Asymptotic results on modified Bergman-Dirichlet spaces and examples of Segal-Bargmann transforms

In this paper, we start by introducing the modified Bergman-Dirichlet space ${\mathcal{D}}_m^2({\mathbb{D}}_R,{\mu }_{\alpha ,\beta }^R)$ and then we study its asymptotic behavior when the parameter $\alpha$ goes to infinity and to $(-1)$ to obtain respectively the modified Bargmann-Dirichlet and the modified Hardy-Dirichlet spaces with their reproducing kernels. Finally, we give some examples of Segal-Bargmann transforms of those spaces.