Journal of Computer Science & Computational Mathematics, Volume 9, Issue 4, December 2019 DOI: 10.20967/jcscm.2019.04.003 Hyperbolic k-Fibonacci Quaternions Fügen Torunbalci Aydin Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Istanbul, TURKEY [email protected] Abstract: In this paper, hyperbolic k-Fibonacci quaternions are de- where fined. Also, some algebraic properties of hyperbolic k-Fibonacci e 2 = e 2 = e 2 = e e e = 1, quaternions which are connected with hyperbolic numbers and k- 1 2 3 1 2 3 − Fibonacci numbers are investigated. Furthermore, d’Ocagne’s iden- e1e2 = e2e1 = e3,e2e3 = e3e2 = e1, tity, the Honsberger identity, Binet’s formula, Cassini’s identity and − − e3e1 = e1e3 = e2, n 1. Catalan’s identity for these quaternions are given. − ≥ In 2015, Ramirez [11] defined the k-Fibonacci and the k- Keywords: Fibonacci number, k-Fibonacci number, k-Fibonacci Lucas quaternions as follows: quaternion, k-Fibonacci dual quaternion, hyperbolic quaternion. D = F + iF + jF + kF k,n { k,n k,n+1 k,n+2 k,n+3 | F , n th k-Fib. number , 1. Introduction k,n − } The quaternions constitute an extension of complex numbers P = L + iL + jL + kL into a four-dimensional space and can be considered as four- k,n { k,n k,n+1 k,n+2 k,n+3 | dimensional vectors, in the same way that complex numbers Lk,n, n th k-Lucas number are considered as two-dimensional vectors. − } Quaternions were first described by Irish mathematician where i, j, k satisfy the multiplication rules (2). Hamilton in 1843. Hamilton [1] introduced a set of quater- In 2015, Polatlı Kızılate¸sand Kesim [12] defined split k- nions which can be represented as Fibonacci and split k-Lucas quaternions (Mk,n) and (Nk,n) respectively as follows: H = q = q0 + iq1 + jq2 + kq3 qi R, i = 0,1,2,3 { | ∈ } M = F + iF + jF + kF (1) k,n { k,n k,n+1 k,n+2 k,n+3 | F , n th k-Fibonacci number k,n − } where where i, j, k are split quaternionic units which satisy the mul- i2 = j2 = k2 = 1, ij = ji = k, jk = kj = i, tiplication rules − − − ki = ik = j. 2 2 2 − i = 1, j = k = ijk = 1, (2) − ij = ji = k, jk = kj = i, ki = ik = j. − − − − Several authors worked on different quaternions and their In 2018, Aydın Torunbalcı [13] defined k-Fibonacci dual generalizations ([2], [3], [4], [5], [6], [7]). quaternions as follows: Horadam [8] defined complex Fibonacci and Lucas quater- nions as follows DFk,n = DFk,n = Fk,n + iFk,n+1 + jFk,n+2 + kFk,n+3 { | Fk,n, n th k-Fibonacci number Qn = Fn + iFn+1 + jFn+2 + kFn+3 − } where i, j, k are dual quaternionic units which satisy the mul- and tiplication rules Kn = Ln + iLn+1 + jLn+2 + kLn+3 i2 = j2 = k2 = 0, ij = ji = jk = kj = ki = ik = 0. where − − − i2 = j2 = k2 = 1, ij = ji = k, jk = kj = i, In 2018, Kösal [14] defined hyperbolic quaternions (K) as − − − follows: ki = ik = j. − K = q =a0 + ia1 + ja2 + ka3 a0, a1, a2, a3 R, In 2012, Halıcı [9] gave generating functions and Binet’s { | ∈ i,j,k / R formulas for Fibonacci and Lucas quaternions. In 2013, ∈ } Halıcı [10] defined complex Fibonacci quaternions as fol- where i,j,k are hyperbolic quaternionic units which satisy lows: the multiplication rules 2 2 2 HF C = Rn = Cn + e1Cn+1 + e2Cn+2 + e3Cn+3 i = j = k = 1, { | C = F + iF ,i2 = 1 ij = k = ji, jk = i = kj, ki = j = ik. n n n+1 − } − − − 64 Hyperbolic k-Fibonacci Quaternions In this paper, the hyperbolic k-Fibonacci quaternions and Then, the addition and subtraction of two hyperbolic k- the hyperbolic k-Lucas quaternions will be defined respec- Fibonacci quaternions are defined in the obvious way, tively, as follows HFk,n HFk,m = (Fk,n + iFk,n+1 + jFk,n+2 + kFk,n+3) HFk,n = q = Fk,n + iFk,n+1 + jFk,n+2 + kFk,n+3 ± { | (Fk,m + iFk,m+1 + jFk,m+2 + kFk,m+3) Fk,n, nthk-Fib. num. ± } =(Fk,n Fk,m) + i(Fk,n Fk,m ) (3) ± +1 ± +1 + j(Fk,n+2 Fk,m+2) + k(Fk,n+3 Fk,m+3). (10) and ± ± Multiplication of two hyperbolic k-Fibonacci quaternions HLk,n = q = Lk,n + iLk,n+1 + jLk,n+2 + kLk,n+3 { | is defined by L , nthk-Lucas num. k,n } (4) HFk,nHFk,m = (Fk,n + iFk,n+1 + jFk,n+2 + kFk,n+3) where (Fk,m + iFk,m+1 + jFk,m+2 + kFk,m+3) =(Fk,nFk,m + Fk,n Fk,m + Fk,n Fk,m i2 = j2 = k2 = 1, +1 +1 +2 +2 (5) + Fk,n Fk,m ) ij = k = ji, jk = i = kj, ki = j = ik. +3 +3 − − − + i(Fk,nFk,m+1 + Fk,n+1Fk,m + Fk,n+2Fk,m+3 The aim of this work is to present in a unified manner Fk,n+3Fk,m+2) a variety of algebraic properties of both the hyperbolic k- − + j(F F + F F F F Fibonacci quaternions as well as the k-Fibonacci quater- k,n k,m+2 k,n+2 k,m − k,n+1 k,m+3 nions and hyperbolic quaternions. In accordance with these + Fk,n+3Fk,m+1) definitions, we given some algebraic properties and Binet’s + k(Fk,nFk,m+3 + Fk,n+3Fk,m + Fk,n+1Fk,m+2 formula for hyperbolic k-Fibonacci quaternions. More- Fk,n+2Fk,m+1) over, some sums formulas and some identities such as − =HFk,mHFk,n. (11) d’Ocagne’s, Honsberger, Cassini’s and Catalan’s identities 6 for these quaternions are given. The scaler and the vector parts of hyperbolic k-Fibonacci quaternion HFk,n are denoted by 2. Hyperbolic k-Fibonacci quaternions S F = Fk,n The k-Fibonacci sequence Fk,n n N [11] is defined as H k,n { } ∈ (12) VHFk,n = iFk,n+1 + jFk,n+2 + kFk,n+3. Fk,0 = 0,Fk,1 = 1 Fk,n+1 = kFk,n + Fk,n 1, n 1 Thus, hyperbolic k-Fibonacci quaternion HFk,n is given − ≥ or by Fk,n = S F + V F . The conjugate of hyperbolic H H k,n D k,n 2 3 4 2 k F F Fk,n n N = 0,1,k,k + 1,k + 2k,k + 3k + 1,... -Fibonacci quaternion H k,n is denoted by H k,n and it is { } ∈ { } (6) HF k,n = Fk,n iFk,n+1 jFk,n+2 kFk,n+3. (13) Here, k is a positive real number. In this section, firstly hy- − − − perbolic k-Fibonacci quaternions will be defined. Hyperbolic The norm of hyperbolic k-Fibonacci quaternion HFk,n is k-Fibonacci quaternions are defined by using the k-Fibonacci defined as follows numbers and hyperbolic quaternionic units as follows 2 HFk,n =HFk,nHF k,n k k (14) HFk,n = q = Fk,n + iFk,n+1 + jFk,n+2 + kFk,n+3 2 2 2 2 { | =Fk,n Fk,n+1 Fk,n+2 Fk,n+3. F , n th k-Fib. num. , − − − k,n − } (7) In the following theorem, some properties related to hy- perbolic k-Fibonacci quaternions are given. where Theorem 1: Let Fk,n and HFk,n be the n-th terms of 2 2 2 i = j = k = 1, k-Fibonacci sequence (Fk,n) and hyperbolic k-Fibonacci quaternion (HFk,n), respectively. In this case, for n 1 we ij = k = ji, jk = i = kj, ki = j = ik. ≥ − − − can give the following relations: Let HFk,n and HFk,m be two hyperbolic k-Fibonacci quaternions such that HFk,n+2 =kHFk,n+1 + HFk,n (15) 2 HFk,n =2Fk,nHFk,n HFk,nHF k,n (16) HFk,n = Fk,n + iFk,n+1 + jFn+2 + kFk,n+3 (8) − and HFk,n iHFk,n+1 jHFk,n+2 kHFk,n+3 F F F F F (9) − − − (17) H k,m = k,m + i k,m+1 + j k,m+2 + k k,m+3 = F F F F k,n − k,n+2 − k,n+4 − k,n+6 Fugen¨ Torunbalci Aydin 65 Proof: (15): By the equation (8) we get, Theorem 3: For n,m 0 the Honsberger identity for hyper- ≥ bolic k-Fibonacci quaternions HFk,n and HFk,m is given by HFk,n + kHFk,n+1 =(F + iF + jF + kF ) HFk,n+1HFk,m + HFk,nHFk,m 1 k,n k,n+1 k,n+2 k,n+3 − (19) + k(Fk,n+1 + iFk,n+2 + jFk,n+3 + kFk,n+4) = 2HFk,n+m + kFk,n+m+1 + Lk,n+m+5. Proof: (19) By using (11) =(Fk,n + kFk,n+1) + i(Fk,n+1 + kFk,n+2) + j(F + kF ) + k(F + kF ) k,n+2 k,n+3 k,n+3 k,n+4 HFk,n+1HFk,m =(Fk,n+1Fk,m + Fk,n+2Fk,m+1 =F + iF + jF + kF k,n+2 k,n+3 k,n+4 k,n+5 + Fk,n+3Fk,m+2 + Fk,n+4Fk,m+3) = F . H k,n+2 + i(Fk,n+1Fk,m+1 + Fk,n+2Fk,m + F F F F ) (16): By the equation (7) we get, k,n+3 k,m+3 − k,n+4 k,m+2 + j(Fk,n+1Fk,m+2 Fk,n+2Fk,m+3 2 2 2 2 2 − HFk,n =(Fk,n + Fk,n+1 + Fk,n+2 + Fk,n+3) + Fk,n+3Fk,m + Fk,n+4Fk,m+1) + 2Fk,n(iFk,n+1 + jFk,n+2 + kFk,n+3) + k(Fk,n+1Fk,m+3 + Fk,n+2Fk,m+2 2 =2Fk,nHFk,n 2F Fk,n+3Fk,m+1 + Fk,n+4Fk,m), − k,n − 2 2 2 2 HFk,nHFk,m 1 =(Fk,nFk,m 1 + Fk,n+1Fk,m + (Fk,n + Fk,n+1 + Fk,n+2 + Fk,n+3) − − + Fk,n+2Fk,m+1 + Fk,n+3Fk,m+2) =2Fk,nHFk,n HFk,nHF k,n.