  We introduce the so-called Clifford-Gegenbauer polynomials in the framework
of Dunkl operators, as well on the unit ball B(1), as on the Euclidean space
$R^m$. In both cases we obtain several properties of these polynomials, such as
a Rodrigues formula, a differential equation and an explicit relation
connecting them with the Jacobi polynomials on the real line. As in the
classical Clifford case, the orthogonality of the polynomials on $R^m$ must be
treated in a completely different way than the orthogonality of their
counterparts on B(1). In case of $R^m$, it must be expressed in terms of a
bilinear form instead of an integral. Furthermore, in this paper the theory of
Dunkl monogenics is further developed.
