# The orthogonal complement of the Hilbert space associated to holomorphic Hermite polynomials

## TL;DR

This paper investigates the orthogonal complement of a specific Hilbert space linked to holomorphic Hermite polynomials, providing explicit bases, kernels, and integral transforms to deepen understanding of its structure.

## Contribution

It introduces a polyanalytic orthonormal basis and explicit formulas for reproducing kernels and Segal--Bargmann transforms for the space.

## Key findings

- Explicit polyanalytic orthonormal basis constructed
- Reproducing kernel functions derived
- Segal--Bargmann integral transforms formulated

## Abstract

We study the orthogonal complement of the Hilbert subspace considered by by van Eijndhoven and Meyers in [J. Math. Anal. Appl. 146 (1990), no. 1, 89--98} and associated to holomorphic Hermite polynomials. A polyanalytic orthonormal basis is given and the explicit expressions of the corresponding reproducing kernel functions and Segal--Bargmann integral transforms are provided.