# The Dark Matter Halo of M54

Raymond G. Carlberg Department of Astronomy & Astrophysics University of
Toronto Toronto, ON M5S 3H4, Canada<EMAIL_ADDRESS>Carl J.
Grillmair IPAC, California Institute of Technology, Pasadena, CA 91125
<EMAIL_ADDRESS>
###### Abstract

M54 is a prototype for a globular cluster embedded in a dark matter halo. Gaia
EDR3 photometry and proper motions separate the old, metal-poor stars from the
more metal rich and younger dwarf galaxy stars. The metal poor stars dominate
the inner 50 pc, with a velocity dispersion profile that declines to a minimum
around 30 pc then rises back to nearly the central velocity dispersion, as
expected for a globular cluster at the center of a CDM cosmology dark matter
halo. The Jeans equation mass analysis of the three separate stellar
populations gives consistent masses that rise approximately linearly with
radius to 1 kpc. These data are compatible with an infalling CDM dark matter
halo reduced to $\simeq 3\times 10^{8}M_{\sun}$ at the 50 kpc apocenter 2.3
Gyr ago, with a central globular cluster surrounded by the remnant of a dwarf
galaxy. Tides gradually remove material beyond 1 kpc but have little effect on
the stars and dark matter within 300 pc of the center. M54 appears to be a
“transitional” system between globular clusters with and without local dark
halos whose evolution within the galaxy depends on the time of accretion and
orbital pericenter.

##  1 INTRODUCTION

The relationship between the very old globular clusters and sub-galactic mass
cosmological dark matter halos (Doroshkevich et al., 1967; Peebles & Dicke,
1968; Peebles, 1984), is important to understand the earliest phases of star
and galaxy formation. The observed internal kinematics of globular cluster
stars over a range of stellar masses combined with point mass n-body modeling
of individual clusters finds that the mass density of the clusters is
consistent with the stellar mass of an old, evolved, stellar population inside
2-3 half-mass radii (Zonoozi et al., 2011; Baumgardt & Hilker, 2018; Ebrahimi
et al., 2020; Vasiliev & Baumgardt, 2021). At larger radii the declining
density of cluster stars makes comtamination by foreground and background
stars an increasing problem for velocity dispersion measurements. However, the
growing number of radial velocity measurements and the increasing time
baseline of the Gaia data (Gaia Collaboration et al., 2016) is allowing the
kinematics of stars at larger radii to be examined.

There is evidence that some, but certainly not all, halo globular clusters
have velocity dispersion profiles at large cluster-centric radii above the
predictions from the stellar mass of the cluster alone. Comparisons of the
observed profiles to detailed n-body velocity dispersion profiles find that
$\sim$10% of currently measured clusters have velocity dispersion profiles
above the values predicted from the stars (Baumgardt & Hilker, 2018; Bianchini
et al., 2019; Wan et al., 2021). For a set of 19 halo clusters measured to at
least 5 half mass radii about 10% have a rising velocity dispersion profile
and another 30% are approximately flat (Carlberg & Grillmair, 2021). The
current measurements are still somewhat noisy but will improve significantly
with future Gaia releases.

There are a number of possible explainations for some or all of the excess
velocity dispersion, including single epoch observations of binary stars (Wan
et al., 2021), the complexity of orbits between the cluster and the tidal
surface (Fukushige & Heggie, 2000; Claydon et al., 2017), particularly for
highly elliptical orbits near the galactic center (Carlberg & Grillmair,
2021), stripped galactic nuclei (Kuzma et al., 2018; Wirth et al., 2020), and
dark matter halos (Mashchenko & Sills, 2005a, b; Peñarrubia et al., 2017;
Boldrini et al., 2020; Vitral & Boldrini, 2021; Carlberg & Keating, 2022).
Dynamical measurements of potential progenitor systems can be used to test
these ideas.

The M54/NGC6715 cluster is at the center of the Sagittarius (Sgr) dwarf galaxy
(Ibata et al., 1994; Bellazzini et al., 2008). M54, with a mass of
$1.4-1.8\times 10^{6}M_{\sun}$, is one of the most massive globular clusters
within the Milky Way halo. M54 satisfies the definition of nuclear star
cluster (NSC), one of four within smaller galaxies that have been identified
as having accreted into the Milky Way (Pfeffer et al., 2021). As a nuclear
star cluster M54 is around the median NSC mass and one of the most metal poor
known (Neumayer et al., 2020). The stars in M54 are predominantly old and
metal poor (Bellazzini et al., 2008; Alfaro-Cuello et al., 2019) as opposed to
several nearby NSCs whose central regions are predominantly young stars
(Hannah et al., 2021). M54 has no compelling evidence for a central black hole
(Ibata et al., 2009; Wrobel et al., 2011). The Sagittarius dwarf luminosity,
$L_{V}=1.8\times 10^{6}L_{\sun}$ (Mateo, 1998) is a factor of 10 less luminous
than expected, relative to the mean relation for NSC hosts (Neumayer et al.,
2020). The Sgr dwarf has prominent tidal tails (Ibata et al., 1997; Belokurov
et al., 2006; Ibata et al., 2020) which help constrain the recent orbital
history of the system. Spectroscopic observations of the line of sight
velocity profile of M54 find that the old, metal poor, blue stars of M54 have
a velocity dispersion profile that declines from a central value of
approximately 10 km s-1 to a minimum of about 5 km s-1 at 22 pc and then rises
back up to nearly 10 km s-1 again at 55 pc, where it becomes essentially the
same as the velocity dispersion of the younger, more metal rich, redder stars
of the dwarf galaxy surrounding the cluster (Bellazzini et al., 2008; Alfaro-
Cuello et al., 2020).

This paper uses Gaia data to separate the stars within about 4 kpc of M54 into
blue, red and an intermediate color (“green”) populations. The proper motions
give the projected radial and tangential velocities of the stars. The
resulting radial velocity dispersion profiles and the degree of radial
anisotropy of each distinct stellar population are used with the Jeans
equation to measure the total mass profile with radius. The results are
compared to n-body models to constrain the dark mass near and into the star
cluster and within the surrounding dwarf galaxy. The models are also used to
assess the long term evolution of the M54 star cluster-dark halo systems and
its dependence on the system’s orbital pericenter.

##  2 Data

We use the Early Data Release 3 (Gaia Collaboration et al., 2021) of the Gaia
mission (Gaia Collaboration et al., 2016) to select $B_{P}-R_{P}$ colors, G
magnitudes, parallaxes, proper motions, and their estimated errors, for stars
within 10 degrees of the globular cluster M54/NGC6715, or 4.587 kpc at our
adopted distance of 26.28 kpc (Baumgardt & Vasiliev, 2021). The Gaia
photometry is corrected for extinction using the reddening maps of (Schlegel
et al., 1998), themselves corrected using the prescription of Schlafly &
Finkbeiner (2011), and using the Gaia DR2 coefficients derived by Gaia
Collaboration et al. (2018). A $G_{0},(G_{BP}-G_{RP})_{0}$ color-magnitude
locus for the red giant branch (RGB), subgiant branch, and main sequence is
constructed using stars between one and three arcminutes from the cluster
center. A theoretical isochrone with [Fe/H] = -1.59 (Harris, 1996) generated
in the Gaia passbands at http://stev.oapd.inaf.it/cgi-bin/cmd (Girardi et al.,
2004) is used as a guide at fainter magnitudes. This isochrone is not used
directly due to significant and variable offsets between the observed and
theoretical giant branches. Color offsets from this sampled color-magnitude
locus are then computed using the tabulated photometric uncertainties in Gaia
EDR3. The proper motion uncertainties lead to undertainties in each velocity
component of 5 and 10 km s-1 at $G_{0}$=15.5 and 17.0 mag, respectively. The
data are limited to an extinction corrected brightness of $G_{0}$=17.5
magnitudes. Hereafter we drop the 0 subscript for the extinction corrected
magnitudes and colors.

There are several distinct age and metallicity stellar populations within the
M54 cluster and Sagittarius dwarf (Bellazzini et al., 2008; Ibata et al.,
2009; Vasiliev & Belokurov, 2020). Detailed spectroscopic studies find that
the cluster stars are predominantly an old metal poor population with a
continuous spread to a younger metal rich population (Alfaro-Cuello et al.,
2019). Along the lines of previous studies (Bellazzini et al., 2008; del Pino
et al., 2021) we use the Gaia photometry to separate the stars into three
groups: the blue population, which is old and metal poor, a red population
which is young and metal rich, and a population with intermediate colors that
we designate “green”. The blue stars are those having a color within 1 sigma
of the cluster sequence. The red stars are defined as those between
$B_{P}-R_{P}>1.45+0.15(16-G)$ and $B_{P}-R_{P}<1.57+0.15(16-G)$ for $18>G>16$,
and for $G<16$ colors between $B_{P}-R_{P}>1.45+0.5(16-G)$ and
$B_{P}-R_{P}<1.57+0.5(16-G)$. The intermediate color, “green”, stars are those
that are at least 10 sigma to the red of the cluster sequence and bluer than
the blue edge of the red sequence. The resulting color-magnitude diagram is
shown in Figure 1. The sample shown in Figure 1 includes cuts on velocity
relative to the cluster center of mass, as discussed below.

Figure 1: The extinction corrected color-magnitude diagram for the velocity
selected star sub-samples.

Figure 2: The unit area normalized distribution of relative stellar velocities
parallel to the galactic plane for the three color sub-samples. The top panel
is for stars within 1 kpc of the cluster, the bottom for 4 kpc. The velocities
perpendicular to the plane have similar distributions.

The M54 cluster is located close to the Galactic plane, at Galactic latitude
of -14.1° and longitude 5.6° resulting in a large and variable foreground of
Galactic stars. The sample also has systematic errors with position (del Pino
et al., 2021). The parallax errors are sufficiently large for most of the
stars that a parallax limit is not helpful. Velocities are calculated assuming
that the stars are at the distance of the cluster and placed in the center of
mass frame of the cluster. Velocity cuts are important to reduce both the
background and systematic errors.

The radial velocities in the plane of the sky are projection corrected, using
the center of mass velocity of M54, 143.06 km s-1 (Baumgardt et al., 2019) at
a consensus distance of 26.28 kpc (Baumgardt & Vasiliev, 2021). We will
designate the two coordinates in the plane of the sky Y and Z, which are close
but not quite identical to the Galactic Y and Z. Relative to the Galactic
frame, our Z coordinate is tilted 5.5° backward from us and our Y coordinate
is rotated 14° counterclockwise, as seen from above. The distribution of the
velocities parallel to the Galactic plane, relative to the cluster, is shown
in Figure 2 for stars within 1 and 4 kpc of the cluster (top and bottom
panels, respectively). There is a clear excess of stars above 30 km s-1 at
large radii. The velocities perpendicular to the plane show a similar excess
of blue stars with velocities larger than about 30 km s-1 beyond 1 kpc, most
of the stars being those close to the plane. Gaia data has various sky
location selection effects in the region of M54 (see del Pino et al. (2021)
for a discussion of the DR2 data). To produce an acceptably uniform sample in
the M54 region we select stars that have a projected velocity relative to the
cluster of 30 km s-1 or less. The distribution of the selected stars on the
sky is shown in Figure 3. While Galactic plane stars intrude at the top of the
figure, background contamination has been greatly reduced. Using a larger
velocity cut of 50 km s-1 makes little difference for densities but boosts
velocity dispersions at all radii, with a strong rise beyond 300 pc. We adopt
the 30 km s-1 limit for our measurements.

Figure 3: The sky distribution of the M54/Sgr $|v_{XY}|\leq
30{\,\hbox{km\,s}^{-1}}$ stars plotted in our longitude and latitude
coordinates converted to kiloparsecs at the adopted distance of M54. The
Galactic mid-plane is at y=6.46 kpc in these coordinates. The inner $\pm$0.1
kpc region, marked with a square at the center, is shown in the inset. At the
distance of M54, 1 kpc is 2.18° or 7.64 pc per arc-minute. Crowding causes an
increasingly large under-sampling in the central region, where the M54 stars
dominate those of the Sgr dwarf.

##  3 M54-Sgr Density and Velocity Dispersion Profiles

The density and velocity dispersion profiles of the star cluster are measured
as projected azimuthally averaged quantities, which is appropriate for the
M54/NGC6715 as it has an ellipticity of 0.06 (Harris, 1996). Beyond 100 pc the
dwarf galaxy stars are in a significantly elliptical distribution, as shown in
Figure 3. The Sagittarius dwarf galaxy is modeled as a prolate triaxial dwarf
galaxy with its axis of rotation tilted approximately 40-60° from the sky
plane (del Pino et al., 2021) with gradients of both density and velocity
dispersion so shallow that radial averaging is acceptable. The radially
averaged surface densities of the three color samples are shown in Figure 4.

The red and green stars dominate the surface density beyond 50 pc in the Sgr
dwarf, with blue stars comprising about 5% of the combined surface density
beyond 100 pc. The central part of the field is so crowded that the density
profile of M54 is severely undersampled inside 10 pc (Pancino et al., 2017),
nearly twice M54’s half-mass radius of 5.6 pc. The green and red subsamples
show only a weak rise towards the center in these data. Better resolved data
for the young and metal rich red stars finds that their density rises within
the cluster approximately in proportion to the blue stars, but with about 10%
of the surface density (Bellazzini et al., 2008). Sgr is unusual in having a
dominant old, metal poor population, whereas young, metal rich stars often
dominate the central regions of well resolved nearby nuclear star clusters
(Hannah et al., 2021).

Figure 4: The azimuthally averaged surface density versus projected radius for
the color selected stars with absolute velocities less than 30 km s-1. Image
crowding greatly reduces completeness inside approximately 20 pc. The central
M54 cluster has a 3D half mass radius of 5.6 pc.

Figure 5: The error corrected projected radial (top panel) and tangential
(bottom panel) velocity dispersion profile of M54/Sgr for the blue, green, and
red subsamples. The 2D velocity anisotropy,
$\beta_{2D}=1-\sigma_{t}^{2}/\sigma_{r}^{2}$, evaluated in wider radial bins,
is shown in the inset using the same line colors. The tangential velocity
dispersion between 50 and 100 pc, where there are less than 10 data points per
smaller bin, is consistent with zero

The 2D velocity dispersion in the plane of the sky is
$\sigma_{2D}=\sqrt{\sigma_{r}^{2}+\sigma_{t}^{2}}$, where $\sigma_{r}$ and
$\sigma_{t}$ are the velocity dispersion measured in the plane of the sky
relative to the cluster center. The velocity dispersions have the mean proper
motion velocity error in each bin, which range from about 8 to 12 km s-1,
subtracted in quadrature. The resulting projected radial velocity dispersion,
$\sigma_{r}(r)$ and tangential velocity dispersion, $\sigma_{t}$, of the three
color subsamples are shown in Figure 5. The tangential velocity dispersion in
the range of 50 to 100 pc is consistent with zero, with less than 10 data
points per bin in this radial range making the measurement of a small velocity
dispersion difficult. The combined 2D velocity dispersion is shown in Figure 8
compared with the n-body velocity dispersions.

The red and green stars, primarily in the dwarf galaxy, have similar velocity
dispersion profiles, with a shallow decline toward the center inside 1000
parsecs. The blue stars, primarily in the central star cluster, have a
distinctive velocity profile, with a minimum at 30 parsecs and beyond that a
rise back up to nearly the central value. The velocity dispersion dip in the
blue stars and the flat profile of the red stars was noted in the line-of-
sight radial velocity data of Bellazzini et al. (2008); the data here confirm
that behavior using the orthogonal components provided by the Gaia proper
motions. The lower velocity dispersion of the red stars relative to the
intermediate green stars was seen in the line-of-sight velocity analysis of
Alfaro-Cuello et al. (2020).

The inset in Figure 5 shows the 2D velocity anisotropy,
$\beta_{2D}=1-\sigma_{t}^{2}/\sigma_{r}^{2}$, in somewhat wider radial bins to
increase the signal-to-noise ratio. The velocity ellipsoid of the blue stars
becomes nearly completely radial around 80 pc. At larger radii all three color
subsamples have nearly isotropic velocity ellipsoids. The blue stars in the
central 50 pc are kinematically distinct from stars of all colors at larger
radii, which belong to the Sagittarius dwarf galaxy and have ages ranging from
the last 0.5 Gyr to a Hubble age (Alfaro-Cuello et al., 2019).

##  4 Total Mass Profile within 1 kpc

The radial Jeans equation measures the total mass interior to radius $r$,

$M(r)=-{{r\sigma_{r}^{2}}\over{G}}\left[\frac{d\ln{\nu}}{d\ln{r}}+\frac{d\ln{\sigma_{r}^{2}}}{d\ln{r}}+2\beta(r)\right],$
(1)

where $\nu(r)$ is the density of a tracer population, $\sigma_{r}(r)$ is its
radial velocity dispersion, and $\beta(r)$ is its 3D velocity dispersion
anisotropy. The data displayed in Figures 4 and 5 are used in the Jeans
equation to estimate total interior masses shown in Figure 6. The Jeans
equation assumes that the stars are in equilibrium and we are making the
assumption of a spherical underlying potential. These assumptions are examined
with models below.

The power-law slope, $s_{\nu}=d\ln{\nu}/d\ln{r}$, of the 3D stellar density
profiles, $\nu(r)$, is estimated from the slope of the surface density profile
of the population, using the power law relation that the 3D slope is the 2D
slope plus one. The 2D $\beta$ values from the Figure 5 inset are close to the
3D values, $\beta_{3D}=1-(\sigma_{t1}^{2}+\sigma_{t2}^{2})/2\sigma_{r}^{2}$,
for our assumed spherically symmetric system in which the two components,
$\sigma_{t1}$ and $\sigma_{t2}$, of the 3D tangential velocity dispersion, are
equal. For the two special cases of the velocity ellipsoid being isotropic or
completely radial, $\beta_{2D}=\beta_{3D}$, with values of 0 and 1,
respectively.

Figure 6 shows the results of the Jeans mass analysis. An important outcome is
that the three separate star samples give consistent results at 30 pc and
beyond. The mass inside 30 pc is estimated to be 1.4, 1.9 and 1.1$\times
10^{6}M_{\sun}$ for blue, green and red stars, respectively. The 30 pc mass is
consistent with the stellar dynamical mass of the star cluster alone, though
it does not rule out a small fraction of dark matter. Between 30 and 95 pc the
total interior mass rises from the average value of $1.5\times 10^{6}$ to
$3.5\times 10^{6}M_{\sun}$, a factor 2.3. There is little increase in the
stellar mass in this region, so the bulk of this mass increase must be made up
of dark matter. The mass rises another factor of about 17 from 95 to 950 pc to
$89\times 10^{6}M_{\sun}$. Again, this must be made up mostly of dark matter,
though the stars of the Sagittarius dwarf (with about 70% of the luminosity
$L_{V}=1.8\times 10^{7}L_{\sun}$ within 950 pc) will contribute about 15% of
the mass for a stellar $M/L_{V}\simeq 1$.

Figure 6: Mass estimates from the Jeans equation for the three independent
color subsamples. The mass points partially overlap at the largest radii. The
star symbol indicates the Baumgardt & Hilker (2018) stellar mass of the
cluster which is placed at 3 half-mass radii.

##  5 Comparison with N-body models

Figure 7: The n-body model cluster stars (blue) and one of every 50 dark
matter particles in the $3\times 10^{8}M_{\sun}$ halo simulation.

Our model for the M54-Sgr system begins when the system has settled into its
current orbit, likely having shed the outer part of a more massive dark matter
halo (Łokas et al., 2010). A King model star cluster (King, 1966) is inserted
into a Hernquist model dark halo (Hernquist, 1990). The two populations
equilibrate in 0.02 Gyr. The combined star cluster-dark halo system is placed
at the apocenter of M54’s orbit and integrated backward in time in a MW2014
potential (Bovy, 2015) with the bulge component modeled as a 0.5 kpc Plummer
sphere. The large mass-to-light ratio of the Sagittarius dwarf, 22
$M_{\sun}/L_{\sun}$ (Mateo, 1998) or the 15% within 1 kpc we found in our
Jeans mass analysis means that the stellar mass of the dwarf galaxy
contributes little to the overall potential and can be ignored. The initial
system is spherically symmetric. We select an inner, flattened, tilted, subset
of dark matter particles on relatively circular orbits as proxies for the
dwarf galaxy disk stars to follow their evolution with time.

###  5.1 The N-body Model Star Cluster

We adopt a star cluster mass of $1.78\times 10^{6}M_{\sun}$ and a half-mass
radius of 5.6 pc for M54 (Baumgardt & Hilker, 2018)
(https://people.smp.uq.edu.au/ HolgerBaumgardt/globular). The star cluster
along with the surrounding dark matter is integrated with Gadget-4 (Springel
et al., 2021) modified to include a collisional star cluster (Carlberg &
Keating, 2021). Here we have included stellar “kicks” (Meiron et al., 2021) to
mimic binary encounters, modeled as the addition of Gaussian random velocities
of $\sigma\sqrt{3/2}$ added to each component of the velocity for a random
fraction of 0.001 of the stars within half of the virial radius, where
$\sigma$ is the 1D velocity dispersion. The cluster expands with time, so we
start the simulation with half mass radius of 5.0 pc. The cluster expands to a
half-mass radius of 6.6 pc at 2.29 Gyr although this makes little difference
to the dynamics beyond 10 pc.

The dark halo is modeled as a Hernquist sphere (Hernquist, 1990) specified
with a mass and scale radius with values motivated by the sub-galactic halos
found in our cosmological simulations (Carlberg & Keating, 2021). The pre-
infall mass of the entire Sagittarius galaxy to its virial radius may have
been quite large, $1.6\times 10^{10}M_{\sun}$ (Łokas et al., 2010) but we are
interested in the mass remaining on the current orbit. The dark matter mass
within 10 pc of the globular cluster center ( about two half-mass radii) is
observationally constrained to be well below the stellar mass, otherwise the
velocity dispersion of the cluster stars would be larger than observed. The
dark matter mass within 100 pc must rise to be at least the mass of the
stellar cluster in order to have any dynamical effect. We run models with
$M_{h}$= $1\times 10^{8}M_{\sun}$ and $3\times 10^{8}M_{\sun}$, both $a=0.4$
kpc, and $M_{h}=7\times 10^{8}M_{\sun}$, $a=0.6$ kpc. The star particles have
mass 20 $M_{\sun}$ and the dark matter particles 40 $M_{\sun}$ with softenings
of 2 and 5 pc, respectively, which is sufficient to provide an accurate star
cluster model at the half mass radius and beyond. The current location and
velocity of M54 (Baumgardt et al., 2019) is used to integrate the orbit
backward to the orbital apocenter around 2.29 Gyr ago (Vasiliev & Belokurov,
2020).

The star cluster is placed at the center of the dark halo. It is possible that
the cluster formed at some distance from the center of the dark halo and that
dynamical friction subsequently dragged it inwards. Dynamical friction would
have caused the massive, dense, M54 to spiral to the center of the dark matter
halo with minimal tidal mass loss. For instance, a straightforward simulation
shows that if started on a circular orbit at 0.5 kpc, the star cluster spirals
into the center in less than 0.5 Gyr, heating the central dark matter
particles to create a 30 pc core in their distribution.

The star cluster and its dark halo adjust quickly to the new equilibrium with
each other and their particle gravitational softening (2 and 5 pc,
respectively). In the first 0.01 Gyr the star cluster half mass radius
increases about 10% and the dark halo density at 10 pc approximately doubles,
with its velocity dispersion going up about 50%. The simulation responds
dynamically on the radial orbit timescale, 0.86 Gyr. Successive pericenter
passages heat the outer dark matter halo, which expands over the orbital
period and is pulled away at the next pericenter passage. Tides and internal
dynamical heating of stars moves about 1% of the cluster mass into the region
between 30 and 200 pc. If there were no dark halo these stars would be unbound
from the star cluster and would form a thin tidal stream of stars. Internal
relaxation is very slow, as expected for M54 whose relaxation time has been
estimated to be about 6 Gyr (Baumgardt & Hilker, 2018). By comparison, our
model has a relaxation time of 6.3 Gyr.

Figure 8: The projected 2D (top) and radial (bottom) velocity dispersion of
M54’s stars compared to the same quantity measured for the star particles in
the simulations that start with 1, 3, and 7$\times 10^{8}$ dark halos, shown
as black dashed, dotted and solid lines, respectively.  Figure 9: The 2D
velocity anisotropy, $\beta_{2D}=1-\sigma_{t}^{2}/\sigma_{r}^{2}$, for the
color subsamples compared to the simulated star cluster’s initial dark halo.
The velocity dispersions have been corrected for the proper motion velocity
errors. The values in the 1, 3, and 7$\times 10^{8}$ dark halos are shown as
black dashed, dotted and solid lines, respectively.

###  5.2 Evolution of the Simulation

The model runs for 2.29 Gyr to reach a Galactic position close to that of the
observed cluster. The star particle positions and velocities are projected
onto the sky and analyzed in the same was as the M54 cluster stars. The 2D
velocity dispersion of M54’s blue stars are compared to the star particles in
the model in Figure 8. The models predict a velocity minimum around 30 pc and
then a rise back to nearly the central velocity dispersion at 100 pc. Our blue
subsample shows the same trend.

The model star cluster particles gradually extend to somewhat more than 100
pc. The M54/Sgr stars selected to be in the blue subsample extend over the
entire surface of the dwarf galaxy, with cluster stars in the center and dwarf
galaxy stars at larger radii. A kinematic discriminant between cluster and
Galaxy stars is the velocity anisotropy. Defining
$\beta_{2D}=1-\sigma_{t}^{2}/\sigma_{r}^{2}$ the model values are compared to
the color subsamples in Figure 9. The cluster stars in the model are expected
to be radially anisotropic in the 30 to 100 pc range, as their orbits are
generally perturbed only in the central region of the cluster and become
highly elliptical. The green and red samples are essentially isotropic at all
radii. The M54/Sgr data show that the blue sample is radially anisotropic in
the range 30-100pc, becoming isotropic at larger radii. The change in velocity
anisotropy with radius is also consistent with the change in density profile
shown in Figure 4, with stars inside of 100 pc largely belonging to the
cluster while stars beyond 100 pc largely belong to the dwarf galaxy.

##  6 The Mean Velocity Field of the Sagittarius Dwarf

The Sagittarius Dwarf galaxy is being shredded in the Milky Way’s
gravitational field (Ibata et al., 1994) to produce a tidal stream wrapped
around the galaxy at least twice (Ibata et al., 2020). Our interest is to use
the velocity profile of the inner few kiloparsecs of Sagittarius stars to
estimate the density profile of the dark matter that remains bound to
Sagittarius and place some limits on the pre-accretion mass of the system and
how it is related to the M54 cluster. The stars in this region have motions
that transition from bound to flowing out into the stream. The velocity field
of the Sagittarius Dwarf has been extensively studied to understand the shape
of the stellar system, the total mass profile and the velocity at which stars
are fed into the stellar stream (Law & Majewski, 2010; Peñarrubia et al.,
2010; Łokas et al., 2010; Frinchaboy et al., 2012; Vasiliev & Belokurov, 2020;
del Pino et al., 2021; Ramos et al., 2021). A point of controversy is the
amount of rotation in the inner 0.5-2 kpc of the system.

Figure 10: The green sample mean velocity field (top panel) and the corner
plot of the fit to a rotation plus shear model. The model removes 73% of the
variance and has a $\chi^{2}/{\rm dof}=2.06$

Figure 11: The blue sample mean velocity field (top panel) and the corner plot
of the fit to a rotation plus shear model. The model removes 55% of the
variance. $\chi^{2}/{\rm dof}=1.10$

Figure 12: The red sample mean velocity field (top panel) and the corner plot
of the fit to a rotation plus shear model. The model removes 66% of the
variance. $\chi^{2}/{\rm dof}=2.02$

###  6.1 The Sagittarius Galaxy Velocities

Our color subsamples are separately binned into 0.4 kpc square pixels in a 8
kpc frame centered on the M54 cluster. The mean velocity fields on the plane
of the sky (YZ Galactic coordinates relative to the cluster center) are shown
in the upper panels of Figures 10, 11 and 12. The motions are modeled as the
sum of a rotation and a shear,

$\displaystyle v_{t}(r)$ $\displaystyle=$ $\displaystyle
v_{rot}{r\over{\sqrt{a^{2}+r^{2}}}},$ (2) $\displaystyle v_{r}(r)$
$\displaystyle=$ $\displaystyle A_{r}r$ (3)
$\displaystyle\vec{v}_{s}(r,\theta)$ $\displaystyle=$ $\displaystyle
A_{s}r[\cos{(\theta-\theta_{0})},\sin{(\theta-\theta_{0})}],$ (4)

which are projected into the x and y components of the velocity. The
parameters of the model are $A_{r}$ the linear expansion rate, $v_{rot}$ the
peak of the maximum of the rotation velocity, $a$ the scale radius of the
rotation velocity, $A_{s}$ the velocity shear, and $\theta_{0}$ the direction
of the shear. The model is fit to the data using the Monte Carlo Markov Chain
algorithm emcee (Foreman-Mackey et al., 2013). The logarithm of the likelihood
function is
$-\text@frac{1}{2}\sum[\Delta(v_{x})^{2}+\Delta(v_{y})^{2}]/\sigma_{v}^{2}+\log{(\sigma_{v}^{2})}$,
where $\Delta(v_{x})$ is the difference between the observed $v_{x}$ and the
value that the model predicts. The $\sigma_{v}$ is taken from the measured 2D
velocity dispersion which is essentially uniform beyond 0.2 kpc from the
cluster, nearly independent of color. We set $\sigma_{v}$ to be constant at 20
km s-1. The priors on the model quantities are set to be flat with such a
large range that the results have no dependence on the range of the prior.

The resulting corner plots showing the fits are displayed in the lower panels
of Figures 10, 11 and 12. The model accounts for 44 to 61% of the variance in
the data overall so the fits are reasonable, with $\chi^{2}/{\rm dof}$ values
ranging from 1.1 to 2.1 assuming a random velocity of 20 km s-1 in each
component. Nevertheless, the model does indicate that all three subsamples
have similar mean motions, with an expansion of
$A_{r}=1.6-2.0{\,\hbox{km\,s}^{-1}}\,{\rm kpc}^{-1}$ and a shear of
$A_{s}=2-2.5{\,\hbox{km\,s}^{-1}}\,{\rm kpc}^{-1}$, at an angle of 40-45° with
respect to the Galactic plane.

###  6.2 An Artificial Star Dwarf Galaxy

Figure 13: The M54-Sgr stellar surface densities with radius (blue, green,
red), the n-body star cluster surface density (blue dashed), the artificial
dwarf galaxy star profiles (purple) and the dark matter surface density in
units of $10^{3}\,{\rm M}_{\sun}{\rm pc}^{-2}$ (gray), where the 1, 3 and
$7\times 10^{8}M_{\sun}$ simulations have dashed, dotted and solid lines,
respectively. Figure 14: The mean 3D density inside radius $r$ inferred from
the Jeans equation in Figure 6 and the 3D dark matter densities of the $3$ and
$7\times 10^{8}M_{\sun}$ simulations as dotted and solid lines, respectively.
The solid grey line is the local 3D density in the $7\times 10^{8}M_{\sun}$
simulation. All simulation measurements are at 2.29 Gyr.  Figure 15: The
starting and final radii of the artificial dwarf galaxy stars for the $3\times
10^{8}M_{\sun}$ simulation. No particles are selected as stars inside 0.5 kpc,
but the particles are on sufficiently elliptical orbits that over time the
central hole fills.  Figure 16: The stream that the artificial dwarf galaxy
star particles produce in the $3\times 10^{8}M_{\sun}$ simulation. The
Galactic center is located at (0,0) and the star cluster is located close to
the observed position of M54, at Galactic coordinates [-14.7, 2.2,-6.2] kpc.
The simulation began at the orbital apocenter 2.29 Gyr in the past. A dwarf
galaxy rotating in the opposite direction does not produce a bifurcated
stream.

To approximate the kinematics of the stars of the Sagittarius dwarf we select
a thick disk-like set of particles from the simulation at early times. We
select from the simulation near the outset, at an age of 0.05 Gyr, identifying
a subset of the dark matter particles around the star cluster which are in a
flattened distribution, tilted to the line of sight. Sagittarius is a low
surface brightness galaxy in which the stars provide relatively little mass
compared to the dark matter (Ibata et al., 1997; Mateo, 1998). The model dwarf
galaxy is constructed 2.24 Gyr ago shortly after the system passed apocenter.
The initial disk orientation is taken to be the same as the current
orientation, with a unit axis [-0.13, 0.96, 0.26] (del Pino et al., 2021). The
artificial stars are selected as dark matter particles within 4 kpc of the
center in the plane of the disk, with a height above the disk less than 0.3 of
their radius and a vertical velocity of 15 km s-1or less. The 0.05 Gyr time
mass profile of the 1-3 and $7\times 10^{8}M_{\sun}$ halos are reasonably
approximated with Hernquist spheres with scale radii of 0.35 and 0.55 kpc,
respectively, slightly smaller scale radii than the initial values. The dark
matter halo mass model defines a circular velocity with radius and the energy
and angular momentum of circular orbits with radius. At 1 kpc the circular
velocity in the halo is 23 km s-1. Particles with energies between 0.8 and 1.2
of the circular velocity energy and 0.8 to 1.2 of the circular velocity energy
within $r_{max}$ of 4.0 kpc are selected at an initial time of 0.05 Gyr as an
artificial first approximation to the distribution of dwarf galaxy stars. To
create a flat density profile we select particles with $r_{min}>$ 0.5 (1.0)
kpc, for the 3 and $7\times 10^{8}M_{\sun}$ models, respectively. At the time
of selection the particles are in a tilted, rotating disk of 4.0 kpc extent
and an aspect ratio of about 1:3. The mean rotation of the stars is
counterclockwise around the angular momentum vector, but clockwise when viewed
in the plane of the sky.

Figure 17: The radial (top) and tangential (bottom) velocities with radius for
the M54/Sgr data and the artificial stars, selected with positive plane-of-
the-sky tangential velocities. The central model cluster star particles are
shown as dotted blue lines. The other lines are as in Figure 13. Figure 18:
The mean velocity vectors in the plane of the sky of the artificial dwarf
galaxy particles in the $3\times 10^{8}M_{\sun}$ simulation. The plot is is
comparable to velocity map panels of Figures 10, 11 and 12. The arrow in the
upper left has a length of 30 km s-1.

The projected surface density profile of the n-body stellar cluster (dashed
blue line) is shown in Figure 13. The comparison with the density profile of
the old, metal-poor stars of our Gaia subsample illustrates the tremendous
undersampling of stars in the central region of the star cluster and how the
stars of the cluster overlap with the old, metal poor, stars of the dwarf
galaxy beyond 50 pc. Three halo models, with initial masses of 1, 3 and
7$\times 10^{8}M_{\sun}$, were evolved along with the star cluster, with the
projected surface density of the dark matter halos at the time that matches
the current orbital location are shown as the dashed, dotted and solid gray
lines, respectively. Randomly selecting particles from the entire dark matter
halo will not mimic the Sgr dwarf density profile, which requires that the
fraction of dark matter particles identified as stars decline inward about a
factor of 100 from 1 kpc to 10 pc. The simple identification procedure here
leads to the surface densities shown as purple lines in Figure 13. The
procedure with an initial central hole in the selected stars leads to a
somewhat too flat density profile which could be fixed by allowing some of the
stars in the central region. The main outcome is that the dwarf galaxy stars
are a decreasing fraction of the dark matter towards its center.

In Figure 14 the mean interior density,
$\overline{\rho}(r)=M(r)/(4\pi/3r^{3})$ of the dark matter (without the star
cluster mass) derived from the three independent color subsamples is compared
to the three n-body dark matter mean interior halo densities at the current
location of the M54-Sgr system. Within this limited range of these three
models the initial dark matter halo of $3\times 10^{8}M_{\sun}$ and scale
radius 0.4 kpc gives results closest to the observed values.

The distributions of the artificial dwarf galaxy star particles are shown in
Figure 16 for the simulation with a total halo mass of $3\times
10^{8}M_{\sun}$. The Galactic XZ plane is shown since the orbit is nearly in
this plane. The stream in the $M_{h}=7\times 10^{8}M_{\sun}$ model is slightly
wider, longer, and fuzzier. The lower mass model produces a significantly
narrower stream. A similar plot, with the direction of the X axis reversed, is
shown in Figure 10 of Vasiliev & Belokurov (2020). Their simulations are based
on dark halos with circular velocity curves that peak in the range 2-4 kpc,
whereas here it peaks at around 0.4 kpc.

It is interesting to note that the selection of positive tangential velocities
(projected into the plane of the sky) leads to stream bifurcations, as seen in
the Sagittarius stream (Belokurov et al., 2006; Koposov et al., 2012). The
same selection with negative velocities does not have prominent bifurcations,
confirming the importance of rotation in the dwarf galaxy for their origin
(Ramos et al., 2021).

One of the uses of the artificial dwarf galaxy stars is to identify effective
tidal radii, within which stars remain bound to the dwarf and beyond which
they eventually join the massive tidal stream. The radii of stars relative to
the star cluster at 0.05 Gyr and 2.29 Gyr are displayed in Figure 15. The
artificial star particles that join the stream have a clearly defined minimum
radius, about 0.5 kpc, for $3\times 10^{8}M_{\sun}$ halo. Beyond the minimum
radius an increasing fraction of particles join the stream with radius, with
almost all joining at 2 kpc for the $3\times 10^{8}M_{\sun}$ halo. The absence
of a crisp tidal radius is a consequence of the elliptical Galactic orbit of
the system and the eccentricity of the particle orbits.

The velocities and positions of the artificial dwarf galaxy stars are
projected onto the sky as seen from the location of the Sun in the Milky Way
at time 2.29 Gyr for the two halo simulations. The velocities are measured in
the same way as for the observational data. The resulting, radially averaged
velocity fields are shown in Figure 17. Figure 17 shows the mean radial
velocities in the top panel and the corresponding model measurements. The
positive bump in the measured mean radial velocity of the blue stars has a
large uncertainty but is also seen at a somewhat larger radius in the n-body
model star cluster particles, where it is likely a result of the Galactic
tidal field for the system which passed pericenter 0.04 Gyr before the time
being analyzed. Figure 17 shows the mean tangential velocities in the bottom
panel. The $3\times 10^{8}M_{\sun}$ model has tangential velocities closer to
the observed values over the 100-1000 pc range, leading us to favor it as the
better velocity model inside 1 kpc.

Beyond 1 kpc the strong shear seen in Sgr shows up as the outer tangential
velocity decline in Figure 17, which is not present in the projection of any
of the models. The artificial disk star velocities plotted over the face of
system shown in Figure 18 are similar in the inner 1-2 kpc but not a good fit
to the velocities of Figures 10, 11 and 12 beyond 2 kpc. The model system has
a smaller shear velocity from top to bottom than observed in Sagittarius.

The model system has a large velocity gradient along the line-of-sight
velocity at the current orbital phase as can be seen in Figure 16, where the
segment of the stream below M54 is moving nearly horizontally away from the
galactic center, whereas the part of the system at the top, closer to the
plane, is turning upward with a reduced line of sight velocity. To increase
the shear in the model one solution would be to rotate the orbit about 5°, to
project more of the strong velocity shear into the plane of the sky. Another
possibility is that the cylindrically symmetric potential used here does not
allow for the Galactic bar, although the asymmetry of the potential due to the
bar will be small at a pericenter distance of 14 kpc.

##  7 Alternative Futures Evolution

Figure 19: Top: The 3D velocity dispersion at 10 Gyr for M54’s orbit (solid)
and the orbit started at apocenter with 1/2 (dashed) and 1/4 (dotted) of the
angular momentum. The black line is for the dark matter, the blue for the
cluster stars. Bottom: the density profiles for the same models with the same
line designation.

Our radial velocity dispersion measurements, their population density slopes,
and $\beta$ values lead to the mass estimates shown in Figure 6. Our Jeans
equation mass analysis finds 1.4, 1.9 and 1.1$\times 10^{6}M_{\sun}$ within 30
pc, for the blue, green and red samples, respectively. The Jeans masses are
essentially consistent with the cluster’s stellar mass of $1.41-1.78\times
10^{6}M_{\sun}$ (Baumgardt & Hilker, 2018). At 95 pc the Jeans analysis masses
are 6.3, 5.9 and 3.7$\times 10^{6}M_{\sun}$, and at 300 pc the masses are
15.9, 18.2, and 13.8$\times 10^{6}M_{\sun}$, for the blue, green and red
samples, respectively. The implied rise of mass with radius is $r^{1.20}$
between 30 and 95 pc and $r^{1.08}$ between 95 and 300 pc, with a total
increase in mass a factor of 10.9 between 30 and 300 pc. The mean density of
the dark matter is close to what is expected for a CDM dark matter halo of
around $3\times 10^{8}M_{\sun}$ initiated at the apocenter of the frictionless
M54 orbit 2.29 Gyr ago.

The mass and extent of the remnant globular cluster dark matter halo depends
on how long ago the system fell into the Milky Way and how close the orbit
comes to the center of the galaxy, where the tidal field dynamically heats the
system’s particles and pulls them away in a tidal stream. Over the 2.29 Gyr of
evolution the models lose about 16% of the mass within 1 kpc to tidal fields.
It is not unlikely that systems like Sagittarius with a massive globular
cluster near their centers fell into the Milky Way much earlier when the bulk
of the dark matter mass was built up, which will lead to more tidal mass loss.
Simply running the simulation to 10 Gyr we find that the $3\times
10^{8}M_{\sun}$ halo loses 38% of its mass inside 1 kpc, but only 9% inside
0.1 kpc. Had the orbit come closer to the Galactic center even more dark
matter mass would have been lost. Two further simulations start at the same
apocenter as used for M54, but reduce the initial orbital velocities so that
the system has 1/2 and 1/4 of the angular momentum. The simulations use the
same static Galactic potential with no allowance for dynamical friction. The
resulting star and dark matter velocity and density dispersion profiles are
shown in Figure 19. As expected, smaller pericenter passages remove more dark
matter more quickly. The velocity dispersion profile of the system on the 1/4
angular momentum orbit is always declining to 100 pc. Tidal fields remove 90%
of the mass inside 1 kpc within the first 2 Gyr. If gas were present, it is
likely that ram pressure would remove much of the gas as well (Tepper-García &
Bland-Hawthorn, 2018). Therefore, it appears that if M54 had fallen in earlier
on a smaller pericenter orbit it would have formed few if any surrounding
dwarf galaxy stars and lost so much of its dark halo that it would be observed
today as a completely old globular cluster with no detectable dark halo. On
the other hand, if M54’s and the surrounding halo was still in the galactic
outskirts, it could have grown into a substantial dwarf galaxy and the central
star cluster could well have grown into a much larger nuclear star cluster,
possibly even into one containing a detectable black hole. In that sense, M54
can be considered a “transition” object.

##  8 Discussion and Conclusions

Our photometric selection using isochrones adjusted to the Gaia data separates
the stars into blue, red (metal rich and younger) and an intermediate, “green”
population. Each population is kinematically and spatially distinct, although
the red and green populations are similar. A Jeans equation mass analysis
using the three populations finds that the mass inside 95 pc is approximately
3.5 times the stellar mass of the M54 star cluster. Beyond 100 pc the mass
continues to grow approximately linearly with distance. The overall rise of
mass from 30 to 1000 pc is nearly linear, $M\propto r^{1.1}$, implying an
approximately $r^{-2}$ density profile, with a core radius no larger than
30-50 pc. N-body simulations that start 2.29 Gyr in the past with a star
cluster in a Hernquist sphere of masses $1-3\times 10^{8}M_{\sun}$ and a scale
radius of 400 pc provide a good match to most of the properties of the system.
The system may have been much more massive when it first encountered the Milky
Way, before it spiraled into the current orbit.

Most of the Milky Way halo globular clusters have no significant dark matter
detected in their outskirts. However, $\sim$10% of the accessible halo
clusters have been tentatively identified as being consistent with having a
local dark matter halo (Carlberg & Grillmair, 2021). Simulations show that
approximately 25-35% of clusters started in the centers of local halos retain
significant dark matter as the Galactic halo assembles (Carlberg & Keating,
2021). The M54-Sagittarius system appears to be an intermediary system that
would have become a standard globular cluster had it fallen in earlier and its
orbit taken it into regions of somewhat stronger tides. On the other hand, if
the system had stayed in the outskirts the surrounding dwarf galaxy would
likely have continued to form stars to become a more prominent dwarf galaxy
today.

The Sagittarius galaxy and its stream contains 8 other high confidence
globular clusters (Da Costa & Armandroff, 1995; Bellazzini et al., 2020)
including the distant, high mass NGC 2419 cluster (Ibata et al., 2013). Four
of the eight are old, metal poor clusters, the other 4 are more metal rich and
younger, along the lines of the disk and halo clusters of the Milky Way
(Searle & Zinn, 1978). An additional 10-20 lower luminosity clusters are
proposed members of Sagittarius (Minniti et al., 2021). If the Milky Way is
assembled out of systems similar to Sagittarius, then its one cluster (M54) in
a dark halo of the nine luminous clusters somewhat weakly bolsters the idea
that roughly 10% of disk galaxy clusters retain pre-galactic dark matter
halos. If the M54/Sagittarius system is a late-time accretion event that is
representative of earlier ones (Massari et al., 2019; Malhan et al., 2022),
then the globular clusters in those accretion events may still have one or two
clusters with local dark matter halos orbiting freely.

This research was supported by NSERC of Canada.

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# A note on the Frobenius-Euler Numbers and polynomials associated with
Bernstein polynomials

Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>and  Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>
###### Abstract.

The present paper deals with Bernstein polynomials and Frobenius-Euler numbers
and polynomials. We apply the method of generating function and fermionic
$p$-adic integral representation on $\mathbb{Z}_{p}$, which are exploited to
derive further classes of Bernstein polynomials and Frobenius-Euler numbers
and polynomials. To be more precise we summarize our results as follows, we
obtain some combinatorial relations between Frobenius-Euler numbers and
polynomials. Furthermore, we derive an integral representation of Bernstein
polynomials of degree $n$ on $\mathbb{Z}_{p}$. Also we deduce a fermionic
$p$-adic integral representation of product Bernstein polynomials of different
degrees $n_{1},n_{2},\cdots$ on $\mathbb{Z}_{p}$ and show that it can be
written with Frobenius-Euler numbers which yields a deeper insight into the
effectiveness of this type of generalizations. Our applications possess a
number of interesting properties which we state in this paper

###### Key words and phrases:

Frobenius-Euler numbers and polynomials, Bernstein polynomials, fermionic
$p$-adic integral on $\mathbb{Z}_{p}$.

###### 2000 Mathematics Subject Classification:

05A10, 11B65, 28B99, 11B68, 11B73.

##  1\. Introduction and Notations

Let $p$ be a fixed odd prime number. Throughout this paper we use the
following notations. By $\mathbb{Z}_{p}$ we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$. The $p$-adic absolute
value is defined by

$\left|p\right|_{p}=\frac{1}{p}\text{.}$

In this paper, we assume $\left|q-1\right|_{p}<1$ as an indeterminate. In
[17-19], let $UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly
differentiable functions on $\mathbb{Z}_{p}$. For $f\in
UD\left(\mathbb{Z}_{p}\right)$, the fermionic $p$-adic integral on
$\mathbb{Z}_{p}$ is defined by T. Kim:

(1.1)
$I_{-1}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{-1}\left(\xi\right)=\lim_{N\rightarrow\infty}\sum_{\xi=0}^{p^{N}-1}f\left(\xi\right)\left(-1\right)^{\xi}\text{.}$

From (1.1), we have well known the following equality:

(1.2) $I_{-1}\left(f_{1}\right)+I_{-1}\left(f\right)=2f\left(0\right)$

here $f_{1}\left(x\right):=f\left(x+1\right)$ (for details, see[3-24]).

Let $C\left(\left[0,1\right]\right)$ be the space of continuous functions on
$\left[0,1\right]$. For $C\left(\left[0,1\right]\right)$, the Bernstein
operator for $f$ is defined by

$B_{n}\left(f,x\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)B_{k,n}\left(x\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)\binom{n}{k}x^{k}\left(1-x\right)^{n-k}$

where $n,$ $k\in\mathbb{Z}_{+}:=\left\\{0,1,2,3,...\right\\}$. Here
$B_{k,n}\left(x\right)$ is called Bernstein polynomials, which are defined by

(1.3) $B_{k,n}\left(x\right)=\binom{n}{k}x^{k}\left(1-x\right)^{n-k}\text{,
}x\in\left[0,1\right]$

(for more informations on this subject, see [1-6, 11, 14, 15, 17, 21-24])

In [7], as is well known, Frobenius-Euler polynomials are defined by means of
the following generating function:

(1.4)
$\sum_{n=0}^{\infty}H_{n}\left(u,x\right)\frac{t^{n}}{n!}=e^{H\left(u,x\right)t}=\frac{1-u}{e^{t}-u}e^{xt}\text{.}$

where the usual convention about replacing $H^{n}\left(u,x\right)$ by
$H_{n}\left(u,x\right)$. For $x=0$ in (1.4), we have to
$H_{n}\left(u,0\right):=H_{n}\left(u\right)$, which is called Frobenius-Euler
numbers. Then, we can write the following

(1.5)
$e^{H\left(u\right)t}=\sum_{n=0}^{\infty}H_{n}\left(u\right)\frac{t^{n}}{n!}=\frac{1-u}{e^{t}-u}\text{.}$

By (1.4) and (1.5), we easily see the following applications:

$\displaystyle e^{\left(H\left(u\right)+1\right)t}-ue^{H\left(u\right)t}$
$\displaystyle=$ $\displaystyle 1-u$
$\displaystyle\sum_{n=0}^{\infty}\left[\left(H\left(u\right)+1\right)^{n}-uH_{n}\left(u\right)\right]\frac{t^{n}}{n!}$
$\displaystyle=$ $\displaystyle 1-u$

After these applications, we derive the following Lemma.

######  Lemma 1.

For $\left|u\right|>1$ and
$n\in\mathbb{Z}_{+}:=\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$,
we have

(1.6)
$\left(H\left(u\right)+1\right)^{n}-uH_{n}\left(u\right)=\left\\{\QATOP{1-u\text{,
if }n=0}{0\text{, \ \ \ \ \ \ if }n\neq 0.}\right.$

In this paper, we obtained some relations between the Frobenius-Euler numbers
and polynomials and the Bernstein polynomials. From these relations, we derive
some interesting identities on the Frobenius-Euler numbers.

##  2\. On the Frobenius-Euler numbers and polynomials

Let us take $f\left(x\right)=u^{x}e^{tx}$ in (1.1), by (1.2), we see that

(2.1) $\int_{\mathbb{Z}_{p}}u^{\eta}e^{\eta
t}d\mu_{-1}\left(\eta\right)=\frac{2}{1+u}H_{n}\left(-u^{-1}\right)\text{.}$

By (1.4) and (2.1), we have the following theorem.

######  Theorem 1.

(2.2)
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(x+\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}H_{n}\left(-u^{-1},x\right)\text{.}$

By applying some combinatorial techniques in (2.2), we derive the following

$\int_{\mathbb{Z}_{p}}u^{\eta}\left(x+\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}\left\\{\int_{\mathbb{Z}_{p}}u^{\eta}\eta^{k}d\mu_{-1}\left(\eta\right)\right\\}\text{.}$

So, from above, we have the well known identity

(2.3)
$H_{n}\left(-u^{-1},x\right)=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}H_{k}\left(-u^{-1}\right)=\left(H\left(-u^{-1}\right)+x\right)^{n}\text{.}$

by using the $umbral$(symbolic) convention
$H^{n}\left(u\right):=H_{n}\left(u\right)$.

The Frobenius-Euler polynomials have to symmetric properties, which is shown
by Choi $et$ $al$. in [7], as follows:

$H_{n}\left(-u^{-1},1-x\right)=\left(-1\right)^{n}H_{n}\left(-u^{-1},x\right)\text{.}$

For $n\in\mathbb{N}$, by (2.3), Choi $et$ $al$. derived the following
equality:

(2.4)
$u^{2}H_{n}\left(-u^{-1},2\right)=u^{2}+u+H_{n}\left(-u^{-1}\right)\text{.}$

From (2.2) and (2.4), we easily see that

$\displaystyle\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\int_{\mathbb{Z}_{p}}u^{\eta}\left(\eta-1\right)^{n}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\left(-1\right)^{n}H_{n}\left(-u^{-1},-1\right)$
$\displaystyle=$ $\displaystyle\frac{2}{u+1}H_{n}\left(-u^{-1},2\right).$

Thus, we obtain the following Theorem.

######  Theorem 2.

The following identity

(2.6)
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}H_{n}\left(-u^{-1},2\right)$

is true.

Let $n\in\mathbb{N}$. By expression of (2.4) and (2.6), we get

(2.7)
$\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n}\left(-u^{-1}\right).$

From (2.7), we procure the following corollary.

######  Corollary 1.

For $n\in\mathbb{N}$, we have

$\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n}d\mu_{-1}\left(\eta\right)=\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n}\left(-u^{-1}\right).$

##  3\. Some identities on the Frobenius-Euler numbers

In this section, we develop Frobenius-Euler numbers, that is, we derive some
interesting and worthwhile relations for studying in Theory of Analytic
Numbers.

Now also, for $x\in\left[0,1\right]$, we rewrite definition of Bernstein
polynomials as follows:

(3.1) $B_{k,n}\left(x\right)=\binom{n}{k}x^{k}\left(1-x\right)^{n-k}\text{,
where }n,k\in\mathbb{Z}_{+}\text{.}$

By expression of (3.1), we have the properties of symmetry of Bernstein
polynomials as follows:

(3.2) $B_{k,n}\left(x\right)=B_{n-k,n}\left(1-x\right)\text{, (for detail, see
\cite[cite]{[\@@bibref{}{kim 19}{}{}]}).}$

Thus, from Corollary 1, (3.1) and (3.2), we see that

$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}B_{n-k,n}\left(1-\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\int_{\mathbb{Z}_{p}}u^{\eta}\left(1-\eta\right)^{n-l}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n-l}\left(-u^{-1}\right)\right)\text{.}$

For $n$, $k\in\mathbb{Z}_{+}$ with $n>k$, we compute

$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n-l}\left(-u^{-1}\right)\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n-l}\left(-u^{-1}\right)\right),\text{
if }k>0.}\right.$

Let us take the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ on the
Bernstein polynomials of degree $n$ as follows:

$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\int_{\mathbb{Z}_{p}}\eta^{k}\left(1-\eta\right)^{n-k}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\binom{n}{k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}H_{l+k}\left(-u^{-1}\right)\text{.}$

Consequently, by expression of (3) and (3), we state the following Theorem:

######  Theorem 3.

The following identity holds true:

$\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}H_{l+k}\left(-u^{-1}\right)=\left\\{\QATOPD..{1+u^{-1}+u^{-2}H_{n}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0\text{,}}{\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(1+u^{-1}+u^{-2}H_{n-l}\left(-u^{-1}\right)\right),\text{
if }k>0\text{.}}\right.$

Let $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k$. Then, we derive
the followings

$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\int_{\mathbb{Z}_{p}}\left(1-\eta\right)^{n_{1}+n_{2}-l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left(\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}}\left(-u^{-1}\right)\text{,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ if
}k=0,}{\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right),\text{
\ if }k\neq 0.}\right.$

Therefore, we obtain the following Theorem:

######  Theorem 4.

For $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k,$ we have

$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}}\left(-u^{-1}\right)\text{,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ if
}k=0,}{\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right),\text{
\ if }k\neq 0.}\right.$

By using the binomial theorem, we can derive the following equation.

$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}\eta^{2k+l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}H_{2k+l}\left(-u^{-1}\right).$

Thus, we can obtain the following Corollary:

######  Corollary 2.

For $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k,$ we have

$\displaystyle\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}H_{2k+l}\left(-u^{-1}\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{1+u^{-1}+u^{-2}H_{n_{1}+n_{2}}\left(-u^{-1}\right)\text{,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(1+u^{-1}+u^{-2}H_{n_{1}+n_{2}-l}\left(-u^{-1}\right)\right),\text{
\ if }k\neq 0\text{.}}\right.$

For $\eta\in\mathbb{Z}_{p}$ and $s\in\mathbb{N}$ with $s\geq 2,$ let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk$. Then
we take the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ for the
Bernstein polynomials of degree $n$ as follows:

$\displaystyle\int_{\mathbb{Z}_{p}}\underset{s-times}{\underbrace{B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)...B_{k,n_{s}}\left(\eta\right)}}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\int_{\mathbb{Z}_{p}}\eta^{sk}\left(1-\eta\right)^{n_{1}+n_{2}+...+n_{s}-sk}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{l+sk}\int_{\mathbb{Z}_{p}}\left(1-\xi\right)^{n_{1}+n_{2}+...+n_{s}-l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ if
}k=0,}{\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}-l}\left(-u^{-1}\right)\right),\text{
\ \ if }k\neq 0.}\right.$

So from above, we have the following Theorem:

######  Theorem 5.

For $s\in\mathbb{N}$ with $s\geq 2$, let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk$. Then
we have

$\displaystyle\int_{\mathbb{Z}_{p}}u^{\eta}\mathop{\displaystyle\prod}\limits_{i=1}^{s}B_{k,n_{i}}\left(\eta\right)u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ if
}k=0,}{\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(\frac{2}{u+1}+\frac{2}{u^{2}+u}+\frac{2}{u^{3}+u}H_{n_{1}+n_{2}+...+n_{s}-l}\left(-u^{-1}\right)\right),\text{
\ \ if }k\neq 0.}\right.$

From the definition of Bernstein polynomials and the binomial theorem, we
easily get

(3.6)
$\displaystyle\int_{\mathbb{Z}_{p}}\underset{s-times}{\underbrace{B_{k,n_{1}}\left(\eta\right)B_{k,n_{2}}\left(\eta\right)...B_{k,n_{s}}\left(\eta\right)}}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}\xi^{sk+l}u^{\eta}d\mu_{-1}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\frac{2}{u+1}\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}H_{sk+l}\left(-u^{-1}\right).$

Therefore, by (3.6), we get novel properties of Frobenius-Euler numbers with
the following corollary:

######  Corollary 3.

For $s\in\mathbb{N}$ with $s\geq 2$, let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk.$
Then, we have

$\displaystyle u^{2}$
$\displaystyle\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}H_{sk+l}\left(-u^{-1}\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{u^{2}+u+H_{n_{1}+n_{2}+...+n_{s}}\left(-u^{-1}\right),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(u^{2}+u+H_{n_{1}+n_{2}+...+n_{s}-l}\left(-u^{-1}\right)\right),\text{
\ \ if }k\neq 0.}\right.$

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