  The Fueter mapping theorem gives a constructive way to extend holomorphic
functions of one complex variable to monogenic functions, i.e., null solutions
of the generalized Cauchy-Riemann operator in $\mathbb{R}^4$, denoted by
$\mathcal{D}$. This theorem is divided in two steps. In the first step a
holomorphic function is extended to a slice hyperholomorphic function. The
Cauchy formula for these type of functions is the starting point of the
$S$-functional calculus. In the second step a monogenic function is obtained by
applying the Laplace operator in four real variables, namely $ \Delta$, to a
slice hyperholomorphic function. The polyanalytic functional calculus, that we
study in this paper, is based on the factorization of $\Delta= \mathcal{D}
\mathcal{\overline{D}}$. Instead of applying directly the Laplace operator to a
slice hyperholomorphic function we apply first the operator $
\mathcal{\overline{D}}$ and we get a polyanalytic function of order 2, i..e, a
function that belongs to the kernel of $ \mathcal{D}^2$. We can represent this
type of functions in an integral form and then we can define the polyanalytic
functional calculus on $S$-spectrum. The main goal of this paper is to show the
principal properties of this functional calculus. In particular, we study a
resolvent equation suitable for proving a product rule and generate the Riesz
projectors.
