Slice regular holomorphic Cliffordian functions of order k

Holomorphic Cliffordian functions of order k are functions in the kernel of the differential operator ∂̄Δᵏ. When ∂̄Δᵏ is applied to functions defined on the paravector space of some Clifford Algebra Rₘ with an odd number of imaginary units, the Fueter-Sce construction establish a critical index k=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 m-1/2. In this paper we analyze the case k<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.