  We find a new formula for the orthonormal polynomials corresponding to a
measure mu on the unit circle whose Verblunsky coefficients are periodic. The
formula is presented using the Chebyshev polynomials of the second kind and the
discriminant of the periodic sequence. We present several applications
including a resolution of a problem suggested by Simon in 2006 regarding the
existence of singular points in the bands of the support of the measure and a
universality result at all points of the essential support of mu.
