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On H(X)-Fibonacci Octonion Polynomials, Ahmet Ipek & Kamil

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Alabama Journal of Mathematics ISSN 2373-0404 39 (2015) On h(x)-Fibonacci octonion polynomials Ahmet Ipek˙ Kamil Arı Karamanoglu˘ Mehmetbey University, Karamanoglu˘ Mehmetbey University, Science Faculty of Kamil Özdag,˘ Science Faculty of Kamil Özdag,˘ Department of Mathematics, Karaman, Turkey Department of Mathematics, Karaman, Turkey In this paper, we introduce h(x)-Fibonacci octonion polynomials that generalize both Cata- lan’s Fibonacci octonion polynomials and Byrd’s Fibonacci octonion polynomials and also k- Fibonacci octonion numbers that generalize Fibonacci octonion numbers. Also we derive the Binet formula and and generating function of h(x)-Fibonacci octonion polynomial sequence. Introduction and Lucas numbers (Tasci and Kilic (2004)). Kiliç et al. (2006) gave the Binet formula for the generalized Pell se- To investigate the normed division algebras is greatly a quence. important topic today. It is well known that the octonions The investigation of special number sequences over H and O are the nonassociative, noncommutative, normed division O which are not analogs of ones over R and C has attracted algebra over the real numbers R. Due to the nonassociative some recent attention (see, e.g.,Akyigit,˘ Kösal, and Tosun and noncommutativity, one cannot directly extend various re- (2013), Halici (2012), Halici (2013), Iyer (1969), A. F. Ho- sults on real, complex and quaternion numbers to octonions. radam (1963) and Keçilioglu˘ and Akkus (2015)). While ma- The book by Conway and Smith (n.d.) gives a great deal of jority of papers in the area are devoted to some Fibonacci- useful background on octonions, much of it based on the pa- type special number sequences over R and C, only few of per of Coxeter et al. (1946). Octonions made further appear- them deal with Fibonacci-type special number sequences ance ever since in associative algebras, analysis, topology, over H and O (see, e.g., Akyigit˘ et al. (2013), Halici (2012), and physics. Nowadays octonions play an important role in Halici (2013), Iyer (1969), A. F. Horadam (1963) and Keçil- computer science, quantum physics, signal and color image ioglu˘ and Akkus (2015)), notwithstanding the fact that there processing, and so on (e.g. Adler and Finkelstein (2008)). are a lot of papers on various types of Fibonacci-type spe- Many references may be given for Fibonacci-like se- cial number sequences over R and C (see, e.g., Cureg and quences and numbers and related issues. The readers are Mukherjea (2010), Falcon and Plaza (2007), A. Horadam referred to Koshy (n.d.) and Vajda (1989) for some other (1961), Kilic (2008), Öcal et al. (2005), Yang (2008)-Yilmaz sequences which can be obtained from the Fibonacci num- and Taskara (2013)). bers. One may find many applications of this type sequences In this paper, we introduce h(x)-Fibonacci octonion poly- of numbers in various branches of science like pure and nomials that generalize both Catalan’s Fibonacci octonion applied mathematics, in biology or in phyllotaxies, among polynomials Ψn(x) and Byrd’s Fibonacci octonion polyno- many others. Hence, many researches are performed on the mials and also k-Fibonacci octonion numbers that generalize various type of Fibonacci sequence; for example see Cureg Fibonacci octonion numbers. Also we derive the Binet for- and Mukherjea (2010), Falcon and Plaza (2007), A. Horadam mula and and generating function of h(x)-Fibonacci octonion (1961), Kilic (2008), Öcal, Tuglu, and Altini¸sik(2005), Yang polynomial sequence. (2008)-Yilmaz and Taskara (2013). The rest of the paper is structured as follows. Some pre- The connection between the Fibonacci-type numbers and liminaries which are required are given in Section 2. In Sec- the Golden Section is expressed by the well-known math- tion 3 we introduce Catalan, Byrid, Fibonacci octonion poly- ematical formula, so-called Binet’s formulas. Many refer- nomials and numbers and provide their some properties. Sec- ences may be given for Fibonacci-like sequences and num- tion 4 ends with our conclusion. bers and Binet’s formulas. By using a generating function Levesque (1985) gave a Binet formula for the Fibonacci se- Some Preliminaries quence. Kilic and Tasci (2006) gave the generalized Binet formulas for the generalized order-k Fibonacci (Er (1984)) We start this section by introducing some definitions and notations that will help us greatly in the statement of the re- sults. We begin this section by giving the following definition Corresponding Author Email: [email protected] for the classic Fibonacci sequence. The classic Fibonacci 1 2 IPEK˙ & ARI f g Fn n2N sequence is a very popular integer sequence. Catarino (2015) introduced h(x)-Fibonacci quaternion polynomials. The basic properties of Fibonacci quaternion Definition 1. The classic Fibonacci fF g sequence is de- n n2N numbers can be found in Akyigit˘ et al. (2013), Halici (2012), fined by Halici (2013), Iyer (1969) and A. F. Horadam (1963). The octonions in Clifford algebra C are a normed division algebra with eight dimensions over the real numbers larger 4 F0 = 0; F1 = 1 and Fn = Fn−1 + Fn−2 for n ≥ 2; (1) than the quaternions. The field O C of octonions th where Fn denotes the n classic Fibonacci number (Dunlap X7 (n.d.), Koshy (n.d.), Vajda (1989)). α = αses, ai(i = 0; 1; :::; 7) 2 R (4) The books written by Hoggat (1969), Koshy (n.d.) and s=0 Vajda (1989) collects and classifies many results dealing with is an eight-dimensional non-commutative and non- the these number sequences , most of them are obtained quite associative -field generated by eight base elements recently. R e ; e ; :::; e and e . The multiplication rules for the basis of Nalli and Haukkanen (2009) introduced the h(x)- 0 1 6 7 O are listed in the following table Fibonacci polynomials. Then, Ramírez (2014) introduced the convolved h(x)-Fibonacci polynomials and obtain new × 1 e1 e2 e3 e4 e5 e6 e7 identities. 1 1 e1 e2 e3 e4 e5 e6 e7 Definition 2. Let h(x) be a polynomial with real coefficients. e1 e1 −1 e3 −e2 e5 −e4 −e7 e6 1 e2 e2 −e3 −1 e1 e6 e7 −e4 −e5 The h(x)-Fibonacci polynomials fFh;n(x)gn=0 are defined by the recurrence relation e3 e3 e2 −e1 −1 e7 −e6 e5 −e4 e4 e4 −e5 −e6 −e7 −1 e1 e2 e3 Fh;n+1(x) = h(x)Fh;n(x) + Fh;n−1(x), n ≥ 1, (2) e5 e5 e4 −e7 e6 −e1 −1 −e3 e2 e6 e6 e7 e4 −e5 −e2 e3 −1 −e1 with initial conditions Fh;0(x) = 0,Fh;1(x) = 1 (Nalli and e7 e7 −e6 e5 e4 −e3 −e2 e1 −1 Haukkanen (2009)). Table 1. The multiplication table for the basis of O. For h(x) = x, it is obtained Catalan’s Fibonacci polynomi- (5) P7 als n(x), and for h(x) = 2x it is obtained Byrd’s Fibonacci The scalar and the vector part of a octonion α = αses 2 s=0 polynomials φn(x). If for k real number h(x) = k, it is ob- −! P7 tained the k-Fibonacci numbers Fk;n. For k = 1 and k = 2 O are denoted by S α = α0 and Vα = αses, respectively. it is obtained the usual Fibonacci numbers Fn and the Pell s=1 7 numbers P , respectively. P n Let α and β be two octonions such that α = αses and The quaternion, which is a type of hypercomplex num- s=0 bers, was formally introduced by Hamilton in 1843. P7 β = βses. We now are going to list some properties, with- s=0 Definition 3. The real quaternion is defined by out proofs, for α and β octonions. q = qr + qii + q j j + qkk P7 • Property 1. α ± β = (αs ± βs)es s=0 where qr; qi; q j and qk are real numbers and i; j and k are complex operators obeying the following rules • Property 2. αβ = S αS β+S αVβ+VαS β−Vα·Vβ+Vα×Vβ, 7 2 2 2 −! P i = j = k = −1; i j = − ji = k; jk = −k j = i; ki = −ik = j: where S α = α0 and Vα = αses. s=1 In recent years there has been a flurry of activity for doing P7 research with Fibonacci quaternion and octonion numbers. • Property 3. α = α0e0 − αses We can start from the Fibonacci quaternion numbers intro- s=1 duced by Horadam in 1963. • Property 4. (αβ) = βα. Definition 4. The Fibonacci quaternion numbers that are 7 th • α+α α−α P given for the n classic Fibonacci Fn number are defined Property 5. 2 = α0e0 and 2 = αkek: k=1 by the following recurrence relations: • Property 6. αα = αα Qn = Fn + iFn+1 + jFn+2 + kFn+3 (3) 7 • P 2 ± ± Property 7. αα = αk: where n = 0; 1; 2; :::(A. F. Horadam (1963)). k=0 FIBONACCI OCTONION POLYNOMIALS 3 p The norm of an octonion can be defined as kαk = αα. • If for k real number h(x) = k, it is obtained the k- Fibonacci numbers Fk;n from the h(x)-Fibonacci poly- • kαβk kαk kβk Property 8. = . nomials Fh;n(x), and thus for h(x) = k we obtain k-Fibonacci octonion numbers Q from the h(x)- P7 k;n • Property 9. k.α = (kαi) ei: Fibonacci octonion polynomials Qh;n(x). Also, for i=0 k = 1 and k = 2 we have the Fibonacci octonion P7 numbers Qn and the Pell octonion numbers Υn, respec- • Property 10. hα, βi = αiβi: i=0 tively. The Fibonacci octonion numbers are given by the follow- The ordinary generating function (OGF) of the sequence 1 ing definition.
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