Fueter regularity and slice regularity: meeting points for two function theories

We present some meeting points between two function theories, the Fueter theory of regular functions and the recent theory of quaternionic slice regular functions, which includes polynomials and power series with quaternionic coefficients. We show that every slice regular function coincides up to the first order with a unique regular function on the three-dimensional subset of reduced quaternions. We also characterize the regular functions so obtained. These relations have a higher dimensional counterpart between the theory of monogenic functions on Clifford algebras and the one of slice regular functions of a Clifford variable. We define a first order differential operator which extends the Dirac and Weyl operators to functions that can depend on all the coordinates of the algebra. The operator behaves well both w.r.t. monogenic functions and w.r.t. the powers of the (complete) Clifford variable. This last property relates the operator with the recent theory of slice monogenic and slice regular functions of a Clifford variable.