Bivariate Vieta-Fibonacci and Bivariate Vieta-Lucas Polynomials

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1 IOSR Joural of Mathematcs (IOSR-JM) e-issn: 78-78, p-issn: 19-76X. Volume 1, Issue Ver. II (Jul. - Aug.016), PP -0 Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals E. Gokce KOCER 1 1 (Departmet of Mathematcs-Computer Sceces, Uverst of Necmett Erbaka, Turke) Abstract: I ths paper, we cosder the bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals whch are geeralzed of Veta-Fboacc, Veta-Lucas, Veta-Pell, Veta-Pell-Lucas polomals. Also, we gve the some propertes. Afterwards, we obta the some dettes for the bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals b usg the kow propertes of bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals. Kewords: Bet s Formula, Fboacc Polomals, Lucas Polomals, Veata Polomals. I. Itroducto I [1], Horadam cosder the Veta-Fboacc ad Veta-Lucas polomals whch are defed b the followg recurrece relatos V V V wth ad wth 1 V 0, V 1 v v v 1 v, v. Also, the author gve the relatoshps amog Veta, Jacobsthal ad Morga-Voce polomals b usg the kow coectos wth Fboacc, Lucas ad Chebshev polomals. I [, ], the author defed the bvarate Fboacc ad bvarate Lucas polomals ad gve the some propertes of these polomals. Also, Catal obta the some dettes for Bvarate Fboacc ad Bvarate Lucas polomals derved from a book of Gould. I [], Catal defed the geeralzed bvarate Fboacc polomal ad gve the summato ad verso formulas. Swamm gve the geeralzed Fboacc ad Lucas polomals ad ther assocated dagoal polomals []. I [6], Adre-Jea defe a geeral class of polomals b the followg recurrece relatos U p, q; pu 1 p, q; qu p, q;, wth U p, q; 0, U p, q; 1. Partcular cases of U, ; DOI: / Page p q are Fboacc polomals, Pell polomals, the frst Fermat polomals ad the Morga-Voce polomals of the secod kd. Also, the author gve the combatoral propertes of the polomals U p, q;. I [7], Djordjevc defe a geeralzato of the polomals U, ; Jea. Also, the author obta the some propertes of the geeralzed polomals. p q whch s gve b Adre- I [8], the author defe the ew polomals b usg the polomal U, ; p q ad vestgate the propertes of a ew polomals. I [9], Robbs developed some propertes of a specal fte tragular arra whch was dscovered b Veta. Also, the author prove some rreducblt propertes of Veta polomals. I [10], the authors defe the Veta-Pell ad Veta-Pell-Lucas polomals ad gve the propertes of these polomals. I [11], the authors troduce the geeralzed Veta-Jacobsthal ad Veta-Jacobsthal-Lucas polomals. Also, varous famles of multlear ad multlateral geeratg fuctos for these polomals are derved. I lght of the foregog, we ca cosder the bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals. Clearl, these polomals are a geeralzato of the Veta-Fboacc, Veta-Lucas, Veta-Pell ad Veta-Pell-Lucas polomals.

2 Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals I the rest sectos of ths paper, we gve the defto of bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals. Afterwards, we obta some propertes of these polomals ad gve the Pascal arras geeratg these polomals. II. Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals Defto.1. Let be teger. The recurrece relatos of the bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals are V, V, V, (.1) 1 wth the tal codtos ad V, 0, V, 1 v, v 1, v, where v0,, v1,. The frst few terms of V, ad v, V, v, (.) polomals are as followg table The characterstc equato of the bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals s 0 (.) Let ad be the roots of the characterstc equato (.). ad satsf the followg equatos as ad,,. Usg the stadart tecques, we have the Bet s formulas of V, ad v, V, polomals v,. (.) The geeratg fuctos for the fte sets of polomals V, ad, the usual wa to be ad (.) v are foud, V 1 1 t t t t (.6) 0, v 1 1 t t t t. (.7) 0 We ca also eted the defto of V, ad v, to the egatve de DOI: / Page

3 ad 1 V V,, Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals (.8) 1 v v,,. (.9) Some Iterrelatoshp The relatoshps betwee V, ad v, V, V, v, 1 1 III. Elemetar Propertes,,, v v V 1 1,,, V, V, v, V V V 1 1,,, v v V 1 1 v, v, v, ca be gve as follows,,,,,, 1,, 1,,,,,, v V v v V v v V V v v V V v Summato Formulas polomals as ad Usg the Bet s formulas (.)-(.), we ca gve the summato of the V, ad v, V 1, V, 1 Vk, (.1) k 0 1 v 1, v, vk, (.) k 0 where 1. 1 Eplct Formulas Iducto ca be used, wth a lttle effort, to establsh the eplct formulas of bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals as ad (.), 1 V 0. (.), 1 v DOI: / Page

4 Dfferetato Formulas Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals If we use the eplct formula (.) of bvarate Veta-Lucas Polomal v, v, v, V,, V, Usg (.) ad recurreces (.1)-(.), we ca gve the followg equatos,, 1 v v v, V, V, 1 V, v 1, v, 1 v 1, v, V, 0, we have. (.) Some Idettes If we use the Bet s formulas (.) - (.) ad the eplct formulas (.)-(.), we have the followg dettes for the bvarate Veta-Fboacc ad bvarate Veta-Lucas polomals. 1 V V, 0 1 v 1, 1 v1, v 1, v, 1 1, 1 (.6) (.7) (.8) (.9) Cass ad Hosberger Formulas Bvarate Veta-Fboacc polomals are geerated b the matr Q, 1 Q 0. It ca be proved b mathematcal ducto o V 1,, V Q V, V 1, V, s th bvarate Veta-Fboacc polomal. Usg the determats of the matrces Q where ad Q, we ca gve the Cass dett for the bvarate Veta-Fboacc polomals as,,, V 1 1 V 1 V. (.10) Smlarl, the Cass dett for the bvarate Veta-Lucas polomals s,,, v 1 1 v 1 v. (.11) Also, From the matr Q, we have the other dett whch s called Hosberger Formula as V, V, V, V, V,. (.1) m m1 m 1 DOI: / Page

5 Takg m the formula (.1), we have the followg dett,,,, 1 Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals V V V V (.1) Usg 1 stead of m the formula (.1), we have the other dett as follows,,, V V V (.1) 1 1 IV. Pascal Arras Geeratg Bvarate Veta-Fboacc Polomals ad Bvarate Veta-Lucas Polomals We cosder the followg table. Deote the coeffcet of the power of ad the th row ad th colum b F, DOI: / Page 10 Table1: Bvarate Veata-Fboacc Polomals from Rsg Dagoals Defe the etres the row as the terms the epaso 1, that s 1 1 F,,. (.1) 1 From (.1), we obta 1 F, 1. (.) Now, usg the rsg dagoal les Table 1, we have F 1,. 1 Usg (.), we obta F , 1 1 From the eplct formula (.), we have F 1, V, Namel, the sum of the elemets o the rsg dagoal les the Table 1 s the bvarate Veta- Fboacc polomal, V.

6 Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals Now, we cosder the Table. Deote the coeffcet of the power of ad the th row ad th colum b L, Table : Bvarate Veata-Lucas Polomals from Rsg Dagoals 6 We ma defe the etres the row as the terms the epaso of that s L,. (.) 1 Usg (.) ad Pascal s formula, we obta 1 L, 1. (.) Now, usg the rsg dagoal les Table, we have 1 L 1,. 1 Usg (.), we obta L 1 1 1, From the eplct formula (.), we have 1 1 L 1, v,. 1. It clearl, the sum of the elemets o the rsg dagoal les the Table s the bvarate Veta- Lucas polomal, v. V. Cocluso I a future paper, we shall vestgate the sequeces V ad defed b the recurreces relatos DOI: / Page p,, p,, v of polomals,

7 ,,, p, p, 1 p, p1 Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals V V V for p wth the tal codtos ad 1 V, 0, V,, 1,,, p p,0 p,,,, v v v for p p, p, 1 p, p1 wth the tal codtos v, p 1, v,, 1,,, p. p,0 p, Refereces [1] A. F. Horadam, Veta Polomals, The Fboacc Quarterl, 0(), 00, -. [] M. Catal, Some Formulae for Bvarate Fboacc ad Lucas Polomals, arxv: math.co/006 v1, 16 Ju 00. [] M. Catal, Idettes for Fboacc ad Lucas Polomals derved from a book Gould, arxv: math.co/00710 v1, 7 Jul 00. [] M. Catal, Geeralzed Bvarate Fboacc Polomals, arxv: math.co/01166 v, Ju 00. [] M. N. S. Swamm, Geeralzed Fboacc ad Lucas Polomals ad Ther Assocated Dagoal Polomals, The Fboacc Quarterl, 7(), 1999, 1-. [6] R. Adre-Jea, A Note o a Geeral Class of Polomals, The Fboacc Quarterl, (), 199, -. [7] G. B. Djordjevc, Some Propertes of a Class of Polomals, Math. Ves., 9, 1997, [8] M. N. S. Swamm, O a Class of Geeralzed Polomals, The Fboacc Quarterl,, 1997, 9-. [9] N. Robbs, Veta s Tragular Arra ad a Related Faml of Polomals, Iterat. J. Math. & Math. Sc., 1(), 1991, 9-. [10] D. Tasc, F. Yalc, Veta-Pell ad Veta-Pell-Lucas Polomals, Advaces Dfferece Equatos,,01, 1-8. [11] N. F. Yalc, D. Tasc, E. Erkus-Duma, Geeralzed Veta-Jacobsthal ad Veta-Jacobsthal-Lucas Polomals, Mathematcal Commucatos, 0,01, 1-1. DOI: / Page