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, 42075, Kampus, Koya, Turkey kokefikri@gmailcom, dbozkurt@selcukedutr Abstract I this study, we defie the Jacobsthal Lucas E-matrix ad R-matrix alike to the Fiboacci Q-matrix Usig this matrix represatatio we have foud some equalities ad Biet-like formula for the Jacobsthal ad Jacobsthal-Lucas umbers Mathematis Subject Classificatio: 11B39; 11K31; 15A24; 40C05 Keywords: Jacobsthal umbers; Jacobsthal-Lucas umbers; Matrix method 1 Itroductio Fiboacci ad Lucas umbers ad their geeralizatio have may iterestig properties ad applicatios i almost every field of sciece ad art For the prettiess ad rich applicatios of these umbers ad their relatives oe ca see sciece ad the ature 3-7 I 1960, Charles H Kig studied o the followig Q-matrix Q 1 1 1 0 i his Ms thesis He showed below Q F+1 F F Moreover, it is clearly show below F 1 ad det(q) 1 F +1 F 1 F 2 ( 1) (Cassii s formula) 1 This study is the part of Ms Thesis of F Köke
1630 F Köke ad D Bozkurt The above equalities demostrate that there is a very close lik betwee the matrices ad Fiboacci umbers 7 We have defied Jacobsthal F -matrix give by F 1 2 1 0 i our article 2 ad have demostrated a very close lik betwee this matrix ad Jacobsthal umbers More geerally, there are some relatios betwee the iteger sequeces ad matrices 5-10 I 1, the Jacobsthal ad the Jacobsthal-Lucas sequeces J ad j are defied by the recurrece relatios J 0 0,J 1 1,J J 1 +2J 2 for 2 j 0 2,j 1 1,j j 1 +2j 2 for 2 respectively I this study, we have defied Jacobsthal-Lucas E-matrix by 5 2 E (2) 1 4 (1) It is feasible that, it ca be writte j+2 j +1 E J+1 J ad 9 J+2 J +1 1 9 E j+1 j where J ad j are the th Jacobsthal ad Jacobsthal Lucas umbers, respectively Furthermore, we have defied Jacobsthal-Lucas R-matrix by R 1 4 2 1 (3) Throughout this paper, J ad j deote the th Jacobsthal ad Jacobsthal-Lucas umbers 2 The Matrix Represetatio I the sectio, we will get some properties of the Jacobsthal-Lucas E-matrix ad R-matrix Moreover, usig these matrices, we have obtaied the Cassiilike ad the Biet-like formulas for the Jacobsthal ad Jacobsthal Lucas umbers
Jacobsthal-Lucas umbers by matrix method 1631 Theorem 1 Let E be a matrix as i (2) The, for all positive itegers, the followig matrix power is held below 3 J+1 2J, if eve E J 2J 1 (4) 3 1 j+1 2j, if odd j 2j 1 Proof We will use the Priciple of Mathematical Iductio (PMI) o odd ad eve, seperately (For odd ) Whe 1, 5 2 E 1 j2 2j 1, 1 4 j 1 2j 0 is correct We assume that it is correct for odd k Now, we show that it is correct for k + 2 We ca write E k+2 E k E 2 3 k+1 jk+3 2j k+2 j k+2 2j k+1 ad the result is easily see immediately (For eve ) Whe 2, 27 18 E 2 3 2 J3 2J 2 9 18 J 2 2J 1 is correct We assume it is correct for eve k correct for k + 2 We calculate E k+2 E k E 2 3 k+2 Jk+3 2J k+2 J k+2 2J k+1 Fially, we show that it is Corollary 2 Let E be as i (4) For all positive itegers, the followig determiatal equalities are held: i) det(e )3 2 2, ii) J +1 J 1 J 2 ( 1) 2 1, iii) j +1 j 1 j 2 ( 1) +1 3 2 2 1 Proof We will use the PMI o Whe 1, it is see to be true We assume that it is true for k We will show that it is true for k +1 det(e k+1 ) det(e k ) det(e) 3 2k+2 2 k+1 ad the proof of (i) is completed The proof of (ii) ad (iii) are easily show from (i) ad (4)
1632 F Köke ad D Bozkurt Theorem 3 Let be a itegerthe Biet-like formulas of the Jacobsthal ad Jacobsthal Lucas umbers are J 2 ( 1) 3 ad j 2 +( 1) Proof If we calculate the eigevalues ad eigevectors of the E-matrix, they are allotted λ 1 6,λ 2 3 ad v 1 ( 1, 1), v 2 (2, 1), respectively We ca diagoalize of the matrix E by V U 1 EU where U (v T 1,vT 2 ) 1 2 1 1 ad V diag(λ 1,λ 2 ) 3 0 0 6 Thus, applyig from the properties of similar matrices, we ca write that E UV U 1 (5) By (4) ad (5) matrix equatios, desired results are obtaied via E 3 1 2 +1 + 1 2(2 1) 2 1 2(2 1 +1) Theorem 4 Let m ad be itegers The, the followig equalities are valid: i) 9J m+ j m j +1 +2j m 1 j, ii) J m+ J m J +1 +2J m 1 J, iii) j m+ j J m+1 +2j 1 J m, iv) 9 ( 1) +1 2 1 J m j 1 j m+1 j j m, v) ( 1) 2 1 J m J m J 1 J m 1 J, vi) ( 1) +1 2 1 j m J m j 1 J m 1 j, vii) ( 1) +1 2 J j 1 J 1 j Proof Let the matrix E as i (2) Sice E m+ E m E Thus, equalities (i), (ii) ad (iii) are easily see
Jacobsthal-Lucas umbers by matrix method 1633 If we compute E, we obtai ( 1) 2J 1 2J, for eve, E 6 J J +1 ( 1) +1 2j 1 2j for odd, 3 6 j j +1 Sice it is that E m E m E, equalities (iv), (v) ad (vi) are clearly see For proof of (vii), it is obtaied by takig m i (i) of the Theorem 4, this completes the proof Theorem 5 Let R be i (3) The, there are followig equalities, RF j+1 2j j+1 2j ad R 9F j 2j 1 where F is the Jacobsthal F -matrix i (1) j 2j 1 Proof We have recalled that j +1 J +1 +4J, j 2J +1 J,9J +1 j +1 +4j ad 9J 2j +1 j i 1 Usig these equatios, we ca easly see that this relatio betwee R-matrix ad E-matrix Refereces 1 A F Horadam, Jacobsthal Represetatio Numbers, Fib Quart34,40-54,1996 2 F Koke ad D Bozkurt, O The Jacobsthal Numbers by Matrix Methods, It J Cotemp Math Scieces, Vol3,2008, o13,605 614 3 A Stakhov, Fiboacci matrices, a geeralizatio of the Cassii formula, ad a ew codig theory, Chaos, Solitos & Fractals 2006;30:56 66 4 D Kalma, Geeralized Fiboacci umbers by matrix methods, Fib Quart 20(1)(1982) 73 76 5 E Karaduma, O determiats of matrices with geeral Fiboacci umbers etries, Appl Math ad Comp 167(2005)670 676 6 E Karaduma, A applicatio of Fiboacci umbers i matrices, Appl Math ad Comp 147 (2003) 903 908 7 T Koshy, Fiboacci ad Lucas Numbers with Applicatios, Joh Wiley ad Sos, NY, 2001 Received: March 25, 2008