Representations of the Quantum Algebra $\mathrm{su}_q(1,1)$ and Discrete $q$-Ultraspherical Polynomials

We derive orthogonality relations for discrete $q$-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra $\mathrm{su}_q(1,1)$. Spectra and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthonormal basis in the representation space.