The moment problem associated with the q-Laguerre polynomials

We consider the indeterminate Stieltjes moment problem associated with the q-Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes-Wigert polynomials is studied; we show how to come from q-Laguerre solutions to Stieltjes-Wigert solutions by letting the parameter alpha --> infinity, and we explain how to lift a Stieltjes-Wigert solution to a q-Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.