  We introduce and study a generalization of the classical weighted Bergman and
Dirichlet spaces on the unit ball in high dimension, the Bergman-Dirichlet
spaces. Their counterparts on the whole $n$-complex space, the
Bargmann-Dirichlet spaces, are also introduced and studied. Mainly, we give a
complete description of the considered spaces, including orthonormal basis and
the explicit formulas for their reproducing kernel functions. Moreover, we
investigate their asymptotic behavior when the curvature goes to $0$.
