= Solution
A hydrogen <recombination line> is a discrete <spectral line> produced when <radiative recombination> captures an <electron> into an excited bound level and a subsequent <radiative cascade> emits <photons>. The free-to-bound capture itself produces a continuum; the later bound-to-bound transitions produce the <recombination lines>. For example, <H-alpha> is the $n=3\to2$ transition in the <Balmer series>.
For the neutral hydrogen absorbing an ionizing stellar continuum, the principal bound-free transition is <photoionization> from the $1s$ <ground state> into the continuum. Most neutral hydrogen is in its <ground state>, and the <photoionization> <cross section> is largest near its $13.6\ \mathrm{eV}$ threshold, falling approximately as $\nu^{-3}$ above it. Dominance of the \i[population-weighted opacity] is intended here; this does not mean that every excited-state microscopic <cross section> is smaller at every <frequency>.
In an ionization-bounded, optically thick nebula, direct <radiative recombination> into $1s$ emits an ionizing <photon> that promptly photoionizes another hydrogen atom. One ion disappears and another is created: there is no net removal of an ion from the global budget. <Case B recombination> therefore uses $\alpha_B=\sum_{n\geq2}\alpha_n$, with local recycling of ground-state recombination <photons>. This is the on-the-spot approximation, rather than an unconditional property of <radiative recombination>: if those <photons> escape, direct capture does produce net neutralization and a Case A treatment is appropriate.
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