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30 results on '"CAÇÃO, I."'

1. On generalized Vietoris' number sequences - origins, properties and applications

Author

Cação, I., Falcão, M. I., and Malonek, H. R.

Subjects
Mathematics - Classical Analysis and ODEs, 15A16, 65F60, 11B83
Abstract

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., Comment: 13 pages

Published
2016

2. Relating certain coefficients of special hyperholomorphic polynomials with a Horadam sequence.

Author

Cação, I., Malonek, H. R., Falcão, M. I., and Tomaz, G.

Subjects
*POLYNOMIALS, *PARTIAL sums (Series)
Abstract

Hyperholomorphic polynomials are generalizations of holomorphic polynomials to higher dimensions in the framework of hypercomplex function theory. We consider a special sequence of hyperholomorphic polynomials whose coefficients can be arranged as triangular arrays of the Pascal type. Our main focus is a one-parameter sequence formed by partial alternating sums of their main diagonal elements that in the limit case is an extended Horadam sequence of rational numbers. [ABSTRACT FROM AUTHOR]

Published
2024
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3. Generalized Vietoris' number sequences from real and hypercomplex points of view.

Author

Cação, I., Malonek, H. R., Falcão, M. I., and Tomaz, G.

Subjects
*JACOBI polynomials, *POLYNOMIALS, *RATIONAL numbers, *OPTIMISM
Abstract

We revisit a special rational number sequence, introduced by L. Vietoris in 1958 in the study of the positivity of some trigonometric sums and used in other contexts by several authors. The aim of the present paper is to embrace and explore real and hypercomplex analytical methods to obtain generalizations of that rational number sequence, where Jacobi polynomials and generalized Appell polynomials are involved. [ABSTRACT FROM AUTHOR]

Published
2023
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4. Starting with the differential: Representation of monogenic functions by polynomials of non-monogenic variables.

Author

Malonek, H. R., Cação, I., Falcão, M. I., and Tomaz, G.

Subjects
*MONOGENIC functions, *POLYNOMIALS, *CLIFFORD algebras, *SET functions, *POWER series
Abstract

This paper deals with different power series expansions of generalized holomorphic (monogenic) functions in the setting of Clifford Analysis. Our main concern are generalized Appell polynomials as a special class of monogenic polynomials which have been introduced in 2006 by two of the authors using several monogenic hypercomplex variables. We clarify the reasons why a particular pair of non-monogenic variables allows to obtain a power series expansion by those generalized Appell polynomials. The approach is based on the differential of a function. Some other monogenic polynomials as well as applications are mentioned. [ABSTRACT FROM AUTHOR]

Published
2023
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7. On an Hypercomplex Generalization of Gould-Hopper and Related Chebyshev Polynomials

Author

Cação, I., Malonek, H. R., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Murgante, Beniamino, editor, Gervasi, Osvaldo, editor, Iglesias, Andrés, editor, Taniar, David, editor, and Apduhan, Bernady O., editor

Published
2011
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8. On Generalized Hypercomplex Laguerre-Type Exponentials and Applications

Author

Cação, I., Falcão, M. I., Malonek, H. R., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Murgante, Beniamino, editor, Gervasi, Osvaldo, editor, Iglesias, Andrés, editor, Taniar, David, editor, and Apduhan, Bernady O., editor

Published
2011
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13. Matrices related to orthogonal hypercomplex polynomial systems

Author

Cação, I., Malonek, H. R., and Tomaz, G.

Subjects
Hypercomplex polynomials, Shift matrix, Matrix representation, Creation matrix
Abstract

We show how special well known matrices, namely, the creation and shift matrices play an important role on a matrix representation of orthogonal systems of polynomials with a hypercomplex variable and values in a Cli ord algebra.

Published
2017

14. On Vietoris' number sequence and combinatorial identities with quaternions

Author

Cação, I., Falcão, M. I., Malonek, H. R., and Universidade do Minho

Subjects
Vietoris' number sequence, Combinatorial identities, Quaternions, Ciências Naturais::Matemáticas
Abstract

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/publishedVersion

Published
2017

16. Recurrence relations for Clifford algebra-valued orthogonal polynomials

Author

Cação, I., Falcão, M. I., Malonek, H. R., and Universidade do Minho

Subjects
special functions, Recurrence relations, Clifford analysis, Generalized Appell polynomials, Ciências Naturais::Matemáticas
Abstract

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.

Published
2015

17. Monogenic generalized Laguerre and Hermite polynomials and related functions

Author

Falcão, M. I., Cação, I., Malonek, H. R., and Universidade do Minho

Subjects
Exponential operators, Generalize Laguerre polynomials, Generalized Hermite polynomials, Clifford analysis
Abstract

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., FCT

Published
2011

27. An introduction to the quaternionic Zernike spherical polynomials.

Author

Cação, I. and Morais, J.

Abstract

In continuation of [4], this paper discusses the quaternionic Zernike spherical polynomials (QZSPs), which refine and extend the Zernike polynomials or radial polynomials introduced in the early thirties by F. Zernike's Nobel prize. In particular, the underlying polynomials are of three real variables and take on values in the quaternions (identified with R4). QZSPs are complete and orthonormal in the unit ball. The representation of these functions are explicitly given, 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. [ABSTRACT FROM AUTHOR]

Published
2013
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28. Monogenic Generalized Hermite Polynomials and Associated Hermite-Bessel Functions.

Author

Cação, I.

Subjects
*HERMITE polynomials, *GENERATING functions, *BESSEL functions, *MONOGENIC functions, *HANKEL functions
Abstract

A large range of generalizations of the ordinary Hermite polynomials of one or several real or complex variables has been considered by several authors, using different methods. We construct monogenic generalizations of ordinary Hermite polynomials starting from a hypercomplex analogue to the real valued Lahiri exponential generating function. By using specific operational techniques, we derive some of their properties. As an application of the constructed polynomials, we define associated monogenic Hermite-Bessel functions. [ABSTRACT FROM AUTHOR]

Published
2010
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29. Special monogenic polynomials and L-approximation.

Author

Cação, I., Gürlebeck, K., and Malonek, H.

Abstract

The paper features some observations about the behaviour of special monogenic polynomials in the unit ball of ℝ which are relevant to the development of general approximation methods in the framework of Clifford Analysis. It focuses on an estimate for the best approximation of monogenic functions by a set of polynomials in Sobolev spaces. We illustrate some rather surprising orthogonality properties between symmetric homogeneous polynomials of the same degree and also an elementary method of obtaining a variety of linearly independent homogeneous polynomials of such type for the case of functions defined in ℝ and with values in the algebra of quaternions. The proof of their generalizations to higher dimensions in a suitable way is possible but needs more complicated combinatorical calculations which exceed the scope of this communication. [ABSTRACT FROM AUTHOR]

Published
2001
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30. Combinatorial identities associated with a multidimensional polynomial sequence

Author

Cação, I., Helmuth Malonek, Falcão, M. I., Tomaz, G., and Universidade do Minho

Subjects
Science & Technology, Binomial coefficient, Special sequence, Hypercomplex appell polynomial, Matemáticas [Ciências Naturais], Ciências Naturais::Matemáticas
Abstract

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., 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-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013.

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