On the b-function with respect to weights of annihilating ideals in the Weyl algebra

Given a polynomial $f\in \mathbb{C}[x_1,\dots ,x_n]$ and an integer $\mathrm{\ell }\in \mathbb{Z}$, we study some properties of the b-function with respect to weights of the annihilating ideal Ann$(f^{\mathrm{\ell }})$. In some particular cases the expression of the b-function is given explicitly.