Hypercomplex Polynomials, Vietoris’ Rational Numbers and a Related Integer Numbers Sequence

Cação, Isabel; Falcão, M. I.; Malonek, Helmuth R.

doi:10.1007/s11785-017-0649-5

Cited by 26 publications

(42 citation statements)

References 17 publications

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“…In fact such sequences, in particular the n = 2 case, have important applications in harmonic analysis, theory of stable holomorphic functions and combinatorics. We refer the interested reader to [45,46] and references therein.…”

Section: Definition and Some Propertiesmentioning

confidence: 99%

Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

Malonek

Cação

Falcão

et al. 2019

Springer Proceedings in Mathematics &Amp; Statistics

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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.

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Section: Definition and Some Propertiesmentioning

confidence: 99%

Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

Malonek

Cação

Falcão

et al. 2019

Springer Proceedings in Mathematics &Amp; Statistics

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“…with values [15] by the hypercomplex approach allows now to establish a different expansion of a rational function of type (4) with c k , k = 0, . .…”

Section: Remarkmentioning

confidence: 99%

“…It is a sequence that can be considered on the crossroad of positivity of trigonometric sums (see [2,3,4,33]), stable behavior of some classes of holomorphic functions (see [31]), and a set of Appell polynomials in several hypercomplex variables (see e.g. [12,15,16,20,28]).…”

Section: Introductionmentioning

confidence: 99%

“…The paper is structured in the following way. Starting in Section 2 from some reminiscence of Vietoris' theorem from 1958 and using the generating function of S already directly obtained in [15] we arrive at the end to the compact representation of the elements of S(n) in form of a quotient of two Pochhammer symbols. This leads immediately to the determination of the corresponding generating function in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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On generalized Vietoris’ number sequences

Cação

Falcão

Malonek

2019

Discrete Applied Mathematics

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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.

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“…the generalization of (2), can be achieved by combining several results previously obtained in other contexts and not as a direct generalization with the same methods as (2) was obtained in [6]. We start with recalling a theorem proved in [9].…”

Section: A Combinatorial Identity In Terms Of Generators Of a Clifformentioning

confidence: 98%

Preface: International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2017)

2018

AIP Conference Proceedings

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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.

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Hypercomplex Polynomials, Vietoris’ Rational Numbers and a Related Integer Numbers Sequence

Cited by 26 publications

References 17 publications

Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

On generalized Vietoris’ number sequences

Preface: International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2017)

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