  Since 2006 the theory of slice hyperholomorphic functions and the related
spectral theory on the S-spectrum have had a very fast development. This new
spectral theory based on the S-spectrum has applications, for example, in the
formulation of quaternionic quantum mechanics, in Schur analysis and in
fractional diffusion problems. In this paper we introduce and study the theory
of poly slice monogenic functions, also proving some Cauchy type integral
formulas. Then we introduce the associated functional calculus, called
PS-functional calculus, which is the polyanalytic version of the S-functional
calculus and which is based on the notion of S-spectrum. We study some
different formulations of the calculus and we prove some of its properties,
among which the product rules.
