Applications of a new formula for OPUC with periodic Verblunsky coefficients (Q2319812)

The author finds a formula for the orthonormal polynomials corresponding to a measure \(\mu\) on the unit circle whose Verblunsky coefficients are periodic. If \(\{\Phi_n\}_{n=0}^\infty\) denotes the sequence of monic orthogonal polynomials obtained by applying Gram-Schmidt orthogonalization to the sequence \(\{z^n\}_{n=0}^\infty\), he defines the sequence of Verblunsky coefficients \(\{\alpha_n\}_{n=0}^\infty\) by \(-\bar \alpha_n=\Phi_{n+1}(0)\). Let \(\{\varphi_n\}_{n=0}^\infty\) denote the sequence of orthonormal polynomials for the measure \(\mu\). The author presents formulas for \(\{\varphi_n\}_{n=0}^\infty\) in Theorem 2.1. After deriving these formulas, the remainder of the paper is devoted to applications.