  Classical Cepheids (DCEPs) are the most important primary indicators for the
extragalactic distance scale, but they are also important objects per se,
allowing us to put constraints on the physics of intermediate-mass stars and
the pulsation theories. We have investigated the peculiar DCEP HD 344787, which
is known to exhibit the fastest positive period change among DCEPs along with a
quenching amplitude of the light variation. We have used high-resolution
spectra obtained with HARPS-N@TNG for HD 344787 and the more famous Polaris
DCEP, to infer their detailed chemical abundances. Results from the analysis of
new time-series photometry of HD 344787 obtained by the TESS satellite are also
reported. The double mode nature of HD344787 pulsation is confirmed by analysis
of the TESS light curve, although with rather tiny amplitudes of a few tens of
millimag. This is an indication that HD344787 is on the verge of quenching the
pulsation. Analysis of the HARPS-N@TNG spectra reveals an almost solar
abundance and no depletion of carbon and oxygen. Hence, the star appears to
have not gone through the first dredge-up. Similar results are obtained for
Polaris. Polaris and HD344787 are confirmed to be both most likely at their
first crossing of the instability strip (IS). The two stars are likely at the
opposite borders of the IS for first overtone DCEPs with metal abundance
Z=0.008. A comparison with other DCEPs which are also thought to be at their
first crossing allows us to speculate that the differences we see in the
Hertzsprung-Russell diagram might be due to differences in the properties of
the DCEP progenitors during the main sequence phase.
