On block matrices of pascal type in clifford analysis

Since the 90-ties the Pascal matrix, its generalizations and applications have been in focus of a great amount of publications. As it is well known, the Pascal matrix, the symmetric Pascal matrix and other special matrices of Pascal type play an important role in many scientific areas, among them Numerical Analysis, Combinatorics, Number Theory, Probability, Image processing, Sinal processing, Electrical enginneering, etc. We present a unified approach to matrix representations of special polynomials in several hypercomplex variables (new Bernoulli, Euler etc. polynomials), extending results of H. Malonek, G.Tomaz: Bernoulli polynomials and Pascal matrices in the context of Clifford Analysis, Discrete Appl. Math. 157(4) (2009) 838-847. The hypercomplex version of a new Pascal matrix with block structure, which resembles the ordinary one for polynomials of one variable will be discussed in detail

ON BLOCK MATRICES OF PASCAL TYPE IN CLIFFORD ANALYSIS

Since the 90-ties the Pascal matrix, its generalizations and applications have been in the focus of a great amount of publications. As it is well known, the Pascal matrix, the symmetric Pascal matrix and other special matrices of Pascal type play an important role in many scientific areas, among them Numerical Analysis, Combinatorics, Number Theory, Probability, Image processing, Sinal processing, Electrical engineering, etc. We present a unified approach to matrix representations of special polynomials in several hypercomplex variables (new Bernoulli, Euler etc. polynomials), extending results of H. Malonek, G.Tomaz: Bernoulli polynomials and Pascal matrices in the context of Clifford Analysis, Discrete Appl. Math. 157(4)(2009) 838-847. The hypercomplex version of a new Pascal matrix with block structure, which resembles the ordinary one for polynomials of one variable will be discussed in detail

Bernoulli polynomials and Pascal matrices in the context of Clifford analysis

AbstractThis paper describes an approach to generalized Bernoulli polynomials in higher dimensions by using Clifford algebras. Due to the fact that the obtained Bernoulli polynomials are special hypercomplex holomorphic (monogenic) functions in the sense of Clifford Analysis, they have properties very similar to those of the classical polynomials. Hypercomplex Pascal and Bernoulli matrices are defined and studied, thereby generalizing results recently obtained by Zhang and Wang (Z. Zhang, J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math. 154 (11) (2006) 1622–1632)

Pascal’s triangle and other number triangles in Clifford analysis

The recent introduction of generalized Appell sequences in the framework of Clifford Analysis solved an open question about a suitable construction of power-like monogenic polynomials as generalizations of the integer powers of a complex variable. The deep connection between Appell sequences and Pascal’s triangle called also attention to other number triangles and, at the same time, to the construction of generalized Pascal matrices. Both aspects are considered in this communication.FC

Laguerre polynomials in several hypercomplex variables and their matrix representation

Recently the creation matrix, intimately related to the Pascal matrix and its generalizations, has been used to develop matrix representations of special polynomials, in particular Appell polynomials. In this paper we describe a matrix approach to polynomials in several hypercomplex variables based on special block matrices whose structures simulate the creation matrix and the Pascal matrix. We apply the approach to hypercomplex Laguerre polynomials, although it can be used for other Appell sequences, too

(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

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j} together with the Lie-algebraic representation of raising and lowering operators acting on the lattice h\BZ^n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous improvements throughout the paper was don

Matrix approach to hypercomplex Appell polynomials

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy–Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras