Recursive formula for $\psi ^g-\lambda _1\psi ^{g-1}+\cdots +(-1)^g\lambda _g$ in $\overline {\mathcal {M}}_{g,1}$

Abstract:

Mumford proved that $\psi ^g-\lambda _1\psi ^{g-1}+\cdots +(-1)^g\lambda _g=0$ in the Chow ring of $\overline {\mathcal {M}}_{g,1}$. We find an explicit recursive formula for $\psi ^g-\lambda _1\psi ^{g-1}+\cdots + (-1)^g\lambda _g$ in the tautological ring of $\overline {\mathcal {M}} _{g,1}$ as a combination of classes supported on boundary strata.