  This paper studies properties of q-Jacobi polynomials and their duals by
means of operators of the discrete series representations for the quantum
algebra U_q(su_{1,1}). Spectrum and eigenfunctions of these operators are found
explicitly. These eigenfunctions, when normalized, form an orthogonal basis in
the representation space. The initial U_q(su_{1,1})-basis and the bases of
these eigenfunctions are interconnected by matrices, whose entries are
expressed in terms of little and big q-Jacobi polynomials. The orthogonality by
rows in these unitary connection matrices leads to the orthogonality relations
for little and big q-Jacobi polynomials. The orthogonality by columns in the
connection matrices leads to an explicit form of orthogonality relations on the
countable set of points for {}_3\phi_2 and {}_3\phi_1 polynomials, which are
dual to big and little q-Jacobi polynomials, respectively. The orthogonality
measure for the dual little q-Jacobi polynomials proves to be extremal, whereas
the measure for the dual big q-Jacobi polynomials is not extremal.
