The first and second moments of reversed Dickson polynomials over finite fields

Let n and k be nonnegative integers. In 2010, Hou and Ly evaluated the first and second moments of the n-th reversed Dickson polynomial of the first kind. In 2016, Hong, Qin and Zhao presented a recursive formula for the first moment of the n-th reversed Dickson polynomial of the second kind. In this paper, we introduce a new method to investigate the moments of the n-th reversed Dickson polynomial of (k+1)-th kind. In fact, we first show an extension of the famous Lucas' congruence and then study arithmetic properties of some two-variable linear congruences. Finally, with more efforts, we arrive at the explicit formulas for the first and second moments of the n-th reversed Dickson polynomial of the (k+1)-th kind.