  We investigate the Wightman function, the vacuum expectation values of the
field squared and the energy-momentum tensor for a massive scalar field with
general curvature coupling in the generalized cosmic string geometry with a
compact dimension along its axis. The boundary condition along the compactified
dimension is taken in general form with an arbitrary phase. The vacuum
expectation values are decomposed into two parts. The first one corresponds to
the uncompactified cosmic string geometry and the second one is the correction
induced by the compactification. The asymptotic behavior of the vacuum
expectation values of the field squared, energy density and stresses are
investigated near the string and at large distances. We show that the
nontrivial topology due to the cosmic string enhances the vacuum polarization
effects induced by the compactness of spatial dimension for both the field
squared and the vacuum energy density. A simple formula is given for the part
of the integrated topological Casimir energy induced by the planar angle
deficit. The results are generalized for a charged scalar field in the presence
of a constant gauge field. In this case, the vacuum expectation values are
periodic functions of the component of the vector potential along the compact
dimension.
