Context : 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 .

Aims : 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 .

Methods : We construct white-dwarf sequences from low-metallicity progenitors by means of full evolutionary calculations that take into account the entire history of progenitor stars , including the thermally pulsing and the post-asymptotic giant branch phases , and analyze their pulsation stability by solving the linear , nonadiabatic , nonradial pulsation equations for the models in the range of effective temperatures T _ { eff } \sim 15 000 - 8 000 K .

Results : We demonstrate that , for white dwarf models with masses M _ { \star } \lesssim 0.71 M _ { \sun } and effective temperatures 8 500 \lesssim T _ { 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 .

Conclusions : 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 .