COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun

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1 Korea J. Math ) No. 3 pp COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized Fiboacci sequece {q }. The give a coectio betwee the sequece {q } ad the Chebyshev polyomials of the secod kid U x). With the aid of factorizatio of Chebyshev polyomials of the secod kid U x) we derive the complex factorizatios of the sequece {q }. 1. Itroductio For ay iteger 0 the well-kow Fiboacci sequece {F } is defied by the secod order liear recurrece relatio F +2 = F +1 + F where F 0 = 0 ad F 1 = 1. The Fiboacci sequece has bee geeralized i may ways for example by chagig the recurrece relatio see [8]) by chagig the iitial values see [4 5]) by combiig of these two techiques see [3]) ad so o. I [2] Edso ad Yayeie defied a further geeralized Fiboacci sequece {q } depedig o two real parameters used i a o-liear piecewise liear) recurrece relatio amely 1) q = a 1 ξ) b ξ) q 1 + q 2 2) Received May Revised September Accepted September Mathematics Subject Classificatio: 11B39. Key words ad phrases: geeralized Fiboacci sequeces tridiagoal matrices Chebyshev polyomials complex factorizatio. Fudig for this paper was provided by Namseoul Uiversity. c The Kagwo-Kyugki Mathematical Society This is a Ope Access article distributed uder the terms of the Creative commos Attributio No-Commercial Licese -c/3.0/) which permits urestricted o-commercial use distributio ad reproductio i ay medium provided the origial work is properly cited.

2 372 Sag Pyo Ju with iitial values q 0 = 0 ad q 1 = 1 where a ad b are positive real umbers ad { 0 if is eve 2) ξ) = 1 if is odd is the parity fuctio. Also the authors showed that the terms of the sequece {q } are give by the exteded Biet s formula 3) q = a 1 ξ) ) ξ) 2 ) α β α β where α ad β are roots of the quadratic equatio x 2 x = 0 ad α > β. These sequeces arise i a atural way i the study of cotiued fractios of quadratic irratioals ad combiatorics o words or dyamical system theory. Some well-kow sequeces are special cases of this geeralizatio. The Fiboacci sequece is a special case of {q } with a = b = 1. Whe a = b = 2 we obtai the Pell s sequece {P }. Eve further if we set a = b = k for some positive iteger k we obtai the k-fiboacci sequece {F k }. Usig the exteded Biet s formula 3) Edso ad Yayeie [2] derived a umber of mathematical properties icludig geeralizatios of Cassii s Catala s ad d Ocage s idetities for the Fiboacci sequece Yayeie [11] obtaied umerous ew idetities of {q } ad Zhag ad Wu [12] studied the partial ifiite sums of reciprocal of {q }. Jag ad Ju [7] give liearlizatio of the sequece {q }. I [9] the authors obtaied complex factorizatio formulas for the Fiboacci Pell ad k-fiboacci umbers by usig the determiats of sequeces of tridiagoal matrices. They used the tridiagoal matrices 1 2i i 1 i i i 2i 1 2 2i i i i 2i 2 k i i k i i k i i k

3 Complex factorizatios of the geeralized Fiboacci sequeces {q } 373 respectively to prove that 1 F = 1 2i cos πk ) P = 1 1 F k = k 2i cos πj ) j=1 for ay iteger 2 where i = i cos πk ) I this paper we give a coectio betwee the sequece {q } ad the Chebyshev polyomials of the secod kid. With the aid of factorizatio of Chebyshev polyomials of the secod kid we derive the complex factorizatios of the sequece {q }. 2. Chebyshev polyomials of the secod kid Chebyshev polyomials are of great importace i may areas of mathematics particularly approximatio theory. Chebyshev polyomials of the secod kid U x) defied by settig U 0 x) = 1 U 1 x) = 2x ad the recurrece relatio 4) U x) = 2xU 1 x) U 2 x) = 2 3. Hsiao [6] gave a complete factorizatio of Chebyshev polyomials of the first kid. Rivli [10] adapts Hsiao s proof for the Chebyshev polyomials of the secod kid U x) as follows 5) U x) = si + 1) cos 1 x) sicos 1 x) or 6) U x) = 2 )) kπ x cos. + 1

4 374 Sag Pyo Ju Now the first few umbers q ad Chebyshev polyomials of the secod kid U x) are q 0 = 0 : U 0 x) = 1 q 1 = 1 : U 1 x) = 2x q 2 = a : U 2 x) = 4x 2 1 q 3 = + 1 : U 3 x) = 8x 3 4x q 4 = a + 2a : U 4 x) = 16x 4 12x q 5 = a : U 5 x) = 32x 5 32x 3 + 6x q 6 = a 3 b 2 + 4a + 3a : U 6 x) = 64x 6 80x x Complex factorizatios of the sequece {q } I this sectio we give a coectio betwee the sequece {q } ad the Chebyshev polyomials of the secod kid U x). With the aid of factorizatio 5) ad 6) of Chebyshev polyomials of the secod kid we derive the complex factorizatios of the sequece {q }. Lemma 3.1. The sequece {q } satisfies 7) q +1 = a ξ) ξ) U 1 where i = 1 ad a b are positive real umbers. 8) 9) Proof. First ote that ξm + ) = ξm) + ξ) 2ξm)ξ) ξ + 1) = ξ 1). We prove the idetity 7) by iductio o. Whe = 1 we have a ξ1) ξ1) U1 = a 1 12 i2 = a = q 2. Next we assume the idetity 7) holds for all positive itegers less tha or equal to that is 10) q k = a ξk 1) ξk 1) k 1 U k 1 1 k ).

5 Complex factorizatios of the geeralized Fiboacci sequeces {q } 375 The we have a ξ) ξ) U = a ξ) ξ) { 2 U 1 } U 2 ) = a 1 ξ 1) b ξ 1) a ξ 1) ξ 1) 1 U 1 +a ξ 2)) ξ 2)) 2 U 2 2) 8)) = a 1 ξ 1) b ξ 1) q + q 1 10)) = a 1 ξ+1) b ξ+1) q + q 1 9)) = q +1 1)). Therefore the idetity 7) holds for all itegers 1. Theorem 3.2. The sequece {q } satisfies 4)) si + 1) cos 1 ) i 11) q +1 = a ξ) ξ) 2 si cos 1 ) 0 i 2 or 12) q +1 = a ξ) ξ) 2 )) kπ 2i cos where i = 1 ad a b are positive real umbers. Proof. Usig 7) i Lemma 3.1 5) ad 6) we obtai 11) ad 12).

6 376 Sag Pyo Ju Ackowledgmets. At first the author obtaied Theorem 3.2 similar to [9] usig the determiats of sequeces of tridiagoal matrices bi ai bi. 13) M a b) = ai bi ai The the referee suggested to simplify the proof by usig the coectio betwee the sequece {q } ad the Chebyshev polyomials of the secod kid U x). His advice gave a ice perspective. The author is very grateful to the referee. Refereces [1] N. D. Cahill J. R. D Errico ad J. P. Spece Complex factorizatios of the Fiboacci ad Lucas umbers The Fiboacci Quarterly 41 1) 2003) [2] M. Edso ad O. Yayeie A ew geeralizatio of Fiboacci sequece ad exteded Biet s formula Iteger ) [3] Y. K. Gupta Y. K. Pawar ad O. Sikhwal Geeralized Fiboacci Sequeces Theoretical Mathematics ad Applicatios 2 2) 2012) [4] Y. K. Gupta M. Sigh ad O. Sikhwal Geeralized Fiboacci-Like Sequece Associated with Fiboacci ad Lucas Sequeces Turkish Joural of Aalysis ad Number Theory 2 6) 2014) [5] A. F. Horadam A geeralized Fiboacci sequeces Amer. Math. Mothly ) [6] H.J. Hsiao O factorizatio of Chebyshev s polyomials of the first kid Bulleti of the Istitute of Mathematics Academia Siica 12 1) 1984) [7] Y. H. Jag ad S. P. Ju Liearizatio of geeralized Fiboacci sequeces Korea J. Math ) 3) [8] D. Kalma ad R. Mea The Fiboacci umbers - Exposed. The Mathematical Magazie ). [9] A. Oteles ad M. Akbulak Positive iteger power of certai complex tridiagoal matrices Applied Mathematics ad Computatio ) 2013) [10] T.J. Rivli The Chebyshev Polyomials From Approximatio Theory to Algebra ad Number Theory Wiley-Itersciece Joh Wiley 1990). [11] O. Yayeie A ote o geeralized Fiboacci sequeces Applied Mathematics ad Computatio ) [12] H. Zhag ad Z. Wu O the reciprocal sums of the geeralized Fiboacci sequeces Adv. Differ. Equ. 2013) Article ID ).

7 Complex factorizatios of the geeralized Fiboacci sequeces {q } 377 Sag Pyo Ju Iformatio Commuicatio Namseoul Uiversity Cheoa Korea spju7129@aver.com