On Jacobsthal and Jacobsthal-Lucas Octonions

DOI 10.1007/s00009-017-0873-2 1660-5446/17/020001-13

published online February 28, 2017 c

 Springer International Publishing 2017

On Jacobsthal and Jacobsthal–Lucas

Octonions

Cennet Bolat C

¸ imen and Ahmet ˙Ipek

Abstract. Various families of quaternion and octonion number sequences

(such as Fibonacci quaternion, Fibonacci octonion, and so on) have been established by a number of authors in many different ways. In addition, formulas and identities involving these number sequences have been p-resented. In this paper, we aim at establishing new classes of octonion numbers associated with the familiar Jacobsthal and Jacobsthal–Lucas numbers. We introduce the Jacobsthal octonions and the Jacobsthal– Lucas octonions and give some of their properties. We derive the rela-tions between Jacobsthal octonions and Jacobsthal–Lucas octonions.

Mathematics Subject Classification. Primary 11B39; Secondary 11R52;

Tertiary 05A15.

Keywords. Jacobsthal numbers, Jacobsthal–Lucas numbers, Jacobsthal

octonions, Jacobsthal–Lucas octonions, octonion algebra.

1. Introduction

In recent years, the topic of number sequences in real normed division alge-bras has attracted the attention of several researchers. It is worth noticing that there are exactly four real normed division algebras: real numbers (R), complex numbers (C), quaternions (H) and octonions (O). Baez [3] gives a comprehensive discussion of these algebras.

The real quaternion algebra

H = {q = q0+q1i + q2j + q3k : qs∈ R, s = 0, 1, 2, 3}

is a 4-dimensionalR-vector space with basis {1  e0, i  e1, j  e2, k  e3}

× 1 e1 e2 e3

1 1 e1 e2 e3

e1 e1 −1 e3 −e2

e2 e2 −e3 −1 e1

e3 e3 e2 −e1 −1

Table 1: The multiplication table for the basis ofH

The real octonion algebra denoted byO is an 8-dimensional real linear space with basis

{e0= 1, e1=i, e2=j, e3=k, e4=e, e5=ie, e6=je, e7=ke}. (1.1)

The spaceO becomes an algebra via multiplication rules listed in the follow-ing table [17]:

× 1 e1 e2 e3 e4 e5 e6 e7

1 1 e1 e2 e3 e4 e5 e6 e7

e1 e1 −1 e3 −e2 e5 −e4 −e7 e6

e2 e2 −e3 −1 e1 e6 e7 −e4 −e5

e3 e3 e2 −e1 −1 e7 −e6 e5 −e4

e4 e4 −e5 −e6 −e7 −1 e1 e2 e3

e5 e5 e4 −e7 e6 −e1 −1 −e3 e2

e6 e6 e7 e4 −e5 −e2 e3 −1 −e1

e7 e7 −e6 e5 e4 −e3 −e2 e1 −1

Table 2: The multiplication table for the basis ofO

(1.2)

A variety of new results on Fibonacci-like quaternion and octonion num-bers can be found in several papers [2,5–7,10,11,14,15]. The origin of the top-ic of number sequences in division algebra can be traced back to the works by Horadam in [10] and by Iyer in [11]. Horadam [10] defined the quaternions with the classic Fibonacci and Lucas number components as

Qfn=fne0+fn+1e1+fn+2e2+fn+3e3

and

Qln =lne0+ln+1e1+ln+2e2+ln+3e3,

respectively, wheref_n andl_n are the nth classic Fibonacci and Lucas num-bers, respectively, and the author studied the properties of these quaternions. Several interesting and useful extensions of many of the familiar quaternion numbers (such as the Fibonacci and Lucas quaternions [2,6,14], Pell quater-nion [4,5,15], and so on) have been considered by several authors.

In this paper, we define two families of the octonions, where the coeffi-cients in the terms of the octonions are determined by the classic Jacobsthal and Jacobsthal–Lucas numbers. These two families of the octonions are called as the Jacobsthal and Jacobsthal–Lucas octonions, respectively. We mention some of their properties, and apply them to the study of some identities and formulas of the Jacobsthal and Jacobsthal–Lucas octonions. Here, our approach for obtaining some fundamental properties and characteristics of Jacobsthal and Jacobsthal–Lucas octonions is to apply the properties and

characteristics of the classic Jacobsthal and Jacobsthal–Lucas numbers. This approach was originally proposed by Horadam and Iyer in the articles [10,11] for Fibonacci quaternions. The methods used by Horadam and Iyer in that papers have been recently applied to the other familiar quaternion and octo-nion numbers by several authors [2,4–6,8,14,16].

This paper has four main sections. In Sect.2, we provide the basic def-initions of the octonions and the classic Jacobsthal and Jacobsthal–Lucas numbers. Section 3 is devoted to introducing Jacobsthal and Jacobsthal– Lucas octonions, and then to obtaining some fundamental properties and characteristics of these numbers. This paper closes with a section on “Con-clusions”.

2. Preliminaries

In this section, we will start with giving the definitions of the classic Jacob-sthal and JacobJacob-sthal–Lucas numbers and listing the known properties that we will use to present the main results.

The classic Jacobsthal numbers in [9] are defined, for all nonnegative integers, by

Jn=Jn−1+ 2Jn−2, J0= 0, J1= 1. (2.1)

The classic Jacobsthal–Lucas numbers in [9] are defined, for all nonnegative integers, by

jn=jn−1+ 2jn−2, j0= 2, j1= 1. (2.2)

For convenience initial Jacobsthal numbers and Jacobsthal–Lucas numbers are presented in the following table.

n 0 1 2 3 4 5 6 7 8 9 10

Jn 0 1 1 3 5 11 21 43 85 171 341

jn 2 1 5 7 17 31 65 127 257 511 1025

The following properties given for Jacobsthal numbers and Jacobsthal–Lucas numbers play important roles in this paper (see [9]).

jnJn =J2n, (2.3) Jn+jn = 2Jn+1, (2.4) 3J_n+j_n = 2n+1, (2.5) jn+1+ 2jn−1= 9Jn, (2.6) Jmjn+Jnjm= 2Jm+n, (2.7) Jn = 1₃(2n− (−1)n), (2.8) jn = 2n+ (−1)n, (2.9) Jmjn− Jnjm= (−1)n2n+1Jm−n, (2.10) jn+1+jn = 3 (Jn+1+Jn) = 3.2n, (2.11) jn+r− jn−r = 3 (Jn+r− Jn−r) = 2n+r+ 2n−r, (2.12) jn+1− jn = 3 (Jn+1− Jn) + 4 (−1)n+1= 2n+ 2 (−1)n+1, (2.13)

jn+r+jn−r = 3 (Jn+r+Jn−r) + 4 (−1)n−r

= 2n+r+ 2n−r+ 2 (−1)n−r. (2.14) The formulas (2.8) and (2.9) are named as the Binet formula for Jacobsthal numbers and Jacobsthal–Lucas numbers, respectively.

In the following we will study the important properties of the octonions. We refer to [3] for a detailed analysis of the properties of the octonionsp = 7

s=0pses and q =7s=0qses where the coefficients ps andqs, s = 0, . . . 7,

are real. We recall here only the following facts: • The sum and subtract of p and q is defined as

p ± q =

7



s=0

(p_s± q_s)e_s. (2.15) • Every p ∈ O can be written as p = Rep + Imp, and Rep = p0 and

Imp =7_s=1pses are called the real and imaginary parts, respectively. • The conjugate of p is defined by

p = Rep − Imp, (2.16)

and this operation satisfies

p = p, (p + q) = p + q, (pq) = qp (2.17)

for allp, q ∈ O.

• The norm of an octonion, which agrees with the standard Euclidean norm onR8 _{is defined as} N(p) = pp = pp = 7  s=0 p2 s. (2.18)

• The inverse of p = 0 is given by p−1₌ p

N(p). (2.19)

For allp, q ∈ O, from the above two definitions it is deduced that

N(pq) = N(p)N(q) (2.20)

and

(pq)−1=q−1p−1. (2.21)

• O is non-commutative and non-associative but it is alternative

3. Jacobsthal Octonions and Jacobsthal–Lucas Octonions

In this section, we define new kinds of sequences of octonion number called as Jacobsthal octonions and Jacobsthal–Lucas octonions. We study some properties of these octonions. We obtain various results for these classes of octonion numbers included recurrence relations, summation formulas, Binet’s formulas and generating functions.

In [16], the authors introduced the so-called Jacobsthal quaternions, which are a new class of quaternion sequences. They are given by the following recurrence relation ˆ Jn = 3  s=0 Jn+ses (3.1)

whereJ_n is thenth Jacobsthal number.

We now consider the usual Jacobsthal and Jacobsthal–Lucas numbers, and based on the definition (3.1) we give definitions of new kinds of octo-nion numbers, which we call the Jacobsthal octoocto-nions and Jacobsthal–Lucas octonions. Now, in the following, we define thenth Jacobsthal octonion and Jacobsthal–Lucas octonion numbers, respectively, by the following recurrence relations: ˆ Jn = 7  s=0 Jn+ses =J_ne0+Jn+1e1+Jn+2e2+Jn+3e3 +J_n+4e4+Jn+5e5+Jn+6e6+Jn+7e7 (3.2) and ˆ jn= 7  i=s jn+ses =j_ne0+jn+1e1+jn+2e2+jn+3e3 +j_n+4e4+jn+5e5+jn+6e6+jn+7e7 (3.3)

whereJ_nandj_n are thenth Jacobsthal number and Jacobsthal–Lucas num-ber, respectively.

The equalities in (2.15) gives ˆ Jn± ˆjn= 7  s=0 (J_s± j_s)e_s. (3.4) From (2.16), (3.2) and (3.3) an easy computation gives

ˆ Jn =Jne0− Jn+1e1− Jn+2e2− Jn+3e3 − Jn+4e4− Jn+5e5− Jn+6e6− Jn+7e7, (3.5) and ˆ jn =jne0− jn+1e1− jn+2e2− jn+3e3 − jn+4e4− jn+5e5− jn+6e6− jn+7e7. (3.6)

By definition (2.18) it is easily seen from, (3.2) and (3.3), that N( ˆJn) = ˆJn. ˆJn =J_n2+J_n+12 +J_n+22 +J_n+32 +J_n+42 +J_n+52 +J_n+62 +J_n+72 , (3.7) and N(ˆjn) = ˆjn.ˆjn =j_n2+j_n+12 +j_n+22 +j_n+32 +j_n+42 +j_n+52 +j2_n+6+j_n+72 . (3.8) By some elementary calculations we find the following recurrence rela-tions for the Jacobsthal and Jacobsthal–Lucas octonions from (3.2), (3.3), (3.4), (2.1) and (2.2): ˆ Jn+ 2 ˆJn−1= 7  s=0 Jn+ses+ 2 7  s=0 Jn−1+ses = 7  s=0 (J_n+s+ 2J_n−1+s)e_s = 7  s=0 Jn+1+ses = ˆJ_n+1 (3.9) and similarly ˆ jn+ 2ˆjn−1= ˆjn+1.

The characteristic equation of the classic Jacobsthal and Jacobsthal– Lucas numbers is

x2_{− x − 2 = 0.}

It is known that this equation has two real roots: α = 2 and β = −1.

Thus, Binet’s formulas given in (2.8) and (2.9) are obtained for the classic Jacobsthal and Jacobsthal–Lucas numbers such that

Jn= 1₃(2n− (−1)n)

and

jn= 2n+ (−1)n,

respectively. Now, we will state Binet’s formulas for the Jacobsthal and Jacobsthal–Lucas octonions. Repeated use of (2.8) in (3.2) enables one to write forα∗=7_s=02se_sandβ∗=7_s=0(−1)se_s

ˆ Jn = 7  s=0 Jn+ses = 7  s=0 1 3  2n+s− (−1)n+s  es = 1 3(α ∗_αn_{− β}∗_βn_{) (3.10)}

and similarly making use of (2.9) in (3.3) yields ˆ jn= 7  i=s jn+ses = 7  s=0  2n+s+ (−1)n+s  es = (α∗αn+β∗βn). (3.11)

The formulas in (3.10) and (3.11) are called as Binet’s formulas for the Ja-cobsthal and JaJa-cobsthal–Lucas octonions, respectively.

The recurrence relations for the Jacobsthal octonions and the norm of then th Jacobsthal octonion are expressed in the following theorem.

Theorem 3.1. For n ≥ 1, r ≥ 1, we have the following identities:

ˆ Jn+1+ ˆJn=α∗αn, (3.12) ˆ Jn+1− ˆJn= 1 3(α ∗_αn_+αβ∗_βn_{), (3.13)} ˆ Jn+r+ ˆJn−r= 1 3  α∗_αn−r_α2r_{+ 1}_+β∗_αβn−r+2_{, (3.14)} ˆ Jn+r− ˆJn−r= 1 3  α∗_αn+2_{− α}n−2 _{, (3.15)} N_Jˆ_n₌ 1 9  21845.22n+ 170.2nβn+ 8 , (3.16) whereα∗=7_s=02ses andβ∗=7_s=0(−1)ses

Proof. Consider (3.2) we can write ˆ

Jn+1+ ˆJn = (Jn+1+Jn)e0+ (Jn+2+Jn+1)e1+ (Jn+3+Jn+2)e2

+ (Jn+4+Jn+3)e3+ (Jn+5+Jn+4)e4+ (Jn+6+Jn+5)e5

+ (J_n+7+J_n+6)e6+ (Jn+8+Jn+7)e7.

Using the identities

jn+1+jn= 3. (Jn+1+Jn) = 3.2n (3.17)

in (2.11), the above sum can be calculated as ˆ

Jn+1+ ˆJn= 2ne0+ 2n+1e1+ 2n+2e2+ 2n+3e3+ 2n+4e4+ 2n+5e5

+2n+6e6+ 2n+7e7,

which can be simplified as ˆ

whereα∗ =7_s=02se_s. Repeating same steps as in the proof of (3.12), the proofs of (3.13), (3.14) and (3.15) can be given by referring to (3.17), (2.12) and (2.13), and is omitted here. Now, observing that

N_Jˆ_n₌ 7  s=0 J2 n+s, (3.18)

from the Binet formula (2.8) we have

N_Jˆ_n₌1 9

(2n− (−1)n)2+2n+1− (−1)n+12+· · · +2n+7− (−1)n+72  from which, substituting (3.10) in (3.18), we obtain the desired result

(3.16). 

The recurrence relations for the Jacobsthal–Lucas octonions and the norm of thenth Jacobsthal–Lucas octonion are given in the following the-orem. The following theorem follows from the properties and definitions of the classic Jacobsthal and Jacobsthal–Lucas numbers and Jacobsthal and Jacobsthal–Lucas octonions.

Theorem 3.2. Let n ≥ 1, r ≥ 1 be integer. Then

ˆ jn+1+ ˆjn = 32nα∗, (3.19) ˆ jn+1− ˆjn = 2nα∗+ 2 (−1)n+1β∗, (3.20) ˆ jn+r+ ˆjn−r = 2n−r22r+ 1α∗+ 2 (−1)n−rβ∗, (3.21) ˆ jn+r− ˆjn−r = 2n−r22r+ 1α∗, (3.22) N (ˆjn) = 22n21845 + 2n(−1)n(−170) + 8. (3.23) whereα∗=7_s=02ses andβ∗=7_s=0(−1)ses.

The proofs of the identities (3.19)–(3.23) of this theorem are similar to the proofs of the identities (3.12)–(3.16) of Theorem 3.1, respectively, and are omitted here.

The following theorem deals with two relations between the Jacobsthal and Jacobsthal–Lucas octonions.

Theorem 3.3. Let n ≥ 1 be integer. Then

ˆ Jn+ ˆjn = 2 ˆJn+1, (3.24) 3 ˆJ_n+ ˆj_n = 2n+1α∗, (3.25) ˆ jn+1+ 2ˆjn−1= 9 ˆJn. (3.26) whereα∗=7_s=02ses.

Proof. The following recurrence relation ˆ

Jn+ ˆjn = (Jn+jn)e0+ (Jn+1+jn+1)e1

+ (J_n+2+j_n+2)e2+· · · + (Jn+7+jn+7)e7 (3.27)

can be readily written considering that ˆ

and

ˆ

jn =jne0+jn+1e1+jn+2e2+· · · + jn+7e7.

Notice that J_n +j_n = 2J_n+1 from (2.4) (see [10]), whence it follows that (3.27) can be rewritten as

ˆ

Jn+ ˆjn = 2 ˆJn+1

from which the desired result (3.24) of Theorem3.3. In a similar way we can show the second equality. By using the identity 3J_n+j_n= 2n+1we have

3 ˆJ_n+ ˆj_n = 2n+1e0+ 2e1+ 22e2+· · · + 27e7

 , which is the assertion (3.25) of theorem.

By using the identityjn+1+ 2jn−1= 9Jn from (2.6) (see [10]) we have ˆ

jn+1+ 2ˆjn−1= (jn+1+ 2jn−1)e0+ (jn+2+ 2jn)e1

+ (j_n+3+ 2j_n+1)e2+· · · + (jn+8+ 2jn+6)e7

= 9J_ne0+ 9Jn+1e1+ 9Jn+2e2+· · · + 9Jn+7e7,

which is the assertion (3.26) of theorem. 

The following theorem investigates the multiplications ˆj_nJˆ_n and ˆJ_njˆ_n in the octonion algebraO, which is a real non-commutative normed division algebra.

Theorem 3.4. Let n ≥ 1 be integer. Then

ˆ jnJˆn = 1₃  22n+1+1 32 2n_{1− 2}16_{+ 6}  e0 +22n+2− 264 (−2)n+ 2e1 +22n+3− 84 (−2)n− 2e2 +22n+4+ 468 (−2)n+ 2e3 +22n+5+ 180 (−2)n− 2e4 +22n+6− 444 (−2)n+ 2e5 +22n+7− 180 (−2)n− 2e6 +22n+8+ 156 (−2)n+ 2e7  (3.28) and ˆ Jnjˆn = 1₃  22n+1+1 32 2n_{1− 2}16_{+ 6}  e0 +22n+2+ 264 (−2)n+ 2e1 +22n+3+ 84 (−2)n− 2e2 +22n+4− 468 (−2)n+ 2e3 +22n+5− 180 (−2)n− 2e4 +22n+6+ 444 (−2)n+ 2e5

+22n+7+ 180 (−2)n− 2e6

+22n+8− 156 (−2)n+ 2e7



(3.29) Proof. In view of the multiplication table (1.2) and the definitions (3.2) and (3.3) we get ˆ jnJˆn= (Jnjn− Jn+1jn+1− Jn+2jn+2− Jn+3jn+3− Jn+4jn+4 −Jn+5jn+5− Jn+6jn+6− Jn+7jn+7)e0 + (Jnjn+1+Jn+1jn+Jn+2jn+3− Jn+3jn+2+Jn+4jn+5 −Jn+5jn+4− Jn+6jn+7+Jn+7jn+6)e1 + (J_nj_n+2+J_n+2j_n+J_n+3j_n+1− J_n+1j_n+3+J_n+4j_n+6 −Jn+6jn+4+Jn+5jn+7− Jn+7jn+5)e2 + (J_nj_n+3+J_n+3j_n+J_n+1j_n+2− J_n+2j_n+1+J_n+4j_n+7 −Jn+7jn+4+Jn+6jn+5− Jn+5jn+6)e3 + (J_nj_n+4+J_n+4j_n+J_n+5j_n+1− J_n+1j_n+5+J_n+6j_n+2 −Jn+2jn+6+Jn+7jn+3− Jn+3jn+7)e4 + (J_nj_n+5+J_n+5j_n+J_n+1j_n+4− J_n+4j_n+1+J_n+7j_n+2 −Jn+2jn+7+Jn+3jn+6− Jn+6jn+3)e5 + (J_nj_n+6+J_n+6j_n+J_n+1j_n+7− J_n+7j_n+1+J_n+2j_n+4 −Jn+4jn+2+Jn+5jn+3− Jn+3jn+5)e6 + (J_nj_n+6+J_n+6j_n+J_n+1j_n+7− J_n+7j_n+1+J_n+2j_n+4 −Jn+4jn+2+Jn+5jn+3− Jn+3jn+5)e7

Using the following formula ˆ

jn+pJˆn+q = 1₃22n+p+q− 2n+p(−1)n+q+ 2n+q(−1)n+p− (−1)2n+p+q,

we get the required result (3.28). In the same way, from the multiplication ˆ

Jnˆjn and the formula

ˆ Jn+pjˆn+q= 1 3  22n+p+q+ 2n+p(−1)n+q− 2n+q(−1)n+p− (−1)2n+p+q we reach (3.29). 

Based on the Binet’s formulas given in (3.10) and (3.11) for the Ja-cobsthal and JaJa-cobsthal–Lucas octonions, now we give some mathematical identities named as Cassini’s and Catalan identities for these octonions.

Theorem 3.5 [Cassini’s identities]. For every nonnegative integer numbers n

andr such that r ≤ n we get ˆ Jn+1Jˆn−1− ˆJn2= (−2)n 3 α∗_β∗₊1 2β ∗_α∗ _(3.30)

and ˆ jn+1ˆjn−1− ˆj2n=−3(−2)n α∗_β∗₊1 2β ∗_α∗ _(3.31) whereα∗=7_s=02se_s andβ∗=7_s=0(−1)se_s.

Proof. Let α∗ = 7_s=02se_s and β∗ = 7_s=0(−1)se_s. Using the relation in (3.10) for the Jacobsthal octonions, ˆJ_n+1Jˆ_n−1− ˆJ2

n can be written as ˆ Jn+1Jˆn−1− ˆJn2= 1 9(α ∗_αn+1_{− β}∗_βn+1_)(α∗_αn−1_{− β}∗_βn−1₎ −1 9(α ∗_αn_{− β}∗_βn₎2 _(3.32)

which on making use of some mathematical computations on the r.h.s. of Eq. (3.32) and after simplification yields assertion (3.30). Similarly, using the relation in (3.11) for the Jacobsthal–Lucas octonions, ˆj_n+1jˆ_n−1− ˆj2

n can be

calculated as ˆ

jn+1ˆjn−1− ˆj2n = (α∗αn+1+β∗βn+1)(α∗αn−1+β∗βn−1)

−(α∗_αn_+β∗_βn₎2

which can be simplified as ˆ jn+1ˆjn−1− ˆj2n=−3(−2)n α∗_β∗₊1 2β ∗_α∗_.

Thus, we get the required result in (3.31). 

The next theorem gives us Catalan’s identities for the Jacobsthal and Jacobsthal–Lucas octonions. It is well-known that Cassini’s identity is a spe-cial case of Catalan’s identity. If we take r = 1 in Catalan’s identities for the Jacobsthal and Jacobsthal–Lucas octonions, we get Cassini’s identities for the Jacobsthal and Jacobsthal–Lucas octonions.

Theorem 3.6 [Catalan’s identities]. For every nonnegative integer numbers n

andr such that r ≤ n we get ˆ Jn+rJˆn−r− ˆJn2= (−2)n 9  α∗_β∗_{(1− (−2)}r_{) +β}∗_α∗_{1− (−2)}−r and ˆ jn+rjˆn−r− ˆj2n = (−2)n  α∗_β∗₍₍₋₂₎r_{− 1) + β}∗_α∗₍₋₂₎−r_{− 1} whereα∗=7_s=02se_s andβ∗=7_s=0(−1)se_s.

The proofs of Catalan’s identities for the Jacobsthal and Jacobsthal– Lucas octonions in this theorem are similar to the proofs of the identities (3.30), (3.31) of Theorem3.5respectively, and are omitted here.

4. Conclusions

In this study, we presented the results of some new researches on new classes of octonion numbers associated with the familiar Jacobsthal and Jacobsthal– Lucas numbers. Also, we obtained various results including recurrence rela-tions, summation formulas and Binet’s formulas for these classes of octonion numbers.

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Cennet Bolat C¸ imen

Hacettepe Ankara Chamber of Industry 1st Organized Industrial Zone Vocational School Hacettepe University Ankara Turkey e-mail: cennet.cimen@hacettepe.edu.tr Ahmet ˙Ipek

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

Karaman Turkey e-mail: ahmetipek@kmu.edu.tr Received: June 7, 2016. Revised: December 31, 2016. Accepted: January 31, 2017.