  In the absence of a third dredge-up episode during the asymptotic giant
branch phase, white dwarf models evolved from low-metallicity progenitors have
a thick hydrogen envelope, which makes hydrogen shell burning be the most
important energy source. We investigate the pulsational stability of white
dwarf models with thick envelopes to see whether nonradial $g$-mode pulsations
are triggered by hydrogen burning, with the aim of placing constraints on
hydrogen shell burning in cool white dwarfs and on a third dredge-up during the
asymptotic giant branch evolution of their progenitor stars. We construct
white-dwarf sequences from low-metallicity progenitors by means of full
evolutionary calculations, and analyze their pulsation stability for the models
in the range of effective temperatures $T_{\rm eff} \sim 15\,000\,-\, 8\,000$
K. We demonstrate that, for white dwarf models with masses $M_{\star} \lesssim
0.71\,\rm M_{\sun}$ and effective temperatures $8\,500 \lesssim T_{\rm eff}
\lesssim 11\,600$ K that evolved from low-metallicity progenitors ($Z= 0.0001$,
$0.0005$, and $0.001$) the dipole ($\ell= 1$) and quadrupole ($\ell=2$) $g_1$
modes are excited mostly due to the hydrogen-burning shell through the
$\varepsilon$-mechanism, in addition to other $g$ modes driven by either the
$\kappa-\gamma$ or the convective driving mechanism. However, the $\varepsilon$
mechanism is insufficient to drive these modes in white dwarfs evolved from
solar-metallicity progenitors. We suggest that efforts should be made to
observe the dipole $g_1$ mode in white dwarfs associated with low-metallicity
environments, such as globular clusters and/or the galactic halo, to place
constraints on hydrogen shell burning in cool white dwarfs and the third
dredge-up episode during the preceding asymptotic giant branch phase.
