We prove a general Fueter Theorem over real alternative *-algebras. We show that a suitable power of the Laplacian maps Dunkl-regular functions to Dunkl monogenic functions with axial symmetries. Using the embedding of hypercomplex function theories in the class of Dunkl monogenic functions, we subsume several Fueter-type results known in the literature and obtain the most general form for the action of the Laplacian on function spaces over hypercomplex subspaces. We show that Fueter Theorems are in a one-to-one correspondence with a class of graphs, the Fueter trees, that describe the interactions between Dunkl-regular function spaces and the relation with the iterated Laplacian. We obtain that the number of distinct Fueter trees on a hypercomplex space of dimension $n+1$ is equal to the number of partitions in odd parts of the integer $n$.
Slice Fueter- regular functions , originally called slice Dirac- regular functions , are generalized holomorphic functions defined over the octonion algebra $\mathbb{O}$ , recently introduced by M. Jin, G. Ren and I. Sabadini. A function $f:Ω_D\subset\mathbb{O}\to\mathbb{O}$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra $\mathbb{H}_\mathbb{I}$ of $\mathbb{O}$ generated by a pair $\mathbb{I}=(I,J)$ of orthogonal imaginary units $I$ and $J$ ( $\mathbb{H}_\mathbb{I}$ is a `quaternionic slice' of $\mathbb{O}$ ), the restriction of $f$ to $Ω_D\cap\mathbb{H}_\ma…
A Pair of Multiplication-Type Operators in Quaternionic Analysis and the 2-Cauchy-Fueter Equation
School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
In this paper we study left and right n- regular functions that originally were introduced in [FL4]. When n=1, these functions are the usual quaternionic left and right regular functions . We show that n- regular functions satisfy most of the properties of the usual regular functions , including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy-Fueter type reproducing formulas. Arguably, these Cauchy-Fueter type reproducing formulas for n- regular functions are quaternionic analogues of Cauchy's integral formula for the n-th order pole e…
This treatise investigates holomorphic functions defined on the space of bicomplex numbers introduced by Segre. The theory of these functions is associated with Fueter's theory of regular, quaternionic functions. The algebras of quaternions and bicomplex numbers are developed by making use of so-called complex pairs. Special attention is paid to singular bicomplex numbers that lack an inverse. The elementary bicomplex functions are defined and their properties studied. The derivative of a bicomplex function is defined as the limit of a fraction with nonsingular denominator. The existence of th…
A Koszul complex in quaternionic analysis and its applications
School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）
