Dual Recurrence and Christoffel-Darboux-Type Formulas for Orthogonal Polynomials

Let a sequence ϕ n n≥0 of polynomials be given which are orthogonal on the unit circle with respect to an inner product. By a k-shift of their reflection coefficients we obtain the associated polynomials ϕ(k) n n≥0 of order k ≥ 0, analogously to the associated polynomials on the real line.Using these polynomials we derive a dual recurrence formula for polynomials orthogonal ont the unit circle. Modifying the associated polynomials by starting the Geronimus recurrence earlier we derive a Christoffel-Darboux-type formula for some classes of orthogonal polynomials. This formula expresses derivatives of orthogonal polynomials in terms of orthogonal polynomials and their (modified) associated polynomials.