Solving the linear difference equation with periodic coefficients via Fibonacci sequences

Periodic matrix difference equations and companion matrices in blocks: some applications

“…This class of equations has been studied in [5]. The method used consists in transforming equation ( 5 .…”

“…(1.1) (see, for example, [15,17], and references therein). Recently, the homogeneous linear difference equations (1.1) with periodic coefficients, i.e., a j (n + p) = a j (n), have been solved in [4,5], using properties of the generalized Fibonacci sequences in the algebra of square matrices. More precisely, in [4], Eq.…”

“…In the last years, the product of companion matrices has attracted much attention, because this product occurs in various fields of mathematics and applied sciences, such that the Floquet system theory related to the linear difference equations (see [4,5,16,17]). Diverse methods for computing the product of companion matrices have been proposed in the literature.…”

“…Another expression for the product of companion matrices was obtained in [17], based on the study of solutions of non-homogeneous and homogeneous linear difference equations of order N, with variable coefficients. Recently, it was shown in [4,5] that the product of companion matrices plays a central role, for investigating a large class of periodic-discrete homogeneous difference equations via generalized Fibonacci sequences. Moreover, through the key of generalized Fibonacci sequences, there are still some interesting and relevant problems that can be examined.…”