The Fueter Mittag-Leffler Bargmann transform

In this paper we continue exploring the Mittag-Leffler Bargmann (MLB) transform, which maps the Hilbert space L2(R) onto the Mittag-Leffler-Fock (MLF) space. The MLF space is a reproducing kernel Hilbert space that extends the classic Fock space and its reproducing kernel is given by the Mittag-Leffler function. We study the MLB transform and its main properties in the quaternionic setting. In this noncommutative setting there are two function theories that are prominent: the slice hyperholomorphic theory and the Fueter regular theory. The connection between the slice hyperholomorphic functions and the Fueter regular functions is given by the Fueter mapping theorem. The Mittag-Leffler Bargmann transform investigated in this paper maps the quaternionic-valued L2(R, H) space onto a counterpart of the MLF space in the Fueter regular setting. Finally the creation, annihilation, backward-shift and integration operators are studied in the case of the Fueter-MLF space. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).