Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Some Popeties of the K-Jacobsthal Lucas Sequece Deepia Jhala School of studies i Mathematics Viam Uivesity Uai Idia hala.deepia8@gmail.com G.P.S. Rathoe Depatmet of Mathematical Scieces College of Hoticultue Madsau Idia gps_athoe0@yahoo.co.i Abstact I this pape we peset the sequece of the -Jacobsthal-Lucas umbes that geealizes the Jacobsthal-Lucas sequece. We establish a explicit fomula fo the tem of ode the well-ow the Biet s fomula fo - Jacobsthal-Lucas umbes ad obtai some popeties ad givig the geeatig fuctio fo the -Jacobsthal- Lucas sequeces. Keywods: Jacobsthal umbes Jacobsthal-Lucas umbes - Jacobsthal umbes - Jacobsthal Lucas umbes Biet s fomula Geeatig Fuctios.. INTRODUCTION: The Fiboacci sequece is a iexhaustible souce of may iteestig idetities. It is oe of the most famous umeical sequeces i mathematics ad costitutes a itege sequece. Fiboacci umbes ae a popula topic fo mathematical eichmet ad populaizatio. The Fiboacci sequece is famous fo possessig wodeful ad amazig popeties. The Fiboacci appea i umeous mathematical poblems. Fiboacci composed a umbe text i which he did impotat wo i umbe theoy ad the solutio of algebaic equatios. The boo fo which he is most famous is the Libe abaci published i 0. Falco ad Plaza [4] show the elatio betwee the 4TLE patitio ad the Fiboacci umbes as aothe example of the elatio betwee geomety ad umbes. The use of the cocept of atecedet of a tiagle is used to deduce a pai of complex vaiable fuctios. These fuctios i matix fom allow us too diectly ad i a easy way peset may of the basic popeties of some of the best ow ecusive itege sequeces lie the Fiboacci umbes ad the Pell umbes. Dodevic [5] itoduce ad ivestigate some popeties ad elatios ivolvig sequeces of umbes F () fo m ; 3; 4 ad is some eal umbe. These sequeces ae geealizatios of the Jacobsthal ad Jacobsthal Lucas umbes. Auuma Kaa ad Siath [8] detemie a elatio betwee Jacobsthal umbe ad pime Jacobsthal umbe fo twi pime umbes ad a iequality is foud betwee Fiboacci Jacobsthal umbes. Koe ad Bozut [3] deduce some popeties ad Biet lie fomula fo the Jacobsthal umbe by matix method. Falco ad Plaza [4 7] itoduced a ew geealizatio of the classical Fiboacci sequece ad Catio [6] itoduced geealizatio of Pell sequece. It should be oted that the ecuece fomula of these umbes depeds o oe eal paamete. Jhala Sisodiya ad Rathoe [] defie the -Jacobsthal umbe i a explicit way ad may popeties ae poved by easy agumets fo the -Jacobsthal umbe. Hoadam [] defie the. J ad the Jacobsthal-Lucas sequeces { } The Jacobsthal sequece { J } is defie ecuetly by Jacobsthal { } J J J- ; fo³ with iitial coditio J0 0 J. (.) The Jacobsthal-Lucas sequece { } is defie ecuetly by - ; fo³ with iitial coditio 0. (.) m IJMSET-Advaced Scietific Reseach Foum (ASRF) All Rights Reseved ASRF pomotes eseach atue Reseach atue eiches the wold s futue 87
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Jhala [] defie the -Jacobsthal sequece { J } is defie ecuetly by J J J - ; fo³ with iitial coditio J0 0 J. (.3) I this pape we focused o -Jacobsthal-Lucas umbe ad obtai Biet fomula fo the - Jacobsthal-Lucas umbe. Moeove we deived some idetities ad geeatig fuctio fo the - Jacobsthal-Lucas.. -JACOBSTHAL LUCAS NUMBER: Î N Fo ay positive eal umbe the -Jacobsthal-Lucas sequece say { } by - IJMSET-Advaced Scietific Reseach Foum (ASRF) All Rights Reseved ASRF pomotes eseach atue Reseach atue eiches the wold s futue is defied ecuetly ; fo ³ (.) with iitial coditio 0. (.) Next we fid the explicit fomula fo the tem of ode of the -Jacobsthal-Lucas sequece usig the well-ow esults ivolvig ecusive ecueces. Coside the followig chaacteistic equatio associated to the ecuece elatio (.) - - 0 (.3) has two distict oots ad 8-8 beig ad whee is a eal positive umbe. Sice > 0 the < 0 < ad <.. Also we obtai - ad -. 8 3. PROPERTIES OF K-JACOBSTHAL LUCAS NUMBER: 3. Biet s Fomula I the 9th cetuy the Fech mathematicia Biet deived two emaable aalytical fomulas fo the Fiboacci ad Lucas umbes [9]. I ou case Biet s fomula allows us to expess the - Jacobsthal Lucas umbes i fuctio of the oots & of the followig chaacteistic equatio associated to the ecuece elatio (.) - - 0 Theoem : (Biet s Fomula) The th - Jacobsthal-Lucas umbe is give by. (3.) whee ae the oots of the chaacteistic equatio (.3) ad >. Poof: Sice the chaacteistic equatio has two distict oots the sequece c c (3.) is the solutio of the equatio (.). Givig to the values 0 ad ad solvig this system of liea equatios we obtai a uique value fo c ad c. So we get the followig distict values c ad c. Now usig (3.) we obtai (3.) as equied. 3.Covolutio Poduct Idetity Theoem : (Covolutio poduct idetity) J J (3.3) m m m - Poof: Taig R.H.S. ad applyig the Biet s fomula fo both -Jacobsthal umbes [] ad - Jacobsthal Lucas umbes (3.). 88
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; m m m- m- æ - æ - Jm Jm - ( ) ç ( ) ç è - ø è - ø {( )( m m ) ( )( m- m- )} - - - m m m m - - - - { ( - ) ( - ) ( - ) ( )} m m - { ( ) ( )} - - - ( m m ) m ( Q. -) [Fom (3.)] J J That is m m m - 3.3 Summatio Fomula Theoem 3 (Summatio fomula) Summatio fomula fo -Jacobsthal Lucas umbe is give by --4 i (3.4) i Poof: Usig the Biet fomula (3.) ad the fact that - we get i i i ( ) i i - - i i - - - - - - (-)(- ) -( ) ( ) ( )- (- - ) --4 3.4 Relatio Betwee The K-Jacobsthal Ad K-Jacobsthal Lucas Numbe Theoem 4 (Fist Relatio) ( ) 8 J 4( - ) (3.5) Poof: Taig Right had side ad usig the Biet fomula (3.) IJMSET-Advaced Scietific Reseach Foum (ASRF) All Rights Reseved ASRF pomotes eseach atue Reseach atue eiches the wold s futue 89
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; æ - è - ø - 4 ( 8) J 4( - ) ( 8) ç 4( -) That is ( ) ( ) 8 J 4( - ) Theoem 5 (Secod Relatio) Betwee the -Jacobsthal Lucas umbe ad the -Jacobsthal umbe it is veified - J J J J fo ³ (3.6) Poof: By iductio if the 0 Fomula (3.6) is tue fo let us suppose thal fomula is tue util - the - J -3 J -üï ý - J - J ïþ So - - That is ( - ) ( - - ) ( J - J -3) ( J J -) J J J J [Fom (.)] 3 J - J J J - Theoem 6 (Thid Relatio) Fo Î N J J (3.8) Poof: we ow that covolutio poduct idetity fo -Jacobsthal umbe is give by Now let m 3.5 Catala s Idetity Theoem 7: (Catala s idetity) J J J J J m m m - J J J J J - ( J J -) J ( J J - )( J J - ) - J [Fom (.3) & (3.6)] IJMSET-Advaced Scietific Reseach Foum (ASRF) All Rights Reseved ASRF pomotes eseach atue Reseach atue eiches the wold s futue (3.7) 90
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; { } - ( ) 4( ) - - - - - Poof: Usig the Biet fomula (3.) ad the fact that - we obtai the idetity (3.9) (3.9). 3.6 Cassii s Idetity Theoem 8: (Cassii s idetity) - ( ) ( ) - - - 8 (3.0) Poof: fo i Catala s idetity we obtai the Cassii s idetity fo the -Jacobsthal-Lucas sequece. 3.7 d Ocage s Idetity Theoem 9: (d Ocage s idetity) m- ì ü ï æ 8 ï If m > the m - m (- ) 8ím -- ý (3.) ç ïî è ø ïþ Poof: Oce moe usig the Biet s fomula (3.) the fact that - ad m> we obtai the idetity (3.). 3.8 Geeatig Fuctio Fo The -Jacobsthal-Lucas Numbe Next we shall give the geeatig fuctios fo the -Jacobsthal-Lucas sequece. We will show that - Jacobsthal-Lucas sequece ca be cosideed as the coefficiets of the powe seies of the coespodig geeatig fuctio. Theoem 8: Geeatig fuctio fo -Jacobsthal Lucas umbe is give by - x Á ( x) -x-x ( ) IJMSET-Advaced Scietific Reseach Foum (ASRF) All Rights Reseved ASRF pomotes eseach atue Reseach atue eiches the wold s futue (3.) Poof: Let us suppose that the -Jacobsthal-Lucas umbes of ode ae the coefficiets of a powe seies ceteed at the oigi ad let us coside the coespodig aalytic fuctio Á ( x) defied by Á ( x) x x x... x (3.3) 3 0 3 ad called the geeatig fuctio of the -Jacobsthal-Lucas umbes. Usig the iitial coditios (.) we get Á ( x) x x (3.4) Now fom (.) we ca wite (3.4) as follows Á ( ) ( ) x x - - x (3.5) Coside the ight side of the equatio (3.5) ad doig some calculatios we obtai that - - - - - - x x - x x -x x ( ) x x x x (3.6) 9
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Coside that - ad p -. The (3.6) ca be witte by æ p x xç p x 0 x x è p 0 ø 0 p - x x x x x p p 0 0 Theefoe Á ( x) - x x Á ( x) x Á ( x) which is equivalet to Á( x) ( -x - x ) -x ad the the odiay geeatig fuctio of the-jacobsthal-lucas sequece ca be witte as - x Á ( x) -x-x ( ) 4. CONCLUSION: I this pape -Jacobsthal Lucas sequece is itoduced. Some stadad idetities of -Jacobsthal Lucas sequece have bee obtaied ad deived usig geeatig fuctio ad Biet s fomula. 5. ACKNOWLEDGEMENTS: The authos ae thaful to the eviewes fo thei valuable suggestios to ehace the quality of ou aticle. 6. REFERENCES:. A. F. Hoadam Jacobsthal Repesetatio Numbe The Fiboacci Quately (996) 34() 40-54.. D. Jhala K. Sisodiya G. P. S. Rathoe O Some Idetities fo -Jacobstha Numbes It. Joual of Math. Aalysis 7() 03 55-556. 3. F. Koe D. Bozut O the Jacobsthal umbes by matix methods It. J. Cotemp. Math. Scieces 30(3) 008 605-64. 4. Falco S. ad Plaza A. O the Fiboacci -umbes Chaos Solitos ad Factals 3 (5) 007 65-64. 5. G. B. Dodevic Some Geealizatios of the Jacobsthal Numbes Faculty of Scieces ad Mathematics Uivesity of Ni s Sebia 4() 00 43-5. 6. P. Cataio O some idetities ad geeatig fuctios fo -Pell umbes It. Joual of Math. Aalysis 7(38) 03 877-884. 7. S. Falco Plaza A. The -Fiboacci sequece ad the Pascal -tiagle Chaos Solitos & Factals 33 () 007 38-49. 8. S. Auuma V. Kaa ad R. Siath Relatios o Jacobsthal umbes Notes o Numbe Theoy ad Discete Mathematics 9(3) 03 3. 9. S. Vada Fiboacci ad Lucas umbes ad the golde sectio Theoy ad applicatios Chicheste: Ellis Howood limited 989. IJMSET-Advaced Scietific Reseach Foum (ASRF) All Rights Reseved ASRF pomotes eseach atue Reseach atue eiches the wold s futue 9