Measures connected with Bargmann's representation of the q-commutation relation for q > 1

Classical Bargmann's representation is given by operators acting on the space of holomorphic functions with scalar product ${\displaystyle 〈z^n,z^k{〉}_q={\delta }_{n,k}{\left[n\right]}_q!=F\left(z^n{\overline{z}}^k\right)}$. We consider the problem of representing the functional F as a measure. We prove the existence of such a measure for q > 1 and investigate some of its properties like uniqueness and radiality.