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The Fueter-Sce mapping and the Clifford–Appell polynomials

Part of: Generalized function theory Spaces and algebras of analytic functions

Published online by Cambridge University Press:  28 June 2023

Antonino De Martino ,
Kamal Diki  and
Ali Guzmán Adán
Antonino De Martino
Affiliation:
Schmid College of Science and Technology, Chapman University, Orange, CA, USA (ademartino@chapman.edu; ademartino@chapman.edu)
Kamal Diki
Affiliation:
Schmid College of Science and Technology, Chapman University, Orange, CA, USA (ademartino@chapman.edu; ademartino@chapman.edu)
Ali Guzmán Adán
Affiliation:
Clifford Research Group, Department of Mathematical Analysis, Ghent University, Ghent, Belgium (aliguzmanadan@gmail.com
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Abstract

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.

Keywords

Fueter-Sce mapping theoremgeneralized CK extensionClifford–Appell polynomialsreproducing kernelsFock spaceHardy space

MSC classification

Primary: 30G35: Functions of hypercomplex variables and generalized variables 30H20: Bergman spaces, Fock spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 3 , August 2023 , pp. 642 - 688
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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