Gaussian Modified Pell Sequence and Gaussian Modified Pell Polynomial Sequence

Transcription

1 Aksaray University Journal of Science and Engineering e-issn: Aksaray J. Sci. Eng. Volume 2, Issue 1, pp doi: /asujse Available online at Research Article Gaussian Modified Pell Sequence and Gaussian Modified Pell Polynomial Sequence Tulay Yagmur 1,*, Nusret Karaaslan 2 1,2 Department of Mathematics, Aksaray University, Aksaray, Turkey 1 Program of Occupational Health and Safety, Aksaray University, Aksaray, Turkey Received Date: 3 Jan 2018 Revised Date: 06 Apr 2018 Accepted Date: 10 Apr 2018 Published Online: 25 June 2018 Abstract In this paper, we first define the Gaussian modified Pell sequence, for n 2, by the relation Gq n = 2Gq n 1 + Gq n 2 with initial conditions Gq 0 = 1 i and Gq 1 = 1 + i. Then we give the definition of the Gaussian modified Pell polynomial sequence, for n 2, by the relation Gq n (x) = 2xGq n 1 (x) + Gq n 2 (x) with initial conditions Gq 0 (x) = 1 xi and Gq 1 (x) = x + i. We give Binet s formulas, generating functions and summation formulas of these sequences. We also obtain some well-known identities such as Catalan s identities, Cassini s identities and d Ocagne s identities involving the Gaussian modified Pell sequence and Gaussian modified Pell polynomial sequence. Keywords Modified Pell sequence, Modified Pell polynomial sequence, Gaussian modified Pell sequence, Gaussian modified Pell polynomial sequence. * Corresponding Author: Tulay Yagmur, tulayyagmurr@gmail.com; tulayyagmur@aksaray.edu.tr Published by AksarayUniversity 63

2 1. INTRODUCTION The complex Fibonacci numbers have been introduced by Horadam [1] in Then Berzsenyi [2] and Jordan [3] studied on Gaussian Fibonacci and Lucas numbers. The Gaussian Fibonacci numbers {GF n } n=0 are defined recursively by the relation GF n = GF n 1 + GF n 2 with initial conditions GF 0 = i and GF 1 = 1. Similarly, the Gaussian Lucas numbers {GL n } n=0 are defined as GL n = GL n 1 + GL n 2 where GL 0 = 2 i and GL 1 = 1 + 2i. Moreover, many authors studied on these numbers and their properties. For a little part of these studies, one can see, for example [4-6]. Halıcı and Öz [7] introduced the Gaussian Pell and Pell-Lucas numbers respectively by GP 0 = i, GP 1 = 1; GP n = 2GP n 1 + GP n 2, GQ 0 = 2 2i, GQ 1 = 2 + 2i; GQ n = 2GQ n 1 + GQ n 2. Moreover, Pell and Pell-Lucas polynomials are defined as P 0 (x) = 0, P 1 (x) = 1; P n (x) = 2xP n 1 (x) + P n 2 (x), Q 0 (x) = 2, Q 1 (x) = 2x; Q n (x) = 2xQ n 1 (x) + Q n 2 (x) respectively. Furthermore, Horadam and Mahon [8] gave some properties of these polynomials. Additionally, the modified Pell polynomials are defined recursively by the relation q n (x) = 2xq n 1 (x) + q n 2 (x) where q 0 (x) = 1 and q 1 (x) = x. The generating function of the modified Pell polynomials is M(t, x) = Also, the Binet s formula of these polynomials is 1 xt 1 2xt t 2. q n (x) = x αn (x)+β n (x) where α = x + x and β = x x are the roots of the equations r 2 2xr 1 = 0. Then Halıcı and Öz [9] defined Gaussian Pell polynomial sequence as follow: GP 0 (x) = i, GP 1 (x) = 1; GP n (x) = 2xGP n 1 (x) + GP n 2 (x). The main objective of this paper is to define and study Gaussian modified Pell sequence and Gaussian modified Pell polynomial sequence. 2. GAUSSIAN MODIFIED PELL SEQUENCE In this section, we first give the definition of the Gaussian modified Pell sequence, and then we obtain Binet s formula and generating function for this sequence. Moreover, we give some results related with the Gaussian modified Pell sequence. Definition 2.1 The Gaussian modified Pell numbers {Gq n } n=0 are defined, for n 2, recursively by Aksaray J. Sci. Eng. 2:1 (2018)

3 Gq n = 2Gq n 1 + Gq n 2 with initial conditions Gq 0 = 1 i and Gq 1 = 1 + i. Also, it is clear that where q n is the n-th modified Pell numbers. Gq n = q n + iq n 1 Now, we give the generating function for the Gaussian modified Pell sequence by the following theorem. Theorem 2.2 The generating function of the Gaussian modified Pell sequence is Proof. Let us write and Thus, we have Hence, we obtain G(x) = (1 x)+ i( 1+3x) 1 2x x 2. G(x) = n=0 Gq n x n = Gq 0 + Gq 1 x + Gq 2 x Gq n x n +, 2xG(x) = 2Gq 0 x + 2Gq 1 x 2 + 2Gq 2 x Gq n 1 x n +, x 2 G(x) = Gq 0 x 2 + Gq 1 x 3 + Gq 2 x Gq n 2 x n +. G(x)(1 2x x 2 ) = Gq 0 + (Gq 1 2Gq 0 )x. G(x) = (1 x)+ i( 1+3x) 1 2x x 2. The Binet s formula for the Gaussian modified Pell sequence is given by the following theorem. Theorem 2.3 The n-th term of the Gaussian modified Pell sequence is by Gq n = αn + β n + αβ n i(βαn ) α+ β where α and β are the roots of the equation r 2 2r 1 = 0. Proof. We know that the general solution for the recurrence relation is given by Gq n = cα n + dβ n Aksaray J. Sci. Eng. 2:1 (2018)

4 for some coefficients c and d. The initial conditions imply that Gq 0 = c + d and Gq 1 = cα + dβ. Solving the system, we obtain c = 1 βi 2 and d = 1 αi 2. Thus, we get Gq n = αn + β n + αβ n i(βαn ). α+ β We now investigate some identities and properties of the Gaussian modified Pell sequence. Theorem 2.4 Let n and r be two positive integers. Then Catalan s identity for the Gaussian modified Pell sequence is Gq n+r Gq n r Gq 2 n = 2( 1) n+1 (1 i) [1 + ( 1)r 1 (α r +β r ) 2 ]. Proof. By using the Binet s formula of the Gaussian modified Pell sequence, we get Gq n+r Gq n r Gq n 2 = [ αn+r + β n+r Since α + β = 2, we obtain [ αn + β n i ( βαn+r +αβ n+r [ αn r + β n r i (βαn +αβ n 2 4 i ( βαn r +αβ n r = [ 4( 1)n +2( 1) n r (α 2r +β 2r ) () 2 ] + i[ 4( 1)n ( 1) n r (α 2r+1 +β 2r+1 +βα 2r +αβ 2r ) () 2 ] = [ 4( 1)n +2( 1) n r (α 2r +β 2r ) () 2 ] +i[ 4( 1)n 2( 1) n r (α 2r +β 2r ) () 2 ]. Gq n+r Gq n r Gq n 2 = ( 1) n+1 (1 i)+(1 i)( 1) n r [ (αr +β r ) 2 ( 1) r ] 2 = 2(1 i)( 1) n+1 [1 ( 1) r (αr +β r ) 2 ] = 2( 1) n+1 (1 i) [1 + ( 1)r 1 (α r +β r ) 2 ]. By setting r = 1 in Theorem 2.4, we obtain the following corollary which gives Cassini s identity of the Gaussian modified Pell sequence. Corollary 2.5 For positive integer n, we have Gq n+1 Gq n 1 Gq n 2 = 4( 1) n+1 (1 i). 4 4 Aksaray J. Sci. Eng. 2:1 (2018)

5 The following theorem gives d Ocagne s identity involving the Gaussian modified Pell sequence. Theorem 2.6 For positive integers m and n, we have where P n is the n-th Pell number. Gq m Gq n+1 Gq n Gq m+1 = 4( 1) n+1 (1 i)p m n Proof. By using the Binet s formula, we have Gq m Gq n+1 Gq n Gq m+1 = [ αm + β m [ αn + β n i +αβ m (βαm [ αn+1 + β n+1 i +αβ n (βαn [ αm+1 + β m+1 = [ 2(α β)(αn β m α m β n ) () 2 ] + i[ (α2 β 2 )(α m β n α n β m ) () 2 ] = (αn β m α m β n )[2(α β) i(α 2 β 2 () 2. Since α + β = 2 and α β = 2 2, we obtain Gq m Gq n+1 Gq n Gq m+1 = 2(1 i)(α n β m α m β n ) = 2(1 i)( 1) n+1 (α m n β m n ) = 4( 1) n+1 (1 i)p m n. i ( βαn+1 +αβ n+1 i ( βαm+1 +αβ m+1 Theorem 2.7 The sum of the Gaussian modified Pell numbers is n k=1 Gq k = 1 (Gq 2 n+1 + Gq n ) 1. Proof. From the recursive relation related with the Gaussian modified Pell sequence, we can write Then we have Gq n 1 = 1 2 Gq n 1 2 Gq n 2. Gq 1 = 1 2 Gq Gq 0 Gq 2 = 1 2 Gq Gq 1 Aksaray J. Sci. Eng. 2:1 (2018)

6 Hence, we obtain Gq n = 1 2 Gq n Gq n 1. n k=1 Gq k = 1 (Gq 2 n+1 + Gq n ) 1 (Gq Gq 1 ) which completes the proof. = 1 2 (Gq n+1 + Gq n ) 1 The following corollary immediately follows from Theorem 2.7. Corollary 2.8 For n 1, we have i) n k=1 Gq 2k = 1 (Gq 2 2n+1 1 i), ii) n k=1 Gq 2k 1 = 1 (Gq 2 2n 1 + i). 3. GAUSSIAN MODIFIED PELL POLYNOMIAL SEQUENCE In this section, we first define the Gaussian modified Pell polynomials and then we give Binet s formula and generating function of this type polynomials. We also obtain some identities and properties of these polynomials. Definition 3.1 The Gaussian modified Pell polynomials {Gq n (x)} n=0 are defined, for n 2, by the recurrence relation Gq n (x) = 2xGq n 1 (x) + Gq n 2 (x) with initial conditions Gq 0 (x) = 1 xi and Gq 1 (x) = x + i. It is obvious that if we take x = 1, we obtain the Gaussian modified Pell sequence. Also, it is easy to see that Gq n (x) = q n (x) + iq n 1 (x) where q n (x) is the n-th modified Pell polynomial. Now, we aim to give generating function and Binet s formula for the Gaussian modified Pell polynomials. For this purpose, we shall prove the following theorems: Theorem 3.2 The generating function of the Gaussian modified Pell polynomials is P(t, x) = (1 xt)+i( x+t+2x2 t) 1 2xt t 2. Aksaray J. Sci. Eng. 2:1 (2018)

7 Proof. The generating function can be written as P(t, x) = n=0 Gq n (x)t n. Then we have, and P(t, x) = Gq 0 (x) + Gq 1 (x)t + Gq 2 (x)t Gq n (x)t n +, 2xt P(t, x)= 2x Gq 0 (x)t + 2xGq 1 (x)t 2 + 2xGq 2 (x)t xGq n 1 (x)t n +, So, we get Thus, we obtain t 2 P(t, x) = Gq 0 (x)t 2 + Gq 1 (x)t 3 + Gq 2 (x)t Gq n 2 (x)t n +. P(t, x)(1 2xt t 2 ) = Gq 0 (x) + [Gq 1 (x) 2xGq 0 (xt. P(t, x) = (1 xt)+i( x+t+2x2 t) 1 2xt t 2. Theorem 3.3 The n-th term of the Gaussian modified Pell polynomials is Gq n (x) = x[ αn (x)+β n (x) i β(x)αn (x)+α(x)β n (x) ] where α and β are the roots of the equations r 2 2xr 1 = 0. Proof. It is well-known that the general solution for the recurrence relation is given by for some coefficients c and d. Gq n (x) = cα n (x) + dβ n (x) By considering the initial conditions, we get Gq 0 (x) = c + d and Gq 1 (x) = cα + dβ. Solving the system above, we obtain Thus, we have c = 1 2 i β 2 and d = 1 2 i α 2. Gq n (x) = x[ αn (x)+β n (x) i β(x)αn (x)+α(x)β n (x) ]. We now investigate some identities and properties of the Gaussian modified Pell polynomials. Theorem 3.4 Let n and r be two positive integers. Then Catalan s identity for the Gaussian modified Pell polynomials is Aksaray J. Sci. Eng. 2:1 (2018)

8 Gq n+r (x)gq n r (x) Gq 2 n (x) = 2( 1) n+1 (1 xi) [1 + ( 1)r 1 (α r (x)+β r (x)) 2 ]. Proof. By using the Binet s formula of the Gaussian modified Pell polynomial sequence, we get Gq n+r (x)gq n r (x) Gq n 2 (x) =[x αn+r (x)+ β n+r (x) ix ( β(x)αn+r (x)+α(x)β n+r (x) 4 [x αn r (x)+ β n r (x) ix ( β(x)αn r (x)+α(x)β n r (x) [x αn (x)+ β n (x) ix ( β(x)αn (x)+α(x)β n (x) 2 = x 2 [ 4( 1)n (1 xi)+2( 1) n r (α 2r (x)+β 2r (x))(1 xi) () 2 ]. Since α + β = 2x, we obtain Gq n+r (x)gq n r (x) Gq n 2 (x)= ( 1) n+1 (1 xi) + (1 xi)( 1)n r [2x 2 (α r (x)+β r (x)) 2 4x 2 ( 1) r ] () 2 = 2(1 xi)( 1) n [1 ( 1) r x 2 (α r (x)+β r (x)) 2 4x 2 ] = 2( 1) n+1 (1 xi) [1 + ( 1)r 1 (α r (x)+β r (x)) 2 ]. 4 By taking r = 1 in Theorem 3.4, Cassini s identity involving the Gaussian modified Pell polynomials, which is given in the following corollary, is obtained. Corollary 3.5 For positive integer n, we have Gq n+1 (x)gq n 1 (x) Gq n 2 (x) = 2(x 2 + 1)( 1) n+1 (1 xi). d Ocagne s identity involving the Gaussian modified Pell polynomials is given in the following theorem. Theorem 3.6 For positive integers m and n, we get Gq m (x)gq n+1 (x) Gq n (x)gq m+1 (x) = 2(x 2 + 1)( 1) n+1 (1 xi)p m n (x), where P n (x) is the n-th Pell polynomial. Proof. By using the Binet s formula, we have Gq m (x)gq n+1 (x) Gq n (x)gq m+1 (x) Aksaray J. Sci. Eng. 2:1 (2018)

9 = x 2 [ αm (x)+ β m (x) x 2 [ αn (x)+ β n (x) i ( β(x)αm (x)+α(x)β m (x) [ αn+1 (x)+ β n+1 (x) i ( β(x)αn (x)+α(x)β n (x) [ αm+1 (x)+ β m+1 (x) = x 2 [αn (x)β m (x) α m (x)β n (x[2(α(x) β(x)) i(α 2 (x) β 2 (x) () 2 = x 2 + 1[α n (x)β m (x) α m (x)β n (x(1 xi) = 2(x 2 + 1)( 1) n+1 (1 xi)p m n (x). i ( β(x)αn+1 (x)+α(x)β n+1 (x) i ( β(x)αm+1 (x)+α(x)β m+1 (x) Theorem 3.7 The sum of the Gaussian modified Pell polynomials is n k=1 Gq k (x) = 1 [Gq 2x n+1 (x) + Gq n (x) x 1 + i(x 1. Proof. From the recursive relation related with the Gaussian modified Pell sequence, we can write Then we have Hence, we obtain Gq n 1 (x) = 1 2x Gq n (x) 1 2x Gq n 2 (x). Gq 1 (x) = 1 2x Gq 2 (x) 1 2x Gq 0 (x) Gq 2 (x) = 1 2x Gq 3 (x) 1 2x Gq 1 (x) Gq 3 (x) = 1 2x Gq 4 (x) 1 2x Gq 2 (x) Gq n (x) = 1 2x Gq n+1 (x) 1 2x Gq n 1 (x). n k=1 Gq k = 1 (Gq 2x n+1 (x) + Gq n (x)) 1 (Gq 2x 0 (x) + Gq 1 (x)) which completes the proof. = 1 [Gq 2x n+1 (x) + Gq n (x) x 1 + i(x 1 From Theorem 3.7, we can give the following corollary. Corollary 3.8 For n 1, we have i) n k=1 Gq 2k (x) = 1 (Gq 2x 2n+1 (x) x i), Aksaray J. Sci. Eng. 2:1 (2018)

10 ii) n k=1 Gq 2k 1 (x) = 1 (Gq 2x 2n (x) 1 + xi). CONCLUSION In this study, we introduce the concept of the Gaussian modified Pell sequence and Gaussian modified Pell polynomial sequence. We also give some results, such as Binet s formulas, generating functions, summation formulas for these sequences. Moreover, we obtain some well-known identities, such as Catalan s, Cassini s, d Ocagne s identities involving these sequences. REFERENCES [1] A.F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions. American Math. Monthly 70 (1963) [2] G. Berzsenyi, Gaussian Fibonacci Numbers. Fibonacci Quarterly 15(3) (1977) [3] J.H. Jordan, Gaussian Fibonacci and Lucas Numbers. Fibonacci Quarterly 3 (1965) [4] J.J. Good, Complex Fibonacci and Lucas Numbers, Continued Fractions, and the Square Root of the Golden Ratio. Fibonacci Quaterly 31(1) (1981) [5] C.J. Harman, Complex Fibonacci Numbers. Fibonacci Quaterly 19(1) (1981) [6] S. Pethe, A.F. Horadam, Generalized Gaussian Fibonacci Numbers. Bull. Austral. Math. Soc. 33(1) (1986) [7] S. Halıcı, S. Öz, On Some Gaussian Pell and Pell-Lucas Numbers. Ordu Univ. Science and Technology Journal 6(1) (2016) [8] A. F. Horadam, J. M. Mahon, Pell and Pell-Lucas polynomials. Fibonacci Quarterly 23(1) (1985) [9] S. Halıcı, S. Öz, On Gaussian Pell Polynomials and Their Some Properties. Palestine Journal of Mathematics 7(1) (2018) Aksaray J. Sci. Eng. 2:1 (2018)