# On the Bargmann-Fock-Fueter and Bergman-Fueter integral transforms

## TL;DR

This paper explores quaternionic integral transforms of Bargmann-Fock type, establishing new representations and generating functions within quaternionic hyperholomorphic function spaces, based on the Fueter mapping theorem.

## Contribution

It introduces novel quaternionic integral transforms and constructs quaternionic regular polynomial systems using Hermite functions, expanding the theory of quaternionic function spaces.

## Key findings

- Derived new integral representations for quaternionic Fock and Bergman spaces
- Constructed quaternionic regular polynomial systems from Hermite functions
- Established reproducing kernel Hilbert space structures for these transforms

## Abstract

This paper deals with some special integral transforms of Bargmann-Fock type in the setting of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. The construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions we can construct an Appell system of quaternionic regular polynomials. The ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic setting are obtained in both the Fock and Bergman cases.