Abstract
In the algebra of complex quaternions \(\mathbb {H(C)}\) we consider the left– and right–\(\psi \)–hyperholomorphic functions, and left–\(\Lambda -\psi \)–hyperholomorphic functions. We justify the transition in left– and right–\(\psi \)–hyperholomorphic functions to a simpler basis i.e., to the Cartan basis. Using Cartan’s basis we find the solution of Cauchy–Fueter equation. By the same method we find representations of left– and right–\(\psi \)–hyperholomorphic functions, and representation of left–\(\Lambda -\psi \)–hyperholomorphic functions.
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Acknowledgements
The authors express their gratitude to Professors M. E. Luna-Elizarraras and M. Shapiro for the discussion of the results and valuable advice. This work was supported by a grant from the Simons Foundation (1030291,V.S.Sh.). The main ideas of the study belong to V. Sh. The authors received all the results of the article together. The authors declare no competing interest.
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Communicated by Juan Bory Reyes.
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Kuzmenko, T., Shpakivskyi, V. Representations of Some Classes of Quaternionic Hyperholomorphic Functions. Complex Anal. Oper. Theory 18, 116 (2024). https://doi.org/10.1007/s11785-024-01561-x
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DOI: https://doi.org/10.1007/s11785-024-01561-x
Keywords
- Complex quaternions
- Cartan basis
- Left– and right–\(\psi \)–hyperholomorphic function
- Weighted Dirac operator
- Cauchy–Fueter type equation
- Left–\(\Lambda -\psi \)–hyperholomorphic function.