Bull. Math. Soc. Sci. Math. Roumaie Tome 55103 No. 1, 2012, 95 103 Some idetities ivolvig Fiboacci, Lucas polyomials ad their applicatios by Wag Tigtig ad Zhag Wepeg Abstract The mai purpose of this paper is to study some sums of powers of Fiboacci polyomials ad Lucas polyomials, ad give several iterestig idetities. Fially, usig these idetities we shall prove a cojecture proposed by R. S. Melham i [4]. Key Words: Fiboacci polyomials, Lucas polyomials, combiatorial method, idetity, R. S. Melham s cojectures. 2010 Mathematics Subject Classificatio: Primary 11B39 Secodary 11B37. 1 Itroductio For ay variable quatity x, the Fiboacci polyomial F x ad the Lucas polyomial L x are defied as F 0 x 0, F 1 x 1, ad F +1 x xf x + F 1 x for all 1 L 0 x 2, L 1 x x, ad L +1 x xl x + L 1 x for all 1. If x 1, the F 1 F ad L 1 L, the famous Fiboacci sequece ad Lucas sequece, respectively. It is clear that these two polyomial sequeces are the secod-order liear recurrece sequeces. Lettig α x+ x 2 +4 2, β x x 2 +4 2, the from the properties of the secod-order liear recurrece sequeces, we have F x α β α β ad L x α + β. Cocerig F x ad L x, various authors studied them ad obtaied may iterestig results. For example, E. Lucas s classical work [3] first studied the arithmetical properties of L, ad obtaied may importat results. Y. Yua ad Z. Wepeg [6] proved some idetities ivolvig F x. This work is supported i part by the N.S.F.11071194 of P.R.Chia.
96 Wag Tigtig ad Zhag Wepeg Recetly, several authors studied the sums of powers of Fiboacci umbers ad Lucas umbers, ad obtaied a series importat idetities, see [1], [2] ad [5]. At the same time, R. S. Melham [4] also proposed the followig two cojectures: Cojecture 1. Let m 1 be a iteger. The the sum L 1 L 3 L 5 L 2m+1 F 2m+1 ca be expressed as F 2+1 1 2 R 2m 1 F 2+1, where R 2m 1 x is a polyomial of degree 2m 1 with iteger coefficiets. Cojecture 2. Let m 0 be a iteger. The the sum L 1 L 3 L 5 L 2m+1 L 2m+1 ca be expressed as L 2+1 1 Q 2m L 2+1, where Q 2m x is a polyomial of degree 2m with iteger coefficiets. The mai purpose of this paper is to obtai some idetities ivolvig Fiboacci polyomials ad Lucas polyomials. As applicatios, we use these idetities to prove that the above Cojecture 2 is true. That is, we shall prove the followig coclusios: Theorem 1. For ay positive itegers h ad, we have the idetities a. F 2 2m+1x { 1 x 2 + 4 h + 1 2!! 2 + 2 } F 4kh+1 x F x b. L 2 2m+1x h + 1 1 2!! 2 + 2 1 F 4kh+1x F x c. F 2+1 2m+1 x 1 x 2 + 4 F 2+1h+1 x L +1 x d. L 2+1 Theorem 2. 2m+1 x 1 L 2+1h+1x. L +1 x For ay positive itegers h ad, we have the idetities A. L 2 2mx h 2!! 2 + 2 F 2h+1 x F x F x
Idetities ivolvig Fiboacci, Lucas polyomials 97 B. F 2 2mx [ 1 x 2 + 4 h 2!! 2 + 2 1 k F ] 2h+1x F x F x C. L 2+1 2m x L +12h+1 x L +1 x L +1 x D. F 2+1 2m x 1 x 2 + 4 1 F +12h+1x F +1 x L +1 x As several applicatios of Theorem 2, we ca deduce the followig: Corollary 1. Let h 1 ad 0 be two itegers. The the sum L 1 xl 3 xl 5 x L 2+1 x L 2+1 2m x ca be expressed as L 2h+1 x x Q 2 x, L 2h+1 x, where Q 2 x, y is a polyomial i two variables x ad y with iteger coefficiets ad degree 2 of y.. Corollary 2. Let h 1 ad 0 be two itegers. The the sum L 1 xl 3 xl 5 x L 2+1 x F2m 2+1 x ca be expressed as F 2h+1 x 1 H 2 x, F 2h+1 x, where H 2 x, y is a polyomial i two variables x ad y with iteger coefficiets ad degree 2 of y. Takig x 1 i Corollary 1 ad Corollary 2, the we have the followig coclusios for Fiboacci ad Lucas umbers: Corollary 3. Let h 1 be a positive iteger. The the sum L 1 L 3 L 5 L 2+1 F 2+1 ca be expressed as F 2h+1 1 H 2 F 2h+1, where H 2 x is a polyomial of degree 2 with iteger coefficiets. Corollary 4. Let h 1 be a iteger. The the sum L 1 L 3 L 5 L 2+1 L 2+1
98 Wag Tigtig ad Zhag Wepeg ca be expressed as L 2h+1 1 Q 2 L 2h+1, where Q 2 x is a polyomial of degree 2 with iteger coefficiets. Takig x 2 i Corollary 1, ote that L 2 P, the th Pell umber, P 0 0, P 1 1 ad P +2 2P +1 + P for 0. The we also have the followig: Corollary 5. Let h 1 be a iteger. The the sum P 1 P 3 P 5 P 2+1 P 2+1 ca be expressed as P 2h+1 2 R 2 P 2h+1, where R 2 x is a polyomial of degree 2 with iteger coefficiets. It is clear that our Corollary 1 proves a geeralizatio of Melham s Cojecture. Our Corollary 3 make some substatial progress for the Melham s Cojecture 1. Corollary 5 give some ew idetities for the Pell umbers. 2. Proof of the theorems I this sectio, we shall give the proofs of our Theorems. First we prove Theorem 1. I fact, for ay positive iteger ad real umber x 0, by usig the familiar biomial expasio x + x 1 k x we get x + 1 2 2! x! 2 + 2 x + 1 x, 1.1 x 1 2 1 2! x! 2 + 2 1 x + 1 x, 1.2 ad x + 1 x 2+1 x 1 x 2+1 x +1 + 1 x +1, 1.3 1 x +1 1 x +1. 1.4 Now takig x α 2m+1 i 1.1, 1.2, 1.3, ad 1.4, the 1 x β2m+1. From the defiitios of F x ad L x, we may immediately deduce the idetities
Idetities ivolvig Fiboacci, Lucas polyomials 99 [ ] F2m+1x 2 1 2! x 2 + 4! 2 + 2 L2m+1 x, 1.5 L 2 2m+1x 1 2!! 2 + 2 1 L 2m+1 x, 1.6 ad F 2+1 2m+1 x 1 x 2 + 4 L 2+1 2m+1 x F2m+1+1 x, 1.7 1 L 2m+1+1 x. 1.8 Now takig x α 2m i 1.1,1.2, 1.3, ad 1.4, we deduce the idetities L 2 2mx 2!! 2 + 2 L4km x, 1.9 [ F2mx 2 1 x 2 + 4 1 2! ]! 2 + 2 1 L 4km x, 1.10 L 2+1 2m x L2m+1 x, 1.11 ad F 2+1 2m x 1 x 2 + 4 1 F 2m+1 x. 1.12
100 Wag Tigtig ad Zhag Wepeg For ay iteger h > 0, we sum o m i 1.5, F 2 2m+1x h + 1 x 2 + 4 [ 1 x 2 + 4 h + 1 2! { 2!! 2 + 2 h + 1! 2 + [ α α 4kh+1 1 2 h L 2m+1 x α 4k + 1 β β 4kh+1 1 ]} β 4k 1 { h + 1 2! x 2 + 4! 2 + 2 } α 4kh+ α 4kh+6k + β 4kh+ β 4kh+6k h + 1 2 α 4k β 4k. 1.13 ] Note that the idetities α 4kh+ α 4kh+6k + β 4kh+ β 4kh+6k α 4kh+4k β 4kh+4k α β x 2 + 4F 4kh+4k F ad 2 α 4k β 4k α β 2 x 2 + 4F 2, from 1.13 we may immediately deduce the idetity F 2 2m+1x { 1 x 2 + 4 h + 1 2!! 2 + 2 } F 4kh+1 x. F x This proves the idetity a of Theorem 1. Similarly, from formulae 1.6, 1.7 ad 1.8 we ca deduce the other three idetities of Theorem 1.
Idetities ivolvig Fiboacci, Lucas polyomials 101 Now we prove Theorem 2. From 1.9, we have L 2 2mx h 2!! 2 + 2 h α 4km + β 4km h 2!! 2 + 2 α 4kh+1 α 4k α 4k + β4kh+1 β 4k 1 β 4k 1 h 2!! 2 + 2 α 4kh α 4kh+1 2 + α 4k + β 4kh β 4kh+1 + β 4k 2 α 4k β 4k h 2!! 2 + 2 α 4kh+ β 4kh+ α β α β 2 α β 2 h 2!! 2 + 2 F 2h+1 x F x. F x This proves the idetity A of Theorem 2. Similarly, from formulae 1.10, 1.11 ad 1.12 we ca also deduce the other three idetities of Theorem 2. Now we use C of Theorem 2 to prove Corollary 1. It is clear that if P x Zx, the a b divides P a P b. From this properties ad ote that the idetity L +1 L 2+1 x L 2+1+1 x we ca deduce L 2h+1 x x L +1 L 2h+1 x L +1 x L 2h+1+1 x L +1 x. 1.14 Combiig C of Theorem 2, 1.14 ad ote that L +1 x x, L +1 x 1 we may immediately deduce the idetity L 1 xl 3 xl 5 x L 2+1 x L 1 xl 3 xl 5 x L 2+1 x L 2h+1 x x Q 2 x, L 2h+1 x, L 2+1 2m x L 2h+1+1 x L +1 x L +1 x where Q 2 x, y is a polyomial i two variables x ad y with iteger coefficiets ad degree 2 of y. This proves Corollary 1. To prove Corollary 2, from D of Theorem 2 we kow that we oly to prove the polyomials x 2 + 4 ad F 2h+1 x 1 satisfyig F 2h+1 x 1, x 2 + 4 1 ad F 2h+1 x 1 F 2h+1+1 x F +1 x for all itegers k 0.
102 Wag Tigtig ad Zhag Wepeg First from the defiitio of F x ad biomial expasio we ca easy to prove F 2h+1 x 1, x 2 + 4 1. Therefore, F 2h+1 x 1, x 2 + 4 1. Next, we prove that the polyomial F 2h+1 x 1 divide F 2h+1+1 x F +1 x. I fact ote the fact that F a x F b x F a ɛb/2 xl a+ɛb/2 x valid for all a b mod 2 with ɛ {1, 1} give by ɛ 1 if a b mod 4 ad ɛ 1 if a b 2 mod 4. Take a 2h + 1, b 1 so a b 2h ad a 1 + 1a, b 1 + 1. The a 1 b 1 + 1a b, so ɛ is the same for a, b as for a 1, b 1 amely it is 1 if h is eve ad 1 if h is odd. Thus, F 2h+1 x 1 F 2h+1 x F 1 x F h xl h+1 x or F h+1 xl h x accordig to whether h is eve or odd, respectively, ad also or F 2h+1+1 x F +1 x F +1h xl +1h+1 x F +1h+1 xl +1h x agai accordig to whether h is eve or odd respectively. Now the claim follows from the fact that F u x F v x wheever u v ad if additioally v/u is odd, the also L u x L v x. This completes the proof of Corollary 2. It seems that usig our method we ca ot solve the Melham s Cojecture 1 completely. But we believe that it is true. Ackowledgmet The authors would like to thak the referee for his very helpful ad detailed commets, which have sigificatly improved the presetatio of this paper. Refereces [1] H. Prodiger, O a sum of Melham ad its variats, The Fiboacci Quarterly, 46/47 2008/2009, 207-215. [2] K. Ozeki, O Melham s sum, The Fiboacci Quarterly. 46/47 2008/2009, 107-110. [3] E. Lucas, Théorie des foctios umériques simplemet périodiques, Amer. J. Math. 11878, 184-240, 289-321. [4] R. S. Melham, Some cojectures cocerig sums of odd powers of Fiboacci ad Lucas umbers, The Fiboacci Quarterly, 46/47 2008/2009, 312-315.
Idetities ivolvig Fiboacci, Lucas polyomials 103 [5] M. Wiema ad C. Cooper, Divisibility of a F-L Type Covolutio, Applicatios of Fiboacci Numbers, Vol. 9, Kluwer Acad. Publ., Dordrecht, 2004, 267-287. [6] Y. Yua ad Z. Wepeg, Some idetities ivolvig the Fiboacci polyomials, The Fiboacci Quarterly, 40 2002, 314-318. Received: 24.11.2010, Revised: 28.04.2011, 15.12.2011, Accepted: 19.12.2011. Departmet of Mathematics, Northwest Uiversity, Xi a, Shaaxi, P.R.Chia E-mails: tigtigwag126@126.com wpzhag@wu.edu.c