Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences

Cação, Isabel; Malonek, Helmuth R.; Tomaz, Graça

doi:10.1007/978-3-319-62395-5_28

Cited by 2 publications

(2 citation statements)

References 16 publications

(24 reference statements)

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“…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%

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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Section: Introductionmentioning

confidence: 99%

On generalized Vietoris’ number sequences

Cação

Falcão

Malonek

2019

Discrete Applied Mathematics

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“…[9][10][11] Much of the older theory of special monogenic polynomials has been given a different interpretation. A new light has been shed upon the study of elementary functions, [12][13][14][15][16][17][18] the computation of combinatorial identities, [19][20][21] and the study of a generalized Joukowski transformation in Euclidean space of arbitrary higher dimension. 22 Earlier results in the theory of special polynomial bases in hypercomplex analysis and its counterpart in the function theory of several complex variables can be found in previous studies 12,[23][24][25][26][27][28][29][30][31][32][33][34] and elsewhere.…”

Section: Introductionmentioning

confidence: 99%

On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

Zayed

Morais

2022

Math Methods in App Sciences

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This paper shows a hypercomplex function theory emerging in the representation of paravector-valued monogenic functions over the (m + 1)-dimensional Euclidean space through a basic set (or basis) of hypercomplex monogenic polynomials. We derive the properties of the arising hypercomplex Cannon function and present an extension of the well-known Whittaker-Cannon theorem to special monogenic functions defined in an open hyperball in R m+1 . More precisely, we determine what conditions should be applied to a basic set of special monogenic polynomials to attain the effectiveness property in an open hyperball employing Hadamard's three-hyperballs theorem. We also provide a necessary and sufficient condition for a special monogenic Cannon series to represent every function near the origin that is special monogenic there. Additionally, we investigate the effectiveness of a non-Cannon basis and show that the underlying hypercomplex Cannon function maintains similar properties in both cases, the Cannon basis and the non-Cannon basis.

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Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences

Cited by 2 publications

References 16 publications

On generalized Vietoris’ number sequences

On generalized Vietoris’ number sequences

On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

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