  We present a pulsational stability analysis of hot post-AGB H-deficient
pre-white dwarf stars with active He-burning shells. The stellar models
employed are state-of-the-art equilibrium structures representative of PG1159
stars derived from the complete evolution of the progenitor stars. On the basis
of fully nonadiabatic pulsation computations, we confirmed theoretical evidence
for the existence of a separate PG1159 instability strip in the $\log T_{\rm
eff} - \log g$ diagram characterized by short-period $g$-modes excited by the
$\epsilon$-mechanism. This instability strip partially overlaps the already
known GW Vir instability strip of intermediate/long period $g$-modes
destabilized by the classical $\kappa$-mechanism acting on the partial
ionization of C and/or O in the envelope of PG1159 stars. We found that PG1159
stars characterized by thick He-rich envelopes and located inside this
overlapping region could exhibit both short and intermediate/long periods
simultaneously. we study the particular case of VV 47, a pulsating planetary
nebula nucleus that has been reported to exhibit a series of unusually short
pulsation periods. We found that the long periods exhibited by VV 47 can be
readily explained by the classical $\kappa$-mechanism, while the observed
short-period branch below $\approx 300$ s could correspond to modes triggered
by the He-burning shell through the $\epsilon$-mechanism, although more
observational work is needed to confirm the reality of these short-period
modes. Were the existence of short-period $g$-modes in this star convincingly
confirmed by future observations, VV 47 could be the first known pulsating star
in which both the $\kappa$-mechanism and the $\epsilon$-mechanism of mode
driving are simultaneously operating.
