$\alpha'_i+\alpha'_{n+1-i}=-2$. From theorem \[theo:BernsteinEqualResidueGM\] and proposition \[prop:recalldGMS\] we deduce that $\alpha_j=(j-1)-\alpha'_j-1$ for any $j\in\{1,\ldots,n\}$, hence, $\alpha_i+\alpha_{n+1-i}=((i-1)-\alpha'_i-1)+((n+1-i)-1-\alpha'_{n+1-i}-1)=n-1$.

We outline another consequence of the theorem \[theo:BernsteinEqualResidueGM\]. Its interest is motivated by comparing the situation considered here with the one where $f$ is still a generic linear form, but $h$ is supposed to by an arbitrary monomial $h=\prod x_i^{w_i}$, i.e., non-reduced. The corresponding Gauß-Manin-systems resp. Brieskorn lattices have been studied in [@DS2], [@Dou3] and [@DM]. It is known that they are closely related to the *Mirror symmetry* phenomenon, i.e., one constructs a Frobenius structure on the semi-universal unfolding of $f_{|h^{-1}(t)}$, $t\neq 0$ which is known to be isomorphic to the orbifold quantum cohomology of the weighted projective spaces. For a linear free divisor $D$, a similar construction of a Frobenius manifold has been carried out in [@dGMS]. Although these are not a priori mirrors of some variety or orbifold, the following corollary shows an interesting similarity with the case $h=\prod x_i^{w_i}$.

The spectrum at $\theta=\infty$ of both $(G_0(h),\nabla)$ and $(G(*D),\nabla)$ contains a (non-trivial) block of integer numbers $k,k+1,\ldots,n-1-k$ for some $k\in\{0,\ldots,n-1\}$.

For the spectrum of $(G_0(h),\nabla)$, this is obvious as this block corresponds to the root $-1$ of the Bernstein polynomial $b_h(s)$. For 