  We present some new relations between the Cauchy-Riemann operator on the real
Clifford algebra $\mathbb R_n$ of signature $(0,n)$ and slice-regular functions
on $\mathbb R_n$. The class of slice-regular functions, which comprises all
polynomials with coefficients on one side, is the base of a recent function
theory in several hypercomplex settings, including quaternions and Clifford
algebras. In this paper we present formulas, relating the Cauchy-Riemann
operator, the spherical Dirac operator, the differential operator
characterizing slice regularity, and the {spherical derivative} of a slice
function. The computation of the Laplacian of the spherical derivative of a
slice regular function gives a result which implies, in particular, the
Fueter-Sce Theorem. In the two four-dimensional cases represented by the
paravectors of $\mathbb R_3$ and by the space of quaternions, these results are
related to zonal harmonics on the three-dimensional sphere and to the Poisson
kernel of the unit ball of $\mathbb R^4$.
