Applied Mathematical Sciences, Vol. 5, 2011, no. 64, 3173 - 3181
On the Pell, Pell-Lucas and Modified Pell Numbers By Matrix Method
Ahmet Dasdemir
Department of Mathematics Faculty of Arts and Sciences, Sakarya University 54100 Serdivan, Sakarya, Turkey [email protected]
Abstract In this paper, we consider the usual Pell, Pell-Lucas and Modified Pell sequences, and we define some new matrices, which are based on Modified Pell and Pell-Lucas numbers, with determinants equal to 2n or 23n.This unique properties provide very important formulas like the Simpson F ormula of Modified Pell and Pell-Lucas numbers. Also we show that the Modified Pell and Pell-Lucas numbers have the generating matrices.Further, we present some elementary identities between them by these matrices.
Mathematics Subject Classification: 11B37; 15A15
Keywords: Pell Number, Modified Pell Number, Pell-Lucas Number, Generating Matrix, Simpson Formula
1 Introduction
In the literature, especially in mathematics and physics, there are a lot of in- teger sequences, which are used in almost every field modern sciences. Admit- tedly, the Fibonacci sequence is one of the most famous and curious numerical sequences in mathematics and have been widely studied from both algebraic and combinatorial prospectives.Also, there is the Pell sequence, which is as important as the Fibonacci sequence.The Pell sequence {Pn} are defined by recurrence Pn =2Pn−1 + Pn−2 ,n≥ 2 with P0 = 0 and P1 = 1 and the Pell-Lucas sequence {Qn} and Modified Pell sequence {qn} by the same re- currence but with initial conditions Q0 = Q1 = 2 and q0 = 1 and q1 =1, respectively.Infact, Qn =2qn. (1) 3174 A. Dasdemir
Consequently, the known properties of {Qn} can be easily written for {qn}. Thereby, a study of {qn} involves inevitably familiarity with {Qn}.Further details about Pell, Pell-Lucas and Modified Pell sequences can be found in [3, 5, 9, 13].For instance, in [3], Horadam gave some results as follows:
qn = Pn+1 − Pn, (2)
qn+1 = Pn+1 + Pn (3) and qn+1 + qn Pn+1 = . (4) 2 Explicit Binet forms for {Pn}, {Qn} and {qn} are αn − βn Pn = , (5) α − β n n Qn = α + β , (6) and αn + βn qn = (7) α + β where α and β are the roots of the characteristic equation x2 −2x−1=0.[1, 11] The some of elements of the Pell, Pell-Lucas and Modified Pell sequences are given by the following table; n 012 3 4 5 6 7 8 9 ··· Pn 0 1 2 5 12 29 70 169 408 985 ··· Qn 2 2 6 14 34 82 198 478 1154 2786 ··· qn 1 1 3 7 17 41 99 239 577 1393 ···
Table 1. The values of {Pn}, {Qn} and {qn}. In the last decades, some mathematicians have studied to find miscella- neous affinities between the matrices and linear recurrences.They want to ob- tain always the terms of linear recurrences by matrices, which are said to be the Generating Matrices.Comparable matrix generators have been considered previously for the Fibonacci and Pell sequences[7, 8, 12]. In [12], the following property of the nth power of the Q-matrix is found n Fn+1 Fn Q = , (8) Fn Fn−1 11 where Q = .Similarly, in [8] author showed that the members of the 10 Pell sequence can be derived by a matrix representation: n Pn+1 Pn M = , (9) Pn Pn−1 Pell, Pell-Lucas and modified Pell numbers 3175