(Discrete) Almansi type decompositions: an umbral calculus framework based on osp(1|2) symmetries

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials IR[x¯] shall be described in terms of the generators of the Weyl–Heisenberg algebra. The extension of IR[x¯] to the algebra of Clifford‐valued polynomials P gives rise to an algebra of Clifford‐valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra osp(1|2). This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of Almansi decomposition in Clifford analysis obtained by Ryan (Zeitschrift für Analysis und ihre Anwendungen 1990) and Malonek & Ren (Mathematical Methods in the Applied Sciences 2002;2007) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces ker(D′)k. We will discuss afterwards how the symmetries of (even part of osp(1|2)) are ubiquitous on the recent approach of RENDER (Duke Mathematical Journal 2008) showing that they can be interpreted in terms of the method of separation of variables for the Hamiltonian operator in quantum mechanics. Copyright © 2011 John Wiley & Sons, Ltd.