On h(x)-fibonacci octonion polynomials

Transcription

1 Alabama Journal of Mathematics 39 (05) ISSN On h(x)-fibonacci octonion polynomials Ahmet İpek Karamanoğlu Mehmetbey University, Science Faculty of Kamil Özdağ, Department of Mathematics, Karaman, Turkey Kamil Arı Karamanoğlu Mehmetbey University, Science Faculty of Kamil Özdağ, Department of Mathematics, Karaman, Turkey In this paper, we introduce h(x)-fibonacci octonion polynomials that generalize both Catalan s Fibonacci octonion polynomials Byrd s Fibonacci octonion polynomials also k- Fibonacci octonion numbers that generalize Fibonacci octonion numbers. Also we derive the Binet formula generating function of h(x)-fibonacci octonion polynomial sequence. Introduction To investigate the normed division algebras is greatly a important topic today. It is well known that the octonions O are the nonassociative, noncommutative, normed division algebra over the real numbers R. Due to the nonassociative noncommutativity, one cannot directly extend various results on real, complex quaternion numbers to octonions. The book by Conway Smith (n.d.) gives a great deal of useful background on octonions, much of it based on the paper of Coxeter et al. (946). Octonions made further appearance ever since in associative algebras, analysis, topology, physics. Nowadays octonions play an important role in computer science, quantum physics, signal color image processing, so on (e.g. Adler Finkelstein (008)). Many references may be given for Fibonacci-like sequences numbers related issues. The readers are referred to Koshy (n.d.) Vajda (989) for some other sequences which can be obtained from the Fibonacci numbers. One may find many applications of this type sequences of numbers in various branches of science like pure applied mathematics, in biology or in phyllotaxies, among many others. Hence, many researches are performed on the various type of Fibonacci sequence; for example see Cureg Mukherjea (00), Falcon Plaza (007), A. Horadam (96), Kilic (008), Öcal, Tuglu, Altinişik (005), Yang (008)-Yilmaz Taskara (03). The connection between the Fibonacci-type numbers the Golden Section is expressed by the well-known mathematical formula, so-called Binet s formulas. Many references may be given for Fibonacci-like sequences numbers Binet s formulas. By using a generating function Levesque (985) gave a Binet formula for the Fibonacci sequence. Kilic Tasci (006) gave the generalized Binet formulas for the generalized order-k Fibonacci (Er (984)) Corresponding Author ahmetipek@kmu.edu.tr Lucas numbers (Tasci Kilic (004)). Kiliç et al. (006) gave the Binet formula for the generalized Pell sequence. The investigation of special number sequences over H O which are not analogs of ones over R C has attracted some recent attention (see, e.g.,akyiğit, Kösal, Tosun (03), Halici (0), Halici (03), Iyer (969), A. F. Horadam (963) Keçilioğlu Akkus (05)). While majority of papers in the area are devoted to some Fibonaccitype special number sequences over R C, only few of them deal with Fibonacci-type special number sequences over H O (see, e.g., Akyiğit et al. (03), Halici (0), Halici (03), Iyer (969), A. F. Horadam (963) Keçilioğlu Akkus (05)), notwithsting the fact that there are a lot of papers on various types of Fibonacci-type special number sequences over R C (see, e.g., Cureg Mukherjea (00), Falcon Plaza (007), A. Horadam (96), Kilic (008), Öcal et al. (005), Yang (008)-Yilmaz Taskara (03)). In this paper, we introduce h(x)-fibonacci octonion polynomials that generalize both Catalan s Fibonacci octonion polynomials Ψ n (x) Byrd s Fibonacci octonion polynomials also k-fibonacci octonion numbers that generalize Fibonacci octonion numbers. Also we derive the Binet formula generating function of h(x)-fibonacci octonion polynomial sequence. The rest of the paper is structured as follows. Some preliminaries which are required are given in Section. In Section 3 we introduce Catalan, Byrid, Fibonacci octonion polynomials numbers provide their some properties. Section 4 ends with our conclusion. Some Preliminaries We start this section by introducing some definitions notations that will help us greatly in the statement of the results. We begin this section by giving the following definition for the classic Fibonacci sequence. The classic Fibonacci

2 İPEK & ARI {F n } n N sequence is a very popular integer sequence. Definition. The classic Fibonacci {F n } n N sequence is defined by F 0 = 0, F = F n = F n + F n for n, () where F n denotes the n th classic Fibonacci number (Dunlap (n.d.), Koshy (n.d.), Vajda (989)). The books written by Hoggat (969), Koshy (n.d.) Vajda (989) collects classifies many results dealing with the these number sequences, most of them are obtained quite recently. Nalli Haukkanen (009) introduced the h(x)- Fibonacci polynomials. Then, Ramírez (04) introduced the convolved h(x)-fibonacci polynomials obtain new identities. Definition. Let h(x) be a polynomial with real coefficients. The h(x)-fibonacci polynomials {F h,n (x)} are defined by the recurrence relation F h,n+ (x) = h(x)f h,n (x) + F h,n (x), n, () with initial conditions F h,0 (x) = 0, F h, (x) = (Nalli Haukkanen (009)). For h(x) = x, it is obtained Catalan s Fibonacci polynomials ψ n (x), for h(x) = x it is obtained Byrd s Fibonacci polynomials φ n (x). If for k real number h(x) = k, it is obtained the k-fibonacci numbers F k,n. For k = k = it is obtained the usual Fibonacci numbers F n the Pell numbers P n, respectively. The quaternion, which is a type of hypercomplex numbers, was formally introduced by Hamilton in 843. Definition 3. The real quaternion is defined by q = q r + q i i + q j j + q k k where q r, q i, q j q k are real numbers i, j k are complex operators obeying the following rules i = j = k =, i j = ji = k, jk = k j = i, ki = ik = j. In recent years there has been a flurry of activity for doing research with Fibonacci quaternion octonion numbers. We can start from the Fibonacci quaternion numbers introduced by Horadam in 963. Definition 4. The Fibonacci quaternion numbers that are given for the n th classic Fibonacci F n number are defined by the following recurrence relations: Q n = F n + if n+ + jf n+ + kf n+3 (3) where n = 0, ±, ±,...(A. F. Horadam (963)). Catarino (05) introduced h(x)-fibonacci quaternion polynomials. The basic properties of Fibonacci quaternion numbers can be found in Akyiğit et al. (03), Halici (0), Halici (03), Iyer (969) A. F. Horadam (963). The octonions in Clifford algebra C are a normed division algebra with eight dimensions over the real numbers larger than the quaternions. The field O C 4 of octonions α = 7 α s e s, a i (i = 0,,..., 7) R (4) is an eight-dimensional non-commutative nonassociative R-field generated by eight base elements e 0, e,..., e 6 e 7. The multiplication rules for the basis of O are listed in the following table e e e 3 e 4 e 5 e 6 e 7 e e e 3 e 4 e 5 e 6 e 7 e e e 3 e e 5 e 4 e 7 e 6 e e e 3 e e 6 e 7 e 4 e 5 e 3 e 3 e e e 7 e 6 e 5 e 4 e 4 e 4 e 5 e 6 e 7 e e e 3 e 5 e 5 e 4 e 7 e 6 e e 3 e e 6 e 6 e 7 e 4 e 5 e e 3 e e 7 e 7 e 6 e 5 e 4 e 3 e e Table. The multiplication table for the basis of O. (5) The scalar the vector part of a octonion α = 7 α s e s O are denoted by S α = α 0 V α = 7 α s e s, respectively. s= Let α β be two octonions such that α = 7 α s e s β = 7 β s e s. We now are going to list some properties, without proofs, for α β octonions. Property. α ± β = 7 (α s ± β s )e s Property. αβ = S α S β +S α V β +V α S β V α V β +V α V β, where S α = α 0 V α = 7 α s e s. s= Property 3. α = α 0 e 0 7 α s e s Property 4. (αβ) = βα. s= Property 5. α+α = α 0 e 0 α α = 7 Property 6. αα = αα Property 7. αα = 7 α k. k=0 k= α k e k.

3 FIBONACCI OCTONION POLYNOMIALS 3 The norm of an octonion can be defined as α = αα. Property 8. αβ = α β. Property 9. k.α = 7 (kα i ) e i. i=0 Property 0. α, β = 7 α i β i. i=0 The Fibonacci octonion numbers are given by the following definition. Definition 5. For n 0, the Fibonacci octonion numbers that are given for the n th classic Fibonacci F n number are defined by the following recurrence relations: Q n = 7 F n+s e s. The basic properties of Fibonacci octonion numbers can be found in Keçilioğlu Akkus (05). The h(x)-fibonacci octonion polynomials In this section, we introduce h(x)-fibonacci octonion polynomials that generalize both Catalan s Fibonacci octonion polynomials Ψ n (x) Byrd s Fibonacci octonion polynomials also k-fibonacci octonion numbers. Also we derive the Binet formula generating function of h(x)- Fibonacci octonion polynomial sequence. Let e i, i = 0,,, 3, 4, 5, 6, 7, be a basis of O which satisfy the non-commutative non-associative multiplication rules are listed in Table in (5). Let h(x) be a polynomial with real coefficients. Definition 6. The h(x)-fibonacci octonion polynomials {Q h,n (x)} are defined by the recurrence relation Q h,n (x) = 7 F h,n+s (x)e s (6) where F h,n (x) is the n th h(x)-fibonacci polynomial. Particular cases of the previous definition are: For h(x) = x, it is obtained Catalan s Fibonacci polynomials ψ n (x) from the h(x)-fibonacci polynomials F h,n (x), thus for h(x) = x we have Catalan s Fibonacci octonion polynomials Ψ n (x) from the h(x)- Fibonacci octonion polynomials Q h,n (x). For h(x) = x, it is obtained Byrd s Fibonacci polynomials φ n (x) from the h(x)-fibonacci polynomials F h,n (x), thus for h(x) = x we get Byrd s Fibonacci octonion polynomials Φ n (x) from the h(x)-fibonacci octonion polynomials Q h,n (x). If for k real number h(x) = k, it is obtained the k- Fibonacci numbers F k,n from the h(x)-fibonacci polynomials F h,n (x), thus for h(x) = k we obtain k-fibonacci octonion numbers Q k,n from the h(x)- Fibonacci octonion polynomials Q h,n (x). Also, for k = k = we have the Fibonacci octonion numbers Q n the Pell octonion numbers Υ n, respectively. The ordinary generating function (OGF) of the sequence {a n } is defined by g(x) = a n x n (Rosen (999)) the exponential generating function (EGF) of a sequence {a n } is defined by g(x) = (Kincaid Cheney (00)). a n x n n! For general material on generating functions the interested reader may refer to the books written by Lo (003)?. The generating function g Q (t) of the sequence {Q h,n (x)} is defined by Q h,n (x)t n. (7) We know that the power series (7) converges if only if t < min(/ α, / β ) for α β given (). We consider g Q (t) a formal power series which is needed not take care of the convergence. For convenience, we use the following notations: F h,n = F h,n (x) Q h,n = Q h,n (x). Equipped with the definitions properties above, we can present the fundamental theorems of this paper as follows. Theorem. The generating function for the h(x)-fibonacci octonion polynomials Q h,n (x) is Q h,0 + ( Q h, h(x)q h,0 ) t h(x)t t (8) h(x)t t 7 ( Fh,s + F h,s t ) e s. (9) Proof. The generating function g Q (t) of the h(x)-fibonacci octonion polynomials is Q h,0 + Q h, t + Q h, t Q h,n t n +... (0) The orders of Q h,n Q h,n, respectively, are less than the order of Q h,n. Thus, we obtain h(x)g Q (t)t = h(x)q h,0 t + h(x)q h, t () +h(x)q h, t h(x)q h,n t n +... g Q (t)t = Q h,0 t + Q h, t 3 + Q h, t Q h,n t n +... ()

4 4 İPEK & ARI From Definition 6, (0), () (), we have ( h(x)t t ) Q h,0 + ( Q h, h(x)q h,0 ) t (3) we thus obtain (8). From Definition 6, it follows that Q h, h(x)q h,0 = 7 F h,s e s. (4) Combining (3) (4), then (9) is evident. Similarly, we have the following result. Theorem. Suppose that h(x) is an odd polynomial. Then for g Q (t) = Q h,n ( x)( t) n we have g Q (t) = Q h,0( x) ( Q h, ( x) + h(x)q h,0 ( x) ) t h(x)t t (5) g Q (t) = h(x)t t Proof. We now consider 7 ( ) s+ ( F h,s (x) + F h,s (x)t ) e s. g Q (t) = Q h,n ( x)( t) n. (6) g Q (t) is a formal power series. Therefore, we need not take care of the convergence of the series. Thus, we write g Q (t) = Q h,0( x) Q h, ( x)t + Q h, ( x)t (7)... + Q h,n ( x)( t) n +... The orders of Q h,n ( x) Q h,n ( x), respectively, are less than the order of Q h,n ( x). Thus, since h( x) = h(x) we obtain h(x)g Q (t)t = h(x)q h,0( x)t + h(x)q h, ( x)t (8) h(x)q h, ( x)t ( ) n h(x)q h,n ( x)t n +... g Q (t)t = Q h,0 ( x)t Q h, ( x)t 3 + Q h, ( x)t 4 (9)... + ( ) n Q h,n ( x)(t) n +... From (7), (8) (9) we get (5). On the other h, by using Definition (6), we can compute LHS = Q h, ( x) + h(x)q h,0 ( x) LHS = 7 ( Fh,s+ ( x) + h(x)f h,s ( x) ) e s. (0) Combining Definition () (0) gives the following equality Q h, ( x) + h(x)q h,0 ( x) = 7 F h,s ( x)e s. () Since F h,n ( x) = ( ) n+ F h,n (x) from Theorem. in Nalli Haukkanen (009) from (5) () we have (6). Binet s formulas are well known in the theory of the Fibonacci numbers. These formulas can also be carried out for the h(x)-fibonacci octonion polynomials. The characteristic equation associated with the recurrence relation () is v = h(x)v +. The roots of this equation are α(x) = h(x) + h(x) + 4, β(x) = h(x) h(x) + 4. () The following basic identities is needed for our purpose in proving. α(x) + β(x) = h(x), α(x) β(x) = h(x) + 4, α(x).β(x) =. (3) For convenience of representation, we adopt the following notations: α = α(x) β = β(x). The following lemma is directly useful for stating our next main results. Lemma. For the generating function g Q (t) in (7) of the h(x)-fibonacci octonion polynomials Q h,n (x), we have [ Qh, βq h,0 Q ] h, αq h,0. αt βt Proof. The proof can be obtained easily from (3). We obtain following Binet s formula for Q h,n (x). Theorem 3. For n 0, the Binet s formula for the h(x)- Fibonacci octonion polynomials Q h,n (x) is as follows Q h,n (x) = α α n β β n where α = 7 α s e s β = 7 β s e s. (4) Proof. From Lemma, we obtain [ Qh, βq h,0 Q ] h, αq h,0 αt βt = (Q h, βq h,0 ) α n t n (5) (Q h, αq h,0 ) β n t n.

5 FIBONACCI OCTONION POLYNOMIALS 5 By taking (6) into (5), we can get 7 ( ) Fh,s+ βf h,s es α n t n (6) 7 ( ) Fh,s+ αf h,s es β n t n. Since F h,s+ βf h,s = α s F h,s+ αf h,s = β s from (6) we have 7 7 α s e s α n t n β s e s β n t n. (7) For α = 7 α s e s β = 7 β s e s, from (7) we obtain (α α n β β n ) t n. (8) Consequently, by the equality of generating function in (7) (8), we have Binet s formula for Q h,n (x) in (4) Theorem 4. For m Z, n N, the generating function of the sequence {Q h,m+n (x)} is as follows Q h,m+n (x)t n = Q h,m(x) + Q h,m (x)t h(x)t t. Proof. By using the Binet formula for Q h,n (x), we write Q h,m+n (x)t n (α α m+n β β m+n ) = t n. Therefore, we get Q h,m+n (x)t n = = α α m α n t n β β m β n t n [ α α m αt β β m βt So, from this (3) we obtain Q h,m+n (x)t n = α α m β β m + ( α α m β β m ) t h(x)t t. Consequently, if we recall the Binet s formula for Q h,n (x), we get the result. Conclusions In this paper, we consider the h(x)-fibonacci octonion polynomials that generalize both Catalan s Fibonacci octonion polynomials Ψ n (x) Byrd s Fibonacci octonion polynomials also k-fibonacci octonion numbers that generalize Fibonacci octonion numbers. Then we give the generating function Binet formula of the h(x)-fibonacci octonion polynomials. We predict that in which part of science ]. the above-introduced generating function Binet formula for the h(x)-fibonacci octonion polynomials numbers will have the most effective application. Acknowledgements We would like to thank the editor the reviewers for their helpful comments on our manuscript which have helped us to improve the quality of the present paper. References Adler, S. L., & Finkelstein, D. R. (008). Quaternionic quantum mechanics quantum fields. Physics Today, 49(6), Akyiğit, M., Kösal, H. H., & Tosun, M. (03). Split fibonacci quaternions. Advances in Applied Clifford Algebras, 3(3), Catarino, P. (05). A note on h (x)- fibonacci quaternion polynomials. Chaos, Solitons & Fractals, 77, 5. Conway, J. H., & Smith, D. A. (n.d.). On quaternions octonions: their geometry, arithmetic, symmetry. ak peters, natick, massachusetts (canada), 003 (Tech. Rep.). ISBN Coxeter, H., et al. (946). Integral cayley numbers. Duke Mathematical Journal, 3(4), Cureg, E., & Mukherjea, A. (00). Numerical results on some generalized rom fibonacci sequences. Computers & mathematics with applications, 59(), Dunlap, R. A. (n.d.). The golden ratio fibonacci numbers. Er, M. (984). Sums of fibonacci numbers by matrix methods. Fibonacci Quart, (3), Falcon, S., & Plaza, A. (007). On the fibonacci k-numbers. Chaos, Solitons & Fractals, 3(5), Halici, S. (0). On fibonacci quaternions. Advances in Applied Clifford Algebras, (), Halici, S. (03). On complex fibonacci quaternions. Advances in Applied Clifford Algebras, 3(), 05. Hoggat, V. (969). Fibonacci lucas numbers. palo alto. CA: Houghton. Horadam, A. (96). A generalized fibonacci sequence. American Mathematical Monthly, Horadam, A. F. (963). Complex fibonacci numbers fibonacci quaternions. American Mathematical Monthly, Iyer, M. (969). A note on fibonacci quaternions. The Fibonacci Quarterly, 3, 5 9. Keçilioğlu, O., & Akkus, I. (05). The fibonacci octonions. Advances in Applied Clifford Algebras, 5(), Kilic, E. (008). The binet formula, sums representations of generalized fibonacci p-numbers. European Journal of combinatorics, 9(3), Kiliç, E., et al. (006). The generalized binet formula, representation sums of the generalized order-k pell numbers. Taiwanese Journal of Mathematics, 0(6), pp 66. Kilic, E., & Tasci, D. (006). On the generalized order-k fibonacci lucas numbers. Rocky Mountain J. Math., to appear. Kincaid, D. R., & Cheney, E. W. (00). Numerical analysis: mathematics of scientific computing (Vol. ). American Mathematical Soc.

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