# The slice hyperholomorphic Bergman space on $\mathbb{B}_R$: Integral representation and asymptotic behavior

## TL;DR

This paper explores the properties of the weighted hyperholomorphic Bergman space on quaternionic balls, providing explicit kernel formulas, introducing a quaternionic Bargmann transform, and analyzing asymptotic behavior as the radius grows.

## Contribution

It completes the study of the hyperholomorphic Bergman space, derives explicit kernel expressions, and establishes the asymptotic connection to Bargmann-Fock spaces in the quaternionic setting.

## Key findings

- Explicit Bergman kernel expressed via hypergeometric functions.
- Introduction of a quaternionic Bargmann transform and analysis of its properties.
- Asymptotic behavior linking the Bergman kernel to the Bargmann-Fock space as radius tends to infinity.

## Abstract

The aim of the present paper is three folds. Firstly, we complete the study of the weighted hyperholomorphic Bergman space of the second kind on the ball of radius $R$ centred at the origin. The explicit expression of its Bergman kernel is given and can be written in terms of special hypergeometric functions of two non-commuting (quaternionic) variables. Secondly, we introduce and study some basic properties of an associated integral transform, the quaternionic analogue of the so-called second Bargmann transform for the holomorphic Bergman space. Finally, we establish the asymptotic behavior as $R$ goes to infinity. We show in particular that the reproducing kernel of the weighted slice hyperholomorphic Bergman space gives rise to its analogue for the slice hyperholomorphic Bargamann-Fock space.