  Unlike in ordinary metals, in graphene, phonon structure can be seen in the
quasiparticle electronic density of states, because the latter varies on the
scale of the phonon energy. In a magnetic field, quantization into Landau
levels creates even more significant variations. We calculate the density of
states incorporating electron-phonon coupling in this case and find that the
coupling has pronounced new effects: shifting and broadening of Landau levels,
creation of new peaks, and splitting of any Landau levels falling near one of
the new peaks. Comparing our calculations with a recent experiment, we find
evidence for a phonon with energy similar to but somewhat greater than the
optical $E_{2g}$ mode and a coupling corresponding to a mass enhancement
parameter $\lambda \simeq 0.07$.
