A POLYANALYTIC FUNCTIONAL CALCULUS OF ORDER 2 ON THE S-SPECTRUM

. The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator in R4, denoted by 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 operator Delta in four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. Delta = DD 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 of D2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the S-spectrum.