Complex Factorizations of the Lucas Sequences via Matrix Methods

Firstly, we show a connection between the first Lucas sequence and the determinants of some tridiagonal matrices. Secondly, we derive the complex factorizations of the first Lucas sequence by computing those determinants with the help of Chebyshev polynomials of the second kind. Furthermore, we also obtain the complex factorizations of the second Lucas sequence by the similar matrix method using Chebyshev polynomials of the first kind. 1. Introduction Given two nonzero integers and satisfying . The first Lucas sequence？？ and the second Lucas sequence？？ are defined by the recurrence relations respectively. By assigning and some special values, we will see some well-known Lucas sequences, which are important historically and for their own sake. The numbers are called the Fibonacci numbers while the numbers are called the Lucas numbers, the numbers and are the Pell numbers and the Pell-Lucas numbers, and are Jacobsthal numbers and Jacobsthal-Lucas numbers, respectively, are Mersenne numbers, and so on. There is a long tradition of using matrix methods to study Lucas sequences [2–5]. In 2003, Cahill et al. obtained complex factorizations for Fibonacci numbers and Lucas numbers by using the determinants of two slightly different sequences of tridiagonal matrices [2]. They used the tridiagonal matrix with entries , and to prove that where is the th Fibonacci number and , and also used the tridiagonal matrix with entries , and to prove that where is the th Lucas number and . In 2011, Burcu Bozkurt et al. obtained the complex factorization of the second Lucas sequence [4]: by using the tridiagonal matrix with entries , , and , where . In this study, number theory and linear algebra (with the help of orthogonal polynomials) are similarly intertwined to yield the complex factorizations of the first and second Lucas sequences. Now we give the following lemma which will be needed later. Lemma 1 ([2]). Let be a sequence of tridiagonal matrices of the form Then the successive determinants of are given by the recursive formula: 2. Complex Factorizations of the First Lucas Sequence First of all, we introduce the tridiagonal matrix sequence and express the first Lucas sequence by the determinant of for . Then we use this connection to prove the explicit formula for which is a generalization of the Binet Form for . Theorem 2. Let be a sequence of tridiagonal matrices of the form with , where and are nonzero integers satisfying . Then where denotes the determinant and are roots of . Proof. According to Lemma 1, successive determinants of are given by the recursive