Reversed Dickson polynomials of the $(k+1)$-th kind over Finite Fields, II

Neranga Fernando

Published 2017-06-05Version 1

Let $p$ be an odd prime and $q$ a power of $p$. The properties and the permutation behavior of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over finite fields have been an interesting topic as shown in the predecessor of the present paper. In this paper, we explain the permutation behaviour of $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$, where $l_1, l_2$, $l_3$, and $l_4$ are non-negative integers. A generalization $n=p^{l_1}+p^{l_2}+\cdots +p^{l_i}$ is also explained. We also present some algebraic and arithmetic properties of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$.