Slice Regular Holomorphic Cliffordian Functions of Order k

Abstract

Holomorphic Cliffordian functions of order k are functions in the kernel of the differential operator \(\overline{\partial }\Delta ^k\). When \(\overline{\partial }\Delta ^k\) is applied to functions defined in the paravector space of some Clifford Algebra \(\mathbb {R}_m\) with an odd number of imaginary units, the Fueter–Sce construction establishes a critical index \(k=\frac{m-1}{2}\) (sometimes called Sce exponent) for which the class of slice regular functions is contained in the one of holomorphic Cliffordian functions of order \(\frac{m-1}{2}\). In this paper, we analyze the case \(k<\frac{m-1}{2}\) and find that the polynomials of degree at most 2k are the only slice regular holomorphic Cliffordian functions of order k.