Helmuth R. Malonek’s research while affiliated with University of Aveiro and other places

... 15,40 A short overview can also be found in a very recent survey. 41 The simplest example of complexification for working with the algebra of complex numbers in the sense mentioned by V. I. Arnold is the identification of R 2 with C, formally expressed by the two conjugate variables z = x + iy andz = x − i . For a system of two real differentiable functions u = u(x, y) and v = v(x, y) and demanding that = (z,z) is complex differentiable with respect to z, this leads to the usual system of Cauchy-Riemann equations. ...

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A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

... Motivated by a similar perspective to the studies [1,9,10], we consider the Brousseau sum for Vietoris' numbers in this paper. Vietoris' sequence is a sequence of rational numbers that can be considered on the crossroad of positivity of trigonometric sums, stable behavior of some classes of holomorphic functions and a set of Appell polynomials in several hypercomplex variables as mentioned in [11]. Vietoris' sequence firstly appeared in a theorem stated by L. Vietoris in 1958 [12]. ...

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A note on weighted sums of Vietoris’ sequence

... Sequences of polynomials have also been considered by several researchers. For instance, Cação, Malonek and Tomaz, in [12], work with shifted generalized Pascal matrices in the context of Appell sequences. ...

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On the Leonardo quaternions sequence

... are axially Fueter regular functions, first studied in [23,24] and further investigated in [50,54]. ...

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New Taylor and Laurent series of axially harmonic, Fueter regular and polyanalytic functions

... The first few values of the Apostol-Frobenius-Euler polynomials of order α are determined explicitly as follows: The Apostol-Frobenius-Euler polynomials of order αH z u λ ; ; n α ( )are λ extensions of the Frobenius-Euler polynomials. Frobenius-Euler polynomials and numbers are named after the great German mathematician Ferdinand Georg Frobenius [13], who made essential works on the context of these polynomials in number theory and the relation of their divisibility properties with the Stirling numbers of the second kind [14,15]. ...

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Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions

... [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, 12-18 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. ...

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On the Hadamard's Three-hyperballs Theorem and its Applications to Whittaker-Cannon Hypercomplex Theory

... To answer the last question of the previous subsection we use induction to prove Theorem 6. The solution of the recurrence (60)- (61), with nonconstants coefficients, is given by the generalized Vietoris numbers c k (n) written in the form ...

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A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

... The idea is to introduce a symmetric product that preserves the monogenicity of the factors. To start with, some relations used in [36] for the generalization of power series in terms of ...

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Clifford analysis and harmonic polynomials

... A matrix approach to paravector valued Appell sequences can also advantageously be used, as it was shown in [11]. As one of its application a matrix recurrence for the paravector valued building blocks of those Gelfand-Tsetlin bases can be found in [12]. A central question like the construction of generating functions for spherical harmonics and spherical monogenics was answered in the paper [16]. ...

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Three-Term Recurrence Relations for Systems of Clifford Algebra-Valued Orthogonal Polynomials

... The flexibility introduced by structural sets ensembles makes it possible to seek new perspectives in investigations concerning the Fischer decompositions, mapping properties of multi-dimensional Ahlfors-Beurling transforms and additive decompositions of harmonic functions, see [1][2][3]11,12,19,23,25,28,29,34,43]. ...

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On Second Order Elliptic Systems of Partial Differential Equations in Clifford Analysis