Two-loop amplitudes for Higgs plus jet production involving a modified trilinear Higgs coupling

We calculate the contributions to the two-loop scattering amplitudes $h \to gg$, $h \to ggg$ and $h \to q \bar q g$ that arise from a modified trilinear Higgs coupling $\lambda$. Analytic expressions are obtained by performing an asymptotic expansion near the limit of infinitely heavy top quark. The calculated amplitudes are necessary to study the impact of the ${\cal O} (\lambda)$ corrections to the transverse momentum distributions ($p_{T,h}$) in single-Higgs production at hadron colliders for low and moderate values of $p_{T,h}$.