On the global operator and Fueter mapping theorem for slice polyanalytic functions

TL;DR

This paper extends the theory of slice polyanalytic functions on quaternions by linking them to a global operator, and generalizes the Fueter mapping theorem to this setting, including integral representations.

Contribution

It introduces a global operator framework for slice polyanalytic functions and extends the Fueter mapping theorem to this broader class of functions.

Findings

Slice polyanalytic functions are solutions to powers of a global operator.

Extension of Fueter mapping theorem to polyanalytic functions under axially symmetric conditions.

Integral representations of these functions on the quaternionic unit ball.

Abstract

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.