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

Abstract

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 by Q 2. The starting point for each trajectory is fixed by R 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-called r-Bonacci polynomials. © 2002 Springer.