A Christoffel-Darboux-Type Formula for Szegö Polynomials and Polynomial Evaluation

Polynomials Φ n , n ≥ 0, orthogonal on the complex unit circle satisfy a recurrence relation. If we shift their recurrence coefficients, we obtain the associated polynomials, which can be modified by changing the initialization. We prove a dual recurrence relation and a mixed Christoffel-Darboux-type formula, which expresses the derivative of an orthogonal polynomial in terms of orthogonal polynomials and the modified associated polynomials. We exhibit connections between polynomial evaluation and these modified associated polynomials by showing, for example, how a polynomial q n expanded in terms of the Φ ν , ν = 0, …, n, and its derivatives can be evaluated.