  This paper studies a particular class of higher order conformally invariant
dif- ferential operators and related integral operators acting on functions
taking values in particular finite dimensional irreducible representations of
the Spin group. The differential operators can be seen as a generalization to
higher spin spaces of kth- powers of the Euclidean Dirac operator. To construct
these operators, we use the framework of higher spin theory in Clifford
analysis, in which irreducible represen- tations of the Spin group are realized
as polynomial spaces satisfying a particular system of differential equations.
As a consequence, these operators act on functions taking values in the space
of homogeneous harmonic or monogenic polynomials de- pending on the order.
Moreover, we classify these operators in analogy with the quantization of
angular momentum in quantum mechanics to unify the terminology used in studying
higher order higher spin conformally invariant operators: for integer and
half-integer spin, these are respectively bosonic and fermionic operators.
Fundamental solutions and their conformal invariance are presented here.
