  In the present paper, we prove a resolvent equation for the
$\mathcal{S}$-resolvent operator in the quaternionic framework. Exploiting this
resolvent equation, we find a series expansion for the $\mathcal{S}$-resolvent
operator in an open neighborhood of any given quaternion belonging to the
$\mathcal{S}$-resolvent set. Some consequences of the series expansion are
deduced. In particular, we describe a property of the geometry of the
$\mathcal{S}$-resolvent set in terms of the Cassini pseudo-metric on
quaternions. The concept of vector-valued real analytic function of several
variables plays a crucial role in the proof of the mentioned series expansion
for the $\mathcal{S}$-resolvent operator.
