 1, λ2= 1, λ3=−1,then gives us

the algebra of split quaternions.

(iv) If λ1= 1, λ2=−1, λ3= 0,then we get the

algebra of split semiquaternions.

Any 3P GQ p =x0+x1e1+x2e2+x3e3consists

of two parts, the vector part and the scalar part:

p=S(p) + V(p) such as:

S(p) = x0and V(p) = x1e1+x2e2+x3e3

The rules of addition, scalar multiplication and

multiplication are deﬁned on kas follows:

Let p=x0+x1e1+x2e2+x3e3and q=y0+y1e1+

y2e2+y3e3be 3PGQs and αbe a real number.

•Addition:

p+q= (S(p) + S(q)) + (V(p) + V(q))

= (x0+y0)+(x1+y1)e1+

(x2+y2)e2+ (x3+y3)e3.

•Multiplication by scalar:

αp =αx0+αx1e1+αx2e2+αx3e3

for all α∈R.

•Multiplication: from the multiplication table,

Table 1, we have

pq = (x0y0−λ1λ2x1y1−λ1λ3x2y2−

λ2λ3x3y3) + e1(x0y1+y0x1+λ3x2y3

−λ3x3y2) + e2(x0y2+y0x2+λ2x3y1

−λ2x1y3) + e3(x0y3+y0x3+λ1x1y2

−λ1x2y1)

•The norm of a quaternion pis:

N(p) = x2

0+λ1λ2x2

1+λ1λ3x2

2+λ2λ3x2

Deﬁnition 2.2 A subring O⊆kλ1,λ2,λ3is an

order in kλ1,λ2,λ3if Ois a Z-lattice of kλ1,λ2,λ3.

That is, Ois a ﬁnitely generated Z-submodule of

kλ1,λ2,λ3(which is also a subring of kλ1,λ2,λ3by

[17]).

2.2 2-Parameter Generalized

Quaternions

Let H(α, β) be the generalized real quaternion

algebra, the elements of H(α, β) are written in

the form p=x0+x1e1+x2e2+x3e3,where xi∈R,

1=α,e2

2=β,e3=e1e2=−e2e1.The following

expressions represent the norm and the trace of

a generalized quaternion p:

N(p) = x2

0−αx2

1−βx2

2+αβx2

3and t(a)=2x0.

As is well known, we have

p2−t(p)p+N(p) = 0 for all p∈H(α, β).

Deﬁnition 2.3 The quaternion algebra Ais said

to be a division algebra if for all p∈A∗,N(p)6=

0, otherwise Ais called a split algebra.

Deﬁnition 2.4 For a∈HQ(α, β).The central-

izer of the element ais

C(a) = {x∈HQ(α, β)|ax =xa}.

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2.3 Properties of the Pell and

Pell-Lucas Numbers

•Let (Pn)n≥0be a sequence of Pell numbers:

Pn= 2Pn−1+Pn−2where n≥2 (1)

with P0= 0, P1= 1.

•Let (Qn)n≥0be a sequence of Pell-Lucas

numbers:

Qn= 2Qn−1+Qn−2where n≥2 (2)

with Q0= 2, Q1= 2.

•The Binets formulas of the nth Pell and Pell-

Lucas numbers are:

Pn=γn−δn

γ−δfor all n∈N,

Qn=γn+δnfor all n∈N

with γ= 1 + √2, δ = 1 −√2.(3)

The following properties of Pell and Pell-Lucas

numbers are known from [18], [19], [20].

Proposition 2.5 Let (Pn)n≥0be a sequence of

Pell numbers and (Qn)n≥0be the Pell-Lucas se-

quence. Then the following properties hold:

1)- P2

n+P2

n+1 =P2n+1 ∀n∈N,

2)- Q2

n+Q2

n+1 = 8P2n+1 ∀n∈N,

3)- P2

n+1 −P2

n=Q2n+1 + 2(−1)n

4∀n∈N,

4)- Q2

n+1 −Q2

n= 8P2n+1 −4(−1)n∀n∈N,

5)- P2

n=Q2n+ 2(−1)n+1

8∀n∈N,

6)- Q2

n=Q2n+ 2(−1)n∀n∈N,

7)- Pn+1Pn=Q2n+1 −2(−1)n

8∀n∈N,

8)- QnQn+1 −Q2n+1 = 2(−1)n∀n∈N,

9)- P2n+1 =PnQn+1 + (−1)n∀n∈N,

10)- PnQm=Pn+m+ (−1)mPn−m∀n, m ∈Z,

11)- PnPn+k=1

8(Q2n+k+ (−1)n+1Qk)∀n, k ∈

12)- Qn+2 +Qn−2= 6Qn∀n≥2,

13)- Pn+1 +Pn−1=Qn∀n∈N∗.

We will introduce some other properties of Pell

and Pell-Lucas numbers. These properties will

be useful later.

Proposition 2.6 Let (Pn)n≥0be the Pell se-

quence and (Qn)n≥0be the Pell-Lucas sequence,

then we have

QnQn+k=Q2n+k+ (−1)nQk∀n, k ∈N.(4)

Proof. If we denote γ= 1 + √2 and δ= 1 −√2,

by Binets formula, we have

Pn=γn−δn

γ−δfor all n∈Nand

Qn=γn+δnfor all n∈N

i) Let m, p ∈R,p6m, thus

QmQp−8PmPp= (γm+δm)(γp+δp)

−(γm−δm)(γp−δp)

= 2γpδp(γm−p+δm−p)

= 2(−1)pQm−p.

It results QmQp−8PmPp= 2(−1)pQm−p. So,

QmQp= 8PmPp+ 2(−1)pQm−p.We use the

Proposition 2.5 (11), we obtain

QmQp=Qm+p+ (−1)pQm−p∀n∈N, p 6m.

From this it follows that

QnQn+k=Q2n+k+ (−1)nQkfor all n, k ∈N.

3 Generalized Pell-Pell Lucas

numbers and generalized

Pell-Pell Lucas Quaternions

Let r,tbe two arbitrary integers, and let nbe

an arbitrary positive integer. The numbers in a

sequence (gn)n>1,where

gn+1 =rPn+t Qn+1 where n>0

are called generalized Pell-Pell-Lucas numbers.

To emphasize the presence of the integers rand

t, we will use gr,t

ninstead of the notation gn. Let

λ1,λ2,λ3be the generalized quaternion algebra

over the rational ﬁeld. We deﬁne the nth general-

ized Pell-Pell-Lucas quaternion to be an element

of the form

Gr,t

n=gr,t

ne0+gr,t

n+1e1+gr,t

n+2e2+gr,t

n+3e3

In the following proposition, we compute the

norm for the nth generalized Pell-Pell-Lucas

quaternions.

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Proposition 3.1 Let n,rbe two positive inte-

gers and tbe an arbitrary integer. Let Gr,t

nbe

the nth generalized Pell-Pell-Lucas quaternion.

Then the norm of Gr,t

nin the quaternion algebra

λ1,λ2,λ3is given by:

N(Gr,t

n) =g(2rt, 6+λ1λ2

8r2+t2)

+g(2rλ1λ2t, −1

8r2+λ1λ3

8+λ1λ2t2)

2n+2

+g(2rλ1λ3t,λ1λ3t2+λ2λ3

8r2)

2n+4 +g(2rλ2λ3t,λ2λ3t2)

2n+6

+g((1−λ1λ2+λ1λ3−λ2λ3)( r2

4+2t2+2rt)(−1)n,0)

Proof. We have

N(Gr,t

n) = g2

n+λ1λ2g2

n+1 +λ1λ3g2

n+2 +λ2λ3g2

n+2

= (rPn−1+tQn)2+λ1λ2(rPn+t Qn+1)2

+λ1λ3(rPn+1 +tQn+2)2+λ2λ3(rPn+2

+tQn+3)2

=r2P2

n−1+t2Q2

n+ 2rtPn−1Qn+

λ1λ2(r2P2

n+t2Q2

n+1 + 2rtPnQn+1)

+λ1λ3(r2P2

n+1 +t2Q2

n+2 + 2rtPn+1Qn+2)

+λ2λ3(r2P2

n+2 +t2Q2

n+3 + 2rtPn+2Qn+3)

=r2P2

n−1+λ1λ2r2P2

n+λ2λ3r2P2

n+2

+λ1λ3r2P2

n+1 +t2Q2

n+λ1λ2t2Q2

n+1

+λ1λ3t2Q2

n+2 +λ2λ3t2Q2

n+3 + 2rtPn−1Qn

+ 2rtλ1λ2PnQn+1 + 2rtλ1λ3Pn+1Qn+2

+ 2rtλ2λ3Pn+2Qn+3

Using Proposition 2.5(5-6-10-12), we obtain

N(Gr,t

n) = r2

8(6Q2n−Q2n+2 + 2(−1)n)

+λ1λ2r2

8(Q2n+ 2(−1)n+1)

+λ1λ3r2

8(Q2n+2 + 2(−1)n+2)

+λ2λ3r2

8(Q2n+4 + 2(−1)n+3)

+t2(Q2n+ 2(−1)n)

+λ1λ3t2(Q2n+4 + 2(−1)n+2)

+λ2λ3t2(Q2n+6 + 2(−1)n+3)

+ 2rt(P2n−1+ (−1)n)+

+λ1λ22rt(P2n+1 + (−1)n+1)

+λ1λ32rt(P2n+3 + (−1)n+2)

+λ2λ32rt(P2n+5 + (−1)n+3)

= 2rtP2n−1+λ1λ22rtP2n+1

+λ1λ32rtP2n+3 +λ2λ32rtP2n+5

+ (6r2

8+λ1λ2r2

8+t2)Q2n

+ (−r2

8+λ1λ3

8+λ1λ2t2)Q2n+2

+ (λ2λ3r2

8+λ1λ3t2)Q2n+4

+λ2λ3t2Q2n+6

+r2

4(−1)n+λ1λ2r2

4(−1)n+1

+λ1λ3r2

4(−1)n+2 +λ2λ3r2

4(−1)n+3

+ 2t2(−1)n+ 2t2λ1λ2(−1)n+1

+ 2t2λ1λ3(−1)n+2 + 2t2λ2λ3(−1)n+3

+ 2rt(−1)n+ 2rtλ1λ2(−1)n+1

+ 2rtλ1λ3(−1)n+2 + 2rtλ2λ3(−1)n+3

=g(2rλ1λ2t, −1

8r2+λ1λ3

8+λ1λ2t2)

2n+2

+g(2rλ1λ3t,λ1λ3t2+λ2λ3

8r2)

2n+4 +g(2rλ2λ3t,λ2λ3t2)

2n+6

+r2

4(−1)n+λ1λ2r2

4(−1)n+1 + 2t2(−1)n

+λ1λ3r2

4(−1)n+λ2λ3r2

4(−1)n+1

+ 2t2λ1λ2(−1)n+1 + 2t2λ1λ3(−1)n

+ 2t2λ2λ3(−1)n+1 + 2rt(−1)n

+ 2rtλ1λ2(−1)n+1 + 2rtλ1λ3(−1)n

+ 2rtλ2λ3(−1)n+1

=g(2rt, 6

8r2+λ1λ2

8r2+t2)

+g(2rλ1λ2t, −1

8r2+λ1λ3

8+λ1λ2t2)

2n+2

+g(2rλ1λ3t,λ1λ3t2+λ2λ3

8r2)

2n+4

+g(2rλ2λ3t,λ2λ3t2)

2n+6

+r2

4(−1)n(1 −λ1λ2+λ1λ3−λ2λ3)

+ 2t2(−1)n(1 −λ1λ2+λ1λ3−λ2λ3)

+ 2rt(−1)n(1 −λ1λ2+λ1λ3−λ2λ3)

=g(2rt, 6+λ1λ2

8r2+t2)

2n+g(2rλ1λ2t, −1

8r2+λ1λ3

8+λ1λ2t2)

2n+2

+g(2rλ1λ3t,λ1λ3t2+λ2λ3

8r2)

2n+4 +g(2rλ2λ3t,λ2λ3t2)

2n+6

+ (1 −λ1λ2+λ1λ3−λ2λ3)(−1)n(r2

4+ 2t2+ 2rt)

so,

N(Gr,t

n) = g(2rt, 6+λ1λ2

8r2+t2)

+g(2rλ1λ2t, −1

8r2+λ1λ3

8+λ1λ2t2)

2n+2

+g(2rλ1λ3t,λ1λ3t2+λ2λ3

8r2)

2n+4

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+g(2rλ2λ3t,λ2λ3t2)

2n+6

+g((1−λ1λ2+λ1λ3−λ2λ3)( r2

4+2t2+2rt)(−1)n,0)

Using the generalized Pell-Pell-Lucas quater-

nions, we can construct an order of quaternion al-

gebras, and we show that Pell-Pell-Lucas quater-

nions can also have an algebraic structure over Q.

The following remarks will help us.

Remark 3.1 Let r,tbe two arbitrary integers

and nbe an arbitrary positive integer. Let

(gr,t

n)n≥1be the generalized Pell-Pell-Lucas num-

bers. Then rPn+1 +tQn=gr,t

n+g2r,0

n+1 for all

n∈N∗.

Proof.

rPn+1 +tQn=r(2Pn+Pn−1+tQn)

=rPn−1+tQn+ 2rPn

=gr,t

n+g2r,0

n+1.

Remark 3.2 Let r,tbe two arbitrary inte-

gers and nbe an arbitrary positive integer.

Let (Gr,t

n)n>1be the generalized Pell-Pell Lucas

quaternion elements. Then Gr,t

n= 0 if and only

r=t= 0.

Proof. ⇐).It is trivial.

⇒).If Gr,t

n= 0,since {e0, e1, e2, e3,}is a basis in

kλ1,λ2,λ3,we obtain that

gr,t

n=gr,t

n+1 =gr,t

n+2 =gr,t

n+3 = 0

It results

gr,t

n−1=gr,t

n+1 −gr,t

n= 0, . . . , gr,t

2= 0, gr,t

1= 0,

therefore, t= 0. From gr,t

2= 0,we obtain r= 0.

Theorem 3.2 Let Mbe the set

i=1

8Gri,ti

ni|n∈N∗, ri, ti∈Z∀i= 1 . . . n)∪{1}.

Then

(i) The set Mwith addition and multiplication

of quaternions has a ring structure.

(ii) The set Mis an order of the quaternion al-

gebra kλ1,λ2,λ3.

(iii) n

i=1

8Gr0

i,t0

ni|n∈N∗, r0

i, t0

i∈Q∀i= 1 . . . n∪

{1}is a Q-algebra.

Proof. (i) Obviously it is.

(ii) Using Remark 3.2, we ﬁrst note that 0 ∈

M. We now show that Mis a Z-submodule of

kλ1,λ2,λ3.

Let n, m ∈N∗, a, b, r, t, r0, t0∈Z.It is easy to

prove that

agr,t

n+bgr0,t0

m=gar,at

n+gbr0,bt0

This implies that aGr,t

n+bGr0,t0

m=Gar,at

n+Gbr0,bt0

It is clear that Mis a Z-submodule of the quater-

nion algebra kλ1,λ2,λ3. Since this submodule basis

is {e0, e1, e2, e3}, M is a free Z-module of rank 4.

Now we prove that Mis a subring of kλ1,λ2,λ3. It

is suﬃcient to show that 8Gr,t

n.8Gr0,t0

m∈M. For

this, if m < n, we compute

8gr,t

n.8gr0,t0

m= 8(rPn−1+tQn).8(r0Pm−1+t0Qm)

= 64rr0Pn−1Pm−1+ 64rt0Pn−1Qm

+ 64tr0PnQm−1+ 64tt0QnQm(5)

Using Proposition 2.5 (10, 11), Proposition 2.6,

Remark 3.1 and the equality (5), we obtain:

8gr,t

n.8gr0,t0

m= 8rr0(Qn+m−2+ (−1)mQn−m)

+ 64rt0(Pm+n−1+ (−1)mPn−m−1)

+ 64tr0(Pm+n−1+ (−1)mPn−m+1)

+ 64tt0(Qn+m+ (−1)mQn−m)

= 8(rr0Qn+m−2+ 8r0tPn+m−1)

+ 8(8r0t(−1)mPn−m+1 +rr0(−1)mQn−m)

+ 64(rt0)Pn+m−1+tt0Qn+m)

+ 64rt0(−1)mPn−m−1+tt0(−1)mQn−m

= 8g8r0t,rr0

m+n−2+ 8g16r0t,0

m+n−1

+g8r0t(−1)m,rr0(−1)m

n−m+ 8g8r0t(−1)m,0

n−m+1

+ 8g8rt0,8tt0

n+m+ 8g8rt0(−1)m,8tt0(−1)m

n−m

Therefore, 8Gr,t

n.8Gr0,t0

m∈M. Consequently, Mis

an order of the quaternion algebra kλ1,λ2,λ3.

(iii) is obvious.

It is known that if ris an odd prime positive

integer, the algebra HQ(−1, r) is a split algebra if

and only if r≡1(mod4) (see, [21], [22]). In the

following, we will show that this algebra contains

an inﬁnite number of invertible generalized Pell-

Pell-Lucas quaternion elements. In this part we

replace (λ1, λ2, λ3) by (1,1,−r), i.e. we focus on

1,1,−r=HQ(−1, r).

Proposition 3.3 Let tbe any integer and n,r

be two positive integers. Let Gr,t

nbe the nth gen-

eralized Pell-Pell Lucas quaternion. The norm of

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Gr,t

nin the quaternion algebra kQ

1,1,−rhas the form

(i)N(Gr,t

n) = (r2+2rt+r3+14r2t+48rt2)P2n−1

+ (8t2−6r3+ 2rt −72r2t−240rt2)P2n+1.(6)

(ii)N(Gr,t

n) = gu,v

2n, where

u=r2+ 7r3+ 86r2t+ 288rt2

v= 8t2−6r3+ 2rt −72r2t−240rt2.(7)

Proof. (i)

N(Gr,t

n) =g2

n+g2

n+1 −rg2

n+2 −rg2

n+2

=(rPn−1+tQn)2+ (rPn+tQn+1)2−

r(rPn+1 +tQn+2)2−r(rPn+2 +tQn+3)2

=r2P2

n−1+t2Q2

n+ 2rtPn−1Qn

+(r2P2

n+t2Q2

n+1 + 2rtPnQn+1)

−r(r2P2

n+1 +t2Q2

n+2 + 2rtPn+1Qn+2)

−r(r2P2

n+2 +t2Q2

n+3 + 2rtPn+2Qn+3)

=r2P2

n−1+r2P2

n−r3P2

n+1 −r3P2

n+2

+t2Q2

n+t2Q2

n+1 −rt2Q2

n+2 −rt2Q2

n+3

+2rtPn−1Qn+ 2rtPnQn+1

−2r2tPn+1Qn+2 −2r2tPn+2Qn+3

=r2(P2

n−1+P2

n)−r3(P2

n+1 +P2

n+2)

+t2(Q2

n+Q2

n+1)+2rt(Pn−1Qn+PnQn+1)

−2r2t(Pn+1Qn+2 +Pn+2Qn+3)

−rt2(Q2

n+2 +Q2

n+3)

Using Proposition 2.5, we obtain:

N(Gr,t

n) = r2P2n−1+ 8t2P2n+1

+ 2rt(P2n−1+ (−1)n+P2n+1 + (−1)n+1)

−r3P2n+3 −8rt2P2n+5

−2r2t(P2n+3 + (−1)n+2 +P2n+5 + (−1)n+3)

= (r2+ 2rt)P2n−1+ (8t2+ 2rt)P2n+1

+ (−r3−2r2t)P2n+3 + (−8rt2−2r2t)P2n+5

Using Pell recurrence, we obtain:

P2n+3 =6P2n+1 −P2n−1and

P2n+5 =30P2n+1 −6P2n−1

Thus, we conclude that

N(Gr,t

n) =(r2+ 2rt +r3+ 14r2t+ 48rt2)P2n−1

+(8t2−6r3+ 2rt −72r2t−240rt2)P2n+1.

(ii) according to i) we have:

N(Gr,t

n) =(r2+ 2rt +r3+ 14r2t+ 48rt2)P2n−1

+(8t2−6r3+ 2rt −72r2t−240rt2)P2n+1.

Using Proposition 2.5 (13), we obtain :

N(Gr,t

n) =(r2+ 7r3+ 86r2t+ 288rt2)P2n−1

+(8t2−6r3+ 2rt −72r2t−240rt2)Q2n

=uP2n−1+vQ2n

=gu,v

2n,

where

u=r2+ 7r3+ 86r2t+ 288rt2and

v=8t2−6r3+ 2rt −72r2t−240rt2.

Proposition 3.4 Let nbe an arbitrary positive

integer. Let (Pn)n≥0be the Pell sequence and

(Qn)n≥0be the Pell Lucas sequence. Let rbe

an odd prime positive integer, r≡1(mod4),t

be an arbitrary integer. Let Gr,t

nbe the nth gen-

eralized Pell-Pell Lucas quaternion and k1,1,−rbe

the quaternion algebra. Then,

N(Gr,t

n)6= 0 for all (n, t)∈N∗×N.

Proof. From Proposition 3.3, we know that

N(Gr,t

n) =(r2+ 2rt +r3+ 14r2t+ 48rt2)P2n−1

+(8t2−6r3+ 2rt −72r2t−240rt2)P2n+1.

Since r∈N∗, it follows that

r2+ 2rt +r3+ 14r2t+ 48rt2<

−8t2+ 6r3−2rt + 72r2t+ 240rt2.

Using the inequality P2n−1< P2n+1, we ob-

tain that N(Gr,t

n)<0 so N(Gr,t

n)6= 0.From [23,

Proposition 2.13], We are aware that the equation

ax =bx, a, b ∈HK(α, β),(8)

where Kis an arbitrary ﬁeld of char(K)6= 0,

a, b /∈K,a6=¯

b, has the solutions of the form

x=λ[a−a0+b−b0]+µ[N(a−a0)−(a−a0)(b−b0)],

(9)

where, λ, µ ∈K.

If HK(α, β) is a division quaternion algebra or

if HK(α, β)) is a split quaternion algebra and

N(a)6= 0, N(b)6= 0.

Proposition 3.5 Let nbe a positive integer. Let

(Pn)n≥0be the Pell sequence and (Qn)n≥0be the

Pell-Lucas sequence. Let rbe an odd prime pos-

itive integer, r≡1(mod4),tbe an arbitrary in-

teger. Therefore, the centralizer of the element

Gr,t

n∈HK(−1, r)is the set

C(Gr,t

n) = {Gε,σ

n+χ, χ ∈Q},

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where

ε=2λr,

σ=2λt

χ=g−2λr,−2λt

n+g2µ(u−r2−2rt),2µ(v+r2

8−t2)

2n+ 2µϕ,

with λ, µ ∈Qand ϕ= 2(−1)n+1(r2

8+t2+rt).

Proof. Since

C(Gr,t

n) = {x∈HQ(−1, r)|xGr,t

n=Gr,t

nx},

using relations (8) and (9) for a=b, we obtain

that the equation xGr,t

n=Gr,t

nxhas the solutions

of the form

x= 2λ[a−a0]+2µ[N(a−a0)], λ, µ ∈Q

x= 2λ[Gr,t

n−gr,t

n]+2µ[N(Gr,t

n−gr,t

n)].(10)

For Gr,t

n=gr,t

ne0+gr,t

n+1e1+gr,t

n+2e2+gr,t

n+3e3,we

have N(Gr,t

n) = gu,v

2nwith uand vas in Proposi-

tion 3.3. From here, we have that

N(Gr,t

n−gr,t

n)) = gu,v

2n−(gr,t

n)2

=gu,v

2n−(rPn−1+tQn)2.

Using Proposition 2.5, relations 6), 5) and 10), it

results

N(Gr,t

n−gr,t

n) = gu,v

2n−(rPn−1+tQn)2

=gu,v

2n−r2P2

n−1−t2Q2

−2rtPn−1Qn.

We have

n=Q2n+ 2(−1)n,

n−1=1

8Q2n−2+ 2(−1)n),

Pn−1Qn=P2n−1+ (−1)n,

N(Gr,t

n−gr,t

n)) =gu,v

2n−r2(1

8(Q2n−2+ 2(−1)n))

−t2(Q2n+ 2(−1)n)

−2rt(P2n−1+ (−1)n)

=gu,v

2n−r2

8Q2n−2−t2Q2n

−2rtP2n−1−2r2

8(−1)n

−2t2(−1)n−2rt(−1)n,

we have Pn=1

8(Qn+1 +Qn−1) so Q2n−2=

8P2n−1−Q2n,we obtain:

N(Gr,t

n−gr,t

n) =gu,v

2n−r2

8(8P2n−1−Q2n)−t2Q2n

−2rtP2n−1+ 2(−1)n+1(r2

8+t2+rt)

=gu,v

2n−r2P2n−1+r2

8Q2n−t2Q2n

−2rtP2n−1+ 2(−1)n+1(r2

8+t2+rt)

=gu,v

2n+ (−r2−2rt)P2n−1+ (r2

8−t2)Q2n

+2(−1)n+1(r2

8+t2+rt)

=gu,v

2n+g−r2−2rt, r2

8−t2

+2(−1)n+1(r2

8+t2+rt)

=gu−r2−2rt,v+r2

8−t2

2n+ϕ,

where ϕ= 2(−1)n+1(r2

8+t2+rt).

Using relation (10), we obtain

x= 2λ[Gr,t

n−gr,t

n]+2µ[gu−r2−2rt,v+r2

8−t2

2n+ϕ]

= 2λ[Gr,t

n−gr,t

n]+2µ[gu−r2−2rt,v+r2

8−t2

2n+ϕ]

=G2λr,2λt

n+g−2λr,−2λt

+g2µ(u−r2−2rt),2µ(v+r2

8−t2)

2n+ 2µϕ.

4 Conclusions

In this study we introduced a special set of el-

ements, called Pell and Pell-Lucas quaternions,

and showed that this set is an order of the quater-

nion algebra kλ1,λ2,λ3in the sense of ring theory.

The determination of all properties of this alge-

bra, as well as the circumstances under which

it is a split algebra or a division algebra, will

be highly intriguing, especially in the practical

ﬁeld. More precisely, the applications of Bell-

Lucas quaternions are not as extensive as the

applications of more familiar mathematical con-

cepts such as complex numbers or quaternions

themselves. Therefore, we will try to focus fu-

ture work on studying the applied aspect of Bell-

Lucas quaternions by addressing the following fu-

ture works:

1)- Cryptographic algorithms,

2)- Image processing.

Acknowledgment:

The authors thank the reviewer for valuable

suggestions and comments.

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self:

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Conﬂict of Interest:

There is no conﬂict of interest.

Contribution of individual authors:

The authors contributed equally to this work.

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