  Helium core white dwarfs (WDs) with mass M <~ 0.20 Msun undergo several Gyrs
of stable hydrogen burning as they evolve. We show that in a certain range of
WD and hydrogen envelope masses, these WDs may exhibit g-mode pulsations
similar to their passively cooling, more massive carbon/oxygen core
counterparts, the ZZ Cetis. Our models with stably burning hydrogen envelopes
on helium cores yield g-mode periods and period spacings longer than the
canonical ZZ Cetis by nearly a factor of 2. We show that core composition and
structure can be probed using seismology since the g-mode eigenfunctions
predominantly reside in the helium core. Though we have not carried out a fully
nonadiabatic stability analysis, the scaling of the thermal time in the
convective zone with surface gravity highlights several low-mass helium WDs
that should be observed in search of pulsations: NLTT 11748, SDSS J0822+2753,
and the companion to PSR J1012+5307. Seismological studies of these He core WDs
may prove especially fruitful, as their luminosity is related (via stable
hydrogen burning) to the hydrogen envelope mass, which eliminates one model
parameter.
