We prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of a global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. Then, we introduce a 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 discuss also some integral representations of these results based on the quaternion poly-Cauchy formula. This talk is based on a research project in progress joint with Prof. Alpay Daniel and Prof. Sabadini Irene.