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 matrix recurrence for systems of Clifford algebra-valued orthogonal polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory ("Matrix representations of a basic polynomial sequence in arbitrary dimension". Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Guerlebeck, Bock, Lavicka, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.Fundação para a Ciência e a Tecnologia (FCT

On generalized hypercomplex laguerre-type exponentials and applications

In hypercomplex context, we have recently constructed Appell sequences with respect to a generalized Laguerre derivative operator. This construction is based on the use of a basic set of monogenic polynomials which is particularly easy to handle and can play an important role in applications. Here we consider Laguerre-type exponentials of order m and introduce Laguerre-type circular and hyperbolic functions.Financial support from Center for research and development in Mathematics and Applications of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), is gratefully acknowledged

Monogenic generalized Laguerre and Hermite polynomials and related functions

In recent years classical polynomials of a real or complex variable and their generalizations to the case of several real or complex variables have been in a focus of increasing attention leading to new and interesting problems. In this paper we construct higher dimensional analogues to generalized Laguerre and Hermite polynomials as well as some based functions in the framework of Clifford analysis. Our process of construction makes use of the Appell sequence of monogenic polynomials constructed by Falcão/Malonek and stresses the usefulness of the concept of the hypercomplex derivative in connection with the adaptation of the operational approach, developed by Gould et al. in the 60's of the last century and by Dattoli et al. in recent years for the case of the Laguerre polynomials. The constructed polynomials are used to define related functions whose properties show the application of Special Functions in Clifford analysis.FC

On Vietoris' number sequence and combinatorial identities with quaternions

Ruscheweyh and Salinas showed in 2004 the relationship of a celebrated theorem of Vietoris (1958) about the positivity of certain sine and cosine sums with the function theoretic concept of stable holomorphic functions in the unit disc. The present paper reveals that the coefficient sequence in Vietoris' theorem is identical to a number sequence obtained by a new combinatorial identity which involves generators of quaternions. In this sense Vietoris' sequence of rational numbers combines seemingly disperse subjects in Real, Complex and Hypercomplex Analysis. Thereby we show that a non-standard application of Clifford algebra tools is able to reveal new insights in objects of combinatorial nature.The work of the first and third authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEstOE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Laguerre derivative and monogenic Laguerre polynomials : an operational approach

Hypercomplex function theory generalizes the theory of holomorphic functions of one complex variable by using Clifford Algebras and provides the fundamentals of Clifford Analysis as a refinement of Harmonic Analysis in higher dimensions. We define the Laguerre derivative operator in hypercomplex context and by using operational techniques we construct generalized hypercomplex monogenic Laguerre polynomials. Moreover, Laguerre-type exponentials of order $m$ are defined.Financial support from "Center for research and development in Mathematics and Applications'' of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT) is gratefully acknowledged

Combinatorial identities in the context of hypercomplex function theory

Recently, the authors have shown that a certain combinatorial identity in terms of generators of quaternions is related to a particular sequence of rational numbers (Vietoris' number sequence). This sequence appeared for the first time in a theorem by Vietoris (1958) and plays an important role in harmonic analysis and in the theory of stable holomorphic functions in the unit disc. We present a generalization of that combinatorial identity involving an arbitrary number of generators of a Clifford algebra. The result reveals new insights in combinatorial phenomena in the context of hypercomplex function theory.The work of the first and third authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Recurrence relations for Clifford algebra-valued orthogonal polynomials

The theory of orthogonal polynomials of one real or complex variable is well established as well as its generalization for the multidimensional case. Hypercomplex function theory (or Clifford analysis) provides an alternative approach to deal with higher dimensions. In this context, we study systems of orthogonal polynomials of a hypercomplex variable with values in a Clifford algebra and prove some of their properties.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 Research Centre of Mathematics of the University of Minho and the Portuguese Foundation for Science and Technology (“FCT - Fundação para a Ciência e a Tecnologia”), within projects PEst-OE/MAT/UI4106/2014 and PEstOE/MAT/UI0013/2014

Vietoris' number sequence and its generalizations through hypercomplex function theory

The so-called Vietoris' number sequence is a sequence of rational numbers that appeared for the first time in a celebrated theorem by Vietoris (1958) about the positivity of certain trigonometric sums with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/ Salinas, 2004). In the context of hypercomplex function theory those numbers appear as coefficients of special homogeneous polynomials in R^3 whose generalization to an arbitrary dimension n lead to a n-parameter generalized Vietoris' number sequence that characterizes hypercomplex Appell polynomials in R^n.The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013. The work of the other authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013

Intrinsic properties of a non-symmetric number triangle

Several authors are currently working on generalized Appell polynomials and their applications in the framework of hypercomplex function theory in Rn+1. A few years ago, two of the authors of this paper introduced a prototype of these generalized Appell polynomials, which heavily draws on a one-parameter family of non-symmetric number triangles T (n), n ≥ 2. In this paper, we prove several new and interesting properties of finite and infinite sums constructed from entries of T (n), similar to the ordinary Pascal triangle, which is not a part of that family. In particular, we obtain a recurrence relation for a family of finite sums, analogous to the ordinary Fibonacci sequence, and derive its corresponding generating function.UA - Universidade de Aveiro(UIDB/00013/2020)This work was supported by Portuguese funds through the CMAT - Research Centre of Mathematics of University of Minho - and through the CIDMA - Center of Research and De velopment in Mathematics and Applications (University of Aveiro) and the Portuguese Foun dation for Science and Technology (“FCT - Funda¸c˜ao para a Ciˆencia e Tecnologia”), within projects UIDB/00013/2020, UIDP/00013/2020, UIDB/04106/2020 and UIDP/04106/2020