Ramanujan's trigonometric sums and orthogonal polynomials on the unit circle (Q2097508)

Ramanujan's trigonometric sums \(c_M(n)\) defined as \[ c_M(n)=\sum_{(s,M)=1}\exp\Big(\frac{2\pi isn}M\Big) \] where \(n\) is an integer, \(M\) is a positive integer and summation is taken over \(1\le s< M\) coprime with \(M\) are considered. In this paper, new explicit examples of finite systems of orthogonal polynomials on the unit circle (OPUC) with equal concentrated masses located at primitive roots of unity are proposed. For a given positive integer \(M\) it is taken \(N + 1 =\varphi(M)\) (\(\varphi(n)\) is Euler's totient function) and it is defined the system \(\Phi_0(z)=1, \Phi_1(z),\dotsc,\Phi_N (z),\Phi_{N+1}(z)\) of OPUC by identifying their trigonometric moments \(\sigma_n\) with the Ramanujan's sum \(\sigma_n = (N + 1)^{-1}c_M(n)\), \(n=0,1,\dotsc,N+1\). The corresponding OPUC \(\Phi_n(z)\) are called the Ramanujan orthogonal polynomials. The main problem of the paper is to find explicitly the OPUC \(\Phi_n(z)\) and corresponding Verblunsky parameters \(a_n\). An important result concerning relations between ``Sturmian'' OPUC and Ramanujan OPUC is obtained in Section 2. It is shown that these two systems are mirror-dual one with respect to another. In Sections 3 and 4, several explicit examples of such dual systems are presented. They are connected either with the cyclotomic or with the Kronecker polynomials.