A New Generalization of Fibonacci and Lucas Type Sedenions

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

A NEW GENERALIZATION OF FIBONACCI AND LUCAS TYPE SEDENIONS

CAN KIZILATE¸SAND SELIHAN KIRLAK

Abstract. In this paper, by using the q integer, we introduce a new generalization of Fi- bonacci and Lucas sedenions called q Fibonacci− and q Lucas sedenions. We present some fundamental properties of these type− of sedenions such− as Binet formulas, exponential generating fuctions, summation formulas, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity.

1. Introduction The set of Sedenions, denoted by S, are 16 dimensional algebra. Sedenions are noncom- mutative and nonassociative algebra over the− set of real numbers, obtained by applying the Cayley—Dickson construction to the octonions. Like octonions, multiplication of sedenions are not neither commutative and associative. Sedenions appear in many areas of science, such as electromagnetic theory, linear gravity and the field of quantum mechanics [1—6]. A sedenion is defined by 15 p = qiei, i=0 X where q0, q1, ..., q15 R and e0, e1, ..., e15 are called unit sedenion such that e0 is the unit ∈ element and e1, e2, ..., e15 are imaginaries satisfying, for i, j, k = 1, 2,..., 15 the following multiplication rules: 2 e0ei = eie0 = ei, (ei) = e0, (1.1) − eiej = ejei, i = j, (1.2) − 6 ei (ejek) = (eiej) ek, i = j, eiej = ek. (1.3) − 6 6 ± The addition of sedenions is defined as componentwise and for p1, p2 S, the multiplication is defined as follows: ∈ 15 15 p1p2 = aiei bjej   i=0 ! j=0 X X 15   = aibj (eiej) , i,j=0 X where eiej satisfies the identities (1.1), (1.2) and (1.3). Interestingly, in [4] the authors defined and studied an effi cient algorithm for the multiplication of sedenions.

Key words and phrases. Sedenion algebra, Horadam number, q integer, Binet-Like formula, exponential generating function. − 2000 Math. Subject. Classiﬁcation: 11B39, 11R52, 05A15, 05A30.

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For a, b, p, q R, the Horadam numbers hn = hn(a, b; p, q) are defined by the following recurrence relations∈ hn = phn 1 + qhn 2, h0 = a, h1 = b. − − Some special cases of the Horadam sequence hn(a, b; p, q) are as following table: a b p q Sequence 0 1 1 1 Fibonacci sequence; Fn 2 1 1 1 Lucas sequence; Ln 0 1 2 1 Pell sequence; Pn 2 2 2 1 Pell-Lucas sequence; PLn 0 1 k 1 k Fibonacci sequence; Fk,n − 2 1 k 1 k Lucas sequence; Lk,n − 0 1 1 2 Jacobsthal sequence; Jn 2 1 1 2 Jacobsthal-Lucas sequence; jn 0 1 2 k k Pell sequence; Pk,n − 2 2 2 k k Pell-Lucas sequence; PLk,n − Table 1: Special cases of the Horadam sequence The Binet formula of the Horadam number is given by, for n 0, ≥ Aαn Bβn h = − , n α β − where A = b aβ, B = b aα, α and β are the roots of the x2 px q = 0. We note that the above numbers− have been− studied in the literature widely and− extensively− (see, for example, [7—10]). Now, we talk about some notations related to q calculus. For x N0, we give the − ∈ q integer [x]q − x 1 q 2 x 1 [x] = − = 1 + q + q + ... + q − . (1.4) q 1 q − From (1.4), for all x, y Z, we can easily find that ∈ x [x + y]q = [x]q + q [y]q. Many of the papers on quantum (q ) calculus which has been provided by the mathematicians in the past for the purpose of application− to the branches of mathematics such as combinatorics, differential equations as well as in other areas in physics, probability theory and so on. For further information, we specially refer to books in [11, 12]. Several generalizations of the well-known sedenions such as Fibonacci sedenions, Lucas sedenions, k Pell and k Pell-Lucas sedenions, Jacobshtal and Jacobshtal-Lucas sedenions, and so on have− been studied− by several researchers. For example, in [13], the authors defined the Fibonaccci and Lucas sedenions 15 Fˆn = Fn+ses, s=0 X and 15 Lˆn = Ln+ses. s=0 X Then they also obtained the generating functions, Binet-Like formulas and some interesting identities related to Fibonacci and Lucas sedenions. For more detailed information, please refer to the closely related to the paper [13—16]. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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Akkus and Kizilaslan [17], defined a more general quaternion sequence by receiving com- ponents from complex sequences. Then, they gave some properties and identities related to these quaternions. In [18] Kızılate¸sdefined the another generalization of hybrid numbers which called the q Fibonacci hybrid numbers and q Lucas hybrid numbers. Moreover, the author gave some important− algebraic properties of these− numbers. Motivated by some of the above-cited recent papers, by the help of the q integers, we define here a new family of sedenions called q Fibonacci sedenions and q Lucas− sedenions. We obtain some special cases of the q Fibonacci− sedenions and q Lucas sedenions− studied by many researchers before. We get a number− of results for q Fibonacci− sedenions and q Lucas sedenions included Binet-Like formulas, exponential generating− functions, summation− formulas, Catalan’sidentities, Cassini’sidentities and d’Ocagne’sidentities.

2. q Fibonacci Sedenions and q Lucas Sedenions − − In this part of the our paper, we will introduce the q Fibonacci sedenions and the q Lucas sedenions. Then we also give some algebraic properties− of these sedenions. Throughout− the paper, we take n N and 1 q = 0. ∈ − 6 Deﬁnition 2.1. The q Fibonacci sedenions and the q Lucas sedenions are deﬁned by the following: − − 15 n+s 1 SFn(α; q) = α − [n + s]q es, (2.1) s=0 X and 15 [2(n + s)] (α; q) = αn+s q e . (2.2) SLn [n + s] s s=0 q X Some special cases of q Fibonacci sedenions SFn(α; q) or shortly SFn and q Lucas sede- − − nions SLn(α; q) or shortly SLn are as following table: α q q Fibonacci Sedenions q Lucas sedenions 1+√5 1 − − 2 −α2 Fibonacci sedenion; SFn Lucas sedenion; SLn √ 1 1 + 2 −α2 Pell sedenion; SPn Pell-Lucas sedenion; SPLn k+√k2+4 1 2 −α2 k Fibonacci sedenion; SFk,n k Lucas sedenion; SLk,n 1 − − 2 −2 Jacobsthal sedenion; SJn Jacobsthal-Lucas sedenion; Sjn k 1 + √1 + k −2 k Pell sedenion; SPk,n k Pell-Lucas sedenion; SPLk,n α − − Table 2: Special cases for q Fibonacci Sedenions and q Fibonacci sedenions − − Theorem 2.2. The Binet formulas for the q Fibonacci sedenions SFn and the q Lucas sede- − − nions SLn are αnα˜ (αq)nβ˜ = − , (2.3) SFn α(1 q) − and n n ˜ SLn = α α˜ + (αq) β, (2.4) 15 s ˜ 15 s where α˜ = s=0 α es and β = s=0 β es. P P Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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Proof. Owing to (2.1) and (1.4), we ﬁnd that

15 n+s 1 SFn = α − (n + s)qes s=0 Xn 1 n n+14 = α − (n)qe0 + α (n + 1)qe1 + ... + α (n + 15)qe15 n n+1 n+15 n 1 1 q n 1 q n+14 1 q = α − − e + α − e + ... + α − e 1 q 0 1 q 1 1 q 15 − − − 1 n n n+1 n+1 n+15 n+15 = (α (αq) )e0 + (α (αq) )e1 + ... + (α (αq) )e15 α(1 q) − − − − 1 n 15 n 15 = α (e0 + αe1 + ... + α e15) (αq) (e0 + (αq)e1 + ... + (αq) e15 α(1 q) − − αnα˜ (αq )nβ˜ = − . α(1 q) − Similarly, equality (2.4) can be obtained.

Corollary 2.3. The Binet formulas for the Fibonacci sedenions SFn and the Lucas sedenions SLn are n n α α β∗β SF = ∗ − , (2.5) n α β − n n SLn = α∗α + β∗β , (2.6) 15 s 15 s where α∗ = s=0 α es and β∗ = s=0 β es, respectively. P P 1+√5 1 1 Proof. This follows from substituting α = 2 , β = −α and q = −α2 in the Equation (2.3) and (2.4), respectively.

Corollary 2.4. The Binet formulas for the Pell sedenions SPn and the Pell—Lucas sedenions SPLn, are n n r1∗r1 r2∗r2 SPn = − , (2.7) r1 r2 − n n SPLn = r1∗r1 + r2∗r2 , (2.8) 15 s 15 s where r1∗ = s=0 r1es and r2∗ = s=0 r2es, respectively. 1 1 Proof. ThisP follows from substitutingP α = r = 1 + √2, r = and q = 2 in the Equation 1 2 −r1 −α (2.3) and (2.4), respectively.

Corollary 2.5. The Binet formulas for the Jacobsthal sedenions SJn and the Jacobsthal— Lucas sedenions Sjn, are n n α 2 β∗( 1) SJ = ∗ − − , (2.9) n 3 n n Sjn = α∗2 + β∗( 1) , (2.10) − 15 s 15 s where α∗ = s=0 α es and β∗ = s=0 β es, respectively. 1 Proof. ThisP follows from substitutingP α = 2, β = 1 and q = − in the Equation (2.3) and − 2 (2.4), respectively. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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Theorem 2.6. The exponential generating functions for the q Fibonacci sedenions and q Lucas sedenions are − − ∞ xn αe˜ αx βe˜ αqx = − , (2.11) SFn n! α(1 q) n=0 X − and ∞ xn =αe ˜ αx + βe˜ αqx. (2.12) SLn n! n=0 X Proof. From the Binet formula for the q Fibonacci sedenions, we obtain − ∞ xn ∞ αnα˜ (αq)nβ˜ xn = − SFn n! α(1 q) n! n=0 n=0 − ! X X n n α˜ ∞ (αx) β˜ ∞ (αqx) = α(1 q) n! − α(1 q) n! n=0 n=0 − X − X αe˜ αx βe˜ αqx = − . α(1 q) − Equality (2.12) can be similarly derived. Corollary 2.7. The exponential generating functions for the Fibonacci sedenions and the Lucas sedenions are n αx βx ∞ x α e β∗e SF = ∗ − , (2.13) n n! α β n=0 X − and ∞ xn SL = α eαx + β eβx. (2.14) n n! ∗ ∗ n=0 X 1+√5 1 1 Proof. This follows from substituting α = 2 , β = −α and q = −α2 in the Equation (2.11) and (2.12), respectively. Corollary 2.8. The exponential generating functions for the Pell sedenions and the Pell— Lucas sedenions are ∞ xn r er1x r er2x SP = 1∗ − 2∗ , (2.15) n n! r r n=0 1 2 X − and ∞ xn SP = r er1x + r er2x. (2.16) n n! 1∗ 2∗ n=0 X 1 1 Proof. This follows from substituting α = r = 1 + √2, r = and q = 2 in the Equation 1 2 −r1 −α (2.11) and (2.12), respectively. Corollary 2.9. The exponential generating functions for the Jacobsthal sedenions and the Jacobsthal—Lucas sedenions are n 2x x ∞ x α e β∗e SJ = ∗ − − , (2.17) n n! 3 n=0 X Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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and ∞ xn Sj = α e2x + β e x. (2.18) n n! ∗ ∗ − n=0 X 1 Proof. This follows from substituting α = 2, β = 1 and q = − in the Equation (2.11) and − 2 (2.12), respectively. Theorem 2.10. For n, k 0, we have ≥ n n n 2 n i ∆ 2 SFn+k if n is even, α q − SF2i+k = n 1 (2.19) i − ∆ −2 if n is odd, i=0 ( SLn+k X n n n 2 n i ∆ 2 SLn+k if n is even, α q − SL2i+k = n+1 (2.20) i − ∆ 2 if n is odd, i=0 ( SFn+k X n n n i 2 n i ( 1) α q − SF2i+k = α [2] SFn+k, (2.21) i − − − q i=0 Xn n n i 2 n i ( 1) α q − SL2i+k = α [2] SLn+k. (2.22) i − − − q i=0 X where ∆ = (α αq)2 . − Proof. Let’sfirst prove the equality (2.19). Similarly (2.20), (2.21) and (2.22) can be obtained. From Binet formula (2.11), we find that n n 2i+k 2i+k ˜ n 2 n i n 2 n i α α˜ (αq) β α q − SF2i+k = α q − − i − i − α(1 q) i=0 i=0 − X X 1 n n = α2 α2q αkα˜ α2q2 α2q (αq)kβ˜ α αq − − − − n α√∆ αkα˜ ( αq√∆)n(αq)kβ˜ = − − . (2.23) α αq − If n is even in (2.23), we have n n α√∆ αkα˜ (αq√∆)n(αq)kβ˜ n 2 n i − α q − SF2i+k = i − α αq i=0 − X n αn+kα˜ (αq)n+kβ˜ = √∆ − α(1 q) n − = ∆ 2 SFn+k. If n is odd in (2.23), we obtain n n α√∆ αkα˜ + (αq√∆)n(αq)kβ˜ n 2 n i α q − SF2i+k = i − α αq i=0 − X n 1 = √∆ − (αn+kα˜ + (αq)n+kβ˜) n 1 − = ∆ 2 SLn+k. Thus the proof is finished. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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Corollary 2.11. For n 0, we have ≥ m m m 5 2 SFm+k if m is even, SF2n+k = m 1 , n 5 2− SL if m is odd, n=0 ( m+k X m m m 5 2 SLm+k if m is even, SL2n+k = m+1 , n 5 2 SF if m is odd, n=0 ( m+k X m m n m ( 1) SF2n+k = ( 1) SFm+k, n − − n=0 and X m m n m ( 1) SL2n+k = ( 1) SLm+k. n − − n=0 X 1+√5 1 Proof. This follows from substituting α = 2 and q = −α2 in the Equation (2.19), (2.20), (2.21) and (2.22), respectively. Corollary 2.12. For n 0, we ﬁnd that ≥ m m m 8 2 SPm+k if m is even, SP2n+k = m 1 , n 8 2− SPL if m is odd, n=0 ( m+k X m m m 8 2 SPLm+k if m is even, SPL2n+k = m+1 , n 8 2 SP if m is odd, n=0 ( m+k X m m n m ( 1) SP2n+k = ( 1) SPm+k, n − − n=0 and X m m n m ( 1) SPL2n+k = ( 1) SPLm+k. n − − n=0 X √ 1 Proof. This follows from substituting α = r1 = 1 + 2 and q = −α2 in the Equation (2.19), (2.20), (2.21) and (2.22), respectively. Corollary 2.13. For n 0, we obtain ≥ m m m 9 2 SJm+k if m is even, SJ2n+k = m 1 , n 9 2− Sj if m is odd, n=0 ( m+k X m m m 9 2 Sjm+k if m is even, Sj2n+k = m+1 , n 9 2 SJ if m is odd, n=0 ( m+k X m m n m n m ( 1) 2 − SJ2n+k = ( 1) SJm+k, n − − n=0 and X m m n m n m ( 1) 2 − Sj2n+k = ( 1) Sjm+k. n − − n=0 X 1 Proof. This follows from substituting α = 2 and q = −2 in the Equation (2.19), (2.20), (2.21) and (2.22), respectively. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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Theorem 2.14. For n 0, we get ≥ n n i 2 n i (α + αq) α q − SFi = SF2n, (2.24) i − i=0 X n n i 2 n i (α + αq) α q − SLi = SL2n. (2.25) i − i=0 X Proof. Using the Binet formula for the the q Fibonacci sedenions (2.3), we have − n n i 2 n i (α + αq) α q − SFi i − i=0 X n i i ˜ n i 2 n i α α˜ (αq) β = (α + αq) α q − − i − α(1 q) i=0 X − n i n i n (α + αq) α2q − αiα˜ = − i α(1 q) i=0 X − n i n i n (α + αq) α2q − (αq)iβ˜ − − i α(1 q) i=0 X − (α2(1 + q) α2q)nα˜ = − α(1 q) − (α2q(1 + q) α2q)nβ˜ − − α(1 q) − = SF2n. Equality (2.25) can be similarly derived. Theorem 2.15. For n 0, we get ≥ m m SF = SF , n n 2m n=0 X and m m SL = SL . n n 2m n=0 X 1+√5 1 Proof. This follows from substituting α = 2 and q = −α2 in the Equation (2.24), (2.25). Corollary 2.16. For n 0, we obtain ≥ m m SP = SP , n n 2m n=0 X and m m SPL = SPL . n n 2m n=0 X √ 1 Proof. This follows from substituting α = r1 = 1 + 2 and q = −α2 in the Equation (2.24), (2.25). Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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Corollary 2.17. For n 0, we have ≥ m m 2m nSJ = SJ , n − n 2n n=0 X ve m m 2m nSj = Sj . n − n 2n n=0 X 1 Proof. This follows from substituting α = 2 and q = −2 in the Equation (2.24), (2.25). Theorem 2.18. (Catalan’s Identity). For positive integers n and r, with n r, then the following identity is true: ≥ 2n 2 n r ˜ r ˜ 2 α − q (1 q )(βα˜ q− α˜β) SFn+rSFn r SFn = − − , (2.26) − − (1 q)2 − 2 2n n r r ˜ r ˜ SLn+rSLn r SLn = α q − (1 q ) α˜β q βα˜ . (2.27) − − − − Proof. From the Binet formula of the q Fibonacci sedenions, we have the LHS of the equality (2.26), − 2 SFn+rSFn r SFn − − αn+rα˜ (αq)n+rβ˜ αn rα˜ (αq)n rβ˜ = − − − − α(1 q) α(1 q) − ! − ! 2 αnα˜ (αq)nβ˜ − . − α(1 q) − ! After some calculations, we get 2n 2 n r ˜ r ˜ 2 α − q (1 q )(βα˜ q− α˜β) SFn+rSFn r SFn = − − . − − (1 q)2 − The result (2.27) can be similarly obtained. Theorem 2.19. (Cassini’s Identity). For n 1, the following equality holds: ≥ 2n 2 n ˜ 1 ˜ 2 α − q (βα˜ q− α˜β) SFn+1SFn 1 SFn = − , (2.28) − − (1 q) − 2 2n n 1 ˜ ˜ SLn+1SLn 1 SLn = α q − (1 q) α˜β qβα˜ (2.29) − − − − Proof. If we take r = 1, in (2.26) and (2.27), we obtain the assertions of the theorem. Theorem 2.20. (d’Ocagne’s Identity). Suppose that n is a non-negative integer number and m natural number. If m > n + 1, then the expression of the d’Ocagne’s identities are given by the following: m+n 1 n ˜ m ˜ α − (q α˜β q βα˜) SFmSFn+1 SFm+1SFn = − , (2.30) − (1 q) − m+n+1 n ˜ m ˜ SLmSLn+1 SLm+1SLn = α (q 1)(q α˜β q βα˜). (2.31) − − − Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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Proof. Using the Binet formula of the q Fibonacci sedenions, we obtain − SFmSFn+1 SFm+1SFn − αmα˜ (αq)mβ˜ αn+1α˜ (αq)n+1β˜ = − − α(1 q) α(1 q) − − αm+1α˜ (αq)m+1β˜ αnα˜ (αq)nβ˜ − − . − α(1 q) α(1 q) − − Thus m+n 1 n ˜ m ˜ α − (q α˜β q βα˜) SFmSFn+1 SFm+1SFn = − . − (1 q) − Equality (2.31) can be similarly derived.

3. Conclusion and Discussion In our present research, we have defined and examined systematically q Fibonacci sedenions and q Lucas sedenions which are defined by means of the q integer.− We have obtained several− quirky properties of q Fibonacci sedenions and q Lucas− sedenions such as Binet-Like formulas, exponential generating− functions, summation formulas,− Cassini’sidentities, Catalan’s identities and d’Ocagne’s identities. According to the special cases of α and q, all the results given in Section 2 are applicable to all Fibonacci-type sedenions and Lucas-type sedenions mentioned in Table 2. On the other hand, a trigintaduonion is defined by 31 t = a0 + aiei, i=0 X where ai , i = 0, 1,..., 31 are real numbers and ei , i = 0, 1,..., 31 are imaginary units. Detailed{ information} about the trigintaduonion have{ } been presented in the literature. (for example, see [19]). Indeed, for the interested readers of this work, results presented here have the potential to motivate further researches of the subject of the q Fibonacci Trigintaduonions and q Lucas Trigintaduonions. − − References [1] Baez, J.C., The octonions. Bull. Am. Math. Soc. 39, 145—205 (2002). [2] Carmody, K., Circular and Hyperbolic Quaternions, Octonions and Sedenions, Appl. Math.Comput., 28, 47—72, (1988). [3] Cawagas, R. E., On the structure and zero divisors of the Cayley-Dickson sedenion algebra, Discuss. Math. Gen. Algebra Appl. 24, 251—265, (2004). [4] Cariow, A. and Cariowa, G., An algorithm for fast multiplication of sedenions, Inform. Process. Lett.113, 324—331, (2013). [5] V. L. Mironov and S. V. Mironov, Associative space-time sedenions and their application in relativistic quantum mechanics and field theory, Appl. Math. 6(1), 46—56, (2015). [6] Imaeda, K. and Imaeda, M., Sedenions: algebra and analysis, Appl. Math. Comput. 115, 77—88, (2000). [7] Horadam, A.F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3, 161—176, (1965). [8] Horadam,A.F., Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J., 32, 437—446, (1965). Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2020

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[9] Horzum,T., Kocer, E.G., On Some Properties of Horadam Polynomials, Int. Math. Forum, 4, 25, 1243—1252, (2009). [10] Cimen, C.B., Ipek, A. On Jacobsthal and Jacobsthal—Lucas Octonions, Mediterr. J. Math., 14:37 (2017). [11] Kac V, Cheung P. Quantum Calculus Universitext, Springer-Verlag, New York, (2002). [12] Andrews GE, Askey R, Roy R. Special functions. Cambridge Univ. Press, New York, (1999). [13] Bilgici, G., Tokeser, U. and Unal, Z., Fibonacci and Lucas Sedenions, J. Integer Seq., Vol. 20, Article 17.1.8, (2017). [14] Çimen, C., Ipek, A., On Jacobsthal and Jacobsthal-Lucas Sedenios and Several Identities Involving These Numbers. Math. Aeterna, 7, 447—454, (2017). [15] Catarino, P. k Pell, k-Pell-Lucas and modiﬁed k Pell sedenions. Asian-Eur. J. Math., 12 (1), (2019).− − [16] Çimen, C., Ipek, A., Çimen, E., On Horadam Sedenions, Journal of Science and Arts, 4, (49), 887—898, (2019). [17] Akkus, I., Kizilaslan, G., Quaternions: Quantum calculus approach with applications, Kuwait Journal of Science, 46 (4), 1—13, (2019). [18] Kızılate¸sC., A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos Solitons Fractals, 130, (2020). [19] Cawagas, R. E., Carrascal, A. S., Bautista, L. A., Maria, J. P. Sta., Urrutia J. D., Nobles, B., The Basic Subalgebra Structure Of The Cayley-Dickson Algebra Of Dimension 32 (Trigintaduonions), arXiv:0907.2047v3, (2009).

BULENT ECEVIT UNIVERSITY, DEPARTMENT OF MATHEMATICS, ZONGULDAK, 67100, TURKEY E-mail address: [email protected] and [email protected]