The Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices

Transcription

1 Internatonal Mathematcal Forum, Vol 11, 2016, no 11, HIKARI Ltd, wwwm-hkarcom The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan Department of Mathematcs, Ahmet Keleşoğlu Faculty of Educaton Necmettn Erbakan Unversty, Konya, Turkey Fkr Köken Ereğl Kemal Akman Vocatonal School Necmettn Erbakan Unversty, Konya, Turkey Copyrght c 2016 Saadet Arslan and Fkr Köken Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted Abstract In ths paper, the Jacobsthal and Jacobsthal-Lucas numbers wth specalzed ratonal subscrpts are studed by usng square roots of the matrces F n and S n We also reveal the denttes nvolvng these numbers derved by the matrces F n/2 and S n/2 Further we show that the matrces F n and S n are generalzed to ratonal powers by usng the Abel s functonal equaton Mathematcs Subject Classfcaton: 11B39; 15A24 Keywords: Jacobsthal and Jacobsthal-Lucas numbers; Square roots of matrces 1 Introducton We summarze basc nformaton about the defnton and the propertes assocated wth Jacobsthal and Jacobsthal-Lucas numbers The Jacobsthal {J n } n=0 and Jacobsthal-Lucas {j n } n=0 sequences are defned by the recurrence relatons J n+2 = J n+1 + 2J n, j n+2 = j n+1 + 2j n, n 0,

2 514 Saadet Arslan and Fkr Köken where J 0 = 0, J 1 = 1, j 0 = 2 and j 1 = 1 Also, The Jacobsthal and Jacobsthal-Lucas numbers can easly be computed usng the closed-forms alke J n = 2 n 1 n /3 and j = 2 n + 1 n 1,2 And so, the J n and j n numbers can be derved by matrces snce by takng successve powers of matrces F and S 3,4; Jn+1 2J n jn 9J n F n = J n 2J n 1, S n = 1 2 There are less theory related to complex value Jacobsthal and Jacobsthal- Lucas sequences from other nteger sequences In 1 study, Horadam derved from cartesan coordnates x, y of a pont n the plane, x = 2 θ cos θπ /3 and y = 2 θ sn θπ/3 for J n, 1 x = 2 θ + cos θπ and y = 2 θ sn θπ for j n, θ R 2 The author denomnated equatons 1 and 2 as the modfed Bnet forms for J n and j n, respectvely Settng θ = n Z, x = J n n 1 and x = j n n 2 are the ordnary Bnet form for J n and j n The equatons 1 and 2 yeld the Jacobsthal and Jacobsthal-Lucas curves Thers statonary ponts le on the sutable branches of the rectangular hyperbolas y x ± k log 2 = kπ and y x ± k log 2 = kπ 3 9 Also, for J θ + j θ = 2J θ+1, a composte curve s gven to beng equvalent to the Jacobsthal curve 1 It s the object of ths artcle to reveal the correspondng complex value sequences assocated wth real ndex {J x } and {j x }, for x R va computng the square roots of the matrces F n and S n 2 Man Results Our am s not to calculate the square roots of a 2x2 matrx The certan methods have been studed for computng the square roots of arbtrary 2x2 matrces 5,6, and a matrx may be have several square roots or no square roots If a matrx generate any sequence and have several square roots, we nvestgate propertes whch new sequences generated by square roots matrces have, and the man results of the paper are gven the followng Theorem 21 Let F n/2 F n Then, F n/2 1,2 = ± Jn+2/2 2J n/2 J n/2 J n j n = 1, 2, 3, 4 denote the square roots of the matrx 2J n 2/2, F n/2 3,4 = ±1 jn+2/2 2j n/2 3 j n/2 2j n 2/2 3

3 Jacobsthal and Jacobsthal-Lucas numbers 515 Proof When square roots of the matrx F n are computed by the Cayley- Hamlton method 5, t s seen that the matrx F n has four square roots n the forms F n ±1 = T ± 2 F n ± det F n I, 4 det F n where I s the dentty matrx T = j n s the trace of the matrx F n and as det F n = 2 n e nπ, we have det F n = 2 n/2 e nπ/2 Thus, t s obtaned that T ± 2 det F n = 2 n/2 ± e nπ/2 We select frstly as F n = ±1 Jn n/2 e nπ/2 2J n 2 n/2 + e nπ/2 J n 2J n n/2 e nπ/2 Usng the modfed Bnet form n 1, we can wrte the values of elements 1,1 and 2,1 for the rght sde matrx as ± 2 n/2 e nπ/2 J n n/2 e nπ/2 = ± 2 n+2/2 e n+2π/2 = ±J 2 n e nπ n+2/2, 3 and ±J n 2 n/2 e nπ/2 n = ±J n 2 n/2 e nπ/2 = ±J 2 n e nπ n/2 3J n If the elements 1,2 and 2,2 are founded n the smlar way, we get two square roots of the matrx F n F n/2 Jn+2/2 2J 1 = n/2, F n/2 Jn+2/2 2J 2 = n/2 J n/2 2J n 2/2 J n/2 2J n 2/2 When we choose the other state of the matrx equaton 4, we have F n = ± 2 n/2 + e nπ/2 Jn+1 2 n/2 e nπ/2 2J n 2 n e nπ 2J n 1 2 n/2 e nπ/2 J n If all elements are gven n the same way, then the other two square roots matrces are possessed Moreover, t s F n = F n/2 F n/2 because the square roots of the matrx F n are the matrces F n/2 = 1, 2, 3, 4 If we admt the matrces F n/2 = 1, 2, then the equatons J 2 n+2/2 + 2J 2 n/2 = J n+1, J n+2/2 J n/2 + 2J n/2 J n 2/2 = 2J n are determned by equatng of correspondng elements for equal matrces In addton, consderng the elements of the matrces for F n/2 = 3, 4 cause j 2 n+2/2 + 2j 2 n/2 = 9J n+1, j n+2/2 j n/2 + 2j n/2 j n 2/2 = 9J n

4 516 Saadet Arslan and Fkr Köken And also, takng the determnant of the matrces F n/2 = 1, 2, 3, 4 gve J n+2/2 J n 2/2 J 2 n/2 = 2 n 2/2 e nπ/2, j n+2/2 j n 2/2 j 2 n/2 = 9e nπ/2 2 n 2/2, whch are a general case of the Cassn-lke formula for the Jacobsthal and Jacobsthal-Lucas numbers wth specalzed ratonal subscrpts, respectvely Hence, the matrces F n/2 = 1, 2, 3, 4 n 3 are nonsngular, the nverses of the matrces F n/2 are denomnated notaton F n/2 by gven; F n/2 1,2 = ±e nπ/2 2Jn 2/2 2J n/2, F n/2 2 n/2 3,4 = e nπ/2 2jn 2/2 2j n/2 J n/2 J n+2/2 32 n/2 j n/2 j n+2/2 A lot of elementary propertes for these numbers can be found by equatng of correspondng elements for the equal matrces such as F k/2 F n+1/2 = F k+n+1/2, F n F 1/2 = F 2n+1/2 and F n/2 F n+1/2 = F 2n+1/2 for all nteger n and k For equal matrces F k/2 F n+1/2 = F k+n+1/2 1 = 1, 2 or = 3, 4, equatng of correspondng elements gves followng equaltes J k+n+1/2 = J k/2 J n+3/2 +2J k 2/2 J n+1/2 = 1 9 jk+2/2 j n+1/2 + 2j k/2 j n 1/2 Consderng the equaton F n F 1/2 F 2n+1/2 1 = 1, 2, 3, 4 derve = F 2n+1/2 1 = 1, 3 and F n/2 F n+1/2 = J 2n+1/2 = J n J 3/2 + 2J n 1 J 1/2 = 1 Jn+1 j 1/2 + 2J n j 1/2, 3 J 2n+1/2 = J n+2/2 J n+1/2 + 2J n/2 J n 1/2 = 1 jn/2 j n+3/2 + 2j n 2/2 j n+1/2 9 Computng the equaltes F k n/2 = F k/2 equaltes F n/2 = 1, 2, 3, 4, we have the J k n/2 = 2 n/2 e nπ/2 Jk/2 J n+2/2 J k+2/2 J n/2 = e nπ/2 92 n/2 jk+2/2 j n/2 j k/2 j n+2/2 The next goal s to fnd dfferent relatons between the Jacobsthal and Jacobsthal-Lucas numbers wth specalzed ratonal subscrpts by usng the square roots of the matrx S n Theorem 22 Let S n/2 = 1, 2, 3, 4 denote a square root of the matrx S n Then, S n/2 1,2 = ±1 jn/2 9J n/2 2 J n/2 j n/2, S n/2 3,4 = ±3 Jn/2 j n/2 2 j n/2 /9 J n/2

5 Jacobsthal and Jacobsthal-Lucas numbers 517 Proof If we apply for the Cayley-Hamlton method for computng the square roots of the matrx S n, square root matrces are obtaned n form ±1 Sn = T ± 2 S n ± det S n I 5 det S n where I s the dentty matrx T = j n s the trace of the matrx S n and snce det S n = 2 n e nπ, we have det S n = 2 n/2 e nπ/2 Therefore, t s obtaned that T ± 2 det S n = 2 n/2 ± e nπ/2 From ths pont of vew, we select as S n/2 = ±1 jn + 2 n+2/2 e nπ/22n/2 9J n 2 2 n/2 + e nπ/2 J n j n + 2 n+2/2 e nπ/2 Usng the modfed Bnet forms n 1 and 2, we wrte down the values of elements 1, 1 and 1, 2 for the rght sde matrx ± 2 n/2 e nπ/2 j n + 2 n+2/2 e nπ/2 2 2 n e nπ = ± 2n e nπ 2 n/2 + e nπ/2 = ±j n/2, 6J n 2 and ±9J n 2 n/2 e nπ/2 2 2 n e nπ = ±9J n 2 n/2 e nπ/2 = ±9J n/2 3J n When other elements are gven n the same way, we determne two square root matrces S n/2 1 = 1 jn/2 9J n/2, S n/2 2 = 1 jn/2 9J n/2 2 J n/2 j n/2 2 J n/2 j n/2 If we prefer the other crcumstance n the matrx equaton 5, then the other two matrces are procured by computng the above mentoned values of all elements for the rght sde matrx We suppose that the matrx S n/2 = 1, 2, 3, 4 s one of the square roots of the matrx S n Equatng correspondng elements of the equal matrces S n/2 S n/2 = S n gves followng denttes j 2 n/2 + 9J 2 n/2 = 2j n, J n/2 j n/2 = J n, and so takng the determnant of the matrces S n/2 = 1, 2, 3, 4 yelds j 2 n/2 9J 2 n/2 = 2 n+4/2 e nπ/2 In addton to above mentoned equaltes, dfferent equaltes for the Jacobsthal and Jacobsthal-Lucas numbers wth specalzed ratonal subscrpts can be found by equatng of correspondng elements for the equal matrces

6 518 Saadet Arslan and Fkr Köken S k/2 S n+1/2 = S k+n+1/2, S n S 1/2 = S 2n+1/2 and S n/2 S n+1/2 = S 2n+1/2 Also, snce the matrces S n/2 = 1, 2, 3, 4 are nonsngular, f the nverses of the matrces S n/2 are shown wth S n/2, we get S n/2 1,2 = ±e nπ/2 jn/2 9J n/2, S n/2 = ±3e nπ/2 Jn/2 j n/2 2 n+2/2 3,4 J n/2 j n/2 2 n+2/2 j n/2 /9 J n/2 For all ntegers n and k, equatng of correspondng elements for the equal matrces S n+1/2 S k/2 = S n+k+1/2 1 = 1, 2, 3, 4, we get 2j n+k+1/2 = j n+1/2 j k/2 + 9J n+1/2 J k/2, 2J n+k+1/2 = J n+1/2 j k/2 + j n+1/2 J k/2 The equatons of the matrces S n S 1/2 = S 2n+1/2 1 = 1, 3 and S n/2 S 2n+1/2 1 = 1, 2, 3, 4 gve to equaltes 2J 2n+1/2 = J n j 1/2 + j n J 1/2, 2j 2n+1/2 = j n j 1/2 + 9J n J 1/2, 2j 2n+1/2 = j n/2 j n+1/2 + 9J n/2 J n+1/2, 2J 2n+1/2 = J n/2 j n+1/2 + j n/2 J n+1/2 The equatons S k/2 S n/2 = S k n/2 1 = 1, 2, 3, 4 generate to equaltes 2e nπ/2 j k n/2 = j k/2 j n/2 9J k/2 J n/2, 2e nπ/2 J k n/2 = J k/2 j n/2 j k/2 J n/2 S n+1/2 = Also, the computng square roots of dfferent matrx generators of the Jacobsthal and Jacobsthal-Lucas numbers 3,4 can be carred out n the above mentoned fashon Lastly, we establsh extended matrces F r/q and S r/q for q Z + and r Z, whch generate the Jacobsthal and Jacobsthal-Lucas number wth ratonal subscrpts To do ths, we use the connecton between the equaton p x = J r x 2 + 2J r 1 J r+1 x 2J r or p x = J r 2 x 2 + j r j r 2 x 9J r 2 and the roots matrces F r/q or S r/q va the Abel s functonal equaton 5 The polynomal p x = J r x 2 x 2 s related to the matrces F r Let Ax = dx px Ax = dx J r x 2 x + 1 = 1 ln 3J r x 2 x + 1 be, then We defne a functon Φ F rx = J r+1x+2j r J rx+2j r 1 for the matrx F r The Abel s functonal equaton A Φ F rx = Ax + k s satsfed for the certan real constant k Then, Jr+1 x + 2J r A J r x + 2J r 1 = 1 3J r ln = A x + 1 3J r ln x Jr+1 2J r + J r 2J r 1 x J r+1 + J r + J r + J r 1 e rπ 2 r

7 Jacobsthal and Jacobsthal-Lucas numbers 519 A closed form for the q th roots of the matrx F r s derved from the functonal equaton Φ F r/qx = A A 1 x + k, where the nverse functon of A x s q shown wth A 1 x gven by A 1 x = e3xjr e 3xJr It follows that Φ F r/qx = 1 x 2 A ln 1 3Jr 1 e rπ/q + ln x + 1 3J r 2 r/q = erπ/q x x r/q 2 r/q x + 1 e rπ/q x 2 = x 2 r/q 2 + e rπ/q r/q e rπ/q x 2 r/q e rπ/q + 2e rπ/q + 2 r/q = xj r/q+1 + 2J r/q xj r/q + 2J r/q 1 Hence the roots matrces F r/q s related to the functon Φ F r/qx, we have F r/q Jr+q/q 2J = ± r/q 6 J r/q 2J r q/q If we make out above mentoned calculaton for the matrx S r, that s, f the functon Φ S r/qx s computed, we have S r/q = ±1 jr/q 9J r/q 7 2 J r/q j r/q Takng the determnant of the matrx equatons 6 and 7 yelds J r+q/q J r q/q J 2 r/q = 2 r q/q e rπ/q, j 2 r/q 9J 2 r/q = 4e rπ/q 2 r/q Consderng dfferent ratonal powers of the matrx F and S, and usng the followng matrx equatons, F rs+qt/qs = F r/q F t/s, r, t Z and q, s Z +, S rs+qt/qs = S r/q S t/s, we determne the denttes nvolvng terms of the Jacobsthal and Jacobsthal- Lucas numbers wth ratonal subscrpts by gven J rs+qt/qs = J r/q+1 J t/s + 2J r/q J t/s 1 = J r/q J t/s+1 + 2J r/q 1 J t/s, 2J rs+qt/qs = J r/q j t/s + j r/q J t/s, 2j rs+qt/qs = j r/q j t/s + 8J r/q J t/s It shows that, the Jacobsthal and Jacobsthal-Lucas numbers wth ratonal subscrpts hold analogous denttes for the classcal Jacobsthal and Jacobsthal- Lucas numbers

8 520 Saadet Arslan and Fkr Köken References 1 AF Horadam, Jacobsthal and Pell Curves, The Fb Quart, , no 1, AF Horadam, Jacobsthal representaton numbers, The Fb Quart, , no 1, F Koken, D Bozkurt, On the Jacobsthal Numbers by Matrx Methods, Int Journal Contemp Math Scences, , no 13, F Köken, D Bozkurt, On the Jacobsthal-Lucas Numbers by Matrx Method, Int Journal Contemp Math Scences, , no 33, S Northsheld, Square Roots of 2x2 Matrces, Contemporary Mathematcs, , D Sullvan, The Square Roots of 2x2 Matrces, Math Magazne, , no 5, Receved: Aprl 24, 2016; Publshed: May 30, 2016