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)

A note on a generalized Joukowski transformation

AbstractIt is well known that the Joukowski transformation plays an important role in physical applications of conformal mappings, in particular in the study of flows around airfoils. We present, for n≥2, an n-dimensional hypercomplex analogue of the Joukowski transformation and describe in some detail the 3D case. A generalized 3D Joukowski profile, produced with Maple, is included

A note on totally regular variables and Appell sequences in hypercomplex function theory

Series title : Lecture notes in computer science, vol. 7971, ISSN 0302-9743The aim of our contribution is to call attention to the relationship between totally regular variables, introduced by R. Delanghe in 1970, and Appell sequences with respect to the hypercomplex derivative. Under some natural normalization condition the set of all paravector valued totally regular variables defined in the three dimensional Euclidean space will be completely characterized. Together with their integer powers they constitute automatically Appell sequences, since they are isomorphic to the complex variables.Fundação para a Ciência e a Tecnologia (FCT

Three-term recurrence relations for systems of Clifford algebra-valued orthogonal polynomials

Recently, systems of Clifford algebra-valued orthogonal polynomials have been studied from different points of view. We prove in this paper that for their building blocks there exist some three-term recurrence relations, similar to that for orthogonal polynomials of one real variable. As a surprising byproduct of own interest we found out that the whole construction process of Clifford algebra-valued orthogonal polynomials via Gelfand-Tsetlin basis or otherwise relies only on one and the same basic Appell sequence of polynomials.This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications of the University of Aveiro, the CMAT - Research Centre of Mathematics of the University of Minho and the FCT - Portuguese Foundation for Science and Technology (“Fundação para a Ciˆencia e a Tecnologia”), within projects PEst-OE/MAT/UI4106/2014 and PEst-OE/MAT/UI0013/2014.info:eu-repo/semantics/publishedVersio

A unified matrix approach to the representation of Appell polynomials

In this paper, we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix – a special matrix which has only the natural numbers as entries and is closely related to the well-known Pascal matrix. By this means, we stress the arithmetical origins of Appell polynomials. The approach also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations