Bargmann representation of q-commutation relations for q > 1 and associated measures

The classical Bargmann representation is given by operators acting on the space of holomorphic functions with the scalar product < z(n)|z(k)>(q) = delta(n,k)[n](q)! = F(z(n)z(-k)). We consider the problem of representing the functional F as a measure for q > 1. We prove the existence of such a measure and investigate some of its properties like uniqueness and radiality. The above problem is closely related to the indeterminate Stieltjes moment problem.