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

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.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

On multivariable cumulant polynomial sequences with applications

A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending on what is plugged in the indeterminates, either sequences of moments either sequences of cumulants can be recovered. The main tool is a formal generalization of random sums, also with a multivariate random index and not necessarily integer-valued. Applications are given within parameter estimations, L\'evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in characterizing these models. Some open problems end the paper.Comment: 17 pages, In pres

Josef Meixner: his life and his orthogonal polynomials

This paper starts with a biographical sketch of the life of Josef Meixner. Then his motivations to work on orthogonal polynomials and special functions are reviewed. Meixner's 1934 paper introducing the Meixner and Meixner-Pollaczek polynomials is discussed in detail. Truksa's forgotten 1931 paper, which already contains the Meixner polynomials, is mentioned. The paper ends with a survey of the reception of Meixner's 1934 paper.Comment: v4: 18 pages, generating function for Krawtchouk polynomials on p.10 correcte

On a representation of time space-harmonic polynomials via symbolic L\'evy processes

In this paper, we review the theory of time space-harmonic polynomials developed by using a symbolic device known in the literature as the classical umbral calculus. The advantage of this symbolic tool is twofold. First a moment representation is allowed for a wide class of polynomial stochastic involving the L\'evy processes in respect to which they are martingales. This representation includes some well-known examples such as Hermite polynomials in connection with Brownian motion. As a consequence, characterizations of many other families of polynomials having the time space-harmonic property can be recovered via the symbolic moment representation. New relations with Kailath-Segall polynomials are stated. Secondly the generalization to the multivariable framework is straightforward. Connections with cumulants and Bell polynomials are highlighted both in the univariate case and in the multivariate one. Open problems are addressed at the end of the paper

Multivariate time-space harmonic polynomials: a symbolic approach

By means of a symbolic method, in this paper we introduce a new family of multivariate polynomials such that multivariate L\'evy processes can be dealt with as they were martingales. In the univariate case, this family of polynomials is known as time-space harmonic polynomials. Then, simple closed-form expressions of some multivariate classical families of polynomials are given. The main advantage of this symbolic representation is the plainness of the setting which reduces to few fundamental statements but also of its implementation in any symbolic software. The role played by cumulants is emphasized within the generalized Hermite polynomials. The new class of multivariate L\'evy-Sheffer systems is introduced.Comment: In pres