Generalization of Horadam s Sequence

Transcription

1 Tuish Joual of Aalysis ad Nube Theoy 6 Vol No 3-7 Available olie at Sciece ad Educatio Publishig DOI:69/tjat---5 Geealizatio of Hoada s Sequece CN Phadte * YS Valaulia Depatet of Matheatics GVM s College of Coece & Ecooics Poda Goa 3 Idia Depatet of Matheatics Goa Uivesity Taleigao Plateau 36 Goa Idia *Coespodig autho: dbyte9@gailco Received July 8 6; Revised August 3 6; Accepted Septebe 6 Abstact I this pape a ew class of Fiboacci lie sequece is itoduced Hee we coside o-hoogeeous ecuece elatio to obtai geealizatio of Hoada s Sequece Soe idetities coceig this ew sequece ae obtaied ad poved Soe exaples ae give i suppot of the esults Keywods: pseudo fiboacci ubes o-hoogeeous ecuece elatio Cite This Aticle: CN Phadte ad YS Valaulia Geealizatio of Hoada s Sequece Tuish Joual of Aalysis ad Nube Theoy vol o (6: 3-7 doi: 69/tjat---5 Itoductio Fiboacci sequece is geealized i ay ways Oe of the ost widely used extesio is the oe give by AF Hoada [] He defied the geealized sequece of ubes as follows Let { W } be a sequece defied by W W( a b; p q pw qw ( fo whee p q a ad b ae iteges with W a ad W b Oe ca see that { W } educes to geealized Fiboacci sequece { U } whe a ad b ad subsequetly to the Fiboacci sequece { F } whe p q That is U W(; pq ad F W ( ; I a seies of papes [358] vaious popeties of the geealized sequece { W } have bee developed Usig Hoada Sequece ad the cocept of Pseudo Fiboacci sequece [A58] defied ealie i [6] ad studied futhe i [7] we ow defie a ew extesio of the sequece { W } Defiitio The geealized Pseudo Fiboacci (GPF Sequece { G } is defied as the sequece satisfyig the followig o-hoogeeous ecuece elatio ( G pg qg At fo A ad t α β with G a ad G b Hee ab pq ae iteges ad α βae distict oots of chaacteistic equatio x px + q of the coespodig hoogeeous equatio Fist few GPF ubes ae give below: G ag b G ( pb qa + A 3 G ( p b pqa qb + pa + At 3 G ( p b p qa pqb + q a + ( p q A + pat + At 5 3 G ( p b p qa 3p qb + q a + q b ( p pq A + ( p q pat + pat + At Obseve that each GPF ube G cosist of two pats The fist pat is a expessio i p q a ad b while the secod is a polyoial i t whose coefficiets ae A ties tes i p ad q This is show i the followig tables: Table Fist pat of Expessio i p q a b pb qa 3 p b pqa qb p b pqb p qa + q a G p b 3p qb + q b p qa + q a p b p qb p qa + 3q a + q Table Secod pat of G A At 3 p p At q p 3 At 5 3 p pq p q p Fo the above tables we have the followig elatio betwee GPF ad Hoada ubes W Theoe Fo the te G W( abpq ; + A ( ; pqt

2 Tuish Joual of Aalysis ad Nube Theoy of the sequece { G } satisfy the o-hoogeeous ecuece elatio G pg qg + At + + Poof Coside + + (; pqt Wt + Wt + Wt W+ Wt + ( pw qw t + ( pw qw t + + ( pw qw Wt + pwt ( + Wt + + W qw ( t + Wt + + W Wt + p (; p qt q ( ; p qt That is Now + + W (; pqt W(; pqt + p W ( ; pqt q W ( ; pqt + + G+ W+ ( abpq ; + A ( ; pqt Usig equatios ( ad (3 we wite G+ pw+ ( a b; p q qw( a b; p q + Ap (; p q t qa W ( ( pqt ; + AW ; pqt pw [ ( ab pq A t + ; + ] qw [ ( abpq ; + A ( ; pqt ] + At pg+ qg + At Hece the theoe Soe Idetities fo G (3 I this sectio we obtai soe fudaetal idetities fo GPF sequece { G } Biet type Foula: Let z z( t At pt + q The the Biet fo of G is give by whee α + β + ( G c c zt c c ( baβ z( tβ α β ( aα b z( α t α β ( αβ ae the distict oots of the equatio x px + q give by p + d p α β d (3 witig d p p Note that α + β p αβ q α β d ( We ca deduce fo ( that c+ c a z c c (( b ap z( t p d (5 cc ed whee c abp b a q z{ bp bt aq + atp + A} (6 Geeatig fuctio: Geeatig fuctio G ( x fo ( px qx G is give by Ax G ( x + ( a + bx apx + tx We have the followig esult fo su of fist GPF ubes Poofs follow fo ecuece elatio ad Biet foula Popositio 3 Fo pq ± i ii G ( pq G+ b+ ( p( ag A t + ( G ( p q+ [( G+ ( ( + ( ( ( ] + b p+ a+ G + A t t Usig the ecuece elatio ( we have the followig esult The sae ca also be obtaied by iductio Popositio a / t ap + b Gt t ( G qtg + ( pt + qt + + A t Poof Usig the ecuece elatio ( we have

3 Tuish Joual of Aalysis ad Nube Theoy 5 ie ie [ + ] 3 [ 3 + ] G t pg qg At t G t pg qg At t + G t [ pg qg At ] t O suig both sides of these ( equatios we get + + Gt pgt q Gt + A t Gt Gt G p Gt pg + Gt q Gt + qg ( t + Gt + A t Hece ( G t ( pt + qt G / t + G pg + + ( pg q( G t + qg t + A t Gt ( pt + qt ( + at ap + b t G+ qtg + A t We ow obtai the su of the squaes of GPFs ad su of the poduct of two cosecutive GPFs S X G Y GG + Gt S t v p q ad v + p q Fo siplicity let ad Let Futhe let ( [ + ] + ( + AGt Gt G+ t G+ t P p G G G G + ( + P ( + p [ G G+ ] + ( G G+ pg ( + G+ GG We have the followig esults: Popositio 5 Fo p v+ qv P+ qv + ( t q t AS i + ( q A S + qp X G v + ( t v + tqv AS ( v+ v A S p( v+ qv Y GG + v P vp ii Poof Coside G+ pg+ ( G+ pg+ ( G+ + pg+ AG+ pqgg+ + q G A t Hece suig up to + tes o both sides we get G+ p G A G+ + q G pq GG+ A t Adjustig the vaiables of suatio ad siplifyig we get vx + pqy P+ At S A S Siilaly statig with G+ qg pg+ G+ AqGt p G+ + A t ad siplifyig as above we get vx py P qats+ A S Solvig these two equatios fo X ad Y we get the equied esults Next esult deals with su of eve ad odd tes of GPF sequece Agai fo siplicity let E Gi ad i i O Gi OA A t E A A t so that E i A + OA A t Popositio 6 The su of the eve (odd idexed tes of { G } is give by (i G [ pg ( + G { p ( + q } + ( + q( qg + G+ pg+ pat ( + q t ] ad (ii G p + q { ( } [ pqg pg p G ( q( G G ( + q A t Ap t ]

4 6 Tuish Joual of Aalysis ad Nube Theoy Povided p + q Poof Fo the ecuece elatio ( we have + + pg G qg At Suig up to tes we get pg G+ + qg A t which o siplifyig yields pe ( + q O + G+ G OA (7 Siilaly usig the elatio + + pg G + qg At ad suig up to tes we get po ( + q E + G+ pg+ + qg EA (8 Solvig (7 ad (8 we get the equied esults We have followig idetity Popositio 7 Fo > G G qg G ( b zt G + + ( z a qg+ ( aq bt zt z Gt Gt Gqt Gqt [ + ] Poof Usig Biets foula ( cα + cβ + zt cα + cβ + zt α β α β LHS ( ( q( c + c + zt ( c + c + zt O siplificatio we get + baβ ztβ cα + + α α β + LHS [( ( ] [( a b z( t] c zt + zt G + zt G qz{ t G + t G zt } Sice α β d cd ( baβ zt ( β ad cd ( aα b z( α t GG qgg ( b zt G+ + ( z a qg+ + ( aq bt zt+ + z[ G t + Gt G qt G qt ] By lettig we have the followig esult Coollay 8 ( + ( ( aq bt zt [ z t G qt G ] G qg b z G z a qg + + Note that the above coollay alog with Popositio 3( i ca be used to fid G obtaied i Popositio 5( i Next we pove a vesio of Catala s Idetity fo GPF ubes Popositio 9 + zt t G t G+ G G G G eq u + [ + ] whee u W( p; pq ad e is as defied by (6 Poof Usig ( 3 LHS ( cα + cβ + zt ( cα + cβ + zt ( cα + cβ + zt + + cα + cβ + z t + cc ( α β + α β zt ( cα + cβ + zt ( cα + cβ { cα + cβ + zt + ( cc α β + zceta t + zct α } + cc α β ( α β + α β + zt G zt + zt G+ zt + ztg + zt cc ( αβ ( α β + zt [( t G + t G+ + G] ( α β eq + zt [( t G ] ( + t G+ + G αβ eq u + zt [( t G + t G+ + G] Fo this esult we iediately have a vesio of Cassii s idetity fo GPF ubes Coollay + + G G G eq + zt [ tg + t G G ] Next we have a expessio fo G i tes of bioial coefficiets Popositio G i ( q ( p q G i z [( pt q t ] i / Poof We have i q i p / q Gi z pt q t i i i i i α β RHS ( ( [( ] ( q ( p / q ( c + c + zt z[( pt q t ] i i i i ( α ( ( β ( i i i i ( ( [( ] i pt q z pt q t ( α ( β ( c p q + c p q + z c p q + c p q + z pt q z[( pt q t ] Sice α pα q ad β pβ q we get Hece α β RHS c + c + zt G G i ( q ( p q G i z [( pt q t ] i / which is the equied esult

5 Tuish Joual of Aalysis ad Nube Theoy 7 3 Exaples I this sectio we peset soe exaples i suppot of soe esults obtaied i Sectio Exaples: Coside G+ G+ G + ( with G G Hee p q t A Fist few tes of G ae G G G G 3 G G 5 3 G 6 6 G 7 G 8 G 9 3 G ( Veificatio of Popositio 5(i Whe 5 we have S S 6 P 8 P 6 v+ qv 8 The 5 LHS G G + G + + G5 3 RHS Result is veified ( Veificatio of Popositio 5 (ii Hee let 6 We have P 9 P S 7 S 7 The 6 LHS GG + GG + GG + + GG RHS Result is veified (3 Veificatio of Popositio 7 Let ad 3 The z LHS GG3 GG ( ( 6 RHS Result is veified ( Veificatio of Popositio 9 Let 6 ad z e u 8 G6 LHS GG 36 RHS 3 X ( Result is veified (5 Veificatio of Popositio Let 5 LHS G RHS ( q ( p q i i [( ] i / G z pt q t 3(83 / 3 / ( Result is veified Coclusio The well ow Hoada sequece is geealized via o hoogeeous ecuece elatio to obtai a Fiboacci lie sequece All the usual idetities ad popeties of Fiboacci lie sequeces ae obtaied fo the ew geealizatio of Hoada sequece Refeeces [] A F Hoada A Geealized Sequece of Nubes The Aeica Matheatical Mothly 68 No 5(96 pp55-59 [] A F Hoada Basic Popeties of a cetai Geealized Sequece of Nubes The Fiboacci Quately 3 No3(965 pp6-76 [3] A F Hoada Geeatig fuctios fo powe of a cetai Geealized Sequece of ubes Due Math J 3 No3(965 pp 37-6 [] A F Hoada Special Popeties of the Sequece W (ab;pq The Fiboacci Quately 5 No 5 (967 pp -3 [5] C N Phadte SP Pethe Geealizatio of the Fiboacci Sequece Applicatios of Fiboacci Nubes5 Kluwe Acadeic Pub [6] C N Phadte S P Pethe O Secod Ode No-Hoogeeous Recuece Relatio Aales Matheaticae et Ifoaticae vol (3 pp5- [7] C N Phadte Tigooetic Pseudo Fiboacci Sequece Notes o Nube Theoy ad Discete Matheatics No3 (5 pp7-76 [8] J E Walto A F Hoada Soe Aspect of Fiboacci Nubes The Fiboacci Quately