关于Fibonacci数偶次幂的恒等式

关于Fibonacci数偶次幂的恒等式 第20卷第1期 2007年3月 纺织高校基 BASICSCIENCESJOURNAL 础科学 oFTEXTILEUNIVERSn'IES VoJ.2O,No.1 March,2007 ArticleID:1006-8341(2007)O1—0060—04 SomeidentitiesinvolvingFibonacci numbersinevenpower MAJin—ping',LIJie, (1.DepartmentofMathematics,NorthwestUniversity,Xian710069,China; 2.SchoolofStatistics,xianUniversityofFinanceandEconomics,Xian710061,China) Abstract:StudyingpropertiesofthefirstandsecondChebyshevpolynomials,according relationofChebyshevpolynomialsandFibonaccinumbers,usingtheelementarymethod identitiesinvolvingFibonaccinumbersinevenprowerareobtained. Keywords:Chebyshevpolynomial;fibonaccinumbers;Gegenbauerpolynomial;identity CLCnumber:O156.4Documentcode:A. 1Introductionandresults tothe 'SOme Asusual,theFibonaccisequenceF(一0,1,2,…)aredifinedbysecond— orderlinearrecurrence sequence F2一F1+F(1) for?0,F0—0,F11,…Thesesequenceplayaveryimportantroleinthestudiesofthetheoryandap— plicationofmathematics.Therefore,thevariouspropertiesofFwereinvestigatedbymanyaut hors.For example,paperE1—53obtainedsomeidentitiesinvolvingtheFibounaccinumber.Forconvenience,Cheby— shevpolynomialsofthefirstandsecondkind,T(z)一{(z)}and【厂(z)一<(z)}(一O,1,2,…)are definedbythesecond—orderlinearrecurrencesequences 丁2(z)一2zT十1(z)一T()(2) and? 【2(z)=2zU】(z)一L()(3) for?0,To()一1,T1(z):==z,Uo(z)一0andU1(z)一1.GegenbauerpolynomialC(z)(一0,1, 2,…)isdefinedbythefollowinggeneratefunction (1--2矗一a(础(>专,<z<'f,f<)?(4) Inthispaper,accordingtotherelationsbetweenChebyshevpolynomialsandFibonacciseque nces, weshallgiveanidentityfor ?FF…吃,(5)dd…dp where,thesummationistakenoverallP—dimensionnonnegativeintegercoordinates(1,d2,…,d), ReceivedDate:2006-06—22 Foundationitem:SupportedbyNSFC(10671155) Biogr~phy:MAJin—ping(1970一),female,ar~dveofGaolingcounty,Shaanxiprovince,MasterofNorthwesternUniversi — ty.E-mail:jinping-lO18@163.corn 第1期SomeidentitiesinvolvingFibonaccinumbersinevenpower61 suchthatdl-+-d2+…+dp一,Pandqareanypositiveintegers,beanynonnegativeinteger.Infact,we shalluseelementarymethodsandthepropertiesofChebyshevpolynomialstOprovethefollo wing Theorem1Letbedefinedby(3),a(z)bedefinedby(4).Then~oranypositiveintegersm,/-/, Pandq,wehavethecalculatingformula dd,一吃…F一蚤善:(aoa一1口)×,+o+…一H.~,d.+d+…+咀一6^+6.+…+^=,…口m, 鱼c'一(2.kffio\rt]1c(T2+ 一 (), where,(口.口口.)一Tand(2)=2rn!五!(2m一志)!' Takingm一1inTheorem1.andnotethatthefact : 警c(?n--一l0)(2,a():?(,1)f一,,'1(2z)删,f=,_,l一',,', wemayimmediatedlyobtainthefollwingCorollary ). (6) Corollary1LetUbedfinedby(3).Thenforanypositiveintegers,P,q,wehavethecalculating formula …, F . … , 一 (一)('妻壹(j?--s)/',r-?j-t)脚++…+一H'\/k=O一os=Ot=Of=or—o 2SeveralLemmas c,抖件2r2卜2()(户五)(五+jm—s~一l--)(一) n+p-- 恕 k 一 -- j 一 -- 一 t-- ,.r一)(一'一r52+)'一.. WeshallgiveseveralIemmaswhicharenecessaryintheproofofTheorem1.FirstweneedFo — bonaccisequences 去[()一()] andtWOexactexpressionsandgeneratingfunctionson()and().Thatis and ()一(1/2)[(+~/+(一~/『二_)一] 【,一()[(+冈)一(一厕]. WecaneasilydeducethatthegeneratingfunctionofT()is :一 ?(),,I--1<<1,I,I<1).H=0 Applyingthisgeneratingfunction,wecaneasilydeducethefollowing (7) (8) (9) (10) Lemma1Let()bedefinedby(2),thenforanypositiveintegerand.wehavetheidentity ProofSeepaperI]6-1. (丁m())一丁删().(11) Lemma2Let(),a()isdefinedby(3)and(4),thenforanypositiveinteger/72,andP,we havetheidentity 62纺织高校基础科学第2O卷 ?'叼叼…叼(z)'d1+2+…+H. 一 (…)(2× r 壹t=0\rt (c(酾)?(12) ProofToprovethis . Lemma,Wenotethat()(二足),(z+~/)(z一~/):1and 丁_(z)一(z),then )一 2 [(z+_==)"一(z一~/.==)"]一 2×4(z一1)((2+一]? 加 k=0 (--1)~-t(2, z)一(_1)(2T)). Let厂(z,,)一妻(z),c.nsideringtheab.vef.rmula,wehave. . 胁一[客(--1)t(2c ' 41X.dc(k)c一(z一),一\/一" 41c(k)gc(z一),,\厂, fP(x,t)-[c(2…卜 P)垂ck)ak(2c, Form(4)and(1O)wehave. ? g(T2^(z),,)=(1-丁2^(z),)(T2^(z))=== ??1(一1)T(z)(T2(z)) Combiningtwoaboveformulas,wemayget 一 善,置P) 垂跏?(2壹()c?-rk((13) and 厂(z,)一?(z)),一'?((z)(z)…;(z)t"?(M) 第1期SomeidentitiesinvolvingFibonaccinumbersinevenpower63 Comparingthecoefficientsoft"in(13)and(14),wecanget Notingthat ?叼(z)叼(z)…:(z)=do+d1+…+dr," . 置=,l(口o)×4呻(一1)呻.铬6o+6】一?口/ 鱼c'一(2)善ak()ccz,曙 (z)一,=而, from(15)wemayimmediatelydeduce(12).ThisprovesLemma2. 3ProofoftheTheorem1 Firstfromthedefinitionof(z),wehave SOtakingz—i/2.wecanget U(i/2)一, .,, 吨(/2)哦(i/Z)…(i/2)一dl++…+d=" 一 蚤:屹…毫? UsingtheaboveformulaandLemma2,wemayimmediatelyobtain ?F磁…磁一dl+d2+…+d6;n mp ao+a1蚤w置;(a0apl….1/×II,+…+nm6o+6l+…+6m辜n\…?m 重c'…(2耋()c__5Fz)(r2).ThiscompletestheproofofTheorem1. References: (15) (16) [1]ZHANGWem~ng.OnChebyshevpolynomialsandFibonaeeinumbers[J].1eFibonacd Quarterly,2002,4O(5):424.428. E2]ZHANGWen-peng.SomeidentitiesinvolvingtheFibonaecinumbers[J].TheFibonaeci Quarterly,1997,35(3):225.229. [3]ZHAOFeng-zhen,WANGTian-ming.GeneralizationsofsomeidentitiesinvolvingtheF ibonaeeinumbers[J].TheFi. bonaeeiQuarterly,2001,39(2):165—167. [4]徐哲峰,王晓瑛.一些关于Chebyshev多项式和Fibonaeei数的恒等式[J].咸阳师范学院,2003,18(2):11.13. [53刘端林,李超,杨存典.Fibonaeei数奇次幂乘积之和EJ]..纺织高校基础科学,2004,17(3)187.189. [6]刘端森,李超.一些关于Gegenbauer多项式,Fibonaeei数和Lueas数的恒等式[J]..延安大学,2003,22(1):7-9. 关于Fibonacci数偶次幂的恒等式 马金萍.,李洁 (1.西北大学数学系,陕西西安710069;2.西安财经学院统计学院,陕西西安710061) 摘要:根据Chebyshev多项式和Fibonacci数的关系以及第一类第二类Chebyshev多项式的性质,用初等 的 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 得到了Fibonacci数偶次幂积和式的计算公式. 关键词:Chebyshev多项式Fibonacci数;Gegenbauer多项式;恒等式 编辑,校对:黄燕萍