Fibonacci-type polynomial as a trajectory of a discrete dynamical system

Families of polynomials which obey the Fibonacci recursion relation can be generated by repeated iterations of a 2×2 matrix,Q ₂, acting on an initial value matrix,R ₂. One matrix fixes the recursion relation, while the other one distinguishes between the different polynomial families. Each family of polynomials can be considered as a single trajectory of a discrete dynamical system whose dynamics are determined byQ ₂. The starting point for each trajectory is fixed byR ₂(x). The forms of these matrices are studied, and some consequences for the properties of the corresponding polynomials are obtained. The main results generalize to the so-calledr-Bonacci polynomials.