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The goal of this paper is to give a converse to the main result of myprevious paper \cite{[B.22]}, so to prove the existence of a pole with anhypothesis on the Bernstein polynomial of the $(a,b)$-module generated by thegerm $\omega \in \Omega^{n+1}_0$. A difficulty to prove such a result comesfrom the use of the formal completion in $f$ of the Brieskorn module of theholomorphic germ $f: (\mathbb{C}^{n+1}, 0) \to (\mathbb{C}, 0)$ which does notgive access to the cohomology of the Milnor's fiber of $f$, which bydefinition, is outside $\{f=0\}$. This leads to introduce convergent$(a,b)$-modules which allow this passage. In order to take in account Jordanblocs of the monodromy in our result we introduce the semi-simple filtration ofa (convergent) geometric $(a,b)$-module and define the higher order Bernsteinpolynomials in this context which corresponds to a decomposition of the''standard'' Bernstein polynomial in the case of frescos. Our main result is toshow that the existence of a root in $-\alpha - \mathbb{N}$ for the $p$-thBernstein polynomial of the fresco generated by a holomorphic form $\omega \in\Omega^{n+1}_0$ in the (convergent) Brieskorn $(a,b)$-module $H^{n+1}_0 $associated to $f$, under the hypothesis that $f$ has an isolated singularity atthe origin relative to the eigenvalue $\exp(2i\pi\alpha)$ of the monodromy,produces poles of order at least $p$ for the meromorphic extension of the(conjugate) analytic functional, for some $h \in \mathbb{Z}$: $$\omega' \in\Omega^{n+1}_0 \mapsto \frac{1}{\Gamma(\lambda)}\int_{\mathbb{C}^{n+1}} \vertf\vert^{2\lambda} \bar f^{-h} \rho\omega\wedge \bar \omega' $$ at points$-\alpha - N$ for $N$ and $h$ well chosen integers. This result is new, evenfor $p = 1$. As a corollary, this implies that in this situation the existenceof a root in $-\alpha -\mathbb{N}$ for the $p$-th Bernstein polynomial of thefresco generated by a holomorphic form $\omega \in \Omega^{n+1}_0$ implies theexistence of at least $p$ roots (counting multiplicities) for the usual reducedBernstein polynomial of the germ $(f, 0)$. In the case of an isolatedsingularity we obtain that for each $\alpha \in ]0, 1] \cap \mathbb{Q}$ thebiggest root $-\alpha - m$ of the reduced Bernstein polynomial of $f$ in$-\alpha - \mathbb{N}$ produces a pole at $-\alpha - m$ for some $h \in\mathbb{Z}$ for the meromorphic extension of the distribution $$\square\longrightarrow \frac{1}{\Gamma(\lambda)}\vert f\vert^{2\lambda} \barf^{-h}\square.$$

Higher Bernstein Polynomials and Multiple Poles of  $\frac{1}{\Gamma(\lambda)} \int_X \vert f \vert^{2\lambda} \bar  f^{-h}\rho\omega\wedge \bar \omega'$