  We study representations of $U_q(su(1,1))$ that can be considered as quantum
analogs of tensor products of irreducible *-representations of the Lie algebra
$su(1,1)$. We determine the decomposition of these representations into
irreducible *-representations of $U_q(su(1,1))$ by diagonalizing the action of
the Casimir operator on suitable subspaces of the representation spaces. This
leads to an interpretation of the big $q$-Jacobi polynomials and big $q$-Jacobi
functions as quantum analogs of Clebsch-Gordan coefficients.
