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 $\int_\gamma P^i(z)q(z)\,dz$, $i\geq0$, where $P(z),q(z)$ are rational functions of one complex variable and $\gamma$ is a curve in $\mathbb C$. 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 $\gamma$. As an application, we prove a stronger form of the Wermer theorem, describing analytic functions satisfying the system of equations $\int_{S^1}h^i(z)g^j(z)g'(z)\,dz=0$, $i\geq0$, $j\geq0$, 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 $\int_{S^1}L^i(z)m(z)\,dz=0$, $i\geq i_0$.