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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 theproperties of the RRds in Fornax with those in the Magellanic Clouds and innearby dSphs, with special focus on Sculptor. We found that the ratio of RRdsover the total number of RRLs decreases with metallicity. Typically, dSphs havevery few RRds with 0.49$\ltsim P_0 \ltsim $0.53 days, but Fornax fills thisperiod gap in the Petersen diagram (ratio between first overtone overfundamental period versus fundamental period). We also found that thedistribution 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 pulsationdiagnostics independent of distance and reddening. We found that LMC RRds inthis plane are distributed along a short- and a long-period sequence that weidentified as the metal-rich and the metal-poor component. These two groups arealso clearly separated in the Petersen and Bailey (luminosity amplitude versuslogarithmic period) diagrams. These circumstantial evidence indicates that thetwo groups have different evolutionary properties. All the pulsationdiagnostics adopted in this investigation suggest that old stellar populationsin Fornax and Sculptor dSphs underwent different chemical enrichment histories.Fornax RRds are similar to SMC RRds, while Sculptor RRds are more similar tothe metal-rich component of the LMC RRds.

On the Use of Field RR Lyrae as Galactic Probes. VI. Mixed mode RR Lyrae variables in Fornax and in nearby dwarf galaxies