Review Article Incomplete Bivariate Fibonacci and Lucas p-polynomials

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1 Discrete Dyamics i Nature ad Society Volume 2012, Article ID , 11 pages doi: /2012/ Review Article Icomplete Bivariate Fiboacci ad Lucas p-polyomials Dursu Tasci, 1 Mirac Ceti Firegiz, 2 ad Naim Tuglu 1 1 Departmet of Mathematics, Faculty of Sciece, Gazi Uiversity, Teioullar, Aara, Turey 2 Departmet of Mathematics, Faculty of Educatio, Başet Uiversity, Baglica, Aara, Turey Correspodece should be addressed to Mirac Ceti Firegiz, mceti@baset.edu.tr Received 26 November 2011; Accepted 8 February 2012 Academic Editor: Gerald Teschl Copyright q 2012 Dursu Tasci et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited. We defie the icomplete bivariate Fiboacci ad Lucas p-polyomials. I the case x 1, y 1, we obtai the icomplete Fiboacci ad Lucas p-umbers. If x 2, y 1, we have the icomplete Pell ad Pell-Lucas p-umbers. O choosig x 1, y 2, we get the icomplete geeralized Jacobsthal umber ad besides for the icomplete geeralized Jacobsthal-Lucas umbers. I the case x 1, y 1,, we have the icomplete Fiboacci ad Lucas umbers. If x 1, y 1,, 1/, we obtai the Fiboacci ad Lucas umbers. Also geeratig fuctio ad properties of the icomplete bivariate Fiboacci ad Lucas p-polyomials are give. 1. Itroductio Dordević itroduced icomplete geeralized Fiboacci ad Lucas umbers usig explicit formulas of geeralized Fiboacci ad Lucas umbers i 1.I2 icomplete Fiboacci ad Lucas umbers are give as follows: F L 1 1, 0, 2, 0 2, 1.1 where 1, 2, 3,... Note that for the case 1/2 icomplete Fiboacci umbers are reduced to Fiboacci umbers ad for the case /2 icomplete Lucas umbers are

2 2 Discrete Dyamics i Nature ad Society reduced to Lucas umbers i 2. Also the authors cosidered the geeratig fuctios of the icomplete Fiboacci ad Lucas umbers i 3. I4 Dordević ad Srivastava defied icomplete geeralized Jacobsthal ad Jacobsthal-Lucas umbers. The geeralized Fiboacci ad Lucas p-umbers were studied i 5, 6. Icomplete Fiboacci ad Lucas p-umbers are defied by F p L p p 1, 0 p, 0 p 1,, 1.2 for 1i7.I8 the authors itroduced icomplete Pell ad Pell-Lucas p-umbers. The geeralized bivariate Fiboacci p-polyomials F p, x, y ad geeralized bivariate Lucas p-polyomials L p, x, y are defied the recursio for p 1 F p, ) xfp, 1 ) yfp, p 1 ), > p, 1.3 with F p,0 ) 0, Fp, ) x 1 for 1, 2,...p, 1.4 ad L p, ) xlp, 1 ) ylp, p 1 ), > p, 1.5 with L p,0 ), Lp, ) x for 1, 2,...p 1.6 i 5. Whe x y 1, F p, 1, 1 F p. I5, the authors obtaied some relatios for these polyomials sequeces. I additio, i 5, the explicit formula of bivariate Fiboacci p-polyomials is 1/p1 p 1 F p, x, y x p1 1 y, 0, p 1, 1.7 ad the explicit formula of bivariate Lucas p-polyomials is /p1 L p, x, y p x p1 y, 0, p p

3 Discrete Dyamics i Nature ad Society 3 I this paper, we defied icomplete bivariate Fiboacci ad Lucas p-polyomials. We geeralize icomplete Fiboacci ad Lucas umbers, icomplete geeralized Fiboacci umbers, icomplete geeralized Jacobsthal umbers, icomplete Fiboacci ad Lucas p-umbers, icomplete Pell ad Pell-Lucas p-umbers. 2. Icomplete Bivariate Fiboacci ad Lucas p-polyomials Defiitio 2.1. For p 1, 1, icomplete bivariate Fiboacci p-polyomials are defied as F p, p 1 x, y x p1 1 y, For x 1, y 1, Fp,x, y Fp, we get icomplete Fiboacci p-umbers 7. If x 2, y 1, Fp,x, y Pp, we obtaied icomplete Pell p-umbers 8. O choosig x 1, y 2, Fp,x, y J,p1, we have icomplete geeralized Jacobsthal umbers 4. If x 1, y 1, p 1, Fp,x, y F, we get icomplete Fiboacci umbers 2. For x 1, y 1, p 1, 1/,Fp,x, y F, we obtaied Fiboacci umbers 9. Defiitio 2.2. For p 1, 1, icomplete bivariate Lucas p-polyomials are defied as L p p, x, y x p1 y, p If x 1, y 1, L p,x, y L p, we obtaied icomplete Lucas p-umbers 7. For x 2, y 1, L p,x, y Qp, we have icomplete Pell-Lucas p-umbers 8. O choosig x 1, y 2,, L p,x, y,p1, we get icomplete geeralized Jacobsthal-Lucas umbers 4. If x 1, y 1, p 1, L p,x, y L, we obtaied icomplete Lucas umbers 2. For x 1, y 1,, /, L p,x, y L, we have Lucas umbers 9. Propositio 2.3. The icomplete bivariate Fiboacci p-polyomials satisfy the followig recurrece relatio: F 1 p, ) xf 1 p, 1 ) yf p, p 1 ), 0 p

4 4 Discrete Dyamics i Nature ad Society Proof. Usig 2.1, weobtai xf 1 p, 1 x, y yf p, p 1 x, y 1 p 2 x x p1 2 y y p p 2 x p p1 2 y 1 p 2 p p 2 x p1 1 y x p p1 2 y 1 1 p 2 1 p 2 x p1 1 y x p1 1 y 1 [ ] 1 p 2 p 2 2 x p1 1 y p 1 x p1 1 y 0 1 x Fp, 1 x, y. Taig x y 1i2.3, we could obtai a formula for icomplete Fiboacci p- umbers see 7,Propositio3. Taig x y i2.3, we could obtai a formula for icomplete Fiboacci umbers see 2, Propositio1. Propositio 2.4. The ohomogeeous recurrece relatio of icomplete bivariate Fiboacci p- polyomials is Fp, x, y xf p, 1 x, y yf p 1 2 p, p 1 x, y x p1 2 y Proof. It is easy to obtai from 2.1 ad 2.3. Propositio 2.5. For 0 h p 1/, oe has h h y h x F x, y F h p,p 1 p,p1h p ). 2.6

5 Discrete Dyamics i Nature ad Society 5 Proof. Equatio 2.6 clearly holds for h 0. Suppose that the equatio holds for h>0. We show that the equatio holds for h 1. We have h1 h 1 y h1 x F x, y p,p 1 [ ] h1 h h y h1 x F x, y 1 p,p 1 h1 h y h1 x F h1 h x, y p,p 1 1 ( yf h h p,p1h p x, y h 1 h h y h x 1 F 1 p,p 1 ) ) ( yf h h h p,p1h p x, y x x h1 F h1 p,ph x, y h y h1 F p, p x, y 1 yf h p,p1h p x, y xf h1 p,p1h x, y ) y h x F 1 p,p x, y y h1 x F x, y p,p F h1 p,p1h1 x, y. Propositio 2.6. For p 2, h 1 y x F p, p 1 x, y F1 xh 1 p,h x, y xf 1 x, y. 2.8 p, Proof. Equatio 2.8 ca be easily proved by usig 2.3 ad iductio o h. We have the followig propositio i which the relatioship betwee the icomplete bivariate Fiboacci ad Lucas p-polyomials is preserved as foud i 5 before. Propositio 2.7. Oe has L p, x, y F p,1 x, y pyf 1 p, p x, y,

6 6 Discrete Dyamics i Nature ad Society Proof. By 2.1, rewrite the right-had side of 2.9 as F p,1 x, y pyf 1 p 1 p p 1 p, p x, y x p1 y py x p p1 1 y p p 1 x p1 y py 1 x p1 y 1 [ ] p p 1 1 x p1 y 1 1 p x p1 y p 1 x L p,( x, y ) Propositio 2.8. The icomplete bivariate Lucas p-polyomials satisfy the followig recurrece relatio: L 1 p, x, y xl 1 p, 1 x, y yl p 2 p, p 1 x, y, Proof. We write by usig 2.3 ad 2.9 L 1 p, x, y F 1 p,1 x, y pyf p, p x, y xfp, 1 [ x, y yf p, p x, y py xf ] p, p 1 x, y yf 1 p, 2p 1 x, y [ x F 1 p, ) x, y pyf p, p 1( ] [ x, y y F p, p ] x, y pyf 1 p, 2p 1 x, y 2.12 xl 1 ) p, 1 x, y yl p, p 1( x, y. Propositio 2.9. The ohomogeeous recurrece relatio of icomplete bivariate Lucas p- polyomials is L p, x, y xl p, 1 x, y yl p, p 1 x, y p 1 p 1 1 x p11 y 1. p Proof. The proof ca be doe by usig 2.2 ad 2.11.

7 Discrete Dyamics i Nature ad Society 7 Propositio For 0 p h/, oe has h h x y h L x, y L h p,p 1 p,p1h p ) Proof. Proof is similar to the proof of Propositio 2.5. Propositio For 1, oe has h 1 y x L p, p 1 x, y L1 xh 1 p,h x, y xl 1 x, y p, Proof. Proof is obtaied immediately by usig 2.11 ad iductio h. Propositio Oe has /p1 L p, 0 ( x, y ) 1 L p, x, y [ ] xfp, x, y Lp, x, y Proof. We ca write from 2.2 /p1 L p, 0 x, y L 0 p, x, y L 1 p, x, y L 2 /p1 p, x, y L x, y [ x x ] p x p1 y 0 0 p 1 [ x ( ] p x p1 2p y )x 2p1 y 2 0 p 1 2p 2 x 0 / ( ) p p x p1 /p1 y /p1 p,

8 8 Discrete Dyamics i Nature ad Society ( ) ( ) p 1 x 1 1 x p1 y 0 p 1 ( ) 2p 1 2 x 2p1 y 2 2p 1 / ( ) p p x p1 /p1 y /p1 /p1 ( ) p 1 x p1 y p ( ) /p1 p 1 x p1 y p /p1 p x p1 y. p 2.17 Equatio 2.17 is calculated usig the formula L p, x, y ad L p, x, y/ x F p, x, y5 /p1 L p, 0 ( x, y ) /p1 1 p F p, x, y x 1 L p, x, y x 1 p x p1 y 2.18 ( ) [ ] 1 L p, x, y xfp, x, y Lp, x, y. The we have the followig coclusio. Coclusio 1. Whe x y i2.16, weobtai L /2 0 ( ) 1 L 2 2 F L 2.19 which is Propositio 11 i 2.

9 Discrete Dyamics i Nature ad Society 9 3. Geeratig Fuctios of the Icomplete Bivariate Fiboacci ad Lucas p-polyomials Lemma 3.1 see 3. Let {s } 0 be a complex sequece satisfyig the followig ohomogeeous recurrece relatio: s xs 1 ys p 1 r, > p, 3.1 where {r } is a give complex sequece. The the geeratig fuctio S px, y; t of the sequece {s } is [ S p x, y; t s 0 r 0 p [ s i xs i 1 r i t i Gt] 1 xt yt p1] 1, 3.2 i1 where Gt deotes the geeratig fuctio of {r }. Theorem 3.2. The geeratig fuctio of the icomplete bivariate Fiboacci p-polyomials is [ R p p x, y; t t p11 F p,p11 x, y t i( ) F p,p11i x, y xfp,p1i x, y i1 ] 3.3 y1 t p1 [ 1 xt yt p1] 1. 1 xt 1 Proof. From 2.1 ad 2.5, F p,x, y 0for0 <1, F p,p11( x, y ) Fp,p11 ), F p,p12( x, y ) Fp,p12 ),. 3.4 F p,p1p1( x, y ) Fp,p1p1 ), ad for p 2 Fp, x, y xf p, 1 x, y yf p 1 2 p, p 1 x, y ( ) x p1 2 y 1. p Now let s 0 F p,p11 x, y, s1 F p,p12 x, y,..., sp F p,p1p1 x, y, 3.6

10 10 Discrete Dyamics i Nature ad Society ad s F p,p11( x, y ). 3.7 Also r 0 r 1 r p 0, p 1 r p 1 x p 1 y We obtaied that Gt y 1 t p1 /1 xt 1 is the geeratig fuctio of the sequece {r }. From Lemma 3.1, we get that the geeratig fuctio S px, y; t of sequece {s } is [ S p x, y; t F p ( p,p11 x, y t i i1 F p,p11i ) xf p,p1i ) ) ] 3.9 y1 t p1 [ 1 xt yt p1] 1. 1 xt 1 Therefore, R p; t ) t p11 S p; t ) Theorem 3.3. The geeratig fuctio of the icomplete bivariate Lucas p-polyomials is [ Wp p x, y; t t p1 L p,p1 x, y t i( ) L p,p1i x, y xlp,p1i 1 x, y i1 tp1 y 1[ p1 xt 1 ] 1 xt 1 ] [ 1 xt yt p1] Proof. From 2.9 ad 3.3, W p ; t ) L p, 0 ) t [ F ] p,1 x, y pyf 1 p, p x, y t 0 Fp,1( ) x, y t py F 1 x, y t 0 0 p, p 3.12 t 1 R p x, y; t pyt p R 1 p x, y; t.

11 Discrete Dyamics i Nature ad Society 11 For the geeral case i Theorems 3.2 ad 3.3, we fid the geeratig fuctios of some special umbers by the special cases x, y, p. For example, x y 1i3.3 we obtai the geeratig fuctio of icomplete Fiboacci p-umbers. Refereces 1 G. B. Dordević, Geeratig fuctios of the icomplete geeralized Fiboacci ad geeralized Lucas umbers, The Fiboacci Quarterly, vol. 42, o. 2, pp , P. Filippoi, Icomplete Fiboacci ad Lucas umbers, Redicoti del Circolo Matematico di Palermo. Serie II, vol. 45, o. 1, pp , Á. Pitér ad H. M. Srivastava, Geeratig fuctios of the icomplete Fiboacci ad Lucas umbers, Redicoti del Circolo Matematico di Palermo. Serie II, vol. 48, o. 3, pp , G. B. Dordević ad H. M. Srivastava, Icomplete geeralized Jacobsthal ad Jacobsthal-Lucas umbers, Mathematical ad Computer Modellig, vol. 42, o. 9-10, pp , N. Tuglu, E. G. Kocer, ad A. Stahov, Bivariate Fiboacci lie p-polyomials, Applied Mathematics ad Computatio, vol. 217, o. 24, pp , E. G. Kocer ad N. Tuglu, The Biet formulas for the Pell ad Pell-Lucas p-umbers, Ars Combiatoria, vol. 85, pp. 3 17, D. Tasci ad M. C. Firegiz, Icomplete Fiboacci ad Lucas p-umbers, Mathematical ad Computer Modellig, vol. 52, o. 9-10, pp , D. Tasci, M. Ceti Firegiz, ad G. B. Dordević, Icomplete Pell ad Pell-Lucas p-umbers, Submit. 9 T. Koshy, Fiboacci ad Lucas Numbers with Applicatios, Pure ad Applied Mathematics, Wiley, New Yor, NY, USA, 2001.

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