# The polyanalytic reproducing kernels

## TL;DR

This paper studies polyanalytic reproducing kernels for spaces of square-integrable polyanalytic functions under rotation-invariant measures, providing a general formula, new examples, and higher-dimensional generalizations.

## Contribution

It establishes a general formula for polyanalytic reproducing kernels and introduces new examples, extending known cases to higher dimensions.

## Key findings

- Proved that the space of square-integrable q-analytic functions is the closure of q-analytic polynomials.
- Derived a general formula for the polyanalytic reproducing kernel.
- Extended results to higher-dimensional rotation-invariant measures.

## Abstract

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.