# S--polyregular Bargmann spaces

## TL;DR

This paper introduces two new classes of quaternionic Hilbert spaces related to slice polyregular functions, providing their properties, explicit kernels, Segal--Bargmann transforms, and spectral descriptions as subspaces of eigenfunctions.

## Contribution

It generalizes existing hyperholomorphic Bargmann spaces to the slice polyregular setting and explores their structure, kernels, transforms, and spectral properties.

## Key findings

- Explicit formulas for reproducing kernels
- Introduction of associated Segal--Bargmann transforms
- Spectral characterization as eigenspaces of a differential operator

## Abstract

We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit formulas of their reproducing kernels are given and associated Segal--Bargmann transforms are also introduced and studied. The spectral description as special subspaces of $L^2$-eigenspaces of a second order differential operator involving the slice derivative is investigated.