# Holomorphic Hermite functions and ellipses

## TL;DR

This paper explores holomorphic Hermite functions linked to ellipses on the complex plane, connecting them to Segal-Bargmann spaces and extending prior work on these functions and their applications.

## Contribution

It demonstrates that holomorphic Hermite functions associated with ellipses are characterized by specific ellipse cases and relate to Segal-Bargmann spaces via Bargmann-type transforms.

## Key findings

- Holomorphic Hermite functions are determined by certain ellipses.
- Reproducing kernel Hilbert spaces correspond to Segal-Bargmann spaces.
- Connections established between ellipse-based functions and Bargmann transforms.

## Abstract

In 1990 van Eijnghoven and Meyers introduced systems of holomorphic Hermite functions and reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreover they studied the relationship between the family of all their Hilbert spaces and a class of Gelfand-Shilov functions. After that, their systems of holomorphic Hermite functions have been applied to studying quantization on the complex plane, combinatorics, and etc. On the other hand, the author recently introduced systems of holomorphic Hermite functions associated with ellipses on the complex plane. The present paper shows that their systems of holomorphic Hermite functions are determined by some cases of ellipses, and that their reproducing kernel Hilbert spaces are some cases of the Segal-Bargmann spaces determined by the Bargmann-type transforms introduced by Sjoestrand.