A manifold Fueter-Sce phenomenon in one hypercomplex variable
Riccardo Ghiloni, Caterina Stoppato
TL;DR
This work broadens the Fueter–Sce phenomenon from classical Clifford and quaternionic contexts to the wider arena of strongly $T$-regular functions over general associative $*$-algebras, introducing a multi-axial symmetry framework. It defines global differential operators $ar ullpartial_T$, ullpartial_T, and $ riangle_T$ on $T$-functions, and establishes their interplay with $T$-regularity and $T$-harmonicity through a $T$-stem framework. The main result shows that, under odd step differences $t_h-t_{h-1}=2n_h+1$, iterated Laplacians yield a strongly $T_ au$-regular (and monogenic in the Clifford case) function, revealing that Fueter–Sce phenomena are the final step of a multi-stage process and suggesting further mixed phenomena beyond the current oddness hypotheses. The paper thus provides a unified, multi-axial generalization of classical results and opens avenues for new hypercomplex phenomena in one variable across diverse algebras.
Abstract
Fueter's theorem states, in modern terms, that the Laplacian maps slice-regular quaternionic functions into Fueter-regular functions with axial symmetry. This phenomenon is also present in the Clifford setting, where both slice-monogenic functions and generalized partial-slice monogenic are mapped by the Laplacian into monogenic functions with axial symmetry. These results are due, respectively, to Sce and Qian and to Xu and Sabadini. The present work puts the Fueter-Sce phenomenon into context for the wider class of strongly $T$-regular functions. It shows that the phenomenon appears over general associative $*$-algebras. Moreover, the symmetry considered here is multi-axial in a sense introduced by Eelbode. Additionally, but more surprisingly, the phenomenon studied by Fueter, Sce, Xu and Sabadini turns out to be the last step in a multi-step process. A new phenomenon in one hypercomplex variable is therefore discovered.