Notes o Nuber Theory d Dscrete Mthetcs Prt ISSN 30-53, Ole ISSN 367-875 Vol. 3, 07, No., 9 03 O the Pell -crcult sequeces Yeş Aüzü, Öür Devec, d A. G. Sho 3 Dr., Fculty of Scece d Letters, Kfs Uversty 3600, Turey Assocte Professor, Fculty of Scece d Letters, Kfs Uversty 3600, Turey 3 Eertus Professor, Fculty of Egeerg & IT, Uversty of Techology Sydey, 007, Austrl Receved: Jury 06 Acceted: 3 Jury 07 Abstrct: I ths er, we defe the geerlzed Pell -crcult sequece d the Pell - crcult sequece by usg the crcult trces whch re obted fro the chrcterstc olyol of the geerlzed Pell (, ) -sequece d the, we obt scelleous roertes of these sequeces. Also, we cosder the cyclc grous whch re geerted by the geertg trces d the uxlry equtos of the defed recurrece sequeces d the, we study the orders of these grous. Furtherore, we exted the Pell -crcult sequece to grous. Flly, we obt the legths of the erods of Pell -crcult sequeces the sedhedrl grou SD for 4 s lctos of the results obted. Keywords: Crcult Mtrx, Sequece, Grou, Perod. AMS Clssfcto: B50, 0F05, 5A36, 0D60. Itroducto Klc [7] defed the geerlzed Pell (, ) -ubers s follows: for y gve ( =,,3, ) > d 0 () ( () P P ( ) P ( ) ( wth tl codtos ) ( P () = = P ) = 0 d = (.) 9
( ) ( ) ( P ) = P = = P ( ) =. It s cler tht the chrcterstc olyol of the geerlzed Pell (, ) -sequece s f x = x x. Dvs [4] defed the crcult trx C = c ssocted wth the ubers c0, c,, c s follows: The ( th ) C c0 c c c c c0 c3 c = c c 3 c0 c c c c c 0. P x = c c x c x s clled the ssocted degree olyol olyol of the crcult trx 0 C [cf.,5,0,,4]. Suose tht the ( )th ter of sequece s defed recursvely by ler cobto of the recedg ters: = c0 c c, where c0, c,, c re rel costts. Kl [6] derved uber of closed-for foruls for the geerlzed sequece by the coo trx ethod s follows: 0 0 0 0 0 0 0 0 0 0 0 0 0 A = 0 0 0 0 c0 c c c c. He lso showed tht A 0 =. These hve bee used by Gry the develoet of the relted theory Toeltz trces [3], d by Sho d Berste exteso of geerlzto of cotued frctos to rbtrry order recursve sequeces [3]. I Secto, we defe the geerlzed Pell -crcult sequece d the Pell -crcult 9
sequece such tht these sequeces re obted fro the crcult trx C P whch s defed by usg the chrcterstc olyol of the geerlzed Pell (, ) -sequece. The we obt ther scelleous roertes. I [5,6,7,8,9,9], the uthors obted the cyclc grous d the segrous v soe secl trces. I Secto 3, we cosder the ultlctve orders of the crcult trx C P d the Pell -crcult trx M P worg to odulo d the, we obt the cyclc grous whch re geerted by reducg these trces odulo. Also ths secto, we study the defed recurrece sequeces odulo. The we derve the reltoshs betwee the orders of the obted cyclc grous d the erods of the defed sequeces ccordg to odulo. The study of recurrece sequeces grous beg wth the erler wor of Wll [6] where the ordry Fbocc sequeces cyclc grous were vestgted. The cocet exteded to soe secl ler recurrece sequeces by severl uthors; see for exle, [,3,5,7,9,0,,8,9,,5]. I Secto 4, we defe the Pell -crcult sequece by es of the eleets of the grous whch hve two or ore geertors, d the we exe ths sequece fte grous. Furtherore, we exe the behvours of the legths of the erods of the Pell -crcult sequeces the sedhedrl grou SD for 4. The Geerlzed Pell -Crcult d The Pell -Crcult Sequeces We c wrte the followg crcult trx for the olyol f ( x ) s s follows: C P ( ) ( = ) ( = = ) ( ) ( ) ( ) f =, f d,, = C = P P f =, =, = d =, =, 0 other wse. For exle, the trx C 4 s s follows: 0 0 C4 = 0 0. Defe the geerlzed Pell -crcult sequece by usg the trces C s show: 93
x x x, od, x = x x x, od( ), x x x, od( ), x x x 3 0od( ) where x = x = = x = 0 d x =. For 0, by ductve rguet, we y wrte x x x x x x x x x x x x x x ( C ) = x x x x x x( ) x ( ) x ( ) x ( ) It s esy to see tht ( ) 3 3 ( ) x C s crcult trx of order. for >, (.). (.) We ext defe the Pell -crcult sequece s = for > 0 (.3) where = = = = 0, = d. We the obt tht the geertg fucto of the Pell -crcult sequece { } s s follows: x g( x) = x x x. By (.3), we c wrte the followg coo trx: 0 0 0 0 0 0 0 0 0 0 MP = = 0 0 0 0 0 0 0 0 The trx M P s sd to be the Pell -crcult trx. It s cler tht = M For, by ductve rguet, we y wrte 94
( M ) 3 =. 3 (.4) Note tht det M =. It s well-ow tht the Sso detty for recursve sequece c be obted fro the detert of ts geertg trx. Le.. The chrcterstc equto of the Pell -crcult sequece x x x = 0 does ot hve ultle roots. Proof. Let d let μ be ultle root of h( z ). The 0 h z = z z z h μ = d h ( μ ) 0 root of h( z ). So, we obt ' Thus d hece, h d h ( μ ) ( μ ) Sce =. We frst ote tht 0 s ot h μ = μ μ μ = μ μ μ = 0. μ, = ± 7 4 4 ( ) 7 4 ( ) 7 4 ( ) 7 4 = = 4 4 4 ( ) 7 4 ( ) 7 4 ( ) 7 4 = = 4 4 4, by ducto o, t s see tht h( μ ) d h( μ ), whch re cotrdctos. Thus, the equto h( μ ) = 0 does ot hve ultle roots. h ε be the chrcterstc olyol of the Pell -crcult trx M. If Let ε, ε,, ε re egevlues of the trx M, the by Le., t s ow tht ε, ε,, ε re dstct. Let V be ( ) ( ) Vderode trx: 95
Suose tht d (, ) V s ( ) ( ) ε ε ε ε ε ε V = ε ε ε. W ε ε = ε trx obted fro V by relcg the th colu of V by W. We c ow estblsh the Bet forul for the Pell -crcult sequece wth the followg Theore. Theore.. Let be the th ter of the Pell -crcult sequece. The where det = det (, ) ( V ) ( V ) ( M ) =. Proof. Sce the egevlues of the trx M re dstct, the trx M s D = dg ε, ε,, ε, the t s esy to see tht M V = VD. Sce dgolzble. Let the trx V s vertble, get ( ) = for M V VD So, we obt V M V = D. Hece, the trx M s slr to D. So we. The we wrte the followg ler syste of equtos for: ε ε = ε ε ε = ε ε ε = ε. (, ) ( V ) det = for, =,,,. det ( V ) 96
3 The Cyclc Grous v the trces C P d M P For gve trx A= wth s tegers, A ( od ) reduced odulo, tht s, A( od ) ( ( od ) ) A = { A ( od ) 0 }. If gcd (, det A ) =, the the set otto A. Sce det A deotes the order of the set es tht ech eleet of A s =. Let us cosder the set A s cyclc grou. Let the M =, t s cler tht the set M s cyclc grou for every ostve teger. Slrly, the set C s cyclc grou f ( C ) trces C P d M P. gcd, =. We ext cosder the cyclc grous geerted by these Theore 3.. Let be re d let G be y of the cyclc grous C d M such tht N. If u s the lrgest ostve teger such tht G = G u, the G v v u = G for every v u. I rtculr, f G G, the G v = G for every v. Proof. Let us cosder the cyclc grou d let M P M be deoted by h( ) ( h ) MP I ( od ), the ( h ) MP I ( od ) d I s ( ) ( ) detty trx. Thus we fd tht h( ) dvdes h( ) wrtg M I. If where s ostve teger ( ) = ( ( )) = ( ) ( od ) h = we get fro the bol exso tht ( ) h P = 0 M I I whch yelds tht h( ) dvdes h( ). Thus, h( ) = h( ) or h h It s cler the tht h( ) = h( ) holds f d oly f there exsts teger u s ot dvsble by. Sce u s the lrgest ostve teger such tht h( ) h( ) u u h( ) h( ( u ) ). The, there exsts teger u u tht h( ) h( ). To colete the roof we y use ductve ethod o u.,. Also, =. whch =, we hve whch s ot dvsble by. So we get There s slr roof for the cyclc grou C. λ 97
Theore 3.. Let t = ( t ) e =, G be y of the cyclc grous C P d M d let where s re dstct res. The G = lc G, G,, G e e e t t Proof. Let us cosder the cyclc grou C P. Suose tht C e = α for t d let C d α =. The by (.), we obt α ( ) α ( ) e 0 od for, α ( ) α ( ) e od 0od for, od. Ths les tht = α α ( ) for d N α for ll vlues of. Thus t s verfed tht for C P. tht s, α C P s of the C = lc C,,, e e e C C t t. There s slr roof for the set M. It s well-ow tht sequece s erodc f, fter cert ot, t cossts oly of reettos of fxed subsequece. The uber of eleets the reetg subsequece s the erod of the sequece. A sequece s sly erodc wth erod f the frst eleets the sequece for reetg subsequece. Reducg the geerlzed Pell -crcult sequece d the Pell -crcult sequece of the by odulus, we c get the reetg sequeces, resectvely deoted by x = x, x, x,, x, d { } { } 3 {,,,,, } { } 3 where x = x ( od ), ( od ) =, =. They hve the se recurrece relto s the deftos of the geerlzed Pell -crcult sequece d the Pell -crcult sequece, resectvely. Theore 3.3. The sequece Slrly, the sequece { } { x } s sly erodc sequece f ( C ) s sly erodc for every ostve teger. gcd det,. = 98
{ } Proof. Let us cosder the Pell -crcult sequece {(,,, ) 0 } Q= q q q q, the Q P =. Sce there re. Let P dstct -tules { } of eleets of, t lest oe of the. Thus, the subsequece followg ths -tule reets; tht s the sequece s erodc. So f -tules ers twce the sequece,,, od. Fro the defto, we c esly derve tht d >, the,,, =. Thus we get tht the sequece s sly erodc. x. { } There s slr roof for the sequece { } We deote the legths of the erods of the sequeces x ( ) d ( ) by x { } l, l, resectvely. The, we hve the followg useful results fro (.) d (.4), resectvely. Corollry 3.. Let be re. The. If ( C ) l gcd det,. l =. C. =, the x P = M. Let be re d let such tht { od :, } ( ) ( ) N. The, t s cler tht the set A = x x = x x A s cyclc grou. Now we c gve reltosh betwee the chrcterstc equto of the Pell -crcult sequece d the erod l by the followg Corollry. Corollry 3.. Let be re d let N. The, the cyclc grou A( ) s soorhc to the cyclc grou M. 4 The Pell -Crcult Sequece Grous Let G be fte -geertor grou d let X be the subset of (,,, ) x x x X f d oly f G s geerted by x, x,, x geertg -tule for G. 99 G G G G such tht. We cll ( x, x,, x )
Defto 4.. Let G = X be ftely geerted grou such tht X = { x, x,, x}. The we defe the Pell -crcult sequece the grou G s follows: ( ) ( ) ( ) ( ) ( ) f, = f > for, wth tl codtos For -tule (,,, ) PC ( G ) x, x,, x. = x for. x x x X, the Pell -crcult sequece grou G s deoted by Theore 4.. A Pell -crcult sequece fte grou s sly erodc. Proof. Suose tht s the order of G. Sce there dstct G, t lest oe of the -tules ers twce the sequece -tules of eleets of PC ( G ) x, x,, x. Thus, cosder the subsequece followg ths -tule. Becuse of the reetg, the sequece s erodc. Sce the sequece wth u v, such tht PC By the defto of the sequece s erodc, there exst turl ubers u d v, ( G ) x, x,, x =, =,, =. u v u v u v PC ( G ) x, x,, x ( ) ( ) ( ) ( )( ) Therefore, we obt u = v, d hece,, we ow tht f, =. f > =, =,, =, u v u v u v whch les tht the sequece s sly erodc sequece. Let LPC ( G ) x, x,, x deote the legth of the erod of the sequece the defto of the sequece PC ( G ) x, x,, x PC ( G ) x, x,, x. Fro t s cler tht the erod of ths sequece fte grou deeds o the chose geertg set d the order whch the ssgets of x, x,, x re de. 00
We shll ow ddress the legths of the erods the Pell -crcult sequeces the sedhedrl grou SD. A grou SD s sedhedrl grou of order f SD =, b = b = e, b b = for every 4. Note tht the orders d b re d, resectvely [cf.,4]. Theore 4.. The legths of the erods the Pell -crcult sequeces the sedhedrl grou SD re s follows: 3. ( ) LPC SD ;, b = l for =,. ( ) LPC SD ;, b = l for 3. Proof.. We rove ths by drect clculto. Note tht the grou SD s defed by the resetto b, = b = eb, b=. =, b = d l = 5. The, It s cler tht b b =. The sequece ( ) PC SD ;, b s 3 ( 3)( ) 5 ( 7) 3 ( 7) ( ), b, b, b, b,, b,, b, ( ) 3 ( 7) ( 4) b,,, b,, e. Usg the bove, the sequece becoes: x =, x = b, x = b, x = b,, 3 4 x =, x = b, x = b, x = b,,, 5 4 6 7 8 9 x =, x = b, x = b, x = b,. 4 4 5 5 5 3 5 4 So we eed the sllest teger such tht 4 = for N. It s esy to see tht the legth of the erod of the sequece s. For 3, the sequece ( ) Fro bove, the sequece becoes: 3 5. PC SD ;, b hs the followg for: x =, x = b, x3 = b, x4 = b,, x = b,, ( 4) ( 4) λ x =, x = b, x = b, l l l 3 ( 4) λ ( 4) λ x = b,, x = b,, l 4 l 0
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