International Journal of Advanced and Applied Sciences

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Itratioal Joural of Advacd ad Applid Scics x(x) xxxx Pags: xx xx Cotts lists availabl at Scic Gat Itratioal Joural of Advacd ad Applid Scics Joural hompag: http://wwwscic gatcom/ijaashtml Symmtric

mrada@yahoofr 3 Dpartmt of Ecoomics Faculty of Ecoomics Admiistrativ ad Social Scics Hasa Kalyocu Uivrsity TR-740 Gaiatp Tury; mtsr@hotmailcom 4 Dpartmt of Mathmatics Faculty of Arts ad ScicUivrsity

fuctios; Fiboacci polyomials A B S T R A C T I this papr w itroduc a w oprator i ordr to driv som w symmtric proprtis of -Fiboacci ad -Lucas umbrs ad Fiboacci polyomials By maig us of th w oprator

1 Itratioal Joural of Advacd ad Applid Scics x(x) xxxx Pags: xx xx Cotts lists availabl at Scic Gat Itratioal Joural of Advacd ad Applid Scics Joural hompag: gatcom/ijaashtml Symmtric Fuctios of th -Fiboacci ad -Lucas umbrs Ali Boussayoud M Krada Sra Araci 3* Mhmt Acigo 4 LMAM Laboratory ad Dpartmt of Mathmatics Mohamd Sddi B Yahia Uivrsity Jijl Algria; aboussayoud@yahoofr mrada@yahoofr 3 Dpartmt of Ecoomics Faculty of Ecoomics Admiistrativ ad Social Scics Hasa Kalyocu Uivrsity TR-740 Gaiatp Tury; mtsr@hotmailcom 4 Dpartmt of Mathmatics Faculty of Arts ad ScicUivrsity of Gaiatp TR-730 Gaiatp Tury; acigo@gatpdutr A R T I C L E I N F O Articl history: Rcivd April 5 07 Rcivd i rvisd form Ju 0 07 Accptd Ju 07 Availabl oli Kywords: -Fiboacci umbrs; -Lucas umbrs; Gratig fuctios; Fiboacci polyomials A B S T R A C T I this papr w itroduc a w oprator i ordr to driv som w symmtric proprtis of -Fiboacci ad -Lucas umbrs ad Fiboacci polyomials By maig us of th w oprator dfid i this papr w giv som w gratig fuctios for -Fiboacci ad Pll umbrs ad Fiboacci polyomials 04 IASE Publishr All rights rsrvd Itroductio ad Notatios Fiboacci umbrs ad thir graliatios hav may itrstig proprtis ad applicatios to almost vry fild of scic ad art (g s [7]) Fiboacci umbrs F ar dfid by th rcurrc rlatio * Corrspodig Author Addrss: mtsr@hotmailcom (S Araci) F0 F F F F Thr xist a lot of proprtis about Fiboacci umbrs I particular thr is a bautiful combiatorial idtity to Fiboacci umbrs [7] F i i 0 i () From () Filippoi [Phlypoi] itroducd th icomplt Fiboacci umbrs F ( s ) ad th icomplt Lucas umbrs L ( s ) Thy ar dfid by s j F ( s ) 0 s ; 0 j 0 j s j L ( s ) 0 s ; 0 j 0 j I [7] Djordjvic gav th icomplt gralid Fiboacci ad Lucas umbrs I [8] Djordjvic ad Srivastava dfid icomplt gralid Jacobsthal ad Jacobsthal-Lucas umbrs I [6] th authors dfi th icomplt Fiboacci ad Lucas umbrs O th othr had may ids of graliatios of Fiboacci umbrs hav b prstd i th litratur I particular a graliatio is th - Fiboacci Numbrs For ay positiv ral umbr th -Fiboacci squc say ( F ) is dfid rcurrtly by F 0 F F F F I [4] -Fiboacci umbrs wr foud by studyig th rcursiv applicatio of two gomtrical trasformatios usd i th four-triagl logst-dg (4TLE) partitio Ths umbrs hav b studid i svral paprs; s [4 5] For ay positiv ral umbr th -Lucas

2 ( ) L squc say is dfid rcurrtly by L 0 L L L L If w hav th classical Pll-Lucas umbrs appars: Q0 Q ad Q Q Q for [3] I this cotributio w shall dfi a w usful oprator dotd by for which w ca formulat xtd ad prov w rsults basd o our prvious os [3 5] I ordr to dtrmi w gratig fuctios of th products of som ow umbrs ad polyomails w combi btw our idicatd past tchiqus ad ths prstd polishig approachs Lt ad b two positiv itgr ad { a a a } ar st of giv variabls rcall [0] that th -th lmtary symmtric fuctio ( a a a ) ad th -th complt homogous symmtric fuctio h( a a a ) ar dfid rspctivly by i i i ( a a a ) a a a 0 ii i with i i i 0 or h ( a a a ) a a a 0 i i i ii i with i i i 0 First w st 0( a a a ) ad h ( a a a ) (by covtio) For or 0 0 w st ( a a a) 0 ad h ( a a a ) 0 Dfiitio Lt E { } a alphabt w dfi th symmtric fuctio S associatd with th alphabt E by with S ( ) h( ) S ( ) h ( ) 0 0 S ( ) h ( ) S ( ) h ( ) Dfiitio Lt A ad B b ay two alphabts th w giv S ( A B ) by th followig form: b B( b ) S ( A B ) () ( a ) aa 0 with th coditio S ( A B ) 0 for 0 (s []) Corollary3 Taig A 0 i () givs ( b ) S ( B ) bb 0 Furthr i th cas A 0 or 0 B w hav S A B S A S B ( ) ( ) ( ) Dfiitio4 Lt g b ay fuctio o th w cosidr th dividd diffrc oprator as th followig form g ( x x x x ) g ( x x x x ) i i i i x ( ) ix g i x i x i (s [9] ) Dfiitio5 [9] Th symmtriig oprator dfid by f ( ) f ( ) is ( f ) for all N Rmar6 Lt E { } a alphabt w hav h ( ) S ( ) ( ) Th Fiboacci Polyomials Not that if is a ral variabl x th F F ad thy corrspod to th Fiboacci x polyomials dfid by [4] if 0 F ( x ) x if xf( x ) F ( x ) if from whr th first Fiboacci polyomials ar F x F x x F 3 x x F 4 x x 3 x F 5 x x 4 3x F 6 x x 5 4x 3 3x F 7 x x 6 5x 4 6x F 8 x x 7 6x 5 0x 3 4x Ad from ths xprssio as for th umbrs w ca writ [4]:?? i i ( ) for 0 i 0 i F x x () -Fiboacci

3 Not that F (0) 0 ad x 0 is th oly ral root whil (0) with o ral roots Also for x F w obtai th lmts of th - Fiboacci umbrs By itratig rcurrc rlatio of formula () th followig proprty is straightforwardly dducd Propositio [6] For r holds: F ( x ) F ( x ) F ( x ) F ( x ) F ( x ) r ( r ) r ( r ) Propositio (Bit's formula) Th th Fiboacci polyomial may b writt as ( ) x x 4 F ( x) big Proof Not that th charactristic quatio for - Fiboacci polyomials is r x roots r ad x x 4 r 0 with r from whr Formula () is dducd Propositio3 [5] (Asymptotic bhaviour of th quotit of coscutiv trms) If x x 4 F th lim ( x ) F ( x ) As a cosquc th quotit btw two coscutiv trms of th -Fiboacci umbrs 3 { F; } {0 } tds to th positiv charactristic root For ach itgr is calld th th mtallic ratio [8]: Gold Ratio for ad Bro Ratio for 3 Propositio4 [4] (Hosbrgr's formula) for m itgrs it holds: F ( x ) F ( x ) F ( x ) F ( x ) F ( x ) m m m 3 O th Symmtric Fuctios of Som - Fiboacci Numbrs ad Fiboacci Polyomials I this part w ar ow i a positio to provid Thorm 33 Also w driv th w gratig fuctios of th products of som ow umbrs ad polyomails Dfiitio3 Th symmtriig oprator is dfid by [7] f ( ) f ( ) ( f ) for all ( ) ( ) Lmma3 [7] Lt E { } w dfi th oprator as follows: h ( ) f ( ) f ( ) f ( ) Thorm33 Lt E ad A b two alphabts rspctivly ad a a th w hav 0 aa a a a a aa h a a ( ) 0 for all Proof By applyig th oprator a f ( ) w hav aa 0 h a a ( ) f a a a ( ) 0 O th othr had (3) to th sris a a a a a a 0 0 a a a a 0 0 a a aa aa a a a 0 a a aa aa ( f ) h a a a h a a h a a h a a a 0 h a a 0 h a a h a a a ( ) 0 h a a a ( ) 0 This complts th proof W ow driv th w gratig fuctios of th products of som ow polyomials Idd w cosidr Thorm 33 i ordr to driv -Fiboacci umbrs ad Fiboacci polyomials if Thorm34 Lt E ad A b two alphabts

4 rspctivly 0 ad a a th w hav a a a a h ( ) h ( a a ) ( a a) aa h ( a a ) (3) I th cas aa a a aa A a basd o Thorm 34 w dduc th followig Lmmas Lmma35 Giv two alphabts E A a w hav ad a 0 a ( ) a (33) Lmma36 Giv two alphabts E A a w hav ad a 0 a ( ) a (34) Assumig that ad a i (33) ad (34) w obtai th gratig fuctios giv by Boussayoud t al [ 5] wich rprst: ) Th gratig fuctio of th Fiboacci umbrs F ) Th gratig fuctio of th Lucas umbrs L Choosig ad such that ad substitutig i (33) ad (34) w d up with [3] (35) 0 (36) 0 w dduc th followig thorm Thorm37 [] For th gratig fuctio of th -Fiboacci umbrs is giv by F 0 Multiplyig th quatio (35) by ( ) ad subtract it from (36) by ( ) w obtai 0 ad w hav th followig thorm Thorm38 [] For th gratig fuctio of th -Lucas umbrs is giv by L 0 Put i th rlatioship (37) w hav Q 0 which rprsts a gratig fuctio for Pll-Lucas umbrs [5] Choosig ad such that substitutig i (33) w d up with 0 x ad x x 4 with x Thus w gt th followig thorm Thorm39 W hav th followig a w gratig fuctio of th Fiboacci polyomials as F ( x ) x 0 For th cas E ad A a a with rplacig by a by a i (3) w hav h ( a [ a ]) h ( [ ]) 0 a a a a a a aa h( ) h( a a ) ( a a) aa (38) This cas cosists of thr rlatd parts Firstly th substitutios i (38) giv a a x ad aa x x ( x ) x h( a a ) h( ) F 0 F ( x ) From which w hav th followig thorm Thorm30 W hav th followig a w gratig fuctio of th product of -Fiboacci umbrs ad Fiboacci polyomials as x F F ( x ) (39) 3 4 x ( x ) x i th rlatioship (39) w hav x F F( x ) x ( x 3) x 0 Put which rprsts a w gratigs fuctios of th product of Fiboacci umbrs ad Fiboacci polyomials Put i th rlatioship (39) w hav 5x P F( x ) 3 4 x ( x 6) x 0 which rprsts a w gratigs fuctiosof th product of Pll umbrs ad Fiboacci polyomials Scodly by maig th followig rstrictios:

5 i (38) giv 0 0 a a ad aa h ( a a ) h ( ) ( ) 3 4 F F (30) rprstig a w gratig fuctio of - Fiboacci umbrs F O th othr had w cosidr ( ) 0 0 Sic 0 F F F F F 0 0 F F F F ( ) 3 4 w hav ( ) ( ) F F ( ) 3 4 (s 3 ) from thos applicatios w dduc th followig thorm Thorm3 W hav th followig a w gratig fuctio of th product of two coscutiv -Fiboacci umbrs as ( ) F F (3) ( ) Put i th rlatioship (30) ad (3) w obtai th followig rsults Corollary3 W hav th followig a w gratig fuctio of th product of Fiboacci umbrs as F F Corollary33 For th gratig fuctio of th product of two coscutiv Fiboacci umbrs is giv by F F 0 3 Put i th rlatioship (30) ad (3) w obtai th followig rsults Corollary34 W hav th followig a w gratig fuctio of th product of Pll umbrs as 54 P P 3 4 Corollary35 W hav th followig a w gratig fuctio of th product of two coscutiv Pll umbrs as 4 P P Coclusios I this papr w hav drivd w thorms i ordr to dtrmi gratig fuctios of -Fiboacci umbrs - Lucas umbrs ad Fiboacci polyomials Th drivd thorms ad Lmmas ar basd o symmtric fuctios ad products of ths umbrs ad polyomials Rfrcs AAbdrra Gééralisatio d la trasformatio d'eulr d'u séri formll Adv Math (994) A Boussayoud M Boulyr M Krada O Som Idtitis ad Symmtric Fuctios for Lucas ad Pll Numbrs ElctroJMathAalysisAppl (07) A Boussayoud N Harrouch Complt Symmtric Fuctios ad - Fiboacci Numbrs Commu Appl Aal (06) A Abdrra M Krada A Boussayoud Graliatio of Som Hadamard Product Commu Appl Aal (06) A Boussayoud M Krada M Boulyr A simpl ad accurat mthod for dtrmiatio of som gralid squc of umbrs It J Pur Appl Math (06) A Boussayoud A Abdrra M Krada Som applicatios of symmtric fuctios Itgrs 5 A#48-7 (05) A Boussayoud R Sahali Th applicatio of th oprator L j j bb i th sris j 0a jb J Adv Rs Appl Math (05) A Boussayoud M Krada R Sahali W Rouibah Som Applicatios o Gratig Fuctios J Cocr Appl Math (04) A Boussayoud M Krada A Abdrra A Graliatio of Som Orthogoal Polyomials Sprigr Proc Math Stat (03) AT Bjami JJ Qui Proofs That Rally Cout: Th Art of Combiatorial Proof Mathmatical Associatio of Amrica Washigto DC 003 AF Horadam Basic proprtis of a crtai gralid squc of umbrsfiboacci Q (965) AF Horadam Gratig fuctios for powrs of a crtai gralid squc of umbrs Du Math J (965) AF Horadam JM Maho Pll ad Pll-Lucas Polyomials Fiboacci Q (985) A Lascoux AM Fua Partitio aalysis ad symmtriig oprators J Comb Thory A 09

6 (005) C Bolat H Kos O th Proprtis of -Fiboacci Numbrs It J Cotmp Math Scics (00) D Tasci M Cti Firgi Icomplt Fiboacci ad Lucas p-umbrs Math Comput Modllig (00) G B Djordjvic Gratig fuctios of th icomplt gralid Fiboacci ad gralid Lucas umbrs Fiboacci Q (004) G B Djordjvic H M Srivastava Icomplt gralid Jacobsthal ad Jacobsthal-Lucas umbrs Math Comput Modllig (005) IG Macdoald Symmtric fuctios ad Hall polyomias Oxford Uivrsity Prss 979 M Mrca A Graliatio of th symmtry btw complt ad lmtary symmtric fuctios Idia J Pur Appl Math (04) P A Pao O th gratig fuctios of Mrs ad Frmat prims Collct Math (0) P Filippoi Icomplt Fiboacci ad Lucas umbrs Rd Circ Mat Palrmo (996) S Araci Novl idtitis ivolvig Gocchi umbrs ad polyomials arisig from applicatios of umbral calculus Appl Math Comput (04) S Falco A Plaa O th Fiboacci - umbrs Chaos Sulutios & Fractals (007) S Falco A Plaa Th - Fiboacci squc ad th Pascal -triagl Chaos Sulutios & Fractals (008) S Falco A Plaa O - Fiboacci squcs ad Polyomials ad thir drivativs Chaos Sulutios & Fractals (009) T Koshy Fiboacci ad Lucas Numbrs with Applicatios Wily-Itrscic 00 VW Spiadl Thmtallic mas family ad forbidd symmtris It Math J (00)

SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C

Joural of Mathatical Aalysis ISSN: 2217-3412, URL: www.ilirias.co/ja Volu 8 Issu 1 2017, Pags 156-163 SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C BURAK