  A generalized Brans-Dicke (GBD) theory in the framework of Palatini formalism
are proposed in this paper. We derive the field equations by using the
variational approach and obtain the linearized equations by using the
weak-field approximation method. We show various properties of the geometrical
scalar field in the Palatini-formalism of GBD theory: it is massless and
source-free, which are different from the results given in the metric-formalism
of GBD theory. Also, we investigate the polarization modes of gravitational
waves (GWs) by using the the geodesic deviation method and the Newman-Penrose
method in the Palatini-GBD theory. It is observed that there are three
polarizations modes and four oscillation in the Palatini-GBD theory.
Concretely, they are the two transverse tensor ($+$) and ($\times$) standard
polarization modes, and one breathing mode (with two oscillation). The results
of GWs polarization in the Palatini-GBD theory are different from that in the
metric-GBD theory, where there are four polarizations modes: the two standard
tensorial modes ($+$ and $\times$), a scalar breathing mode, and a massive
scalar mode that is a mix of the longitudinal and the breathing polarization.
Comparing with the Palatini-$f(\tilde{R})$ theory and the General Relativity,
we can see that the extra breathing mode of GWs polarization can be found in
the Palatini-GBD theory. At last, the expression of the parameterized post
Newton (PPN) parameter is derived, which could pass through the experimental
test.
