# The Open Cluster Chemical Abundances and Mapping Survey: V.
Chemical Abundances of CTIO/Hydra Clusters using The Cannon

Amy E. Ray  Department of Physics and Astronomy, Texas Christian University,
TCU Box 298840
Fort Worth, TX 76129, USA (a.e.ray, p.frinchaboy, j.donor<EMAIL_ADDRESS>Peter M. Frinchaboy Department of Physics and Astronomy, Texas Christian
University, TCU Box 298840
Fort Worth, TX 76129, USA (a.e.ray, p.frinchaboy, j.donor<EMAIL_ADDRESS>John Donor  Department of Physics and Astronomy, Texas Christian University,
TCU Box 298840
Fort Worth, TX 76129, USA (a.e.ray, p.frinchaboy, j.donor<EMAIL_ADDRESS>S. D. Chojnowski  Department of Astronomy, New Mexico State University, Las
Cruces, NM, 88001, USA  Matthew Melendez  Department of Physics and Astronomy,
Texas Christian University, TCU Box 298840
Fort Worth, TX 76129, USA (a.e.ray, p.frinchaboy, j.donor<EMAIL_ADDRESS>
(Received August 4, 2021; Accepted January 31, 2020)

###### Abstract

Open clusters are key chemical and age tracers of Milky Way evolution. While
open clusters provide significant constraints on galaxy evolution, their use
has been limited due to discrepancies in measuring abundances from different
studies. We analyze medium resolution ($R\sim 19,000$) CTIO/Hydra spectra of
giant stars in 58 open clusters using The Cannon to determine [Fe/H], [Mg/Fe],
[Si/Fe], [Al/Fe], and [O/Fe]. This work adds an additional 55 primarily
southern hemisphere open clusters calibrated to the SDSS/APOGEE DR16
metallicity system. This uniform analysis is compared to previous studies
[Fe/H] measurements for 23 clusters and we present spectroscopic metallicities
for the first time for 35 open clusters.

Open star clusters (1160), Galactic abundances (2002), Milky Way evolution
(1052), Chemical abundances (224)

††journal: AJ††facilities:  Sloan (APOGEE), FLWO:2MASS, Gaia,
CTIO:Hydra††software: Astropy, The Cannon

##  1 Introduction

Open clusters provide reliable ages, a key constraint needed to study Galactic
chemical evolution since their stars formed at the same time out of similar
material. Many studies over the past few decades (e.g., Janes, 1979; Yong et
al., 2005; Bragaglia et al., 2008; Jacobson et al., 2009; Friel et al., 2010;
Carrera & Pancino, 2011; Magrini et al., 2017; Casamiquela et al., 2019; Donor
et al., 2018, 2020) have used open clusters to explore the radial metallicity
gradient in the Galactic disk, which has shown that clusters closer to the
center are generally more metal rich than clusters in the outer Galaxy.

However, there are several problems in using open clusters to study the
Galactic abundance gradient. One is the number of open clusters used in
individual studies. A way to improve the current knowledge in this area is to
increase the number of clusters with known chemical abundances. There are
roughly 2000 known open clusters (Cantat-Gaudin et al., 2020), but only a
small portion of them have been analyzed chemically. Even the ones with
measured abundance values can have substantial uncertainties from study to
study (Yong et al., 2005). A few reasons for such large uncertainties are due
to varying data quality, the type of data, and different data analysis methods
between studies. Another source of uncertainty arises depending on which
catalog each survey chose for open cluster distances, as there are several
that have determined substantially different distance results as discussed in
Donor et al. (2018). This difference translates into widely varying results
when attempting to determine a chemical abundance gradient across the disk of
the Milky Way. Yong et al. (2012) and Donor et al. (2018, 2020) highlight this
problem in their abundance gradient research. Netopil et al. (2016) compiled a
homogenized sample of open cluster abundances, but there are still large
uncertainties due to different types of observations and resolutions.

In this study, we put together a large uniform sample open clusters observed
with CTIO/hydra spectra. For this work we are using The Cannon, developed by
Ness et al. (2015), which offers a unique way to find stellar parameters
without having to use any models. Instead, this machine learning method takes
a subset of stars with known parameters or “labels” and creates a model based
on pixel-to-pixel variations. This model can be applied to the rest of the set
of stars to infer labels for them. Our training set is based on the Apache
Point Observatory Galactic Evolution Experiment Data Release 16 (APOGEE DR16)
system in order to correct the problems with current surveys that were listed
above. This sample trained with stars from Donor et al. (2020, also on the
APOGEE DR16 system) is designed to form a more extensive dataset for Galactic
abundance studies. The sections in this paper are as follows: Section 2
describes APOGEE DR16 and the observations taken for the sample of 58 open
clusters. The Cannon and the training set that we used are described in
Section 3. In Section 4, the results of this study are presented and we
discuss comparison to similar studies, and finally conclusions in Section 5.

##  2 Data and Observations

The primary data for this study comes from optical data near the Calcium
infrared triplet (7745–8730 Å) taken with the Hydra spectrograph using the
Cerro Tololo Inter-American Observatory (CTIO) 4m telescope. Data was observed
on UT 2002 March, 2003 March, 2003 July, and 2003 August. The data reduction
and radial velocity and membership analysis of this data was conducted using
IRAF111IRAF is distributed by the National Optical Astronomy Observatory,
which is operated by the Association of Universities for Research in
Astronomy, Inc., under cooperative agreement with the National Science
Foundation. standard ccdproc, dohydra, and fxcor routines, which is fully
described in Frinchaboy & Majewski (2008).

We have chosen to train The Cannon using data from the Sloan Digital Sky
Survey’s (SDSS; Blanton et al., 2017; Eisenstein et al., 2011) sixteenth data
release (DR16; Ahumada et al., 2020; Holtzman et al., 2018; Jönsson et al.,
2020) taken as part of the APOGEE (APOGEE 2; Majewski et al., 2017). The
APOGEE/DR16 dataset includes about 430,000 stars, collected using two APOGEE
spectrographs (Wilson et al., 2012) at Apache Point Observatory (APO; New
Mexico Gunn et al., 2006) and Las Campanas Observatory (LCO; Chile Bowen &
Vaughan, 1973) . For this study data from LCO provides a key overlap with our
available training data. The APOGEE data reduction pipeline (Nidever et al.,
2015; Holtzman et al., 2018) provides stellar atmospheric parameters and
radial velocity measurements, while elemental abundances are provided from the
ASPCAP pipeline (García Pérez et al., 2016; Holtzman et al., 2018; Jönsson et
al., 2020)

For this study, we have applied The Cannon to derive chemical abundances for
selected available elements, after applying a cut based on the radial velocity
cross-correlation quality Tonry-Davis Ratio ($TDR\geq 11$ Tonry & Davis,
1979). After this cut we were able to use 25 stars from 3 clusters that are
also observed with APOGEE DR16 in the training set. The Galactic distribution
of the clusters in our sample are shown in Figure 1 using distances from
(Cantat-Gaudin et al., 2020) and also showing the positions of the high
quality clusters from (Donor et al., 2020).

Additionally, to verify the results from Frinchaboy & Majewski (2008), we also
used updated proper motion data from Gaia DR2 (Gaia Collaboration et al.,
2018) to re-check membership and found no change in membership between the
Tycho-2 and Gaia-based proper motion selection.

Figure 1: Spatial Galactic distribution of open clusters in this survey color-
coded with ages and $R_{GC}$ distance from Cantat-Gaudin et al. (2020). Also
shown are the location of OCCAM-IV DR16-based cluster from Donor et al. (2020)
also updated with ages and $R_{GC}$ distance from (Cantat-Gaudin et al., 2020,
; red crosses).

In addition to the data collected as part of Frinchaboy & Majewski (2008),
observations of the the Sagittarius dwarf galaxy were also obtained on the
same observing runs with the same instrument setup (Frinchaboy et al., 2012).
Many of these stars were also observed as part of the APOGEE survey and so
these additional 365 spectra are included in The Cannon training set described
below ($TDR\geq 11$). Data reduction and membership is discussed fully in
Frinchaboy et al. (2012), which is the same as described in Frinchaboy &
Majewski (2008). The Cannon analysis of these Sagittarius data will be
presented in Frinchaboy et al. (in prep.).

##  3 The Cannon

Figure 2: A comparison of effective temperature (Teff) and surface gravity
(log $g$) values obtained by The Cannon and values obtained using APOGEE DR16
data. Most of the values fall within the scatter which is illustrated by the
gray shaded areas. Figure 3: A comparison of [Fe/H], [Mg/Fe], [Si/Fe],
[Al/Fe], and [O/Fe] values obtained by The Cannon and values obtained using
APOGEE DR16 data. The shaded areas represent the acceptable scatter as in 2.

###  3.1 Background

Many high resolutions abundance studies use the curve of growth, the Boltzmann
and Saha equations, and stellar models to determine chemical abundances.
However, for this medium resolution, lower-S/N data from CTIO Hydra, we have
chosen to use The Cannon (Ness et al., 2015) which has been shown to work well
for these types of data. The models, however, usually do not account for many
distinctive factors in a star’s atmosphere which means research groups can get
opposing values for the same star (Ness et al., 2015). For this study, the set
of parameters used were Teff, $\log g$, [Fe/H], [Si/Fe], [Al/Fe], [Mg/Fe], and
[O/Fe] following the methodology from Ho et al. (2017).

###  3.2 Training Set

The training set was comprised of 390 stars in total with 25 stars from open
clusters and 365 from the Sagittarius dwarf galaxy. When this set was used to
train The Cannon, a check was performed to determine if the output values were
accurate. We first validated the output of the model by training The Cannon
using 90% of the training sample, then deriving labels for the remaining 10%.
We conducted this 10 times and analyze how the input labels compare to the
output labels from this cross-validation test.

Figure 2 illustrates the full training set one-to-one plots for Teff and $\log
g$, and Figure 3 shows similar plots for [Fe/H], [Si/Fe], [Al/Fe], [Mg/Fe],
and [O/Fe]. Values obtained from The Cannon for the reference set are listed
in Table 1 along with the values from APOGEE DR16. We find insignificant
offset from DR16 for our parameters (Table 2), except maybe Si, but due to the
lower signal-to-noise (S/N) for training set stars in the study, we obtain
less accurate parameters in comparison to some larger studies (e.g., Ho et
al., 2017; Hasselquist et al., 2020). Due to the small offset, and large
scatter, we find that the errors are dominated by the scatter from low-S/N
which are significantly larger that the uncertainties from the APOGEE DR1
survey input labels.

Table 1: APOGEE DR16 Training Set Stars

2MASS ID | S/N | Temp |  $\log$ g | [Fe/H] | [Si/Fe] | [Al/Fe] | [Mg/Fe] | [O/Fe]
---|---|---|---|---|---|---|---|---
| (pixel-1) | (K) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex)
2M18503032-2855221 | 9.2 | 3843 |  $-$0.4391 |  $-$0.8314 |  $-$0.2329 |  $-$0.5243 |  $-$0.1327 |  $-$0.1655
2M19040855-3103228 | 4.6 | 3523 |  $-$0.2544 |  $-$0.5450 |  $-$0.2667 |  $-$0.4611 |  $-$0.0940 |  $-$0.1646
2M18472966-2957462 | 7.7 | 3584 |  $-$0.2492 |  $-$0.5612 |  $-$0.1955 |  $-$0.3655 |  $-$0.0364 |  $-$0.0801
2M18554328-2937095 | 5.7 | 3558 |  $-$0.2422 |  $-$0.6150 |  $-$0.1624 |  $-$0.4713 |  $+$0.0011 |  $-$0.1635
2M18492810-2841275 | 9.0 | 3669 |  $-$0.2399 |  $-$0.6806 |  $-$0.1299 |  $-$0.5335 |  $-$0.0565 |  $-$0.0406
2M18514730-2857148 | 9.5 | 3570 |  $-$0.2336 |  $-$0.5483 |  $-$0.1026 |  $-$0.3219 |  $-$0.0369 |  $-$0.0277
2M18585375-3151470 | 7.3 | 3905 |  $-$0.1646 |  $-$0.7033 |  $-$0.1113 |  $-$0.4253 |  $+$0.0892 |  $-$0.0574
2M18470776-3017243 | 5.6 | 3685 |  $-$0.0545 |  $-$0.5985 |  $-$0.1252 |  $-$0.5417 |  $+$0.0497 |  $-$0.0252
2M18355640-2917489 | 5.8 | 3631 |  $-$0.0424 |  $-$0.7777 |  $-$0.1671 |  $-$0.4961 |  $-$0.0492 |  $-$0.0848
2M18575903-3203421 | 10.2 | 3678 |  $+$0.0109 |  $-$0.2958 |  $-$0.1947 |  $-$0.4098 |  $-$0.1184 |  $-$0.1434
$\cdots$ |

Note. — This table is available in its entirety in machine-readable form in
the online journal. A portion is shown here for guidance regarding its form
and content.

Table 2: APOGEE DR16/The Cannon Comparison for the Training Set  Parameter | Mean Offset | Std. Dev.
---|---|---
$T_{eff}$ |  $+$17 K | 117 K
$\log$ g |  $+$0.069 dex |  0.32 dex
[Fe/H] |  $+$0.001 dex | 0.27 dex
[Si/Fe] |  $+$0.010 dex | 0.09 dex
[Al/Fe] |  $+$0.004 dex | 0.16 dex
[Mg/Fe] |  $-$0.004 dex | 0.10 dex
[O/Fe] |  $+$0.004 dex | 0.07 dex

##  4 Results & Discussion

Our resultant sample in this study consists of 58 open clusters with 237
member stars of which 35 have no previous spectroscopically determined
metallicity measurements. The individual stellar abundance measurements for Fe
as well as the $\alpha$-elements magnesium, silicon, and oxygen, plus the
odd-$Z$ element aluminum are presented in Table 3. Signal-to-noise (S/N) for
all cluster stars is computed at 8000 Å. We find that all of the elements
scale relative to iron, in solar abundance ratios, as expected for clusters
nearby the Sun and are within the normal of solar neighborhood cluster mean
abundances as seen in Donor et al. (2020). The resultant membership and
clusters averages are for all clusters studied are presented in Table 4, which
are calculated as in (Donor et al., 2020).

Table 3: The Cannon results star results for all clusters

Cluster | 2MASS ID | S/N | Temp |  $\log$ g | [Fe/H] | [Si/Fe] | [Al/Fe] | [Mg/Fe] | [O/Fe]
---|---|---|---|---|---|---|---|---|---
name |  | (pixel-1) | (K) | (dex) | (dex) | (dex) | (dex) | (dex) | (dex)
Collinder 205 | 09001428-4901040 | 47.4 | 5174 | 3.45 |  $-$0.22 |  $+$0.22 |  $+$0.27 |  $+$0.20 |  $+$0.12
Collinder 205 | 09003203-4858199 | 28.0 | 4975 | 2.75 |  $-$0.45 |  $+$0.13 |  $+$0.21 |  $+$0.03 |  $+$0.16
Collinder 258 | 12271397-6043435 | 14.5 | 4896 | 2.59 |  $-$0.37 |  $+$0.09 |  $+$0.16 |  $-$0.03 |  $+$0.11
Collinder 258 | 12265230-6047561 | 20.1 | 5157 | 3.53 |  $-$0.17 |  $+$0.22 |  $+$0.28 |  $+$0.20 |  $+$0.12
Collinder 258 | 12291805-6031347 | 70.1 | 5213 | 3.88 |  $-$0.11 |  $+$0.29 |  $+$0.38 |  $+$0.25 |  $+$0.15
Harvard 10 | 16183313-5454573 | 47.6 | 4890 | 2.53 |  $-$0.30 |  $+$0.05 |  $+$0.12 |  $+$0.02 |  $+$0.09
Harvard 10 | 16183125-5457397 | 40.3 | 5138 | 3.28 |  $-$0.18 |  $+$0.21 |  $+$0.26 |  $+$0.19 |  $+$0.11
Harvard 10 | 16200896-5452364 | 21.3 | 4949 | 2.57 |  $-$0.44 |  $+$0.14 |  $+$0.25 |  $+$0.03 |  $+$0.14
Harvard 10 | 16195235-5503168 | 21.8 | 4933 | 2.68 |  $-$0.29 |  $+$0.05 |  $+$0.09 |  $+$0.02 |  $+$0.08
Harvard 10 | 16183216-5502259 | 44.9 | 5138 | 3.31 |  $-$0.18 |  $+$0.21 |  $+$0.25 |  $+$0.19 |  $+$0.11
$\cdots$ |

Note. — This table is available in its entirety in machine-readable form in
the online journal. A portion is shown here for guidance regarding its form
and content.

Table 4: The Cannon -based cluster parameters.  Cluster | Num |  $\log(Age)$aaCluster ages taken from Cantat-Gaudin et al. (2020) | [Fe/H] | [Si/Fe] | [Al/Fe] | [Mg/Fe] | [O/Fe] | New? |
---|---|---|---|---|---|---|---|---|---
name | Stars | (yrs) | (dex) | (dex) | (dex) | (dex) | (dex) |  |
Collinder 205 | 2 | 6.66 |  $-$0.07 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.04 $\pm$ 0.11 |  $-$0.03 $\pm$ 0.07 |  $+$0.00 $\pm$ 0.05 | Y |
Collinder 258 | 2 | 7.99 |  $-$0.03 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.03 $\pm$ 0.11 |  $-$0.03 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.05 | Y |
Harvard 10 | 4 | 8.30 |  $-$0.04 $\pm$ 0.14 |  $+$0.02 $\pm$ 0.05 |  $+$0.03 $\pm$ 0.08 |  $-$0.03 $\pm$ 0.05 |  $+$0.00 $\pm$ 0.04 | N |
IC 2395 | 1 | 7.31 |  $+$0.05 $\pm$ 0.27 |  $+$0.01 $\pm$ 0.09 |  $+$0.01 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.10 |  $+$0.00 $\pm$ 0.07 | N |
IC 2488 | 3 | 8.21 |  $+$0.06 $\pm$ 0.16 |  $+$0.02 $\pm$ 0.05 |  $+$0.00 $\pm$ 0.09 |  $-$0.01 $\pm$ 0.06 |  $+$0.00 $\pm$ 0.04 | N |
IC 2581 | 1 | 7.01 |  $-$0.10 $\pm$ 0.27 |  $-$0.01 $\pm$ 0.09 |  $+$0.01 $\pm$ 0.16 |  $-$0.05 $\pm$ 0.10 |  $-$0.01 $\pm$ 0.07 | N |
IC 4651 | 9 | 9.22 |  $+$0.04 $\pm$ 0.09 |  $+$0.00 $\pm$ 0.03 |  $+$0.01 $\pm$ 0.05 |  $+$0.01 $\pm$ 0.03 |  $+$0.00 $\pm$ 0.02 | N |
IC 4756 | 3 | 9.11 |  $-$0.05 $\pm$ 0.16 |  $+$0.06 $\pm$ 0.05 |  $-$0.15 $\pm$ 0.09 |  $+$0.02 $\pm$ 0.06 |  $+$0.01 $\pm$ 0.04 | N |
Lynga 1 | 2 | 8.22 |  $-$0.01 $\pm$ 0.19 |  $+$0.01 $\pm$ 0.06 |  $+$0.02 $\pm$ 0.11 |  $-$0.04 $\pm$ 0.07 |  $+$0.00 $\pm$ 0.05 | N |
Lynga 2 | 3 | 8.01 |  $+$0.06 $\pm$ 0.16 |  $+$0.03 $\pm$ 0.05 |  $+$0.03 $\pm$ 0.09 |  $+$0.01 $\pm$ 0.06 |  $+$0.02 $\pm$ 0.04 | N |
NGC 1662 | 3 | 8.89 |  $+$0.04 $\pm$ 0.16 |  $+$0.06 $\pm$ 0.05 |  $-$0.07 $\pm$ 0.09 |  $+$0.05 $\pm$ 0.06 |  $+$0.04 $\pm$ 0.04 | N |
NGC 2215 | 7 | 8.84 |  $-$0.07 $\pm$ 0.10 |  $+$0.02 $\pm$ 0.03 |  $+$0.03 $\pm$ 0.06 |  $-$0.02 $\pm$ 0.04 |  $+$0.01 $\pm$ 0.03 | N |
NGC 2301 | 3 | 8.33 |  $+$0.09 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.05 |  $+$0.02 $\pm$ 0.09 |  $+$0.01 $\pm$ 0.06 |  $+$0.01 $\pm$ 0.04 | N |
NGC 2323 | 3 | 8.12 |  $+$0.11 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.05 |  $+$0.05 $\pm$ 0.09 |  $+$0.03 $\pm$ 0.06 |  $+$0.02 $\pm$ 0.04 | N |
NGC 2353 | 5 | 8.01 |  $-$0.16 $\pm$ 0.12 |  $+$0.03 $\pm$ 0.04 |  $+$0.04 $\pm$ 0.07 |  $-$0.03 $\pm$ 0.04 |  $+$0.02 $\pm$ 0.03 | N |
NGC 2354 | 10 | 9.15 |  $-$0.12 $\pm$ 0.09 |  $+$0.02 $\pm$ 0.03 |  $+$0.03 $\pm$ 0.05 |  $-$0.03 $\pm$ 0.03 |  $+$0.00 $\pm$ 0.02 | N |
NGC 2423 | 8 | 9.04 |  $+$0.10 $\pm$ 0.10 |  $-$0.01 $\pm$ 0.03 |  $-$0.01 $\pm$ 0.06 |  $+$0.00 $\pm$ 0.04 |  $-$0.02 $\pm$ 0.02 | N |
NGC 2437 | 3 | 8.48 |  $-$0.08 $\pm$ 0.16 |  $+$0.02 $\pm$ 0.05 |  $+$0.02 $\pm$ 0.09 |  $-$0.02 $\pm$ 0.06 |  $-$0.01 $\pm$ 0.04 | Y |
NGC 2447 | 17 | 8.76 |  $-$0.06 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.02 |  $+$0.02 $\pm$ 0.04 |  $-$0.03 $\pm$ 0.02 |  $-$0.02 $\pm$ 0.02 | T |
NGC 2482 | 5 | 8.54 |  $+$0.07 $\pm$ 0.12 |  $-$0.01 $\pm$ 0.04 |  $-$0.01 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.04 |  $-$0.01 $\pm$ 0.03 | N |
NGC 2516 | 3 | 8.38 |  $-$0.09 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.05 |  $+$0.05 $\pm$ 0.09 |  $-$0.04 $\pm$ 0.06 |  $-$0.01 $\pm$ 0.04 | N |
NGC 2527 | 3 | 8.84 |  $-$0.11 $\pm$ 0.16 |  $+$0.02 $\pm$ 0.05 |  $+$0.03 $\pm$ 0.09 |  $-$0.04 $\pm$ 0.06 |  $+$0.00 $\pm$ 0.04 | N |
NGC 2539 | 9 | 8.84 |  $-$0.08 $\pm$ 0.09 |  $+$0.02 $\pm$ 0.03 |  $+$0.02 $\pm$ 0.05 |  $-$0.03 $\pm$ 0.03 |  $-$0.01 $\pm$ 0.02 | N |
NGC 2546 | 1 | 8.15 |  $+$0.06 $\pm$ 0.27 |  $+$0.01 $\pm$ 0.09 |  $+$0.04 $\pm$ 0.16 |  $+$0.04 $\pm$ 0.10 |  $+$0.02 $\pm$ 0.07 | Y |
NGC 2547 | 3 | 7.51 |  $-$0.05 $\pm$ 0.16 |  $+$0.02 $\pm$ 0.05 |  $+$0.03 $\pm$ 0.09 |  $-$0.02 $\pm$ 0.06 |  $-$0.01 $\pm$ 0.04 | N |
NGC 2548 | 3 | 8.59 |  $+$0.12 $\pm$ 0.16 |  $-$0.01 $\pm$ 0.05 |  $+$0.00 $\pm$ 0.09 |  $+$0.00 $\pm$ 0.06 |  $-$0.02 $\pm$ 0.04 | N |
NGC 2567 | 6 | 8.50 |  $-$0.07 $\pm$ 0.11 |  $+$0.01 $\pm$ 0.04 |  $+$0.02 $\pm$ 0.07 |  $-$0.03 $\pm$ 0.04 |  $-$0.01 $\pm$ 0.03 | N |
NGC 2579 | 2 |  $\cdots$ |  $-$0.08 $\pm$ 0.19 |  $+$0.01 $\pm$ 0.06 |  $+$0.05 $\pm$ 0.11 |  $-$0.03 $\pm$ 0.07 |  $-$0.02 $\pm$ 0.05 | Y |
NGC 2669 | 1 | 8.13 |  $-$0.06 $\pm$ 0.27 |  $+$0.00 $\pm$ 0.09 |  $+$0.03 $\pm$ 0.16 |  $-$0.04 $\pm$ 0.10 |  $-$0.03 $\pm$ 0.07 | Y |
NGC 2670 | 2 | 8.01 |  $-$0.05 $\pm$ 0.19 |  $+$0.03 $\pm$ 0.06 |  $+$0.03 $\pm$ 0.11 |  $-$0.02 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.05 | N |
NGC 2682 | 10 | 9.63 |  $+$0.09 $\pm$ 0.09 |  $-$0.02 $\pm$ 0.03 |  $-$0.01 $\pm$ 0.05 |  $+$0.00 $\pm$ 0.03 |  $-$0.02 $\pm$ 0.02 | T |
NGC 2925 | 1 | 8.11 |  $+$0.10 $\pm$ 0.27 |  $+$0.02 $\pm$ 0.09 |  $+$0.03 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.10 |  $+$0.01 $\pm$ 0.07 | N |
NGC 3680 | 6 | 9.34 |  $-$0.05 $\pm$ 0.11 |  $-$0.01 $\pm$ 0.04 |  $+$0.01 $\pm$ 0.07 |  $-$0.03 $\pm$ 0.04 |  $-$0.02 $\pm$ 0.03 | N |
NGC 5138 | 2 | 7.68 |  $-$0.01 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.03 $\pm$ 0.11 |  $-$0.01 $\pm$ 0.07 |  $+$0.00 $\pm$ 0.05 | N |
NGC 5281 | 7 | 7.60 |  $-$0.02 $\pm$ 0.10 |  $+$0.01 $\pm$ 0.03 |  $+$0.02 $\pm$ 0.06 |  $-$0.04 $\pm$ 0.04 |  $-$0.01 $\pm$ 0.03 | Y |
NGC 5316 | 4 | 8.22 |  $-$0.01 $\pm$ 0.14 |  $+$0.01 $\pm$ 0.05 |  $+$0.03 $\pm$ 0.08 |  $-$0.03 $\pm$ 0.05 |  $-$0.02 $\pm$ 0.04 | N |
NGC 5460 | 2 | 8.20 |  $-$0.03 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.03 $\pm$ 0.11 |  $-$0.02 $\pm$ 0.07 |  $-$0.01 $\pm$ 0.05 | N |
NGC 5617 | 6 | 8.02 |  $-$0.04 $\pm$ 0.11 |  $+$0.02 $\pm$ 0.04 |  $+$0.03 $\pm$ 0.07 |  $-$0.03 $\pm$ 0.04 |  $+$0.00 $\pm$ 0.03 | N |
NGC 5662 | 2 | 8.30 |  $+$0.07 $\pm$ 0.19 |  $+$0.03 $\pm$ 0.06 |  $+$0.02 $\pm$ 0.11 |  $+$0.00 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.05 | N |
NGC 5822 | 3 | 8.96 |  $+$0.01 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.05 |  $+$0.02 $\pm$ 0.09 |  $-$0.03 $\pm$ 0.06 |  $-$0.02 $\pm$ 0.04 | N |
NGC 5823 | 2 | 7.90 |  $+$0.00 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.03 $\pm$ 0.11 |  $-$0.03 $\pm$ 0.07 |  $-$0.01 $\pm$ 0.05 | N |
NGC 6025 | 3 | 8.02 |  $+$0.19 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.05 |  $+$0.04 $\pm$ 0.09 |  $+$0.02 $\pm$ 0.06 |  $+$0.00 $\pm$ 0.04 | N |
NGC 6031 | 2 | 7.95 |  $-$0.07 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.03 $\pm$ 0.11 |  $-$0.03 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.05 | N |
NGC 6067 | 1 | 8.10 |  $+$0.03 $\pm$ 0.27 |  $+$0.03 $\pm$ 0.09 |  $+$0.06 $\pm$ 0.16 |  $-$0.02 $\pm$ 0.10 |  $+$0.02 $\pm$ 0.07 | N |
NGC 6124 | 19 | 8.28 |  $+$0.07 $\pm$ 0.06 |  $+$0.02 $\pm$ 0.02 |  $+$0.02 $\pm$ 0.04 |  $+$0.00 $\pm$ 0.02 |  $+$0.01 $\pm$ 0.02 | Y |
NGC 6134 | 2 | 8.99 |  $+$0.00 $\pm$ 0.19 |  $+$0.01 $\pm$ 0.06 |  $+$0.02 $\pm$ 0.11 |  $-$0.02 $\pm$ 0.07 |  $+$0.00 $\pm$ 0.05 | N |
NGC 6167 | 3 | 8.19 |  $+$0.05 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.05 |  $+$0.02 $\pm$ 0.09 |  $+$0.01 $\pm$ 0.06 |  $+$0.01 $\pm$ 0.04 | Y |
NGC 6250 | 1 | 7.38 |  $-$0.15 $\pm$ 0.27 |  $+$0.07 $\pm$ 0.09 |  $-$0.17 $\pm$ 0.16 |  $+$0.03 $\pm$ 0.10 |  $+$0.00 $\pm$ 0.07 | Y |
NGC 6281 | 7 | 8.71 |  $-$0.06 $\pm$ 0.10 |  $+$0.02 $\pm$ 0.03 |  $+$0.03 $\pm$ 0.06 |  $-$0.04 $\pm$ 0.04 |  $+$0.00 $\pm$ 0.03 | N |
NGC 6405 | 6 | 7.54 |  $-$0.06 $\pm$ 0.11 |  $+$0.02 $\pm$ 0.04 |  $+$0.03 $\pm$ 0.07 |  $-$0.03 $\pm$ 0.04 |  $+$0.01 $\pm$ 0.03 | N |
NGC 6416 | 6 | 8.36 |  $-$0.07 $\pm$ 0.11 |  $+$0.02 $\pm$ 0.04 |  $+$0.03 $\pm$ 0.07 |  $-$0.03 $\pm$ 0.04 |  $+$0.01 $\pm$ 0.03 | N |
NGC 6603 | 1 | 8.34 |  $+$0.05 $\pm$ 0.27 |  $-$0.01 $\pm$ 0.09 |  $-$0.01 $\pm$ 0.16 |  $-$0.04 $\pm$ 0.10 |  $-$0.01 $\pm$ 0.07 | N |
NGC 6705 | 3 | 8.49 |  $+$0.01 $\pm$ 0.16 |  $+$0.06 $\pm$ 0.05 |  $-$0.08 $\pm$ 0.09 |  $+$0.01 $\pm$ 0.06 |  $+$0.01 $\pm$ 0.04 | T |
NGC 6885 | 2 |  $\cdots$ |  $+$0.02 $\pm$ 0.19 |  $+$0.06 $\pm$ 0.06 |  $-$0.08 $\pm$ 0.11 |  $+$0.01 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.05 | Y |
Roslund 3 | 1 | 7.73 |  $+$0.03 $\pm$ 0.27 |  $+$0.01 $\pm$ 0.09 |  $-$0.06 $\pm$ 0.16 |  $+$0.01 $\pm$ 0.10 |  $+$0.00 $\pm$ 0.07 | N |
Ruprecht 119 | 2 | 7.68 |  $+$0.08 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.03 $\pm$ 0.11 |  $+$0.01 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.05 | Y |
Trumpler 10 | 2 | 7.51 |  $-$0.06 $\pm$ 0.19 |  $+$0.02 $\pm$ 0.06 |  $+$0.04 $\pm$ 0.11 |  $-$0.02 $\pm$ 0.07 |  $+$0.00 $\pm$ 0.05 | N |
Trumpler 18 | 2 | 7.68 |  $+$0.14 $\pm$ 0.19 |  $+$0.01 $\pm$ 0.06 |  $+$0.04 $\pm$ 0.11 |  $+$0.03 $\pm$ 0.07 |  $+$0.01 $\pm$ 0.05 | N |

###  4.1 Direct Comparison to Donor et al. (2020)

To verify our results, we first compare to previous work using APOGEE DR16
directly (Donor et al., 2020). We have 3 clusters in common with some stars in
the The Cannon training set. While only a small subset of clusters, the
studies agree within the uncertainties as shown in Table 5 .

Table 5: Average in common open cluster iron abundance from Donor et al. (2020) compared to (this study). | This Study |  | Donor
---|---|---|---
Cluster | Number | [Fe/H] |  | Number | [Fe/H]
Name | of Stars | (dex) |  | of Stars | (dex)
NGC 2447 | 17 |  $-$0.06 $\pm$ 0.07 |  | 3 |  $-$0.08 $\pm$ 0.01
NGC 2682 | 10 |  $+$0.09 $\pm$ 0.09 |  | 32 |  $+$0.01 $\pm$ 0.03
NGC 6705 | 3 |  $+$0.01 $\pm$ 0.16 |  | 12 |  $+$0.12 $\pm$ 0.04

###  4.2 Comparison to Other Surveys

The Santos et al. (2009) survey provided iron abundances for 13 open clusters
using high-resolution spectra. There were 8 clusters that overlapped with this
survey, and their values are listed in Table 6. To better illustrate how the
two surveys compare, we constructed a one-to-one plot of iron abundances from
The Cannon vs. values from three significant studies, including Santos et al.
(2009), which is shown in Figure 4. All of the cluster values from Santos et
al. (2009) lie within the average uncertainties, although our values were
slightly more metal-poor.

The next comparison was to the Reddy et al. (2013a) and Reddy et al. (2015a)
studies. Both examined a total of 12 clusters using high-resolution spectra.
Here, there was also an overlap of 7 clusters between this survey and the
combined Reddy surveys. The values for each are shown in Table 6. These
clusters are also shown on the one-to-one plot Figure 4. The values obtained
by Reddy et al. (2013a, 2015a) were more metal-poor than the values determined
using The Cannon. Reddy et al. (2013a) and Reddy et al. (2015a) have been
found to be slightly more metal-poor with respect to most measurements other
high-resolution measurements, as summarized in Donor et al. (2018) and seen in
Figure 2 from Reddy et al. (2016) in their comparison to previous literature
results.

The last spectroscopic survey compared to was by Netopil et al. (2016). They
examined 172 clusters with a variety of data including low, medium, and high-
resolution spectra as well as photometric data. The 12 clusters that
overlapped had iron abundances determined from high-resolution spectra. All of
the iron abundances are listed in Table 6 and the one-to-one plot Figure 4,
which shows that the majority of the iron abundances are consistent and most
clusters lie within the scatter range from The Cannon. There are two outliers
which are discussed in §4.3.

Table 6: Average open cluster iron abundance for open clusters in common between this study and literature studies.  | This Study |  | Other Studies |
---|---|---|---|---
Cluster | Stars | [Fe/H] |  | Stars | [Fe/H] | Citation
Name |  | (dex) |  |  | (dex) |
IC 2581 | 1 |  $-$0.10 $\pm$ 0.27 |  | 1 |  $-$0.34 $\pm$ $\cdots$ | Luck (1994)
IC 4651 | 9 |  $+$0.04 $\pm$ 0.09 |  | 3 |  $+$0.01 $\pm$ 0.01 | Santos et al. (2009)
|  |  |  | 18 |  $+$0.12 $\pm$ 0.04 | Netopil et al. (2016)
|  |  |  | 5 |  $+$0.12 $\pm$ 0.05 |  Pace et al. (2008).
|  |  |  | 3 |  $+$0.11 $\pm$ 0.01 | Carretta et al. (2004a)
IC 4756 | 3 |  $-$0.05 $\pm$ 0.16 |  | 3 |  $+$0.02 $\pm$ 0.02 | Santos et al. (2009)
|  |  |  | 9 |  $-$0.02 $\pm$ 0.01 | Bagdonas et al. (2018)
|  |  |  | 12 |  $-$0.01 $\pm$ 0.10 | Ting et al. (2012)
|  |  |  | 2 |  $+$0.01 $\pm$ $\cdots$ | Pace et al. (2010a)
NGC 1662 | 3 |  $+$0.04 $\pm$ 0.16 |  | 2 |  $-$0.10 $\pm$ 0.06 | Reddy et al. (2013a, 2015a)
|  |  |  | 2 |  $-$0.11 $\pm$ 0.01 |  Netopil et al. (2016).
NGC 2354 | 10 |  $-$0.12 $\pm$ 0.09 |  | 2 |  $-$0.19 $\pm$ 0.04 | Reddy et al. (2013a, 2015a)
|  |  |  | 2 |  $-$0.18 $\pm$ 0.02 | Netopil et al. (2016)
NGC 2423 | 8 |  $+$0.10 $\pm$ 0.10 |  | 3 |  $+$0.14 $\pm$ 0.06 | Santos et al. (2009)
|  |  |  | 3 |  $+$0.08 $\pm$ 0.05 | Netopil et al. (2016)
NGC 2447 | 17 |  $-$0.06 $\pm$ 0.07 |  | 3 |  $-$0.08 $\pm$ 0.01 | Donor et al. (2020)
|  |  |  | 3 |  $-$0.10 $\pm$ 0.03 | Santos et al. (2009)
|  |  |  | 3 |  $-$0.13 $\pm$ 0.05 | Reddy et al. (2013a, 2015a)
|  |  |  | 4 |  $+$0.07 $\pm$ 0.03 | Netopil et al. (2016)
|  |  |  | 12 |  $-$0.17 $\pm$ 0.05 | da Silveira et al. (2018)
NGC 2482 | 5 |  $+$0.07 $\pm$ 0.12 |  | 1 |  $-$0.07 $\pm$ 0.04 | Reddy et al. (2013a, 2015a)
|  |  |  | 1 |  $-$0.07 $\pm$ $\cdots$ | Netopil et al. (2016)
NGC 2527 | 3 |  $-$0.11 $\pm$ 0.16 |  | 2 |  $-$0.11 $\pm$ 0.04 | Reddy et al. (2013a, 2015a)
NGC 2516 | 3 |  $-$0.09 $\pm$ 0.16 |  | 2 |  $+$0.05 $\pm$ 0.11 | Netopil et al. (2016)
NGC 2539 | 9 |  $-$0.08 $\pm$ 0.09 |  | 3 |  $+$0.13 $\pm$ 0.03 | Santos et al. (2009)
|  |  |  | 2 |  $-$0.06 $\pm$ 0.04 | Reddy et al. (2013a, 2015a)
|  |  |  | 4 |  $-$0.02 $\pm$ 0.08 | Netopil et al. (2016)
|  |  |  | 12 |  $-$0.03 $\pm$ 0.07 | Martinez et al. (2020)
NGC 2548 | 3 |  $+$0.12 $\pm$ 0.16 |  | 95 |  $-$0.06 $\pm$ 0.01 | Sun et al. (2020)
NGC 2567 | 6 |  $-$0.07 $\pm$ 0.11 |  | 3 |  $-$0.04 $\pm$ 0.08 | Netopil et al. (2016)
NGC 2682 | 10 |  $+$0.09 $\pm$ 0.09 |  | 32 |  $+$0.01 $\pm$ 0.03 | Donor et al. (2020)
|  |  |  | 3 |  $+$0.00 $\pm$ 0.01 | Santos et al. (2009)
|  |  |  | 3 |  $-$0.08 $\pm$ 0.04 | Reddy et al. (2013a, 2015a)
|  |  |  | 27 |  $+$0.03 $\pm$ 0.05 | Netopil et al. (2016)
|  |  |  | 6 |  $+$0.03 $\pm$ 0.04 | Pace et al. (2008)
NGC 3680 | 6 |  $-$0.05 $\pm$ 0.11 |  | 3 |  $-$0.04 $\pm$ 0.01 | Santos et al. (2009)
|  |  |  | 10 |  $-$0.01 $\pm$ 0.06 | Netopil et al. (2016)
|  |  |  | 6 |  $-$0.06 $\pm$ 0.07 | Peña Suárez et al. (2018)
|  |  |  | 11 |  $-$0.03 $\pm$ 0.02 | Mitschang et al. (2012)
|  |  |  | 2 |  $+$0.04 $\pm$ 0.03 | Pace et al. (2008)
NGC 5617 | 6 |  $-$0.04 $\pm$ 0.11 |  | 2 |  $-$0.18 $\pm$ 0.02 | De Silva et al. (2015)
NGC 5822 | 3 |  $+$0.01 $\pm$ 0.16 |  | 3 |  $+$0.05 $\pm$ 0.04 | Santos et al. (2009)
|  |  |  | 7 |  $+$0.08 $\pm$ 0.08 | Netopil et al. (2016)
|  |  |  | 11 |  $-$0.09 $\pm$ 0.06 | Peña Suárez et al. (2018)
|  |  |  | 3 |  $+$0.15 $\pm$ 0.08 | Pace et al. (2010b)
|  |  |  | 7 |  $+$0.08 $\pm$ 0.08 | Luck (1994)
NGC 6067 | 1 |  $+$0.03 $\pm$ 0.27 |  | 5 |  $+$0.19 $\pm$ 0.05 | Alonso-Santiago et al. (2017)
NGC 6134 | 2 |  $+$0.00 $\pm$ 0.19 |  | 8 |  $+$0.11 $\pm$ 0.07 | Netopil et al. (2016)
|  |  |  | 6 |  $+$0.15 $\pm$ 0.07 | Carretta et al. (2004b)
NGC 6281 | 7 |  $-$0.06 $\pm$ 0.10 |  | 2 |  $+$0.06 $\pm$ 0.06 | Netopil et al. (2016)
NGC 6405 | 6 |  $-$0.06 $\pm$ 0.11 |  | 44 |  $+$0.07 $\pm$ 0.03 | Kılıçoğlu et al. (2016)
NGC 6603 | 1 |  $+$0.05 $\pm$ 0.27 |  | 7 |  $+$0.34 $\pm$ 0.15 | Carrera et al. (2015)
NGC 6705 | 3 |  $+$0.01 $\pm$ 0.16 |  | 12 |  $+$0.12 $\pm$ 0.04 | Donor et al. (2020)
|  |  |  | 21 |  $+$0.12 $\pm$ 0.09 | Netopil et al. (2016)

The remaining clusters with iron abundances from smaller high-resolution
studies were also examined for inconsistencies. Table 6 lists all of the
values for [Fe/H] determined by The Cannon, the [Fe/H] values from the
literature, and the type of data that was used. Most of the clusters fall
within the range of scatter again (Table 6); however, 3 clusters were not.
Reasons for why these values do not appear to agree are discussed in §4.3.

Figure 4: A comparison of average [Fe/H] values for in common open clusters
from this survey and from the literature compilation in Tables
LABEL:table:Sample3, & 6. Magenta clusters are from Santos et al. (2009), blue
clusters are from Netopil et al. (2016), and orange clusters are from Reddy et
al. (2013a, 2015a).

The cluster NGC 2682 is one of the most well studied open clusters, therefore
it was used as a calibration cluster. It also has an average [Fe/H] value
determined from APOGEE DR16 data from Donor et al. (2020) making it a
significant check on how well The Cannon produced values. The average [Fe/H]
from Donor et al. (2020) was 0.01$\pm$0.03, and this survey found a value of
0.09 $\pm$ 0.16. Additionally, the five other studies compared to in this
paper with [Fe/H] determined for NGC 2682 (Santos et al., 2009; Reddy et al.,
2013a, 2015a; Netopil et al., 2016; Pace et al., 2008) agree within the
uncertainties.

###  4.3 Discrepancies

There were 3 clusters that were almost outside of the acceptable scatter when
compared to the literature values, which are significant given our larger
uncertainties due to the lower S/N of this study. This section discusses
possible reasons why there were differences in iron abundance values.

The value obtained for NGC 6705 was within the uncertainty of Donor et al.
(2020), however it was more metal-poor. This is likely due to Donor et al.
(2020) having four times the number of member stars used for average abundance
analysis as well as higher resolution, higher-S/N spectra than this study. The
same is also true when compared to the results of (Netopil et al., 2016)

For NGC 1662, both Reddy et al. (2015b) and Netopil et al. (2016) had values
that were in agreement and for reference, these values are listed in Tables 6
and 7. The value for NGC 1662 from this study was more metal-rich than both
Reddy et al. (2015b) and Netopil et al. (2016) and the comparison is shown in
Figure 4. Each of these studies used high-resolution spectra of two stars
where we used medium-resolution spectra of three stars.

The cluster NGC 2482 showed similar discrepancies to NGC 1662. Reddy et al.
(2013b) and Netopil et al. (2016) found more metal-poor values for this
cluster and each only used one star for the average abundance analysis
compared to five stars used in this study, therefore cluster membership of the
other studies may be the reason for such a large offset. This comparison is
also shown in Figure 4.

##  5 Conclusion

Using CTIO/Hydra medium-resolution ($R\sim 19,000$) spectroscopy of clusters
stars with RV and proper motion membership determinations, we determined the
mean [Fe/H] values for a set of 58 open clusters using The Cannon.

With this study:

  * •

We measured the first spectroscopic metallicity [Fe/H] for 35 open clusters
with member stars verified by Frinchaboy & Majewski (2008) and re-verified
with updated proper motions from Gaia Collaboration et al. (2018).

  * •

We confirm the overall abundance scale, based on APOGEE DR16 (Ahumada et al.,
2020) is generally consistent with other study for the 22 clusters that have
spectroscopic metallicity determinations.

  * •

We find the clusters in this study have abundance ratios for oxygen, silicon,
magnesium, and aluminum consistent with solar values, which is reasonable for
clusters near the solar neighborhood.

These clusters add to the work of Donor et al. (2020) yielding a combined
dataset of over 150 clusters, all on the APOGEE DR16 abundance scale.

We would like to thank Anna Ho for help in assisting in setting up The Cannon
for this work. A.E.R, P.M.F., J.D., and M.M. acknowledge support for this
research from the National Science Foundation (AST-1311835 & AST-1715662).
P.M.F. also acknowledges some of this was performed at the Aspen Center for
Physics, which is supported by National Science Foundation grant PHY-1607611.
P.M.F. also acknowledges travel support from NOAO/NOIRLab from the original
collection of this data. Funding for the Sloan Digital Sky Survey IV has been
provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy
Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at the
University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by
the Astrophysical Research Consortium for the Participating Institutions of
the SDSS Collaboration including the Brazilian Participation Group, the
Carnegie Institution for Science, Carnegie Mellon University, the Chilean
Participation Group, the French Participation Group, Harvard-Smithsonian
Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns
Hopkins University, Kavli Institute for the Physics and Mathematics of the
Universe (IPMU) / University of Tokyo, Lawrence Berkeley National Laboratory,
Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für
Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA
Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National
Astronomical Observatories of China, New Mexico State University, New York
University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio
State University, Pennsylvania State University, Shanghai Astronomical
Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma
de México, University of Arizona, University of Colorado Boulder, University
of Oxford, University of Portsmouth, University of Utah, University of
Virginia, University of Washington, University of Wisconsin, Vanderbilt
University, and Yale University. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research made use of
Astropy, a community-developed core Python package for Astronomy (Astropy
Collaboration, 2018). .

## References

  * Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3
  * Alonso-Santiago et al. (2017) Alonso-Santiago, J., Negueruela, I., Marco, A., et al. 2017, MNRAS, 469, 1330
  * Bagdonas et al. (2018) Bagdonas, V., Drazdauskas, A., Tautvaišienė, G., Smiljanic, R., & Chorniy, Y. 2018, A&A, 615, A165
  * Blanton et al. (2017) Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28
  * Bowen & Vaughan (1973) Bowen, I. S., & Vaughan, A. H., J. 1973, Appl. Opt., 12, 1430
  * Bragaglia et al. (2008) Bragaglia, A., Sestito, P., Villanova, S., et al. 2008, A&A, 480, 79
  * Cantat-Gaudin et al. (2020) Cantat-Gaudin, T., Anders, F., Castro-Ginard, A., et al. 2020, A&A, 640, A1
  * Carrera et al. (2015) Carrera, R., Casamiquela, L., Ospina, N., et al. 2015, A&A, 578, A27
  * Carrera & Pancino (2011) Carrera, R., & Pancino, E. 2011, A&A, 535, A30
  * Carretta et al. (2004a) Carretta, E., Bragaglia, A., Gratton, R. G., & Tosi, M. 2004a, A&A, 422, 951
  * Carretta et al. (2004b) —. 2004b, A&A, 422, 951
  * Casamiquela et al. (2019) Casamiquela, L., Blanco-Cuaresma, S., Carrera, R., et al. 2019, MNRAS, 490, 1821
  * da Silveira et al. (2018) da Silveira, M. D., Pereira, C. B., & Drake, N. A. 2018, MNRAS, 476, 4907
  * De Silva et al. (2015) De Silva, G. M., Carraro, G., D’Orazi, V., et al. 2015, MNRAS, 453, 106
  * Donor et al. (2018) Donor, J., Frinchaboy, P. M., Cunha, K., et al. 2018, AJ, 156, 142
  * Donor et al. (2020) —. 2020, AJ, 159, 199
  * Eisenstein et al. (2011) Eisenstein, D. J., Weinberg, D. H., Agol, E., et al. 2011, AJ, 142, 72
  * Friel et al. (2010) Friel, E. D., Jacobson, H. R., & Pilachowski, C. A. 2010, AJ, 139, 1942
  * Frinchaboy & Majewski (2008) Frinchaboy, P. M., & Majewski, S. R. 2008, AJ, 136, 118
  * Frinchaboy et al. (2012) Frinchaboy, P. M., Majewski, S. R., Muñoz, R. R., et al. 2012, ApJ, 756, 74
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * García Pérez et al. (2016) García Pérez, A. E., Allende Prieto, C., Holtzman, J. A., et al. 2016, AJ, 151, 144
  * Gunn et al. (2006) Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332
  * Hasselquist et al. (2020) Hasselquist, S., Zasowski, G., Feuillet, D. K., et al. 2020, ApJ, 901, 109
  * Ho et al. (2017) Ho, A. Y. Q., Ness, M. K., Hogg, D. W., et al. 2017, ApJ, 836, 5
  * Holtzman et al. (2018) Holtzman, J. A., Hasselquist, S., Shetrone, M., et al. 2018, AJ, 156, 125
  * Jacobson et al. (2009) Jacobson, H. R., Friel, E. D., & Pilachowski, C. A. 2009, AJ, 137, 4753
  * Janes (1979) Janes, K. A. 1979, ApJS, 39, 135
  * Jönsson et al. (2020) Jönsson, H., Holtzman, J. A., Allende Prieto, C., et al. 2020, AJ, 160, 120
  * Kılıçoğlu et al. (2016) Kılıçoğlu, T., Monier, R., Richer, J., Fossati, L., & Albayrak, B. 2016, AJ, 151, 49
  * Luck (1994) Luck, R. E. 1994, ApJS, 91, 309
  * Magrini et al. (2017) Magrini, L., Randich, S., Kordopatis, G., et al. 2017, A&A, 603, A2
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94
  * Martinez et al. (2020) Martinez, C. F., Holanda, N., Pereira, C. B., & Drake, N. A. 2020, MNRAS, 494, 1470
  * Mitschang et al. (2012) Mitschang, A. W., De Silva, G. M., & Zucker, D. B. 2012, MNRAS, 422, 3527
  * Ness et al. (2015) Ness, M., Hogg, D. W., Rix, H.-W., Ho, A. Y. Q., & Zasowski, G. 2015, ApJ, 808, 16
  * Netopil et al. (2016) Netopil, M., Paunzen, E., Heiter, U., & Soubiran, C. 2016, A&A, 585, A150
  * Nidever et al. (2015) Nidever, D. L., Holtzman, J. A., Allende Prieto, C., et al. 2015, AJ, 150, 173
  * Pace et al. (2010a) Pace, G., Danziger, J., Carraro, G., et al. 2010a, A&A, 515, A28
  * Pace et al. (2010b) —. 2010b, A&A, 515, A28
  * Pace et al. (2008) Pace, G., Pasquini, L., & François, P. 2008, A&A, 489, 403
  * Peña Suárez et al. (2018) Peña Suárez, V. J., Sales Silva, J. V., Katime Santrich, O. J., Drake, N. A., & Pereira, C. B. 2018, ApJ, 854, 184
  * Reddy et al. (2013a) Reddy, A. B. S., Giridhar, S., & Lambert, D. L. 2013a, MNRAS, 431, 3338
  * Reddy et al. (2013b) Reddy, A. B. S., Giridhar, S., & Lambert, D. L. 2013b, MNRAS, 431, 3338
  * Reddy et al. (2015a) Reddy, A. B. S., Giridhar, S., & Lambert, D. L. 2015a, MNRAS, 450, 4301
  * Reddy et al. (2015b) Reddy, A. B. S., Giridhar, S., & Lambert, D. L. 2015b, MNRAS, 450, 4301
  * Reddy et al. (2016) Reddy, A. B. S., Lambert, D. L., & Giridhar, S. 2016, MNRAS, 463, 4366
  * Santos et al. (2009) Santos, N. C., Lovis, C., Pace, G., Melendez, J., & Naef, D. 2009, A&A, 493, 309
  * Sun et al. (2020) Sun, Q., Deliyannis, C. P., Steinhauer, A., Twarog, B. A., & Anthony-Twarog, B. J. 2020, AJ, 159, 220
  * Ting et al. (2012) Ting, Y.-S., De Silva, G. M., Freeman, K. C., & Parker, S. J. 2012, MNRAS, 427, 882
  * Tonry & Davis (1979) Tonry, J., & Davis, M. 1979, AJ, 84, 1511
  * Wilson et al. (2012) Wilson, J. C., Hearty, F., Skrutskie, M. F., et al. 2012, in Proc. SPIE, Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV, 84460H
  * Yong et al. (2012) Yong, D., Carney, B. W., & Friel, E. D. 2012, AJ, 144, 95
  * Yong et al. (2005) Yong, D., Carney, B. W., & Teixera de Almeida, M. L. 2005, AJ, 130, 597


# The Vertical Recursive Relation of Riordan Arrays and Their Matrix
Representation

Tian-Xiao He
Department of Mathematics
Illinois Wesleyan University
Bloomington, IL 61702-2900, USA

###### Abstract

A vertical recursive relation approach to Riordan arrays is induced, while the
horizontal recursive relation is represented by $A$\- and $Z$-sequences. This
vertical recursive approach gives a way to represent the entries of a Riordan
array $(g,f)$ in terms of a recursive linear combinations of the coefficients
of $g$. A matrix representation of the vertical recursive relation is also
given. The set of all those matrices forms a group, called the quasi-Riordan
group. The extensions of the horizontal recursive relation and the vertical
recursive relation in terms of $c$\- and $C$\- Riordan arrays are defined with
illustrations by using the rook triangle and the Laguerre triangle. Those
extensions represent a way to study nonlinear recursive relations of the
entries of some triangular matrices from linear recursive relations of the
entries of Riordan arrays. In addition, the matrix representation of the
vertical recursive relation of Riordan arrays provides transforms between
lower order and high order finite Riordan arrays, where the $m$th order
Riordan array is defined by $(g,f)_{m}=(d_{n,k})_{m\geq n,k\geq 0}$.
Furthermore, the vertical relation approach to Riordan arrays provides a
unified approach to construct identities.

AMS Subject Classification: 05A15, 05A05, 15B36, 15A06, 05A19, 11B83.

Key Words and Phrases: Riordan array, Riordan group, quasi-Riordan array,
vertical recursive relation, $A$-sequence, $Z$-sequence, rook triangle,
Laguerre triangle, Fuss-Catalan numbers and matrix, Catalan numbers and
matrix, $(c)$-Riordan arrays, $(C)$-Riordan arrays.

##  1 Introduction

Riordan matrices are infinite, lower triangular matrices defined by the
generating function of their columns. With the matrix multiplication, they
form a group, called the Riordan group (see Shapiro, Getu, Woan and Woodson
[25]).

More formally, let us consider the set of formal power series ring
$\mbox{$\mathcal{F}$}=K[\\![$$t$$]\\!]$, where ${{\mathbb{K}}}$ is the field
of ${{\mathbb{R}}}$ or ${{\mathbb{C}}}$. The _order_ of
$f(t)\in\mbox{$\mathcal{F}$}$, $f(t)=\sum_{k=0}^{\infty}f_{k}t^{k}$
($f_{k}\in{{\mathbb{K}}}$), is the minimum number $r\in{\mathbb{N}}_{0}$ such
that $f_{r}\neq 0$, where ${{\mathbb{N}}}_{0}={{\mathbb{N}}}\cup\\{0\\}$ and
${{\mathbb{N}}}$ is the positive integer set. We denote by
$\mbox{$\mathcal{F}$}_{r}$ the set of formal power series of order $r$. Let
$g(t)\in\mbox{$\mathcal{F}$}_{0}$ and $f(t)\in\mbox{$\mathcal{F}$}_{1}$; the
pair $(g,\,f)$ defines the (proper) Riordan matrix
$D=(d_{n,k})_{n,k\in{\mathbb{N}}_{0}}=(g,f)$ having

$d_{n,k}=[t^{n}]g(t)f(t)^{k}$ (1)

or, in other words, having $gf^{k}$ as the generating function of the $k$th
column of $(g,f)$. The first fundamental theorem of Riordan matrices  means
the action of the proper Riordan matrices on the formal power series presented
by

$(g(t),f(t))h(t)=g(t)(h\circ f)(t),$

which can be abbreviated as $(g,f)h=gh(f)$. Thus we immediately see that the
usual row-by-column product of two Riordan matrices is also a Riordan matrix:

$(g_{1},\,f_{1})(g_{2},\,f_{2})=(g_{1}g_{2}(f_{1}),\,f_{2}(f_{1})).$ (2)

The Riordan matrix $I=(1,\,t)$ is the identity matrix because its entries are
$d_{n,k}=[t^{n}]t^{k}=\delta_{n,k}$.

Let $(g\left(t\right),\,f(t))$ be a Riordan matrix. Then its inverse is

$(g\left(t\right),\,f(t))^{-1}=\left(\frac{1}{g(\overline{{f}}(t))},\overline{f}(t),\right)$
(3)

where $\overline{f}(t)$ is the compositional inverse of $f(t)$, i.e.,
$(f\circ\overline{f})(t)=(\overline{f}\circ f)(t)=t$. In this way, the set
$\mathcal{R}$ of all proper Riordan matrices forms a group (see [25]) called
the Riordan group,

Here is a list of six important subgroups of the Riordan group (see [25, 27,
1, 26]).

  * •

the Appell subgroup $\\{(g(z),\,z)\\}$.

  * •

the Lagrange (associated) subgroup $\\{(1,\,f(z))\\}$.

  * •

the $k$-Bell subgroup $\\{(g(z),\,z(g(z))^{k})\\}$, where $k$ is a fixed
positive integer.

  * •

the hitting-time subgroup $\\{(zf^{\prime}(z)/f(z),\,f(z))\\}$.

  * •

the derivative subgroup $\\{(f^{\prime}(z),\,f(z))\\}$.

  * •

the checkerboard subgroup $\\{(g(z),\,f(z))\\},$ where $g$ is an even function
and $f$ is an odd function.

The $1$-Bell subgroup is referred to as the Bell subgroup for short, and the
Appell subgroup can be considered as the $0$-Bell subgroup if we allow $k=0$
to be included in the definition of the $k$-Bell subgroup.

Let $G$ be a group, and let $H$ and $N$ be two subgroups of $G$ with $N$
normal. Then the following statements are equivalent:

  * •

$NH=G$ and $N\cap H={e}$.

  * •

Every $g\in G$ can be written uniquely as $g=nh$, where $n\in N$ and $h\in H$.

  * •

Define $\psi:H\to G/N$ by $\psi(h)=Nh$, $h\in H$. Then $\psi$ is an
isomorphism.

If these conditions hold, we write $G=N\rtimes H$ and say that $G$ is
expressed as an semidirect product of $N$ and $H$. Since for every
$(g,f)\in\mathcal{R}$

$(g,f)=(g,t)(1,f),$

where $(g,t)\in\mathcal{A}$ and $(1,f)\in\mathcal{L}$ and $\mathcal{A}$ and
$\mathcal{L}$ are Appell subgroup, the normal subgroup of $\mathcal{R}$, and
the Lagrange subgroup, respectively.

An infinite lower triangular matrix $[d_{n,k}]_{n,k\in{{\mathbb{N}}_{0}}}$ is
a Riordan matrix if and only if a unique sequence
$A=(a_{0}\not=0,a_{1},a_{2},\ldots)$ exists such that for every
$n,k\in{{\mathbb{N}}_{0}}$

$d_{n+1,k+1}=a_{0}d_{n,k}+a_{1}d_{n,k+1}+\cdots+a_{n-k}d_{n,n}.$ (4)

This is equivalent to

$f(t)=tA(f(t))\quad\text{or}\quad t=\bar{f}(t)A(t).$ (5)

Here, $A(t)$ is the generating function of the $A$-sequence. The first formula
of (5) is also called the second fundamental theorem of Riordan matrices.
Moreover, there exists a unique sequence $Z=(z_{0},z_{1},z_{2},\ldots)$ such
that every element in column $0$ can be expressed as the linear combination

$d_{n+1,0}=z_{0}d_{n,0}+z_{1}d_{n,1}+\cdots+z_{n}d_{n,n},$ (6)

or equivalently,

$g(t)=\frac{1}{1-tZ(f(t))},$ (7)

in which and thoroughly we always assume $g(0)=g_{0}=1$, a usual hypothesis
for proper Riordan matrices with normalization. From (7), we may obtain

$Z(t)=\frac{g(\bar{f}(t))-1}{\bar{f}(t)g(\bar{f}(t))}.$

$A$\- and $Z$-sequence characterizations of Riordan matrices were introduced,
developed, and/or studied in Merlini, Rogers, Sprugnoli, and Verri [20], Roger
[24], Sprugnoli and the author [13], Cheon and Jin [2], Cheon, Luzón, Morón,
Prieto Martinez, and Song [4], [10], Jean-Louis and Nkwanta [15], Luzón,
Morón, and Prieto-Martinez [17, 18], etc. In [13] the expressions of the $A$\-
and $Z$-sequences of the product depend on the analogous sequences of the two
given factors.

The Catalan numbers, $C_{n}=\binom{2n}{n}/(n+1)$, are a sequence of integers
that occur in many counting situations. The books Enumerative Combinatorics,
Volume 2 and the more recent Catalan Numbers by Richard Stanley [29] and [30]
give a wealth of information. The generating function, denoted by $C(z)$, of
the Catalan numbers can be written as

$C(t)=\sum_{n=0}^{\infty}C_{n}t^{n}=\frac{1-\sqrt{\mathstrut 1-4t}}{2t},$

which is equivalent to the functional equation $C(t)=1+tC(t)^{2}$. It can be
shown that [8]

$C(t)^{k}=\sum_{n=0}^{\infty}\frac{k}{2n+k}\binom{2n+k}{n}t^{n}.$

A Dyck path of length $2n$ is a path in the plane lattice
$\mathbb{Z}\times\mathbb{Z}$ from the origin $(0,0)$ to $(2n,0)$, made up with
steps $(1,1)$ and $(1,-1).$ The other requirement is that a path can never go
below the $x$-axis. We refer to $n$ as the semilength of the path. It is well
known that the number of Dyck paths of semilength $n$ is the $n$-th Catalan
number $C_{n}$. A partial Dyck path is a Dyck path without requiring that end
point be on the $x$-axis. Hence,

$C(n,k):=[t^{n}]C(t)^{k}=\frac{k}{2n+k}\binom{2n+k}{n}$ (8)

is the number of the partial Dyck paths from $(0,0)$ to $(2n+k,k)$.

For a positive integer $m$, an $m$-Dyck path is a path from the origin to
$(mn,0)$ using the steps $(1,1)$ and $(1,1-m)$ and again not going below the
$x$-axis. We refer to $mn$ as the length of the path. A partial $m$-Dyck path
is defined as an $m$-partial Dyck path. It is well known that the number of
$m$-Dyck paths of length $mn$ is (see, for example, [8])

$F_{m}(n,1)=\frac{1}{mn+1}\binom{mn+1}{n},$

the Fuss-Catalan numbers. For $m=2$, the Fuss-Catalan numbers are the Catalan
numbers $F_{2}(n,1)$. More generally, the Fuss-Catalan numbers are

$F_{m}(n,r):=\frac{r}{mn+r}\binom{mn+r}{n},$ (9)

which are named after N. I. Fuss and E. C. Catalan (see [6, 8, 16, 21, 22, 9,
14]). The Fuss-Catalan numbers have many combinatorial applications (see, for
example, Shapiro and the author [14]).

The generating function $F_{m}(t)$ for the Fuss-Catalan numbers, $\\{F_{m}(n,$
$1)\\}_{n\geq 0}$ is called the generalized binomial series in [8], and it
satisfies the function equation $F_{m}(t)=1+tF_{m}(t)^{m}$. Hence from
Lambert’s formula for the Taylor expansion of the powers of $F_{m}(t)$ (see
[8]), we have that

$F_{m}^{r}\equiv F_{m}(t)^{r}=\sum_{n\geq
0}\frac{r}{mn+r}\binom{mn+r}{n}t^{n}$ (10)

for all $r\in{{\mathbb{R}}}$. The key case(10) leads the following formula for
$F_{m}(t)$:

$F_{m}(t)=1+tF_{m}^{m}(t).$ (11)

Actually,

$\displaystyle 1+tF_{m}^{m}(t)$ $\displaystyle=$ $\displaystyle 1+\sum_{n\geq
0}\frac{m}{mn+m}\binom{mn+m}{n}t^{n+1}$ $\displaystyle=$ $\displaystyle
1+\sum_{n\geq 1}\frac{m}{mn}\binom{mn}{n-1}t^{n}$ $\displaystyle=$
$\displaystyle\sum_{n\geq 0}\frac{1}{mn+1}\binom{mn+1}{n}t^{n}=F_{m}(t).$

For the cases $m=1$ and $2$, we have $F_{1}=1/(1-t)$ and $F_{2}=C(z)$,
respectively. When $m=3$, the Fuss-Catalan numbers $\left(F_{3}\right)_{n}$
form the sequence $A001764$ (see [28]), $1,1,3,12,55,273,1428,\ldots$, the
ternary numbers. The ternary numbers count the number of $3$-Dyck paths or
ternary paths. The generating function of the ternary numbers is denoted as
$T(t)=\sum_{n=0}^{\infty}T_{n}t^{n}$ with
$T_{n}=\frac{1}{3n+1}\binom{3n+1}{n}$, and is given equivalently by the
equation

$T(t)=1+tT(t)^{3}.$

##  2 A vertical recursive relation view of Riordan arrays

In a Riordan array $(g,f)=(d_{n,k})_{n,k\geq 0}$, the first column ($0$th
column), with its generating function $g(t)$, usually possesses an interesting
combinatorial interpretation or represents an important sequence, while the
other columns might be considered as the compositions of the first column in
the view of the following recurrence relation:

$d_{n,k}=[t^{n}]gf^{k}=\sum^{n}_{j=1}f_{j}[t^{n-j}]gf^{k-1}=\sum^{n-k+1}_{j=1}f_{j}d_{n-j,k-1}$
(12)

for $n,k\geq 1$, where $f(t)=\sum_{j\geq 1}f_{j}t^{j}$. For instance,

$\displaystyle d_{1,1}=f_{1}d_{0,0},$ $\displaystyle
d_{2.1}=f_{2}d_{0,0}+f_{1}d_{1,0},\,\,d_{2,2}=f_{1}d_{1,1},$ $\displaystyle
d_{3,1}=f_{3}d_{0,0}+f_{2}d_{1,0}+f_{1}d_{2.0},\,\,d_{3,2}=f_{2}d_{2,1}+f_{1}d_{1,1},\,\,d_{3,3}=f_{1}d_{2,2},\ldots$

######  Example 2.1.

For the Pascal triangle $(1/(1-t),t/(1-t))$, from (12) we obtain the well-
known recursive relation

$\binom{n}{k}=\sum^{n-k+1}_{j=1}\binom{n-j}{k-1}.$ (13)

Recurrence relation (12) provides a resource for constructing identities and
an algorithm for computing powers and multiplications of formal power series.
The latter approach may simplifies the corresponding computation process by
using Faà di Bruno’s formula.

######  Theorem 2.2.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array, and let $C(n,k)$ and
$F_{m}(n,r)$ be defined by (8) and (9), respectively. Then we obtain the
recurrence relation (12). Particularly, for $g=F_{m}$ we have

$\displaystyle\frac{k+1}{m(n-k)+k+1}\binom{m(n-k)+k+1}{n-k}$ $\displaystyle=$
$\displaystyle\sum^{n-k}_{j=0}\frac{k}{(mj+1)(m(n-j-k)+k)}\binom{mj+1}{j}\binom{m(n-j-k)+k}{n-k-j}.$
(14)

If $m=2$, then $g=F_{2}=C$ and (2.2) becomes

$\displaystyle\frac{k+1}{2n-k+1}\binom{2n-k+1}{n-k}$ $\displaystyle=$
$\displaystyle\sum^{n-k}_{j=0}\frac{k}{(2j+1)(2(n-j-k)+k)}\binom{2j+1}{j}\binom{2(n-j-k)+k}{n-k-j},$
(15)

or equivalently,

$C(n-k,k+1)=\sum^{n-k}_{j=0}C(j,1)C(n-j-k,k),$ (16)

where $C(m,\ell)$ are defined by (8), and $C(j,1)=[t^{j}]C(t)$, the $j$th
Catalan number.

######  Theorem 2.3.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array. Then its entries
$d_{n,k}$, $n,k\geq 0$, can be represented recursively by

$\displaystyle d_{n,k}=$
$\displaystyle\sum^{n-(k-1)}_{i_{k}=1}f_{i_{k}}\sum^{n-(k-2)-i_{k}}_{i_{k-1}=1}f_{i_{k-1}}\sum^{n-(k-3)-i_{k}-i_{k-1}}_{i_{k-2}=1}f_{i_{k-2}}\cdots$
$\displaystyle\sum^{n-i_{k}-i_{k-1}-i_{2}}_{i_{1}=1}f_{i_{1}}g_{n-i_{1}-i_{2}-\cdots-
i_{k}}.$ (17)

Let $A$ and $B$ be $m\times m$ and $n\times n$ matrices, respectively. The we
define the direct sum of $A$ and $B$ by

$A\oplus B=\left[\begin{matrix}A&0\\\
0&B\end{matrix}\right]_{(m+n)\times(m+n)}.$ (18)

######  Definition 2.4.

Let $g\in\mbox{$\mathcal{F}$}_{0}$ with $g(0)=1$ and
$f\in\mbox{$\mathcal{F}$}_{1}$. We call the following matrix a quasi-Riordan
array and denote it by $[g,f]$.

$[g,f]:=(g,f,tf,t^{2}f,\ldots),$ (19)

where $g$, $f$, $tf$, $t^{2}f\cdots$ are the generating functions of the
$0$th, $1$st, $2$nd, $3$rd, $\cdots$, columns of the matrix $[g,f]$,
respectively. It is clear that $[g,f]$ can be written as

$[g,f]=\left(\begin{matrix}g(0)&0\\\ (g-g(0))/t&(f,t)\end{matrix}\right),$
(20)

where $(f,t)=(f,tf,t^{2}f,t^{3}f,\ldots)$. Particularly, if $f=tg$, then the
quasi-Riordan array $[g,tg]=(g,t)$, a Appell-type Riordan array.

Note that $[g,f]$ defined by (19) is not the Riordan-Krylov matrix defined in
[3], which relationship is worth being investigated.

######  Theorem 2.5.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array, and let
$([1]\oplus(g,f))$ and $[g,f]$ be defined by (2.3) and (19), respectively.
Then $(g,f)$ has the expression

$(g,f)=[g,f]([1]\oplus(g,f)).$ (21)

###### Proof.

We write the formal power series $g$ and $f$ in the Riordan array $(g,f)$ as
$g=\sum_{n\geq 0}g_{n}t^{n}$ and $f=\sum_{n\geq 1}f_{n}t^{n}$. Then

$\displaystyle\left(\begin{matrix}g(0)&0\\\
(g-g(0))/t&(f,t)\end{matrix}\right)([1]\oplus(g,f))$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}d_{0,0}&&&&\\\ d_{1,0}&f_{1}&&&\\\
d_{2,0}&f_{2}&f_{1}&&\\\ d_{3,0}&f_{3}&f_{2}&f_{1}&\\\
d_{4,0}&f_{4}&f_{3}&f_{2}&f_{1}\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right)\left(\begin{matrix}1&&&&\\\
0&d_{0,0}&&&\\\ 0&d_{1,0}&d_{1,1}&&\\\ 0&d_{2,0}&d_{2,1}&d_{2,2}&\\\
0&d_{3,0}&d_{3,1}&d_{3,2}&d_{3,3}\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{lllll}d_{0,0}&&&&\\\
d_{1,0}&f_{1}d_{0,0}&&&\\\ d_{2,0}&f_{2}d_{0,0}+f_{1}d_{1,0}&f_{1}d_{1,1}&&\\\
d_{3,0}&f_{3}d_{0,0}+f_{2}d_{1,0}+f_{1}d_{2,0}&f_{2}d_{1,1}+f_{1}d_{2,1}&f_{1}d_{2,2}&\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)$ (27) $\displaystyle=$
$\displaystyle\left(\begin{matrix}d_{0,0}&&&&\\\ d_{1,0}&d_{1,1}&&&\\\
d_{2,0}&d_{2,1}&d_{2,2}&&\\\ d_{3,0}&d_{3,1}&d_{3,2}&d_{3,3}&\\\
d_{4,0}&d_{4,1}&d_{4,2}&d_{4,3}&d_{4,4}\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right),$ (28)

where the last step follows (12), which completes the proof of (19). ∎

For an integer $r\geq 0$ denote the $r$-th truncations of a power series
$h=\sum_{n\geq 0}h_{n}t^{n}$ by $h|_{r}:=\sum^{r}_{n=0}h_{n}t^{n}$.

######  Corollary 2.6.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array with $g=\sum_{n\geq
0}g_{n}t^{n}$ and $f=\sum_{n\geq 1}f_{n}t^{n}$, and let $(g,f)_{n}$ be the
$n$th order leading principle submatrix of $(g,f)$. Then we have the recursive
relation of $(g,f)_{n}$ in the following form:

$(g,f)_{n}=[g,f]_{n}([1]\oplus(g,f)_{n-1}),$ (29)

where $[g,f]_{n}$ is the $n$th order leading principle submatrix of the quasi-
Riordan array $[g,f]$ defined by (2.3), namely

$[g,f]_{n}=\left(\begin{matrix}g(0)&0\\\
((g-g(0))/t)|_{n-1}&(f,t)_{n-1}\end{matrix}\right),$ (30)

where the $n-1$st truncation of $(g-g(0)/t)|_{n-1}$ is
$(g-g(0)/t)|_{n-1}=\sum^{n-1}_{k=1}g_{k}t^{k-1}$, and
$(f,t)_{n-1}=(f,tf,t^{2}f,\ldots,t^{n-1}f).$ We call $[g,f]_{n}$ the recursive
matrix of the Riordan array $(g,f)$.

Mao, Mu, and Wang [19] uses (30) gives another interesting criterion for the
total positivity of Riordan arrays.

In the next section, we will show that the set of all quasi-Riordan arrays
forms a group, called the quasi-Riordan group.

##  3 The quasi-Riordan group

Denote by ${\cal R}_{r}$ the set of all quasi-Riordan arrays defined by (19).
In this section, we will show ${\cal R}_{r}$ is a group with respect to
regular matrix multiplication.

######  Proposition 3.1.

Let $[g,f]$, $[d,h]\in{\cal R}_{r}$, and let $u=\sum_{n\geq
0}u_{n}t^{n}\in\mbox{$\mathcal{F}$}_{0}$. Then there holds the first
fundamental theorem for quasi-Riordan arrays (FFTQRA)

$[g,f]u=gu(0)+\frac{f}{t}(u-u(0)),$ (31)

which implies

$[g,f][d,h]=\left[g+\frac{f}{t}(d-1),\frac{fh}{t}\right].$ (32)

Hence, $[1,t]$ is the identity of ${\cal R}_{r}$.

###### Proof.

The FFTQRA (31) can be proved as follows.

$\displaystyle[g,f]u=(g,f,tf,t^{2}f,\cdots)\left(\begin{matrix}u_{0}\\\
u_{1}\\\ u_{2}\\\ \vdots\end{matrix}\right)$ $\displaystyle=$ $\displaystyle
gu_{0}+f\sum_{n\geq 1}u_{n}t^{n-1}=gu_{0}+\frac{f}{t}(u-u_{0}).$

By using FFTQRA and noting $d(0)=1$ and $h(0)=0$, we have

$\displaystyle[g,f][d,h]=(g,f,tf,t^{2}f,\cdots)(d,h,th,t^{2}h,\cdots)$
$\displaystyle=$
$\displaystyle\left(g+\frac{f}{t}(d-1),\frac{f}{t}h,\frac{f}{t}th,\frac{f}{t}t^{2}h,\cdots\right),$

which implies (32).

Substituting $[g,f]=[1,t]$ and $[d,h]=[1,t]$ into (32), we obtain,
respectively,

$\displaystyle[1,t][g,f]=[1+(g-1),f]=[g,f]\quad\mbox{and}$
$\displaystyle[g,f][1,t]=\left[g+\frac{f}{t}(1-1),\frac{f}{t}t\right]=[g,f],$

which implies $[1,t]$, the identity matrix, is the identity of ${\cal R}_{r}$.
∎

######  Theorem 3.2.

The set of all quasi-Riordan arrays ${\cal R}_{r}$ is a group, called the
quasi-Riordan group, with respect to the multiplication represented in (32).

###### Proof.

From (32) of Proposition 3.1, ${\cal R}_{r}$ is closed with respect to the
multiplication. (32) also shows $[1,t]$ is the identity of ${\cal R}_{r}$. For
any $[g,f]\in{\cal R}_{r}$, its inverse is

$[g,f]^{-1}=\left[1+\frac{t}{f}(1-g),\frac{t^{2}}{f}\right],$ (33)

since

$\displaystyle\left[1+\frac{t}{f}(1-g),\frac{t^{2}}{f}\right][g,f]=\left[1+\frac{t}{f}(1-g)+\frac{t^{2}/f}{t}(g-1),\frac{t^{2}}{f}\frac{f}{t}\right]=[1,t].$
(34)

Finally, the associative law is satisfied for any $[g,f],[d,h]$, and
$[u,v]\in{\cal R}_{r}$ because

$\displaystyle([g,f][d,h])[u,v]=\left[g+\frac{f}{t}(d-1),\frac{fh}{t}\right][u,v]$
$\displaystyle=$
$\displaystyle\left[g+\frac{f}{t}(d-1)+\frac{fh}{t^{2}}(u-1),\frac{fhv}{t^{2}}\right]$

and

$\displaystyle[g,f]([d,h])[u,v]=[g,f]\left[d+\frac{h}{t}(u-1),\frac{hv}{t}\right]$
$\displaystyle=$
$\displaystyle\left[g+\frac{f}{t}\left(d+\frac{h}{t}(u-1)-1\right),\frac{fhv}{t^{2}}\right]$
$\displaystyle=$
$\displaystyle\left[g+\frac{f}{t}(d-1)+\frac{fh}{t^{2}}(u-1),\frac{fhv}{t^{2}}\right]$

shows $([g,f][d,h])[u,v]=[g,f]([d,h][u,v])$. ∎

######  Example 3.3.

Consider the quasi-Riordan array $[1/(1-t),t/(1-t)]$, which is a Appell-type
Riordan array $(1/(1-t),t)$ with its inverse $(1-t,t)$. From (33) its inverse
in ${\cal R}_{r}$ is

$\displaystyle\left[\frac{1}{1-t},\frac{t}{1-t}\right]^{-1}=\left[1+\frac{t}{f}(1-g),\frac{t^{2}}{f}\right]$
$\displaystyle=$
$\displaystyle\left[1+\frac{t}{t/(1-t)}\left(1-\frac{1}{1-t}\right),\frac{t^{2}}{t/(1-t)}\right]=[1-t,t(1-t)]$
$\displaystyle=$ $\displaystyle(1-t,t)=\left(\begin{matrix}1&&&\\\ -1&1&&\\\
0&-1&1&\\\ 0&0&-1&1\\\ \vdots&\vdots&\vdots&\ddots\end{matrix}\right).$

######  Theorem 3.4.

A quasi-Riordan array $[g,f]$ is a Riordan array if and only if $f=tg$, i.e.,
when $[g,f]$ is an Appell-type Riordan array. Hence, ${\cal
A}_{r}:=\\{[g,tg]:g\in\mbox{$\mathcal{F}$}_{0},g(0)=1\\}$ is a subgroup of
${\cal R}_{r}$, which is called the Apell quasi-Riordan subgroup.

###### Proof.

Let $[g,tg]$, $[d,td]\in{\cal R}_{r}$. Then

$[g,tg][d,td]=\left[g+\frac{tg}{t}(d-1),\frac{t^{2}gd}{t}\right]=[gd,tgd].$

${\cal A}$ is closed under the multiplication. In addition, the inverse of
$[g,f]$

$[g,tg]^{-1}=\left[1+\frac{t}{tg}(1-g),\frac{t^{2}}{tg}\right]=\left[\frac{1}{g},\frac{t}{g}\right]$

is also in ${\cal A}$. ∎

######  Theorem 3.5.

Let ${\cal R}_{r}$ be the quasi-Riordan group. Then every conjugate of an
element $[g,f]\in{\cal R}_{r}$ is in the set ${{\cal
R}(f)}_{r}:=\\{[d,f]:d\in\mbox{$\mathcal{F}$}_{0},d(0)=1\\}$. Hence, ${\cal
A}=\\{[g,t]:g\in\mbox{$\mathcal{F}$}_{0},g(0)=1\\}$ is a normal subgroup of
${\cal R}_{r}$.

###### Proof.

Let $[d,h]\in{\cal R}_{r}$. Then for an arbitrarily fixed $[g,f]\in{\cal
R}_{r}$, we have

$\displaystyle[d,h][g,f][d,h]^{-1}=[d,h][g,f]\left[1+\frac{t}{h}(1-d),\frac{t^{2}}{h}\right]$
$\displaystyle=$
$\displaystyle\left[d+\frac{h}{t}(g-1)-\frac{f}{t}(d-1),f\right]\in{{\cal
R}(f)}_{r}.$

Particularly, if $f=t$ in $[g,f]$, then

$\displaystyle[d,h][g,t][d,h]^{-1}=\left[1+\frac{h}{t}(g-1),t\right],$

which implies ${\cal A}=\\{[g,t]:g\in\mbox{$\mathcal{F}$}_{0},g(0)=1\\}$ is a
normal subgroup of ${\cal R}_{r}$. ∎

##  4 An vertical recursive relation view of $(c)$\- and $(C)$-Riordan arrays

In this section, we will extend the vertical recursive relation to the $(c)$
Riordan arrays. We start from the following definition of $(c)$-class of
Riordan arrays.

######  Definition 4.1.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array, and let
$c=(c_{0},c_{1},c_{2},...)$ be a sequence satisfying $c_{0}=1$ and
$c_{k}\not=0$ for all $k=1,2,\ldots$. Denote

$(g,f)_{c}=\left(\frac{c_{n}}{c_{k}}d_{n,k}\right)_{n,k\geq 0}.$ (35)

Then the set

$\\{{{\cal R}}\\}_{c}:=\left\\{(g,f)_{c}:(g,f)=(d_{n,k})_{n,k\geq 0}\in{{\cal
R}}\right\\}$ (36)

is called the $(c)$-class of ${{\cal R}}$ or the set of the $(c)$-Riordan
arrays. Since we may change $c_{n}$ and $c_{k}$ respectively to $1/c_{n}$ and
$1/c_{k}$,

$\left(\frac{c_{k}}{c_{n}}d_{n,k}\right)_{n,k\geq
0}=\left(\frac{1/c_{n}}{1/c_{k}}d_{n,k}\right)_{n,k\geq 0}$

is in the $(1/c)$-class of ${{\cal R}}$, where
$(1/c)=(1/c_{0},1/c_{1},1/c_{2},\ldots)$.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array, and let
$C=(c_{n,k})_{n,k\geq 0}$ be a lower triangle matrix satisfying $c_{n,0}=1$
and $c_{n,k}\not=0$ for all $0\leq k\leq n$ and $c_{n,k}=0$ for all $k>n$.
Denote

$(g,f)_{C}=\left(\frac{c_{n,n}}{c_{n,k}}d_{n,k}\right)_{n,k\geq 0}.$ (37)

The set

$\\{{{\cal R}}\\}_{C}:=\left\\{(g,f)_{C}:(g,f)=(d_{n,k})_{n,k\geq 0}\in{{\cal
R}}\right\\}$ (38)

is called the $(C)$-class of ${{\cal R}}$, or the set of the $(C)$-Riordan
arrays. Similarly, $\left(\frac{c_{n,k}}{c_{n,n}}d_{n,k}\right)_{n,k\geq 0}$
is in the $(1/C)$-class of ${{\cal R}}$, where $(1/C)=(1/c_{n,k})_{n,k\geq
0}$.

Wang and Wang [31] defines the $(c)$-Riordan arrays by using the generalized
formal power series. Gould and the author [7] claims that for a sequence $(c)$
the $(c)$-class of ${{\cal R}}$, or the set of the $(c)$-Riordan arrays
$\\{{{\cal R}}\\}_{c}$, forms a group. More precisely, we have the following
theorem.

######  Theorem 4.2.

[7] Let $c=(c_{0},c_{1},c_{2},...)$ be a sequence satisfying $c_{0}=1$ and
$c_{k}\not=0$ for all $k=1,2,\ldots$, and let $\\{{{\cal R}}\\}_{c}$ be the
$(c)$-class defined by (36). Then $\\{{{\cal R}}\\}_{c}$ forms a group with
respect to the regular matrix multiplication. This group is also denoted by
${{\cal R}}_{c}$ and called the $(c)$-Riordan group with respect to the
sequence $(c)$.

###### Proof.

If $(g_{1},f_{1})_{c},(g_{2},f_{2})_{c}\in\\{(g,f)\\}_{c}$, and denote
$(g_{1},f_{1})=(d_{n,k})_{n,k\geq 0}$, $(g_{2},f_{2})=(e_{n,k})_{n,k\geq 0}$,
and
$(g_{1},f_{1})(g_{2},f_{2})=(g_{1}g_{2}(f_{1}),f_{2}(f_{1}))=(h_{n,k})_{n,k\geq
0}$, then $(d_{n,k})_{n,k\geq 0}(e_{n,k})_{n,k\geq 0}=(h_{n,k})_{n,k\geq 0}$,
and

$\displaystyle(g_{1},f_{1})_{c}(g_{2},f_{2})_{c}=$
$\displaystyle\left(\sum^{\min{n,k}}_{j=0}\left(\frac{c_{n}}{c_{j}}d_{n,j}\right)\left(\frac{c_{j}}{c_{k}}e_{j,k}\right)\right)_{n,k\geq
0}$ $\displaystyle=$
$\displaystyle\left(\frac{c_{n}}{c_{k}}\sum^{\min{n,k}}_{j=0}(d_{n,j}e_{j,k})\right)_{n,k}$
$\displaystyle=$
$\displaystyle\left(\frac{c_{n}}{c_{k}}h_{n,k}\right)_{n,k\geq
0}=(g_{1}g_{2}(f_{1}),f_{2}(f_{1}))_{c}.$

Hence, we may find $(1.t)_{c}$ is the identity of ${{\cal R}}_{c}$ and the
inverse of an element $(g,f)_{c}\in{{\cal R}}_{c}$ is
$(1/g(\overline{f}),\overline{f})$, where $\overline{f}$ is the compositional
inverse of $f$. Finally, we have

$\displaystyle\left((g,f)_{c}(d,h)_{c}\right)(u,v)_{c}=$
$\displaystyle\left(gd(f)u(h(v)),v(h(f))\right)$ $\displaystyle=$
$\displaystyle(g,f)_{c}\left((d,h)_{c}(u,v)_{c}\right).$

completing the proof. ∎

More materials on $(c)$-Riordan arrays can be found in [11, 26].

######  Example 4.3.

Let $(g,f)=(1/(1-t),t/(1-t))$. Then
$(1/(1-kt),t/(1-kt))\in\\{(1/(1-t),t/(1-t))\\}_{c}$ with $c=(c_{n})_{n\geq
0}=(k^{n})_{n\geq 0}$. The $(c)$-Riordan array $(1/(1-kt),t/(1-kt))_{c}$
begins

$\left[\begin{array}[]{cccccccc}1&0&0&0&0&0&0&\\\ k&1&0&0&0&0&0&\\\
k^{2}&k^{2}&1&0&0&0&0&\\\ k^{3}&k^{2}+k^{3}&k^{2}+k&1&0&0&0&\\\
k^{4}&2k^{3}+k^{4}&2k^{2}+2k^{3}&k^{2}+2k&1&0&0&\\\
k^{5}&3k^{4}+k^{5}&4k^{3}+3k^{4}&4k^{2}+3k^{3}&k^{2}+3k&1&0&\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right]$

######  Example 4.4.

Let $(g,f)=(1/(1-t),t/(1-t))=(\binom{n}{k})_{n,k\geq 0}$, and let
$c=(1/n!)_{n\geq 0}$. Then

$(r_{n,k})_{n,k\geq 0}=\left(\frac{n!}{k!}\binom{n}{k}\right)_{n,k\geq
0}=\left((n-k)!\left(\binom{n}{k}\right)^{2}\right)\in\\{(1/(1-t),t/(1-t))\\}_{c},$
(39)

where $c=(c_{n})_{n\geq 0}=(n!)_{n\geq 0}$. The $(c)$-Riordan array
$(r_{n,k})_{n,k\geq 0}$ begins

$(r_{n,k})_{n,k\geq 0}=\left[\begin{array}[]{cccccccc}1&0&0&0&0&0&0&\\\
1&1&0&0&0&0&0&\\\ 2&4&1&0&0&0&0&\\\ 6&18&9&1&0&0&0&\\\ 24&96&72&16&1&0&0&\\\
120&600&600&200&5&1&0&\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right].$

The matrix $(r_{n,k})_{n,k\geq 0}$ is called a rook matrix. The polynomial
$r_{n}(x)=\sum^{n}_{k=0}r(n,k)x^{n-k}$ are called the rook polynomial of $n$th
order (cf. Fielder [5] and Riordan [23]).

In general, consider a board that represents a full or a part chess board. Let
$m$ be the number of squares present in the board. Two pawns or rooks placed
on a board are said to be in non capturing positions if they are not in same
row or same column. For $2\leq k\leq m$, let $r_{k}$ denote the number of ways
in which $k$ rooks can be placed on a board such that no two rooks capture
each other. Then the polynomial $1+r_{1}x+r_{2}x^{2}+\cdots+r_{m}x^{m}$ is
called the (general) rook polynomial for the board considered. If the board is
denoted by $C$, then the corresponding polynomial is denoted by $r(C,x)$. The
rook polynomials are defined for $m\geq 2$. If $m=1$, then the board contains
only one square so that $r_{k}=0$ for $k\geq 2$, and $r(C,x)=1+x$. Hence, the
coefficient $r_{n,k}$ of the roof polynomial
$r_{n}(x)=\sum^{n}_{k=0}r(n,k)x^{n-k}$ is referred to as the number of ways in
which $k$ rooks can be placed on a $n\times n$ full chess board such that no
two rooks capture each other. The rook polynomials for $1\times 1$, $2\times
2$, $3\times 3$, $4\times 4$, and $5\times 5$ full boards are, respectively,

$\displaystyle r_{1}(x)=x+1,$ $\displaystyle r_{2}(x)=2x^{2}+4x^{1}+1,$
$\displaystyle r_{3}(x)=6x^{3}+18x^{2}+9x+1,$ $\displaystyle
r_{4}(x)=24x^{4}+96x^{3}+72x^{2}+16x+1,$ $\displaystyle
r_{5}(x)=120x^{5}+600x^{4}+600x^{3}+200x^{2}+5x+1.$

In a given board C, suppose we choose a particular square denoted as $(\ast)$.
Let $D$ denote the board obtained from $C$ by deleting the row and column
containing the square $(\ast)$, and let $E$ be the board obtained from $C$ by
deleting only the square $(\ast)$. Then the rook polynomial for the board $C$
is given by $r(C,x)=xr(D,x)+r(E,x)$. This is known as Expansion formula for
$r(C,x)$.

Denote the $(n+1)\times(n+1)$ full board by $C_{n+1}$. Let $n\times n$ full
board $C_{n}$ be located at the left upper corner of $C_{n+1}$, and let
$E_{n}$ be the board obtained from the board $C_{n+1}$ by deleting only the
square at the right lower corner of $C_{n+1}$. Then, we have

$\displaystyle r(E_{n},x)=$ $\displaystyle
r(C_{n+1},x)-xr(C_{n},x)=r_{n+1}(x)-xr_{n}(x)$ $\displaystyle=$
$\displaystyle\sum^{n+1}_{k=0}\frac{(n+1)!}{k!}\binom{n+1}{k}x^{n+1-k}-x\sum^{n}_{k=0}\frac{n!}{k!}\binom{n}{k}x^{n-k}$
$\displaystyle=$
$\displaystyle\sum^{n+1}_{k=0}\frac{n!}{k!}\left((n+1)\binom{n+1}{k}-\binom{n}{k}\right)x^{n+1-k}$
$\displaystyle=$
$\displaystyle\sum^{n}_{k=0}\frac{n!}{k!}\binom{n}{k}\left(\frac{(n+1)^{2}}{n-k+1}-1\right)x^{n+1-k}+1$
$\displaystyle=$
$\displaystyle\sum^{n}_{k=0}\frac{n!}{k!}\frac{n^{2}+n+k}{n-k+1}\binom{n}{k}x^{n+1-k}+1$
$\displaystyle=$
$\displaystyle\sum^{n}_{k=0}\frac{n^{2}+n+k}{n-k+1}(n-k)!\left(\binom{n}{k}\right)^{2}x^{n+1-k}+1.$
(40)

We denote $r(E_{n},x)=\sum^{n+1}_{k=0}E_{n,k}x^{n+1-k}$ and call it the
remainder polynomial, where

$E_{n,k}=\frac{n^{2}+n+k}{n-k+1}(n-k)!\left(\binom{n}{k}\right)^{2}$ (41)

for $0\leq k\leq n$ and $E_{n,n+1}=1$. Then, the lower triangle matrix
$(E_{n,k})_{n,k\geq 0}$, called the remainder triangle, begins

$(E_{n,k})_{n,k\geq 0}=\left[\begin{array}[]{cccccccc}0&1&0&0&0&0&0&\\\
1&3&1&0&0&0&0&\\\ 4&14&8&1&0&0&0&\\\ 18&78&63&15&1&0&0&\\\
96&504&528&184&4&1&0&\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right].$

Here is a recursive relation related to the rook triangle $(r_{n,k})_{n,k\geq
0}$ and the triangle $(E_{n,k})_{n,k\geq 0}$.

######  Proposition 4.5.

Let $(r_{n,k})_{n,k\geq 0}$ and $(E_{n,k})_{n,k\geq 0}$ be the rook triangles
and the remainder triangle defined in Example 4.4, and let
$r_{n}(x)=\sum^{n}_{k=0}r(n,k)x^{n-k}$ and
$r(E_{n},x)=\sum^{n}_{k=0}E_{n,k}x^{n-k}$ be the rook polynomial and the
remainder polynomial, respectively. Then, there exists recursive relation

$r_{n+1}(x)=xr_{n}(x)+r(E_{n},x)$ (42)

for $n\geq 0$, which can be expressed as a matrix form

$\overline{(r_{n,k})}=(r_{n,k})+(E_{n,k}),$ (43)

where $n\geq 0$ and $\overline{(r_{n,k})}$ is obtained from $(r_{n,k})$ by
deleting its first row. Furthermore, we have the expansion formula for
$r_{n+1}(x)$ as

$r_{n+1}(x)=\sum^{n}_{k=0}x^{n-k}r(E_{k},x)+x^{n+1}.$ (44)

###### Proof.

Equation (42) comes from the Expansion formula. Substituting
$r_{n}(x)=\sum^{n}_{k=0}r(n,k)x^{n-k}$ and
$r(E_{n},x)=\sum^{n}_{k=0}E_{n,k}x^{n-k}$ into (42) and comparing the
coefficients of the same powers of $x$ on the both sides yields

$r_{n+1,k}=r_{n,k}+E_{n,k},$

which can be combined as the matrix form (43) for $n,k\geq 0$. From (42) we
have

$\displaystyle r_{n+1}(x)=xr_{n}(x)+r(E_{n},x),$ $\displaystyle
xr_{n}(x)=x^{2}r_{n-1}(x)+xr(E_{n-1},x),$ $\displaystyle
x^{2}r_{n-1}(x)=x^{3}r_{n-2}(x)+x^{2}r(E_{n-2},x),$ $\displaystyle\cdots,$
$\displaystyle x^{n}r_{1}(x)=x^{n+1}r_{0}(x)+x^{n}r(E_{0},x).$

Adding up the above system and cancelling the same terms from the both sides
yields (44). ∎

######  Example 4.6.

Let $(g,f)=(1/(1-t),t/(1-t))=(\binom{n}{k})_{n,k\geq 0}$. Then

$\displaystyle(L_{n,k})_{n,k\geq 0}=$
$\displaystyle\left(\frac{(-1)^{n}/(n)_{n}}{(-1)^{k}/(n)_{k}}\binom{n}{k}\right)_{n,k\geq
0}$ $\displaystyle=$
$\displaystyle\left(\frac{(-1)^{n-k}}{(n-k)!}\binom{n}{k}\right)\in\\{(1/(1-t),t/(1-t))\\}_{C},$
(45)

where $C=(c_{n,k})_{n,k\geq 0}=((-1)^{n}/(n)_{k})_{n,k\geq 0}$. The
$(C)$-Riordan array $(L_{n,k})_{n,k\geq 0}$ begins

$\left[\begin{array}[]{cccccccc}1&0&0&0&0&0&0&\\\ -1&1&0&0&0&0&0&\\\
\frac{1}{2}&-2&1&0&0&0&0&\\\ -\frac{1}{6}&\frac{3}{2}&-3&1&0&0&0&\\\
\frac{1}{24}&-\frac{2}{3}&3&-4&1&0&0&\\\
-\frac{1}{120}&\frac{5}{24}&-\frac{5}{3}&5&-5&1&0&\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right].$

The matrix $(L_{n,k})_{n,k\geq 0}$ is called the Laguerre matrix. The
polynomial $L_{n}(n,k)=\sum^{n}_{k=0}L_{n,k}x^{n-k}$ are called the Laguerre
polynomial of order $n$ (cf., for example, Hsu, Shiue, and the author [12] and
Riordan [23]).

Comparing (39) and (4.6) and noting

$r_{n,k}=\frac{n!}{k!}\binom{n}{k},$

we immediately obtain

$r_{n,n-k}=\frac{n!}{(n-k)!}\binom{n}{k}=\frac{n!}{(n-k)!}(-1)^{n-k}(n-k)!L_{n,k}=(-1)^{n-k}n!L_{n,k},$
(46)

which implies

$\sum^{n}_{k=0}r_{n,n-k}x^{n-k}=n!\sum^{n}_{k=0}L_{n,k}(-x)^{n-k}=n!L_{n}(-x)$
(47)

for $0\leq k\leq n$. Since the left-hand side of (47) can be written as

$\displaystyle\sum^{n}_{k=0}r_{n,n-k}x^{n-k}=$
$\displaystyle\sum^{n}_{k=0}r_{n,k}x^{k}=x^{n}\sum^{n}_{k=0}r_{n,k}\left(\frac{1}{x}\right)^{n-k}=x^{n}r_{n}\left(\frac{1}{x}\right).$

Hence, we have the well-known formula

$r_{n}(x)=n!x^{n}L_{n}\left(-\frac{1}{x}\right)$

for $n\geq 0$.

######  Definition 4.7.

We may extend the rook triangle and the rook polynomials to a general case by
defending the following generalized rook triangle and generalized rook
polynomials:

$(\hat{r}_{n,k})_{n,k\geq 0}=\left(\frac{n!}{k!}d_{n,k}\right)_{n,k\geq
0}=\left((n-k)!\binom{n}{k}d_{n,k}\right)\in\\{(g(t),f(t)\\}_{c},$ (48)

where $c=(c_{n})_{n\geq 0}=(n!)_{n\geq 0}$ and $(g,f)=(d_{n,k})_{n,k\geq 0}$.
In addition, the polynomial

$\hat{r}_{n}(x)=\sum^{n}_{k=0}\hat{r}_{n,k}x^{n-k}$ (49)

is referred to as the generalized rook polynomial of degree $n$.

We may also extend the Laguerre triangle and the Laguerre polynomials to a
general case by defending the following generalized Riordan type Laguerre
triangle and generalized Riordan type Laguerre polynomials:

$\displaystyle(\hat{L}_{n,k})_{n,k\geq 0}=$
$\displaystyle\left(\frac{(-1)^{n}/(n)_{n}}{(-1)^{k}/(n)_{k}}d_{n,k}\right)_{n,k\geq
0}$ $\displaystyle=$
$\displaystyle\left(\frac{(-1)^{n-k}}{(n-k)!}\binom{n}{k}d_{n,k}\right)\in\\{(g(t),f(t)\\}_{C},$
(50)

where $C=(c_{n,k})_{n,k\geq 0}=(n!)_{n\geq 0}$ and $(g,f)=(d_{n,k})_{n,k\geq
0}$. In addition, the polynomial

$\hat{L}_{n}(x)=\sum^{n}_{k=0}\hat{L}_{n,k}x^{n-k}$ (51)

is referred to as the generalized Laguerre polynomial of degree $n$.

Similar to the relationship between the rook triangle and the Laguerre
triangle and the relationship between the rook polynomials and the Laguerre
polynomials, we may extend the relationships to the general cases as follows.

######  Theorem 4.8.

Let the generalized rook triangle and the rook polynomials and the generalized
Riordan type Laguerre triangle and the generalized Riordan type Laguerre
polynomials be defined in Definition 4.7. Then we have

$\hat{r}_{n,n-k}=(-1)^{n-k}n!\hat{L}_{n,k}$ (52)

for $0\leq k\leq n$ and

$\hat{r}_{n}(x)=n!x^{n}\hat{L}_{n}\left(-\frac{1}{x}\right)$

for $n\geq 0$.

From Examples 4.4 and 53, we have a relationship between

######  Theorem 4.9.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array, and let $(g,f)_{c}$ and
$(g,f)_{C}$ be defined in Definition 4.1. Then
$(g,f)_{c}=(d^{(c)}_{n,k})_{n,k\geq 0}$ and $(g,f)_{C}=d^{(C)}_{n,k\geq 0}$
satisfy the horizontal recursive relations, which are extensions of $A$\- and
$Z$-sequence representation (4) and (6) of $(g,f)$ to $(g,f)_{c}$ and
$(g,f)_{C}$, respectively:

$d^{(c)}_{n+1,k+1}=\frac{c_{n+1}}{c_{n}c_{k+1}}\sum^{n-k}_{j=0}a_{j}c_{k+j}d^{(c)}_{n,k+j}$
(53)

for $k\geq 0$ and

$d^{(c)}_{n+1,0}=\frac{c_{n+1}}{c_{n}}\sum^{n}_{j=0}z_{j}c_{j}d^{(c)}_{n,j}$
(54)

as well as

$d^{(C)}_{n+1,k+1}=\frac{c_{n+1,n+1}}{c_{n,n}c_{n+1,k+1}}\sum^{n-k}_{j=0}a_{j}c_{n,k+j}d^{(C)}_{n,k+j}$
(55)

for $k\geq 0$ and

$d^{(C)}_{n+1,0}=\frac{c_{n+1,n+1}}{c_{n,n}}\sum^{n}_{j=0}z_{j}c_{n,j}d^{(C)}_{n,j}.$
(56)

######  Theorem 4.10.

Let $(g,f)=(d_{n,k})_{n,k\geq 0}$ be a Riordan array, and let $(g,f)_{c}$ and
$(g,f)_{C}$ be defined in Definition 4.1. Then
$(g,f)_{c}=(d^{(c)}_{n,k})_{n,k\geq 0}$ and $(g,f)_{C}=d^{(C)}_{n,k\geq 0}$
satisfy the vertical recursive relations, which are extensions of vertical
recursive relation (12) of the entries of $(g,f)$ to $(g,f)_{c}$ and
$(g,f)_{C}$, namely,

$d^{(c)}_{n,k}=\frac{c_{n}}{c_{k}}\sum^{n-k+1}_{j=1}f_{j}\frac{c_{k-1}}{c_{n-j}}d^{(c)}_{n-j,k-1}$
(57)

and

$d^{(C)}_{n,k}=\frac{c_{n,n}}{c_{n,k}}\sum^{n-k+1}_{j=1}f_{j}\frac{c_{n-j,k-1}}{c_{n-j,n-j}}d^{(C)}_{n-j,k-1},$
(58)

respectively.

######  Example 4.11.

It is known that the $A$\- and $Z$\- sequences of $(1/(1-t),t/(1-t))$ are
$(1,1,0,\ldots)$ and $(1,0,0,\ldots)$, respectively. Then for the rook
triangle represented in Example 4.4 with $d_{n,k}=\binom{n}{k}$, from (53) and
(54), we obtain the horizontal recursive relations for the entries of the rook
triangle,

$r_{n,k}=nr_{n-1,k}+\frac{n}{k}r_{n-1,k-1}$ (59)

for $k\geq 1$ and

$r_{n,0}=nr_{n-1,0}.$ (60)

From (57) and noting (13) we obtain the vertical recursive relation for the
entries of the rook triangle

$r_{n,k}=\sum^{n-k+1}_{j=1}\frac{(n)_{j}}{k}r_{n-j,k-1}$ (61)

for $k\geq 1$.

From (39), one may obtain an equivalent identity of (59)

$\binom{n}{k}^{2}=\frac{n}{n-k}\binom{n-1}{k}^{2}+\frac{n}{k}\binom{n-1}{k-1}^{2}.$

Simialrly, from (61)

$\binom{n}{k}^{2}=\sum^{n-k+1}_{j=1}\frac{(n)_{j}}{k(n-k)_{j-1}}\binom{n-j}{k-1}^{2}.$

######  Example 4.12.

Similarly, for the Laguerre triangle represented in Example 4.6 with
$d_{n,k}=\binom{n}{k}$, from (53) and (54), we obtain the horizontal recursive
relations for the entries of the Laguerre triangle,

$L_{n,k}=L_{n-1,k-1}-\frac{1}{n-k}L_{n-1,k}$ (62)

for $k\geq 1$ and

$L_{n,0}=-\frac{1}{n}L_{n-1,0}.$ (63)

From (57) and noting (13) we obtain the vertical recursive relation for the
entries of the Laguerre triangle

$L_{n,k}=\frac{1}{(n-k)!}\sum^{n-k+1}_{j=1}(-1)^{j-1}(n-k-j+1)!L_{n-j,k-1}$
(64)

for $k\geq 1$. It can be seen that (64) is equivalent to (13).

## References

  * [1] P. Barry, Riordan Arrays: A Primer, LOGIC Press, Kilcock, 2016.
  * [2] G.-S. Cheon and S.-T. Jin, Structural properties of Riordan matrices and extending the matrices, Linear Algebra Appl. 435 (2011), 2019–2032.
  * [3] G.-S. Cheon, M. Song , A new aspect of Riordan arrays via Krylov matrices, Linear Algebra Appl., 554 (2018), 329–341.
  * [4] G.-S. Cheon, A. Luzón, M. A. Morón, L. F. Prieto-Martinez, and M. Song, Finite and infinitedimensional Lie group structures on Riordan groups, Adv. Math. 319 (2017), 522–566.
  * [5] D. C. Fielder, A generator of rook polynomials, Mathematica J. 9 (2004), 371–375.
  * [6] P. J. Forrester and D.-Z. Liu, Raney distributions and random matrix theory, J. Stat. Phys. 158 (2015), 1051–1082.
  * [7] H. Gould and T.-X. He, Characterization of (c)-Riordan arrays, Gegenbauer-Humbert-Type polynomial sequences, and (c)-Bell polynomials, J. Math. Research Appl. 33 (2013), No.5, 505-527.
  * [8] R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989.
  * [9] T. -X. He, Parametric Catalan numbers and Catalan triangles. Linear Algebra Appl. 438 (2013), no. 3, 1467 -1484.
  * [10] T. -X. He, Matrix characterizations of Riordan matrices, Linear Algebra Appl. 465 (2015), 15-42.
  * [11] T.-X. He, Methods for the Summation of Series, Chapman and Hall/CRC, New York, 2022.
  * [12] T.-X. He, L. C. Hsu, and P. J.-S. Shiue, The Sheffer Group and the Riordan Group, Discrete Appl. Math. 155 (2007), 1895–1909.
  * [13] T. -X. He and R. Sprugnoli, Sequence Characterization of Riordan Arrays, Discrete Math., 309 (2009), 3962-3974.
  * [14] T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Linear Algebra Appl. 532 (2017), 25–42.
  * [15] C. Jean-Louis and A. Nkwanta, Some algebraic structure of the Riordan group, Linear Algebra Appl. 438 (2013), 2018–2035.
  * [16] J.-G. Liu and R. Pego, On generating functions of Hausdorff moment sequences, to appear in Trans. Am. Math. Soc.
  * [17] A. Luzón, M. A. Morón, and L. F. Prieto-Martinez, A formula to construct all involutions in Riordan matrix groups. Linear Algebra Appl. 533 (2017), 397–417.
  * [18] A. Luzón, M. A. Morón, and L. F. Prieto-Martinez, The group generated by Riordan involutions, Revista Matemática Complutense https://doi.org/10.1007/s13163-020-00382-8, 2020.
  * [19] J. Mao, L. Mu, Y. Wang, Yet another criterion for the total positivity of Riordan arrays, Linear Algebra Appl. 634 (2022), 106–111.
  * [20] D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan matrices, Canadian J. Math., 49 (1997), 301–320.
  * [21] W. Mlotkowski, Fuss-Catalan numbers in noncommutative probability, Documenta Mathematica, 15 (2010), 939–955.
  * [22] K. A. Penson and K. Życzkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions, Phys. Rev. E, 83 (2011) 061118, 9 pp.
  * [23] J. Riordan, An Introduction to Combinatorial Analysis, ”The Problem of the Rooks,” ”Properties of Rook Polynomials,” and ”Rectangular Boards.” $ 7.2, 7.3, and 7.4, New York: Wiley, pp. 164–170, 1958.
  * [24] D. G. Rogers, Pascal triangles, Catalan numbers and renewal matrices, Discrete Math., 22 (1978), 301–310.
  * [25] L. W. Shapiro, S. Getu, W. J. Woan and L. Woodson, The Riordan group, Discrete Appl. Math. 34(1991) 229–239.
  * [26] L. W. Shapiro, R. Sprugnoli, P. Barry, G.-S. Cheon, T.-X. He, D. Merlini, and W. Wang, The Riordan Group and Applications, Springer, New York, 2022.
  * [27] L. W. Shapiro, Bijections and the Riordan group, Random generation of combinatorial objects and bijective combinatorics, Theoret. Comput. Sci. 307 (2003), no. 2, 403–413.
  * [28] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
  * [29] R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999.
  * [30] R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015.
  * [31] W. Wang and T. Wang, Generalized Riordan arrays, Discrete Math. 308 (2008), 6466–6500.