The Fibonacci Polynomials in Rings

Abstract

In this paper, we study the Fibonacci polynomials modulo $m$ such that $x^2=x+1$ and then we obtain miscellaneous properties of these sequences. Also, we extend the Fibonacci polynomials to the ring of complex numbers. We define the Fibonacci Polynomial-type orbits $F_{(a,b)}^R(x)=\{x_i\}$, where $R$ is a 2-generator ring and $(a,b)$ is a generating pair of the ring $R$. Furthermore, we obtain the periods of the Fibonacci Polynomial-type orbits $F_{(a,b)}^R(x)$ in finite 2-generator rings of order $p^2$.