q-Fibonacci bicomplex quaternions a, F. Torunbalcı Aydın ∗ aYildiz Technical University, Faculty of Chemical and Metallurgical Engineering, Department of Mathematical Engineering, Davutpasa Campus, 34220, Esenler, Istanbul,˙ TURKEY Abstract In the paper, we define the q-Fibonacci bicomplex quaternions and the q- Lucas bicomplex quaternions, respectively. Then, we give some algebraic properties of q-Fibonacci bicomplex quaternions and the q-Lucas bicomplex quaternions. Keywords: Bicomplex number; q-integer; Fibonacci number; Bicomplex Fibonacci quaternion; q-quaternion; q-Fibonacci quaternion. 1. Introduction The real quaternions were first described by Irish mathematician William Rowan Hamilton in 1843. The real quaternions constitute an extension of complex numbers into a four-dimensional space and can be considered as four-dimensional vectors, in the same way that complex numbers are consid- ered as two-dimensional vectors. Hamilton [1] introduced the set of quater- nions which can be represented as H = { q = q0 + i q1 + j q2 + k q3 | q0, q1, q2, q3 ∈ R } (1) where arXiv:2108.06351v1 [math.RA] 13 Aug 2021 i2 = j2 = k2 = −1 , i j = −j i = k, jk = −k j = i, ki = −i k = j . (2) Horadam [2, 3] defined complex Fibonacci and Lucas quaternions as follows Qn = Fn + Fn+1 i + Fn+2 j + Fn+3 k (3) ∗Corresponding author Email address: [email protected] (F. Torunbalcı Aydın) Preprint submitted to Chaos, Solitons & Fractals August 17, 2021 and Kn = Ln + Ln+1 i + Ln+2 j + Ln+3 k (4) where Fn and Ln denote the n−th Fibonacci and Lucas numbers, respectively. Also, the imaginary quaternion units ij, k have the following rules i2 = j2 = k2 = −1 , i j = −j i = k, jk = −k j = i, ki = −ik = j There are several studies on quaternions as the Fibonacci and Lucas quater- nions and their generalizations, for example,[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Bicomplex numbers were introduced by Corrado Segre in 1892 [15]. In 1991, G. Baley Price, the bicomplex numbers are given in his book based on multi- complex spaces and functions [16]. In recent years, fractal structures of these numbers have also been studied [17, 18]. The set of bicomplex numbers can be expressed by the basis {1 ,i,j,ij } (Table 1) as, C2 = { q = q1 + i q2 + j q3 + i j q4 | q1, q2, q3, q4 ∈ R} (5) where i,j and i j satisfy the conditions i2 = −1, j2 = −1, i j = j i. (6) In 2018, the bicomplex Fibonacci quaternions defined by Aydın Torunbalcı [19] as follows BFn = Fn + i Fn+1 + j Fn+2 + i j Fn+3 (7) =(Fn + i Fn+1)+(Fn+2 + i Fn+3) j where i, j and i j satisfy the conditions (6). The theory of the quantum q-calculus has been extensively studied in many branches of mathematics as well as in other areas in biology, physics, electro- chemistry, economics, probability theory, and statistics [20, 21]. For n ∈ N0 , the q-integer [n]q is defined as follows n 1 − q 2 n 1 [n] = =1+ q + q + ... + q − . (8) q 1 − q m By (8), for all m, n ∈ Z, can be easily obtained [m + n]q = [m]q + q [n]q. For more details related to the quantum q-calculus, we refer to [22, 23]. 2 In 2019, q-Fibonacci hybrid and q-Lucas hybrid numbers defined by Kızılate¸s [24] as follows n 1 n1 n+1 HFn(α; q)=α − [n]q + α [n + 1]q i + α [n + 2]q ε n+2 (9) + α [n + 3]q h and [2 n] [2 n + 2] [2 n + 4] HL (α; q)=αn q + αn+1 q i + αn+2 q ε n [n] [n + 1] [n + 2] q q q (10) [2 n + 6] + αn+3 q h [n + 3]q where i, ε and h satisfy the conditions 2 2 2 i = −1, ε =0, h =1, i h = h i = ε + i. (11) Also, Kızılate¸sderived several interesting properties of these numbers such as Binet-Like formulas, exponential generating functions, summation formu- las, Cassini-like identities, Catalan-like identities and d’Ocagne-like identities [24]. 2. q-Fibonacci bicomplex quaternions In this section, we define q-Fibonacci bicomplex quaternions and q-Lucas bicomplex quaternions by using the basis {1,i,j,ij}, where i, j and i j satisfy the conditions (6) as follows n 1 n n+1 n+2 BFn(α; q)= α − [n]q + α [n + 1]q i + α [n + 2]q j + α [n + 3]q i j n 1 qn n+1 1 qn+1 = α ( α−αq )+ α ( α− αq ) i − − n+2 1 qn+2 n+3 1 qn+3 +α ( α− αq ) j + α ( α− αq ) i j − − αn 2 3 = α (αq) [1+ α i + α j + α i j ] − n (αq) 2 3 − α (αq) [1+(α q) i +(α q) j +(α q) i j ] − (12) 3 and n [2 n]q n+1 [2 n+2]q n+2 [2 n+4]q n+3 [2 n+6]q BLn(α; q)= α + α i + α j + α i j [n]q [n+1]q [n+2]q [n+3]q 2n 1 q2n 2n+2 1 q2n+2 = α ( αn −(αq)n )+ α ( αn+1− (αq)n+1 ) i − − 2n+4 1 q2n+4 2n+6 1 q2n+6 +α ( αn+2− (αq)n+2 ) j + α ( αn+3− (αq)n+3 ) i j − − = αn (1 + α i + α2 j + α3 i j ) −(α q)n (1+(α q) i +(α q)2 j +(α q)3 i j ) (13) 1+√5 1 BF For α = 2 and (α q)= −α , q-Fibonacci bicomplex quaternion n(α; q) become the bicomplex Fibonacci quaternions BFn. The addition, substraction and multiplication by real scalars of two q-Fibonacci bicomplex quaternions gives q-Fibonacci bicomplex quaternion. Then, the addition, subtraction and multiplication by scalar of q-Fibonacci bicomplex quaternions are defined by n 1 n n+1 BFn(α; q) ± BFm(α; q)= (α − [n]q + α [n + 1]q i + α [n + 2]q j n+2 +α [n + 3]q i j) m 1 m m+1 ±(α − [m]q + α [m + 1]q i + α [m + 2]q j m+2 +α [m + 3]q i j) n 1 qn m 1 qm = [α ( α−αq ) ± α ( α−αq )] n+1− 1 qn+1 −m+1 1 qm+1 +[α ( α− αq ) ± α ( α− αq )] i n+2 1 −qn+2 m+2 1 −qm+2 +[α ( α− αq ) ± α ( α− αq )] j n+3 1 −qn+3 m+3 1 −qm+3 +[α ( α− αq ) ± α ( α− αq )] i j 1 n− m − 2 3 = α αq { (α ± α )(1 + α i + α j + α i j) +((− α q)n ± (α q)m )(1+(α q) i +(α q)2 j +(α q)3 i j ) }. (14) The multiplication of q-Fibonacci bicomplex quaternion by the real scalar λ is defined as λ BF (α; q) = λαn(1 + α i + α2 j + α3 i j ) n (15) +λ (α q)n (1+(α q) i +(α q)2 j +(α q)3 i j ). 4 The scalar and the vector part of BFn(α; q) which is the n − th term of the q-Fibonacci bicomplex quaternion are denoted by n 1 n n+1 n+2 SBFn(α;q) = α − [n]q, VBFn(α;q) = α [n + 1]q i+α [n + 2]q j+α [n + 3]q i j. (16) Thus, the q-Fibonacci bicomplex quaternion BFn(α; q) is given by BF n(α; q)= SBFn(α;q) + VBFn(α;q) . The multiplication of two q-Fibonacci bicomplex quaternions is defined by n 1 n n+1 BFn(α; q) × BFm(α; q)= (α − [n]q + α [n + 1]q i + α [n + 2]q j n+2 +α [n + 3]q i j) m 1 m m+1 × (α − [m]q + α [m + 1]q i + α [m + 2]q j m+2 +α [m + 3]q i j) 1 n+m 2 4 6 = α αq (α ) { (1 − α − α + α ) +2− i (α − α5)+2 j (α2 − α4)+4 i j (α3)} −qm {(1 − α(α q) − α2(α q)2 + α3(α q)3) +i (α +(α q) − α2(α q)3 − α3(α q)2) +j (α2 +(α q)2 − α(α q)3 − α3(α q)) +i j (α3 +(α q)3 − α(α q)2 − α2(α q))} −qn {(1 − α(α q) − α2(α q)2 + α3(α q)3) +i (α +(α q) − α2(α q)3 − α3(α q)2) +j (α2 +(α q)2 − α(α q)3 − α3(α q)) +i j (α3 +(α q)3 + α2(α q)+ α(α q)2)} +qn+m {(1 − (α q)2 − (α q)4 +(α q)6) +2 i ((α q) − (α q)5)+2 j ((α q)2 − (α q)4) +4 i j ((α q)3 )} = BFm(α; q) × BFn(α; q) (17) Here, quaternion multiplication is done using bicomplex quaternionic units (table 1), and this product is commutative. Also, the q-Fibonacci bicomplex quaternion product may be obtained as follows 5 Table 1: Multiplication scheme of bicomplex units x 1 i j ij 1 1 i j ij i i −1 i j −j j j ij −1 −i ij ij −j −i 1 BFn(α; q) × BFm(α; q)= n−1 n n+1 n+2 m−1 α [n]q α [n + 1]q α [n + 2]q α [n + 3]q α [m]q n − n−1 − n+2 n+1 m α [n + 1]q α [n]q α [n + 3]q α [n + 2]q α [m + 1]q n+1 −n+2 − n−1 − n .