  I compute the contributions of the one-loop single-real-emission amplitudes,
$gg\to H g$, $qg\to H q$, etc., to inclusive Higgs boson production through
next-to-next-to-next-to-leading order (N^3LO) in the strong coupling
$\alpha_s$. The next-to-leading (NLO) and next-to-next-to-leading order (NNLO)
terms are computed in closed form, in terms of $\Gamma$-functions and the
hypergeometric functions ${}_{2}F_{1}$ and ${}_{3}F_{2}$. I compute the \nnnlo\
terms as Laurent expansions in the dimensional regularization parameter through
order $(\epsilon^{1})$. To obtain the \nnnlo\ terms, I perform an extended
threshold expansion of the phase space integrals and map the resulting
coefficients onto a basis of harmonic polylogarithms.
