This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
The paper studies discrete structural properties of polynomials that play an important role in th... more The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of Harmonic Analysis by means of generalized holomorphic (monogenic) functions of Hypercomplex Analysis. The Sturm-Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences , first encountered as a special sequence (corresponding to = 2) by L. Vietoris in 1958 in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations we obtain a general recurrence relation for and we derive an exponential generating function of expressed by Kummer's confluent hypergeometric function.
The construction of two different representations of special Appell polynomials in (n + 1) real v... more The construction of two different representations of special Appell polynomials in (n + 1) real variables with values in a Clifford algebra suggested to explore the relation between the respective coefficients. Properties of sequences resulting from such relation and an interesting trapezoidal array of their elements are pointed out.
Using Clifford analysis methods, we provide a unified approach to obtain explicit solutions of so... more Using Clifford analysis methods, we provide a unified approach to obtain explicit solutions of some partial differential equations combining the n-dimensional Dirac and Euler operators, including generalizations of the classical time-harmonic Maxwell equations. The obtained regular solutions show strong connections between hypergeometric functions and homogeneous polynomials in the kernel of the Dirac operator.
In this paper we study the structure of the solutions to higher dimensional Dirac type equations ... more In this paper we study the structure of the solutions to higher dimensional Dirac type equations generalizing the known λ-hyperholomorphic functions, where λ is a complex parameter. The structure of the solutions to the system of partial differential equations (D- λ) f=0 show a close connection with Bessel functions of first kind with complex argument. The more general system of partial differential equations that is considered in this paper combines Dirac and Euler operators and emphasizes the role of the Bessel functions. However, contrary to the simplest case, one gets now Bessel functions of any arbitrary complex order.
Modern Methods in Operator Theory and Harmonic Analysis, 2019
Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over ... more Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over R n+1 are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.
This work was supported by Portuguese funds through the CIDMA - Center for Research and Developme... more 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 - Funda¸c˜ao para a Ciˆencia e a Tecnologia”), within projects PEst-OE/MAT/UI4106/2014 and PEstOE/MAT/UI0013/2014.
The work of the first and third authors was supported by Portuguese funds through the CIDMA - Cen... more 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-Fundacao para a Ciencia 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.
The so-called Vietoris’ number sequence is a sequence of rational numbers that appeared for the f... more 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 whose generalization to an arbitrary dimension n lead to a n-parameter generalized Vietoris’ number sequence that characterizes hypercomplex Appell polynomials in R. 2010 Mathematics Subject Classifications: 30G35, 11B83, 05A10
The work of the third author was supported by Portuguese funds through the CMAT - Centreof Mathem... more The work of the third author was supported by Portuguese funds through the CMAT - Centreof Mathematics and FCT within the Project UID/MAT/00013/2013. The work of the otherauthors was supported by Portuguese funds through the CIDMA - Center for Researchand Development in Mathematics and Applications, and the Portuguese Foundation forScience and Technology (“FCT-Fundacao para a Ciencia e Tecnologia”), within project PEst-OE/MAT/UI4106/2013.
Ruscheweyh and Salinas showed in 2004 the relationship of a celebrated theorem of Vietoris (1958)... more 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 shows that the coefficient sequence in Vietoris' theorem is identical with the number sequence that characterizes generalized Appell sequences of homogeneous Clifford holomorphic polynomials in $\mathbb{R}^3.$ The paper studies one-parameter generalizations of Vietoris' number sequence, their properties as well as their role in the framework of Hypercomplex Function Theory.
The paper shows the role of shifted generalized Pascal matrices in a matrix representation of hyp... more The paper shows the role of shifted generalized Pascal matrices in a matrix representation of hypercomplex orthogonal Appell systems. It extends results obtained in previous works in the context of Appell sequences whose first term is a real constant to sequences whose initial term is a suitable chosen polynomial of n variables.
Recently, the authors have shown that a certain combinatorial identity in terms of generators of ... more 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.
Recently, by using methods of hypercomplex function theory, the authors have shown that a certain... more Recently, by using methods of hypercomplex function theory, the authors have shown that a certain sequence S of rational numbers (Vietoris' sequence) combines seemingly disperse subjects in real, complex and hypercomplex analysis. This sequence appeared for the first time in a theorem by Vietoris (1958) with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/Salinas, 2004). A non-standard application of Clifford algebra tools for defining Clifford-holomorphic sequences of Appell polynomials was the hypercomplex context in which a one-parametric generalization S(n), n ≥ 1, of S (corresponding to n = 2) surprisingly showed up. Without relying on hypercomplex methods this paper demonstrates how purely real methods also lead to S(n). For arbitrary n ≥ 1 the generating function is determined and for n = 2 a particular case of a recurrence relation similar to that known for Catalan numbers is proved.
This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the... more This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by L. Vietoris in 1958. For S a generating function is obtained which allows to derive an interesting relation to a result deduced in 1974 by Askey and Steinig about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.
Journal of Mathematical Analysis and Applications, 2017
This paper deals with a unified matrix representation for the Sheffer polynomials. The core of th... more This paper deals with a unified matrix representation for the Sheffer polynomials. The core of the proposed approach is the so-called creation matrix, a special subdiagonal matrix having as nonzero entries positive integer numbers, whose exponential coincides with the well-known Pascal matrix. In fact, Sheffer polynomials may be expressed in terms of two matrices both connected to it. As we will show, one of them is strictly related to Appell polynomials, while the other is linked to a binomial type sequence. Consequently, different types of Sheffer polynomials correspond to different choices of these two matrices.
Recently, systems of Clifford algebra-valued orthogonal polynomials have been studied from differ... more 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.
Computational Science and Its Applications – ICCSA 2014, 2014
ABSTRACT In this work, we give a brief description of the theory and properties of the three-dime... more ABSTRACT In this work, we give a brief description of the theory and properties of the three-dimensional quaternionic Zernike spherical polynomials (QZSPs). A generalization and refinement of the QZSPs to functions vanishing over the unit sphere leads to the computation of the weighted quaternionic Zernike spherical functions (WQZSFs). In particular, the underlying functions are of three real variables and take on values in the quaternions (identified with $\mathbb{R}^4$). Also, in this work, we prove that the WQZSFs are orthonormal in the unit ball with respect to a suitable weight function. The representation of these functions are given explicitly, and a summary of their fundamental properties is also discussed. To the best of our knowledge, this does not appear to have been done in literature before.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
The paper studies discrete structural properties of polynomials that play an important role in th... more The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of Harmonic Analysis by means of generalized holomorphic (monogenic) functions of Hypercomplex Analysis. The Sturm-Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences , first encountered as a special sequence (corresponding to = 2) by L. Vietoris in 1958 in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations we obtain a general recurrence relation for and we derive an exponential generating function of expressed by Kummer's confluent hypergeometric function.
The construction of two different representations of special Appell polynomials in (n + 1) real v... more The construction of two different representations of special Appell polynomials in (n + 1) real variables with values in a Clifford algebra suggested to explore the relation between the respective coefficients. Properties of sequences resulting from such relation and an interesting trapezoidal array of their elements are pointed out.
Using Clifford analysis methods, we provide a unified approach to obtain explicit solutions of so... more Using Clifford analysis methods, we provide a unified approach to obtain explicit solutions of some partial differential equations combining the n-dimensional Dirac and Euler operators, including generalizations of the classical time-harmonic Maxwell equations. The obtained regular solutions show strong connections between hypergeometric functions and homogeneous polynomials in the kernel of the Dirac operator.
In this paper we study the structure of the solutions to higher dimensional Dirac type equations ... more In this paper we study the structure of the solutions to higher dimensional Dirac type equations generalizing the known λ-hyperholomorphic functions, where λ is a complex parameter. The structure of the solutions to the system of partial differential equations (D- λ) f=0 show a close connection with Bessel functions of first kind with complex argument. The more general system of partial differential equations that is considered in this paper combines Dirac and Euler operators and emphasizes the role of the Bessel functions. However, contrary to the simplest case, one gets now Bessel functions of any arbitrary complex order.
Modern Methods in Operator Theory and Harmonic Analysis, 2019
Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over ... more Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over R n+1 are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.
This work was supported by Portuguese funds through the CIDMA - Center for Research and Developme... more 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 - Funda¸c˜ao para a Ciˆencia e a Tecnologia”), within projects PEst-OE/MAT/UI4106/2014 and PEstOE/MAT/UI0013/2014.
The work of the first and third authors was supported by Portuguese funds through the CIDMA - Cen... more 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-Fundacao para a Ciencia 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.
The so-called Vietoris’ number sequence is a sequence of rational numbers that appeared for the f... more 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 whose generalization to an arbitrary dimension n lead to a n-parameter generalized Vietoris’ number sequence that characterizes hypercomplex Appell polynomials in R. 2010 Mathematics Subject Classifications: 30G35, 11B83, 05A10
The work of the third author was supported by Portuguese funds through the CMAT - Centreof Mathem... more The work of the third author was supported by Portuguese funds through the CMAT - Centreof Mathematics and FCT within the Project UID/MAT/00013/2013. The work of the otherauthors was supported by Portuguese funds through the CIDMA - Center for Researchand Development in Mathematics and Applications, and the Portuguese Foundation forScience and Technology (“FCT-Fundacao para a Ciencia e Tecnologia”), within project PEst-OE/MAT/UI4106/2013.
Ruscheweyh and Salinas showed in 2004 the relationship of a celebrated theorem of Vietoris (1958)... more 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 shows that the coefficient sequence in Vietoris' theorem is identical with the number sequence that characterizes generalized Appell sequences of homogeneous Clifford holomorphic polynomials in $\mathbb{R}^3.$ The paper studies one-parameter generalizations of Vietoris' number sequence, their properties as well as their role in the framework of Hypercomplex Function Theory.
The paper shows the role of shifted generalized Pascal matrices in a matrix representation of hyp... more The paper shows the role of shifted generalized Pascal matrices in a matrix representation of hypercomplex orthogonal Appell systems. It extends results obtained in previous works in the context of Appell sequences whose first term is a real constant to sequences whose initial term is a suitable chosen polynomial of n variables.
Recently, the authors have shown that a certain combinatorial identity in terms of generators of ... more 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.
Recently, by using methods of hypercomplex function theory, the authors have shown that a certain... more Recently, by using methods of hypercomplex function theory, the authors have shown that a certain sequence S of rational numbers (Vietoris' sequence) combines seemingly disperse subjects in real, complex and hypercomplex analysis. This sequence appeared for the first time in a theorem by Vietoris (1958) with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/Salinas, 2004). A non-standard application of Clifford algebra tools for defining Clifford-holomorphic sequences of Appell polynomials was the hypercomplex context in which a one-parametric generalization S(n), n ≥ 1, of S (corresponding to n = 2) surprisingly showed up. Without relying on hypercomplex methods this paper demonstrates how purely real methods also lead to S(n). For arbitrary n ≥ 1 the generating function is determined and for n = 2 a particular case of a recurrence relation similar to that known for Catalan numbers is proved.
This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the... more This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by L. Vietoris in 1958. For S a generating function is obtained which allows to derive an interesting relation to a result deduced in 1974 by Askey and Steinig about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.
Journal of Mathematical Analysis and Applications, 2017
This paper deals with a unified matrix representation for the Sheffer polynomials. The core of th... more This paper deals with a unified matrix representation for the Sheffer polynomials. The core of the proposed approach is the so-called creation matrix, a special subdiagonal matrix having as nonzero entries positive integer numbers, whose exponential coincides with the well-known Pascal matrix. In fact, Sheffer polynomials may be expressed in terms of two matrices both connected to it. As we will show, one of them is strictly related to Appell polynomials, while the other is linked to a binomial type sequence. Consequently, different types of Sheffer polynomials correspond to different choices of these two matrices.
Recently, systems of Clifford algebra-valued orthogonal polynomials have been studied from differ... more 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.
Computational Science and Its Applications – ICCSA 2014, 2014
ABSTRACT In this work, we give a brief description of the theory and properties of the three-dime... more ABSTRACT In this work, we give a brief description of the theory and properties of the three-dimensional quaternionic Zernike spherical polynomials (QZSPs). A generalization and refinement of the QZSPs to functions vanishing over the unit sphere leads to the computation of the weighted quaternionic Zernike spherical functions (WQZSFs). In particular, the underlying functions are of three real variables and take on values in the quaternions (identified with $\mathbb{R}^4$). Also, in this work, we prove that the WQZSFs are orthonormal in the unit ball with respect to a suitable weight function. The representation of these functions are given explicitly, and a summary of their fundamental properties is also discussed. To the best of our knowledge, this does not appear to have been done in literature before.
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