A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

Abstract

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator inR4\mathbb {R}^4, denoted byD\mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operatorΔ\Deltain four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e.Δ=D¯D\Delta = \mathcal {\overline {D}} \mathcal {D}to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions ofD2\mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on theSS-spectrum.