Pseudo-Chebyshev Functions Over Finite Fields

In this paper, we introduce the notion of pseudo-Chebyshev functions over finite fields. In brief, such functions correspond to a generalization of the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-th Chebyshev polynomial, where <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is not restricted to integer values, but can take on any rational value. Our approach is mainly based on concepts of trigonometry over finite fields. Besides defining the referred functions, we derive several of their properties and establish necessary and sufficient conditions for them to permute the elements of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>. Potential applications of the proposed functions are preliminarily discussed.