Some identities involving Fibonacci, Lucas polynomials and their applications
|
|
- Corey Rice
- 5 years ago
- Views:
Transcription
1 Bull. Math. Soc. Sci. Math. Roumaie Tome No. 1, 2012, 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 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 , β x x , 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 of P.R.Chia.
2 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 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 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 h + 1 2!! } F 4kh+1 x F x b. L 2 2m+1x h !! F 4kh+1x F x c. F 2+1 2m+1 x 1 x 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!! F 2h+1 x F x F x
3 Idetities ivolvig Fiboacci, Lucas polyomials 97 B. F 2 2mx [ 1 x h 2!! 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 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
4 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 ! x! x + 1 x, 1.1 x ! x! x + 1 x, 1.2 ad x + 1 x 2+1 x 1 x 2+1 x x +1, x +1 1 x 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
5 Idetities ivolvig Fiboacci, Lucas polyomials 99 [ ] F2m+1x 2 1 2! x 2 + 4! L2m+1 x, 1.5 L 2 2m+1x 1 2!! L 2m+1 x, 1.6 ad F 2+1 2m+1 x 1 x L 2+1 2m+1 x F2m+1+1 x, 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!! L4km x, 1.9 [ F2mx 2 1 x ! ]! L 4km x, 1.10 L 2+1 2m x L2m+1 x, 1.11 ad F 2+1 2m x 1 x F 2m+1 x. 1.12
6 100 Wag Tigtig ad Zhag Wepeg For ay iteger h > 0, we sum o m i 1.5, F 2 2m+1x h + 1 x [ 1 x h + 1 2! { 2!! h + 1! 2 + [ α α 4kh h L 2m+1 x α 4k + 1 β β 4kh+1 1 ]} β 4k 1 { h + 1 2! x 2 + 4! } α 4kh+ α 4kh+6k + β 4kh+ β 4kh+6k h α 4k β 4k ] 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 h + 1 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.
7 Idetities ivolvig Fiboacci, Lucas polyomials 101 Now we prove Theorem 2. From 1.9, we have L 2 2mx h 2!! h α 4km + β 4km h 2!! α 4kh+1 α 4k α 4k + β4kh+1 β 4k 1 β 4k 1 h 2!! α 4kh α 4kh α 4k + β 4kh β 4kh+1 + β 4k 2 α 4k β 4k h 2!! α 4kh+ β 4kh+ α β α β 2 α β 2 h 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 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 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 ad F 2h+1 x 1 satisfyig F 2h+1 x 1, x ad F 2h+1 x 1 F 2h+1+1 x F +1 x for all itegers k 0.
8 102 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 Therefore, F 2h+1 x 1, x 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 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/ /2009, [2] K. Ozeki, O Melham s sum, The Fiboacci Quarterly. 46/ /2009, [3] E. Lucas, Théorie des foctios umériques simplemet périodiques, Amer. J. Math , , [4] R. S. Melham, Some cojectures cocerig sums of odd powers of Fiboacci ad Lucas umbers, The Fiboacci Quarterly, 46/ /2009,
9 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, [6] Y. Yua ad Z. Wepeg, Some idetities ivolvig the Fiboacci polyomials, The Fiboacci Quarterly, , Received: , Revised: , , Accepted: Departmet of Mathematics, Northwest Uiversity, Xi a, Shaaxi, P.R.Chia s: tigtigwag126@126.com wpzhag@wu.edu.c
EVALUATION OF SUMS INVOLVING PRODUCTS OF GAUSSIAN q-binomial COEFFICIENTS WITH APPLICATIONS
EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS WITH APPLICATIONS EMRAH KILIÇ* AND HELMT PRODINGER** Abstract Sums of products of two Gaussia -biomial coefficiets are ivestigated oe of
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationFactors of sums and alternating sums involving binomial coefficients and powers of integers
Factors of sums ad alteratig sums ivolvig biomial coefficiets ad powers of itegers Victor J. W. Guo 1 ad Jiag Zeg 2 1 Departmet of Mathematics East Chia Normal Uiversity Shaghai 200062 People s Republic
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES
Publ. Math. Debrece 8504, o. 3-4, 85 95. ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES QING-HU HOU*, ZHI-WEI SUN** AND HAOMIN WEN Abstract. We cofirm Su s cojecture that F / F 4 is strictly decreasig
More informationON RUEHR S IDENTITIES
ON RUEHR S IDENTITIES HORST ALZER AND HELMUT PRODINGER Abstract We apply completely elemetary tools to achieve recursio formulas for four polyomials with biomial coefficiets I particular, we obtai simple
More informationCOMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun
Korea J. Math. 23 2015) No. 3 pp. 371 377 http://dx.doi.org/10.11568/kjm.2015.23.3.371 COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized
More informationGAMALIEL CERDA-MORALES 1. Blanco Viel 596, Valparaíso, Chile. s: /
THE GELIN-CESÀRO IDENTITY IN SOME THIRD-ORDER JACOBSTHAL SEQUENCES arxiv:1810.08863v1 [math.co] 20 Oct 2018 GAMALIEL CERDA-MORALES 1 1 Istituto de Matemáticas Potificia Uiversidad Católica de Valparaíso
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationMatrix representations of Fibonacci-like sequences
NTMSCI 6, No. 4, 03-0 08 03 New Treds i Mathematical Scieces http://dx.doi.org/0.085/tmsci.09.33 Matrix represetatios of Fiboacci-like sequeces Yasemi Tasyurdu Departmet of Mathematics, Faculty of Sciece
More informationSOME TRIBONACCI IDENTITIES
Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationSome Trigonometric Identities Involving Fibonacci and Lucas Numbers
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 (2009), Article 09.8.4 Some Trigoometric Idetities Ivolvig Fiboacci ad Lucas Numbers Kh. Bibak ad M. H. Shirdareh Haghighi Departmet of Mathematics Shiraz
More informationA NOTE ON PASCAL S MATRIX. Gi-Sang Cheon, Jin-Soo Kim and Haeng-Won Yoon
J Korea Soc Math Educ Ser B: Pure Appl Math 6(1999), o 2 121 127 A NOTE ON PASCAL S MATRIX Gi-Sag Cheo, Ji-Soo Kim ad Haeg-Wo Yoo Abstract We ca get the Pascal s matrix of order by takig the first rows
More informationFLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
More informationConcavity of weighted arithmetic means with applications
Arch. Math. 69 (1997) 120±126 0003-889X/97/020120-07 $ 2.90/0 Birkhäuser Verlag, Basel, 1997 Archiv der Mathematik Cocavity of weighted arithmetic meas with applicatios By ARKADY BERENSTEIN ad ALEK VAINSHTEIN*)
More informationInteresting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces Vol. 9 06 Article 6.. Iterestig Series Associated with Cetral Biomial Coefficiets Catala Numbers ad Harmoic Numbers Hogwei Che Departmet of Mathematics Christopher Newport
More informationThe log-concavity and log-convexity properties associated to hyperpell and hyperpell-lucas sequences
Aales Mathematicae et Iformaticae 43 2014 pp. 3 12 http://ami.etf.hu The log-cocavity ad log-covexity properties associated to hyperpell ad hyperpell-lucas sequeces Moussa Ahmia ab, Hacèe Belbachir b,
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationOn the Jacobsthal-Lucas Numbers by Matrix Method 1
It J Cotemp Math Scieces, Vol 3, 2008, o 33, 1629-1633 O the Jacobsthal-Lucas Numbers by Matrix Method 1 Fikri Köke ad Durmuş Bozkurt Selçuk Uiversity, Faculty of Art ad Sciece Departmet of Mathematics,
More informationBijective Proofs of Gould s and Rothe s Identities
ESI The Erwi Schrödiger Iteratioal Boltzmagasse 9 Istitute for Mathematical Physics A-1090 Wie, Austria Bijective Proofs of Gould s ad Rothe s Idetities Victor J. W. Guo Viea, Preprit ESI 2072 (2008 November
More informationarxiv: v1 [math.nt] 28 Apr 2014
Proof of a supercogruece cojectured by Z.-H. Su Victor J. W. Guo Departmet of Mathematics, Shaghai Key Laboratory of PMMP, East Chia Normal Uiversity, 500 Dogchua Rd., Shaghai 0041, People s Republic of
More informationOn Divisibility concerning Binomial Coefficients
A talk give at the Natioal Chiao Tug Uiversity (Hsichu, Taiwa; August 5, 2010 O Divisibility cocerig Biomial Coefficiets Zhi-Wei Su Najig Uiversity Najig 210093, P. R. Chia zwsu@ju.edu.c http://math.ju.edu.c/
More informationSuper congruences concerning Bernoulli polynomials. Zhi-Hong Sun
It J Numer Theory 05, o8, 9-404 Super cogrueces cocerig Beroulli polyomials Zhi-Hog Su School of Mathematical Scieces Huaiyi Normal Uiversity Huaia, Jiagsu 00, PR Chia zhihogsu@yahoocom http://wwwhytceduc/xsjl/szh
More informationCERTAIN GENERAL BINOMIAL-FIBONACCI SUMS
CERTAIN GENERAL BINOMIAL-FIBONACCI SUMS J. W. LAYMAN Virgiia Polytechic Istitute State Uiversity, Blacksburg, Virgiia Numerous writers appear to have bee fasciated by the may iterestig summatio idetitites
More informationq-lucas polynomials and associated Rogers-Ramanujan type identities
-Lucas polyomials associated Rogers-Ramauja type idetities Joha Cigler Faultät für Mathemati, Uiversität Wie johacigler@uivieacat Abstract We prove some properties of aalogues of the Fiboacci Lucas polyomials,
More informationA LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II 1. INTRODUCTION
A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II C. T. LONG J. H. JORDAN* Washigto State Uiversity, Pullma, Washigto 1. INTRODUCTION I the first paper [2 ] i this series, we developed certai properties
More informationSome Extensions of the Prabhu-Srivastava Theorem Involving the (p, q)-gamma Function
Filomat 31:14 2017), 4507 4513 https://doi.org/10.2298/fil1714507l Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: http://www.pmf.i.ac.rs/filomat Some Extesios of
More informationOn Generalized Fibonacci Numbers
Applied Mathematical Scieces, Vol. 9, 215, o. 73, 3611-3622 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.5299 O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet
More informationApplied Mathematics Letters. On the properties of Lucas numbers with binomial coefficients
Applied Mathematics Letters 3 (1 68 7 Cotets lists available at ScieceDirect Applied Mathematics Letters joural homepage: wwwelseviercom/locate/aml O the properties of Lucas umbers with biomial coefficiets
More informationProc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS
Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of
More informationON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS
Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet
More informationA Note On The Exponential Of A Matrix Whose Elements Are All 1
Applied Mathematics E-Notes, 8(208), 92-99 c ISSN 607-250 Available free at mirror sites of http://wwwmaththuedutw/ ame/ A Note O The Expoetial Of A Matrix Whose Elemets Are All Reza Farhadia Received
More informationApplicable Analysis and Discrete Mathematics available online at ON JENSEN S AND RELATED COMBINATORIAL IDENTITIES
Applicable Aalysis ad Discrete Mathematics available olie at http://pefmathetfrs Appl Aal Discrete Math 5 2011, 201 211 doi:102298/aadm110717017g ON JENSEN S AND RELATED COMBINATORIAL IDENTITIES Victor
More informationAccepted in Fibonacci Quarterly (2007) Archived in SEQUENCE BALANCING AND COBALANCING NUMBERS
Accepted i Fiboacci Quarterly (007) Archived i http://dspace.itrl.ac.i/dspace SEQUENCE BALANCING AND COBALANCING NUMBERS G. K. Pada Departmet of Mathematics Natioal Istitute of Techology Rourela 769 008
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationA Combinatoric Proof and Generalization of Ferguson s Formula for k-generalized Fibonacci Numbers
Jue 5 00 A Combiatoric Proof ad Geeralizatio of Ferguso s Formula for k-geeralized Fiboacci Numbers David Kessler 1 ad Jeremy Schiff 1 Departmet of Physics Departmet of Mathematics Bar-Ila Uiversity, Ramat
More informationarxiv: v1 [math.co] 6 Jun 2018
Proofs of two cojectures o Catala triagle umbers Victor J. W. Guo ad Xiuguo Lia arxiv:1806.02685v1 [math.co 6 Ju 2018 School of Mathematical Scieces, Huaiyi Normal Uiversity, Huai a 223300, Jiagsu, People
More informationThe Binet formula, sums and representations of generalized Fibonacci p-numbers
Europea Joural of Combiatorics 9 (008) 70 7 wwwelseviercom/locate/ec The Biet formula, sums ad represetatios of geeralized Fiboacci p-umbers Emrah Kilic TOBB ETU Uiversity of Ecoomics ad Techology, Mathematics
More informationand each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
More informationMath 2112 Solutions Assignment 5
Math 2112 Solutios Assigmet 5 5.1.1 Idicate which of the followig relatioships are true ad which are false: a. Z Q b. R Q c. Q Z d. Z Z Z e. Q R Q f. Q Z Q g. Z R Z h. Z Q Z a. True. Every positive iteger
More informationThe 4-Nicol Numbers Having Five Different Prime Divisors
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 14 (2011), Article 11.7.2 The 4-Nicol Numbers Havig Five Differet Prime Divisors Qiao-Xiao Ji ad Mi Tag 1 Departmet of Mathematics Ahui Normal Uiversity
More informationSome p-adic congruences for p q -Catalan numbers
Some p-adic cogrueces for p q -Catala umbers Floria Luca Istituto de Matemáticas Uiversidad Nacioal Autóoma de México C.P. 58089, Morelia, Michoacá, México fluca@matmor.uam.mx Paul Thomas Youg Departmet
More informationBINOMIAL PREDICTORS. + 2 j 1. Then n + 1 = The row of the binomial coefficients { ( n
BINOMIAL PREDICTORS VLADIMIR SHEVELEV arxiv:0907.3302v2 [math.nt] 22 Jul 2009 Abstract. For oegative itegers, k, cosider the set A,k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationInternational Journal of Mathematical Archive-3(4), 2012, Page: Available online through ISSN
Iteratioal Joural of Mathematical Archive-3(4,, Page: 544-553 Available olie through www.ima.ifo ISSN 9 546 INEQUALITIES CONCERNING THE B-OPERATORS N. A. Rather, S. H. Ahager ad M. A. Shah* P. G. Departmet
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationSHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE (1 + 1/n) n
SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE + / NECDET BATIR Abstract. Several ew ad sharp iequalities ivolvig the costat e ad the sequece + / are proved.. INTRODUCTION The costat e or
More informationJournal of Ramanujan Mathematical Society, Vol. 24, No. 2 (2009)
Joural of Ramaua Mathematical Society, Vol. 4, No. (009) 199-09. IWASAWA λ-invariants AND Γ-TRANSFORMS Aupam Saikia 1 ad Rupam Barma Abstract. I this paper we study a relatio betwee the λ-ivariats of a
More informationAn enumeration of flags in finite vector spaces
A eumeratio of flags i fiite vector spaces C Rya Viroot Departmet of Mathematics College of William ad Mary P O Box 8795 Williamsburg VA 23187 viroot@mathwmedu Submitted: Feb 2 2012; Accepted: Ju 27 2012;
More informationPellian sequence relationships among π, e, 2
otes o umber Theory ad Discrete Mathematics Vol. 8, 0, o., 58 6 Pellia sequece relatioships amog π, e, J. V. Leyedekkers ad A. G. Shao Faculty of Sciece, The Uiversity of Sydey Sydey, SW 006, Australia
More informationA Simplified Binet Formula for k-generalized Fibonacci Numbers
A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationProof of a conjecture of Amdeberhan and Moll on a divisibility property of binomial coefficients
Proof of a cojecture of Amdeberha ad Moll o a divisibility property of biomial coefficiets Qua-Hui Yag School of Mathematics ad Statistics Najig Uiversity of Iformatio Sciece ad Techology Najig, PR Chia
More informationGeneral Properties Involving Reciprocals of Binomial Coefficients
3 47 6 3 Joural of Iteger Sequeces, Vol. 9 006, Article 06.4.5 Geeral Properties Ivolvig Reciprocals of Biomial Coefficiets Athoy Sofo School of Computer Sciece ad Mathematics Victoria Uiversity P. O.
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationP. Z. Chinn Department of Mathematics, Humboldt State University, Arcata, CA
RISES, LEVELS, DROPS AND + SIGNS IN COMPOSITIONS: EXTENSIONS OF A PAPER BY ALLADI AND HOGGATT S. Heubach Departmet of Mathematics, Califoria State Uiversity Los Ageles 55 State Uiversity Drive, Los Ageles,
More informationEXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES
LE MATEMATICHE Vol. LXXIII 208 Fasc. I, pp. 3 24 doi: 0.448/208.73.. EXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES THOMAS ERNST We preset idetities of various kids for
More information(m) (-I)P-I(ap b. rn+s n. kpn kpn-1 AND LINEAR SECOND ORDER RECURRENCES SOME CONGRUENCE PROPERTIES OF BINOMIAL COEFFICIENTS NEVILLE ROBBINS (1.
Iterat. J. Math. & Math. Sci. VOL. II NO. 4 (1988) 743-750 743 SOME CONGRUENCE ROERTIES OF BINOMIAL COEFFICIENTS AND LINEAR SECOND ORDER RECURRENCES NEVILLE ROBBINS Departmet of athematlcs Sa Fracisco
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More informationCOMMON FIXED POINT THEOREMS VIA w-distance
Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3, Pages 182-189 COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA
More informationON SOME RELATIONSHIPS AMONG PELL, PELL-LUCAS AND MODIFIED PELL SEQUENCES
SAÜ Fe Bilimleri Dergisi, Cilt, Sayı, s-5, 00 O Some Relatioships Amog ell, ell-lucas ad Modified ell Seueces ON SOME RELATIONSHIS AMONG ELL, ELL-LUCAS AND MODIFIED ELL SEQUENCES, Ahmet DAŞDEMİR Sakarya
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationAPPROXIMATION PROPERTIES OF STANCU TYPE MEYER- KÖNIG AND ZELLER OPERATORS
Hacettepe Joural of Mathematics ad Statistics Volume 42 (2 (2013, 139 148 APPROXIMATION PROPERTIES OF STANCU TYPE MEYER- KÖNIG AND ZELLER OPERATORS Mediha Örkcü Received 02 : 03 : 2011 : Accepted 26 :
More informationAsymptotic distribution of products of sums of independent random variables
Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationSeries with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationSOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS
Folia Mathematica Vol. 5, No., pp. 4 6 Acta Uiversitatis Lodziesis c 008 for Uiversity of Lódź Press SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS ROMAN WITU LA, DAMIAN S
More informationThe r-generalized Fibonacci Numbers and Polynomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 2008, o. 24, 1157-1163 The r-geeralized Fiboacci Numbers ad Polyomial Coefficiets Matthias Schork Camillo-Sitte-Weg 25 60488 Frakfurt, Germay mschork@member.ams.org,
More informationVIETA-LIKE PRODUCTS OF NESTED RADICALS
VIETA-IKE PRODUCTS OF ESTED RADICAS Thomas J. Osler athematics Deartmet Rowa Uiversity Glassboro, J 0808 Osler@rowa.edu Itroductio The beautiful ifiite roduct of radicals () π due to Vieta [] i 9, is oe
More informationA NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES
Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 8 Issue 42016), Pages 91-97. A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES ŞEBNEM YILDIZ Abstract.
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationarxiv: v3 [math.nt] 24 Dec 2017
DOUGALL S 5 F SUM AND THE WZ-ALGORITHM Abstract. We show how to prove the examples of a paper by Chu ad Zhag usig the WZ-algorithm. arxiv:6.085v [math.nt] Dec 07 Keywords. Geeralized hypergeometric series;
More informationNew Inequalities For Convex Sequences With Applications
It. J. Ope Problems Comput. Math., Vol. 5, No. 3, September, 0 ISSN 074-87; Copyright c ICSRS Publicatio, 0 www.i-csrs.org New Iequalities For Covex Sequeces With Applicatios Zielaâbidie Latreuch ad Beharrat
More informationRamanujan s Famous Partition Congruences
Ope Sciece Joural of Mathematics ad Applicatio 6; 4(): - http://wwwopescieceoliecom/joural/osjma ISSN:8-494 (Prit); ISSN:8-494 (Olie) Ramauja s Famous Partitio Cogrueces Md Fazlee Hossai, Nil Rata Bhattacharjee,
More informationSum of cubes: Old proofs suggest new q analogues
Sum of cubes: Old proofs suggest ew aalogues Joha Cigler Faultät für Mathemati, Uiversität Wie ohacigler@uivieacat Abstract We show how old proofs of the sum of cubes suggest ew aalogues 1 Itroductio I
More informationThe Asymptotic Expansions of Certain Sums Involving Inverse of Binomial Coefficient 1
Iteratioal Mathematical Forum, 5, 2, o. 6, 76-768 The Asymtotic Easios of Certai Sums Ivolvig Iverse of Biomial Coefficiet Ji-Hua Yag Deartmet of Mathematics Zhoukou Normal Uiversity, Zhoukou 466, P.R.
More informationarxiv: v1 [math.nt] 10 Dec 2014
A DIGITAL BINOMIAL THEOREM HIEU D. NGUYEN arxiv:42.38v [math.nt] 0 Dec 204 Abstract. We preset a triagle of coectios betwee the Sierpisi triagle, the sum-of-digits fuctio, ad the Biomial Theorem via a
More informationA q-analogue of some binomial coefficient identities of Y. Sun
A -aalogue of some biomial coefficiet idetities of Y. Su arxiv:008.469v2 [math.co] 5 Apr 20 Victor J. W. Guo ad Da-Mei Yag 2 Departmet of Mathematics, East Chia Normal Uiversity Shaghai 200062, People
More informationOn Infinite Series Involving Fibonacci Numbers
Iteratioal Joural of Cotemporary Mathematical Scieces Vol. 10, 015, o. 8, 363-379 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijcms.015.594 O Ifiite Series Ivolvig Fiboacci Numbers Robert Frotczak
More informationAN APPLICATION OF HYPERHARMONIC NUMBERS IN MATRICES
Hacettepe Joural of Mathematic ad Statitic Volume 4 4 03, 387 393 AN APPLICATION OF HYPERHARMONIC NUMBERS IN MATRICES Mutafa Bahşi ad Süleyma Solak Received 9 : 06 : 0 : Accepted 8 : 0 : 03 Abtract I thi
More informationSOME FIBONACCI AND LUCAS IDENTITIES. L. CARLITZ Dyke University, Durham, North Carolina and H. H. FERNS Victoria, B. C, Canada
SOME FIBONACCI AND LUCAS IDENTITIES L. CARLITZ Dyke Uiversity, Durham, North Carolia ad H. H. FERNS Victoria, B. C, Caada 1. I the usual otatio, put (1.1) F _
More informationSums Involving Moments of Reciprocals of Binomial Coefficients
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 14 (2011, Article 11.6.6 Sums Ivolvig Momets of Reciprocals of Biomial Coefficiets Hacèe Belbachir ad Mourad Rahmai Uiversity of Scieces ad Techology Houari
More informationA New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem
This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: +852-2766-7410
More informationSome families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions
J. Math. Aal. Appl. 297 2004 186 193 www.elsevier.com/locate/jmaa Some families of geeratig fuctios for the multiple orthogoal polyomials associated with modified Bessel K-fuctios M.A. Özarsla, A. Altı
More informationCLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS
Submitted to the Bulleti of the Australia Mathematical Society doi:10.1017/s... CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS GAŠPER JAKLIČ, VITO VITRIH ad EMIL ŽAGAR Abstract I this paper,
More informationA note on the p-adic gamma function and q-changhee polynomials
Available olie at wwwisr-publicatioscom/jmcs J Math Computer Sci, 18 (2018, 11 17 Research Article Joural Homepage: wwwtjmcscom - wwwisr-publicatioscom/jmcs A ote o the p-adic gamma fuctio ad q-chaghee
More informationA generalization of Morley s congruence
Liu et al. Advaces i Differece Euatios 05 05:54 DOI 0.86/s366-05-0568-6 R E S E A R C H Ope Access A geeralizatio of Morley s cogruece Jiaxi Liu,HaoPa ad Yog Zhag 3* * Correspodece: yogzhag98@63.com 3
More information(6), (7) and (8) we have easily, if the C's are cancellable elements of S,
VIOL. 23, 1937 MA THEMA TICS: H. S. VANDIVER 555 where the a's belog to S'. The R is said to be a repetitive set i S, with respect to S', ad with multiplier M. If S cotais a idetity E, the if we set a,
More informationA GRÜSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN NORMED LINEAR SPACES AND APPLICATIONS
A GRÜSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN NORMED LINEAR SPACES AND APPLICATIONS S. S. DRAGOMIR Abstract. A discrete iequality of Grüss type i ormed liear spaces ad applicatios for the discrete
More informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More informationON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS
Ordu Üiv. Bil. Tek. Derg., Cilt:6, Sayı:1, 016,8-18/Ordu Uiv. J. Sci. Tech., Vol:6, No:1,016,8-18 ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS Serpil Halıcı *1 Sia Öz 1 Pamukkale Ui., Sciece ad Arts Faculty,Dept.
More informationUnimodality of generalized Gaussian coefficients.
Uimodality of geeralized Gaussia coefficiets. Aatol N. Kirillov Steklov Mathematical Istitute, Fotaka 7, St.Petersburg, 191011, Russia Jauary 1991 Abstract A combiatorial proof [ of] the uimodality of
More informationOn Summability Factors for N, p n k
Advaces i Dyamical Systems ad Applicatios. ISSN 0973-532 Volume Number 2006, pp. 79 89 c Research Idia Publicatios http://www.ripublicatio.com/adsa.htm O Summability Factors for N, p B.E. Rhoades Departmet
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationVienna, Austria α n (1 x 2 ) n (x)
ON TURÁN S INEQUALITY FOR LEGENDRE POLYNOMIALS HORST ALZER a, STEFAN GERHOLD b, MANUEL KAUERS c2, ALEXANDRU LUPAŞ d a Morsbacher Str. 0, 5545 Waldbröl, Germay alzerhorst@freeet.de b Christia Doppler Laboratory
More informationA Study on Some Integer Sequences
It. J. Cotemp. Math. Scieces, Vol. 3, 008, o. 3, 03-09 A Study o Some Iteger Sequeces Serpil Halıcı Sakarya Uiversity, Departmet of Mathematics Esetepe Campus, Sakarya, Turkey shalici@sakarya.edu.tr Abstract.
More information