Applied Mathematics Letters. On the properties of Lucas numbers with binomial coefficients

Transcription

1 Applied Mathematics Letters 3 ( Cotets lists available at ScieceDirect Applied Mathematics Letters joural homepage: wwwelseviercom/locate/aml O the properties of Lucas umbers with biomial coefficiets N Taskara, K Uslu, HH Gulec Selçuk Uiversity, Sciece Faculty, Departmet of Mathematics, 47, Campus, Koya, Turkey a r t i c l e i f o a b s t r a c t Article history: Received 13 April 9 Received i revised form 18 August 9 Accepted 18 August 9 I this study, some ew properties of Lucas umbers with biomial coefficiets have bee obtaied to write Lucas sequeces i a ew direct way I additio, some importat cosequeces of these results related to the Fiboacci umbers have bee give 9 Elsevier Ltd All rights reserved Keywords: Lucas umbers Fiboacci umbers Biomial coefficiets 1 Itroductio Fiboacci ad Lucas umbers have log iterested mathematicias for their itrisic theory ad their applicatios For rich applicatios of these umbers i sciece ad ature, oe ca see the citatios i [1 ] For istace, the ratio of two cosecutive of these umbers coverges to the Golde sectio α = 1+ (The applicatios of Golde ratio appears i may research areas, particularly i Physics, Egieerig, Architecture, Nature ad Art Physicists Naschie ad Marek-Crjac gave some examples of the Golde ratio i Theoretical Physics ad Physics of High Eergy Particles [6 9] Therefore, i this paper, we are maily iterested i whether some ew mathematical developmets ca be applied to these umbers I this paper we obtai ew results about Lucas umbers As a remider for the rest of this paper, for >, the well-kow Fiboacci {F } ad Lucas {L } sequeces are defied by F = F 1 + F ad L = L 1 + L, where F 1 = F = 1 ad L 1 =, L = 1, respectively Moreover, for the first Fiboacci umbers, it is well kow that the sum of the squares is F i = F F +1 Also i i = F +1 The sum of the squares formula is our motivatio to look for combiatorial sums related to the square of Lucas umbers Thus, agai for the motivatio of the paper, we should ote that, i [1], Spivey preseted a ew approach for evaluatig combiatorial sums by usig fiite differeces Also, he exteded this ew approach to hadle biomial sums of the form ( ( k= k ( 1 k a k, k ak ad k k kid [ k= ]a k k ad s(, k= ka k { There is also iterest for k-fiboacci polyomials Let F k, } N ( k+1 ak, as well as sums ivolvig usiged ad siged Stirlig umbers of the first be a k-fiboacci sequece Note that if k is a real variable x the F k, = F x, ad they correspod to the Fiboacci polyomials defied by { 1 if =, F +1 (x = x if = 1, xf (x + F 1 (x if > 1, (see [11] Actually may relatios for the derivatives of Fiboacci polyomials proved i that paper As a fial setece of this sectio, we ote that i the referece [1], some ew properties of Fiboacci umbers with biomial coefficiets have bee ivestigated Actually these ew properties will be eeded i the proof of oe of the mai results Correspodig author addresses: taskara@selcukedutr (N Taskara, kuslu@selcukedutr (K Uslu, hhgulec8@gmailcom (HH Gulec /$ see frot matter 9 Elsevier Ltd All rights reserved doi:1116/jaml987

2 N Taskara et al / Applied Mathematics Letters 3 ( Mai results I this study, we ivestigate the ew properties of Lucas umbers i relatio with Fiboacci umbers by usig biomial coefficiets This strategy allows us to obtai i easy form a family of Lucas sequeces i a ew ad direct way Theorem 1 For > ad Z, we have the relatio L +6 = 1L L (1 Proof Let us use the priciple of mathematical iductio o For = 3, it is easy to see that L 9 = 1L + 13L 1 = 47 Assume that it is true for all positive itegers = k, that is, L k+6 = 1L k L k ( Addig L k+ to both sides of (, we have L k+6 + L k+ = 1L k L k + L k+ Sice L k = L k 1 + L k, we first obtai L k+7 = L k+6 + L k+ o the left had side of the above equality, ad for o the right had side of the equality, we ca write L k+ = L k+4 +L k+3 Hece, by iteratig this procedure, we ca write L k+1 = L k +L k 1 Therefore L k+7 = 1L k + 13L k 1, as required I the followig theorem, for special values of Z, we will formulate special Lucas umbers i terms of their differet idices Theorem For ad Z, we have the followig relatios: [ ( / ( ] 1/ (a L 3+4 = i +1 4i i 4, ( / ( (b L +3 = i i Proof (a For ad Z, we kow / F 3(+1 = +1 4i i i from [1] Usig the property i (3 ad the equality L = F 1 4 give i referece [4], i the followig iteratio, we have a geeralizatio L 4 = [ [ ( 1/ F 4] 1/ 3 = 1 4] = 4 L 1 = [ [ ( 1/ F 4] 1/ 1 9 = + 1 4] = 76 L 16 = [ [ ( 1/ F 4] 1/ = ] = L 3+4 = [ F 4] 1/ 3+3 [ ( ( ( 1 1/ = ] (3

3 7 N Taskara et al / Applied Mathematics Letters 3 ( or usig the summatio symbol, we write / L 3+4 = +1 4i i 4 i (b Usig the equality L +3 = F give i referece [4], for Z ad 1, F +1 = 1 i [1], the proof ca be see easily By cosiderig the proof of this above result, we ca obtai the followig theorem Theorem 3 For 1 ad 1 Z, we have the followig relatios: [ ( 1 1/ (a L 3+4 = i +1 4i + 4], i ( 1 (b L +3 = i + i 1/ i as give i Proof Proof of this theorem ca be see easily i a similar maer with Theorem I additio to Theorem, we may also obtai more special Lucas umbers as i the followig Theorem 4 For ad Z, we have the followig relatios: (a L 3+ = / i i 3 i L 4 3+1, (b L 3+3 = / i +1 4i + F i 3+1 Proof (a Let us use the priciple of mathematical iductio o For =, it is easy to see that L = [ ] L 1 = 1 For =, we write [ L 8 = 4 ( + 1 ( 1 ] 3 4 L 7 = 9 Assume that it is true for all positive itegers = k That is, L 6k+ = 4 k k i 4k+1 4i 3 k i 4 L 6k+1 (4 Therefore, we have to show that it is true for = k + I other words, L 6k+8 = k+1 k + i 4k+ 4i 3 4 k + i 4 L 6k+7 Let us rewrite (4 by usig (3, L 6k+ = 4 F 6k L 6k+1 ( Addig 3 L 6k+i+1 to both sides of (, we have L 6k+ + L 6k+i+1 = 4 F 6k L 6k+1 + L 6k+8 = 4 F 6k L 6k+1 + L 6k+i+1 L 6k+i+1

4 N Taskara et al / Applied Mathematics Letters 3 ( Sice F = F + F +1, we first obtai F 6k+3 = F 6k+ F 6k+4 o the right had side of above equality Thus we ca write L 6k+8 = 4 (F 6k+ F 6k L 6k+1 + Hece, by iteratig this procedure, we have L 6k+i+1 L 6k+8 = 4 F 6k+9 + F 6k+9 1F 6k L 6k+1 + Also, it is kow from [4] that L 6k+i+1 (6 F = 1 (L + L + (7 Usig (7, we see F 6k+9 = 1 (L 6k+9 + L 6k+11 ad F 6k+8 = 1 (L 6k+8 + L 6k+1 So oe ca easily rearrage (6 ad have L 6k+8 = 4 F 6k+9 + L 6k+9 + L 6k+11 L 6k+8 L 6k L 6k+1 + L 6k+i+1 Sice L k = L k 1 + L k, we ca write L 6k+1 = L 6k+3 L 6k+ ad L 6k+11 = L 6k+1 + L 6k+9 Hece, by iteratig this procedure, we obtai 3 4 L 6k+7 = L 6k+9 + L 6k+11 L 6k+8 L 6k L 6k+1 + L 6k+i+1 It is obvious that L 6k+8 = 4 F 6k L 6k+7 After all, by usig (3, we obtai L 6k+8 = 4 k+1 4k+ 4i ( k + i k + i 3 4 L 6k+7, which eds up the iductio Therefore we have the required formulate o L 3+ (b The proof ca be see by usig the priciple of iductio o By applyig the same method as i the proof of Theorem 4, we have the followig Theorem For 1 ad 1 Z, we have the followig relatios: (a L 3+ = 1 i i 3 i L 4 3+1, (b L 3+3 = 1 i +1 4i + F i 3+1 Proof The proof is similar to the proof of Theorem 4 I the last part of this paper, we would like to preset the followig two facts other tha the above results about how to obtai some Lucas umbers with biomial coefficiets I fact we thought that this would be eeded for the reader For ad Z, we have relatios 1/ L + = + i 4 i ad L +3 = i i 1/

5 7 N Taskara et al / Applied Mathematics Letters 3 ( Refereces [1] VE Hoggat, Fiboacci ad Lucas Numbers, Palo Alto, CA, Houghto, 1969 [] S Vajda, Fiboacci ad Lucas Numbers, ad the Golde Sectio, Theory ad Applicatios, Joh Wiley ad Sos, New York, 1989 [3] Arthur T Bejami, Jeifer J Qui, Fracis Edward Su, Phased tiligs ad geeralized Fiboacci idetities, Fiboacci Quarterly 38 (3 ( 8 88 [4] T Koshy, Fiboacci ad Lucas Numbers with Applicatios, Joh Wiley ad Sos Ic, NY, 1 [] A Stakhov, Fiboacci matrices, a geeralizatio of the Cassii formula, ad a ew codig theory, Chaos, Solitos & Fractals 3 ( [6] MS El Naschie, The golde mea i quatum geometry, Kot theory ad related topics, Chaos, Solutios & Fractals 1 (8 ( [7] MS El Naschie, The Fiboacci code behid super strigs ad P-Braes, a aswer to M Kakus fudametal questio, Chaos, Solutios & Fractals 31 (3 ( [8] L Marek-Crjac, O the mass spectrum of the elemetary particles of the stadard model usig El Naschie s golde field theory, Chaos, Solutios & Fractals 1 (4 ( [9] L Marek-Crjac, The mass spectrum of high eergy elemetary particles via El Naschie s golde mea ested oscillators, the Dukerly Southwell eigevalue theorems ad KAM, Chaos, Solutios & Fractals 18 (1 ( [1] MZ Spivey, Combiatorial sums ad fiite differeces, Discrete Mathematics 37 ( [11] S Falcoń, A Plaza, The k-fiboacci sequece ad the Pascal -triagle, Chaos, Solitos & Fractals 33 (1 ( [1] HH Gulec, N Taskara, O the properties of Fiboacci umbers with biomial coefficiets, Iteratioal Joural of Cotemporary Mathematical Scieces 4 ( (