Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9 (Pt) Reseach Atcle O The Geeatg uctos of the Powes of the K-boacc Numbes Sego alco Uvesty of Las Palmas de Ga Caaa Depatmet of Mathematcs 07-Las Palmas de Ga Caaa Spa. *Coespodg autho Sego alco Emal: Abstact: I ths pape we wll toduce the boacc factoal fucto! ad the boomal umbes. The we peset the geeatg fucto of the powes ofthe boacc umbes. ally we study the umeatos ad the deomatos of these fuctos. As cosequece of ths study we fd out seveal tege sequeces some of the mae lsted the Ole Ecyclopeda of Itege Sequeces (OEIS) ad we toduce some moe. Keywods: boacc umbes boomal umbes Lucas umbes Geeatg fucto. MSC000: A6; C0; B9 INTRODUCTION -boacc umbes [] ae defed as the sequece 0 0 ad fo The ecuece elato of ths fomula s. The Bet fomula fo these umbes s () m m m. such that 0 whose solutos ae ad the covoluto fomula s om the defto of the -boacc umbes the fsts of them ae peseted Table. ad Table : st -boacc umbes 0 0 6 So the fst -boacc sequeces ae A0000 [] OEIS fom ow.... 08... that s the classcal boacc sequece 669
Sego alco. Sch. J. Eg. Tech. 0; (C):669-67 P 0 970... the Pell sequece A0009 OEIS. 000960... A00690 OEIS etc. I the same way the Lucas umbes [] ae defed by mea of the fomula L L L wth the tal codtos L 0 ad L. The the fst Lucas umbes ae peseted the followg table: Table : st -boacc umbes L L L 0 L L L o = we obta the classcal Lucas umbes LL 78... A0000 ad fo = the sequece PL L 68... A000 s the Pell-Lucas sequece. boacc umbes ad Lucas umbes ae elated by mea of the fomula L K-IBONOMIAL NUMBERS I the developmet of ths atcle we have followed the deas developed [7]. We toduce the boacc factoal! as the poduct of the boacc umbes fom dow to. That s! o example! 09 0 0790 Assume! 0 beg the empty poduct evaluates to [6]. We defe a ew fom of bomal umbes but usg ths defto of factoal stead.! We defe the boomal umbes as boomal( )!! whee ad ae o-egatve teges 0. We wll suppose 0. Tag to accout [] we deduce all boomal umbe s tege. om the defto ad tag to accout! 0 t s easy to pove 0 ad We ca put the boomal umbes tagula fom as follows: 670
Sego alco. Sch. J. Eg. Tech. 0; (C):669-67 Table : The boomal tagle = 0 = = = = I patcula fo = we obta the followg table. Table : boomal tagle o boomal tagle = 0 = = = = 6 = = 6 8 0 60 0 8 Dagoal sequeces ae lsted OEIS. The sequece of ow sums of boomal tagle s A0669. o = we have Table : boomal tagle o Pelloomal tagle = 0 = = = = 0 = 9 7 7 9 = 6 70 0 6 0 70 6... Also these dagoal sequeces ad the sequece of ow sums of Pelloomal tagle 6 08 608... A 09998 ae lsted OEIS. Some popetes of the -boomal umbes I the sequel we peset some popetes of the -boomal umbes Symmety popety: It s obvous. Poof. LHS RHS!! Ths popety ca also be wtte the fom I smla fom we ca pove 67
Sego alco. Sch. J. Eg. Tech. 0; (C):669-67 Addto fomula: ( a) Poof. ( b) ( a) ( b)! sce fom omula (). o stace: 7 6 6 70 0 9 A GENERATING UNCTION OR THE POWERS O THE K IBONACCI NUMBERS ollowg the pocess used [] to fd the geeatg fucto of the boacc umbes we ca also fd the geeatg fucto of the atual powes of these umbes. As a esult we wll see that thee s a elatoshp betwee these geeatg fuctos ad the boomals umbes. Hee s a table summasg these esults whee gf s the geeatg fucto of the th powe of the boacc umbes (wthout 0 0 ): gf x x x gf x x x gf gf x x x x x x ( ) x ( ) x x x x x x x gf wth ( ) x ( )( ) x ( ) x x 6 6 6 6 6 x x x x x x 6 67
Sego alco. Sch. J. Eg. Tech. 0; (C):669-67 6 7 6 6 6 6 8 6 7 6 6 9 7 8 6 o stace fo = the geeatg fuctos of the fst powes of the classcal boacc umbes ae: gf [ ] x x x gf [ ] x x x xx gf [ ] x 6x x x x x x gf [ ] x x x x x Ad the geeatg fuctos of the fst powes of the Pell umbes ae gf [ P ] x x x gf [ P ] x x x xx gf [ P ] x 0x x x x x x gf [ P ] 9x 7x 7x 9x x Study of the deomatos of the geeatg fuctos Wth help of MATHEMATICA below s the polyomal factozato of the deomatos of the above geeatg fuctos whee D ae the cosecutve deomatos: D x x D x ( ) x ( x ) D x ( ) x x x D x x x x x D x x x x x x We ca loo at the coeffcets of these polyomals wthout the sgs ad accodg to Table ae the Lucas umbes. Cosequetly we ca wte accodg the subscpts ae odd o eve D x L x ( ) 67
Sego alco. Sch. J. Eg. Tech. 0; (C):669-67 ( ) D x L x x L x D x L x x L x x L x D x L x x L x x L x x L x 7 7 ad D x L x ( x ) ( ) ( ) D x L x x L x x D x L x x L x x L x x 6 6 D x L x x L x x L x x L x ( x ) 8 8 6 Ad fally we ca summazg these elatos the fomulas D ( ) x ( ) L x ( ) 0 D ( x ( ) ) x ( ) L x ( ) 0 Study of the omatos of the geeatg fuctos Below s a table wth the coeffcets of the umeatos of the successve geeatg fuctos (wthout the sgs): Table 6: Coeffcets of the umeatos of the geeatg fuctos gf 0 6 8 We wll dcate as T the elemets of ths table whee s the ow ad s the umbe of ode of ths elemet ths ow fo = 0 ad = 0. Wth the codtos T 0 T ad T 0 fo > these elemets vefy the elato T T T () o stace the elemets of the ffth ow of Table 6 vefy the elato T 6 T T ad so ts elemets ae T T T T T 0 T 6 6 8 0 T T T T 6 0 6 omula () s smla that fomula used the costucto of the classcal Pascal tagle T T T deduced fom the elato Symmety popety of the coeffcets T The elemets T vefy the symmety popety T T that we wll pove by ducto. o = ths elato s poved Table 6. Let us suppose ths elato s tue utl that s T T 67
Sego alco. Sch. J. Eg. Tech. 0; (C):669-67 T T T ad The ( ) ( ) T T T T T T ( ) ( ) So the successve umeatos of the geeatg fuctos gf ae of the fom N ( ) T x I patcula fo = ad = we obta the espectve tagle Table 7: Coeffcets of the umeatos of the Geeatg ucto of = 0 = = = = 7 6 7 = Oly the fst fve dagoal sequeces ad the ow sums { 0 } ae lsted OEIS. The sxth dagoal sequece s { 0 9 988 60 6606 96969 6766 76900788 07770 } o the Pell umbes we obta the followg table: Table 8: Coeffcets of the umeatos of the Geeatg ucto of = 0 = = = = 8 0 8 = 0 06 06 0 Oly the fst dagoal sequece { 8 0 } s lsted OEIS. The followg fou dagoal sequeces beg as follows: { 0 06 87 67 76 9908 98 8006 } { 8 06 6 670 79686 0776876 699776 668807 } { 0 87 670 89777 76666 8767988 9879890 } { 80 67 79686 76666 880 608680 } CONCLUSIONS We have defed the boomal umbes a smla fom to the Bomal umbes ad obta fomulas fo geeatg these umbes fom the Geeatg fucto. REERENCES. alco S; O the Lucas umbes. It. J. Cotemp. Math. Sceces 0; 6():09 00.. alco S Plaza A; O the boacc umbes. Chaos. Solt. &act. 007; ():6 6.. alco S Plaza A; The boacc sequece ad the Pascal -tagle. Chaos Solt. &act. 007; ():8-9.. alco S Plaza A; boacc sequeces modulo m Chaos Solt. &act. 009; ():97 0.. Avalable ole at http://oes.og/ The Ole Ecyclopeda of Itege Sequeces. 6. Avalable ole at http://e.wpeda.og/w/boomal_coeffcet 7. Avalable ole at http://www.maths.suey.ac.u/hosted stes/r.kott/boacc/boomals.html 0 P 67