On the complex factorization of the Lucas sequence

Abstract

a b s t r a c tIn this paper, firstly we present a connection between determinants of tridiagonal matrices and the Lucas sequence. Secondly, we obtain the complex factorization of Lucas sequence by considering how the Lucas sequence can be connected to Chebyshev polynomials by determinants of a sequence of matrices.

Complex Factorizations of the Lucas Sequences via Matrix Methods

“…), (55), (56), and (50), we derive ( , ) = 2(√ ) cos ( arccos ( For = 1, = −1, (38) and (39) are exactly the formulas for Lucas numbers in [2].Remark 7. For = −1, (38) and (39) are exactly the formulas for { ( , −1)} in[4].…”

“…The complex factorization of { ( , −1)} ≥0 was derived in [4], and we generalize the formula to { ( , )} ≥0 for any integer in the following theorem.…”

“…by Theorem 1 of [4]. Unlike the derivation in Theorem 3, we will not compute the spectrum of ( ) directly.…”

“…There is a long tradition of using matrix methods to study Lucas sequences [2][3][4][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].…”

“…where is the th Lucas number and i = √ −1. In 2011, Burcu Bozkurt et al obtained the complex factorization of the second Lucas sequence { ( , −1)} ≥0 [4]:…”