Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants

Transcription

1 Rochester Isttute of echology RI Scholr Wors Artcles 8-00 bocc d ucs Nubers s rdgol trx Deterts Nth D. Chll Est Kod Copy Drre Nry Rochester Isttute of echology ollow ths d ddtol wors t: Recoeded Ctto N.D. Chll d D.A. Nry. "bocc d ucs Nubers s rdgol trx Deterts" he bocc Qurterly. (00): 6-. hs Artcle s brought to you for free d ope ccess by RI Scholr Wors. It hs bee ccepted for cluso Artcles by uthorzed dstrtor of RI Scholr Wors. or ore forto plese cotct rtscholrwors@rt.edu.

2 IBONACCI AND UCAS NUBERS AS RIDIAGONA ARIX DEERINANS Nth D. Chll* Drre A. Nry** *Est Kod Copy Stte Street Rochester NY 650 **Deprtet of thetcs d Sttstcs Rochester Isttute of echology Oe ob eorl Drve Rochester NY 6 th.chll@od.co ds@rt.edu. INRODUCION here re y ow coectos betwee deterts of trdgol trces d the bocc d ucs ubers. or exple Strg [5 6] presets fly of trdgol trces gve by: where ( ) s ( ) = (). It s esy to show by ducto tht the deterts ( ) re the bocc ubers. Aother exple s the fly of trdgol trces gve by: H ( ) = ()

3 descrbed [] d [] (lso [5] but wth d o the off-dgols sted of ). he deterts H ( ) re ll the bocc ubers strtg wth =. I slr fly of trces [] the ( ) eleet of H ( ) s replced wth. he deterts ow geerte the ucs sequece strtg wth = (the ucs sequece s defed by the secod order recurrece = = = ). I ths rtcle we exted these results to costruct fles of trdgol trces whose deterts geerte y rbtrry ler subsequece or = of the bocc or ucs ubers. We the choose specfc ler subsequece of the bocc ubers d use t to derve the followg fctorzto: = π = cos. () hs fctorzto s geerlzto of oe of the fctorztos preseted []: = ( ) π = cos. I order to develop these results we ust frst preset theore descrbg the sequece of deterts for geerl trdgol trx. et A ( ) be fly of trdgol trces where A( ) =. heore : he deterts A ( ) c be descrbed by the followg recurrece relto:

4 A () = A A ( ) = ( ) A( ) A( ). = Proof: he cses = d = re cler. Now ( ) A = det. By cofctor expso o the lst colu d the the lst row A ( ) = A( ) det 0 A( ) A( ). =. IBONACCI SUBSEQUENCES Usg heore we c geerlze the fles of trdgol trces gve by () d () to costruct for every ler subsequece of bocc ubers fly of trdgol trces whose successve deterts re gve by tht subsequece. heore : he syetrc trdgol fly of trces ( ) eleets re gve by: = whose

5 = = j j = j = = j j = j j = ( ) j < wth Z d N hs successve deterts ( ) =. I order to prove heore we ust frst preset the followg le: e : = ( ) for. Proof: We use the secod prcple of fte ducto o to prove ths le: et =. he the le yelds = whch defes the bocc sequece. Now ssue tht = ( ) for N. he N = N N N N ( ) ( ) = N N N N N ( ) ( ) ( ) = N N N N ( ) ( ) = N N N Now usg heore d e we c prove heore. Proof of heore : We use the secod prcple of fte ducto o to prove ths theore: () = det =. ( ) = det =.

6 Now ssue tht ( ) = for N. he by heore ( ) ( ) ( ) = ( ) ( ) ( ) = ( ) ( ) = = (by e ) ( ) = Aother fly of trces tht stsfes heore c be foud by choosg the egtve root for ll of the super-dgol d sub-dgol etres. Wth heore we c ow costruct fly of trdgol trces whose successve deterts for y ler subsequece of the bocc ubers. or exple the deterts of: d 5 5 re gve by the bocc subsequeces d 5.. UCAS SUBSEQUENCES We c lso geerlze the fles of trdgol trces gve by () d () to show slr result for ler subsequeces of ucs ubers. We stte ths result s the followg theore:

7 heore : he syetrc trdgol fly of trces ( ) eleets re gve by: = whose t = t = t j j = j t = t = t j j = t j j = ( ) t j < wth Z d N hs successve deterts ( ). = Ag we beg wth le; ts proof ttes the proof of e. e : = ( ) for Proof of heore : We use ducto: () = det =.. ( ) = det =. Now ssue tht ( ) for N. he by heore = ( ) = t ( ) t t ( ) ( ) ( ) ( ) = ( ) ( ) = = (by e ) = ( )

8 Wth heore we c ow costruct fly of trdgol trces whose successve deterts for y ler subsequece of the ucs ubers. or exple the deterts of: d 9 re gve by the ucs subsequeces d 5.. A ACORIZAION O HE IBONACCI NUBERS I order to derve the fctorzto () gve by = = cos π we cosder the syetrc trdgol trces: ( ) = B. By e = d 6 ( ) = =. urtherore = = so ( ) ( ) B = s specfc stce of the trdgol fly of trces descrbed heore. herefore by heore ( ) B ( ) =. By usg the property of deterts tht B A AB = d by defg j e to be the j th colu of the detty trx I we hve ( ) ( ) C B = where:

9 C ( ) = e e B ( ). I he detert s the product of the egevlues. herefore let λ egevlues of C ( ) (wth ssocted egevectors x ) so ( ) = ( ) = C ( ) I we see tht G ( ) x C ( ) x Ix G he γ = λ re the egevlues of ( ) C = = x x G. = be the λ. ettg = λ ( ) x = λ. A egevlue γ of G ( ) s root of the chrcterstc polyol ( ) γi = 0 Note tht ( ) γi of the polyol: G ( ( ) e e )( G ( ) I) ( I ( ) e e ) = I G. γ so γ s lso root γ γ γ γ γ = 0. hs polyol s trsfored Chebyshev polyol of the secod d [] wth π roots γ = cos. herefore = ( ) C ( ) ( ) B = = = () follows by sple chge of vrbles. λ = ( cos ) = π. REERENCES. Byrd P.. Proble B-: A ucs Detert bocc Qurterly Vol. No. Deceber 96 p. 8.

10 . Chll N. D. D Errco J. R. Nry D. A. d Nry J. Y. bocc Deterts College thetcs Jourl Vol. No. y 00 pp Chll N. D. D Errco J. R. d Spece J. P. Coplex ctorztos of the bocc d ucs Nubers bocc Qurterly to pper.. Rvl. Chebyshev Polyols: ro Approxto heory to Algebr d Nuber heory d Ed. Joh Wley & Sos Ic Strg G. Itroducto to er Algebr d Ed. Wellesley A Wellesley- Cbrdge Strg G. d Borre K. er Algebr Geodesy d GPS Wellesley A Wellesley-Cbrdge 99 pp AS Clssfcto Nubers: B9 C0