I report the discovery of blueshifted broad absorption line (BAL) troughs in at least six transitions of the Balmer series of hydrogen (Hbeta to H9) and in CaII, MgII and excited FeII in the quasar SDSS J125942.80+121312.6. This is only the fourth active galactic nucleus known to exhibit Balmer absorption, all four in conjunction with low-ionization BAL systems containing excited Fe II. The substantial population in the n=2 shell of H I in this quasars absorber likely arises from Ly-alpha trapping. In an absorber sufficiently optically thick to show Balmer absorption, soft X-rays from the quasar penetrate to large tau_Lyalpha and ionize H I. Recombination then creates Ly-alpha photons that increase the n=2 population by a factor tau_Lyalpha since they require about tau_Lyalpha scatterings to diffuse out of the absorber. Observing Ly-alpha trapping in a quasar absorber requires a large but Compton-thin column of gas along our line of sight which includes substantial H I but not too much dust. Presumably the rarity of Balmer-line BAL troughs reflects the rarity of such conditions in quasar absorbers.
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