（离散）Almansi型分解：基于$\mathfrak{osp}(1|2)$对称性的本影演算框架
(Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries
摘要：
我们介绍了超复变量的影演算形式主义，从多元多项式$\BR[\underline{x}]$的代数应描述的Weyl-Heisenberg代数的生成元的事实。将$\BR[\underline{x}]$推广到Clifford值多项式代数$\mathcal{P}$，得到了Clifford值算子代数，其标准生成元同构于正交辛李代数$\mathfrak{osp}(1|2)$ .这种扩展提供了一个有效的框架，在连续性和离散性，使我们能够建立一个替代公式的阿尔芒西分解克利福德分析（c.f. \cite{Ryan90,MR02,MAGU}），它对应于子空间$\ker (D')^k$的Fischer分解的有意义的推广。我们将在后面讨论$\mathfrak{sl}_2(\BR)$的对称性（甚至是$\mathfrak{osp}(1|2)$的一部分）是如何在最近的\textsc{Render}方法中普遍存在的（c. f. \cite{Render08}），表明它们可以用量子力学中哈密顿算符的分离变量分离变量法来解释。
英文摘要：
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.
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