The Fueter Mittag-Leffler Bargmann transform

Abstract

In this paper we continue exploring the Mittag-Leffler Bargmann (MLB) transform, which maps the Hilbert space $L^2\left(R\right)$ 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 $L^2(R,H)$ 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.