  The transfer matrix of the XXZ open spin-1/2 chain with general integrable
boundary conditions and generic anisotropy parameter (q is not a root of unity
and |q|=1) is diagonalized using the representation theory of the q-Onsager
algebra. Similarly to the Ising and superintegrable chiral Potts models, the
complete spectrum is expressed in terms of the roots of a characteristic
polynomial of degree d=2^N. The complete family of eigenstates are derived in
terms of rational functions defined on a discrete support which satisfy a
system of coupled recurrence relations. In the special case of linear relations
between left and right boundary parameters for which Bethe-type solutions are
known to exist, our analysis provides an alternative derivation of the results
by Nepomechie et al. and Cao et al.. In the latter case the complete family of
eigenvalues and eigenstates splits in two sets, each associated with a
characteristic polynomial of degree $d< 2^N$. Numerical checks performed for
small values of $N$ support the analysis.
