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 {Mn(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 M0(x) = Pk(x) is a given but arbitrary monogenic polynomial of degree k defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{m+1}}$$\end{document}. In particular, we construct an explicit sequence for the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_0(x)=\mathbf{P}_k(\underline x)}$$\end{document} being an arbitrary homogeneous monogenic polynomial defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R^m}$$\end{document}. The connection of this sequence with the so-called Fueter’s theorem will also be discussed.