Quaternionic slice regular functions with some sphere bundles

In the hypercomplex analysis, it is well known that the quaternionic slice regular functions are defined on axially symmetric slice domains and that each quaternion is represented in the form (Formula presented.), where (Formula presented.) and (Formula presented.) but the mapping (Formula presented.) allows to see that any axially symmetric slice domain is the base space of a trivial sphere bundle. Remember that a sphere bundle is a useful concept in topology and it is a fiber bundle in which the fibers are spheres of dimension n. The previous idea is the natural motivation of this work whose purpose is to present a definition of the quaternionic slice regular functions on the total space of some sphere bundles, show two non-trivial examples of these function spaces and establish some important results of this function theory.