Matrix approach to hypercomplex Appell polynomials

Abstract

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.

Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

“…Based on this concept, it is possible to obtain a binomial-type identity for hypercomplex Appell sequences, which extend the identity 10 Hypercomplex Appell polynomials have received a lot of attention in the last decade. They have been studied in detail in several papers by different authors and various applications have been considered [33][34][35][36][37][38][39][40][41]. In what follows we focus on a class of polynomials of the form…”

Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences

“…Finally we emphasize that in this paper the key point was the role of both creation and shift matrices. In fact, we showed the relevance of those matrices in the representation of general hypercomplex sequences unifying and generalizing the approach already considered in [2] and [3].…”

“…[1]). The importance of the creation matrix was also confirmed in [3] where the authors developed the matrix representation of homogeneous Appell polynomials that are solutions of a generalized CauchyRiemann system in Euclidean spaces of arbitrary dimensions. In that work the first term of the considered sequence is a real constant.…”

On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

“…[9][10][11] Much of the older theory of special monogenic polynomials has been given a different interpretation. A new light has been shed upon the study of elementary functions, [12][13][14][15][16][17][18] the computation of combinatorial identities, [19][20][21] and the study of a generalized Joukowski transformation in Euclidean space of arbitrary higher dimension. 22 Earlier results in the theory of special polynomial bases in hypercomplex analysis and its counterpart in the function theory of several complex variables can be found in previous studies 12,[23][24][25][26][27][28][29][30][31][32][33][34] and elsewhere.…”