  In this paper we continue the study of Bergman theory for the class of slice
regular functions.
  In the slice regular setting there are two possibilities to introduce the
Bergman spaces, that are called of the first and of the second kind. In this
paper we mainly consider the Bergman theory of the second kind, by providing an
explicit description of the Bergman kernel in the case of the unit ball and of
the half space. In the case of the unit ball, we study the Bergman-Sce
transform. We also show that the two Bergman theories can be compared only if
suitable weights are taken into account. Finally, we use the Schwarz reflection
principle to relate the Bergman kernel with its values on a complex half plane.
