Slice regularity and harmonicity on Clifford algebras

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 \(\mathbb R^4\).