  The S-functional calculus is a functional calculus for $(n+1)$-tuples of non
necessarily commuting operators that can be considered a higher dimensional
version of the classical Riesz-Dunford functional calculus for a single
operator. In this last calculus, the resolvent equation plays an important role
in the proof of several results. Associated with the S-functional calculus
there are two resolvent operators: the left $S_L^{-1}(s,T)$ and the right one
$S_R^{-1}(s,T)$, where $s=(s_0,s_1,\ldots,s_n)\in \mathbb{R}^{n+1}$ and
$T=(T_0,T_1,\ldots,T_n)$ is an $(n+1)$-tuple of non commuting operators. These
two S-resolvent operators satisfy the S-resolvent equations
$S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal{I}$, and
$sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal{I}$, respectively, where $\mathcal{I}$
denotes the identity operator. These equations allows to prove some properties
of the S-functional calculus. In this paper we prove a new resolvent equation
for the S-functional calculus which is the analogue of the classical resolvent
equation. It is interesting to note that the equation involves both the left
and the right S-resolvent operators simultaneously.
