sign in sign up
<<
Home , Fibonacci polynomials, Real number

Alabama Journal of ISSN 2373-0404 39 (2015)

On h(x)-Fibonacci

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 that generalize Fibonacci octonion numbers. Also we derive the Binet formula and and generating of h(x)-Fibonacci octonion .

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 is greatly a quence. important topic today. It is well known that the The investigation of special over H and O are the nonassociative, noncommutative, normed division O which are not analogs of ones over R and C has attracted 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 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 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, , 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- science, quantum physics, signal and color 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 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

{ } Fn n∈N sequence is a very popular sequence. Catarino (2015) introduced h(x)-Fibonacci quaternion polynomials. The basic properties of Fibonacci quaternion Definition 1. The classic Fibonacci {F } sequence is de- n n∈N 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 with eight over the real numbers larger 4 F0 = 0, F1 = 1 and Fn = Fn−1 + Fn−2 for n ≥ 2, (1) than the . The field O  C of octonions th where Fn denotes the n classic (Dunlap X7 (n.d.), Koshy (n.d.), Vajda (1989)). α = αses, ai(i = 0, 1, ..., 7) ∈ 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 elements recently. R e , e , ..., e and e . The rules for the 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 ∞ e2 e2 −e3 −1 e1 e6 e7 −e4 −e5 The h(x)-Fibonacci polynomials {Fh,n(x)}n=0 are defined by the recurrence 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 ψn(x), and for h(x) = 2x it is obtained Byrd’s Fibonacci The and the vector part of a octonion α = αses ∈ 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 √ The 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 ∞ ing definition. ∞ P n {an}n=0 is defined by g(x) = an x (Rosen (1999)) and the n=0 Definition 5. For n ≥ 0, the Fibonacci octonion numbers ∞ exponential generating function (EGF) of a sequence {an}n=0 th ∞ that are given for the n classic Fibonacci Fn number are P xn is defined by g(x) = an n! (Kincaid and Cheney (2002)). defined by the following recurrence relations: n=0 For general material on generating functions the interested X7 reader may refer to the books written by Lando (2003) and ?. Qn = Fn+ses. ∞ The generating function gQ(t) of the sequence {Qh,n(x)}n=0 s=0 is defined by ∞ The basic properties of Fibonacci octonion numbers can X g (t) = Q (x)tn. (7) be found in Keçilioglu˘ and Akkus (2015). Q h,n n=0 The h(x)-Fibonacci octonion polynomials We know that the power (7) converges if and only if |t| < min(1/ |α| , 1/ |β|) for α and β given (22). We consider In this section, we introduce h(x)-Fibonacci octonion gQ(t) a formal which is needed not take care of polynomials that generalize both Catalan’s Fibonacci octo- the convergence. nion polynomials Ψ (x) and Byrd’s Fibonacci octonion poly- n For convenience, we use the following notations: Fh,n = nomials and also k-Fibonacci octonion numbers. Also we de- Fh,n(x) and Qh,n = Qh,n(x). rive the Binet formula and and generating function of h(x)- Equipped with the definitions and properties above, we Fibonacci octonion polynomial sequence. can present the fundamental theorems of this paper as fol- Let ei, i = 0, 1, 2, 3, 4, 5, 6, 7, be a basis of O which sat- lows. isfy the non-commutative and non-associative multiplication rules are listed in Table 1 in (5). Let h(x) be a polynomial Theorem 1. The generating function for the h(x)-Fibonacci with real coefficients. octonion polynomials Qh,n(x) is  Definition 6. The h(x)-Fibonacci octonion polynomials Qh,0 + Qh,1 − h(x)Qh,0 t g t ∞ Q( ) = 2 (8) {Qh,n(x)}n=0 are defined by the 1 − h(x)t − t

X7 and Qh,n(x) = Fh,n s(x)es (6) + X7 s=0 1  g (t) = F + F − t e . (9) Q 1 − h(x)t − t2 h,s h,s 1 s th s=0 where Fh,n(x) is the n h(x)-Fibonacci polynomial. Proof. The generating function g (t) of the h(x)-Fibonacci Particular cases of the previous definition are: Q octonion polynomials is

• For h(x) = x, it is obtained Catalan’s Fibonacci poly- 2 n gQ(t) = Qh,0 + Qh,1t + Qh,2t + ... + Qh,nt + .... (10) nomials ψn(x) from the h(x)-Fibonacci polynomials

Fh,n(x), and thus for h(x) = x we have Catalan’s Fi- The orders of Qh,n−1 and Qh,n−2, respectively, are 1 and 2 less bonacci octonion polynomials Ψn(x) from the h(x)- than the order of Qh,n. Thus, we obtain Fibonacci octonion polynomials Qh,n(x). 2 h(x)gQ(t)t = h(x)Qh,0t + h(x)Qh,1t (11) • For h(x) = 2x, it is obtained Byrd’s Fibonacci poly- 3 n +h(x)Qh,2t + ... + h(x)Qh,n−1t + ... nomials φn(x) from the h(x)-Fibonacci polynomials Fh,n(x), and thus for h(x) = 2x we get Byrd’s Fibonacci and octonion polynomials Φn(x) from the h(x)-Fibonacci 2 2 3 4 n octonion polynomials Qh,n(x). gQ(t)t = Qh,0t + Qh,1t + Qh,2t + ... + Qh,n−2t + .... (12) 4 IPEK˙ & ARI

From Definition 6, (10) , (11) and (12), we have Combining Definition (2) and (20) gives the following equal- ity  2  1 − h(x)t − t gQ(t) = Qh,0 + Qh,1 − h(x)Qh,0 t (13) X7 and we thus obtain (8). From Definition 6, it follows that Qh,1(−x) + h(x)Qh,0(−x) = Fh,s−1(−x)es. (21) s=0 X7 − n+1 Qh,1 h(x)Qh,0 = Fh,s−1es. (14) Since Fh,n(−x) = (−1) Fh,n(x) from Theorem 2.2 in Nalli s=0 and Haukkanen (2009) from (15) and (21) we have (16).  Combining (13) and (14), then (9) is evident.  Binet’s formulas are well known in the theory of the Fi- Similarly, we have the following result. bonacci numbers. These formulas can also be carried out for the h(x)-Fibonacci octonion polynomials. Theorem 2. Suppose that h(x) is an odd polynomial. Then The associated with the recurrence ∞ p P n relation (2) is v2 = h(x)v + 1. The roots of this equation are for gQ(t) = Qh,n(−x)(−t) we have n=0 p p h(x) + h(x)2 + 4 h(x) − h(x)2 + 4 Q (−x) − Q (−x) + h(x)Q (−x) t α(x) = , β(x) = . gp (t) = h,0 h,1 h,0 (15) 2 2 Q 1 − h(x)t − t2 (22) The following basic identities is needed for our purpose in and proving.  X7 α(x) + β(x) = h(x),  p 1  g (t) = (−1)s+1 F (x) + F (x)t e . p  Q 2 h,s h,s−1 s α x − β x h x 2 , (23) 1 − h(x)t − t ( ) ( ) = ( ) + 4  s=0 −  (16) α(x).β(x) = 1.  For convenience of representation, we adopt the following Proof. We now consider notations: α = α(x) and β = β(x). X∞ The following lemma is directly useful for stating our next p n gQ(t) = Qh,n(−x)(−t) . main results. n=0 Lemma 1. For the generating function gQ(t) in (7) of the p gQ(t) is a . Therefore, we need not take h(x)-Fibonacci octonion polynomials Qh,n(x), we have care of the convergence of the series. Thus, we write " # 1 Qh,1 − βQh,0 Qh,1 − αQh,0 p 2 gQ(t) = − . gQ(t) = Qh,0(−x) − Qh,1(−x)t + Qh,2(−x)t (17) α − β 1 − αt 1 − βt −... + Q (−x)(−t)n + .... h,n Proof. The proof can be obtained easily from (23). 

The orders of Qh,n−1(−x) and Qh,n−2(−x), respectively, are 1 We obtain following Binet’s formula for Qh,n(x). and 2 less than the order of Qh,n(−x). Thus, since h(−x) = −h(x) we obtain Theorem 3. For n ≥ 0, the Binet’s formula for the h(x)- Fibonacci octonion polynomials Qh,n(x) is as follows p 2 −h(x)gQ(t)t = −h(x)Qh,0(−x)t + h(x)Qh,1(−x)t (18) ∗ n ∗ n 3 α α − β β −h(x)Qh,2(−x)t + ... Qh,n(x) = (24) n−1 n α − β −(−1) h(x)Qh,n−1(−x)t + ... 7 7 ∗ P s ∗ P s and where α = α es and β = β es. s=0 s=0 p 2 2 3 4 gQ(t)t = Qh,0(−x)t − Qh,1(−x)t + Qh,2(−x)t (19) Proof. From Lemma 1, we obtain n−2 n −... + (−1) Qh,n−2(−x)(t) + .... " # 1 Qh,1 − βQh,0 Qh,1 − αQh,0 gQ(t) = − From (17) , (18) and (19) we get (15). On the other hand, α − β 1 − αt 1 − βt  ∞ by using Definition (6), we can compute LHS = Qh,1(−x) + 1  X n n − = (Qh,1 − βQh,0) α t (25) h(x)Qh,0( x) α − β  n=0 ∞  X7 X  − −  − − n n LHS = Fh,s+1( x) + h(x)Fh,s( x) es. (20) (Qh,1 αQh,0) β t  . s=0 n=0 FIBONACCI OCTONION POLYNOMIALS 5

By taking (6) into (25), we can get the above-introduced generating function and Binet formula  for the h(x)-Fibonacci octonion polynomials and numbers X7 X∞ 1   n n will have the most effective application. gQ(t) =  Fh,s+1 − βFh,s es α t (26) α − β  s=0 n=0 7 ∞  Acknowledgements X X  − F − αF  e βntn . h,s+1 h,s s  s=0 n=0 We would like to thank the editor and the reviewers for their helpful comments on our manuscript which have helped F − βF αs F − αF βs Since h,s+1 h,s = and h,s+1 h,s = and from us to improve the quality of the present paper. (26) we have   X7 X∞ X7 X∞  References 1  s n n s n n gQ(t) =  α es α t − β es β t  . (27) α − β   s=0 n=0 s=0 n=0 Adler, S. L., & Finkelstein, D. R. (2008). Quaternionic and quantum fields. Physics Today, 49(6), 58–60. 7 7 ∗ P s ∗ P s Akyigit,˘ M., Kösal, H. H., & Tosun, M. (2013). Split fibonacci For α = α es and β = β es, from (27) we obtain s=0 s=0 quaternions. Advances in Applied Clifford Algebras, 23(3), ∞ 535–545. X (α∗αn − β∗βn) g (t) = tn. (28) Catarino, P. (2015). A note on h (x)- fibonacci quaternion polyno- Q α − β n=0 mials. Chaos, Solitons & Fractals, 77, 1–5. Conway, J. H., & Smith, D. A. (n.d.). On quaternions and octo- Consequently, by the of generating function in (7) nions: their , arithmetic, and . ak peters, and (28), we have Binet’s formula for Qh,n(x) in (24)  natick, massachusetts (canada), 2003 (Tech. Rep.). ISBN 1-56881-134-9. Theorem 4. For m ∈ , n ∈ , the generating function of Z N Coxeter, H., et al. (1946). cayley numbers. Duke Mathe- { } the sequence Qh,m+n(x) is as follows matical Journal, 13(4), 561–578. ∞ Cureg, E., & Mukherjea, A. (2010). Numerical results on some gen- X Qh,m(x) + Qh,m−1(x)t Q (x)tn = . eralized random fibonacci sequences. & mathe- h,m+n 1 − h(x)t − t2 n=0 matics with applications, 59(1), 233–246. Proof. By using the Binet formula for Q (x), we write Dunlap, R. A. (n.d.). The golden ratio and fibonacci numbers. h,n Er, M. (1984). Sums of fibonacci numbers by methods. ∞ ∞ ∗ m+n ∗ m+n Fibonacci Quart, 22(3), 204–207. X n X (α α − β β ) n Qh,m+n(x)t = t . Chaos, α − β Falcon, S., & Plaza, A. (2007). On the fibonacci k-numbers. n=0 n=0 Solitons & Fractals, 32(5), 1615–1624. Therefore, we get Halici, S. (2012). On fibonacci quaternions. Advances in Applied Clifford Algebras, 22(2), 321–327. ∞  ∞ ∞  X 1  X X  Halici, S. (2013). On complex fibonacci quaternions. Advances in Q (x)tn = α∗αm αntn − β∗βm βntn h,m+n α − β   Applied Clifford Algebras, 23(1), 105–112. n=0 n=0 n=0 " # Hoggat, V. (1969). Fibonacci and lucas numbers. palo alto. CA: 1 1 1 Houghton. = α∗αm − β∗βm . α − β 1 − αt 1 − βt Horadam, A. (1961). A generalized fibonacci sequence. American Mathematical Monthly, 455–459. So, from this and (23) we obtain Horadam, A. F. (1963). Complex fibonacci numbers and fibonacci     quaternions. American Mathematical Monthly, 289–291. X∞ α∗αm − β∗βm + α∗αm−1 − β∗βm−1 t  n 1   Iyer, M. (1969). A note on fibonacci quaternions. The Fibonacci Qh,m+n(x)t =   . α − β  1 − h(x)t − t2  n=0 Quarterly, 3, 225–229. Keçilioglu,˘ O., & Akkus, I. (2015). The fibonacci octonions. Ad- Consequently, if we recall the Binet’s formula for Qh,n(x), we vances in Applied Clifford Algebras, 25(1), 151–158. get the result.  Kilic, E. (2008). The binet formula, sums and representations of generalized fibonacci p-numbers. European Journal of com- Conclusions binatorics, 29(3), 701–711. In this paper, we consider the h(x)-Fibonacci octonion Kiliç, E., et al. (2006). The generalized binet formula, representa- tion and sums of the generalized order-k pell numbers. Tai- polynomials that generalize both Catalan’s Fibonacci octo- wanese Journal of Mathematics, 10(6), pp–1661. nion polynomials Ψn(x) and Byrd’s Fibonacci octonion poly- Kilic, E., & Tasci, D. (2006). On the generalized order-k fibonacci nomials and also k-Fibonacci octonion numbers that gener- and lucas numbers. Rocky Mountain J. Math., to appear. alize Fibonacci octonion numbers. Then we give the gener- Kincaid, D. R., & Cheney, E. W. (2002). : math- ating function and Binet formula of the h(x)-Fibonacci octo- ematics of scientific computing (Vol. 2). American Mathe- nion polynomials. We predict that in which part of science matical Soc. 6 IPEK˙ & ARI

Koshy, T. (n.d.). Fibonacci and lucas numbers with applications. mathematics. CRC press. 2001. A Wiley-Interscience Publication. Tasci, D., & Kilic, E. (2004). On the order-k generalized lucas Lando, S. K. (2003). Lectures on generating functions (Vol. 23). numbers. Applied mathematics and computation, 155(3), American mathematical society Providence, RI. 637–641. Levesque, C. (1985). On m-th order linear recurrences. In Fi- Vajda, S. (1989). Fibonacci & lucas numbers, and the golden sec- bonacci quarterly. tion. theory and applications, ser. Ellis Horwood Books in Nalli, A., & Haukkanen, P. (2009). On generalized fibonacci and Mathematics and its Applications. Chichester-New York: El- lucas polynomials. Chaos, Solitons & Fractals, 42(5), 3179– lis Horwood Ltd. 3186. Yang, S.-l. (2008). On the k-generalized fibonacci numbers and Öcal, A. A., Tuglu, N., & Altini¸sik,E. (2005). On the representa- high-order linear recurrence relations. Applied Mathematics tion of k-generalized fibonacci and lucas numbers. Applied and Computation, 196(2), 850–857. mathematics and computation, 170(1), 584–596. Yilmaz, N., & Taskara, N. (2013). Binomial transforms of the Ramírez, J. L. (2014). On convolved generalized fibonacci and padovan and perrin matrix sequences. In Abstract and ap- lucas polynomials. Applied Mathematics and Computation, plied analysis (Vol. 2013). 229, 208–213. Rosen, K. H. (1999). Handbook of discrete and combinatorial