Orthogonality on the semicircle: old and new results

Orthogonal polynomials on the semicircle were introduced by Gautschi and Milovanovic in [Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (July 1985), pp. 179-185] and [J. Approx. Theory, 46 (1986), pp. 230-250]. In this paper we give an account of this kind of orthogonality, weighted generalizations mainly oriented to Chebyshev weights of the first and second kinds, including several interesting properties of such polynomials. Moreover, we also present a number of new results including those for Laurent polynomials (rational functions) orthogonal on the semicircle. In particular, we give their recurrence relations and study special cases for the Legendre weight and for the Chebyshev weights of the first and second kind. Explicit expressions for such orthogonal systems with Chebyshev weights are presented, as well as the corresponding zero distributions. Key words. complex orthogonal systems, recurrence relations, zeros, weight function, orthogonal Laurent polynomials AMS subject classifications. 30C10, 30C15, 33C45, 33C47, 42C05