AIMS Mathematics, 2017, 2(4): 586-609. doi: 10.3934/Math.2017.4.586

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Permutational behavior of reversed Dickson polynomials over finite fields II

School of Mathematics and Information, China West Normal University, Nanchong 637009, P.R.China

In this paper, we study the special reversed Dickson polynomial of theform $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1,x)$, where $s,e_1, ..., e_s$are positive integers, $\ell$ is an integer with $0 ≤ \ell  < p$. In fact, by using Hermite criterion we first give an answer to the questionthat the reversed Dickson polynomials of the forms $D_{p^{s}+1, k}(1,x)$,$D_{p^{s}+2, k}(1,x)$, $D_{p^{s}+3, k}(1,x)$, $D_{p^{s}+4, k}(1,x)$,$D_{p^{s}+p^{t}, k}(1,x)$ and $D_{p^{s}+p^{t}+1, k}(1,x)$ are permutationpolynomials of ${\mathbb F}_{q}$ or not. Finally, utilizing the recursiveformula of the reversed Dickson polynomials, we represent$D_{p^{e_1}+...+p^{e_s}+\ell, k}(1,x)$ as the linear combinationof the elementary symmetric polynomials with the power of $1-4x$being the variables. From this, we present a necessary and sufficient conditionfor $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1,x)$ to be a permutation polynomialof ${\mathbb F}_{q}$.
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