Research Article A Method to Construct Generalized Fibonacci Sequences

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1 Applied Mathematic Volume 6, Article ID , 6 page Reearch Article A Method to Contruct Generalized Fibonacci Sequence Adalberto García-Máynez and Adolfo Pimienta Acota Intituto de Matemática, Univeridad Nacional Autónoma de México, Area de la Invetigación Científica Circuito Exterior, Ciudad Univeritaria Coyoacán, 45 México, DF, Mexico Departamento de Matemática, Univeridad Autónoma Metropolitana, Prolongación Canal de Miramonte No 3855, Colonia Ex-Hacienda San Juan de Dio, Delegación Tlalpan, 4387 México, DF, Mexico Correpondence hould be addreed to Adolfo Pimienta Acota; pimienta33@hotmailcom Received 7 September 5; Accepted 6 January 6 Academic Editor: Allan C Peteron Copyright 6 A García-Máynez and A P Acota Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited The main purpoe of thi paper i to tudy the convergence propertie of Generalized Fibonacci Sequence and the erie of partial um aociated with them When the proper value of an real matrix A are real and different, we give a neceary and ufficient condition for the convergence of the matrix equence A, A,A 3,to a matrix B Introduction The Fibonacci Sequence The o-called Golden ratio λ = + 5/ appear in nature very frequently It i alo conidered the mot ethetic ratio between the bai and height of a rectangle:,,,,3,5,8,3,, i an intereting numerical equence that occur quite frequently in many part of nature Thi equence ha a pecial feature; every element of thi equence, tarting from the third, i the um of it two predeceor and can be generated recurively by the formula x n+ =x n +x n+ Iticlearthatweneedthefirttwotermx =, x =and the recurive formula to define the equence If we want to know the term x k without contructing the previou term, we can ue the unexplainable formula ee []: x k = 5 [+ 5 k k 5 ] 3 What do the irrational number 5 have to do with the original equence? B H If we replace the recurive formula by B H = x n+ =x n +x n+, 5 we obtain a new equence,, /, 3/4, 5/8, /6, /3, and thi equence i no longer divergent; in fact, it converge to /3 To define a Generalized Fibonacci Sequence, we fix a natural number and two element x,x,,x, a,a,,a in the Euclidean pace R Therecuriveformulai 6 x k+ =a x k +a x k+ + +a x k+ 7

2 Applied Mathematic The main purpoe of thi paper i to tudy the convergence propertie of Generalized Fibonacci Sequence and the erie of partial um aociated with them When the proper value of an real matrix A arerealanddifferent,wegivea neceary and ufficient condition for the convergence of the matrix equence I, A, A,A 3,to a matrix B:weayA k B if for every ordered pair i, j, wherei,j {,,,},the equence of the i, j-entrie of A k converge to the i, j- entry of B Aaparticularcae,wetudywhendowehave the convergence of the power φ, φ,φ 3, of a Moebiu tranformation to a contant function Then we would like to lit the four publihed monograph about generalized Fibonacci equence [ 4] and everal more pecialized article [5 8] Main Reult Conider a Generalized Fibonacci Sequence GFS with initial term x,x,,x and recurive formula 7 Define the matrice : C=, a a a a x k x k+ x k+ x k+ x k+ x k+ D k =, k N {} x k+ x k+ x k+ The characteritic polynomial of the matrix C i 8 φ x =x a x a x a 9 Suppoealltherootλ,λ,,λ of φx are real and pairwie different; that i, λ i = λ j,fori = j Conider Vandermonde matrix: λ λ λ V= λ λ λ λ λ λ Since = ± i<j λ i λ j,wededucethat =and, hence, V i invertible We need the following matrix relation Theorem V and C are related by the following formula: VEV =C, where E i the diagonal matrix: λ λ E= d λ Proof Let V, V,,V R be the proper vector of the matrix C WehavethenV i C=λ i V i for each i =,,, If w = V + V + +V and k N, wehavewc k = λ k V +λ k V + +λ k V, k =,,, Since =, we deduce that the vector w, wc, wc,,wc are linearly independent and hence they contitute a bai for R Calling B = {V, V,V }, B = {w,wc,,wc },wehave V=mId,B, B On the other hand, conider the linear tranformation T:R R defined by the formula TV = VC Clearly, Therefore, C=mT,B, B C=mT,B, B 3 = m Id, B, BmT, B, B mid,b, B =VEV In the next theorem, we relate V and C with D k Theorem Onehathefollowingformula: a CD k =D k+ and C k D =D k for every k N b D k =VE k V D for every k N Proof a It i traightforward b Uing 36, we have 4 D k =C k D = VEV k D =VE k V D 5 Uing the formula D k =VE k V D,weobtainanymember of the correponding Generalized Fibonacci Sequence Theorem 3 Conider x k = λ k λ k λ k x λ λ λ x d λ λ λ x 6

3 Applied Mathematic 3 Proof The firt row of the matrix VE k i the following: λ k,λk,,λk ThefirtcolumnofthematrixV D i V j x j j= V j x j j=, 7 V j x j j= where V i the cofactor of the entry of V in the i, j poition x k i the entry in the, poition of the matrix D k Therefore, x k = [ [ +λ k j= λ k j= V j x j ] ] V j x j +λ k j= V j x j + The expreion inide the quare bracket coincide with λ k λ k λ k x λ λ λ x λ λ λ x 8 9 To ee thi, develop thi determinant by the firt row and the lat column The coefficient of λ k i x j i then i +j = V ji λ λ i λ i+ λ λ j λ j i λj i+ λ j+ λ j+ i λj+ i+ λ Thi complete the proof d λ i λ i+ In the particular cae k=,weobtain λ j λ j+ λ λ k λk x k = x λ λ λ λ x If we further aume that x =and x =,weobtain x k = λ k λ λ λk In the original Fibonacci equence, we have C= 3 and hence φx = x x Therootofthipolynomial are λ =+ 5/ and λ = 5/ We jutify then the mythical formula x k =/ 5[+ 5/ k 5/ k ] In the cae a =a =/,weobtain C=, x φ x = x =x x 4 The root of φx are λ =and λ = /Hence,x k = /3[ / k ]ItinowclearthatthilatGFSconverge to /3 We give next a ufficient condition for the convergence of the erie of a GFS Theorem 4 main theorem Suppoe the root of the characteritic polynomial φx of a GFS {x k } are pairwie different and all of them lie in the open interval, Then the erie of {x k } converge to λ λ λ x λ λ λ x 5 λ λ λ x Proof Thi i a conequence of the identity x +x + +x k = +λ +λ + +λk +λ +λ + +λk x λ λ x λ λ x and the convergence j= λj i to / λ i We give now two example 6

4 4 Applied Mathematic Example 5 One ha =3, x =x =, x =, a =/6, a =,anda = /6 The characteritic polynomial φx of the correponding GFS i φ x = x x 6 x 6 = x x x 3 =x 3 6 x +x 6 Therefore, λ =, λ =/, λ 3 =/3,and 7 In thi cae, x k,but k= x k = 8 / /3 /3 = Given different real number λ,λ,λ,,λ,wemay contruct, for every x,x,,x R,aGFS;namely, λ k λ k λ k Hence = 4 x k = = k 3 k k 3 k = x k = j<i λ i λ j x λ λ λ x d λ λ λ x 33 The miing equence a,a,,a may be obtained from the coefficient of the polynomial: φ x =x λ x λ x λ 34 For intance, a = λ +λ + +λ and a = j<i λ i λ j With the help of thi remark, we prove the following Theorem 7 Let a be an invertible matrix Suppoe the characteritic polynomial φx of A factor into the form: φ x =x λ x λ x λ, 35 Clearly x k 8 =3 3 3 Example 6 One ha = 3, x = x =, x =, a = /8, a = /8, anda =3/Inthicae,φx = x 3 3/x +/8x /8 = x /x /3x /3 Therefore, λ =/, λ =/3, λ 3 =/3,and = /8 Hence, x k = 8 k 3 k 3 k where λ,λ,,λ are pairwie different real number Let be the entry in the i, j poition of the matrix A k Then a k a k = λ k λ k λ k a λ λ λ a λ λ λ a λ λ λ a 36 Proof Let x k be the GFS determined by 36 It i clear that x k =a k for k Proceeding by induction, uppoe x t =a t for every t k,wherek Weclearlyhave a k+ =a k i,ak i,,ak i a j,a j,,a j =a k i a j +a k i a j + +a k i a j 37

5 Applied Mathematic 5 Be induction, the term a k i,,ak i may be obtained uing determinant of type 36 We conider the lat column of determinant 36 and we obtain Therefore a k+ a j a i a i a i = a a a = Δ +a j a i a i a i + λ k λ k λ k a λ λ λ a λ λ λ a 3 λ λ λ a The GFS on the right ide of thi equation tart with y = a, y =a,,y =a andhatheamepropervalue of the GFS x k Hencea k+ =y k =x k+ and the proof i complete A an exercie, we calculate the power of a matrix: a b A=, where Δ=ak hb= 4 h k In thi cae we have a =, a =, a =, a a =a, a =b, a a t = =h, a λ λ =khence λ t λt a λ λ a =, 4 Therefore a t = λ λ a t = λ λ a t = λ λ a t = λ λ λ t λt ; λ λ a λ t λt ; λ λ b λ t λt ; λ λ h λ t λt, λ λ k 4 where λ = /[a + k a k +4hb] and λ = /[a + k + a k +4hb]Ifλ = and λ <, we have A t a λ b 43 λ h k λ Thi limit matrix ha determinant = Sointhecaeofa Moebiu tranformation φ z = a+bz h+kz, ak =hb 44 we have the following corollary Corollary 8 If i a proper value of the matrix A= ab hk and if ak hb <, then the power φ t converge to the contant map φz = a λ /h λ = b/ λ k λ Conflict of Interet The author declare that there i no conflict of interet regarding the publication of thi paper Reference [] A F Horadam, A generalized Fibonacci equence, The American Mathematical Monthly,vol68,no5,pp ,96 [] D V Jaiwal, On a generalized Fibonacci equence, Labdev: JournalofScienceandTechnologyPartA,vol7,pp67 7,969 [3] S P Pethe and C N Phadte, A generalized Fibonacci equence, Application of Fibonacci Number,vol5,pp465 47,99 [4] D A Wolfram, Solving generalized Fibonacci recurrence, The Fibonacci Quarterly, vol 36, no, pp 9 45, 998 [5] G Sburlati, Generalized Fibonacci equence and linear congruence, The Fibonacci Quarterly,vol4,no5,pp446 45,

6 6 Applied Mathematic [6] G-Y Lee, S-G Lee, J-S Kim, and H-K Shin, The Binet formula and repreentation of k-generalized Fibonacci number, The Fibonacci Quarterly,vol39,no,pp58 64, [7] G Y Lee, S G Lee, and H G Shin, On the k-generalized Fibonacci matrix Q k, Linear Algebra and It Application, vol 5, pp 73 88, 997 [8] S T Klein, Combinatorial repreentation of generalized Fibonacci number, The Fibonacci Quarterly, vol 9, no, pp 4 3, 99

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