# Moscow Mathematical Journal

Volume 13, Issue 4, October–December 2013 pp. 693–731.

On Rational Functions Orthogonal to All Powers of a Given Rational Function on a Curve**Authors**: F. Pakovich (1)

**Author institution:**(1) Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, Israel

**Summary:**

In this paper we study the generating function *f*(*t*) for the sequence
of the moments ∫_{γ} *P ^{i}*(

*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 ∫

_{S1}

*h>*(

^{i}*z*)

*g*(

^{j}*z*)

*g*′(

*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 ∫

_{S1}

*L*(

^{i}*z*)

*m*(

*z*)

*dz*=0,

*i*≥

*i*

_{0}.

2010 Mathematics Subject Classification. Primary: 30E99; Secondary: 34C99

**Keywords:**Moment problem, center problem, Abel equation, periodic orbits, Cauchy type integrals, compositions

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