Let \(\Omega^{[k]}\) be the class of holomorphic functions \(f(z)\) of the complex variable \(z\) which satisfy, with respect to the \(s\)th root of unity \(\varepsilon_k\), \(k=0,1,\dots,n-1\), the symmetry property \(f(\varepsilon_1z)=\varepsilon_kf(z)\). The authors consider expansions of a function \(f^{[k]}(z)\in\Omega^{[k]}\) with respect to orthonormal systems of functions belonging to the same symmetry class \(\Omega^{[k]}\) and, more precisely, with respect to certain general orthonormal systems of polynomials introduced by \textit{P. E. Ricci} [Atti Semin. Mat. Fis. Univ. Modena 40, No. 2, 667-687 (1992; Zbl 0766.42011)]. These polynomials are orthogonal on the unit circle with respect to a suitable symmetry class and are used in order to generalize the discrete Fourier transform and the fast Fourier transform algorithm.