  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 $\mathcal{P}$ gives rise to an algebra of
Clifford-valued operators whose canonical generators are isomorphic to the
orthosymplectic Lie algebra $\mathfrak{osp}(1|2)$.
  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 $\mathfrak{osp}(1|2)$) 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.
