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

Let a sequence {n} n>O of polynomials be given which are orthogonal on the unit circle with respect to an inner product. By a k-shift of their reflec­ tion coefficients we obtain the associated polynomials {~k)} n>O of order k 2: 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 recur­ rence 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) associ­ ated polynomials.