  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.
