  The origin of phosphorus, one of the essential elements for life on Earth, is
currently unknown. Prevalent models of Galactic chemical evolution (GCE)
underestimate the amount of P compared to observations. The recently discovered
P-rich ([P/Fe] > 1 dex) and metal-poor giants further challenge current
theories on stellar nucleosynthesis. Since the observed stars are low-mass
giants, our primary goal is to find clues on their progenitor. By increasing
the number of known P-rich stars, we aim to narrow down a reliable chemical
abundance pattern and to place robust constraints on the responsible
nucleosynthetic mechanism. In the long term, identifying the progenitor of the
P-rich stars may contribute to the search for the source of P in our Galaxy. We
performed a detailed chemical abundance analysis based on the H-band spectra
from APOGEE-2 (DR17). Employing the BACCHUS code, we measured the abundances of
13 elements in the sample, which is mainly composed of a recent collection of
Si-enhanced giants. We also analyzed the orbital motions and compared the
abundance results to possible nucleosynthetic formation scenarios, and also to
detailed GCE models. We enlarged the sample of confirmed P-rich stars from 16
to 78 giants, which represents the largest sample of P-rich stars to date.
Significant enhancements in O, Al, Si and Ce, as well as systematic
correlations among the elements, unveil the chemical fingerprint of the P-rich
stars. The high Mg and C+N found in some of the P-rich stars with respect to
P-normal stars is not confirmed over the full sample. Strikingly, the strong
over-abundance in the $\alpha$-element Si is accompanied by normal Ca and S
abundances. Our analysis of the orbital motion showed that the P-rich stars do
not belong to a specific sub-population. In addition, we confirm that the
majority of the sample stars are not part of binary systems.
