Fibonacci self-reciprocal polynomials and Fibonacci permutation polynomials

TL;DR

This paper classifies self-reciprocal Fibonacci polynomials over various rings and finite fields, and analyzes their permutation properties by computing moments, advancing understanding of their algebraic and permutation characteristics.

Contribution

It provides a complete classification of self-reciprocal Fibonacci polynomials over integers and certain finite fields, and offers conditions for their permutation behavior.

Findings

Complete classification over $\\mathbb{Z}$ and specific finite fields

Computed moments of Fibonacci polynomials over finite fields

Derived necessary conditions for permutation polynomials

Abstract

Let $p$ be a prime. In this paper, we give a complete classification of self-reciprocal polynomials arising from Fibonacci polynomials over $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p=2$ and $p>5$. We also present some partial results when $p=3, 5$. We also compute the first and second moments of Fibonacci polynomials $f_{n}(x)$ over finite fields, which give necessary conditions for Fibonacci polynomials to be permutation polynomials over finite fields.