Research Article Incomplete k-fibonacci and k-lucas Numbers

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1 Hdaw Pubhg Corporato Chee Joura of Mathematc Voume 013, Artce ID , 7 page Reearch Artce Icompete k-fboacc ad k-luca Number Joé L. Ramírez Ittuto de Matemátca y u Apcacoe, Uverdad Sergo Arboeda, Coomba Correpodece houd be addreed to Joé L.Ramírez; joe.ramrez@ma.uergoarboeda.edu.co Receved 10 Juy 013; Accepted 11 September 013 Academc Edtor: B. L ad W. va der Hoek Copyrght 013 Joé L. Ramírez. Th a ope acce artce dtrbuted uder the Creatve Commo Attrbuto Lcee, whch permt uretrcted ue, dtrbuto, ad reproducto ay medum, provded the orga work propery cted. We defe the compete k-fboacc ad k-luca umber; we tudy the recurrece reato ad ome properte of thee umber. 1. Itroducto Fboacc umber ad ther geerazato have may teretg properte ad appcato to amot every fed of cece ad art (e.g., ee [1]). Fboacc umber F are defed by the recurrece reato F 0 0, F 1 1, F +1 F +F 1, 1. (1) There ext a ot of properte about Fboacc umber. I partcuar, there a beautfu combatora detty to Fboacc umber [1] F ( 1 ). () From (),Fppo [] troduced the compete Fboacc umber F () ad the compete Luca umber L ().They are defed by F () j0 ( 1 j ) (1,,3,...;0 1 j ), L () j0 j ( j j ) (1,,3,...;0 ). (3) Further [3], geeratg fucto of the compete Fboacc ad Luca umber are determed. I [4], Djordjevć gave the compete geerazed Fboacc ad Luca umber. I [5], Djordjevć ad Srvatava defed compete geerazed Jacobtha ad Jacobtha-Luca umber. I [6],theauthordefethecompeteFboaccadLuca p-umber. Ao the author defe the compete bvarate Fboacc ad Luca p-poyoma [7]. O the other had, may kd of geerazato of Fboaccumberhavebeepreetedtheterature.I partcuar, a geerazato the k-fboacc Number. For ay potve rea umber k,the k-fboacc equece, ay {F k, } N, defed recurrety by F k,0 0, F k,1 1, F k,+1 kf k, +F k, 1, 1. (4) I [8], k-fboacc umber were foud by tudyg the recurve appcato of two geometrca traformato ued the four-trage oget-edge (4TLE) partto. Thee umber have bee tuded evera paper; ee [8 14]. For ay potve rea umber k, thek-luca equece, ay {L k, } N, defed recurrety by L k,0, L k,1 k, L k,+1 kl k, +L k, 1. (5) If k1,wehavethecacalucaumber.moreover, L k, F k, 1 +F k,+1, 1;ee[15]. I [1], the expct formuato k-fboacc umber F k, ( 1 )k 1, (6) ad the expct formua of k-luca umber L k, (x) / ( )k. (7)

2 Chee Joura of Mathematc From (6) ad (7), we troduce the compete k- Fboacc ad k-luca umber ad we obta ew recurret reato, ew dette, ad ther geeratg fucto.. The Icompete k-fboacc Number Defto 1. The compete k-fboacc umber are defed by F k, ( 1 )k 1, 0 1. (8) I Tabe 1, ome vaue of compete k-fboacc umber are provded. We ote that F 1, F. (9) For k1, we get compete Fboacc umber []. Some peca cae of (8)are F 0 k, k 1 ; ( 1), F 1 k, k 1 + ( ) k 3 ; ( 3) F k, k 1 + ( ) k 3 + ( 4)( 3) k 5 ; ( 5) F k, F k, ; ( 1) k, { F k, k { { F k, 1 F ( 3)/ ( eve) ( 3). ( odd).1. Some Recurrece Properte of the Number F k, (10) Propoto. The recurrece reato of the compete k- Fboacc umber F k, Proof. Ue Defto 1 to rewrte the rght-had de of (11) a +1 k ( +1 )k + ( k ( +1 F k,+. )k ( 1 )k 1 1 ( 1 )k +1 [( )+( 1 )]) k+1 ( 1 ) ( +1 )k +1 0 Propoto 3. Oe ha ( ) F+ k,+ k F + k,+, (13) 0 1. (14) Proof (by ducto o ). Sum (14)cearyhodfor0ad 1 (ee (11)). Now uppoe that the reut true for a j<+1;weprovetfor+1: +1 ( +1 )F + k,+ k [( )+( )] F+ 1 k,+ k ( )F+ +1 k,+ k + ( 1 )F+ k,+ k F + k,+ +( +1 )F++1 k,++1 k ( )F++1 k,++1 k+1 F + k,+ +0+ ( )F++1 k,++1 k+1 +( 1 )F k, F + k,+ +k ( )F++1 k,++1 k +0 F +1 k,+ kf+1 k,+1 +F k,, 0. (11) F + k,+ +kf++1 k,++1 F ++1 k,++. (15) The reato (11) cabetraformedtotheohomogeeou recurrece reato F k,+ kf k,+1 +F k, ( 1 )k 1. (1) Propoto 4. For +, 1 F k,+ k 1 F +1 k,++1 k F +1 k,+1. (16)

3 Chee Joura of Mathematc 3 Tabe 1: The umber F k,,for1 10. \ k 3 k k +1 4 k 3 k 3 +k 5 k 4 k 4 +3k k 4 +3k +1 6 k 5 k 5 +4k 3 k 5 +4k 3 +3k 7 k 6 k 6 +5k 4 k 6 +5k 4 +6k k 6 +5k 4 +6k +1 8 k 7 k 7 +6k 5 k 7 +6k k 3 k 7 +6k k 3 +4k 9 k 8 k 8 +7k 6 k 8 +7k k 4 k 8 +7k k k k 8 +7k k k k 9 k 9 +8k 7 k 9 +8k 7 + 1k 5 k 9 +8k 7 + 1k 5 + 0k 3 k 9 +8k 7 + 1k 5 + 0k 3 +5k Proof(byductoo). Sum (16) ceary hod for 1 (ee (11)). Now uppoe that the reut true for a j<. We prove t for : By dervg to the prevou equato (repect to k), t obtaed F 1 k,+ k k F k,+ k 1 +F k,+ k(f +1 k,++1 k F +1 k,+1 )+F k,+ (kf +1 k,++1 +F k,+ ) k+1 F +1 k,+1 F +1 k,++ k+1 F +1 k,+1. (17) F k, +kf k, F k, ( )( 1 )k 1 ( 1 )k 1. () Note that f k,(4), a rea varabe, the F k, F x, ad they correpod to the Fboacc poyoma defed by 1 { f 0 F +1 (x) x { f 1 { xf (x) +F 1 (x) f >1. Lemma 5. Oe ha F (x) F +1 (x) xf (x) +F 1 (x) x +4 L (x) xf (x) x. +4 See Propoto 13 of [1]. Lemma 6. Oe ha ( 1 Proof. From (6)we have that kf k, (18) (19) )k 1 ((k +4) 4)F k, kl k,. (k +4) (0) ( 1 ) k. (1) From Lemma 5, F k, +k( L k, kf k, k ) +4 F k, ( 1 )k 1. From where, after ome agebra (0) obtaed. Propoto 7. Oe ha 0 4F k, +kl k, F k, { (k +4) (k +8)F { k, +kl k, { (k +4) ( eve) ( odd). (3) (4)

4 4 Chee Joura of Mathematc Tabe : The umber L k,,for1 9. \ k k k + 3 k 3 k 3 +3k 4 k 4 k 4 +4k k 4 +4k + 5 k 5 k 5 +5k 3 k 5 +5k 3 +5k 6 k 6 k 6 +6k 4 k 6 +6k 4 +9k k 6 +6k 4 +9k + 7 k 7 k 7 +7k 5 k 7 +7k k 3 k 7 +7k k 3 +7k 8 k 8 k 8 +8k 6 k 8 +8k 6 + 0k 4 k 8 +8k 6 + 0k k k 8 +8k 6 + 0k k + 9 k 9 k 9 +9k 7 k 9 +9k 7 + 7k 5 k 9 +9k 7 + 7k k 3 k 9 +9k 7 + 7k k 3 +9k Proof F k, 0 F 0 k, +F1 k, + +F k, ( 1 0 )k 1 0 +[( 1 0 )k 1 +( 1 1 )k 3 ] 0 1 ( 1 +1)( 1 ( 1 )k 1 ( 1 +1)F k, From Lemma 6,(4) obtaed. )k 1 ( 1 )k 1. (5) + + ( 1 0 )k 1 +( 1 1 [ 0 1 [ )k ] + +( 1 1 )k ]] ( 1 +1)( ( )k 3 )k ( 1 )k ( 1 +1 )( 1 1 )k 1 ] 3. The Icompete k-luca Number Defto 8. The compete k-luca umber are defed by L k, ( )k, 0. (6) I Tabe,omeumber of competek-luca umber are provded. We ote that L / 1, L. (7) Some peca cae of (6)are L 0 k, k ; ( 1), L 1 k, k +k ; ( ), L k, k +k ( 3) + k 4 ; ( 4), L / k, L k, ; ( 1), L ( )/ k, { L k, ( eve) L k, k ( odd) ( ). (8)

5 Chee Joura of Mathematc Some Recurrece Properte of the Number L k, Propoto 9. Oe ha L k, F 1 k, 1 +F k,+1, 0. (9) Proof. By (8), rewrte the rght-had de of (9) a 1 ( 1 )k + ( 1 1 )k + [( 1 1 ( )k ( )k +0 L k,. ( )k )+( )] k ( 1 1 ) (30) Propoto 10. The recurrece reato of the compete k- Luca umber L k, L +1 k,+ kl+1 k,+1 +L k,, 0. (31) The reato (31) cabetraformedtotheohomogeeou recurrece reato L k,+ kl k,+1 +L k, ( )k. (3) Proof. Ug (9)ad(11), we wrte L +1 k,+ F k,+1 +F+1 k,+3 Propoto 11. Oe ha Proof. By (9), kf k, +F 1 k, 1 +kf+1 k,+ +F k,+1 k(f k, +F+1 k,+ )+F 1 k, 1 +F k,+1 kl +1 k,+1 +L k,. kl k, F k,+ F k,, F k,+ L k,+1 F 1 k,, whece, from (31), (33) 0 1. (34) F k, L 1 k, 1 F 1 k,, (35) F k,+ F k, L k,+1 L 1 k, 1 kl k,. (36) Propoto 1. Oe ha ( )L+ k,+ k L + k,+, Proof. Ug (9)ad(14), we wrte ( )L+ k,+ k ( )[F+ 1 k,+ 1 +F+ k,++1 ]k ( )F+ 1 Propoto 13. For +1, 1 k,+ 1 k + 0. (37) ( )F+ k,++1 k F 1+ k, 1+ +F+ k,+1+ L+ k,+. (38) L k,+ k 1 L +1 k,++1 k L +1 k,+1. (39) The proof ca be doe by ug (31) ad ducto o. Lemma 14. Oe ha / ( )k [L k, kf k, ]. (40) The proof mar to Lemma 6. Propoto 15. Oe ha / 0 L k, { L k, + k F k, ( eve) { 1 { (L k, +kf k, ) ( odd). (41) Proof. A argumet aaogou to that of the proof of Propoto 7 yed / 0 L k, ( +1)L k, / From Lemma 14,(41) obtaed. ( )k. (4) 4. Geeratg Fucto of the Icompete k-fboacc ad k-luca Number I th ecto, we gve the geeratg fucto of compete k-fboacc ad k-luca umber. Lemma 16 (ee [3, page59]).let { } 0 be a compex equece atfyg the foowg ohomogeeou recurrece reato: a 1 +b +r (>1), (43)

6 6 Chee Joura of Mathematc where a ad b are compex umber ad {r } a gve compex equece. The, the geeratg fucto U(t) of the equece { } U (t) G (t) + 0 r 0 +( 1 0 a r 1 )t 1 at bt, (44) where G(t) deote the geeratg fucto of {r }. Theorem 17. The geeratg fucto of the compete k- Fboacc umber F k, gve by R k, (x) F k, t t +1 [F k,+1 +(F k,+ kf k,+1 )t [1 kt t ] 1. t (1 kt) +1 ] (45) Proof. Let be a fxed potve teger. From (8) ad(1), F k, 0for 0 < +1, F k,+1 F k,+1, adf k,+ F k,+,ad F k, kf k, 1 +F k, ( 3 )k 3. (46) Now et 0 F k,+1, 1 F k,+, F k,++1. (47) Ao et r 0 r 1 0, r ( + )k. (48) The geeratg fucto of the equece {r } G(t) t /(1 kt) +1 (ee [16, page355]).thu,fromlemma 16, wegetthe geeratg fucto R k, (x) of equece { }. Theorem 18. The geeratg fucto of the compete k- Luca umber L k, gve by S k, (x) L k, t t [L k, +(L k,+1 kl k, )t t ( t) (1 kt) +1 ] [1 kt t ] 1. (49) Proof. Theproofofththeoremmartotheproofof Theorem 17. Let be a fxed potve teger. From (6) ad (3), L k, 0for 0 <, L k, L k,,adl k,+1 L k,+1, ad L k, kl k, 1 +L k, ( )k. (50) Now et Ao et 0 L k,, 1 L k,+1, L k,+. (51) r 0 r 1 0, r ( + + )k+. (5) The geeratg fucto of the equece {r } G(t) t ( t)/(1 kt) +1 (ee [16, page355]).thu,fromlemma 16, we get the geeratg fucto S k, (x) of equece { }. 5. Cocuo I th paper, we troduce compete k-fboacc ad k-luca umber, ad we obta ew dette. I [17], the author troduced the h(x)-fboacc poyoma. That geeraze Cataa Fboacc poyoma ad the k-fboacc umber. Let h(x) be a poyoma wth rea coeffcet. The h(x)-fboacc poyoma {F h, (x)} N are defed by the recurrece reato F h,0 (x) 0, F h,1 (x) 1, (53) F h,+1 (x) h(x) F h, (x) +F h, 1 (x), 1. It woud be teretg to tudy a defto of compete h(x)-fboacc poyoma ad reearch ther properte. Ackowedgmet The author woud ke to thak the aoymou referee for ther hepfu commet. The author wa partay upported by Uverdad Sergo Arboeda uder Grat o. USA-II Referece [1] T. Kohy, Fboacc ad Luca Number wth Appcato, Wey-Itercece, 001. [] P. Fppo, Icompete Fboacc ad Luca umber, Redcot de Crcoo Matematco d Paermo,vo.45,o.1,pp.37 56, [3] Á. Ptér ad H. M. Srvatava, Geeratg fucto of the compete Fboacc ad Luca umber, Redcot de Crcoo Matematco d Paermo,vo.48,o.3,pp ,1999. [4] G.B.Djordjevć, Geeratg fucto of the compete geerazed Fboacc ad geerazed Luca umber, Fboacc Quartery,vo.4,o.,pp ,004. [5] G. B. Djordjevć ad H. M. Srvatava, Icompete geerazed Jacobtha ad Jacobtha-Luca umber, Mathematca ad Computer Modeg,vo.4,o.9-10,pp ,005. [6] D. Tac ad M. Cet Fregz, Icompete Fboacc ad Luca p-umber, Mathematca ad Computer Modeg,vo. 5, o. 9-10, pp , 010. [7] D. Tac, M. Cet Fregz, ad N. Tugu, Icompete bvarate Fboaccu ad Luca p-poyoma, Dcrete Dyamc Nature ad Socety, vo. 01, Artce ID , 11 page, 01. [8] S. Facó ad Á. Paza, O the Fboacc k-umber, Chao, Soto ad Fracta,vo.3,o.5,pp ,007.

7 Chee Joura of Mathematc 7 [9] C. Boat ad H. Köe, O the properte of k-fboacc umber, Iteratoa Joura of Cotemporary Mathematca Scece, vo., o. 5, pp , 010. [10] S. Faco, The k-fboacc matrx ad the Paca matrx, Cetra Europea Joura of Mathematc,vo.9,o.6,pp , 011. [11] S. Facó ad Á. Paza, The k-fboacc equece ad the Paca -trage, Chao, Soto ad Fracta, vo. 33, o. 1, pp , 007. [1] S. Facó ad Á. Paza, O k-fboacc equece ad poyoma ad ther dervatve, Chao, Soto ad Fracta,vo.39, o. 3, pp , 009. [13] J. Ramírez, Some properte of covoved k-fboacc umber, ISRN Combatorc,vo.013,ArtceID759641,5page, 013. [14] A. Saa, About k-fboacc umber ad ther aocated umber, Iteratoa Mathematca Forum, vo. 50, o. 6, pp , 011. [15] S. Faco, O the k-luca umber, Iteratoa Joura of Cotemporary Mathematca Scece, vo.1,o.6,pp , 011. [16] H. M. Srvatava ad H. L. Maocha, A Treate o Geeratg Fucto, Hated Pre, E Horwood Lmted, Chcheter, UK, Joh Wey & So, New York, NY, USA, [17] A. Na ad P. Haukkae, O geerazed Fboacc ad Luca poyoma, Chao, Soto ad Fracta,vo.4,o.5, pp ,009.

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