  Motivated by the recent detection of single and binary He-core white dwarfs
in metal-rich clusters, we present a full set of evolutionary calculations and
colors appropriate for the study of such white dwarfs. The paper is also aimed
at investigating whether stable hydrogen burning may constitute a main source
of energy for massive He-core white dwarfs resulting from high-metallicity
progenitors. White dwarf sequences are derived by taking into account the
evolutionary history of progenitor stars with supersolar metallicities. We also
incorporate a self-consistent, time-dependent treatment of gravitational
settling and chemical diffusion, as well as of the residual nuclear burning. We
find that the influence of residual nuclear burning during the late stages of
white dwarf evolution is strongly dependent on the occurrence of chemical
diffusion at the base of the hydrogen-rich envelope. When no diffusion is
considered, residual hydrogen burning strongly influences the advanced stages
of white dwarf cooling, introducing evolutionary delays of several Gyr. By
contrast, when diffusion is taken into account the role of residual nuclear
burning is strongly mitigated, and the evolution is dictated only by the
thermal content stored in the ions. In addition, for all of our sequences, we
provide accurate color and magnitudes on the basis of new and improved non gray
model atmospheres which explicitly include Ly$\alpha$ quasi-molecular opacity.
