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We give, using higher Bernstein polynomials defined in our paper [2], astronger version of our previous result in [1] whose converse is proved in [2]and we give some complements to the results in [2] which help to compute thesehigher order Bernstein polynomials. Then we show some non trivial exampleswhere we determine the root of the second Bernstein polynomial which is not adouble root of the full Bernstein polynomial and where the main theorem of [2]applies and localizes where a double pole exists for the meromorphic extensionof the (conjugate) analytic functional given by polar parts of $\omega$ '$\rightarrow$ |f | 2$\lambda$ f --h $\rho$$\omega$ $\land$ $\omega$' when h$\in$ N is large enough.

Complement to Higher Bernstein Polynomials and Multiple Poles of 1 $\Gamma$($\lambda$) X |f | 2$\lambda$ f --h $\rho$$\omega$ $\land$ $\omega$'