On the Relations between Lucas Sequence and Fibonacci-like Sequence by Matrix Methods

Transcription

1 .J. Mathematical Sciece ad Comutig, 07,, 0-36 Publihed Olie November 07 i MECS (htt:// DO: 0.8/ijmc Available olie at htt:// O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method Arfat Ahmad Wai a ad V. H. adhah b a,b School of Studie i Mathematic, Vikram Uiverit Ujjai, dia. Received: Februar 07; Acceted: 08 Jue 07; Publihed: 08 November 07 Abtract the reet aer firt ad foremot we itroduce a geeraliatio of a claical Fiboacci equece which i called a Fiboacci-ike equece ad at hidmot we obtai ome relatiohi betwee uca equece ad Fiboacci-ike equece b uig two cro two matri rereetatio to the Fiboacci-ike equece. he mot worth oticig caue of thi article i our roof method, ice all the idetitie are roved b uig matri method. de erm: Fiboacci Sequece, uca Sequece, Geeraliatio of Fiboacci Sequece ad Matri Method. Mathematic Subject Claificatio: 37, 39, 0C0. 07 Publihed b MECS Publiher. Selectio ad/or eer review uder reoibilit of the Reearch Aociatio of Moder Educatio ad Comuter Sciece. troductio Fiboacci umber have ma alicatio a well a iteretig roertie almot i ever field of ciece uch a i Phic, iolog, Comuter Sciece, Egieerig, Mathematic (Algebra, Geometr ad Number heor itelf). Furthermore Fiboacci ad uca umber have log itereted mathematicia for their itriic theor ad alicatio. Fiboacci umber ad uca umber cotiue to rovide ivaluable oortuitie for eloratio, ad cotribute hadomel to the beaut of mathematic, eeciall umber theor, oe ca ee the citatio [9, 0, 3]. he Fiboacci ad uca equece are defied b the recurrece relatio: * Correodig author. el.: addre: atharalam3@gmail.com

2 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method Defiitio. [3] For the iteger, the Fiboacci equece i defied b the recurrece relatio a F F F,, F 0, F () 0 Defiitio. [3] For the iteger, the uca equece i defied b the recurrece relatio a,,, () 0 he geeralied Fiboacci equece W W a, b; q 0 i defied a follow:, W W qw, W a, W b (3) where a, b, ad q are arbitrar comle umber with q 0. Sice thee umber were firt tudied b Horadam [], the are called Horadam umber. Sigh et al. i [] delieated geeralied idetitie o the relatio betwee Fiboacci ad uca equece. hogmoo i [] gave idetitie about the commo Factor of Fiboacci ad uca umber. Ceri i [] obtaied roertie o the factor of ummatio of coecutive Fiboacci ad uca umber. 960 Charle H. Kig itroduced the matri for claical Fiboacci umber which i kow a Q-matri [9] ad Q-matri i give a Q 0 Cerda [] tudied Horadam equece (.3) b matri method. Here the author coidered two cae of W : U i defied b U 0 0 ad U V i defied b V ad V 0 Keki ad Demirturk [6] obtaied ome ew idetitie for Fiboacci ad uca umber b matri method. Kilic [7] obtaied ome ummatio idetitie for Fiboacci umber b matri method. [8] koke ad okurt tudied ad defied a uca Q -matri which i imilar to the Fiboacci Q -matri [9] ad the Q - matri i defied a 3 Q the Q F F F F,, for eve for odd where F ad are the th Fiboacci ad uca umber, reectivel. Ju ad Choi i [] tudied the roertie of geeralied Fiboacci umber b matri method the defied the geeralied Fiboacci

3 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method equece a q a b q where a ad b are oitive real umber ad q,, q0 0, q 0 if if i eve i odd [] the author defied the matri M for the above equece. ab b M ad a 0 M a a b q b aq a bq b q additio to thi Dademir i [3] obtaied ome idetitie of Pell, Pell-uca ad Modified Pell umber b the matri method, i [3] the author defied ome two cro two matrice a 3 N, 6 R ad R. Fiboacci-ike equece ad it Matri Rereetatio thi ectio we defie a geeraliatio of Fiboacci equece which i called Fiboacci-ike equece alo we itroduce a matri rereetatio for Fiboacci-ike equece. Defiitio 3 For the iteger ad, the Fiboacci-ike equece i defied b the recurrece relatio a,,, () 0 ad a matri rereetatio for Fiboacci-ike equece ad i give b 3. Mai Reult thi ectio we reet ome mai reult of thi article b uig a matri rereetatio to Fiboacci- ike equece defied i equatio ().

4 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method 3 emma f i a quare matri with the, () Proof. o rove the reult we hall ue iductio o et, we get 0 Hece the reult i true for. Aume that the reult i true for. Now we how that that the herefore, a required. Now we how that, et

5 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method Y herefore, Y Y Y Y Y Y hi how that Y Y Sice ad the, we have Hece the reult. heorem et the (6)

6 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method Proof. Sice the b lemma (3.), we have 0 0 Sice the Hece the reult. heorem For the oitive iteger, we have (7) Proof. Sice

7 6 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method Agai, Hece we coclude that Hece the reult. heorem 3,,,, (8) Proof.

8 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method 7 ut, Give, ad, Hece the theorem. heorem,,, (9) Proof. Sice he b theorem (), we have

9 8 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method ut, he, ad, heorem,, (0)

10 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method 9 Proof. Sice ad herefore, O the other had, 0 0 Hece,

11 30 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method ad heorem 6,,,,, where ad () Proof. the defiitio of the matri, we have Agai,

12 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method 3 Equatig correodig term of the the two matrice, we get Uig theorem (3), we have ad, Uig theorem (3), we have Hece the theorem. heorem 7,,,, ()

13 3 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method Proof. et u coider a roduct From theorem (3), we have the et Agai b theorem (), we have for, 0 herefore,

14 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method 33 herefore, heorem 8,,,, (3) Proof. et u coider a roduct 3

15 3 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method 3 From theorem (3), we have 3 the 3 et Agai b theorem (), we have 0 herefore,

16 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method 3 herefore, ad Hece the reult. Cocluio thi article we reeted a geeralied Fiboacci equece called Fiboacci-ike equece ad after that ome relatio have bee obtaied betwee uca equece ad Fiboacci-ike equece b matri method. Referece [] Cerda, G., Matri Method i Horadam Sequece, oletí de Matemática, 9(), 0, [] Čeri, Z., O Factor of Sum of Coecutive Fiboacci ad uca Number, Aale Mathematicae et formaticae,, 03,. 9-. [3] Dademir, A., O the Pell, Pell-uca ad Modified Pell Number b Matri Method, Alied

17 36 O the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method Mathematical Sciece, (6), 0, [] Horadam, A. F., aic Proertie of a Certai Geeralied Sequece of Number, he Fiboacci Quarterl, 3(3), 96, [] Ju, S. P. ad Choi, K. H., Some Proertie of the Geeralied Fiboacci Sequece { q } b Matri Method Method, he Korea Joural of Mathematic, (), 06, [6] Keki, R. ad Demirtürk,., Some New Fiboacci ad uca detitie b Matri Method, teratioal Joural of Mathematical Educatio i Sciece ad echolog, (3), 00, [7] Kilic, E., Sum of Geeralied Fiboacci Number b Matri Method, Ar Combiatoria, 8, 007,.3-3. [8] Koke, F. ad okurt, D., O uca Number b the Matri Method, Hacettee Joural of Mathematic ad Statitic, 39(), 00,.7-7. [9] Koh,., Fiboacci ad uca Number with Alicatio, Joh Wile & So, 0. [0] Poametier, A. S., he Fabulou Fiboacci Number, Prometheu ook, 007. [] Sigh,., hadouria, P. ad Sikhwal, O., Geeralied detitie volvig Commo Factor of Fiboacci ad uca Number, teratioal Joural of Algebra, (3), 0, [] hogmoo, M., 009, detitie for the Commo Factor of Fiboacci ad uca Number, teratioal Mathematical Forum, (7), 009, [3] Vajda, S., Fiboacci ad uca Number ad the Golde Sectio: heor ad Alicatio, Courier Cororatio, 989. Author Profile Arfat Ahmad Wai, ha comleted ot graduatio i mathematic i 03 from Uiverit of Kahmir. After that i 0 he erolled a a Ph. D reearch cholar i School of Studie i Mathematic Vikram Uiverit Ujjai, dia ad curretl he i carrig a reearch work i Number heor o Fiboacci Number. V. H. adhah, Profeor ad Head i School of Studie i Mathematic Vikram Uiverit Ujjai, dia. He i M.c ad Ph.d from alo School of Studie i Mathematic Vikram Uiverit Ujjai, dia. Hi reearch ecialiatio i Number heor a well a i Fuctioal Aali. He ublihed about 00 aer i iteratioal ad atioal reuted joural. How to cite thi aer: Arfat Ahmad Wai, V. H. adhah,"o the Relatio betwee uca Sequece ad Fiboacci-like Sequece b Matri Method", teratioal Joural of Mathematical Sciece ad Comutig(JMSC), Vol.3, No.,.0-36, 07.DO: 0.8/ijmc