  We study the ranks of commutators and generalized semicommutators of Toeplitz
operators with quasihomogeneous symbols on both the harmonic Bergman space and
the Bergman Space. In particular, when one of quasihomogeneous symbols is the
the form of $e^{ik\theta }r^{m}$, we first obtain specific sufficient and
necessary conditions for commutators and generalized semicommutators to be
finite rank. Then we make further efforts to determine the range of each finite
rank commutator and generalized semicommutators, and consequently the explicit
canonical form and the rank are obtained. Thus, the finite rank problem of
commutators and generalized semicommutators of such special Toeplitz operators
is completely solved. As applications, several interesting corollaries and
nontrivial examples are given. Also, we show close connections of the finite
rank problem between the harmonic Bergman space and Bergman space cases.
