Ordu Üniv. Bil. Tek. Derg., Cilt:6, Sayı:1, 2016,8-18/Ordu Univ. J. Sci. Tech., Vol:6, No:1,2016,8-18 ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS Serpil Halıcı*1 Sinan Öz2 1Pamukkale Uni., Science and Arts Faculty,Dept. of Math., KınıklıCampus, Denizli, Turkey 2Pamukkale Uni., Science and Arts Faculty,Dept. of Math., Denizli, Turkey Abstract In this study, we consider firstly the generalized Gaussian Fibonacci and Lucas sequences. Then we define the Gaussian Pell and Gaussian Pell-Lucas sequences. We give the generating functions and Binet formulas of Gaussian Pell and Gaussian Pell- Lucas sequences. Moreover, we obtain some important identities involving the Gaussian Pell and Pell-Lucas numbers. Keywords. Recurrence Relation, Fibonacci numbers, Gaussian Pell and Pell-Lucas numbers. Özet Bu çalışmada, önce genelleştirilmiş Gaussian Fibonacci ve Lucas dizilerini dikkate aldık. Sonra, Gaussian Pell ve Gaussian Pell-Lucas dizilerini tanımladık. Gaussian Pell ve Gaussian Pell-Lucas dizilerinin Binet formüllerini ve üreteç fonksiyonlarını verdik. Üstelik, Gaussian Pell ve Gaussian Pell-Lucas sayılarını içeren bazı önemli özdeşlikler elde ettik. AMS Classification. 11B37, 11B39.1 * [email protected], 8 S. Halıcı, S. Öz 1. INTRODUCTION From (Horadam 1961; Horadam 1963) it is well known Generalized Fibonacci sequence {푈푛}, 푈푛+1 = 푝푈푛 + 푞푈푛−1 , 푈0 = 0 푎푛푑 푈1 = 1, and generalized Lucas sequence {푉푛} are defined by 푉푛+1 = 푝푉푛 + 푞푉푛−1 , 푉0 = 2 푎푛푑 푉1 = 푝, where 푝 and 푞 are nonzero real numbers and 푛 ≥ 1. For 푝 = 푞 = 1, we have classical Fibonacci and Lucas sequences. For 푝 = 2, 푞 = 1, we have Pell and Pell- Lucas sequences. For detailed information about Fibonacci and Lucas numbers one can see (Koshy 2001). Moreover, generalized Fibonacci and Lucas numbers with negative subscript can be defined as −푈 푉 푈 = 푛 and 푉 = 푛 −푛 (−푞)푛 −푛 (−푞)푛 respectively. From the reccurence relation related with these sequences we can write 푝2 + 4푞 > 0, 훼 = (푝 + √푝2 + 4푞)⁄2 and 훽 = (푝 − √푝2 + 4푞)⁄2 . So, the Binet formulas of generalized Fibonacci and Lucas sequences are given by 훼푛−훽푛 푈 = and 푉 = 훼푛 + 훽푛. 푛 훼−훽 푛 Gaussian numbers are complex numbers 푧 = 푎 + 푏,푎, 푏 ∈ ℤ were investigated by Gauss in 1832 and the set of these numbers is denoted by ℤ[]. In Horadam (1963), introduced the concept the complex Fibonacci number called as the Gaussian Fibonacci number. And then, Jordan (1965) considered two of the complex Fibonacci sequences and extented some relationship which are known about the common Fibonacci sequences. Also the author gave many identities related with them. For example, for these sequences some of identities are given by 퐺퐹푛 = 퐹푛 + 퐹푛−1 , 퐺퐹−푛 = 퐺퐹푛 , 2 푛 퐺퐹푛+1퐺퐹푛−1 − 퐺퐹푛 = (−1) (2 − ), 9 On Some Gaussıan Pell And Pell-Lucas Numbers 2 2 퐺퐹푛+1 − 퐺퐹푛−1 = 퐹2푛−1(1 + 2), 2 2 푛 퐺퐹푛 + 퐺퐹푛+1 = 퐹2푛(1 + 2), ∑푘=0 퐺퐹푘 = 퐺퐹푛+2 − 1, for some 푛 ≥ 2. The above identities are known as the relationship between the usual Fibonacci and Gaussian Fibonacci sequences. Horadam investigated also the complex Fibonacci polynomials. In Berzsenyi (1977), presented a natural manner of extension of the Fibonacci numbers into the complex plane and obtained some interesting identities for the classical Fibonacci numbers. Moreover, the author gave a closed form to Gaussian Fibonacci numbers by the Fibonacci 푄 matrix. In Harman 1981, gave an extension of Fibonacci numbers into the complex plane and generalized the methods given by Horadam (1963); Berzsenyi (1977). In Ascı&gurel (2013), the authors studied the generalized Gaussian Fibonacci numbers. Then they gave the sums of generalized Gaussian Fibonacci numbers by the matrix method. The authors studied also the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers. In this study, we define and study the Gaussian Pell and Gaussian Pell-Lucas sequences. We give generating functions and Binet formulas for these sequences. Moreover, we obtain some important identities involving the Gaussian Pell and Pell-Lucas numbers. Now, let we define the generalized Gaussian Fibonacci sequence 푈푛(푝, 푞; 푎, 푏 ) as follows. 퐺푈푛+1 = 푝퐺푈푛+q 퐺푈푛−1 퐺푈0 = 푎, 퐺푈1 = 푏 (2.1) where 푎 and 푏 are initial values. If we take 푝 = 푞 = 1 , 푎 = , 푏 = 1 in the equation (2.1), then we get the Gaussian Fibonacci sequence GUn(1,1; i, 1 ) that is {퐺퐹푛} = {, 1, 1 + , 2 + , 3 + 2, … }. If we take 푝 = 푞 = 1 , 푎 = 2 − , 푏 = 1 + 2 in the equation (2.1), then we get the Gaussian Lucas sequence {퐺퐿푛} = {2 − , 1 + 2, 3 + , 4 + 3, 7 + 4, … }. 10 S. Halıcı, S. Öz If we take 푝 = 2, 푞 = 1 , 푎 = , 푏 = 1 in the equation (2.1), then we get the Gaussian Pell sequence {퐺푃푛} = {, 1, 2 + , 5 + 2, 12 + 5, … }. If we take 푝 = 2, 푞 = 1 , 푎 = 2 − 2, 푏 = 2 + 2 in (2.1), then we get the Gaussian Pell-Lucas sequence {퐺푄푛} = {2 − 2, 2 + 2, 6 + 2, 14 + 6, 34 + 14, … }. Also we have 퐺푃푛 = 푃푛 + 푃푛−1 and 퐺푄푛 = 푄푛 + 푄푛−1, where 푃푛 and 푄푛 are the 푛푡ℎ Pell and Pell-Lucas numbers, respectively. 2. GAUSSIAN PELL AND GAUSSIAN PELL-LUCAS SEQUENCES In this section, we consider Gaussian Pell and Gaussian Pell-Lucas sequences. We give the Binet formulas for these sequences. Then we obtain the generating functions and we give some identities involving these sequences. Binet formulas are well known formulas in the theory Fibonacci numbers . These formulas can also be carried out to the Gaussian Pell numbers. In the following theorem we give the Binet formulas for Gaussian Pell numbers. THEOREM 1. Binet formulas for Gaussian Pell and Gaussian Pell-Lucas sequences are given by 훼푛−훽푛 훼훽푛−훽훼푛 퐺푃 = + , 푛 ≥ 0 푛 훼−훽 훼−훽 and 푛 푛 푛 푛 퐺푄푛 = (훼 + 훽 ) − (훼훽 + 훽훼 ), 푛 ≥ 0 respectively. PROOF. From the theory of difference equations we know the general term of Gaussian Pell numbers can be expressed in the following form 11 On Some Gaussıan Pell And Pell-Lucas Numbers 푛 푛 퐺푃푛(푥) = 푐훼 (푥) + 푑훽 (푥), where 푐 and 푑 are the coefficients. Using the values 푛 = 0, 1 1 ( 2 1)ii 1 ( 2 1) cd , . 2 2 2 2 can be written. Considering the values cd, and making some calculations, we obtain 훼푛−훽푛 훼훽푛−훽훼푛 퐺푃 = + , 푛 훼−훽 훼−훽 In addition to this, we get 푛 푛 푛 푛 퐺푄푛 = (훼 + 훽 ) − (훼훽 + 훽훼 ). For Gaussian Pell-Lucas sequence {퐺푄푛} generating function (푡) is a formal power series. The generating function (푡) of the sequence {퐺푃푛} is defined by ∞ 푛 (푡) = ∑푛=0 퐺푃푛푡 . Then we can give the generating functions for the Gaussian Pell and Gaussian Pell- Lucas sequences in the following theorem. THEOREM 2. The generating functions to the Gaussian Pell and Gaussian Pell-Lucas sequences are 푡+푖(1−2푡) (푡) = , 1−2푡−푡2 and (2−2푡)+푖(6푡−2) (푡) = , 1−2푡−푡2 respectively. PROOF. Let (푡) be the generating function of sequence {퐺푃푛}. Then we can write nn2 g( t ) GPnn t GP0 GPt 1 GP 2 t ... GP t ... n0 12 S. Halıcı, S. Öz If we use the recursive relation of this sequence, then we get 2 g t 1 2 t – t GP0 GP 1 – 2 GP 0 t . Thus, we obtain t(1 2 t ) i gt() , 12tt2 which is desired. Similarly, the generating function of Gaussian Pell-Lucas sequence {퐺푄푛} (2−2푡)+푖(6푡−2) (푡) = . 1−2푡−푡2 can be obtained. Moreover, we have the negatively subscripted terms of the sequences {퐺푃푛} and {퐺푄푛} by using the recursive relation, GPn2 GP()() n 12 GP n and GQn2 GQ ( n 1) GQ ( n 2) respectively. Notice that 퐺푃−푛 = 푃−푛 + 푃−푛−1 and 퐺푄−푛 = 푄−푛 + 푄−푛−1. In the following theorem, we give the relations between the Gaussian Pell and Gaussian Pell-Lucas sequences involving the negative indices. THEOREM 3. For n 1, we have the following identities. 푛−1 ) 퐺푃−푛 = (−1) (푃푛 − 푃푛+1) 푛 ) 퐺푄−푛 = (−1) (푄푛 − 푄푛+1) PROOF. The proof can be seen by the mathematical induction on 푛 . 13 On Some Gaussıan Pell And Pell-Lucas Numbers In the following Corollary, we give some useful identities concerning the Gaussian Pell and Gaussian Pell-Lucas numbers, and also give some sum formulas for these numbers without proof. COROLLARY 1. For 푛 ≥ 1, we have the following equations ) 4퐺푃푛 = 퐺푄푛 + 퐺푄푛−1, 2 2 ) 퐺푃푛 + 퐺푃푛+1 = 2푃2푛(1 + ), 2 2 ) 퐺푄푛 + 퐺푄푛+1 = 16푃푛 푄푛(1 + ). It is well known that the Cassini identity is one of the oldest identities involving the Fibonacci numbers. In the following theorem, we give the Cassini formula related with the Gaussian Pell and Pell-Lucas numbers. THEOREM 4. (Cassini Formula) Let . Then we have 2 푛 ) 퐺푃푛+1퐺푃푛−1 − 퐺푃푛 = (−1) 2(1 − ) 2 푛+1 ) 퐺푄푛+1퐺푄푛−1 − 퐺푄푛 = (−1) 16(1 − ), respectively. PROOF. By using the mathematical induction method we get 2 GPGP2 0– GP 1 2 1– i ; GP 0 i, GP 2 2 i 2 k GPk11 GP k– GP k 1 2 1– i 2211 GPk2 GP k– GP k 1 2 GP k 1 GP k GP k 1 – GP k 1 GP k 1 n 1 22 11 k GP GP GP GP– 1 2 1– i GP 2 22k k11 k k k 11 k1 GP GP– GP() GP GP 1 2 1– i 22k k11 k k k 14 S. Halıcı, S. Öz 1k 1 2 1– i . Similarly, we can prove the other formula by the mathematical induction method.