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

Matrix approach to Frobenius-Euler polynomials

In the last two years Frobenius-Euler polynomials have gained renewed interest and were studied by several authors. This paper presents a novel approach to these polynomials by treating them as Appell polynomials. This allows to apply an elementary matrix representation based on a nilpotent creation matrix for proving some of the main properties of Frobenius-Euler polynomials in a straightforward way

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

Matrix representation of real and hypercomplex Appell polynomials

In a unfied approach to the matrix representation of di erent types of real Appell polynomials was developed, based on a special matrix which has only the natural numbers as entries. This matrix, also called creation matrix, generates the Pascal matrix and allows to consider a set of Appell polynomials as solution of a rst order vector di erential equation with certain initial conditions. Besides a new elementary construction of the monogenic exponential function studied in, we analogously derive examples of di erent sets of non-homogenous hypercomplex Appell polynomials given by its matrix representation

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

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

Combinatorial Identities Associated with a Multidimensional Polynomial Sequence

In this paper we combine the knowledge of different structures of a special Appell multidimensional polynomial sequence with the problem of establishing combinatorial identities. The elements of this special polynomial sequence have values in a Clifford algebra, are homogeneous hypercomplex differentiable functions of different degrees and their coefficients properties can be used to stress interesting matrix and combinatorial relations

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

Polinómios de Appell multidimensionais e sua representação matricial

Doutoramento em MatemáticaNesta dissertação é apresentada uma abordagem a polinómios de Appell multidimensionais dando-se especial relevância à estrutura da sua função geradora. Esta estrutura, conjugada com uma escolha adequada de ordenação dos monómios que figuram nos polinómios, confere um carácter unificador à abordagem e possibilita uma representação matricial de polinómios de Appell por meio de matrizes particionadas em blocos. Tais matrizes são construídas a partir de uma matriz de estrutura simples, designada matriz de criação, subdiagonal e cujas entradas não nulas são os sucessivos números naturais. A exponencial desta matriz é a conhecida matriz de Pascal, triangular inferior, onde figuram os números binomiais que fazem parte integrante dos coeficientes dos polinómios de Appell. Finalmente, aplica-se a abordagem apresentada a polinómios de Appell definidos no contexto da Análise de Clifford.In this thesis an approach to multidimensional Appell polynomials is presented with special relevance for the structure of their generating function. This structure, together with an adequate choice of an ordering for the monomials that are present in the polynomials, gives a unifying nature to our approach and allows the representation of Appell polynomials by means of block matrices. Such matrices are constructed from another matrix with simple structure, called creation matrix, which is a sub-diagonal matrices whose nonzero entries are the successive natural numbers. The exponential of this matrix is the well known lower triangular Pascal matrix, lower triangular, where the binomial numbers appear as part of the coefficients of Appell polynomials. Finally, the presented approach is applied to Appell polynomials defined in the context of Clifford Analysis

Non-symmetric number triangles arising from hypercomplex function theory in R^(n+1)

The paper is focused on intrinsic properties of a one-parameter family of non-symmetric number triangles T(n),n≥2, which arises in the construction of hyperholomorphic Appell polynomials.UA -Universidade de Aveiro(UIDB/00013/2020