  In this paper, we start by introducing the modified Bergman-Dirichlet space
$\mathcal D_m^2(\mathbb D_R,\mu^R_{\alpha,\beta})$ 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.
