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 \(F^R_{(a,b)}(x) = \{x_i\}\), 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 \(F^R_{(a,b)}(x)\) in finite 2-generator rings of order \(p^2\).