 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 $n=3\to 2$ 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 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 ${\alpha }_B={\sum }_{n\ge 2}{\alpha }_n$, 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.