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# On third-order h-Jacobsthal and third-order -Jacobsthal–Lucas sequences, and related quaternions

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## quaternions

### Gamaliel Cerda-Morales

Abstract. In this paper, inspired by recent articles of G. Anatriello and G. Vincenzi (see [1]), and G. Cerda-Morales (see [3, 4]), we will intro-duce the third-orderh-Jacobsthal and third-order h-Jacobsthal–Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in [4].

Mathematics Subject Classification (2010).Primary: 11B39; Secondary: 11R52.

Keywords.recurrence relations, quaternions, third-order Jacobsthal num-bers, third-order Jacobsthal–Lucas numbers.

### 1. Introduction

The quaternions are objects introduced by Hamilton order to attempt a defi-nition of coordinate system different from cartesian one. Hamilton idea’s had a great success by his contemporary scientist, and in particular it had a strong application in physics. Actually, after a period of shadow, they have been re-cently considered in different branches of mathematics (see for example [10] and reference therein), and many researches are devoted to them.

The set of real quaternions is denoted byHand a quaternion number appears in the formq=q0+q1i+q2j+q3k, whereql∈R(l= 0,1,2,3). The basic rules is given by

i2=j2=k2=ijk=−1. (1.1)

For an introduction to quaternions theory see [11] (see also the intro-duction of [4] for basic rules).

(2)

In 1963, A. F. Horadam [9] had the idea to investigate special quater-nion numbers. Precisely, the Fibonacci quaterquater-nionsQn, that are quaternions

of the form Qn = Fn+iFn+1+jFn+2+kFn+3, where Fn is the n-th

Fi-bonacci number. Following him other authors had investigated other kind of quaternions, and many interesting properties of Fibonacci and generalized Fibonacci quaternions had ben obtained (see [7, 8] and reference therein).

Recently, in [3] and [4] the author starting from third-order Jacobsthal sequence{Jn(3)}n≥0 and third-order Jacobsthal–Lucas sequence{j

(3)

n }n≥0 (

J0(3)= 0, J1(3)=J2(3) = 1

Jn(3)=Jn(3)−1+J (3)

n−2+ 2J (3)

n−3, (

j(3)0 = 2, j1(3)= 1, j2(3)= 5

j(3)n =j(3)n−1+j (3)

n−2+ 2j (3)

n−3,

(1.2)

considered the third-order Jacobsthal quaternions and third-order Jacobsthal– Lucas quaternions, obtaining properties and matrix representation of such numbers.

As the author had highlighted in [3], the numbers of Tribonacci type have many applications in distinct area of mathematics (see also [2] and ref-erence therein). Therefore, it seems a natural question to investigate quater-nions connected to either a Tribonacci-like sequence or more generally, to a whichever recursive sequence of third order; but it seems not easy to develop a general theory for quaternions connected to a whichever sequence too much different from Tribonacci sequence.

In this paper, in order to attempt to this kind of investigation, we re-strict our attention to quaternions connected to generalizations of third-order Jacobsthal sequences and third-order Jacobsthal–Lucas sequence (see Eq. (1.4)).

Let h be a complex number, we will define third-order h-Jacobsthal sequence{Jh,n(3)}n≥0 and third-orderh-Jacobsthal–Lucas sequence{j

(3)

h,n}n≥0 the homogenous recursive sequences of third order with constant coefficients, whose characteristic polynomial is

x3−(h−1)x2−(h−1)x−h= (x−h)(x2+x+ 1), (1.3)

and initial conditions are 0, 1, 1 and 2, 1, 5 respectively:

(

Jh,(3)0= 0, Jh,(3)1= 1, Jh,(3)2=h−1

Jh,n(3) = (h−1)Jh,n(3)1+ (h−1)Jh,n(3)2+hJh,n(3)3,

(

j(3)h,0= 2, jh,(3)1=h−1, jh,(3)2=h2+ 1

j(3)h,n= (h−1)jh,n(3)1+ (h−1)jh,n(3)2+hjh,n(3)3.

(1.4)

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to third-orderh-Jacobsthal quaternions and third-orderh-Jacobsthal–Lucas quaternions.

### 2. Basic Properties

In [6], the authors provided many basic identities for third-order Jacobsthal numbers, {Jn(3)}n≥0, and third-order Jacobsthal–Lucas numbers, {j

(3)

n }n≥0 (see also [5] and reference therein):

3Jn(3)+jn(3) = 2n+1, (2.1)

jn(3)−3Jn(3)= 2jn(3)3, (2.2)

Jn(3)+2−4Jn(3)= (

−2 if n≡1 (mod 3)

1 if n6≡1 (mod 3) , (2.3)

jn(3)−4Jn(3) =

  

 

2 if n≡0 (mod 3) −3 if n≡1 (mod 3) 1 if n≡2 (mod 3)

, (2.4)

jn(3)+1+jn(3)= 3Jn(3)+2, (2.5)

jn(3)−Jn(3)+2=   

 

1 if n≡0 (mod 3) −1 if n≡1 (mod 3) 0 if n≡2 (mod 3)

, (2.6)

jn(3)3

2

+ 3Jn(3)jn(3)= 4n, (2.7)

n

X

k=0

Jk(3)= (

Jn(3)+1 if n6≡0 (mod 3)

Jn(3)+1−1 if n≡0 (mod 3) , (2.8)

n

X

k=0

jk(3)= (

jn(3)+1−2 if n6≡0 (mod 3)

jn(3)+1+ 1 if n≡0 (mod 3) (2.9) and

jn(3)

2

−9Jn(3)

2

= 2n+2jn(3)3. (2.10) Using standard techniques for solving recurrence relations, the auxiliary equation, and its roots are given by

x3−x2−x−2 = 0; x= 2, andx= −1±i √

3

2 .

Note that the latter two are the complex conjugate cube roots of unity. Call them ω1 andω2, respectively. Thus the Binet formulas can be written as

Jn(3) =2 7 ·2

n 3 + 2i

√ 3 21

!

ω1n− 3−2i √

3 21

!

(4)

and

jn(3)= 8 7·2

n+ 3 + 2i

√ 3 7

!

ω1n+ 3−2i √

3 7

!

ω2n, (2.12)

respectively.

Remark 2.1. We note that when h=ω1 or h=ω2, then the characteristic polynomial has the unique rootsω1andω2(with multiplicity 2 one of them). In this case the sequences{Jh,n(3)}n≥0 and{j

(3)

h,n}n≥0 are both geometric and their study can be considered trivial.

Leth6=ω1, ω2 be a complex number. In order to extend similar prop-erties to third-order h-Jacobsthal and third-order h-Jacobsthal–Lucas se-quences we will provide the Binet’s formula for these sese-quences.

Lemma 2.2 (Binet’s Formula). Let h 6= ω1, ω2 be a complex number, and let{Jh,n(3)}n≥0 be the third-order h-Jacobsthal sequence and {j

(3)

h,n}n≥0 be the third-orderh-Jacobsthal–Lucas sequence. Then the Binet’s formulas reads:

Jh,n(3) = 1

σh

hn+1−

hω 2

ω1−ω2

ω1n+1+

hω 1

ω1−ω2

ωn2+1

(2.13)

and

jh,n(3) = 1

σh

(h2+h+ 2)hn+ (h+ 1)

h−ω2

ω1−ω2

ω1n+1−

h−ω1

ω1−ω2

ω2n+1

,

(2.14) whereσh=h2+h+ 1.

Proof. The characteristic polynomial of both the sequences {Jh,n(3)}n≥0 and {jh,n(3)}n≥0is (x−h)(x2+x+ 1), and by hypothesis it has three distinct roots, namely ω1, ω2 and h. It follows that there exist suitable a1, a2, a3, b1, b2,

b3 ∈C such that the Binet’s formula for {Jh,n(3)}n≥0 and{j

(3)

h,n}n≥0 is of the

type:

Jh,n(3) =a1hn+a2ω1n+a3ω2n, j (3)

h,n=b1hn+b2ωn1 +b3ω2n. In particular, using the initial conditions we have the relations:

  

 

Jh,(3)0=a1h0+a2ω01+a3ω20= 0

Jh,(3)1=a1h1+a2ω11+a3ω21= 1

Jh,(3)2=a1h2+a2ω21+a3ω22=h−1 and

  

 

jh,(3)0=b1h0+b2ω10+b3ω20= 2

jh,(3)1=b1h1+b2ω11+b3ω21=h−1

jh,(3)2=b1h2+b2ω12+b3ω22=h2+ 1

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Thus we have the following systems     

a1+a2+a3= 0

a1h+a2ω1+a3ω2= 1

a1h2+a2ω21+a3ω22=h−1 and     

b1+b2+b3= 2

b1h+b2ω1+b3ω2=h−1

b1h2+b2ω12+b3ω22=h2+ 1

,

whose solutions are:

a1=

h

h2+h+ 1, a2=−

ω2

(h−ω1)(ω1−ω2)

, a3=

ω1

(h−ω2)(ω1−ω2) and

b1=

h2+h+ 2

h2+h+ 1, b2=

ω2(h+ 1) (h−ω1)(ω1−ω2)

, b3=−

ω1(h+ 1) (h−ω2)(ω1−ω2)

.

Remark 2.3. We highlight that whenh= 2 the above Binet’s formulas (2.13) and (2.14) give the Binet’s formula of the third-order Jacobsthal sequence {Jn(3)}n≥0:

J0(3)= 0, J1(3)= 1, J2(3) = 1, J3(3)= 2, J4(3)= 5, J5(3)= 9,· · ·

and third-order Jacobsthal–Lucas sequence{jn(3)}n≥0:

j0(3)= 2, j1(3)= 1, j2(3)= 5, j3(3)= 10, j4(3)= 17, j5(3)= 37,· · ·.

Precisely, the Binet’s formulas forJn(3) andj

(3)

n are:

Jn(3)=J2(3),n=1 7

2n+1+

2ω 2+ 3

ω1−ω2

ωn12ω

1+ 3

ω1−ω2

ω2n

(2.15)

and

jn(3)=j2(3),n= 1 7

2n+3−3 2ω

2+ 3

ω1−ω2

ωn1 + 3 2ω

1+ 3

ω1−ω2

ω2n

. (2.16)

(see [6, Eq. (3.1)])

For simplicity of notation, we define

Zh,n=

hω 2

ω1−ω2

ωn1+1−

hω 1

ω1−ω2

ωn2+1. (2.17) Next result show that each term of{Zh,n}n≥0 has one more represen-tation. It will be useful for proving condition of the Lemma 2.5.

Corollary 2.4. Let h6=ω1, ω2 be a complex number, and let{Zh,n}n≥0 as in Eq. (2.17). Then the following identities hold:

Zh,n+Zh,n+1+Zh,n+2= 0, (2.18)

Zh,n=

  

 

h if n≡0 (mod 3) −(h+ 1) if n≡1 (mod 3) 1 if n≡2 (mod 3)

(6)

Proof. (2.18): Using the Eq. (2.17), we obtain

Zh,n+Zh,n+1+Zh,n+2=

hω 2

ω1−ω2

ωn1+1(1 +ω1+ω21)

hω 1

ω1−ω2

ω2n+1(1 +ω2+ω22) = 0.

(2.19): Using the Eq. (2.17),ω1ω2= 1 andω1+ω2=−1. Then,

Zh,n=

h −ω2

ω1−ω2

ω1n+1− h

−ω1

ω1−ω2

ωn2+1

=h

ωn+1 1 −ω

n+1 2

ω1−ω2

ωn

1 −ω2n

ω1−ω2

=   

 

h if n≡0 (mod 3) −(h+ 1) if n≡1 (mod 3) 1 if n≡2 (mod 3)

.

Then, using Corollary 2.4 we can write

Jh,n(3) = 1

h2+h+ 1

hn+1−Zh,n (2.20)

and

jh,n(3) = 1

h2+h+ 1

(h2+h+ 2)hn+ (h+ 1)Zh,n

. (2.21)

The next Lemma shows that similar properties to (2.1)–(2.10) also hold for third-orderh-Jacobsthal and third-orderh-Jacobsthal–Lucas sequences:

Lemma 2.5. Let h6= ω1, ω2 be a complex number, and let {J (3)

h,n}n≥0 be the third-order h-Jacobsthal sequence and {jh,n(3)}n≥0 be the h-Jacobsthal–Lucas sequence. Then the following identities hold:

(h+ 1)Jh,n(3) +jh,n(3) = 2hn, (2.22)

Jh,n(3)+2−h2Jh,n(3) =   

 

h−1 if n≡0 (mod 3) −h if n≡1 (mod 3) 1 if n≡2 (mod 3)

, (2.23)

(h+ 1)Jh,n(3)+2−jh,n(3)+1−jh,n(3) = (h−2)(h+ 1)hn, (2.24)

jh,n(3) −Jh,n(3)+2+ (h−2)hn=   

 

1 if n≡0 (mod 3) −1 if n≡1 (mod 3) 0 if n≡2 (mod 3)

, (2.25)

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jh,n(3)

2

−(h+ 1)2Jh,n(3)

2

= 4hnjh,n(3) −hn, (2.27)

Jh,n(3) +Jh,n(3)+1+Jh,n(3)+2=hn+1, (2.28)

n

X

l=0

Jh,l(3)= 1 3(h−1)

h

Jh,n(3)+2−(h−2)Jh,n(3)+1+hJh,n(3)−1i, h6= 1. (2.29)

Proof. By Eqs. (2.18), (2.19), (2.20) and (2.21), we have (2.22):

(h+ 1)Jh,n(3) +jh,n(3) = 1

h2+h+ 1

(h+ 1)hn+1−(h+ 1)Zh,n

+ 1

h2+h+ 1

(h2+h+ 2)hn+ (h+ 1)Zh,n

= 1

h2+h+ 1

2(h2+h+ 1)hn= 2hn.

(2.23):

Jh,n(3)+2−h2Jh,n(3) = 1

h2+h+ 1

hn+3−Zh,n+2

− 1

h2+h+ 1

h2·hn+h2Zh,n

= 1

h2+h+ 1

Zh,n+1+ (h2+ 1)Zh,n

=   

 

h−1 if n≡0 (mod 3) −h if n≡1 (mod 3) 1 if n≡2 (mod 3)

.

(2.24):

(h+ 1)Jh,n(3)+2−jh,n(3)+1−jh,n(3)

= 1

h2+h+ 1

(h+ 1)hn+3−(h+ 1)Zh,n+2

− 1

h2+h+ 1

(h2+h+ 2)hn+1+ (h+ 1)Zh,n+1

− 1

h2+h+ 1

(h2+h+ 2)hn+ (h+ 1)Zh,n

= 1

h2+h+ 1

(h−2)(h+ 1)(h2+h+ 1)

hn

(8)

(2.25):

jh,n(3) −Jh,n(3)+2+(h−2)hn

= 1

h2+h+ 1

(h2+h+ 2)hn+ (h+ 1)Zh,n

− 1

h2+h+ 1

hn+3−Zh,n+2

+ 1

h2+h+ 1

(h2+h+ 1)(h−2)

hn

= 1

h2+h+ 1[(h+ 1)Zh,n+Zh,n+2]

=   

 

1 if n≡0 (mod 3) −1 if n≡1 (mod 3) 0 if n≡2 (mod 3)

.

(2.26):

j(3)h,n−h2Jh,n(3)+ (h−2)hn= 1

h2+h+ 1

(h2+h+ 2)hn+ (h+ 1)Zh,n

− 1

h2+h+ 1

h2·hn+1−h2Zh,n

+ 1

h2+h+ 1

(h−2)(h2+h+ 1)

hn

=Zh,n.

(2.27):

jh,n(3)

2

−(h+ 1)2Jh,n(3)

2

=j(3)h,n−(h+ 1)Jh,n(3) jh,n(3) + (h+ 1)Jh,n(3)

= 2j(3)h,n−hn·2hn

= 4hnjh,n(3) −hn.

(2.28):

Jh,n(3)+Jh,n(3)+1+Jh,n(3)+2= 1

h2+h+ 1

hn+1−Zh,n

+ 1

h2+h+ 1

hn+2−Zh,n+1

+ 1

h2+h+ 1

hn+3−Zh,n+2

= 1

h2+h+ 1

(9)

(2.29):

n

X

l=0

Jh,l(3)=h+ (h−1)

n

X

l=3

Jh,l(3)1+ (h−1)

n

X

l=3

Jh,l(3)2+h

n

X

l=3

Jh,l(3)3

= (3h−2)

n

X

l=0

Jh,l(3)+ 1−(3h−2)Jh,n(3)−(2h−1)Jh,n(3)1−hJh,n(3)2.

Then, we obtain

n

X

l=0

Jh,l(3)= 1 3(h−1)

h

(3h−2)Jh,n(3)+ (2h−1)Jh,n(3)1+hJh,n(3)2−1i

= 1

3(h−1) h

Jh,n(3)+1+ (2h−1)Jh,n(3) +hJh,n(3)1−1i

= 1

3(h−1) h

Jh,n(3)+2−(h−2)Jh,n(3)+1+hJh,n(3)−1i.

### -Jacobsthal–Lucas Quaternions

Let h be a real number. Following [3], we define the n-th third-order h -Jacobsthal quaternion {J Q(3)h,n}n≥0 and the n-th third-order h-Jacobsthal– Lucas quaternion{jQ(3)h,n}n≥0as

J Q(3)h,n=Jh,n(3) +iJh,n(3)+1+jJh,n(3)+2+kJh,n(3)+3, jQ(3)h,n=jh,n(3) +ijh,n(3)+1+jjh,n(3)+2+kjh,n(3)+3.

(3.1)

Example. If we puth= 2 in the Eqns. (3.1), we have the third-order Jacob-stal quaternions sequence {J Q(3)n }n≥0 and the third-order Jacobstal-Lucas quaternions sequence{jQ(3)n }n≥0 studied in [3]:

J Q(3)0 =i+j+ 2k

J Q(3)1 = 1 +i+ 2j+ 5k

J Q(3)2 = 1 + 2i+ 5j+ 9k

J Q(3)3 = 2 + 5i+ 9j+ 18k

.. .

(10)

Theorem 3.1. Let hbe a real number, and let{J Q(3)h,n}n≥0 be the third-order

h-Jacobsthal and {jQ(3)h,n}n≥0 be the third-order h-Jacobsthal –Lucas quater-nions sequence. Then, for every positive integern, we have:

J Q(3)h,n+J Q(3)h,n+1+J Q(3)h,n+1=hn+1(1 +hi+h2j+h3k), (3.2)

(h+ 1)J Q(3)h,n+jQ(3)h,n= 2hn(1 +hi+h2j+h3k), (3.3)

J Q(3)h,n+2−h2J Q(3)h,n=   

 

h−1−hi+j+ (h−1)k if n≡0 (mod 3) −h+i+ (h−1)j−hk if n≡1 (mod 3) 1 + (h−1)i−hj+k if n≡2 (mod 3)

,

(3.4)

(h+1)J Q(3)h,n+2−jQ(3)h,n+1−jQ(3)h,n= (h−2)(h+1)hn(1+hi+h2j+h3k). (3.5)

Proof. (3.2): By definition, we have

J Q(3)h,n+J Q(3)h,n+1+J Q(3)h,n+1=Jh,n(3) +Jh,n(3)+1+Jh,n(3)+1

+i(Jh,n(3)+1+Jh,n(3)+2+Jh,n(3)+3) +j(Jh,n(3)+2+Jh,n(3)+3+Jh,n(3)+4) +k(Jh,n(3)+3+Jh,n(3)+4+Jh,n(3)+5).

Using the property (2.28) in Lemma 2.5, we have

J Q(3)h,n+J Q(3)h,n+1+J Q(3)h,n+1=hn+1+ihn+2+jhn+3+khn+4

=hn+1(1 +hi+h2j+h3k).

(3.3):

(h+ 1)J Q(3)h,n+jQ(3)h,n= (h+ 1)Jh,n(3)+j(3)h,n

+i((h+ 1)Jh,n(3)+1+jh,n(3)+1) +j((h+ 1)Jh,n(3)+2+jh,n(3)+2) +k((h+ 1)Jh,n(3)+3+jh,n(3)+3).

Using the property (2.22) in Lemma 2.5, we have

(h+ 1)J Q(3)h,n+jQ(3)h,n= 2hn+ 2ihn+1+ 2jhn+2+ 2khn+3

(11)

(3.3):

J Q(3)h,n+2−h2J Q(3)h,n=Jh,n(3)+2−h2Jh,n(3)

+i(Jh,n(3)+3−h2Jh,n(3)+1) +j(Jh,n(3)+4−h2Jh,n(3)+2) +k(Jh,n(3)+5−h2Jh,n(3)+3).

Using the property (2.23) in Lemma 2.5 andn≡0(mod 3), we have

J Q(3)h,n+2−h2J Q(3)h,n=h−1 +i(−h) +j(1) +k(h−1).

Finally, we obtain

J Q(3)h,n+2−h2J Q(3)h,n=   

 

h−1−hi+j+ (h−1)k if n≡0 (mod 3) −h+i+ (h−1)j−hk if n≡1 (mod 3) 1 + (h−1)i−hj+k if n≡2 (mod 3)

.

(3.4): By definition, we have

(h+ 1)J Q(3)h,n+2−jQ(3)h,n+1−jQ(3)h,n= (h+ 1)Jh,n(3)+2−j(3)h,n+1−jh,n(3)

+i((h+ 1)Jh,n(3)+3−jh,n(3)+2−jh,n(3)+1) +j((h+ 1)Jh,n(3)+4−jh,n(3)+3−jh,n(3)+2) +k((h+ 1)Jh,n(3)+5−jh,n(3)+4−jh,n(3)+3).

Using the property (2.24) in Lemma 2.5, we have

(h+ 1)J Q(3)h,n+2−jQ(3)h,n+1−jQ(3)h,n= (h−2)(h+ 1)hn

+i((h−2)(h+ 1)hn+1) +j((h−2)(h+ 1)hn+2) +k((h−2)(h+ 1)hn+3)

= (h−2)(h+ 1)hn(1 +hi+h2j+h3k).

Then, the result is obtained.

In addition, some formulae involving sums of terms of the third-order

h-Jacobsthal quaternion sequence will be provided in the following theorem.

Theorem 3.2. Let h ∈R− {1}. For a natural numbers m, n, with n≥ m, if J Q(3)h,n is the n-th third-order h-Jacobsthal quaternion, then the following identities are true:

n

X

l=0

J Q(3)h,l = 1 3(h−1)

(

J Q(3)h,n+1+ (2h−1)J Q(3)h,n+hJ Q(3)h,n1

−J Q(3)h,2+ (2h−3)J Q(3)h,1+ (h−2)J Q(3)h,0

)

(12)

Proof. Using Eqs. (1.4) and (3.1), we obtain

n

X

l=m

J Q(3)h,l =J Q(3)h,m+J Q(3)h,m+1+J Q(3)h,m+2+

n

X

l=m+3

J Q(3)h,l

=J Q(3)h,m+J Q(3)h,m+1+J Q(3)h,m+2

+ (h−1)

n−1 X

l=m+2

J Q(3)h,l+ (h−1)

n−2 X

l=m+1

J Q(3)h,l +h

n−3 X

l=m

J Q(3)h,l

= (3h−2)

n

X

l=m

J Q(3)h,l

+J Q(3)h,m+2−(2h−3)J Q(3)h,m+1−(h−2)J Q(3)h,m

−(3h−2)J Q(3)h,n−(2h−1)J Q(3)h,n1−hJ Q(3)h,n2

= (3h−2)

n

X

l=m

J Q(3)h,l

+J Q(3)h,m+2−(2h−3)J Q(3)h,m+1−(h−2)J Q(3)h,m

−J Q(3)h,n+1−(2h−1)J Q(3)h,n−hJ Q(3)h,n1.

Then, the result in Eq. (3.6) is completed ifm= 0.

### 4. Conclusion

Sequences of quaternions have been studied over several years, including the well-known Tribonacci quaternion sequence [4] and, consequently, on the third-order Jacobsthal quaternion sequence [3]. In this paper we have also contributed for the study of third-order h-Jacobsthal quaternion, deducing some formulae for the sums of such numbers, presenting their Binet-style formula. It is our intention to continue the study of this type of sequences, exploring some their applications in the science domain. For example, a new type of sequences in the octonion algebra with the use of these numbers and their combinatorial properties.

### References

[1] G. Anatriello and G. Vincenzi, On h-Jacobsthal and h-Jacobsthal–Lucas se-quences, and related quaternions, An. St. Univ. Ovidius Constanta, 27(3), 2019, 5–23.

(13)

[3] G. Cerda-Morales, Identities for Third Order Jacobsthal Quaternions, Adv. Appl. Clifford Algebra, 27(2), 2017, 1043–1053.

[4] G. Cerda-Morales, On a Generalization of Tribonacci Quaternions, Mediterr. J. Math., 14(239), 2017, 1–12.

[5] G. Cerda-Morales, On the third-order Jacobsthal and third-order Jacobsthal-Lucas sequences and their matrix representations, Mediterr. J. Math., 16(32), 2019, 1–12.

[6] C. K. Cook and M. R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathemat-icae et InformatMathemat-icae 41, 2013, 27–39.

[7] S. Halici, On Fibonacci Quaternions, Adv. Appl. Clifford Algebras. 22(2), 2012, 321–327.

[8] S. Halici and A. Karatas, On a Generalization for Fibonacci Quaternions, Chaos, Solitons Fractals 98, 2017, 178–182.

[9] A. F. Horadam, Complex Fibonacci numbers and Fibonacci Quaternions, Am. Math. Month. 70, 1963, 289–291.

[10] G. Krishnaswami and S. Sachdev, Algebra and Geometry of Hamilton’s Quater-nions, Resonance-Journal of Science Education, 21(6), 2016, 529–544.

[11] J. P. Ward, Quaternions and Cayley Numbers: Algebra and Applications, Kluwer Academic Publishers, London, 1997.

Gamaliel Cerda-Morales Instituto de Matem´aticas,

Pontificia Universidad Cat´olica de Valpara´ıso, Blanco Viel 596, Valpara´ıso, Chile.

References

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