On rational functions orthogonal to all powers of a given rational function on a curve

In this paper we study the generating function f(t) for the sequence of the moments, where ∫γPⁱ(z), q(z)dz,i≥ 0 where P(z), q(z) are rational functions of one complex variable and γ is a curve in ℂ. We calculate an analytical expression for f(t) and provide conditions implying that f(t) is rational or vanishes identically. In particular, for P(z) in generic position we give an explicit criterion for a function q(z) to be orthogonal to all powers of P(z) on γ. As an application, we prove a stronger form of the Wermer theorem, describing analytic functions satisfying the system of equations, in the case where the functions ∫_S1hⁱ(z)g^j(z)g^l(z)dz=0, i≥0, j≥0 in the case where the functions h(z), g(z) are rational. We also generalize the theorem of Duistermaat and van der Kallen about Laurent polynomials L(z) whose integer positive powers have no constant term, and prove other results about Laurent polynomials L(z), m(z) satisfying ∫_S1Lⁱ(z)m(z)dz=0, i≥i0.