  In our previous paper [1] we have obtained, for the XXX spin-1/2 Heisenberg
open chain, new determinant representations for the scalar products of separate
states in the quantum separation of variables (SoV) framework. In this article
we perform a similar study in a more complicated case: the XXZ open spin-1/2
chain with the most general integrable boundary terms. To solve this model by
means of SoV we use an algebraic Vertex-IRF gauge transformation reducing one
of the boundary K-matrices to a diagonal form. As usual within the SoV
approach, the scalar products of separate states are computed in terms of
dressed Vandermonde determinants having an intricate dependency on the
inhomogeneity parameters. We show that these determinants can be transformed
into different ones in which the homogeneous limit can be taken
straightforwardly. These representations generalize in a non-trivial manner to
the trigonometric case the expressions found previously in the rational case.
We also show that generically all scalar products can be expressed in a form
which is similar to - although more cumbersome than - the well-known Slavnov
determinant representation for the scalar products of the Bethe states of the
periodic chain. Considering a special choice of the boundary parameters
relevant in the thermodynamic limit to describe the half infinite chain with a
general boundary, we particularize these representations to the case of one of
the two states being an eigenstate. We obtain simplified formulas that should
be of direct use to compute the form factors and correlation functions of this
model.
