Permutational behavior of reversed Dickson polynomials over finite fields

TL;DR

This paper investigates the permutation properties of reversed Dickson polynomials over finite fields, extending known results to a broader class and providing explicit evaluations of their moments.

Contribution

It generalizes the permutational behavior analysis of reversed Dickson polynomials to all non-negative k, building on previous methods.

Findings

Explicit evaluation of the first moment of the polynomials.

Extension of permutational behavior results to all k ≥ 0.

Broader understanding of reversed Dickson polynomials over finite fields.

Abstract

In this paper, we use the method developed previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,k}(1,x)$ of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,k}(1,a)$. Our results extend the known results from the case $0\le k\le 3$ to the general $k\ge 0$ case.