ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS

Transcription

1 Ordu Üiv. Bil. Tek. Derg., Cilt:6, Sayı:1, 016,8-18/Ordu Uiv. J. Sci. Tech., Vol:6, No:1,016,8-18 ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS Serpil Halıcı *1 Sia Öz 1 Pamukkale Ui., Sciece ad Arts Faculty,Dept. of Math., KııklıCampus, Deizli, Turkey Pamukkale Ui., Sciece ad Arts Faculty,Dept. of Math., Deizli, Turkey Abstract I this study, we cosider firstly the geeralized Gaussia Fiboacci ad Lucas sequeces. The we defie the Gaussia Pell ad Gaussia Pell-Lucas sequeces. We give the geeratig fuctios ad Biet formulas of Gaussia Pell ad Gaussia Pell- Lucas sequeces. Moreover, we obtai some importat idetities ivolvig the Gaussia Pell ad Pell-Lucas umbers. Keywords. Recurrece Relatio, Fiboacci umbers, Gaussia Pell ad Pell-Lucas umbers. Özet Bu çalışmada, öce geelleştirilmiş Gaussia Fiboacci ve Lucas dizilerii dikkate aldık. Sora, Gaussia Pell ve Gaussia Pell-Lucas dizilerii taımladık. Gaussia Pell ve Gaussia Pell-Lucas dizilerii Biet formüllerii ve üreteç foksiyolarıı verdik. Üstelik, Gaussia Pell ve Gaussia Pell-Lucas sayılarıı içere bazı öemli özdeşlikler elde ettik. AMS Classificatio. 11B37, 11B39. 1 * shalici@pau.edu.tr, 8

2 S. Halıcı, S. Öz 1. INTRODUCTION From (Horadam 1961; Horadam 1963) it is well kow sequece {U }, Geeralized Fiboacci U +1 = pu + qu 1, U 0 = 0 ad U 1 = 1, ad geeralized Lucas sequece {V } are defied by V +1 = pv + qv 1, V 0 = ad V 1 = p, where p ad q are ozero real umbers ad 1. For p = q = 1, we have classical Fiboacci ad Lucas sequeces. For p =, q = 1, we have Pell ad Pell- Lucas sequeces. For detailed iformatio about Fiboacci ad Lucas umbers oe ca see (Koshy 001). Moreover, geeralized Fiboacci ad Lucas umbers with egative subscript ca be defied as U = U ( q) ad V = V ( q) respectively. From the reccurece relatio related with these sequeces we ca write p + 4q > 0, α = (p + p + 4q) ad β = (p p + 4q). So, the Biet formulas of geeralized Fiboacci ad Lucas sequeces are give by U = α β α β ad V = α + β. Gaussia umbers are complex umbers z = a + ib,a, b Z were ivestigated Gauss i 183 ad the set of these umbers is deoted by Z[i]. by I Horadam (1963), itroduced the cocept the complex Fiboacci umber called as the Gaussia Fiboacci umber. Ad the, Jorda (1965) cosidered two of the complex Fiboacci sequeces ad exteted some relatioship which are kow about the commo Fiboacci sequeces. Also the author gave may idetities related with them. For example, for these sequeces some of idetities are give by GF = F + if 1, GF = igf, GF +1 GF 1 GF = ( 1) ( i), 9

3 O Some Gaussıa Pell Ad Pell-Lucas Numbers GF +1 GF 1 = F 1 (1 + i), GF + GF +1 = F (1 + i), k=0 GF k = GF + 1, for some. The above idetities are kow as the relatioship betwee the usual Fiboacci ad Gaussia Fiboacci sequeces. Horadam ivestigated also the complex Fiboacci polyomials. I Berzseyi (1977), preseted a atural maer of extesio of the Fiboacci umbers ito the complex plae ad obtaied some iterestig idetities for the classical Fiboacci umbers. Moreover, the author gave a closed form to Gaussia Fiboacci umbers by the Fiboacci Q matrix. I Harma 1981, gave a extesio of Fiboacci umbers ito the complex plae ad geeralized the methods give by Horadam (1963); Berzseyi (1977). I Ascı&gurel (013), the authors studied the geeralized Gaussia Fiboacci umbers. The they gave the sums of geeralized Gaussia Fiboacci umbers by the matrix method. The authors studied also the Gaussia Jacobsthal ad Gaussia Jacobsthal Lucas umbers. I this study, we defie ad study the Gaussia Pell ad Gaussia Pell-Lucas sequeces. We give geeratig fuctios ad Biet formulas for these sequeces. Moreover, we obtai some importat idetities ivolvig the Gaussia Pell ad Pell-Lucas umbers. Now, let we defie the geeralized Gaussia Fiboacci sequece U (p, q; a, b ) as follows. GU +1 = pgu +q GU 1 GU 0 = a, GU 1 = b (.1) where a ad b are iitial values. If we take p = q = 1, a = i, b = 1 i the equatio (.1), the we get the Gaussia Fiboacci sequece GU (1,1; i, 1 ) that is {GF } = {i, 1, 1 + i, + i, 3 + i, }. If we take p = q = 1, a = i, b = 1 + i i the equatio (.1), the we get the Gaussia Lucas sequece {GL } = { i, 1 + i, 3 + i, 4 + 3i, 7 + 4i, }. 10

4 S. Halıcı, S. Öz If we take p =, q = 1, a = i, b = 1 i the equatio (.1), the we get the Gaussia Pell sequece {GP } = {i, 1, + i, 5 + i, 1 + 5i, }. If we take p =, q = 1, a = i, b = + i i (.1), the we get the Gaussia Pell-Lucas sequece {GQ } = { i, + i, 6 + i, i, i, }. Also we have GP = P + ip 1 ad GQ = Q + iq 1, where P ad Q are the th Pell ad Pell-Lucas umbers, respectively.. GAUSSIAN PELL AND GAUSSIAN PELL-LUCAS SEQUENCES I this sectio, we cosider Gaussia Pell ad Gaussia Pell-Lucas sequeces. We give the Biet formulas for these sequeces. The we obtai the geeratig fuctios ad we give some idetities ivolvig these sequeces. Biet formulas are well kow formulas i the theory Fiboacci umbers. These formulas ca also be carried out to the Gaussia Pell umbers. I the followig theorem we give the Biet formulas for Gaussia Pell umbers. THEOREM 1. Biet formulas for Gaussia Pell ad Gaussia Pell-Lucas sequeces are give by ad respectively. GP = α β α β + i αβ βα α β, 0 GQ = (α + β ) i(αβ + βα ), 0 PROOF. From the theory of differece equatios we kow the geeral term of Gaussia Pell umbers ca be expressed i the followig form 11

5 O Some Gaussıa Pell Ad Pell-Lucas Numbers GP (x) = cα (x) + dβ (x), where c ad d are the coefficiets. Usig the values = 0, 1 1 ( 1) i 1 ( 1) i c, d. ca be writte. Cosiderig the values GP = α β α β + i αβ βα α β, I additio to this, we get GQ = (α + β ) i(αβ + βα ). cd, ad makig some calculatios, we obtai For Gaussia Pell-Lucas sequece {GQ } geeratig fuctio g(t) is a formal power series. The geeratig fuctio g(t) of the sequece {GP } is defied by g(t) = =0 GP t. The we ca give the geeratig fuctios for the Gaussia Pell ad Gaussia Pell- Lucas sequeces i the followig theorem. THEOREM. The geeratig fuctios to the Gaussia Pell ad Gaussia Pell-Lucas sequeces are ad respectively. g(t) = t+i(1 t) 1 t t, g(t) = ( t)+i(6t ) 1 t t, PROOF. Let g(t) be the geeratig fuctio of sequece {GP }. The we ca write ( ) g t GP t GP GPt GP t GP t 1

6 S. Halıcı, S. Öz If we use the recursive relatio of this sequece, the we get Thus, we obtai t (1 t) i gt (), 1 t t which is desired. 1 g t t t GP GP GP t Similarly, the geeratig fuctio of Gaussia Pell-Lucas sequece {GQ } g(t) = ( t)+i(6t ) 1 t t. ca be obtaied. Moreover, we have the egatively subscripted terms of the sequeces {GP } ad {GQ } by usig the recursive relatio, GP GP GP ( 1) ( ) ad GQ GQ GQ ( 1) ( ) respectively. Notice that GP = P + ip 1 ad GQ = Q + iq 1. I the followig theorem, we give the relatios betwee the Gaussia Pell ad Gaussia Pell-Lucas sequeces ivolvig the egative idices. THEOREM 3. For 1, we have the followig idetities. i) GP = ( 1) 1 (P ip +1 ) ii) GQ = ( 1) (Q iq +1 ) PROOF. The proof ca be see by the mathematical iductio o. 13

7 O Some Gaussıa Pell Ad Pell-Lucas Numbers I the followig Corollary, we give some useful idetities cocerig the Gaussia Pell ad Gaussia Pell-Lucas umbers, ad also give some sum formulas for these umbers without proof. COROLLARY 1. For 1, we have the followig equatios i) 4GP = GQ + GQ 1, ii) GP + GP +1 = P (1 + i), iii) GQ + GQ +1 = 16P Q (1 + i). It is well kow that the Cassii idetity is oe of the oldest idetities ivolvig the Fiboacci umbers. I the followig theorem, we give the Cassii formula related with the Gaussia Pell ad Pell-Lucas umbers. THEOREM 4. (Cassii Formula) Let 1. The we have i) GP +1 GP 1 GP = ( 1) (1 i) ii) GQ +1 GQ 1 GQ = ( 1) +6(1 i), respectively. PROOF. By usig the mathematical iductio method we get GP GP GP 1 i ; GP i, GP i k GP GP GP i 1 1 k 1 k 1 k GP GP GP GP GP GP GP GP k k k 1 k 1 k k 1 k 1 k 1 GP GP GP GP 1 1 i GP k k k 1 k k 1 k GPk GPk 1 GPk ( GPk 1 GPk ) 1 1 i k 1 14

8 S. Halıcı, S. Öz k 1 i. Similarly, we ca prove the other formula by the mathematical iductio method. Thus, the proof is completed. THEOREM 5. For the Gaussia Pell umbers, we have the followig formula. 1 [ (1 )] GPj GP 1 GP i. j0 PROOF. From the recursive relatio we ca write ad GP GP GP GP GP GP GP1 GP GP0 GP GP3 GP1 0 The, we obtai GP GP GP j0 1 1 GP 1 GP 1 i GP GP 1 i GPj 1 P 1 GP0 GP GP G 15

9 O Some Gaussıa Pell Ad Pell-Lucas Numbers This completes the proof. THEOREM 6. For all N, we have the followig sum formula GQ = 1 j=0 (GQ +1 + GQ ) i. I the followig corollary, we give some summatio formulas for the Gaussia Pell ad Pell-Lucas umbers. COROLLARY. For 1, we have i) GP j = 1 j=0 (GP +1 + i 1), ii) j=0 GQ j = 1 GQ +1 + (1 3i). iii) j=1 GP j 1 ad iv) j=1 GQ j 1 = 1 (GP i), = 1 GQ (1 i). The above equalities ca be see by Theorem 6. THEOREM 7 (Catala Formulas) For ozero positive itegers, k we have i) GP +k GP k GP = ( 1) (1 i) [1 + ( 1)k+1 (α k + β k ) ], 4 ii) GQ +k GQ k GQ = ( 1) +1 (1 i)[4 + ( 1) k+1 (α k + β k ) ]. 16

10 S. Halıcı, S. Öz PROOF. For the first equality, from the Biet formula k k k k k k k k [ i][ i] [ i] k k k k k k ( ) ( 1)( ) ( ) ( ) ( 1)( ) ( ) 4 [4 ] i ( ) ( ) ( ) ( ) k1 k k ( 1) ( ) ( 1) (1 i)[1 ] 4 ca be writte which is desired. Usig the same method the other formula ca be give easily. THEOREM 8 (d Ocage s Idetity) For all m, Z we have 1 i) GP 1GP GP GP 1 ( 1) (1 i) P m m m where P is the th Pell umber. ii) GQ m+1 GQ GQ m GQ +1 = 16( 1) (1 i)p m PROOF. By usig the Biet formula fort he Gaussia Pell-Lucas sequece, the proof ca be easily see. 3. CONCLUSION I coclusio, we firstly cosider the geeralized Gaussia Fiboacci ad Lucas sequeces. The we itroduce the Gaussia Pell ad Gaussia Pell-Lucas sequeces. We give the geeratig fuctios ad Biet formulas of Gaussia Pell ad Gaussia Pell-Lucas sequeces. Furthermore, we obtai some importat idetities ivolvig the terms of these sequeces. 17

11 O Some Gaussıa Pell Ad Pell-Lucas Numbers REFERENCES A. F. Horadam, Geeralized Fiboacci Sequece, America Math. Mothly, 68(1961), A. F. Horadam, Complex Fiboacci Numbers ad Fiboacci Quaterios. America Math. Mothly, 70 (1963), T. Koshy, Fiboacci ad Lucas Numbers With Applicatios, A Wiley-Itersciece Publicatio, (001). J. H. Jorda, Gaussia Fiboacci ad Lucas Numbers, Fib. Quart., 3(1965), G. Berzseyi, Gaussia Fiboacci Numbers. Fib. Quart., (1977), 15(3), C. J. Harma, Complex Fiboacci Numbers. The Fib. Quart., (1981), 19(1), M. Aşcı, E. Gurel, Gaussia Jacobsthal ad Gaussia Jacobsthal Lucas Numbers. Ars Combiatoria, 111 (013),