SOME IDENTITIES FOR GENERALIZED FIBONACCI AND LUCAS SEQUENCES

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1 Hcettepe Jourl of Mthemtics d Sttistics Volume , SOME IDENTITIES FOR GENERALIZED FIBONACCI AND LUCAS SEQUENCES Nuretti IRMAK, Murt ALP Received 14 : 06 : 01 : Accepted 18 : 0 : 013 Keywords: Astrct I this study, we defie geerliztio of Lucs sequece {p }. The we oti Biet formul of sequece {p }. Also, we ivestigte reltioships etwee geerlized Fiocci d Lucs sequeces. Exteded Biet Formuls, Geerlized Fiocci d Lucs Sequeces 000 AMS Clssifictio: 11B39,11B37 1. Itroductio For, the Fiocci d Lucs umers re defied y followig recurrece reltios d F 0 0, F 1 1, F F 1 + F L 0, L 1 1, L L 1 + L. Ad Fiocci d Lucs umers Biet formuls re kow s, F τ γ τ γ d L τ + γ where 0 d τ, γ re roots of x x 1 0. These sequeces hve ee geerlized i my wys. For exmple, i [1], the uthor geerlized the sequeces {F } d {L } s follows, W AW 1 + BW 1, W 0, W 1 for, where,, A d B re ritrry itegers. Nigde Uiversity, Mthemtics Deprtmet 5141 Nigde Turkey. E-Mil: irmk@igde.edu.tr Nigde Uiversity, Mthemtics Deprtmet 5141 Nigde Turkey. E-Mil: murtlp@igde.edu.tr

2 33 N. Irmk, M. Alp I [] d [3], the uthors itroduced d studied ew kid geerlized Fiocci sequece d its properties tht depeds o two rel prmeters s defied elow, for > 1 { q 1 + q if is eve, q 0 0, q 1 1 q q 1 + q if is odd. Its exteded Biet s formul ws give y 1 ξ α β q, where α + +4, β +4 d ξ :. Note tht α d β re roots of the qudrtic equtio x x 0 d ξ 0 whe is eve, ξ 1 whe is odd. Also, uthors geerlized some idetities s follows; Cssii Idetity 1 ξ ξ q 1q +1 ξ 1 ξ q 1 Ctl s Idetity ξ r 1 ξ r q rq +r ξ 1 ξ q ξr 1 ξr 1 +1 r q r d Ocge s Idetity ξm+m ξm+m q mq +1 ξm+m ξm+m q m+1q 1 ξm q m Additiol Idetities ξm+m ξm+m q mq +1 + ξm+m ξm+m q m 1q ξm+m q m+ ξkm ξkm+k q mq k m+1 + ξkm+k ξkm q m 1q k m ξk q k 1 ξ+k ξ+k q +k+1 + ξ k 1 ξ k q k q +1q k+1 For more detils, we refer to []. Also, i [7], uthor gve the Geli-Cesro idetity s ξ 1 1 ξ q 4 q q 1q +1q ξ q 1 + I [6], uthor defied k periodic secod order lier recurrece s; 0q 1 + 0q if 0 mod k 1q 1 + 1q if 0 mod k 1.1 q.. k 1 q 1 + k 1 q if 0 mod k d ivestigted the comitoril iterprettio of the coefficiets A k d B k pperig i the recurrece reltio q A k q k +B k q k. Ad i [8], we foud 1.1 s exclipt formul for ritrry coefficiet d ritrry iitil coditios. The geerlized Fiocci d Lucs sequeces hve word comitoril iterprettio d they re closely relted to cotiued expsio of qudrtic irrtiols see i []. There re lots of comitoril idetities etwee Fiocci d Lucs umers. For exmple, F L F F mf F m+k F k 1 k F m+k F k F F mf m+1 + F m 1F m L L mf m+1 + L m 1F m F ml + F L m F m+

3 Some Idetities for Geerlized Fiocci d Lucs Sequeces 333 L F +1 + F 1 5F L +1 + L 1 For more idetities, they c e foud i [4]. pge Up to ow, uthors gve some idetities which re oly cotis Fiocci geerliztios. I this study, we defie geerlized Lucs sequeces d give exteded Biet s formul for geerlized Lucs sequeces. Moreover we ivestigte some properties which re ivolvig geerlized Fiocci d Lucs umers.. Mi Results.1. Defiitio. For y two opozitive rel umers d, the geerlized Lucs sequece {p } is defied s follows; { p 1 + p if is eve, p 0, p 1 1, p p 1 + p if is odd. We ote tht, these ew geerliztios is i the fct of fmily sequeces where ech ew choise of d produces distict sequeces. For exmple, whe we tke 1 i {q }, the sequece produce Fiocci umers. Whe tkig 1 i {p }, it produces Lucs umers. Whe we tke i {p }, it produces Pell-Lucs umers. We derive some idetities ivolvig the geerlized Fiocci d Lucs sequeces. From the defiitios of α d β, we ote tht α + 1 β + 1 1, α+β, αβ, α + 1 α, β α + 1 α. Now we give the geerlized Biet formul for the geerlized Lucs sequeces {p } :.. Theorem. For > 1, p 1 ξ α + β + 1 α β where α + +4, β +4 d ξ :. Proof. I order to prove the theorem, we use followig equtio give i [] : ξ Q Dq + C q 1 where { Q 1 + Q if is eve, Q Q 1 + Q if is odd, Q 0 C d Q 1 D re iitil coditios of the sequece {Q }. Whe C d D 1, we oti ξ p q + q 1 1 ξ α β ξ 1 ξ 1 α 1 β α β α 1 β 1 1 ξ + 1 ξ

4 334 1 ξ 1 ξ 1 ξ N. Irmk, M. Alp α 1 + α 1 β 1 + β 1 α α β α + β + 1 β α β + 1, + 1 α β s climed. Whe we tke 1, we oti Biet formul for Lucs sequeces. 3. Severl Idetities Ivolvig the Geerlized Fiocci Ad Lucs Numers I this sectio, we derive severl idetities ivolvig the geerlized Fiocci d Lucs umers. We strt with the followig result: 3.1. Theorem. For 0 p q ξ q + 1 q. Proof. By usig the Biet formuls of {q } d {p }, we hve 1 ξ α + β p q + 1 α β α β 1 ξ α β α β + 1 ξ 1 ξ α β 1 ξ α β + 1 ξ q + 1 q. Thus the proof is complete. Whe 1, we oti the well kow result for the usul Fiocci d Lucs umers : L F F. 3.. Theorem. For 0 q +1 + q 1 ξ p 1 q.

5 Some Idetities for Geerlized Fiocci d Lucs Sequeces 335 Proof. I order to prove the clim, we gi use the exteded Biet formuls of the sequeces {q } d {p } : q +1 + q 1 1 ξ+1 α +1 β ξ 1 α 1 β ξ 1 α +1 β +1 + α 1 β ξ ξ 1 α + β 1 1 ξ 1 1 α β 1 ξ p 1 q, α α β + 1 α β + β α β s climed. Whe 1 i Theorem 3, we oti the well kow the formul: F +1 + F 1 L Theorem. For 0, ξ p +1 + p 1 q + 1 p 1 q. Proof. Cosider p +1 + p 1 1 ξ+1 α +1 + β α+1 β ξ 1 α 1 + β α 1 β ξ 1 1 α 1 α β + 1 β β α + 1 α 1 ξ 1 1 α β 1 α β ξ ξ 1 1 ξ + 1 ξ ξ 1 1 ξ ξ ξ 1 1 β α β + 1 α + β 1 ξ α β α + β α β + 1 α β

6 336 N. Irmk, M. Alp ξ q + 1 p 1 q. Whe we tke 1, we oti L +1 + L 1 5F Theorem. For m, 0 ξm q mp + q p m q m+ + 1 q q m. Proof. Usig the Biet formuls, d the idetity follows esily from defiitio ξ m + ξ m + ξ ξ m ξ. The cosider 1 ξm α m β m 1 ξ α + β q mp + q p m m + 1 α β 1 ξ α β 1 ξm α m + β m αm β m m ξm ξ α m+ β m+ + m 1 ξ α β 1 ξm α m β m + 1 m ξm 1 ξm+ α m+ β m+ m+ 1 ξ α β 1 ξm α m β m + 1 m ξm q m+ 1 q q m. Whe we tke 1, we oti F m+ F ml + F L m Theorem. For, m 0, ξ ξm p mq m+1 + ξm p m 1q m p. Proof. If we use the Biet formuls of geerlized Fiocci d Lucs sequeces d y usig the idetity ξ m + ξ m + ξ ξ m ξ, we oti tht ξ ξm ξm p mq m+1 + p m 1q m 1 ξ α m+1 β m+1 α m + β m + 1 αm β m + 1 ξ α m β m 1 α m 1 + β m αm 1 β m 1

7 s climed. 1 ξ 1 ξ Some Idetities for Geerlized Fiocci d Lucs Sequeces α α + α α + β β β + α α β p, α α + α + β β + β Whe 1 i Theorem ove, we deduce the followig well kow formul: L L mf m+1 + L m 1F m Theorem. For 0 Proof. Cosider 1 p p p +1 1 ξ 1 ξ+1 +1 So the proof is complete. 1 q q q +1 p p +1. α + β α β α +1 + β +1 α +1 + β α+1 β ξ+1 α +1 β ξ+ξ p α β α +1 β +1 α +1 β +1 1 q q q +1. Whe 1 i Theorem ove, we oti well kow idetity: L L +1 L I the followig theorem, we list Biomil sums with {p } sequece. Ad we proved oe of them. The other oe c e prove i the sme wy Theorem. The sequece {p } stisfies the followig idetities. k0 k ξk k pk p k0 k ξk+r k +ξrξk p k+r p +r. Proof. Usig Biet formul of {p }, ξk+r k +ξrξk p k+r k k0 1 ξk+r k+r ξk+r k +ξrξk k k0 α k+r + β k+r 1 αk+r β k+r

8 338 N. Irmk, M. Alp α k+r + β k+r k k0 + 1 αk+r β k+r r 1 ξ α +r + β +r + 1 α+r β +r +r p +r. Whe we tke 1, we oti k0 Lk L k d k0 k Lk+r L +r. Refereces [1] Hordm, A. F. A geerlized Fiocci Sequece, Amer. Mth. Mothly, 68, , [] Edso, M. d Yyeie, O. A ew geerliztio of Fiocci sequeces d exteded Biet s Formul, Iteger 9, #A48, , 009. [3] Yyeie, O. A ote o geerlized Fiocci sequeces, Applied. Mth. Comp. 17 1, , 011. [4] Koshy, T. Fiocci Ad Lucs Numers with Applictios, Wiley, New York, 001. [5] Vjd, S. Fiocci & Lucs Numers d the Golde Sectio, Theory d Applictios, Ellis Horwood Ltd., Chishester, [6] Cooper, C. A idetity for period k secod order lier recurrece systems, Cog. Numer. 00, , 010. [7] Shi, M. The Geli-Cesro idetity i some coditiol Sequeces, Hcettepe Jourl of Mthemtics d Sttistics, 40, , 011. [8] Irmk, N. d Szly, L. O k periodic Biry Recurrece, A. Mth. Iform. 40, 5 35, 01.

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