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 on the paravector space of some Clifford Algebra $\mathbb{R}_m$ with an odd number of imaginary units, the Fueter-Sce construction establish a critical index $k=\frac{m-1}{2}$ (sometimes called Fueter-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 we find that the polynomials of degree at most $2k$ are the only slice regular holomorphic Cliffordian functions of order $k$.