The moment problem associated with the \(q\)-Laguerre polynomials (Q1864177)

The indeterminate Stieltjes moment problem associated with the \(q\)-Laguerre polynomials is considered. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and a method to construct, for example, continuous singular solutions is presented. The connection with the moment problem associated with the Stieltjes-Wigert polynomials is studied. It is shown how to come from \(q\)-Laguerre solutions to Stieltjes-Wigert solutions by letting the parameter \(\alpha \to\infty\). It is also explained how to lift a Stieltjes-Wigert solution to a \(q\)-Laguerre solution at the level of Pick functions. Finally based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.