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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   o  s  p  (  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   o  s  p  (  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.

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