  We investigate the properties of the mixed-mode (RRd) RR Lyrae (RRL)
variables in the Fornax dwarf spheroidal (dSph) galaxy by using $B$- and
$V$-band time series collected over twenty-four years. We compare the
properties of the RRds in Fornax with those in the Magellanic Clouds and in
nearby dSphs, with special focus on Sculptor. We found that the ratio of RRds
over the total number of RRLs decreases with metallicity. Typically, dSphs have
very few RRds with 0.49$\ltsim P_0 \ltsim $0.53 days, but Fornax fills this
period gap in the Petersen diagram (ratio between first overtone over
fundamental period versus fundamental period). We also found that the
distribution in the Petersen diagram of Fornax RRds is similar to SMC RRds,
thus suggesting that their old stars have a similar metallicity distribution.
  We introduce the Period-Amplitude RatioS (PARS) diagram, a new pulsation
diagnostics independent of distance and reddening. We found that LMC RRds in
this plane are distributed along a short- and a long-period sequence that we
identified as the metal-rich and the metal-poor component. These two groups are
also clearly separated in the Petersen and Bailey (luminosity amplitude versus
logarithmic period) diagrams. These circumstantial evidence indicates that the
two groups have different evolutionary properties. All the pulsation
diagnostics adopted in this investigation suggest that old stellar populations
in Fornax and Sculptor dSphs underwent different chemical enrichment histories.
Fornax RRds are similar to SMC RRds, while Sculptor RRds are more similar to
the metal-rich component of the LMC RRds.
