On pell quaternions and pell-lucas quaternions

0188-7009/010039-13

published online June 16, 2015 DOI 10.1007/s00006-015-0571-8

Advances in

Applied Clifford Algebras

On Pell Quaternions and Pell-Lucas

Quaternions

Cennet Bolat C

¸ imen* and Ahmet ˙Ipek

Abstract. The main object of this paper is to present a systematic in-vestigation of new classes of quaternion numbers associated with the familiar Pell and Pell-Lucas numbers. The various results obtained here for these classes of quaternion numbers include recurrence relations, summation formulas and Binet’s formulas.

Mathematics Subject Classification. 11B39, 11R52, 05A15.

Keywords. Pell numbers, Lucas numbers, Pell quaternions, Pell-Lucas quaternions, Quaternion algebra.

1. Introduction and Preliminaries

Quaternion arithmetic can be used in various areas such as computer sciences, physics and applied mathematics. For a survey on quaternion analysis and a list of references we refer to the book [8].

Fibonacci-like numbers are of great interest and have been a central part of number theory. These numbers appear almost everywhere in mathematics and computer science (for more details see [4,11–17]).

An active research is being conducted around the topic of quaternionic number sequences since in general the finding formulas and identities for quaternionic number sequences may be very complicated. The research groups that work on this topic are interested for instance in obtaining generat-ing functions, Binet’s formulas, sum formulas, several identities for defined quaternionic number sequences (see, for example, [1,6,7,9,10,12,15,18,19])

First time the idea to consider Pell quaternions it was suggested by Horadam on the page 31 in the paper [14].

In this paper we define new kinds of sequences of quaternion number called as Pell Quaternions and Pell-Lucas Quaternions. We study some prop-erties of these quaternions. For these classes of quaternion numbers, we give *Corresponding author.

various results including recurrence relations, summation formulas and Bi-net’s formulas, which can be useful also in problems related with these quater-nions.

The outline of this paper is as follows: In the rest of this section, we in-troduce some necessary definitions and mathematical preliminaries of quater-nions, classical Pell and Pell-Lucas numbers. Section2 is devoted to obtain some properties of Pell Quaternions and Pell-Lucas Quaternions.

We start by recalling some basic results concerning quaternion algebra H and Pell and Pell-Lucas numbers, which can be found in classic books on these subjects.

The algebra

H = {a = a0e0+a1e1+a2e2+a3e3: ai∈ R, i = 0, 1, 2, 3} (1)

of real quaternions is defined as the four-dimensional vector space over R having a basis{e₀, e₁, e₂, e₃} which satisfies the following multiplication rules:

Table 1. The multiplication table for the basis of H

× 1 e₁ e₂ e₃ 1 1 e₁ e₂ e₃ e1 e1 −1 e3 −e2 e2 e2 −e3 −1 e1 e3 e3 e2 −e1 −1 (2)

A quaternionα =3_s=0α_se_s∈ H is pieced into two parts with scalar piece S_α = α₀ and vectorial piece −→V_α = 3_s=1α_se_s. We also write α =

Sα+−→Vα. The conjugate ofα = Sα+−→Vα is then defined as

α = Sα−−→Vα=α0e0− 3



s=1

αses.

We call a real quaternion pure if its scalar part vanishes. Letα and β be two quaternions such thatα = S_α+−→V_αandβ = S_β+V−→_β, whereS_α=α₀, S_β=β₀,

−→

Vα =3_s=1αses and V−→β =3_s=1βses. Multiplication ofα and β is defined as

αβ = SαSβ+SαV−→β+−→VαSβ−−→Vα·−V→β+V−→α×−V→β, (3)

where−→V_α·−V→_β =α₁β₁+α₂β₂+α₃β₃and−→V_α×−V→_β= (α₂β₃−α₃β₂)e₁−(α₁β₃−

α3β1)e2+ (α1β2− α2β1)e3. The conjugate of reel quaternionα is denoted by

α and it is

α = α0e0− α1e1− α2e2− α3e3. (4)

The norm ofα is defined as

α = αα = α2

0+α21+α22+α23. (5)

The sequences of Pell numbersp_n, named after the English diplomat and mathematician John Pell, and Pell-Lucas numbersq_n, which satisfy the three-term recurrence relations respectively

pn= 2pn−1+pn−2 (6) and

qn= 2qn−1+qn−2 (7)

for alln = 2, 3, 4, . . . with initial conditions p₀= 0, p₁= 1 [13] andq₀= 1 =

q1, represent most important number sequences. About the properties of these sequences, some authors had studied these, and obtained many interesting results (see, for example, [13,16,20]).

They can also be defined by the Binet-like formulae

pn= α n_{− β}n

α − β and qn=

αn_+βn

2 , (8)

whereα = 1+√2 andβ = 1−√2 are the solutions of the quadratic equation

x2_{= 2x + 1 (see [2,3,5,13]).}

2. Pell Quaternions and Pell-Lucas Quaternions

In this section, we define new kinds of sequences of quaternion number called as Pell Quaternions and Pell-Lucas Quaternions. We study some properties of these quaternions. We obtain various results for these classes of quater-nion numbers included recurrence relations, summation formulas and Binet’s formulas.

We now consider the usual Pell and Pell-Lucas numbers and we give de-finitions of new kinds of quaternion numbers, which we call the Pell Quater-nions and Pell-Lucas QuaterQuater-nions. Definenth Pell quaternion and Pell-Lucas quaternion numbers, respectively, as shown:

QP_n=p_ne₀+p_n+1e₁+p_n+2e₂+p_n+3e₃, (9) and

QPL_n=q_ne₀+q_n+1e₁+q_n+2e₂+q_n+3e₃, (10) wherep_n andq_n arenth Pell and Pell-Lucas numbers.

LetQP_n and QM_n be two Pell quaternions such thatQP_n =p_ne₀+

pn+1e1+pn+2e2+pn+3e3, and QMn=mne0+mn+1e1+mn+2e2+mn+3e3.

The scalar and the vector part of Pell quaternions QP_n and QM_n are de-noted by S_QPn =p_ne₀, −−−→V_QPn =p_n+1e₁+p_n+2e₂+p_n+3e₃, S_QMn =m_ne₀ and−−−→VQMn=mn+1e1+mn+2e2+mn+3e3, respectively. Therefore, the addi-tion, substraction and multiplication of these quaternions, respectively, are obtained as following QPn± QMn = 3  s=0 (p_s± m_s)e_s (11)

and QPn· QMn =SQPnSQMn+SQPn· −−−→ VQMn+ −−−→ VQPn· SQMn −−−−→VQPn· −−−→ VQMn+ −−−→ VQPn× −−−→ VQMn. (12)

The conjugates ofQP_nand QPL_n are defined by

QPn =pne0− pn+1e1− pn+2e2− pn+3e3, (13)

QPL_n =qne0− qn+1e1− qn+2e2− qn+3e3. (14) The norms ofQP_nandQP L_n are defined by

NQPn =QPnQPn=p2n+p2n+1+p2n+2+p2n+3, (15)

NQPLn = QPLnQP Ln=q2n+q2n+1+q2n+2+qn+32 , (16) respectively.

Proposition 1. Forn ≥ 2, we have the following identities:

QPn+QPn = 2pn, (17)

QP2

n+QPn· QPn = 2pn· QPn, (18)

QPn· QPn = 6p2n+3. (19)

Proof. From (11) and (13), we get

QPn+QPn= 3  s=0 pn+ses+pn− 3  s=1 pn+ses = 2pn

which gives (17). On the other hand, from (17) we have

QP2 n =QPn· QPn=QPn2pn− QPn= 2pn· QPn− QPn· QPn and so QP2 n+QPn· QPn= 2pn· QPn. Since QPn· QPn = 3  s=0 p2 n+s

from Table1, (11) and (13), and also

p2

n+p2n+1=p2n+1 [13], we obtain

QPn· QPn = 6p2n+3.

 Proposition 2. Forn ≥ 2, we have the following identities:

QPn+ 2QPn+1=QPn+2, (20)

Proof. It follows from (9) and (11) that QPn+ 2QPn+1= 3  s=0 pn+ses+ 2 3  s=0 pn+1+ses and therefore QPn+ 2QPn+1= 3  s=0 pn+2+ses =QP_n+2. Considering table (6) from (12) and (17) we get

QPn− QPn+1e1− QPn+2e2− QPn+3e3= 3



s=0

pn+2s.

By using the identitiesq_n=p_n+1− p_n andp_n= 2p_n−1+p_n−2 [13], we obtain

QPn− QPn+1e1− QPn+2e2− QPn+3e3= 12qn+3.

Thus, we complete the proof. 

The identities in following theorem are analogous toq_n+1=p_n+1+p_n andq_n =p_n+1− p_n (see [13]).

Theorem 3. Forn ≥ 2, we have the following identities:

QPn+QPn+1=QP Ln+1, (22)

QPn+1− QPn =QP Ln, (23)

QPn−1+QPn+1=QP Ln, (24) 2QP_n+QP L_n =QP L_n+1. (25)

Proof. It follows from (9) and (11) that

QPn+QPn+1= 3  s=0 pn+ses+ 3  s=0 pn+1+ses = 3  s=0 (p_n+s+p_n+1+s)e_s,

and therefore by using the identityq_n+1=p_n+p_n+1[13] and (10) we get

QPn+QPn+1= 3  s=0 qn+1+ses =QP L_n+1. Similarly, from (9) and (11) we have

QPn+1− QPn= 3  s=0 pn+1+ses− 3  s=0 pn+ses = 3  s=0 (p_n+1+s− p_n+s)e_s,

and therefore by using the identityq_n=p_n+1− p_n(see [13]) and (10) we get QPn+1− QPn= 3  s=0 qn+ses =QP L_n.

By (9) and (11), the identity p_n−1+p_n+1 = q_n (see [5]) and some direct computations we obtain QPn−1+QPn+1= 3  s=0 (pn−1+s+pn+1+s)es,

which implies from (10) that

QPn−1+QPn+1=QP Ln.

Based on the definitions (9) and (10) from the identityq_n+1− q_n= 2p_n (see [2]) we can deduce 2QP_n+QP L_n= 2 3  s=0 pn+ses+ 3  s=0 qn+ses = 3  s=0 (2pn+s+qn+s)es =QP Ln+1.  We now will give Binet’s formulas for Pell and Pell-Lucas quaternions. The characteristic equation of the relationQP_n+ 2QP_n+1=QP_n+2is as follows

x2_{− 2x − 1 = 0. (26)}

So, the roots of this characteristic equation areα = 1 +√2 andβ = 1 −√2. Note that

α + β = 2, α − β = 2√2 and αβ = −1. (27) Now, we give the following useful lemma.

Lemma 4. Forn ≥ 1,

αQPn+QPn−1=αnA

and

βQPn+QPn−1=βnB,

whereA =3_s=0αse_sandB =3_s=0βse_s.

Proof. Letn ≥ 1. For the Pell quaternions QP_n andQP_n+1, we obtain

αQPn+QPn−1= 3



s=0

By the identityαn =αp_n+p_n−1, some straightforward computations lead to

αQPn+QPn−1=αnA, (29)

whereA =3_s=0αse_s. In addition, by considering the identityβn =βp_n+

pn−1, the following equation may be indicated in much a similar way as (29)

βQPn+QPn−1=βnB, (30)

whereB =3_s=0βse_s. Thus, we complete the proof.  The following theorem gives Binet’s formulas for Pell and Pell-Lucas quaternions by a function of the rootsα and β of the charactersictic equation associated to the recurrence relation the Eq. (26).

Theorem 5. Binet’s formulas forQP_nandQP L_n, respectively, are as follows QPn =α n_{A − β}n_B α − β (31) and QP Ln =α n_{A + β}n_B 2 , (32) whereA =3_s=0αse_sandB =3_s=0βse_s.

Proof. If we consider (29) and (30), by straightforward calculations we obtain

αn_{A − β}n_{B = αQP} n− βQPn. Therefore, we get QPn =α n_{A − β}n_B α − β .

By applying the similar techniques from (29) and (30) we have

αn_{A + β}n_{B = (α + β) QP}

n+ 2QPn−1.

Therefore, sinceα + β = 2 and QP_n+QP_n+1=QP L_n+1after simple com-putations we have

QP Ln =α

n_{A + β}n_B

2 .

 Theorem 6. The following formulae are true:

n  s=1 QPs= 1 2(QP Ln+1− QP L1), (33) n  s=1 QP2s= 1₂(QP2n+1− QP1), (34) n  s=1 QP2s−1= 1₂(QP2n− QP0). (35)

Proof. Sincem_i=0p_k+i= ₂1(q_k+m+1− q_k) [2], we have n  s=1 QPs=  _n  s=1 ps  e0+  _n  s=1 ps+1  e1+  _n  s=1 ps+2  e2+  _n  s=1 ps+3  e3 =  1 2(qn+1− q0)− p0 e0+  1 2(qn+2− q1)− p1 e1 +  1 2(qn+3− q2)− p2 e2+  1 2(qn+4− q3)− p3 e3 = 1 2  ₃  s=0 qn+1+ses− 3  s=0 qses− 2 3  s=0 pses  = 1 2(QP Ln+1− QP L0− 2QP0), (36) and so from the identityQP Ln+1= 2QPn+QP Ln we can deduce

n



s=1

QPs= 1₂(QP Ln+1− QP L1).

ByQP_n =αnA−β_α−βnB andQP L_n =αnA+β₂ nB, we have

n  s=1 QPs= n  s=1 αs_{A − β}s_B α − β = 1 α − β Aαn s=1 αs−1_{− Bβ}n s=1 βs−1  = 1 α − β  Aαα_{α − 1}n− 1− Bββ_{β − 1}n− 1 ,

where A = 3_s=0αses and B = 3_s=0βses. Therefore, if we consider (27) some straightforward computations lead to

n  s=1 QPs= 1 2  Aαn_{− Bβ}n α − β + Aαn+1_{− Bβ}n+1 α − β − Aα − Bβ α − β − A − B α − β = 1 2[QPn+QPn+1− QP1− QP0],

and so from the identityQP L_n+1=QP_n+QP_n+1 it follows that

n



s=1

QPs= 1

2[QP Ln+1− QP L1].

Thus, the another proof of (33) is completed. Then by the definition of QP_n as well as (9) and the approach in (36) we have

n  s=1 QP2s=e0 n  s=1 p2s+e1 n  s=1 p2s+1+e2 n  s=1 p2s+2+e3 n  s=1 p2s+3,

and therefore sincem_i=0p_2k+2i=1₂(q_2k+2m+2− p_2k+2m+2+q_2k+p_2k) and _m

i=0p2k+2i+1= 1₂(2q2k+2m+ 3p2k+2m− p2k) [2] we have n  s=1 QP2s=1₂(q2n+p2n− q0− p0)e0+1₂(2q2n+ 3p2n− p0− 2p1)e1 +1 2(q2n+2+p2n+2− q2− p2)e2 +1 2(2q2n+2+ 3p2n+2− p2− 2p3)e3.

By the identitiesq_n=p_n+1− p_nandQP_n+ 2QP_n+1=QP_n+2[13] and some direct computations we obtain

n  s=1 QP2s= 1 2 3  s=0 p2n+1+ses−1 2 3  s=0 p1+ses = 1 2(QP2n+1− QP1).

Using of the Binet’s formula for Pell Quaternions and Pell-Lucas Quater-nions, the following direct proof forn_s=1QP_2s is even simpler. By QP_n =

αn_A−βn_B α−β andQP Ln =α n_A+βn_B 2 , we have n  s=1 QP2s= n  s=1 α2s_{A − β}2s_B α − β = 1 α − β Aα2n s=1 α2(s−1)_{− Bβ}2n s=1 β2(s−1)  = 1 α − β  Aα2α2n− 1 α2_{− 1} − Bβ2 β2n_{− 1} β2_{− 1} ,

whereA =3_s=0αse_sandB =3_s=0βse_s. Therefore, if we consider (27) by direct computations we get

n  s=1 QP2s=−1 4  Aα2n_{− Bβ}2n α − β − Aα 2n+2_{− Bβ}2n+2 α − β +Aα 2_{− Bβ}2 α − β − A − Bα − β = 1 4(QP2n+2− QP2n+QP0− QP2)

and so from the identityQP_n+ 2QP_n+1=QP_n+2it follows that

n  s=1 QP2s= 1₄(2QP2n+1− 2QP1) = 1 2(QP2n+1− QP1). Since m  i=0 p2k+2i=1 2(q2k+2m+2− p2k+2m+2+q2k+p2k)

and m  i=0 p2k+2i+1=1 2(2q2k+2m+ 3p2k+2m− p2k), [2] we have the sumn_s=1QP_2s−1as

n  s=1 QP2s−1=  _n  s=1 p2s−1  e0+  _n  s=1 p2s  e1+  _n  s=1 p2s+1  e2+  _n  s=1 p2s+2  e3 =  1 2(2q2n−2+ 3p2n−2− p−2− 2p−1) e0+  1 2(p2n+1− p1) e1 +  1 2(p2n+2− p2) e2+  1 2(p2n+3− p3) e3.

Combining the identities q_n = p_n+1− p_n, QP_n + 2QP_n+1 = QP_n+2 and

p−n= (−1)n+1pn [13] with the last formula yields n  s=1 QP2s−1=  1 2(2q2n−2+ 3p2n−2+p2− 2p1) e0+  1 2(p2n+1− p1) e1 +  1 2(p2n+2− p2) e2+  1 2(p2n+3− p3) e3 =  1 2(p2n+p0) e0+  1 2(p2n+1− p1) e1+  1 2(p2n+2− p2) e2 +  1 2(p2n+3− p3) e3 = 1 2 3  s=0 p2n+ses−1 2 3  s=0 pses = 1 2(QP2n− QP0).

From Binet’s formula for Pell quaternions and (27), we obtain, in the same way as we did for the sumsn_s=1QP_s andn_s=1QP_2sthe result that

n  s=1 QP2s−1= n  s=1 α2s−1_{A − β}2s−1_B α − β = 1 α − β Aα n  s=1 α2(s−1)_{− Bβ}n s=1 β2(s−1)  = 1 α − β  Aαα2n− 1 α2_{− 1} − Bβ β2n_{− 1} β2_{− 1} =−1 4 _{Aα2n−1− Bβ2n−1} α − β − Aα2n+1_{− Bβ}2n+1 α − β + (A − B) (α + β) α − β ,

whereA =3_s=0αse_sandB =3_s=0βse_s. It follows from this that

n



s=1

QP2s−1=1

2(QP2n− QP0).

Theorem 7. (Cassini’s identities) The following identities are hold: QPn+1.QPn−1− QPn2= (−1)n  α2_{+ 2}_{AB + β}2_BA (α − β)2  and QP Ln+1· QP Ln−1− QP L2n= (−1)n+1  α2_{+ 2}_{AB + β}2_BA 4  , whereA =3_s=0αse_sandB =3_s=0βse_s.

Proof. LetA and B be A = 3_s=0αse_s and B = 3_s=0βse_s, respectively. Using the Binet’s formula (31) and by (27) we get

QPn+1.QPn−1− QPn2 =  αn+1_{A − β}n+1_B α − β  αn−1_{A − β}n−1_B α − β −  αn_{A − β}n_B α − β 2 = 1 (α − β)2  α2n_A2_{− α}n+1_βn−1_{AB − β}n+1_αn−1_{BA + β}2n_B2 −α2n_A2_{+ 2α}n_βn_{AB − β}2n_B2 = 1 (α − β)2  −αn+1_βn−1_{AB − β}n+1_αn−1_{BA + 2α}n_βn_AB = (−1)n  α2_{+ 2}_{AB + β}2_BA (α − β)2  .

Similarly, using the Binet’s formula (32) and by (27), we have

QP Ln+1· QP Ln−1− QP L2n =  αn+1_{A + β}n+1_B 2  αn−1_{A + β}n−1_B 2 −  αn_{A + β}n_B 2 ₂ = (−1)n+1  α2_{AB + β}2_{BA + 2AB} 4 = (−1)n+1  α2_{+ 2}_{AB + β}2_BA 4  .

The identities are proved. 

Theorem 8. The following equalities are hold:

QP L2 n− 2QPn2= (−1)nAB (37) and form  1 QP L2 2m+1− QP L2m−1· QP L2m+3= −AB₂ +1₄α4AB + β4BA, (38) whereA =3_s=0αse_sandB =3_s=0βse_s.

Proof. By Binet’s formulas in (31), (32) and the fact thatαβ = −1, it is easy to check QP L2 n−2QPn2=α 2n_A2_+2αn_βn_AB+β2n_B2_−α2n_A2_+2αn_βn_{AB − β}2n_B2 4 which yields the identity

QP L2

n− 2QPn2= (−1)nAB.

that is, the identity (37) required.

The proof of (38) can be found in the same way. 

3. Conclusions

In this study, we presented a systematic investigation of new classes of quater-nion numbers associated with the familiar Pell and Pell-Lucas numbers. Also, we obtained various results including recurrence relations, summation formu-las, Binet’s formulas and generating functions for these classes of quaternion numbers.

Acknowledgements

The authors are deeply grateful to the Editor and referees for their con-structive comments and suggestions, and thank the referees for their careful readings of the paper.

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Cennet Bolat C¸ imen Department of Mathematics Faculty of Art and Science Mustafa Kemal University Tayfur S¨okmen Campus 31000 Hatay

Turkey

e-mail: bolatcennet@gmail.com Ahmet ˙Ipek

Department of Mathematics Faculty of Kamil ¨Ozda˘g Science Karamano˘glu Mehmetbey University 70100 Karaman

Turkey

e-mail: dr.ahmetipek@gmail.com Received: March 24, 2015. Accepted: June 2, 2015.