  We present new H- and K-band spectroscopy for the bulge of M31, taken with
the LUCI spectrograph at the Large Binocular Telescope (LBT). We studied radial
trends of CO absorption features (namely, CO1.58, CO1.60, CO1.64, CO1.66,
CO1.68, CO2.30, CO2.32, CO2.35) in the bulge of M31, out to a galactocentric
distance of 100'' (380pc). We find that most COs do not exhibit a strong radial
gradient, despite the strong metallicity gradient inferred from the optical
spectral range, except for CO1.64, showing a steep increase in the center. We
compared the observed line strengths to predictions of different
state-of-the-art stellar population models, including an updated version of
EMILES models, which also uses the extended IRTF spectral library. The observed
COs are close to models' predictions, but in some models they turn out to be
underestimated. We find that the lack of radial gradients is due to the
combination of increasing CO strength with metallicity and C abundance, and
decreasing CO strength with IMF slope and O abundance. We speculate that the
steep gradient of CO1.64 might be due to Na overabundance. Remarkably, we were
able to fit, at the same time, optical indices and all the NIR COs except for
CO1.68, leaving abundance ratios (i.e., [C/Fe], [O/Fe], and [Mg/Fe]) as
free-fitting parameters, imposing age and metallicity constraints from the
optical, with no significant contribution from intermediate-age populations.
For the majority of the bulge, we find [Mg/Fe]~0.15dex, [O/Fe] larger than
[Mg/Fe] (by ~0.1dex), and C abundance consistent with that of Mg. In the
central (few arcsec) region, we still find an enhancement of O and Mg, but
significantly lower [C/Fe]. We find that the COs' line strengths of the bulge
are significantly lower than those of massive galaxies, possibly because of a
difference in carbon abundance, as well as, to some extent, total metallicity.
