Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash Sihwal,*, Yogesh Kuma Gupta School of Studies i Maematics, Viam Uivesity, Ujjai-456 (M P, Idia Depatmet of Maematics, Madsau Istitute of Techology, Madsau (M P, Idia School of Studies i Maematics, Viam Uivesity, Ujjai, (M P, Idia *Coespodig auo: opbhsihwal@ediffmailcom Received Septembe 5, 4; Revised Octobe 4, 4; Accepted Novembe, 4 Abstact The Fiboacci sequece is a souce of may ice ad iteestig idetities A simila itepetatio exists fo Lucas sequece The Fiboacci sequece, Lucas umbes ad ei geealizatio have may iteestig popeties ad applicatios to almost evey field Fiboacci sequece is defied by e ecuece fomula F F- + F-, ad F, F, whee F is a umbe of sequece The Lucas Sequece is defied by e ecuece fomula L L- + L-, adl, L, whee L is a umbe of sequece I is pape, Geealized Fiboacci-Lucas sequece is itoduced ad defied by e ecuece elatio B B + B, wi B b, B s, whee b ad s ae iteges We peset some stadad idetities ad detemiat idetities of geealized Fiboacci-Lucas sequeces by Biet s fomula ad oe simple meods Keywods: Fiboacci sequece, Lucas sequece, Geealized Fiboacci-Lucas sequece, Biet s fomula Cite This Aticle: Bijeda Sigh, Ompaash Sihwal, ad Yogesh Kuma Gupta, Geealized Fiboacci- Lucas Sequece Tuish Joual of Aalysis ad Numbe Theoy, vol, o 6 (4: -7 doi: 6/tjat--6- Itoductio Fiboacci umbes F ad Lucas umbes L have delighted maematicias ad amateus alie fo cetuies wi ei beauty ad ei popesity to pop up i quite uexpected places [], [] ad [] It is well ow at geealized Fiboacci ad Lucas umbes play a impotat ole i may subjects such as algeba, geomety, ad umbe eoy Thei vaious elegat popeties ad wide applicatios have bee studied by may auos The Fiboacci ad Lucas sequeces ae examples of secod ode ecusive sequeces The Fiboacci sequece [4] is defied by e ecuece elatio: F F + F,, wi F, F ( The simila itepetatio also exists fo Lucas sequece Lucas sequece [4] is defied by e ecuece elatio: L L + L, wi L, L ( Auos [,,,4] ad [6-] have bee geealized secod ode ecuece sequeces by pesevig e ecuece elatio ad alteig e fist two tems of e sequece, while oes have geealized ese sequeces by pesevig e fist two tems of sequece but alteig e ecuece elatio slightly Hoadam [] itoduced ad studied popeties of a geealized Fiboacci sequece{ H} ad defied geealized Fiboacci sequece { H} by e ecuece elatio: H+ H+ + H, H q ad H p,, ( whee p, q ae abitay iteges Hoadam [] itoduced ad studied popeties of w ad defied aoe geealized Fiboacci sequece { } geealized Fiboacci sequece { w } by e ecuece elatio: { } { } w w ( a, b; p, q : w a, w b, w pw qw,, (4 whee a, b, p ad q ae abitay iteges Waddill ad Sacs [] exteded e Fiboacci umbes ecuece elatio ad defied e sequece P by ecuece elatio: { } P P + P + P,, (5 whee P, P ad P ae ot all zeo give abitay algebaic iteges Jaiswal [5] itoduced ad studied popeties of geealized Fiboacci sequece{ T } ad defied it by T T T-, T a ad T b, + + (6 Falco ad Plaza [] itoduced Fiboacci sequece { F, } ad studied its popeties Fo ay N positive itege, by Fiboacci sequece is defied
4 Tuish Joual of Aalysis ad Numbe Theoy F, F ad F F + F, (7,,, +,, - I is pape we peset Geealized Fiboacci-Lucas sequece ad some specific idetities ad some detemiat idetities Geealized Fiboacci-Lucas Sequece Geealized Fiboacci-Lucas sequece { B} itoduced ad defied by ecuece elatio: B B + B, wi B b ad B s, ( whee b ad s ae o egative iteges The fist few tems ae as follows: B, b B s, B b+ s, B b+, s B4 4b+, s B5 6b+ 5, s B6 b+ 8 s, B7 6b + s ad so o The chaacteistic equatio of ecuece elatio ( is t t which has two eal oots + 5 5 α ad β ( Also, αβ, α + β, α β 5, α + β Geeatig fuctio of geealized Fiboacci-Lucas sequece is b+ ( s bt Bt Bt ( t t ( Biet s fomula of Geealized Fiboacci-Lucas sequece is defied by + 5 5 B Cα + Cβ C + C is (4 s bβ bα s Hee, C ad C 5 5 s bs 4b Also, CC, C β + Cα s+ b ad ( α β C+ C b Geealized Fiboacci-Lucas Sequece geeates may classical sequeces o e basis of value of b ad s Idetities of Geealized Fiboacci- Lucas Sequece Now some idetities of geealized Fiboacci-Lucas sequece ae peset usig geeatig fuctio ad Biet s fomula Auos [6,7] have bee descibed such type idetities Theoem ( (Explicit Sum Fomula Let G be e tem of geealized Fiboacci-Lucas sequece The B b + s b ( ( Poof By geeatig fuctio (, we have b+ ( s bt Bt Bt ( t t + (, { b s bt}( t t { b+ ( s bt} ( t+ t Bt b+ ( s bt t+ t, { b+ ( s bt} t ( + t, { } ( { b s bt } t ( t + (, + { b+ ( s bt} + + { b+ ( s bt} { b+ ( s bt} b t + + ( s b t, Equatig e coefficiet of t, we obtai Bt b ( ( s b + ( By taig diffeet values of b ad s i above idetity, explicit fomulas ca be obtaied fo Fiboacci ad Lucas sequeces Theoem ( (Sum of Fist tems Sum of fist tems of Geealized Fiboacci-Lucas sequece is B B+ s ( Poof Usig e Biet s fomula (4, we have
Tuish Joual of Aalysis ad Numbe Theoy 5 Cα + C B, α β C + C α β ( C+ C ( Cβ + Cα ( Cα + Cβ + αβ ( Cα + Cβ ( α + β + αβ Usig subsequet esults of Biet s fomula, we get B B + B s B+ s Theoem ( (Sum of Fist tems wi odd idices: Sum of fist tems (wi odd idices of Geealized Fiboacci-Lucas sequece is B+ B+ B b B b β ( Poof Usig e Biet s fomula (4, we have + + B + C α + Cβ, α β Cα + C β, α β + + ( Cα + Cβ ( Cα + Cβ + αβ ( Cβ + Cα α β ( Cα + Cβ α + β α β Usig subsequet esults of Biet s fomula, we get B+ B+ B b B b Theoem (4 (Sum of Fist tems wi eve idices Sum of fist tems (wi eve idices of geealized Fiboacci-Lucas sequece is give by B B B s+ b B s+ b (4 Poof Usig e Biet s fomula (4, we have B C α + Cβ, α β C + C, α β ( Cα + Cβ ( C+ C + ( Cβ + Cα α β ( Cα + Cβ α + β α β Usig subsequet esults of Biet s fomula, we get B B B s+ b B s+ b Theoem (5 (Catala s Idetity Let B be e tem of Geealized Fiboacci-Lucas sequece The B B+ B ( ( sb, bb+ > s bs 4b Poof Usig Biet s fomula (4, we have B B+ B ( Cα + Cβ + + ( Cα + Cβ ( Cα + Cβ, CC ( αβ ( α β α β CC ( αβ ( α β α β - CC ( αβ ( α β Usig subsequet esults of Biet s fomula, we get ( (5 ( α β B B+ B s bs 4 b ( ( α β α β Sice ( sb, we obtai bb+ α β s bs 4b ( B ( B+ B sb bb+, > s bs 4b Coollay (5 (Cassii s Idetity Let B be e tem of Geealized Fiboacci-Lucas sequece The B B+ B ( ( s bs 4 b, (6 Taig i e Catala s idetity (5, e equied idetity is obtaied Theoem (6 (d Ocage s Idetity Let B be e tem of geealized Fiboacci-Lucas sequece The BmB+ B ( ( sbm bbm +, m > Poof Usig Biet s fomula (4, we have BmB+ B m m + + ( Cα + Cβ ( Cα + Cβ m+ m+ ( Cα + Cβ ( Cα + Cβ m + + m m+ m+ CC ( α β + α β α β α β m m m m CC ( αβ β( α β α( α β m m CC ( αβ ( α β ( α β Usig subsequet esults of Biet s fomula, we get BmB+ B m m ( α β ( ( s bs 4 b ( α β m m α β sbm bbm + Sice, we obtai α β ( s bs 4 b (7
6 Tuish Joual of Aalysis ad Numbe Theoy BmB+ B ( ( sbm bbm +, m > Theoem (7 (Geealized Idetity Let B be e tem of Geealized Fiboacci-Lucas sequece The BmB Bm B+ m ( ( sb bb+ ( sb m+ bb m+ +, > m Poof Usig Biet s fomula (4, we have BmB Bm B+ m m ( Cα + Cβ ( Cα + Cβ m m + + ( Cα + Cβ ( Cα + Cβ m m α β α β CC ( α β α β m + + m CC ( ( α β ( α β α β m m m+ m+ CC ( α β ( α β ( β α m m+ m+ CC ( ( α β ( α β Usig subsequet esults of Biet s fomula, we get BmB Bm B+ (8 ( s bs 4 b ( m ( ( m+ m+ α β α β ( α β α β Sice ( sb bb+ ad α β s bs 4b m+ m+ α β sb m+ bb m+ + α β ( s bs 4 b We obtai, BmB Bm B+ m ( ( sb bb+ ( sb m+ bb m+ +, > m The idetity (8 povides Catala s, Cassii s ad d Ocage s ad oe idetities: (i If m, e Catala s idetity (5 is obtaied (ii If m ad i idetity (8, e Cassii s idetity (5 is obtaied (iii If m, m + ad i idetity (8, e d Ocage s idetity (6 is obtaied 4 Detemiat Idetities Thee is a log taditio of usig matices ad detemiats to study Fiboacci umbes T Koshy [] explaied two chaptes o e use of matices ad detemiats I is sectio, some detemiat idetities ae peseted Theoem(4 Fo ay iteges, pove at B+ B + B+ B+ 4 B + 5 B+ 6 B+ 7 B + 8 B+ (4 Poof Let Applyig C C+ C, we get Let B+ B + B+ B+ 4 B + 5 B+ 6 B+ 7 B + 8 B+ B+ B + B+ B+ 6 B + 5 B+ 6 B+ B + 8 B+ Sice two colums ae idetical,we obtaied equied esult Theoem (4 Fo ay itege, pove at B B+ B + B+ B+ B B+ B+ B + B B B+ (4 B B B B B B Poof + + + + B B+ B + B+ B+ B Let B+ B+ B + B B B+ B+ B B B+ B + B+ By applyig C C + C + C ad expadig alog fist ow, we obtaied equied esult Theoem (4 Fo ay itege, pove at Poof B B + B+ B+ + B+ B + B+ B + B+ (4 Let B B + B + B+ + B+ B + B+ B + B+ Applyig R R + R, we get, Taig commo out B + B B + B + B+ B+ B+ fom id ow, B+ B B + B + Sice two ows ae idetical, us we obtaied equied esult Theoem (44 Fo ay itege, pove at B B + B+ B + B+ + B+ B B + B+ B + B+ + 4B+ B B + 6 B+ B + 6B+ + B+ BB + B+ Poof (44
Tuish Joual of Aalysis ad Numbe Theoy 7 B B + B+ B + B+ + B+ Let B B + B B + B + 4 B + + + B B + 6 B+ B + 6B+ + B+ R R R, R R R Applyig, we get B B +B + B +B+ + B+ B + B + +B + B + B+ + B+ Applyig R R R ad expadig alog fist ow, we obtaied equied esult Theoem (45 Fo ay itege, pove at Poof B B+ BB+ BB + B+ B+ BB + B+ BB+ B + B+ Let (45 B B+ BB+ BB+ B + B+ BB+ B + B+ Taig commo out B, B +, B + fom C, C, C espectively, we get B B B B+ B+ B+ B + B+ B + Taig commo out B, B +, B+ fom R, R, R espectively ad expadig alog fist ow, we obtaied equied esult Theoem (46 Fo ay itege, pove at B F B F [ FB BF ] (46 B F + + + + + + B F Poof: Let B+ F+ B+ F+ Assume B a, B + b, B + a + b ad F p, F + q, F + p + q Now substitutig e above values i detemiat, we get Applyig R R R Applyig R R R a p b q q a bp q b q q a bp q a p ( pb aq q Substitutig e values of a, b, p ad q, we get equied esult Similaly followig idetities ca be deived: Theoem (48 Fo ay itege, pove at B L B L ( LB BL (48 B + + + + + L+ Theoem (4 Fo ay itege, pove at B + B+ B+ + B+ B+ + B B+ B B+ Theoem 4( Fo ay itege, pove at (4 + B B+ B+ B + B+ B+ + B + B+ + B+ B B+ + B+ 5 Coclusios (4 I is pape, Geealized Fiboacci-Lucas sequece is itoduced Some stadad idetities of geealized Fiboacci-Lucas sequece have bee obtaied ad deived usig geeatig fuctio ad Biet s fomula Also some detemiat idetities have bee established ad deived Refeeces [] A F Hoadam: A Geealized Fiboacci Sequece, Ameica Maematical Moly, Vol 68 (5, 6, 455-45 [] A F Hoadam: Basic Popeties of a Cetai Geealized Sequece of Numbes, The Fiboacci Quately, Vol (, 65, 6-76 [] AT Bejami ad D Walto: Coutig o Chebyshev polyomials, Ma Mag 8,, 7-6 [4] B Sigh, O Sihwal ad S Bhataga: Fiboacci-Lie Sequece ad its Popeties, It J Cotemp Ma Scieces, Vol 5 (8,, 85-868 [5] B Sigh, S Bhataga ad O Sihwal: Fiboacci-Lie Polyomials ad some Idetities, Iteatioal Joual of Advaced Maematical Scieces, (, ( 5-57 [6] B Sigh, S Bhataga ad O Sihwal: Fiboacci-Lie Sequece, Iteatioal Joual of Advaced Maematical Scieces, ( ( 45-5 [7] B Sigh, S Bhataga ad O Sihwal: Geealized Idetties of Compaio Fiboacci-Lie Sequeces, Global Joual of Maematical Aalysis, (,, 4- [8] D V Jaiswal: O a Geealized Fiboacci Sequece, Labdev J Sci Tech Pat A 7, 6, 67-7 [] M Edso ad O Yayeie: A New Geealizatio of Fiboacci sequece ad Exteded Biet s Fomula, Iteges Vol,, 6-654 [] M E Waddill ad L Sacs: Aoe Geealized Fiboacci Sequece, The Fiboacci Quately, Vol 5 (, 67, - [] S Falco ad A Plaza: O e Fiboacci K- Numbes, Chaos, Solutios & Factals, Vol (5, 7, 65-64, [] S Vajda, Fiboacci & Lucas Numbes, ad e Golde Sectio, Theoy ad Applicatios, Ellis Howood Ltd, Chicheste, 8 [] T Koshy, Fiboacci ad Lucas Numbes wi Applicatios, Wiley-Itesciece Publicatio, New Yo (