(Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries

Abstract

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 $\ker (D^{\prime }{)}^k$. 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.

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)

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• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• Representations of Monomiality Principle with Sheffer-type Polynomials and Boson Normal Ordering
• On Appell Sets and the Fueter-Sce Mapping
• Fischer Decomposition for Difference Dirac Operators
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Cited by in corpus (6)

• Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle
• A conformal group approach to the Dirac-Kähler system on the lattice
• Relativistic Wave Equations on the lattice: an operational perspective
• Special Functions of Hypercomplex Variable on the Lattice Based on SU(1,1)
• Hypercomplex Fock States for Discrete Electromagnetic Schrödinger Operators: A Bayesian Probability Perspective
• Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle