Dickson polynomials over finite fields (Q439091)

For any element \(a\) of a finite field \({\mathbb F}_q\) and any integers \(n\geq 1\), \(k\geq 0\), the authors define the \(n\)-th Dickson polynomial of the \((k+1)\)-st kind \(D_{n,k}(x,a)\) over \({\mathbb F}_q\) by NEWLINE\[NEWLINE D_{n,k}(x,a) =\sum _{i=0}^{n/2} \frac{n-ki}{n-i} \binom{n-i}{i} (-a)^ix^{n-2i}. NEWLINE\]NEWLINE Moreover, for \(n=0\) one puts \(D_{n,k}(x,a) =2-k\).NEWLINENEWLINEClearly, for \(k=0\) (\(k=1\)) one finds the Dickson polynomials of the first (second) kind. These remarks are particular instances of a general relationship between Dickson polynomials of the \((k+1)\)-st kind and the familiar Dickson polynomial of the first two kinds pointed out in Section~2. Then the authors prove that for fixed \(k\) and any \(n\geq 2\) one has \(D_{n,k}(x,a) =xD_{n-1,k}(x,a) -aD_{n-2,k}(x,a)\), whence the generating function for \(\left( D_{n,k}(x,a) \right)_n\) is readily obtained. Functional expressions, as well as differential recurrence relations, are also derived. The third section is devoted to complete factorization of Dickson polynomials of the third kind. The paper ends with a study of permutation properties. The main result characterizes Dickson polynomials \(D_{n,2}(x,1)\) which permute the prime field of characteristic at least \(5\). The proof combines Hermite's criterion with Gröbner bases computations, in the same way as \textit{S. D. Cohen} and the reviewer did in [Contemp. Math. 461, 79--90 (2008; Zbl 1211.11134)].