SUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES

Transcription

1 Avlble ole t J. Mth. Comput. Sc. 4 (04) No ISSN: SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of Studes Mthemtcs Vrm Uversty Ujj Id Deprtmet of Mthemtcs Mdsur Isttute of Techology Mdsur Id Copyrght 04 B. Sgh et l. Ths s ope ccess rtcle dstrbuted uder the Cretve Commos Attrbuto cese whch permts urestrcted use dstrbuto d reproducto y medum provded the orgl wor s properly cted. Abstrct: bocc umbers re fsctg d ther mpct o the feld of mthemtcs hs bee gret. I ths pper mly preset formuls for the sums of -ucs umbers wth dees rthmetc sequece sy r for fed tegers d r (0 r ). Also the geertg fucto evlutes d presets the ltertg sum for the -ucs umbers wth rthmetc de. Keywords: -bocc umbers -ucs umbers sequeces of prtl sums. 000 AMS Subject Clssfcto: B7 B9.. INTRODUCTION: The feld of bocc s bout more th just sequece. e y sequece oe c lyze ts strtg vlues d the rto of ts terms from whch we obt the geerlzed bocc sequece d the golde rto respectvely. bocc d ucs umbers d ther geerlzto hve my terestg propertes d pplctos to lmost every feld of scece d rt. Besdes the usul bocc umbers my d of geerlztos of these umbers hve bee preseted the lterture [9 0 ]. I [ 4] ew geerlzto of the clsscl * Correspodg uthor Receved Aprl 0 05

2 06 BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA bocc sequece s troduced. It should be oted tht the recurrece formul of these umbers depeds o oe rel prmeter. Defto : or y teger umber the th bocc sequece sy { } N s defed recurretly by: 0 0 d for. (.) Prtculr cses of defto (.) If we obt the clsscl bocc sequece{058...}. If we obt the Pell sequece{ }. If we obt the sequece{ } { }. N ew terms of the -bocc umbers re My propertes of - bocc umbers re obted drectly from elemetry mtr lgebr. I [4 5] severl propertes of these umbers re deduced tht re relted wth ther dervtves d the so clled Pscl trgle. lco d Plz [] gves severl formuls for the sum of these umbers wth dees rthmetc sequece. Also [6] uthors pply the boml d the -boml trsforms to the -bocc sequeces d derves my formuls le geertg fucto d Bet s formul. I [7] uthors defed - bocc hyperbolc fuctos d deduced some propertes of - bocc hyperbolc fuctos relted wth the logous dettes for the - bocc umbers. I [] lco studed the - ucs umbers d proved vrous propertes relted wth the - bocc umbers.

3 SUM PROPERTIES OR THE K-UCAS NUMBERS 07 Some of the terestg propertes tht the -bocc sequece stsfes re summrzed s below [] [] [4]:. Bet s formul: The Bet s formul for the -bocc umbers s (.) 4 where s the postve root of the chrcterstc equto ssocted to the recurrece relto defed (.). r r 0. Ctl s detty: ( ) r r r r (.). D Ocge Idetty: m m ( ) m 4. Covoluto product: m m m (.4) (.5) 5. Sum of the frst terms: (.6) 6. Sum of the frst eve terms: 7. Sum of the frst odd terms: (.7) (.8) 8. Geertg fucto: f (.9) I [] lco geertes the -ucs umbers from the -bocc umbers whose recurrece relto s gve below: Defto : or y teger umber the { } th ucs sequece sy N s gve by: for wth tl codtos 0 d. (.0)

4 08 BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA As prtculr cses: If we obt the ucs sequece{ } f we obt the Pell-ucs sequece { } f we obt the sequece { } N { }. ew terms of the -ucs umbers (.0) re PREIMINARIES: Now some bsc propertes of -ucs umbers re descrbe.. Bet s formul: 4 4. (.) where d These roots verfy. d 4. (See [] for the proof). m m m- (.) Proof: Applyg the Bet s formul o R. H. S. obt m m m m m m m m m m

5 SUM PROPERTIES OR THE K-UCAS NUMBERS 09 m m m m. m m.. [rom (.)] m m m m (.) Proof: Mthemtcl ducto method use for proof of ths. If 0 the m m m m 0 m. m. m m [rom (.)] m [ ] If the [rom (.)] m m m m m m m m m m m m [ ] Now suppose the formul s true utl ( ) : m m m The m m m [rom (.0)] m m m m m m m.

6 0 BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA. ON THE -UCAS NUMBERS O KIND + r : I ths secto some terestg formuls re derve for the sums of the -ucs umbers wth de rthmetc sequece sy r for fed teger d r such tht 0 r. Also descrbe geertg fucto for these umbers wth de rthmetc sequece. et us prove followg lemms tht we wll eed fter some tme: emm : or ll tegers (see [] for the detl of proof) (.). emm : r r r Proof: Applyg Bet s formul d emm o R. H. S. obt r r r r ( ) r r r r r r [rom (.)] r Sce the the bove formul c be rewrtte s. (.) r r r ormul (.) gves the geerl term of the -ucs sequece of the two precedg terms. r. GENERATING UNCTION OR THE SEQUENCE Suppose tht ( ) r Tht s 0 +r =0 l be the geertg fucto of the sequece r s ler combto : wth 0 r.

7 SUM PROPERTIES OR THE K-UCAS NUMBERS l ()... (.) r r r r r Now multplyg both sdes by the lgebrc epresso? ( ) ( ) ) obt l r r r r r r r r r r r r r r r r r Now from equto (.) the summto of rght hd sde of the bove equto vshes.tht s r r r r l (.4) rom (.) we hve r r r - r r r r r [rom (.)] r r r Hece equto (.4) becomes r l r r r l r r r r (.5) Prtculr cses:

8 BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA The geertg fucto of sequece r for vlues of d r re: ) If : l 0 0 whch s the geertg fucto of the -ucs sequece s t s gve []. ) If ) If ) At the l b) At the l ) At the l b) At the l c) At the l 0 0 ( ) SUM O -UCAS NUMBERS WITH ARITHMETIC INDEX : I ths secto preset d derve the sums of -ucs umbers wth rthmetc de r where d r re fed teger such tht 0 r. Theorem : Sum of -ucs umber of d r s ( ) ( ) (.6) r ( ) r r r r r 0 ( ) Proof: Applyg Bet s formul for the -ucs umbers we hve r r r r r

9 SUM PROPERTIES OR THE K-UCAS NUMBERS ( ) ( ) r ( ) r ( ) r r r r r r r r ( ) r r r r r r r r r ( ) ( ) ( ) ( ) ( ) ( ) ( ) r ( ) r r r r ( ) Corollry 4: ormul for sum of odd -ucs umbers: If the equto (.6) gves ( ) r (s)( ) r (s) r r (s) r (s) r 0 s or emple: () If d 0 0 ) If the t s sum formul for ucs sequece tht s [] 0 b) If the the sum formul for the Pell-ucs sequece tht s P P 0 P () If the ( ) r ( ) r r r r r 0

10 4 BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA ) If r the ( ) 0 0 If = the ( ) ( ) 0 4 b) If r the ( ) If the c) If r the ( ) If the If the ( ) r 5( ) r 5 r r 5r 5r 0 5 ) If b) If c) If d) If

11 SUM PROPERTIES OR THE K-UCAS NUMBERS 5 e) If 0 Corollry 5: Sum of eve -ucs umbers If the equto (.6) s ( ) r s( ) r s r r sr sr 0 s or emple: () If ( ) r ( ) r r r r r 0 ) If r : 0 or the ucs sequece t s b) If r : 0 0 or the ucs sequece t s () If 0 ( ) r 4( ) r 4 r r 4r 4r 0 4 ) If b) If c) If d) If

12 6 BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA Now we cosder the ltertg sequece {( ) }. By the prevous method we c lso r fd the sum formul for ths sequece. Moreover geertg fucto for ths ltertg sequece hs bee lso proved by the prevous method. Theorem 6: Altertg sum of the -ucs umbers wth de r s gve by: ( ) ( ) ( ) r ( ) r r r r ( ) r 0 ( ) or dfferet vlues of d r the bove sum c be wrtte s: () or r 0 d () or () or 0 ( ) 0 ) If r 0 the b) If r the 0 ( ) ) If r 0 the b) If r the c) If r the ( ) ( ) ( ) ( ) ( ) ( ) r ( ) r r r r r 0 0 ( ) ( ) 4 ( ) ( ) 4 4 ( ) ( ) 4 ( ) ( ) ( ) r ( ) r r r r r 0 0 ( ) 0 ( ) ( ) ( ) ( ) ( ) 4 ( ) ( ) 5 ( ) CONCUSION: The -ucs umbers re geerted by the - bocc umbers. I ths pper the sum propertes re preseted for - ucs umbers wth de rthmetc sequece. I ddtol geertg fucto d lterted sum formul for the - ucs umbers preseted d derved.

13 SUM PROPERTIES OR THE K-UCAS NUMBERS 7 ACKNOWEDGEMENTS: We would le to th the oymous referee for umerous helpful suggestos. Coflct of Iterests The uthor declres tht there s o coflct of terests. REERENCES [] lco S. d Plz A. O the bocc -umbers Chos Soltos d rctls (5) (007) [] lco S. O the -ucs umbers It. J. Cotemp. Mth. Sceces (6) (0) [] lco S. d Plz A. O -bocc umbers of rthmetc dees Appled Mthemtcs d Computto 08 (009) [4] lco S. d Plz A. The -bocc sequeces d the Pscl -trgle Chos Soltos d rctls () (007) [5] lco S. d Plz A. O - bocc sequeces d polyomls d ther dervtves Chos Soltos d rctls 9 (009) [6] lco S. d Plz A. Boml trsforms of the - bocc sequece Itertol Jourl of Noler Sceces & Numercl Smulto 0 (-) (009) [7] lco S. d Plz A. The -bocc hyperbolc fuctos Chos Soltos d rctls 8 () (008) [8] Hoggtt V. E. Jr. bocc d ucs Numbers Houghto Mffl Co. Bosto (969). [9] Hordm A.. A geerlzed bocc sequeces Mth. Mg. 68 (96) [0] Hordm A.. d Sho A. G.. Geerlzed bocc trples Amer. Mth. Moth. 80 (97) [] Klc E. The Bet formul sums d represetto of geerlzed bocc p-umbers Eur. J. Comb 9 () (008) 70-7.