arXiv:2108.06351v1 [math.RA] 13 Aug 2021 Keywords: quaternions. ioac quaternion; Fibonacci ua iope utrin,rsetvl.Te,w iesm alge some give we Then, of respectively. properties quaternions, bicomplex Lucas rda w-iesoa etr.Hmlo 1 nrdcdtesto set as the represented introduced be can [1] consider which Hamilton a nions be numbers vectors. complex two-dimensional can that as way and extens ered same an space the constitute in four-dimensional vectors, quaternions a four-dimensional real into The numbers complex 1843. in Hamilton Rowan where .Introduction 1. the define we paper, the In Abstract oaa 2 ]dfie ope ioac n ua utrin as quaternions Lucas and Fibonacci complex defined 3] [2, Horadam rpitsbitdt ho,Solitons Chaos, to submitted Preprint eateto ahmtclEgneig aups Campus Davutpasa Engineering, Mathematical of Department i 2 ∗ h elqaenoswr rtdsrbdb rs ahmtca W mathematician Irish by described first were quaternions real The mi address: Email author Corresponding = a idzTcnclUiest,Fclyo hmcladMetal and Chemical of Faculty University, Technical Yildiz j 2 = k H iope number; Bicomplex 2 q = = q Fbncibcmlxqaenosadthe and quaternions bicomplex -Fibonacci Fbncibcmlxquaternions bicomplex -Fibonacci − { [email protected] 1 q j i , = Q n q q 0 -quaternion; = + = F .Trnac Aydın Torunbalcı F. q i n − q 1 + i j Fbncibcmlxqaenosadthe and quaternions bicomplex -Fibonacci + F q & = q j n itgr ioac ubr Bicomplex number; Fibonacci -integer; TURKEY +1 rcasAgs 7 2021 17, August Fractals k j , k 2 F oublıAydın) Torunbalcı (F. q i + Fbnciquaternion. -Fibonacci + q k F n 3 +2 | = j q − + a, 0 q , j k ∗ F 1 n q , +3 = 42,Esenler, 34220, , 2 k i k , i ugclEngineering, lurgical q , q 3 Lcsbicomplex -Lucas ∈ R = } − econsid- re k i quater- f Istanbul, ˙ follows o of ion = illiam das ed braic . j (1) (2) (3) q - 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 . m+1  α [n + 2]q α [n + 3]q α [n]q α [n + 1]q   α [m + 2]q   n+2 − n+1 n − n−1   m+2   α [n + 3]q α [n + 2]q α [n + 1]q α [n]q   α [m + 3]q  (18) Three kinds of conjugation can be defined for bicomplex numbers [17, 18]. Therefore, conjugation of the q-Fibonacci bicomplex quaternion is defined in three different ways as follows

1 n 1 n n+1 n+2 (BFn(α; q))∗ =(α − [n]q − iα [n + 1]q + jα [n + 2]q − ijα [n + 3]q), (19)

2 n 1 n n+1 n+2 (BFn(α; q))∗ =(α − [n]q + iα [n + 1]q − jα [n + 2]q − ijα [n + 3]q), (20)

3 n 1 n n+1 n+2 (BFn(α; q))∗ =(α − [n]q − iα [n + 1]q − jα [n + 2]q + ijα [n + 3]q). (21)

Therefore, the norm of the q-Fibonacci bicomplex quaternion BFn(α; q) is defined in three different ways as follows

1 1 2 N(BFn(α; q))∗ = k(BFn(α; q)) × (BFn(α; q))∗ k , (22)

2 2 2 N(BFn(α; q))∗ = k(BFn(α; q)) × (BFn(α; q))∗ k , (23)

3 3 2 N(BFn(α; q))∗ = k(BFn(α; q)) × (BFn(α; q))∗ k . (24)

6 Theorem 2.1. (Binet’s Formula). Let BFn(α; q) and BLn(α; q)be the q- Fibonacci bicomplex quaternion and the q-Lucas bicomplex quaternion. For n ≥ 1, Binet’s formula for these quaternions respectively, is as follows:

αn γ − (α q)n δ BF (α; q)= , (25) n bα − α q b and n n BLn(α; q)= α γ +(α q) δ (26) b b where 2 3 1+√5 γ =1+ α i + α j + α ij, α = 2 and b 2 3 1 δ =1+(α q) i +(α q) j +(α q) i j, α q = − . b α Proof. (25): Using (8) and (12), we find that

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 n+2 1 qn+2 n+3 1 qn+3 = α α−αq + α α− αq i + α α− αq j + α α− αq i j − − − −

αn [ 1+αi+α2 j+α3 ij ] (αq)n [ 1+(αq) i+(αq)2 j+(αq)3 ij ] − = α (αq) −

αn γ (αq)n δ − = α αq b b −

In a similar way, equality (26) can be derived as follows

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 α2n − 2 3 − = αn (1 + α i + α j + α i j )

n (αq)2 2 3 − (αq)n (1+(α q) i +(α q) j +(α q) i j )

= αn γ − (α q)n δ. b b 7 where γ =1+ α i + α2 j + α3 i j, δ =1+(α q) i +(α q)2 j +(α q)3 i j and γ δ = δ γb. b b b bb

Theorem 2.2. (Exponential generating function) Let BFn(α; q) be the q-Fibonacci bicomplex quaternion. For the exponential generating function for these quaternions is as follows:

tn ∞ tn γ eα t − δ e(αq) t gBF ( )= BF (α; q) = (27) n(α;q) n! X n n! b α − bα q n=0 Proof. Using the definition of exponential generating function, we obtain

∞ tn ∞ αn γ (αq)n δ tn BF (α; q) = ( − ) P n n! P α αq b n! n=0 n=0 b − n n = γ ∞ t − δ ∞ (α q t) (28) α αq P n! α bαq P n! −b n=0 − n=0 γeαt δe(α q) t − . = α bαq b − Thus, the proof is completed.

Theorem 2.3. (Honsberger identity) For n, m ≥ 0 the Honsberger identity for the q-Fibonacci bicomplex quater- nions BFn(α; q) and BFm(α; q) is given by

BFn(α; q) BFm(α; q)+ BFn+1(α; q) BFm+1(α; q)

αn+m 2 n m 2 n+m 2 = α αq { (1 + α ) γ − γ δ (1 + α(α q))(q + q )+(1+(α q) q δ }. − b b b (29) Proof. (29): By using (12) and (25) we get,

BFn(α; q) BFm(α; q)+ BFn+1(α; q) BFm+1(α; q)

αn γ (αq)n δ αm γ (αq)m δ αn+1 γ (αq)n+1 δ αm+1 γ (αq)m+1 δ − − − − =( α αq b)( α αq b)+( α αq b)( α αq b) b − b − b − b −

αn+m n m αn+m+2 n+1 m+1 = (α αq)2 { (γ − q δ)(γ − q δ)} + (α αq)2 { γ − q δ)(γ − q δ)} − b b b b − b b b b αn+m 2 2 n m 2 n+m 2 = (α αq)2 { (1 + α ) γ − γ δ (1 + α(α q))(q + q )+(1+(α q) ) q δ }. − b b b b where γ δ = δ γ. b b bb 8 Theorem 2.4. (d’Ocagne’s identity) For n, m ≥ 0 the d’Ocagne’s identity for the q-Fibonacci bicomplex quater- nions BFn(α; q) and BFm(α; q) is given by

αn+m−1(qn qm) γ δ BF BF BF BF − (30) m(α; q) n+1(α; q) − m+1(α; q) n(α; q)= (1 q) b. − b Proof. (30): By using (12) and (25) we get,

BFm(α; q) BFn+1(α; q) − BFm+1(α; q) BFn(α; q)

αm γ (αq)m δ αn+1 γ (αq)n+1 δ αm+1 γ (αq)m+1 δ αn γ (αq)n δ − − − − =( α αq b)( α αq b) − ( α αq b)( α αq b) b − b − b − b −

αn+m+1 n m = (α αq)2 {(1 − q)(q − q ) γ δ } − b b αn+m−1(qn qm) γ δ − = (1 q) b. − b Here, γ δ = δ γ is used. b b bb

Theorem 2.5. (Cassini Identity) Let BFn(α; q) be the q-Fibonacci bicomplex quaternion. For n ≥ 1, Cassini’s identity for BFn(α; q) is as follows:

α2n 2 qn q 1 γ δ BF BF BF 2 − (1 − − ) n+1(α; q) n 1(α; q) − n(α; q) = b. (31) − (1 − q) b

Proof. (31): By using (12) and (25) we get

n n n− n− 2 α +1 γ (αq) +1 δ α 1 γ (αq) 1 δ BFn+1(α; q) BFn 1(α; q) − BFn(α; q) = ( − )( − ) bα αq b bα αq b − n − n − α γ (αq) δ 2 −( − ) (bα αq) b − − α2n qn (1 q)(1 q 1) γ δ = − −2 (α αq) b b − − − α2n 2 qn (1 q 1) γ δ − = (1 q) b . − b Here, γ δ = δ γ is used. b b bb

9 Theorem 2.6. (Catalan’s Identity) Let BFn(α; q) be the q-Fibonacci bicomplex quaternion. For n ≥ 1, Catalan’s identity for BFn(α; q) is as follows:

α2n 2 qn qr q r γ δ BF BF BF 2 − (1 − )(1 − − ) n+r(α; q) n r(α; q) − n(α; q) = b . (32) − (1 − q)2 b Proof. (32): By using (12) and (25) we get

n r n r n−r n−r 2 α + γ (αq) + δ α γ (αq) δ BFn+r(α; q) BFn r(α; q) − BFn(α; q) = ( − )( − ) − bα αq b bα αq b αn γ −(αq)n δ − −( − 2 (bα αq) b − α2n qn r γ δ α2n qn+r γ δ+2 α2n qn γ δ − − = b (α αq)2 b b b − − b b α2n qn γ δ [ (q r 1)+(qr 1) ] = − − 2 − b b(α αq) − − − α2n 2 qn γ δ (1 q r )(1 qr) − − = (1b q)2 b − Here, γ δ = δ γ is used. b b bb 3. Conclusion In this paper, algebraic and analytic properties of the q-Fibonacci bicom- plex quaternions are investigated.

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