Orthogonal polynomials on rays: properties of zeros, related moment problems and symmetries

We establish some basic properties of zeros for orthonormal polynomials on radial rays. We introduce a moment problem related to these orthonormal polynomials and obtain necessary and sufficient conditions for its solvability. We establish some properties of orthonormal polynomials in the case when the measure of orthogonality has symmetries and show that the moment problem has solutions with some symmetric properties.