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.

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.