  We investigate the Symanzik improvement of the Wilson quark action on
anisotropic lattices. Taking first a general action with nearest-neighbor and
clover interactions, we study the mass dependence of the ratio of the hopping
parameters, the clover coefficients, and an improvement coefficient for
heavy-light vector and axial vector currents. We show how tree-level
improvement can be achieved. For a particular choice of the spatial Wilson
coupling, the results simplify, and $O(m_0a_\tau)$ improvement is possible.
(Here $m_0$ is the bare quark mass and $a_\tau$ the temporal lattice spacing.)
With this choice we calculate the renormalization factors of heavy-light
bilinear operators at one-loop order of perturbation theory employing the
standard plaquette gauge action.
