Bicomplex K-Fibonacci Quaternions 3

arXiv:1810.05003v1 [math.NT] 2 Oct 2018 n21,Rmrz[6 endtetekFbnciadtek-Lucas the and numb k-Fibonacci follows: k-Fibonacci the as the for defined identities [16] some numbe Ramirez 2015, obtained k-Fibonacci In [3] of Catarino [8]. properties 2014, gave [7], KöseIn [2] [6], and [5], Bolat in 2010, matrices In and sequences numbers, Fibonacci Here, CrepnigAuthor. *Corresponding Aydın Fügen Torunbalcı quaternions k-Fibonacci Bicomplex nteltrtr 9.I 07 h -ioac sequence k-Fibonacci the 2007, In been have [9]. sequence literature Fibonacci the the in of generalizations of kinds Many Introduction 1. yFlo n lz 4 sfollows as [4] Plaza and Falcon by        Abstract. ubr -ioac utrin iope -ioac qu k-Fibonacci bicomplex quaternion; k-Fibonacci number; Keywords. given. are Catalan’s quaternions identity, Cassini’s t formula, identity, Binet’s Honsberger identity, the k-Fibonacci q Furthermore, and investigated. numbers k-Fibonacci are bicomplex bicomplex with of connected are properties which algebraic some Also, k { sapstv elnme.Rcnl,Flo n lz okdo k- on worked Plaza and Falcon Recently, number. real positive a is D F k,n k,n F k,n } F = n nti ae,bcmlxkFbnciqaenosaedefine are quaternions k-Fibonacci bicomplex paper, this In k, ∈ +1 iope ubr -ioac ubr iope k-Fibona bicomplex number; k-Fibonacci number; Bicomplex { -ioac number k-Fibonacci N 0 F or k,n = = 0 = + { F k F , F i 0 , k,n k,n 1 k, ,k k, , 1 + +1 1 = F + 2 k,n F j 1 + − } 1 , k,n k , n , +2 3 2 + ≥ + 1 F k ,k k, k,n 4 +3 3 + | F dniyfrthese for identity k { k,n 2 F ed’Ocagne’s he 1 + k,n n , aternion. } uaternions ... , n − numbers ∈ quaternions N } th rs. presented sdefined is ers. d. cci (1.1) 2 Fügen Torunbalcı Aydın and

Pk,n = Lk,n + iLk,n + j Lk,n + kLk,n Lk,n, n th { +1 +2 +3 | − k-Lucas number } where i, j, k satisfy the multiplication rules

i2 = j2 = k2 = 1 , i j = j i = k, jk = k j = i, ki = i k = j . − − − − In 2015, Polatlı gave Catalans identity for the k-Fibonacci quaternions [13]. In 2016, Polatlı, Kızılate¸sand Kesim [14] deﬁned split k-Fibonacci and split k-Lucas quaternions (Mk,n) and (Nk,n) respectively as follows:

Mk,n = Fk,n + i Fk,n + j Fk,n + k Fk,n Fk,n, n th { +1 +2 +3 | − k-Fibonacci number } where i, j, k are split quaternionic units which satisy the multiplication rules i2 = 1, j2 = k2 = i j k =1 , i j = ji = k, jk = k j = i, ki = i k = j. − − − − − In 1892, bicomplex numbers were introduced by Corrado Segre, for the ﬁrst time [20]. In 1991, G. Baley Price, the bicomplex numbers gave in his book based on multicomplex spaces and functions [15]. In recent years, fractal structures of this numbers are studied [17], [18], [19], [11]. In 2015, Karaku¸s, Sıddıka Ozkaldı¨ and Aksoyak, Ferdag Kahraman worked on generalized bicomplex numbers and Lie Groups [10]. The set of bicomplex numbers can be expressed by a basis 1 ,i,j,ij as, { }

C2 = q = q1 + iq2 + jq3 + ijq4 q1, q2, q3, q4 R (1.2) { | ∈ } where i,j and ij satisfy the conditions

i2 = 1, j2 = 1, i j = j i. − −

A set of bicomplex numbers C2 is a real vector space with the addition and scalar multiplication operations. The vector space C2 equipped with bicomplex product is a real associative algebra Table 1. Also, the vector space together with properties of multiplication and product of the bicomplex numbers is an commutative algebra. Furthermore, three diﬀerent conjugations can operate on bicomplex numbers [17],[18] as follows:

q = q + i q + j q + i j q = (q + iq )+ j (q + iq ), q C 1 2 3 4 1 2 3 4 ∈ 2 qi∗ = q1 iq2 + jq3 ijq4 = (q1 iq2)+ j (q3 iq4), − − − − (1.3) qj ∗ = q + iq jq ijq = (q + iq ) j (q + iq ), 1 2 − 3 − 4 1 2 − 3 4 qij ∗ = q iq jq + ijq = (q iq ) j (q iq ). 1 − 2 − 3 4 1 − 2 − 3 − 4 Bicomplex k-Fibonacci quaternions 3

The norm of the bicomplex numbers is deﬁned as

2 2 2 2 Nq = q qi = q + q q q +2 j (q q + q q ) , i k × k q| 1 2 − 3 − 4 1 3 2 4 | 2 2 2 2 Nq = q qj = q q + q q +2 i (q q + q q ) , (1.4) j k × k q| 1 − 2 3 − 4 1 2 3 4 | 2 2 2 2 Nq = q qi j = q + q + q + q +2 i j (q q q q ) . i j k × k q| 1 2 3 4 1 4 − 2 3 |

Table 1. Multiplication scheme of bicomplex numbers

x 1 i j ij 1 1 i j ij i i -1 ij -j j j i j -1 -i ij ij -j -i 1

In 2015, the bicomplex Fibonacci and Lucas numbers deﬁned by Nurkan and G¨uven [12] as follows

BF n = Fn + i Fn+1 + j Fn+2 + k Fn+3 (1.5) and BLn = Ln + iLn+1 + jLn+2 + kLn+3 (1.6) where the basis 1, i, j, k satisfy the conditions { } i2 = j2 = 1, k2 =1, i j = ji = k, jk = k j = i, ik = ki = j. − − − In 2018, the bicomplex Fibonacci quaternions deﬁned by Aydın Torunbalcı [1] as follows

QF n = Fn + i Fn+1 + j Fn+2 + i j Fn+3 (1.7) where the basis 1,i, j, ij satisfy the conditions { } i2 = 1, j2 = 1, i j = j i, (i j)2 =1. − − In this paper, the bicomplex k-Fibonacci quaternions and the bicomplex k- Lucas quaternions will be deﬁned respectively, as follows

Fk,n BC = QF =Fk,n + i Fk,n + j Fk,n + i j Fk,n Fk,n, nth { k,n +1 +2 +3 | k-Fibonacci number } and

Lk,n BC = QL =Lk,n + iLk,n + jLk,n + ijLk,n Lk,n, nth { k,n +1 +2 +3 | k-Lucas number } where i2 = 1, j2 = 1, i j = j i, (i j)2 =1. − − 4 F¨ugen Torunbalcı Aydın

The aim of this work is to present in a uniﬁed manner a variety of algebraic properties of the bicomplex k-Fibonacci quaternions as well as both the bicomplex numbers and k-Fibonacci numbers. In particular, using three types of conjugations, all the properties established for bicomplex numbers and bicomplex k-Fibonacci numbers are also given for the bicomplex k- Fibonacci quaternions. In addition, Binet’s Formula, the Honsberger identity, the d’Ocagne’s identity, Cassini’s identity and Catalan’s identity for these quaternions are given.

2. The bicomplex k-Fibonacci numbers The bicomplex k-Fibonacci and k-Lucas numbers can be deﬁne by with the basis 1,i, j, ij , where i, j and i j satisfy the conditions { } i2 = 1, j2 = 1, i j = j i, (i j)2 =1. − − as follows

BCFk,n =(Fk,n + i Fk,n+1)+ j (Fk,n+2 + i Fk,n+3) (2.1) =Fk,n + i Fk,n+1 + j Fk,n+2 + i j Fk,n+3 and BCLk,n =(Lk,n + iLk,n+1)+ j (Lk,n+2 + iLk,n+3) (2.2) =Lk,n + iLk,n+1 + jLk,n+2 + ijLk,n+3. The addition and subtraction of two bicomplex k-Fibonacci numbers are deﬁned by

BCF k,n BCF k,m = (Fk,n Fk,m)+ i (Fk,n Fk,m ) ± ± +1 ± +1 +j (Fk,n+2 Fk,m+2)+ i j (Fk,n+3 Fk,m+3) ± ± (2.3) The multiplication of two bicomplex k-Fibonacci numbers is deﬁned by

BCFk,n BCFk,m = (Fk,n Fk,m Fk,n Fk,m × − +1 +1 Fk,n Fk,m Fk,n Fk,m ) − +2 +2 − +3 +3 +i (Fk,n Fk,m+1 + Fk,n+1 Fk,m Fk,n Fk,m Fk,n Fk,m ) − +2 +3 − +3 +2 +j (Fk,n Fk,m+2 + Fk,n+2 Fk,m (2.4) Fk,n Fk,m Fk,n Fk,m ) − +1 +3 − +3 +1 +i j (Fk,n Fk,m+3 + Fk,n+3 Fk,m +Fk,n+1 Fk,m+2 + Fk,n+2 Fk,m+1) = BCFk,m BCFk,n . ×

3. The bicomplex k-Fibonacci quaternions In this section, firstly the bicomplex k-Fibonacci quaternions will be defined. The bicomplex k-Fibonacci quaternions are defined by using the bicomplex Bicomplex k-Fibonacci quaternions 5 numbers and k-Fibonacci numbers as follows

Fk,n BC = QF =Fk,n + i Fk,n + j Fk,n + i j Fk,n Fk,n, n th { k,n +1 +2 +3 | − k-Fib. number } (3.1) where i2 = 1, j2 = 1, i j = j i, (i j)2 =1. − − Let QF k,n and QF k,m be two bicomplex k-Fibonacci quaternions such that QF k,n = Fk,n + i Fk,n+1 + j Fk,n+2 + i j Fk,n+3 (3.2) and QF k,m = Fk,m + i Fk,m+1 + j Fk,m+2 + i j Fk,m+3. (3.3) The addition and subtraction of two bicomplex k-Fibonacci quaternions are deﬁned in the obvious way,

QF QF = (Fk,n + i Fk,n + j Fk,n + i j Fk,n ) k,n ± k,m +1 +2 +3 (Fk,m + i Fk,m + j Fk,m + i j Fk,m ) ± +1 +2 +3 (3.4) = (Fk,n Fk,m)+ i (Fk,n Fk,m ) ± +1 ± +1 +j (Fk,n Fk,m )+ i j (Fk,n Fk,m ). +2 ± +2 +3 ± +3 Multiplication of two bicomplex k-Fibonacci quaternions is deﬁned by

QF QF = (Fk,n + i Fk,n + j Fk,n + i j Fk,n ) k,n × k,m +1 +2 +3 (Fk,m + i Fk,m+1 + j Fk,m+2 + i j Fk,m+3) = [Fk,n Fk,m Fk,n Fk,m − +1 +1 Fk,n Fk,m + Fk,n Fk,m ] − +2 +2 +3 +3 +i [Fk,n Fk,m+1 + Fk,n+1 Fk,m Fk,n Fk,m Fk,n Fk,m ] (3.5) − +2 +3 − +3 +2 +j [Fk,n Fk,m Fk,n Fk,m +2 − +1 +3 +Fk,n Fk,m Fk,n Fk,m ] +2 − +3 +1 +i j [Fk,n Fk,m+3 + Fk,n+1 Fk,m+2 +Fk,n+2 Fk,m+1 + Fk,n+3 Fk,m] = QF QF . k,m × k,n The scaler and the bicomplex vector parts of the bicomplex k-Fibonacci quaternion (QF k,n) are denoted by

S QF k,n = Fk,n and V QF k,n = i Fk,n+1 + j Fk,n+2 + i j Fk,n+3. (3.6)

Thus, the bicomplex k-Fibonacci quaternion QF k,n is given by

QF k,n = S QF k,n + V QF k,n . Three kinds of conjugation can be defined for bicomplex numbers [18]. There- fore, conjugation of the bicomplex k-Fibonacci quaternion is defined in five different ways as follows

1 (QF )∗ =Fk,n i Fk,n+1 + j Fk,n+2 i j Fk,n+3, (3.7) k,n − − 2 (QF )∗ =Fk,n + i Fk,n+1 j Fk,n+2 i j Fk,n+3, (3.8) k,n − − 3 (QF )∗ =Fk,n i Fk,n+1 j Fk,n+2 + i j Fk,n+3, (3.9) k,n − − 6 F¨ugen Torunbalcı Aydın

In the following theorem, some properties related to the conjugations of the bicomplex k-Fibonacci quaternions are given.

1 2 3 Theorem 3.1. Let (QF k,n)∗ , (QF k,n)∗ and (QF k,n)∗ , be three kinds of conjugation of the bicomplex k-Fibonacci quaternion. In this case, we can give the following relations:

1 QF (QF )∗ = Fk, n Fk, n +2 j Fk, n , (3.10) k,n k,n 2 +1 − 2 +5 2 +3 2 2 2 2 2 QF (QF )∗ = (F F + F F ) k,n k,n k,n − k,n+1 k,n+2 − k,n+3 (3.11) +2 i (2 Fk,n Fk,n+1 + k Fk,2n+3),

3 n+1 QF (QF )∗ = (Fk, n + Fk, n )+2 i j ( 1) k, (3.12) k,n k,n 2 +1 2 +5 − Proof. (3.10): By the Eq.(3.2) and (3.7) we get,

1 2 2 2 2 QF (QF )∗ = (F + F F F ) k,n k,n k,n k,n+1 − k,n+2 − k,n+3 +2 j ( Fk,n Fk,n+2 + Fk,n+1 Fk,n+3) = Fk, n Fk, n +2 j Fk, n . 2 +1 − 2 +5 2 +3 2 2 where Fk,n + Fk,n = Fk,2n+1 and Fk,n Fk,m 1 + Fk,n+1 Fk,m = Fk,n+m are +1 − used [4].

(3.11): By the Eq.(3.2) and (3.8) we get,

2 2 2 2 2 QF (QF )∗ = (F F + F F ) k,n k,n k,n − k,n+1 k,n+2 − k,n+3 +2 i (Fk,n Fk,n+1 + Fk,n+2 Fk,n+3) = (F 2 F 2 + F 2 F 2 ) k,n − k,n+1 k,n+2 − k,n+3 +2 i (2 Fk,n Fk,n+1 + k Fk,2n+3). (3.12): By the Eq.(3.2) and (3.9) we get,

3 2 2 2 2 QF k,n (QF k,n)∗ = (Fk,n + Fk,n+1 + Fk,n+2 + Fk,n+3) +2 i j (Fk,n Fk,n+3 Fk,n+1 Fk,n+2) − n+1 = Fk, n + Fk, n +2 i j ( 1) k. 2 +1 2 +5 − 2 n+1 where Fk,n Fk,n 1 Fk,n+1 = ( 1) and Fk,n Fk,n+3 Fk,n+1 Fk,n+2 = − − − − k ( 1)n+1 are used [4]. −

Therefore, the norm of the bicomplex k-Fibonacci quaternion QF k,n is deﬁned in three diﬀerent ways as follows

1 1 2 N QF )∗ = QF (QF )∗ ( k,n k k,n × k,n k = (F 2 + F 2 ) (F 2 + F 2 ) | k,n k,n+1 − k,n+2 k,n+3 (3.13) +2 j ( Fk,n Fk,n + Fk,n Fk,n ) +2 +1 +3 | = Fk, n Fk, n +2 j Fk, n , | 2 +1 − 2 +5 2 +3| 2 2 2 N QF )∗ = QF (QF )∗ ( k,n k k,n × k,n k = (F 2 F 2 ) + (F 2 F 2 ) (3.14) | k,n − k,n+1 k,n+2 − k,n+3 +2 i Fk,n Fk,n + k Fk, n , +1 2 +3 | Bicomplex k-Fibonacci quaternions 7

3 3 2 N QF )∗ = QF (QF )∗ ( k,n k k,n × k,n k = (F 2 + F 2 ) + (F 2 + F 2 ) | k,n k,n+1 k,n+2 k,n+3 (3.15) +2 i j (Fk,n Fk,n+3 Fk,n+1 Fk,n+2) − n+1 | = Fk, n + Fk, n +2 i j ( 1) k . | 2 +1 2 +5 − | In the following theorem, some properties related to the bicomplex k- Fibonacci quaternions are given.

Theorem 3.2. Let QF k,n be the bicomplex k-Fibonacci quaternion. In this case, we can give the following relations:

QF k,n + kQF k,n+1 = QF k,n+2, (3.16)

2 2 2 2 2 (QF ) = (F F F F ) k,n k,n − k,n+1 − k,n+2 − k,n+3 +2 i (Fk,n Fk,n Fk,n Fk,n ) +1 − +2 +3 (3.17) +2 j (Fk,n Fk,n Fk,n Fk,n ) +2 − +1 +3 +2 i j (Fk,n Fk,n+3 + Fk,n+1 Fk,n+2), 2 2 (QF ) + (QF ) = QF + (k Fk, n Fk, n ) k,n k,n+1 k,2n+1 2 +6 − 2 +3 +i (Fk, n 2 Fk, n ) 2 +2 − 2 +6 +j (Fk, n 2 Fk, n )+ i j (3 Fk, n ), 2 +3 − 2 +5 2 +4 (3.18) 2 2 (QF k,n+1) (QF k,n 1) = k [ QF k,2n Fk,2n+2 + k Fk,2n+5 − − − +i (Fk, n 2 Fk, n ) 2 +1 − 2 +5 (3.19) +j ( Fk, n 2 k Fk, n ) − 2 +2 − 2 +3 +i j (3 Fk,2n+3)],

QF iQF + jQF ijQF = Fk,n + Fk,n Fk,n k,n − k,n+1 k,n+2 − k,n+3 +2 − +4 Fk,n+6 +2 jLk,n+3. − (3.20) QF iQF jQF ijQF = Fk,n + Fk,n + Fk,n k,n − k,n+1 − k,n+2 − k,n+3 +2 +4 Fk,n +2 i Fk,n − +6 +5 +2 j Fk,n 2 i j Fk,n +4 − +3 = Lk,n k Fk,n +2 i Fk,n +1 − +5 +5 +2 j Fk,n 2 i j Fk,n . +4 − +3 (3.21) Proof. (3.16): By the Eq.(3.2) we get,

QF k,n + kQF k,n+1 = (Fk,n + k Fk,n+1)+ i (Fk,n+1 + k Fk,n+2) +j (Fk,n+2 + k Fk,n+3)+ i j (Fk,n+3 + k Fk,n+4) = Fk,n+2 + i Fk,n+3 + j Fk,n+4 + i j Fk,n+5 = QF k,n+2. (3.17): By the Eq.(3.2) we get, 2 2 2 2 2 (QF ) = (F F F + F ) k,n k,n − k,n+1 − k,n+2 k,n+3 +2 i (Fk,n Fk,n Fk,n Fk,n ) +1 − +2 +3 +2 j (Fk,n Fk,n Fk,n Fk,n ) +2 − +1 +3 +2 i j (Fk,n Fk,n+3 + Fk,n+1 Fk,n+2). 8 F¨ugen Torunbalcı Aydın

(3.18): By the Eq.(3.2) we get,

2 2 (QF ) + (QF ) = (Fk, n Fk, n Fk, n + Fk, n ) k,n k,n+1 2 +1 − 2 +3 − 2 +5 2 +7 +2 i (Fk, n Fk, n ) 2 +2 − 2 +6 +2 j (Fk, n Fk, n ) 2 +3 − 2 +5 +2 i j (2 Fk,2n+4) = (Fk,2n+1 + i Fk,2n+2 + j Fk,2n+3 + i j Fk,2n+4) Fk, n Fk, n + Fk, n − 2 +3 − 2 +5 2 +7 +i (Fk, n 2 Fk, n )+ j (Fk, n 2 Fk, n ) 2 +2 − 2 +6 2 +3 − 2 +5 +i j (3 Fk,2n+4) = QF + (k Fk, n Fk, n ) k,2n+1 2 +6 − 2 +3 +i (Fk, n 2 Fk, n )+ j (Fk, n 2 Fk, n ) 2 +2 − 2 +6 2 +3 − 2 +5 +i j (3 Fk,2n+4).

(3.19): By the Eq.(3.2) we get,

2 2 2 2 2 2 (QF k,n+1) (QF k,n 1) = [ (Fk,n+1 Fk,n 1) (Fk,n+2 Fk,n) − − − − − − (F 2 F 2 ) + (F 2 F 2 )] − k,n+3 − k,n+1 k,n+4 − k,n+2 +2 i [ (Fk,n+1 Fk,n+2 Fk,n 1 Fk,n) − − (Fk,n Fk,n Fk,n Fk,n )] − +3 +4 − +1 +2 +2 j [ (Fk,n+1 Fk,n+3 Fk,n 1 Fk,n+1) − − (Fk,n Fk,n Fk,n Fk,n )] − +2 +4 − +2 +2 i j [ (Fk,n+1 Fk,n+4 Fk,n 1 Fk,n+2) − − +(Fk,n Fk,n Fk,n Fk,n )] +2 +3 − +1 = k ( Fk,2n k Fk,2n+2 k Fk,2n+4 + k Fk,2n+6) − − 2 +2 i (k Fk, n k Fk, n )+2 j ( k Fk, n ) 2 +1 − 2 +5 − 2 +3 +2 i j (2 k Fk,2n+3) = k [ QF Fk, n + k Fk, n k,2n − 2 +2 2 +5 +i (Fk, n 2 Fk, n ) 2 +1 − 2 +5 +j ( Fk, n 2 k Fk, n ) − 2 +2 − 2 +3 +i j (3 Fk,2n+3)].

(3.20): By the Eq.(3.2) we get,

QF iQF jQF ijQF = (Fk,n + Fk,n k Fk,n ) k,n − k,n+1 − k,n+2 − k,n+3 +2 − +5 +2 i Fk,n+5 +2 j Fk,n+4 2 i j Fk,n − +3 = Lk,n k Fk,n +2 i Fk,n +1 − +5 +5 +2 j Fk,n 2 i j 2 j Fk,n . +4 − +3 (3.21): By the Eq.(3.2) we get,

QF iQF + jQF ijQF = (Fk,n + Fk,n Fk,n k,n − k,n+1 k,n+2 − k,n+3 +2 − +4 Fk,n )+2 j (Fk,n + Fk,n ) − +6 +2 +4 = Lk,n (Fk,n + Fk,n ) +1 − +4 +6 +2 jLk,n+3. Bicomplex k-Fibonacci quaternions 9

Theorem 3.3. Let QF k,n = (Fk,n, Fk,n+1, Fk,n+2, Fk,n+3) and QLk,n = (Lk,n,Lk,n+1,Lk,n+2,Lk,n+3) be the bicomplex k-Fibonacci quaternion and the bicomplex k-Lucas quaternion respectively. The following relations are satisﬁed

QF k,n+1 + QF k,n 1 = Lk,n + iLk,n+1 + jLk,n+2 + ijLk,n+3 = QLk,n, − (3.22)

QF k,n+2 QF k,n 2 = Lk,n + iLk,n+1 + jLk,n+2 + ijLk,n+3 = kQLk,n. − − (3.23)

Proof.

QF k,n+1 + QF k,n 1 = (Fk,n+1 + Fk,n 1)+ i (Fk,n+2 + Fk,n) − − +j (Fk,n+3 + Fk,n+1) +i j (Fk,n+4 + Fk,n+2) = (Lk,n + iLk,n+1 + jLk,n+2 + ijLk,n+3) = QLk,n. and

QF k,n+2 QF k,n 2 = (Fk,n+2 Fk,n 2)+ i (Fk,n+3 Fk,n 1) − − − − − − +j (Fk,n Fk,n)+ i j (Fk,n Fk,n ) +4 − +5 − +1 = k (Lk,n + iLk,n+1 + jLk,n+2 + ijLk,n+3) = kQLk,n.

Theorem 3.4. For n,m 0 the Honsberger identity for the bicomplex k- ≥ Fibonacci quaternions QF k,n and QF k,m is given by

QF QF + QF QF = QF Fk,n m + k Fk,n m k,n k,m k,n+1 k,m+1 k,n+m+1 − + +3 + +6 +i (Fk,n m 2 Fk,n m ) + +2 − + +6 +j (Fk,n m 2 Fk,n m ) + +3 − + +5 +i j (3 Fk,n+m+4). (3.24) 10 F¨ugen Torunbalcı Aydın

Proof. (3.24): By the Eq.(3.2) we get,

QF k,n QF k,m + QF k,n+1 QF k,m+1 = [ (Fk,nFk,m + Fk,n+1Fk,m+1) (Fk,n Fk,m + Fk,n Fk,m ) − +1 +1 +2 +2 (Fk,n Fk,m + Fk,n Fk,m ) − +2 +2 +3 +3 +(Fk,n+3Fk,m+3 + Fk,n+4Fk,m+4)] +i [ (Fk,nFk,m+1 + Fk,n+1Fk,m+2) +(Fk,n+1Fk,m + Fk,n+2Fk,m+1) (Fk,n Fk,m + Fk,n Fk,m ) − +2 +3 +3 +4 (Fk,n Fk,m + Fk,n Fk,m )] − +3 +2 +4 +3 +j [ (Fk,nFk,m+2 + Fk,n+1Fk,m+3) +(Fk,n+2Fk,m + Fk,n+3Fk,m+1) (Fk,n Fk,m + Fk,n Fk,m ) − +1 +3 +2 +4 (Fk,n Fk,m + Fk,n Fk,m )] − +3 +1 +4 +2 +i j [ (Fk,nFk,m+3 + Fk,n+1Fk,m+4) +(Fk,n+1Fk,m+2 + Fk,n+2Fk,m+3) +(Fk,n+2Fk,m+1 + Fk,n+3Fk,m+2) +(Fk,n+3Fk,m + Fk,n+4Fk,m+1)] = (Fk,n m Fk,n m Fk,n m + +1 − + +3 − + +5 +Fk,n m )+2 i (Fk,n m Fk,n m ) + +7 + +2 − + +6 +2 j (Fk,n m Fk,n m ) + +3 − + +5 +i j (4 Fk,n+m+4) = (Fk,n+m+1 + i Fk,n+m+2 + j Fk,n+m+3 + i j Fk,n m ) Fk,n m + k Fk,n m + +4 − + +3 + +6 +i (Fk,n m 2 Fk,n m ) + +2 − + +6 +j (Fk,n m 2 Fk,n m ) + +3 − + +5 +i j (3 Fk,n+m+4) = QF Fk,n m + k Fk,n m k,n+m+1 − + +3 + +6 +i (Fk,n m 2 Fk,n m ) + +2 − + +6 +j (Fk,n m 2 Fk,n m ) + +3 − + +3 +i j (3 Fk,n+m+4).

where the identity Fk,nFk,m + Fk,n+1Fk,m+1 = Fk,n+m+1 was used [4].

Theorem 3.5. For n,m 0 the D’Ocagne’s identity for the bicomplex k- ≥ Fibonacci quaternions QF k,n and QF k,m is given by

m 2 3 QF k,n QF k,m+1 QF k,n+1 QF k,m = ( 1) Fk,n m [ 2 (k + 2) j + (k +2 k) i j ]. − − − (3.25) Bicomplex k-Fibonacci quaternions 11

Proof. (3.25): By the Eq.(3.2) we get,

QF QF QF QF = [ (Fk,nFk,m Fk,n Fk,m) k,n k,m+1 − k,n+1 k,m +1 − +1 (Fk,n Fk,m Fk,n Fk,m ) − +1 +2 − +2 +1 (Fk,n Fk,m Fk,n Fk,m ) − +2 +3 − +3 +2 +(Fk,n Fk,m Fk,n Fk,m )] +3 +4 − +4 +3 + i [ (Fk,nFk,m Fk,n Fk,m ) +2 − +1 +1 +(Fk,n Fk,m Fk,n Fk,m) +1 +1 − +2 (Fk,n Fk,m Fk,n Fk,m )] − +3 +3 − +4 +2 + j [ (Fk,nFk,m Fk,n Fk,m ) +3 − +1 +2 +(Fk,n Fk,m Fk,n Fk,m) +2 +1 − +3 (Fk,n Fk,m Fk,n Fk,m ) − +1 +4 − +2 +3 (Fk,n Fk,m Fk,n Fk,m )] − +3 +2 − +4 +1 + i j [ (Fk,nFk,m Fk,n Fk,m ) +4 − +1 +3 +(Fk,n Fk,m Fk,n Fk,m ) +1 +3 − +2 +2 +(Fk,n Fk,m Fk,n Fk,m ) +2 +2 − +3 +1 +(Fk,n+3Fk,m+1Fk,n+4Fk,m)] m 2 = ( 1) Fk,n m [ 2 (k + 2) j −+(k3 +2−k) i j ].

n where the identity Fk,mFk,n+1 Fk,m+1Fk,n = ( 1) Fk,m n is used [5]. − − −

Theorem 3.6. Let QF k,n be the bicomplex k-Fibonacci quaternion.Then, we have the following identities

n 1 QF = ( QF + QF QF QF ), (3.26) X k,s k k,n+1 k,n − k,1 − k,0 s=1

n 1 QF k,2s 1 = (QF k,2n QF k,0), (3.27) X − k − s=1

n 1 QF = (QF QF ). (3.28) X 2s k k,2n+1 − k,1 s=1 12 F¨ugen Torunbalcı Aydın

n 1 Proof. (3.26) Since Fk,i = (Fk,n+1 + Fk,n 1) [4], we get Pi=1 k − n n n n n Q = F + i F + ε F + i j F X F k,s X k,s X k,s+1 X k,s+2 X k,s+3 s=1 s=1 s=1 s=1 s=1 1 = (Fk,n + Fk,n 1)+ i (Fk,n + Fk,n k 1) k { +1 − +2 +1 − − 2 + j [ Fk,n + Fk,n (k + 1) k ] +3 +2 − − 3 2 + i j [ Fk,n + Fk,n (k +2 k) (k + 1) ] +4 +3 − − } 1 = (Fk,n + i Fk,n + j Fk,n + i j Fk,n ) k { +1 +2 +3 +4 + (Fk,n + i Fk,n+1 + j Fk,n+2 + i j Fk,n+3) [1+(k + 1) i + (k2 + k + 1) j + (k3 + k2 +2 k + 1) ] − } 1 = ( QF + QF QF QF ). k k,n+1 k,n − k,1 − k,0 n n 1 1 (3.27): Using Fk,2i+1 = k Fk,2n+2 and Fk,2i = k (Fk,2n+1 1) [4], iP=1 iP=1 − we get n 1 QF k,2s 1 = k (Fk,2n)+ i (Fk,2n+1 1)+ j (Fk,2n+2 k) sP=1 − { − − 2 +i j (Fk,2n+3 (k +1)) 1 − } = k [ Fk,2n + i Fk,2n+1 + j Fk,2n+2 + i j Fk,2n+3] {(0+ i + k j + 2 (k2 + 1) i j ) −1 } = k QF k,2n [Fk,0 + i Fk,1 + j Fk,2 + i j Fk,3] 1 { − } = ( QF QF ) . k k,2n − k,0 n 1 (3.28): Using Fk,2i = k (F2n+1 1) [4], we obtain iP=1 − n 1 2 QF k,2s = k (Fk,2n+1 1)+ i (Fk,2n+2 k)+ j (Fk,2n+3 (k + 1)) sP=1 { − − − 2 +i j (Fk,2n+4 (k +2 k)) 1 − } = k (Fk,2n+1 + i Fk,2n+2 + j Fk,2n+3 + i j Fk,2n+4) { (1 + ki + (k2 + 1) j + (k3 +2 k) i j ) 1 − } = k QF k,2n+1 (Fk,1 + i Fk,2 + j Fk,3 + i j Fk,4) 1 { − } = ( QF QF ) . k k,2n+1 − k,1

Theorem 3.7. Binet’s Formula Let QF k,n be the bicomplex k-Fibonacci quaternion. For n 1, Binet’s formula for these quaternions is as follows: ≥

1 n n QF = αˆ α βˆ β (3.29) k,n α β − − where 2 3 k+√k2 +4 αˆ =1+ i α + j α + i j α , α = 2 , Bicomplex k-Fibonacci quaternions 13

ˆ 2 3 k √k2+4 β =1+ iβ + jβ + ijβ , β = − 2 , α + β = k , α β = √k2 +4 , αβ = 1. − −

Proof. By using the Binet formula for k-Fibonacci number [5], we obtain

QF k,n = Fk,n + i Fk,n+1 + j Fk,n+2 + i j Fk,n+3

αn βn αn+1 βn+1 αn+2 βn+2 αn+3 βn+3 = − + i ( − )+ j ( − )+ i j ( − ) √k2+4 √k2+4 √k2+4 √k2+4

αn (1+i α+j α2+i j α3) βn (1+iβ+jβ2+ijβ3) = − √k2+4

= 1 (ˆα αn ββˆ n). √k2+4 − whereα ˆ = 1+ i α + j α2 + i j α3, βˆ = 1+ iβ + jβ2 + ijβ3.

Theorem 3.8. Cassini’s Identity Let QF be the bicomplex k-Fibonacci quaternion. For n 1, Cassini’s k,n ≥ identity for QF k,n is as follows:

2 n 2 3 QF k,n 1 QF k,n+1 (QF k,n) = ( 1) [ 2(k + 2) j + (k +2 k) i j ]. (3.30) − − − Proof. (3.30): By using (3.2) we get

2 2 QF k,n 1 QF k,n+1 ( QF k,n) = [ (Fk,n 1Fk,n+1 Fk,n) − − − − 2 (Fk,nFk,n+2 Fk,n+1) − − 2 (Fk,n+1Fk,n+3 Fk,n+2) − − 2 +(Fk,n Fk,n F )] +2 +4 − k,n+3 +i [ (Fk,n 1Fk,n+2 Fk,nFk,n+1) − − (Fk,n Fk,n Fk,n Fk,n )] − +1 +4 − +2 +3 +j [ (Fk,n 1Fk,n+3 Fk,nFk,n+2) − − (Fk,nFk,n Fk,n Fk,n ) − +4 − +1 +3 +(Fk,n Fk,n Fk,n Fk,n) +1 +1 − +2 (Fk,n Fk,n Fk,n Fk,n )] − +2 +2 − +3 +1 +i j (Fk,n 1Fk,n+4 Fk,nFk,n+3) n − 2− 3 = ( 1) Fk,n m[ 2(k + 2) j + (k +2 k) i j ]. − − 2 n where the identities of the k-Fibonacci numbers Fk,n 1Fk,n+1 Fk,n = ( 1) − − − [5]. Furthermore;

n Fk,n 1Fk,n+2 Fk,nFk,n+1 = k ( 1)  − − 2− n Fk,n 1Fk,n+3 Fk,nFk,n+2 = (k +1)( 1) ,  − − 2 − n  Fk,n+1Fk,n+3 Fk,nFk,n+4 = (k +1)( 1) , − 3 − n  Fk,n 1Fk,n+4 Fk,nFk,n+3 = (k +2 k) ( 1)  − − − are used. 14 F¨ugen Torunbalcı Aydın

Theorem 3.9. Catalan’s Identity Let QF be the bicomplex k-Fibonacci quaternion. For n 1, Catalan’s k,n+r ≥ identity for QF k,n+r is as follows: 2 n+r 2 3 QF k,n+r 1 QF k,n+r+1 (QF k,n+r) = ( 1) [ 2(k + 2) j + (k +2 k) i j ]. − − − (3.31) Proof. (3.31): By using (3.2) we get 2 2 QF k,n+r 1 QF k,n+r+1 (QF k,n+r) = (Fk,n+r 1Fk,n+r+1 Fk,n+r) − − − − 2 (Fk,n+rFk,n+r+2 Fk,n+r+1) − − 2 (Fk,n+r+1Fk,n+r+3 Fk,n+r+2) − 2 +(Fk,n r Fk,n r F ) + +2 + +4 − k,n+r+3 +i [ (Fk,n+r 1Fk,n+r+2) (Fk,n+rFk,n+r+1) − − (Fk,n r Fk,n r Fk,n r Fk,n r ) − + +1 + +4 − + +2 + +3 +j [ (Fk,n+r 1Fk,n+r+3 Fk,n+rFk,n+r+2) − − (Fk,n rFk,n r Fk,n r Fk,n r ) − + + +4 − + +1 + +3 +(Fk,n r Fk,n r Fk,n r Fk,n r) + +1 + +1 − + +2 + (Fk,n r Fk,n r Fk,n r Fk,n r )] − + +2 + +2 − + +3 + +1 +i j [ (Fk,n+r 1Fk,n+r+4 Fk,n+rFk,n+r+3) − − +(Fk,n rFk,n r Fk,n r Fk,n r ) + + +3 − + +1 + +2 +(Fk,n+r+2Fk,n+r+1 Fk,n+r+3Fk,n+r)] = ( 1)n+r [ 2 (k2 + 2) j −+ (k3 +2 k) i j ] − 2 where the identity of the k-Fibonacci numbers Fk,n+r 1Fk,n+r+1 Fk,n+r = − − ( 1)n+r is used [5]. Furthermore; − n+r Fk,n+r 1 Fk,n+r+2 + Fk,n+r Fk,n+r+1 = ( 1) k,  − − n+r 2 Fk,n+r 1 Fk,n+r+3 Fk,n+r Fk,n+r+2 = ( 1) (k + 1),  − − − n+r 2  Fk,n+r+1 Fk,n+r+3 Fk,n+r Fk,n+r+4 = ( 1) (k + 1),  − − n+r 3  Fk,n+r 1 Fk,n+r+4 Fk,n+r Fk,n+r+3 = ( 1) (k +2 k), − − − n+r+1  Fk,n+r Fk,n+r+3 Fk,n+r+1 Fk,n+r+2 = ( 1) k,  − − n+r  Fk,n+r+2 Fk,n+r+1 Fk,n+r+3 Fk,n+r = ( 1) k .  − − are used.

4. Conclusion In this paper, a number of new results on bicomplex k-Fibonacci quaternions were derived. This study ﬁlls the gap in the literature by providing the bicomplex k-Fibonacci quaternion using deﬁnitions of the bicomplex number [18], k-Fibonacci quaternion [16] and the bicomplex Fibonacci quaternion [1].

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F¨ugen Torunbalcı Aydın Yildiz Technical University Faculty of Chemical and Metallurgical Engineering Department of Mathematical Engineering Davutpasa Campus, 34220 Esenler, Istanbul, TURKEY e-mail: [email protected] ; [email protected]