The dual discrete $q$-ultraspherical polynomials $D_n^{(s)}(��(x;s)|q)$ correspond to indeterminate moment problem and, therefore, have one-parameter family of extremal orthogonality relations. It is shown that special cases of dual discrete $q$-ultraspherical polynomials $D_n^{(s)}(��(x;s)|q)$, when $s=q^{-1}$ and $s=q$, are directly connected with $q^{-1}$-Hermite polynomials. These connections are given in an explicit form. Using these relations, all extremal orthogonality relations for these special cases of polynomials $D_n^{(s)}(��(x;s)|q)$ are found.

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/