Identities of Generalized Fibonacci-Like Sequence

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1 Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol., No. 5, 7-75 Available olie at Sciece ad Educatio Publishig DOI:.69/tjat--5- Idetities of Geealized Fiboacci-Lie Sequece Mamta Sigh, Ompaash Sihwal, Yogesh Kuma Gupta,* Depatmet of Maematical Scieces ad Compute applicatio, Bhudelhad Uivesity, Jhasi (U. P. Idia Depatmet of Maematics, Madsau Istitute of Techology, Madsau (M. P. Idia Schools of Studies i Maematics, Viam Uivesity Ujjai, (M. P. Idia *Coespodig auo: yogeshgupta.88@ediffmail.com Received August 5, 4; Revised Septembe 6, 4; Accepted Octobe 8, 4 Abstact The Fiboacci ad Lucas sequeces ae well-ow examples of secod ode ecuece sequeces. 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. May auos have defied Fiboacci patte based sequeces which ae populaized ad ow as Fiboacci-Lie sequeces. I is pape, Geealized Fiboacci-Lie sequece is itoduced ad defied by e ecuece elatio M M + M, wi M, M s+, whee s beig a fixed iteges. Some idetities of Geealized Fiboacci-Lie sequece ae peseted by Biet s fomula. Also some detemiat idetities ae discussed. Keywods: Fiboacci sequece, Lucas Sequece, Geealized Fiboacci-Lie Sequece, Biet s Fomula Cite This Aticle: Mamta Sigh, Ompaash Sihwal, ad Yogesh Kuma Gupta, Idetities of Geealized Fiboacci-Lie Sequece. Tuish Joual of Aalysis ad Numbe Theoy, vol., o. 5 (4: doi:.69/tjat 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 [4]. It is well ow at 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 [], [] ad [8] to [4] have bee geealized secod ode ecuece sequeces by pesevig e ecuece elatio ad alteig e fist two tems of e sequece o pesevig e fist two tems of sequece ad 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 [8] 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 ad studied its popeties. Fo ay sequece { F, } N positive itege, by Fiboacci sequece is defied

2 7 Tuish Joual of Aalysis ad Numbe Theoy F,, F, ad F, +, F, + F, -,. (.7 May auos have bee defied Fiboacci patte based sequeces which ae ow as Fiboacci-lie sequeces. The Fiboacci-Lie sequece [4] is defied by ecuece elatio, S S + S, wi S, S. (.8 The associated iitial coditios S ad S ae e sum of iitial coditios of Fiboacci ad Lucas sequece espectively. i.e. S F + L ad S F+ L. Fiboacci-Lie sequece [6] is defied by e ecuece elatio, H H + H fo wi H, H. (.9 I is pape, Geealized Fiboacci-Lie sequece is itoduced. The Biet s fomula is peseted ad established some idetities of Geealized Fiboacci- Lie sequece. Also detemiats idetities ae discussed.. Geealized Fiboacci-Lie Sequece Geealized Fiboacci-Lie sequece is itoduced ad defied by e ecuece elatio M M + M, wi M, M s +, (. whee s beig a fixed iteges. The fist few tems ae as follows: M, M s+, M s+, M s+ 4, M4 s+ 7, M5 5s+, M6 8s+ 8, B s + 9 ad so o. 7 The chaacteistic equatio of ecuece elatio (. is t t. which has two eal oots α ad β. (. Also, αβ, α + β, α β 5, α + β. Geeatig fuctio of geealized Fiboacci-Lie sequece is + ( s t Mt Mt (. t t (. Biet s fomula of Geealized Fiboacci-Lie sequece is defied by M Cα + Cβ C + C (.4 s+ 5 5 s Hee, C ad C. Also, s 5 s CC, C β + Cα s+, ( α β 5 Cα + Cβ s+, Cβ + Cα, C+ C M. (.5. Idetities of Geealized Fiboacci- Lie Sequece Now some idetities of Geealized Fiboacci-Lie sequece ae peset usig geeatig fuctio ad Biet s fomula. Auos [6,7] have bee descibed such type idetities. Theoem (.. (Explicit Sum Fomula Let M be e tem of geealized Fiboacci-Lie sequece. The M +. ( s (. Poof. By geeatig fuctio (., we have + ( s t Mt Mt ( t t + ( s t t t { + ( s t} ( t+ t { + ( s t}. t t + { + ( s t} t { }( { ( s t}. ( t t { ( s t}. t ( t ( Replace by { + ( s t} t { + ( s t} t Replace by ( + t + ( s t Equatig e coefficiet of t we obtai M + ( s..

3 Tuish Joual of Aalysis ad Numbe Theoy 7 Fo s i above idetity, explicit fomulas ca be obtaied fo Fiboacci sequece. Theoem (.. (Sum of Fist tems Sum of fist tems of Geealized Fiboacci-Lie sequece is M M+ s. (. Poof. By Biet s fomula (.4, we have M C α + C + + 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 M M+ + M s M+ s.. Theoem (.. (Sum of Fist tems wi odd idices: Sum of fist tems (wi odd idices of Geealized Fiboacci-Lie sequece is β M M+ M. (. Poof. By Biet s fomula (.4, we have B Cα + Cβ 5 C α + α + α α + C β β β β 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 M M+ M. Theoem (.4. (Sum of Fist tems wi eve idices Sum of fist tems (wi eve idices of geealized Fiboacci-Lie sequece is give by M M M +. (.4 Poof. By Biet s fomula (.4, we have M 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 M M M +. Theoem (.5. (Catala s Idetity Let M be e tem of Geealized Fiboacci-Lie sequece. The M M+ M ( ( s M + M+, >. Poof. By Biet s fomula (.4, we have M M+ M ( 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 ( ( α β M M+ M s 5 (. ( α β ( s+ M M+, we obtai ( s+ M M ( s ( s α β + Sice α β M M+ M ( ( s M + M+, >. (.5 Coollay (.5.. (Cassii s Idetity Let M be e tem of Geealized Fiboacci-Lie sequece. The M M+ M ( ( s 5,. (.6 Taig i e Catala s idetity (.5, e equied idetity is obtaied. Theoem (.6. (d Ocage s Idetity Let M be e tem of geealized Fiboacci-Lie sequece. The

4 7 Tuish Joual of Aalysis ad Numbe Theoy MmM+ M ( ( s+ Mm Mm +, m >. Poof. By Biet s fomula (.4, we have MmM+ M 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 MmM+ M 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 ( ( m m α β (.7 MmM+ M ( s 5. α β ( m m α β s+ Mm Mm + Sice, We get α β ( MmM+ M ( ( s+ Mm Mm +, m >. Theoem (.7. (Geealized Idetity Let M be e tem of Geealized Fiboacci-Lie sequece. The MmM Mm M+ m ( {( } ( s+ M m+ s+ M M+, M m+ + ( s 5 > m. Poof. By Biet s fomula (.4, we have MmM Mm M+ 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 MmM Mm M+ ( ( m ( ( m+ m+ α β α β. ( α β α β Sice {( s + M } M + ad α β (.8 α m+ m+ β α β {( s+ M }, m+ M m+ + we obtai MmM Mm M+ m ( {( } ( s+ M m+ s+ M M+, M m+ + ( s 5 > 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 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. Poblems o detemiats of Fiboacci sequece ad Lucas sequece ae appeaed i vaious issues of Fiboacci Quately. T. Koshy [] explaied two chaptes o e use of matices ad detemiats. May detemiat idetities of geealized Fiboacci sequece ae discussed i [4], [6] ad []. I is sectio some detemiat idetities of Geealized Fiboacci-Lie sequece ae peseted. Eties of detemiats ae satisfyig e ecuece elatio of Geealized Fiboacci-Lie sequece ad oe sequeces. Theoem(4.. Fo ay iteges, pove at Poof. M + M + M+ M + 4 M + 5 M+ 6. M + 7 M + 8 M+ 9 M + M + M+ Let M + 4 M + 5 M+ 6 M + 7 M + 8 M+ 9 Applyig C C+ C, we get Let M + M + M+ M + 6 M + 5 M+ 6 M + 9 M + 8 M+ 9 (4. Sice two colums ae idetical, us we obtaied equied esult. Theoem (4.. Fo ay itege, pove at M M+ M + M+ M+ M M+ M+ M + M M M+. M+ M M M+ M + M+ Poof. (4.

5 Tuish Joual of Aalysis ad Numbe Theoy 74 M M+ M + M+ M+ M Let M+ M+ M + M M M+. M+ M M M+ M + M+ By applyig C C + C + C ad expadig alog fist ow, we obtaied equied esult. Theoem (4.. Fo ay itege, pove at Poof. M M + M+. M+ + M+ M + M+ M + M+ Let M M + M+. M+ + M+ M + M+ M + M+ Applyig R R + R, we get, M M + M +. M+ M+ M+ Taig commo out M + fom id ow, M+ M M + M +. (4. Sice two ows ae idetical, us we obtaied equied esult. Theoem (4.4. Fo ay itege, pove at M M + M+ M + M+ + M+ M M + M+ M + M+ + 4M+ M M + 6 M+ M + 6M+ + M+ MM+ M+. Poof. Let (4.4 M M + M+ M + M+ + M+ M M + M+ M + M+ + 4 M+. M M + 6 M+ M + 6M+ + M+ Applyig R R R, R R R, we get M M +M + M +M+ + M+ M + M + +M +. M + M+ + 9M + Applyig R R R ad expadig alog fist ow, we obtaied equied esult. Theoem (4.5. Fo ay itege, pove at M M+ MM+ MM+ M+ M+ MM+ M + M+ MM+ M+. (4.5 Poof. Let M M+ MM+ MM+ M + M+. MM+ M + M+ Taig commo out C, C, C espectively, we get, M, M M + + fom M M M M+ M+ M+ M +. M + M + Taig commo out M, M +, M+ fom R, R, R espectively ad expadig alog fist ow, we obtaied equied esult. Theoem (4.6. Fo ay itege, pove at M F M+ F+ [ FM + MF+ ]. M + F+ M Poof: Let M F. (4.6 F + + M + F+ Assume M a, M + b, F p, F + q e M+ a + b ad F + p+q Now substitutig e above values i detemiat, we get a p b q a+ b p+ q Applyig R R R a b p q b q a+ b p+ q Applyig R R R a b p q a p ( pb aq. a+ b p+ q Substitutig e values of a, b, p ad q, we get equied esult. Similaly followig idetities ca be deived: Theoem (4.7. Fo ay itege, pove at M M+ M+ M+ M M+ ( M M+. M+ M+ M + (4.7 Theoem (4.8. Fo ay itege, pove at M L M+ L+ ( LM + ML+. M+ L+ (4.8

6 75 Tuish Joual of Aalysis ad Numbe Theoy Theoem (4.9. Fo ay itege, pove at M + M+ M+ + M+ M+ + M M+ M M+. Theoem 4.(. Fo ay itege, pove at + M M+ M+ M + M+ M+ M M+ + M+ + M + M+ + M+. Acowledgemet (4.9 (4. We would lie to a e aoymous efeees fo umeous helpful suggestios. Refeeces [] A. F. Hoadam: A Geealized Fiboacci sequece, Ameica Maematical Moly, Vol. 68. (5, 96, [] A. F. Hoadam: Basic Popeties of a Cetai Geealized Sequece of Numbes, The Fiboacci Quately, Vol. (, 965, [] A. T. Bejami ad D. Walto, Coutig o Chebyshev polyomials, Ma. Mag. 8, 9, 7-6. [4] B. Sigh, O. Sihwal ad S. Bhataga: Fiboacci-Lie Sequece ad its Popeties, It. J. Cotemp. Ma. Scieces, Vol. 5 (8,, [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, ( (, [7] B. Sigh, S. Bhataga ad O. Sihwal: Geealized Idetties of Compaio Fiboacci-Lie Sequeces, Global Joual of Maematical Aalysis, (, 4-9. [8] D. V. Jaiswal: O a Geealized Fiboacci sequece, Labdev J. Sci. Tech. Pat A 7, 969, [9] M. Edso ad O. Yayeie: A New Geealizatio of Fiboacci sequece ad Exteded Biet s Fomula, Iteges Vol. 9, 9, [] M. E. Waddill ad L. Sacs: Aoe Geealized Fiboacci sequece, The Fiboacci Quately, Vol. 5 (, 967, 9-. [] M. Sigh, Y. Gupta, O. Sihwal, Geealized Fiboacci-Lucas Sequeces its Popeties, Global Joual of Maematical Aalysis, (, 4, [] O. Sihwal, Geealizatio of Fiboacci Sequece: A Itiguig Sequece, Lap Lambet Academic Publishig GmbH & Co. KG, Gemay (. [] S. Falco ad A. Plaza: O e Fiboacci K- Numbes, Chaos, Solutios & Factals, Vol. (5, 7, [4] S. Vajda, Fiboacci & Lucas Numbes ad e Golde Sectio, Theoy ad Applicatios, Ellis Howood Ltd., Chicheste, 989. [5] T. Koshy, Fiboacci ad Lucas Numbes wi Applicatios, Wiley- Itesciece Publicatio, New Yo (.