  Some globular clusters host multiple stellar populations with different
chemical abundance patterns. This is particularly true for $\omega$ Centauri,
which shows clear evidence of a helium- enriched sub-population characterized
by a helium abundance as high as $Y= 0.4$. We present a whole and consistent
set of evolutionary tracks from the ZAMS to the white dwarf stage appropriate
for the study of the formation and evolution of white dwarfs resulting from the
evolution of helium-rich progenitors. Different issues of the white dwarf
evolution and their helium-rich progenitors have been explored. In particular,
the final mass of the remnants, the role of overshooting during the
thermally-pulsing phase, and the cooling of the resulting white dwarfs differ
markedly from the evolutionary predictions of progenitor stars with standard
initial helium abundance. Finally, the pulsational properties of the resulting
white dwarfs are also explored. We find that, for the range of initial masses
explored in this paper, the final mass of the helium-rich progenitors is
markedly larger than the final mass expected from progenitors with the usual
helium abundance. We also find that progenitors with initial mass smaller than
$M_\star \simeq 0.65\,M_\odot$ evolve directly into helium-core white dwarfs in
less than 14~Gyr, and that for larger progenitor masses the evolution of the
resulting low-mass carbon-oxygen white dwarfs is dominated by residual nuclear
burning. For helium-core white dwarfs, we find that they evolve markedly faster
than their counterparts coming from standard progenitors. Also, in contrast
with what occurs for white dwarfs resulting from progenitors with the standard
helium abundance, the impact of residual burning on the cooling time of white
dwarfs is not affected by the occurrence of overshooting during the
thermally-pulsing phase of progenitor stars.
