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The present paper focuses on the evolution of hydrogen-deficient white dwarfs with the aim of exploring the consequences of different initial envelope structures on the carbon abundances expected in helium-rich, carbon-contaminated DQ white dwarfs. In particular, the evolutionary link between the DQs with low detected carbon abundances and the PG1159, extreme horizontal branch, and helium-rich R Coronae Borealis (RCrB) stars is explored. We present full evolutionary calculations that take a self-consistent treatment of element diffusion into account as well as expectations for the outer layer chemical stratification of progenitor stars upon entering the white dwarf regime. We find that PG1159 stars cannot be related to any DQ white dwarfs with low C abundances. Instead, we suggest that the latter could constitute the progeny of the giant, helium-rich RCrB stars., Comment: 10 pages, 10 figures. Accepted for publication in Astronomy and Astrophysics

This paper is designed to explore the formation and evolution of hydrogen-deficient post-AGB white dwarfs. To this end, we compute the complete evolution of an initially 2.7 M_sun star from the zero-age main sequence through the thermally pulsing and mass-loss phases to the white dwarf stage. Particular attention is given to the chemical abundance changes during the whole evolution. A time-dependent scheme for the simultaneous treatment of abundance changes caused by nuclear reactions, diffusive overshooting, salt fingers and convection is considered. We employed the double-diffusive mixing-length theory of convection for fluids with composition gradients (Grossman & Taam 1996). The study can therefore be considered as a test of its performance in low-mass stars. Also, time-dependent element diffusion for multicomponent gases is taken into account during the white dwarf evolution. The evolutionary stages corresponding to the last helium thermal pulse on the early white-dwarf cooling branch and the following born-again episode are carefully explored. Relevant aspects for PG1159 stars and DB white dwarf evolution are studied in the frame of these new evolutionary models that take into account the history of the white dwarf progenitor. The scope of the calculations is extended to the domain of the helium-rich, carbon-contaminated DQ white dwarfs with the aim of exploring the plausibility of the evolutionary connection PG1159-DB-DQ. In this regard, the implications for the double-layered chemical structure in pulsating DB white dwarfs is investigated. Another aspect of the investigation concerned the consequences of mass-loss episodes during the PG1159 stage for the chemical stratification of the outer layer of DB and DQ white dwarfs., Comment: 20 pages, 15 figures. Accepted by Astronomy & Astrophysics