Shifted Appell Sequences in Clifford Analysis

The aim of this paper is to present a generalization of the Appell sequences within the framework of Clifford analysis called shifted Appell sequences. It consists of sequences {M _n (x)} _n ≥ 0 of monogenic polynomials satisfying the Appell condition (i.e. the hypercomplex derivative of each polynomial in the sequence equals, up to a multiplicative constant, its preceding term) such that the first term M ₀(x) = P _k (x) is a given but arbitrary monogenic polynomial of degree k defined in ${mathbb{R}^{m+1}}$ . In particular, we construct an explicit sequence for the case ${M_0(x)=mathbf{P}_k(underline x)}$ being an arbitrary homogeneous monogenic polynomial defined in ${mathbb R^m}$ . The connection of this sequence with the so-called Fueter’s theorem will also be discussed.