AIMS Mathematics, 2017, 2(2): 244-259. doi: 10.3934/Math.2017.2.244.

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Permutational behavior of reversed Dickson polynomials over finite fields

School of Mathematics and information, China west normal University, Nanchong 637009, P.R. China

In this paper, we develop the method presented previouslyby Hong, Qin and Zhao to obtain several results on the permutationalbehavior 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 thefirst moment $\sum_{a\in {\mathbb F}_{q}}D_{n,k}(1,a)$.Our results extend the results of Hong, Qin and Zhaoto the general $k\ge 0$ case.
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Keywords Permutation polynomial; Reversed Dickson polynomial of the (k + 1)-th kind; Finite field; Generating function

Citation: Kaimin Cheng. Permutational behavior of reversed Dickson polynomials over finite fields. AIMS Mathematics, 2017, 2(2): 244-259. doi: 10.3934/Math.2017.2.244

References

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  • 2. S. Hong, X. Qin andW. Zhao, Necessary conditions for reversed Dickson polynomials of the second kind to be permutational, Finite Fields Appl., 37 (2016), 54-71.    
  • 3. X. Hou, G. L. Mullen, J.A. Sellers and J.L. Yucas, Reversed Dickson polynomials over finite fields, Finite Fields Appl., 15 (2009), 748-773.    
  • 4. R. Lidl and H. Niederreiter, Finite Fields, second ed., Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 20 1997.
  • 5. X. Qin and S. Hong, Constructing permutation polynomials over finite fields, Bull. Aust. Math. Soc., 89 (2014), 420-430.    
  • 6. X. Qin, G. Qian and S. Hong, New results on permutation polynomials over finite fields, Int. J. Number Theory, 11 (2015), 437-449.    
  • 7. Q. Wang and J. L. Yucas, Dickson polynomials over finite fields, Finite Fields Appl., 18 (2012), 814-831.    

 

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Copyright Info: © 2017, Kaimin Cheng, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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