  We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to
a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using
RHP theory we determine new asymptotic results in the limit as the degree of
the polynomials approach infinity. The RHP formulation also enables us to
obtain further properties. In particular, we consider how the class of
polynomials and their asymptotic behaviours change under translations of the
$q$-discrete lattice and determine the asymptotics of related $q$-Painlev\'e
equations.
