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New higher–dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co–ordinates, the "finite part" distribution $$ Fpx^{\mu }_{ + } $$ on the real line and the generalized spherical means involving vector–valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector–valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self–contained, it offers an overview of the results found so far. [ABSTRACT FROM AUTHOR]