In this paper, we introduce trigonometric definitions for Chebyshev polynomials over finite fields Fq, where q = pm, m is a positive integer and p is an odd prime. From such definitions, we derive recurrence relations which are equivalent to those established for real valued Chebyshev polynomials and for Chebyshev polynomials of the first and second kinds over finite fields. Periodicity and symmetry properties of these polynomials are also studied. Such properties are then used to derive sufficient conditions for the Chebyshev polynomials of the second, third and fourth kinds over finite fields to be permutation polynomials.

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Lima, J.B. (Juliano B.), Panario, D, & De Souza, R.M.C. (Ricardo M. Campello). (2014). A trigonometric approach for Chebyshev polynomials over finite fields. In Applied Algebra and Number Theory: Essays in Honor of Harald Niederreiter on the Occasion of his 70th Birthday (pp. 255–279). doi:10.1017/CCO9781139696456.015