  We generalize the multipole expansion and the structure of the Fax\'en
operator in Stokes flows obtained for bodies with no-slip to generic boundary
conditions, addressing the assumptions under which this generalization is
conceivable. We show that a disturbance field generated by a body immersed in
an ambient flow can be expressed as a multipole expansion the coefficients of
which are the moments of the volume forces, independently on the boundary
conditions. We find that the dualism between the operator giving the
disturbance field of an $n$-th order ambient flow and the $n$-th order Fax\'en
operator, referred to as the Hinch-Kim dualism, holds only if the boundary
conditions satisfy a property that we call Boundary-Condition reciprocity
(BC-reciprocity). If this property is fulfilled, the Fax\'en operators can be
expressed in terms of the $(m,n)$-th order geometrical moments of the volume
forces (defined in the article). In addition, it is shown that in these cases,
the hydromechanics of the fluid-body system is completely determined by the
entire set of the Fax\'en operators. Finally, classical boundary conditions of
hydrodynamic practice are investigated in the light of this property: boundary
conditions for rigid bodies, Newtonian drops at the mechanical equilibrium,
porous bodies modeled by the Brinkman equations are BC-reciprocal, while
deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian drops
and porous bodies modeled by the Darcy equations do not have this property.
  For Navier-slip boundary conditions on a rigid body, we find the analytical
expression for low order Fax\'en operators.
