Polynomials Orthogonal on the Semicircle, II

Generalizing previous work [2], we study complex polynomials {pi sub k}, pi sub k (z)=z sup k + ..., orthogonal with respect to a complex-valued inner product (f,g)=sup int sub o f (e sup (i theta)g(e sup (i theta)w (e sup (i theta)d theta. Under suitable assumptions on the "weight function" w, we show that these polynomials exist whenever Re int sup pi sub o w(e sup i theta) d theta is not identical with O, and we express them in terms of the real polynomials orthogonal with respect to the weight function w(x).