  Let $\nu$ be a rotation invariant Borel probability measure on the complex
plane having moments of all orders. Given a positive integer $q$, it is proved
that the space of $\nu$-square integrable $q$-analytic functions is the closure
of $q$-analytic polynomials, and in particular it is a Hilbert space. We
establish a general formula for the corresponding polyanalytic reproducing
kernel. New examples are given and all known examples, including those of the
analytic case are covered. In particular, weighted Bergman and Fock type spaces
of polyanalytic functions are introduced. Our results have a higher dimensional
generalization for measure on ${\mathbb C}^p$ which are in rotation invariant
with respect to each coordinate.
