Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

Martino, Antonino De; Diki, Kamal

doi:10.1142/s0219530522500191

Cited by 4 publications

(4 citation statements)

References 51 publications

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“…x k 0 P 3 j−2 (q) = − 2j(j − 1)x k 0 P 3 j−2 (q), which is exactly the same result obtained in [25].…”

Section: Proposition 72 (Polyanalytic Decomposition)supporting

confidence: 86%

“…Remark 7.10. This is an extension to general Clifford algebras of the result [25,Theorem 3.3], established in the quaternionic setting. Indeed, if we consider n = 3 in Theorem 7.9, for q ∈ H and j ≥ 2, we get…”

Section: Proposition 72 (Polyanalytic Decomposition)mentioning

confidence: 81%

“…The left module of axially polyanalytic monogenic functions of order m + 1 will be denoted by AM m+1 (U ). The polyanalytic monogenic setting holds the following characterization (see [13,26]). Proposition 7.4.…”

Section: Proposition 72 (Polyanalytic Decomposition)mentioning

confidence: 99%

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The Fueter-Sce mapping and the Clifford–Appell polynomials

Martino¹,

Diki²,

Adán³

2023

Proceedings of the Edinburgh Mathematical Society

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The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in ${\mathbb{R}}^{n+1}$ . This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in $ \mathbb{R}^{n+1}$ . The second one is a suitable power of the Laplace operator in n + 1 variables. Another way to get axially monogenic functions is the generalized Cauchy–Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions $\Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$ , where $x \in \mathbb{R}^{n+1}$ . The expressions obtained is related to a well-known class of Clifford–Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension, we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford–Appell–Fock space and the Clifford–Appell–Hardy space. Finally, using the polyanalytic Fueter-Sce theorems, we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford–Appell polynomials.

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“…x k 0 P 3 j−2 (q) = − 2j(j − 1)x k 0 P 3 j−2 (q), which is exactly the same result obtained in [25].…”

Section: Proposition 72 (Polyanalytic Decomposition)supporting

confidence: 86%

Section: Proposition 72 (Polyanalytic Decomposition)mentioning

confidence: 81%

See 1 more Smart Citation

The Fueter-Sce mapping and the Clifford–Appell polynomials

Martino¹,

Diki²,

Adán³

2023

Proceedings of the Edinburgh Mathematical Society

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“…The polynomials found in (7.11) and (7.12) are polyanalytic of order 2 and 3, respectively. However, these polynomials do not coincide to which ones found in [28], in which the authors obtained polyanalytic polynomials by applying the Fueter-polyanalytic maps, see [5].…”

Section: Series Expansion Of the Kernels Of The Fine Structures Spacesmentioning

confidence: 80%

Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

et al. 2022

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The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S-resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F-functional calculus.

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The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

et al. 2023

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Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S-spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum. Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

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Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

Cited by 4 publications

References 51 publications

The Fueter-Sce mapping and the Clifford–Appell polynomials

The Fueter-Sce mapping and the Clifford–Appell polynomials

Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

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