  The $S$-functional calculus is based on the theory of slice hyperholomorphic
functions and it defines functions of $n$-tuples of not necessarily commuting
operators or of quaternionic operators. This calculus relays on the notion of
$S$-spectrum and of $S$-resolvent operator. Since most of the properties that
hold for the Riesz-Dunford functional calculus extend to the S-functional
calculus it can be considered its non commutative version. In this paper we
show that the Taylor formula of the Riesz-Dunford functional calculus can be
generalized to the S-functional calculus, the proof is not a trivial extension
of the classical case because there are several obstructions due to the non
commutativity of the setting in which we work that have to be overcome. To
prove the Taylor formula we need to introduce a new series expansion of the
$S$-resolvent operators associated to the sum of two $n$-tuples of operators.
This result is a crucial step in the proof of our main results,but it is also
of independent interest because it gives a new series expansion for the
$S$-resolvent operators. This paper is devoted to researchers working in
operators theory and hypercomplex analysis.
