On the permutation behaviour of Dickson polynomials of the second kind (Q1867465)

Let \(q = p^e\) be a prime power and let \(\mathbb F_q\) denote the finite field of order \(q\). The Dickson polynomials of the first kind and the Dickson polynomials of the second kind (DPSK) are defined by \(g_k(X,a) = \sum_{i=0}^{\lfloor k/2 \rfloor}\frac{k}{k-i}{k-i \choose i}{-a}^iX^{k-2i}\) and \(f_k(X,a) = \sum_{i=0}^{\lfloor k/2 \rfloor}{k-i \choose i}{-a}^iX^{k-2i}\), \(a \neq 0 \in \mathbb F_q\), respectively. \(g_k(X,a)\) is a permutation polynomial (PP) over \(\mathbb F_q\), i.e., it induces a permutation on \(\mathbb F_q\) if and only if \(\gcd(k,q^2-1) = 1\). It seems to be much more difficult to decide whether or not \(f_k(X,a)\) is a PP over \(\mathbb F_q\). For the case that \(q = 3^e\) and \(a \neq 0 \in \mathbb F_q\) is a non-square the authors describe a class of DPSK which are PPs over \(\mathbb F_q\). The result expands the known PPs among DPSK in characteristic \(3\) and simplifies the description of classes given in [\textit{M. Henderson} and \textit{R. Matthews}, N. Z. J. Math. 27, 227--244 (1998; Zbl 0976.12002)].