By definition, $h_+{\mathcal{O}}_V(*D)e^{-f}$ is represented by the complex $$(h_*\Omega^{n+\bullet}_V(*D)[\partial_t],d-df\wedge-(dh\wedge-\otimes\partial_t)).$$ The same argument as above shows that this is quasi-isomorphic to $$(h_*\Omega^{n-1+\bullet}_{V/T}(*D),d-df\wedge).$$ Now it is clear that $${{\mathbb H}}^0(T,h_*\Omega^{n-1+\bullet}_{V/T}(*D),d-df\wedge) = \frac{H^0(\Omega^{n-1}_{V/T}(*D))}{(d-df\wedge)H^0(V,\Omega^{n-2}_{V/T}(*D)))} = G_1(*D)$$

**Remark:** Direct images of regular holonomic modules by a morphism consisting of two polynomials occur in the work of C. Roucairol (see [@Rouc1], [@Rouc2] and [@Rouc3]). She also studied direct images of twisted modules, i.e., $h_+({\mathcal{M}}e^{-f})$. However, we will not need her results directly as the computations from [@dGMS] (i.e. proposition \[prop:recalldGMS\] above) give already very precise information about these direct images for a pair $(f,h)$, with $h$ reductive linear free and $f$ linear and generic.

Bernstein Polynomials {#sec:Bernstein}
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We first give the definition of the Bernstein polynomial through the classical functional equation. Next we recall how this can be rephrased using the general theory of $V$-filtrations. This will be useful in proving the main result. Finally, we state and prove the relation between the roots of the Bernstein polynomial of a defining equation $h$ for a linear free divisor and the residue eigenvalues of the family of Gauß-Manin-systems introduced in section \[sec:LFDandGM\].

The following classical statement is due to Bernstein (see, [@Bernstein]).

\[theo:BernsteinLFD\] Let $h\in {\mathcal{O}}_V$ 