  We consider the XXX open spin-1/2 chain with the most general non-diagonal
boundary terms, that we solve by means of the quantum separation of variables
(SoV) approach. We compute the scalar products of separate states, a class of
states which notably contains all the eigenstates of the model. As usual for
models solved by SoV, these scalar products can be expressed as some
determinants with a non-trivial dependance in terms of the inhomogeneity
parameters that have to be introduced for the method to be applicable. We show
that these determinants can be transformed into alternative ones in which the
homogeneous limit can easily be taken. These new representations can be
considered as generalizations of the well-known determinant representation for
the scalar products of the Bethe states of the periodic chain. In the
particular case where a constraint is applied on the boundary parameters, such
that the transfer matrix spectrum and eigenstates can be characterized in terms
of polynomial solutions of a usual T-Q equation, the scalar product that we
compute here corresponds to the scalar product between two off-shell Bethe-type
states. If in addition one of the states is an eigenstate, the determinant
representation can be simplified, hence leading in this boundary case to direct
analogues of algebraic Bethe ansatz determinant representations of the scalar
products for the periodic chain.
