q-Stieltjes classes for some families of q-distributions

The Stieltjes classes play a significant role in the moment problem allowing to exhibit explicitly an infinite family of probability densities with the same sequence of moments. In this paper, the notion of $q$-moment determinacy/indeterminacy is proposed and some conditions for a distribution to be either $q$-moment determinate or indeterminate in terms of its $q$-density have been obtained. Also, a $q$-analogue of Stieltjes classes is defined for $q$-distributions and $q$-Stieltjes classes have been constructed for a family of $q$-densities of $q$-moment indeterminate distributions.