We introduce the umbral calculus formalism for hypercomplex variables
starting from the fact that the algebra of multivariate polynomials
\BR[\underline{x}] shall be described in terms of the generators of the
Weyl-Heisenberg algebra. The extension of \BR[\underline{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 (c.f. \cite{Ryan90,MR02,MAGU}) that corresponds to a
meaningful generalization of Fischer decomposition for the subspaces ker(D′)k.
We will discuss afterwards how the symmetries of \mathfrak{sl}_2(\BR) (even
part of osp(1∣2)) are ubiquitous on the recent approach of
\textsc{Render} (c.f. \cite{Render08}), showing that they can be interpreted in
terms of the method of separation of variables for the Hamiltonian operator in
quantum mechanics.Comment: Improved version of the Technical Report arXiv:0901.4691v1; accepted
for publication @ Math. Meth. Appl. Sci
http://www.mat.uc.pt/preprints/ps/p1054.pdf (Preliminary Report December
2010