Summary: Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra \(U_q(su_{1,1})\) is studied. Spectrum and eigenfunctions of this operator are explicitly found. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial \(U_q(su_{1,1})\) basis and the basis of these eigenfunctions are interconnected by a matrix with entries expressed in terms of big \(q\)-Laguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big \(q\)-Laguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of \(q\)-Meixner polynomials \(M_n(x; b, c; q)\) either with positive or negative values of the parameter \(b\). The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for the \(q\)-Meixner polynomials \(M_n(x; b, c; q)\) with \(b<0\). A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.