Affiliation:
1. Institute of Mathematics, Federal University of Mato Grosso do Sul, Campo Grande 79060-300, Brazil
2. Department of Mathematics, University of Trás-os-Montes e Alto Douro, 5001-801 Vila Real, Portugal
Abstract
In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the properties of the eigenvalues of their related 2×2 matrix, we provide a new approach to studying the analytic representations of these numbers. Moreover, the similarity of the associated 2×2 matrix with a companion matrix, allows us to formulate the bi-periodic Fibonacci numbers in terms of a homogeneous linear recursive sequence of the Fibonacci type. Therefore, the combinatorial aspect and other analytic representations formulas of the Binet type for the bi-periodic Fibonacci numbers are achieved. The case of bi-periodic Lucas numbers is outlined, and special cases are exposed. Finally, some illustrative examples are given.