Generalized Fibonacci-Type Sequence and its Properties

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Geelized Fibocci-Type Sequece d is Popeies Ompsh Sihwl shw Vys Devshi Tuoil Keshv Kuj Mdsu (MP Idi Resech Schol Fculy of Sciece Pcific Acdemy of Highe Educio d Resech Uivesiy Udipu (Rj Absc: The Fibocci sequece is souce of my ice d ieesig ideiies Fibocci sequece is fmous fo possessig wodeful d mzig popeies The Fibocci sequece hs bee geelized i my wys some by pesevig e iiil codiios d oes by pesevig e ecuece elio I is ppe geelized Fibocci-Type sequece is ioduced d defied by 0 d b whee d b e iege Fue some sdd ideiies d deemi ideiies of geelized Fibocci sequece e peseed Keywods: Geelized Fibocci sequece Geelized Fibocci-Lie sequece Geeig fucio Bie s Fomul Ioducio I is well-ow e Fibocci d Lucs sequeces e mos pomie exmple of secod ode ecusive sequeces The Fibocci sequece is sequece of umbes sig wi iege 0 d whee ech ex em of e sequece clculed s e sum of e pevious wo The Fibocci sequece [] is defied by e ecuece elio: F F F Wi F 0 0 F ( The simil iepeio lso exiss fo Lucs sequece Lucs sequece [] is defied by e ecuece elio: L L L Wi L 0 0 L ( The secod ode ecuece sequece hs bee geelized i wo wys mily fis by pesevig e iiil codiios d secod by pesevig e ecuece elio Hodm [] defied geelized Fibocci sequece H by H H H H q d H 0 p 0 whee pqe biy ieges ( I [] Hodm ioduced d sudied popeies of oe geelized Fibocci sequece w d defied geelized Fibocci sequece w by e ecuece elio: w pw qw wi w 0 biy ieges (4 w b whee b p d q e Wddill d Scs [7] exeded e Fibocci umbes ecuece elio d defied e sequece P by e ecuece elio: P P P P (5 Whee P P d P e o ll zeo give biy lgebic ieges Jiswl [5] defied geelized Fibocci sequece T by T T T Wi T Flco d Plz [8] ioduced F N d defied i by FF F d T b (6 Fibocci sequece Wi F 0 F 0 (7 Volume 5 Issue Decembe 06 wwwijse Licesed Ude Ceive Commos Aibuio CC B Edso d yeie [6] sudied ew geelizio q wi iiil codiio q 0 0 d q which is geeed by e ecuece elio (whe is eve o q bq q (whe is odd whee e o zeo el umbes Bsigh SBhg d OSihwl [4] defied Fibocci- Lie sequece S by e ecuece elio: S S S 0 Wi S S The ssocied iiil codiios S0 d S e e sum of e Fibocci d Lucs sequeces especively ie S0 F0 L0 d S F L (8 I is ppe we ioduce geelized Fibocci-Type sequece Also we esblish some of e ieesig popeies of geelized Fibocci-Type sequece lie Cl s ideiy Cssii s ideiy d ocges s ideiy Bies fomul Geeig fucio d some deemi ideiies Geelized Fibocci-Type Sequece Geelized Fibocci-Type sequece 0 is ioduced d defied by e ecuece elio 0 wi 0 d b ( Whee d b e ieges The fis few ems of e s follows: b b 4 b b 4 6bb 5 8 b b 6 b The chceisic equio of ecuece elio ( is 0 Which hs wo el oos: 4 4 d Ppe ID: ART06767 DOI: 075/ART06767 04

Also 4 ( Geeig fucio of geelized Fibocci-Type sequece is b ( 0 Hypegeomeic epeseio of geeig fucio of geelized Fibocci-Type sequece b e F ; ; (4 0! By geeig fucio ( we hve b 0 b b 0 b 0 0! b 0 0 b! 0 0 b 0! 0 be 0 0 0 be!! b e (! 0 0 Hece b e F ; ; 0! Bie s fomul of geelized Fibocci-Type sequece is defied by 4 4q A B A + B Hee Also Ad ( b ( b A d B 4 4 4 b ( b AB A B b A B b (5 Geelized Fibocci-Type sequece geelizes my sequeces fo diffee vlues of d b Exmples of such sequeces e Lucs sequece Fibocci-Lie sequeces Jcobsl-Lucs sequece compio Fibocci-Lie sequece ec Fo d b we obi Lucs sequece Fo d b 0 we obi Fibocci-Lie sequece Fo d b we obi Jcobsl-Lucs sequece Fo d b 0 we obi Compio Fibocci-Lie sequece Volume 5 Issue Decembe 06 wwwijse Licesed Ude Ceive Commos Aibuio CC B Fo d b we obi Compio Fibocci- Lie sequece Ideiies of Geelized Fibocci-Type Sequece Now some ideiies of geelized Fibocci-Type sequece e pese usig geeig fucio d Bie s fomul Theoem (Explici Sum Fomul Le be e of geelized Fibocci-Type sequece The b ( 0 0 Poof By geeig fucio ( we hve b 0 b b 0 b 0 0! b 0 0 b! 0 0 em b 0 0 b 0 0 0 0 Equig e coefficie of o bo sides we obi b 0 0 Theoem (Sum of Fis ems sum of fis ems of geelized Fibocci-Type sequece is give by b ( 0 Poof Usig e Bie s fomul (5 we hve A B A B 0 0 ( AB ( A B ( A B ( A B ( Usig subseque esuls of Bie s fomul we ge b 0 Theoem (Sum of Fis ems wi odd idices Sum of fis ems (wi odd idices of geelized Fibocci- Type sequece is give by b b ( Ppe ID: ART06767 DOI: 075/ART06767 044 0

Poof Usig e Bie s fomul (5 we hve ( ( A B A B 0 0 ( A B ( A B ( A B ( ( A B ( ( Usig subseque esuls of Bie s fomul we ge b b 0 Theoem 4 (Sum of Fis ems wi eve idices Sum of fis ems (wi eve idices of geelized Fibocci- Type sequece is give by b (4 o Poof Usig e Bie s fomul (5 we hve ( ( A B A B 0 0 ( AB ( A B ( A B ( ( A B ( ( Usig subseque esuls of Bie s fomul we ge b o Theoem 5 (Cl s Ideiy Le be e em of geelized Fibocci-Type sequece The b b b 4 (5 Poof Usig e Bie s fomul (5 we hve (LHS = A B A B A B AB AB b 4 b b 4 b b Fom we obi b b 4 b b b 4 Cooll 6 (Cssii s Ideiy Le be e em of geelized Fibocci-Type sequece The b b 4 (6 Tig i e Cl s ideiy (5 e equied ideiy is obied Theoem 7 (d Ocge s Ideiy Le be e em of geelized Fibocci-Type sequece The m m bm m m 0 (7 Poof Usig e Bie s fomul (5 we hve b 4 b m m (LHS = A B A B A B A B AB m m b mm Fom we AB b b 4 m m m m m m m m m m m m m m AB obi m m bm m Theoem 8 (Geelized Ideiy Le be e em of geelized Fibocci-Type sequece The m b b mm m m m b b 4 (8 Poof Usig e Bie s fomul (5 we hve b 4 b (LHS = A m B m A B A m B m A B m m m m m AB m m AB m m m AB Fom Ad obi b b b 4 b m we b b 4 m m m Volume 5 Issue Decembe 06 wwwijse Licesed Ude Ceive Commos Aibuio CC B Ppe ID: ART06767 DOI: 075/ART06767 045

m b bmm m m b b 4 The ideiy (8 povides Cl s Cssii s d d Ocge s ideiies s below: If m e Cl s ideiy (5 is obied If m d e Cssii s ideiy (6 is obied If m m d d Ocge s ideiy (7 is obied 4 Deemi Ideiies Thee is log diio of usig mices d deemis o sudy Fibocci umbes T Koshy [9] explied wo chpes o e use of mices d deemis i Fibocci umbes I is secio some deemi ideiies e peseed Theoem : Fo y ieges 0 pove 4 5 6 0 7 8 9 4 5 6 Poof Le Applyig C C we ge 7 8 9 4 5 6 7 8 9 Applyig CCC d usig e defiiio of geelized Fibocci-Type ecuece elio we ge 6 5 6 9 8 9 Sice wo colums e ideicl us we obied equied esuls Theoem Fo y ieges 0 pove =0 Poof Le Le By pplyig C C C C d expdig log fis ow we obied equied esul Theoem Fo y ieges 0 pove 0 Poof Le Applyig R R R we ge Tig commo ou fom id ow we ge Sice wo ows e ideicl us we obied equied esuls Theoem 4 Fo y ieges 0 pove 4 6 6 Poof Le 4 6 6 Applyig R R R R R R we ge 0 0 9 Applyig R R R d expdig log fis owwe obied equied esul Theoem 5 Fo y ieges 0 pove 0 0 0 0 Poof Le 0 0 Tig commo fom C C C especively we ge 0 = 0 0 Volume 5 Issue Decembe 06 wwwijse Licesed Ude Ceive Commos Aibuio CC B Ppe ID: ART06767 DOI: 075/ART06767 046

Now ig commo fom R R R especively d expdig log fis ow we ge equied esul 5 Coclusios I is ppe geelized Fibocci-Type sequece is ioduced Some sdd ideiies of geelized Fibocci-Type sequece hve bee obied d deived usig geeig fucio d Bie s fomul Also some deemi ideiies hve bee esblished d deived 6 Acowledgemes The uos e ful o e eviewes fo ei cosucive suggesios d commes fo impovig e exposiio of e oigil vesio Refeeces []AF Hodm: A Geelized Fibocci Sequece Ameic Memicl Moly 455-459 (5 96 Vol 68 []AF Hodm: Bsic Popeies Of Cei Geelized Sequece Of Numbes The Fibocci Quely 6-76 ( 965 Vol []B Sigh O Sihwl d SBhg: Fibocci- Lie Sequece d i s Popeies I J Coemp MSciece 859-868 (8 00 Vol 5 [4]B Sigh S Bhg O Sihwl: Geelized Fibocci Sequece d i s Popeies Ope Joul Of Memicl Modelig 94-0(6 0 [5]D V Jiswl: ON Geelized Fibocci sequece Lbdev J Sci Tech P A 7 67-7 969 [6]M Edso d Oyeie: A New Geelizio Of Fibocci Sequece d Exeded Bies Fomul Ieges 69-654 009 Vol 9 [7]M E Wddill d L Scs: Aoe Geelized Fibocci Sequece The Fibocci Quely 09- ( 967 Vol 5 [8]S Flco d A Plz: O e Fibocci -Numbes Chos Soluios $ Fcls 65-64 (5 007 Vol [9]T Koshy Fibocci d Lucs Numbes wi Applicios Wiley 00 Volume 5 Issue Decembe 06 wwwijse Licesed Ude Ceive Commos Aibuio CC B Ppe ID: ART06767 DOI: 075/ART06767 047

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