Green functions, Segre numbers, and King's formula

Let $\mathcal{J}$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log \mathrm{\mid }f\mathrm{\mid }+\mathcal{O}(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal{J}$. We give a meaning to the Monge-Amp`{e}re products $(d{d}^{c}G{)}^k$ for $k=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}$, and prove that the Lelong numbers of the currents $M_k^{\mathcal{J}}:={\mathbf{1}}_Z(d{d}^{c}G{)}^k$ at $x$ coincide with the so-called Segre numbers of $\mathcal{J}$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{\mathcal{J}}$ satisfy a certain generalization of the classical King formula.