# On the zero-field quantization of the anomalous quantum Hall effect in moiré
2D layers

Sankar Das Sarma  Condensed Matter Theory Center and Joint Quantum Institute,
Department of Physics, University of Maryland, College Park, Maryland 20742,
USA  Ming Xie  Condensed Matter Theory Center and Joint Quantum Institute,
Department of Physics, University of Maryland, College Park, Maryland 20742,
USA

###### Abstract

In recent breakthrough experiments, twisted moiré layers of transition metal
dichalcogenides are found to manifest both integer (IQAHE) and fractional
(FQAHE) quantum anomalous Hall effects in zero applied magnetic field because
of the underlying flat band topology and spontaneous breaking of the time
reversal invariance. In the current work, we critically analyze the
experimental values of the quantized conductance in each case to emphasize the
role of disorder in the problem, pointing out that obtaining accurate
quantized conductance in future experiments would necessitate better contacts
and lower disorder.

Recent experimental breakthroughs in multilayer moiré transition metal
dichalcogenides (mTMD) Cai2023 ; Zeng2023 ; Park2023 ; Fan2023  as well as in
pentalayer graphene PentaGraphene2023  report the unambiguous observation of
integer and fractional quantization of anomalous Hall effect without any
Landau quantization imposed by an external magnetic field. Although the effect
was in some loose sense predicted theoretically Tang2011 ; Sun2011 ;
Neupert2011 ; Regnault2011 ; Sheng2011 ; Shuo2012a ; Shuo2012b  in
hypothetical flatband topological systems, nevertheless the remarkable
discovery of the IQAHE and FQAHE in relatively dirty two-dimensional (2D)
systems (with mobility $\sim$ 1000 cm2/Vs for the mTMDs) with rather accurate,
but not precise, resistance quantization has come as a huge surprise, and
created considerable interestLi2021 ; ValentinFCI ; ReddyFCI ; ChongFCI ;
NicolasFCI ; ParkerCFL ; FuCFL ; Yu2023 .

While much of the interest is on the origin of the interaction-induced zero-
field fractional states, we focus on the issue of the accuracy of the
quantization itself because of the extremely dirty nature of the underlying 2D
materials. Why is the quantization seen at all in systems dominated by
disorder? Most of the current and past theoretical literature on IQAHE and
FQAHE completely ignore disorder and consider the emergence of topological
quantum Hall states in clean systems. An earlier work by one of the authors
takes into account its influence in a numerical exact diagonalization study of
a topological flatband model with onsite impurity potential Shuo2012a . In the
current work, we quantitatively analyze the quantized resistance reported in
the mTMD systems of Refs. Park2023, ; Fan2023,  inferring that the expected
future progress in sample fabrication and materials development should lead to
(1) extremely accurate IQAHE quantization, possibly leading to a zero field
resistance standard, and (2) observations of many more FQAHE states currently
inhibited by disorder effects in the samples.

As a background, a mobility of $\sim$1000 cm2/Vs in samples of Refs. Park2023,
in twisted MoTe2 homobilayers Xuprivate  corresponds to a disorder induced
broadening of $\sim\,$$0.75\text{\,}\mathrm{meV}$
($\sim\,$$9\text{\,}\mathrm{K}$). By contrast, the first observation in 1982
of the fractional quantum Hall effect (FQHE) at $1/3$ filling (and an applied
field of $15\text{\,}\mathrm{T}$) used a 2D GaAs sample with a mobility of
100,000 cm2/Vs which corresponds to a disorder broadening of 0.1 meV
($\sim\,$$1\text{\,}\mathrm{K}$)—the estimated FQHE gap was roughly
$1.5\text{\,}\mathrm{K}$ in this sample Tsui1982 . It is well-known that FQHE
is strongly adversely affected by disorder dAmbrumenil2011 ; Morf2002 ;
Morfarxiv ; Qian2017 ; Kleinbaum2020 ; Dean2008 ; Boebinger1987 . Typically,
the pristine fractional quantum Hall gap should be larger than the disorder
strength for the fractional phase to be observable, implying that not only are
all the currently observed FQAHE states in Refs. Park2023, ; Fan2023,  likely
to have much larger intrinsic gaps than the reported gaps in the existing
dirty samples with an $\sim\,$$9\text{\,}\mathrm{K}$ disorder, but also many
more FQAHE phases with pristine gaps $<\,$$9\text{\,}\mathrm{K}$ would
manifest in future experiments as the system disorder is suppressed through
materials improvement. In particular, it is possible that FQAHE corresponding
to non-Abelian fractional quantum Hall states at even denominator fillings
would soon be observed in zero magnetic fields as system disorder improves
since the connection between improved sample quality and robust even
denominator FQHE is well-established Ahn2022 ; Chung2021 .

By contrast, the integer quantum Hall effect, being a single-particle
topological effect directly manifesting a Chern number, should be stable to
disorder unless the disorder is much larger than the (typically large)
excitation gap for the integer quantum Hall effect (whence strong overlap
among the disorder broadened Landau levels considerably complicate the
physics). In fact, disorder is essential in producing the quantized Hall
resistance plateaus as it enables the chemical potential to move smoothly
through the gapped localized regions between Landau levels—too much disorder,
much larger than the pristine gap, complicates the situation by producing
overwhelming localization, but with no disorder, there would be no quantum
Hall plateaus whatsoever. In the recent mTMD QAHE experimentsCai2023 ;
Park2023 ; Fan2023 , the zero-field integer excitation gap is estimated to be
$>$40K, much larger than the disorder strength of $\sim$9K, so disorder should
be irrelevant to the observed FQAHE. The central paradigm of the quantum Hall
effect as arising from broadened Landau levels, where the disorder-induced
localization at the edges of the Landau levels gives rise to the quantized
plateaus in the transverse resistance and vanishing resistance in the
longitudinal resistance, should apply to IQAHE as well. At finite
temperatures, the quantization of the Hall resistance (and the vanishing of
the longitudinal resistance) is not perfect because of carrier activation and
variable range hopping transport in the localized regime.

We mention that the original discovery of IQHE already achieved a Hall
resistance quantization close to 1 part in $10^{5}$ at
$1.8\text{\,}\mathrm{K}$ with the longitudinal resistance being essentially
zero ($\sim$$1\text{\,}\mathrm{\SIUnitSymbolOhm}$) Klitzing1980  to be
contrasted with the currently measured longitudinal resistance
$>\,$$100\text{\,}\mathrm{\SIUnitSymbolOhm}$ in the AQHE experiment. We
believe that this difference arises entirely from the contact resistance
problems in mTMD layers, where the noise arising from the contact resistance
makes an accurate quantization a challenge independent of the cleanliness of
the sample. Future improvement in contact fabrication would be the key to
progress in studying IQAHE and FQAHE.

Figure 1:  Thermal activation fitting of the longitudinal resistance $R_{xx}$
at filling factor $\nu=-1$ and electric field $D/\epsilon_{0}=0$. Data is for
device $D(3.9^{\circ})$ taken from the Source Data of Fig.2d in Ref. Park2023,
. (a,b) are fittings without contact resistance and (c,d) are fittings with a
contact resistance $R_{0}$ as a fitting parameter. The red lines are
experiment data and blue lines are the fitting curves. Data in the shaded
region is excluded from fitting.

In the rest of this work, we provide detailed quantitative fits to the
experimental IQAHE and FQAHE finite temperature data to understand the
quantization accuracy and the role of disorder. In carrying out the fitting,
it becomes clear that the contact resistance plays a significant role as is
obvious from the large background resistance as well the large amount of noise
in the measured resistance even at cryogenic temperatures. In fact, the most
dramatic qualitative difference between the observed zero-field IQAHE/FQAHE
and the well-established high-field 2D IQHE/FQHE is that the large residual
values of the longitudinal resistivity in the IQAHE/FQAHE in the quantized
Hall plateaus which tend to be vanishingly small in regular IQHE/FQHE. We show
through our quantitative analysis that this large residual resistance arises
mainly from the uncontrolled contact problem in the current IQAHE/FQAHE
samples. The exact nature of the residual resistance remains unclear and
demands future experimental study.

Our fittings use activation, Mott variable range hopping, and Efros-Shklovskii
(ES) variable range hopping behaviors separately since which of these
localized transport mechanisms is more dominant in a particular temperature
range in the IQAHE/FQAHE measurements of Refs. Park2023, ; Fan2023,  are a
priori unknown. In addition to the activated temperature dependence, we also
carry out fitting including a contact resistance $R_{0}$ as a free fitting
parameter,

Figure 2:  (a) Mott and (b) Efros-Shklovskii variable range hopping fitting of
the longitudinal resistance $R_{xx}$ at filling factor $\nu=-1$ and electric
field $D/\epsilon_{0}=0$. The source data is the same as used in Fig. 1. The
red lines are experiment data and blue lines are the fitting curves. Data in
the shaded region is excluded from fitting. $T_{0}$ is a parameter in variable
range hopping models in Eq. (2) and (3).  Figure 3:  Thermal activation
fitting of the longitudinal resistance $R_{xx}$ at filling factor $\nu=-1$ and
electric field $D/\epsilon_{0}=0$ for device $D(3.5^{\circ})$. Data is taken
from the Source Data of Extended Fig.2b in Ref. Park2023, . (a) is fitting
without contact resistances and (b-c) are fittings with a contact resistance
$R_{0}$ as a fitting parameter. The red lines are experiment data and blue
lines are the fitting curves. Data in the shaded regions is excluded from
fitting.  Figure 4:  Mott variable range fitting of the same data as presented
in Fig. 3. From (a) to (c), the fitting range in $T^{-1}$ is decreased.
$R_{0}$ is set to zero in all three cases.

$\displaystyle R(T)$ $\displaystyle=R_{0}+Ae^{-\Delta/2k_{B}T}$ (1)

where $\Delta$ is the thermal activation gap and $A$ is the proportional
constant which is also a fitting parameter. In the presence of disorder,
variable range hopping mechanisms can also manifest, thereby contributing to
the electrical resistance observed. In two-dimensional systems, the
temperature dependence of the resistance takes the form

$\displaystyle R(T)$ $\displaystyle=R_{0}+Ae^{-(T_{0}/T)^{1/3}}$ (2)

for Mott variable range hopping Mott1969  and

$\displaystyle R$ $\displaystyle=R_{0}+Ae^{-(T_{0}/T)^{1/2}}$ (3)

for ES variable range hoppingES1975 . In both cases, we carry out fit with
both $R_{0}$ set to 0 and $R_{0}$ as a free fitting parameter. $A$ and $T_{0}$
are model dependent constants treated as free fitting parameters as well.

We analyze two situations from Ref. Park2023,  for the IQAHE: two different
devices $D(3.9^{\circ})$ and $D(3.5^{\circ})$ with twist angle
$\theta=3.9^{\circ}$ and $3.5^{\circ}$ respectively. We also analyze the IQAHE
data of Ref. Fan2023, .  Figure 1 show the thermal activation fits for
$\theta=3.9^{\circ}$ with the top two panels without any contact resistance,
and the bottom two with a contact resistance $R_{0}$ which is extracted from
the best fits. The top two panels show that the fits provide very different
activation energies depending on the fit regions, whereas the bottom two
panels, including the contact resistance, achieve decent fits over the whole
regime with mutually consistent residual (contact) resistance and energy gaps.
If we restrict our activation fits to lower temperatures, where activation
typically dominates, we get $R_{0}\sim 300~{}\Omega$ and gap$\sim$ 70K. We
note that the extracted $R_{0}\sim 300~{}\Omega$ is $2-3$ orders of magnitude
larger than in IQHE for 2D semiconductors and graphene, emphasizing the
considerable scope for materials improvement in IQAHE samples. The two panels
of  Fig. 2 show the Mott and ES variable range hopping fits including the
contact resistance, and the fits are both reasonable, but the important point
is that the extracted contact resistance is roughly consistent with that in
Fig. 1. Currently existing experimental results cannot sharply distinguish
among the localized transport mechanisms, but they all lead to similar contact
resistance which is much larger than the corresponding residual resistance in
regular IQHE.

In Fig. 3, we show the activation fits to the longitudinal resistance at
$\theta=3.5^{\circ}$, finding again a gap $\sim$ 40K, However, the fitted
contact resistance can be much larger in this case $R_{0}\sim
0.4\,-$$1.5\text{\,}\mathrm{k\SIUnitSymbolOhm}$, depending on the choice of
cutoff, which cannot really be precisely pinned down given the limitation of
the available experimental data. In Fig. 4, we reinforce the inadequacy of the
current data in precisely pinning down the localized transport mechanism by
showing fits to Mott variable range hopping by modifying the regime of
fitting—all give good fits ($R_{0}$=0 in these fits to reduce the number of
unknowns), but the fitting parameters are different. Similar behavior is also
seen in the ES fits (not shown).

Figure 5:  (a) Thermal activation, (b) Mott variable range hopping, and (c) ES
variable range hopping fittings of the longitudinal resistance $R_{xx}$ at
filling factor $\nu=-1$ in Ref. Fan2023, . The red lines are experiment data
and blue lines are the fitting curves. The data is extracted from SFig.4b for
$D=-15\,$mV/nm in Ref. Fan2023, .

In  Fig. 5, we show resistance fits for the IQFAH data from the Ref. Fan2023,
using activation, Mott, and ES models including a background resistance
$R_{0}$ in all cases. The fitted gap is around $\sim$$47\text{\,}\mathrm{K}$
for the thermal activation with contact resistance, which is 2-3 times of the
value without considering the contact resistance ($\sim
10-$$20\text{\,}\mathrm{K}$ according to Fig.3c of Ref. Fan2023, ). The
contact resistances $R_{0}\sim 5-$$7\text{\,}\mathrm{k\SIUnitSymbolOhm}$ are
similar and significant from all the fitting scenarios, showing that the
physics is likely to be dominated by the contact resistance problem. We
emphasize that similar analyses give vanishing $R_{0}$
($\sim$$1\text{\,}\mathrm{\SIUnitSymbolOhm}$) for regular high field 2D IQHE
phenomena, reiterating the serious sample problem plaguing the zero field
IQAHE/FQAHE physics. The contact resistance in the sample of Ref. Fan2023,  is
an order of magnitude larger than that of the devices in Ref. Park2023, . Our
work establishes that the difference in the degree of quantization between
them is most likely not arising from the intrinsic IQAHE excitation gap
($\sim$40K in both experiments), but can be improved through materials
improvement.

Figure 6:  Thermal activation fitting of the $R_{xx}$ at filling factor
$\nu=-2/3$ and electric field $D/\epsilon_{0}=0$. Data is for device
$D(3.7^{\circ})$ taken from the Source Data of Extended Fig.5a in Ref.
Park2023, . (a) is fittings without contact resistance and (b-d) are fittings
with a contact resistance $R_{0}$. The red lines are experiment data and blue
lines are the fitting curves. Data in the shaded region is excluded from
fitting.

In the Supplemental Material (SM), we provide detailed results on the
deviations of the measured longitudinal resistance from zero and the Hall
resistance from the quantized value (as well as an empirical relationship
between them) as a function of temperature, displacement field to clearly
demonstrate that the IQAHE quantization is quite inaccurate and is very far
from being useful for a resistance standard. But sample improvements should
make the situation better in the future.

Figure 7:  Thermal activation fitting of the $R_{xx}$ at filling factor
$\nu=-3/5$ and electric field $D/\epsilon_{0}=0$. Data is for device
$D(3.7^{\circ})$ taken from the Source Data of Extended Fig.6b in Ref.
Park2023, . (a) is fittings without contact resistance and (b,c) are fittings
with a contact resistance $R_{0}$. The red lines are experiment data and blue
lines are the fitting curves. Data in the shaded region is excluded from
fitting.

Now, we discuss, for the sake of completeness, the corresponding analyses of
the resistivity data for FQAHE, which, we must warn, is hopelessly inadequate
because of the huge fluctuations in the measured resistance. In Fig. 6, we
show the activation fits to the resistivity for $\theta=3.7^{\circ}$ and the
fractional filling of $-2/3$. The four panels show fits to the same data using
$R_{0}=0$ and finite $R_{0}$ varying the fit regimes. The extracted FQAHE gap
is $\sim$$23\text{\,}\mathrm{K}$ without any contact resistance correction,
but with a finite $R_{0}$ the situation becomes intractable because of the
paucity of data with both the gap and $R_{0}$ changing with varying fit
regimes. We emphasize that the available data are so noisy that any fit works
only over a very narrow range of temperature ($\sim$ 1K only!), and not much
significance should be attached to these fits. The fact that in some fitting
windows $R_{0}$ even becomes negative may suggest that the transport behavior
cannot be well captured by the thermal activation model if we were to rely on
the limited data. The only conclusion is that an activation gap
$\sim$$23\text{\,}\mathrm{K}$ is measured over a narrow regime, but the
quantization itself is extremely noisy, most likely because of considerable
measurement noise arising from the contact resistance. In  Fig. 7, we show
similar activation fits to the $-3/5$ FQAHE in Ref. Park2023,  for
$\theta=3.7^{\circ}$, finding again that the resistivity is simply too noisy
for any decisive conclusion. The gap at $-3/5$ varies between
$14\text{\,}\mathrm{K}$ and $32\text{\,}\mathrm{K}$ depending on the details
of the fits, all of which are, however, over very narrow regimes of
temperatures. The extracted contact resistance is large, $R_{0}\sim
0.7-$$1.6\text{\,}\mathrm{k\SIUnitSymbolOhm}$, again showing that the contact
resistance problem is severe in the IQAHE/FQAHE measurements. The
corresponding regular high-field FQHE measurements basically manifest zero
contact resistance, and the activation fits work over at least one decade in
temperature. It is clear that the noise problem arising from contact
resistance is huge in the current IQAHE/FQAHE experiments, and any firm
quantitative conclusions on the quantization accuracy must await much better
samples with much less noise and much smaller contact resistance. Our similar
analysis (not shown) for the FQAHE data at filling $-2/3$ in Ref. Fan2023,
has similar fitting problems, and the extracted contact resistance
$R_{0}>$$1\text{\,}\mathrm{k\SIUnitSymbolOhm}$ (being consistent with the
corresponding IQAHE analysis) with the extracted gap being very small
($\sim$$2\text{\,}\mathrm{K}$), indicating a rather disordered sample.

In conclusion, we have carried out a detailed quantitative analysis of the
recently reported IQAHE and FQAHE resistivity results, finding a large contact
resistance, considerably complicating the physics of quantization even when
the extracted excitation gap is large. We remark that four probe transport
measurements should eliminate the usual contact resistance artifacts, and
therefore our finding of a large residual resistance (of unknown origin)
through a direct analysis of the experimental data is somewhat of a mystery.
In addition, the contact resistance is very noisy, producing large resistance
fluctuations, which compromise the quantization accuracy. Further materials
improvement and better sample fabrication should improve the situation, but
until that happens, the zero field IQAHE/FQAHE will not be useful in the
implementation of zero field resistance standards.

###### Acknowledgements.

Acknowledgment.—The authors acknowledge helpful discussions with Xiaodong Xu,
Jiaqi Cai, Tingxin Li and Long Ju. This work was supported by Laboratory for
Physical Sciences.

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## Supplemental Material

### Deviation of longitudinal and Hall resistances from quantization

In ideal clean systems, the hallmark of quantum (anomalous) Hall effect is the
quantization of the Hall resistance to an integer or rational fractional value
and the vanishing of the the longitudinal resistance. Finite temperature
effect and the presence of disorder cause both transport coefficients to
deviate from their ideal values. Here we examine the temperature dependence of
the deviations, specifically the non-vanishing longitudinal resistance
$R_{xx}$ and the deviation of the Hall resistance from quantized value $\delta
R_{xy}=R_{xy}-h/e^{2}$ for the IQAHE case. Figure 8 plots $R_{xx}$ and $\delta
R_{xy}$ at different displacement field. For $D/\epsilon_{0}$ away from the
critical value $D_{c}/\epsilon_{0}=150\,$mV/nm, $\delta R_{xy}$ is
approximately proportional to $R_{xx}^{2}$. For $D/\epsilon_{0}$ close to the
critical value, this relation breaks down. The above behavior does not hold
for $R_{xx}\lesssim\,$$1\text{\,}\mathrm{k\SIUnitSymbolOhm}$, where the
temperature dependence of both $R_{xx}$ and $\delta R_{xy}$ are noisy.

Figure 8:  Deviation of $R_{xx}$ and $R_{xy}$ from quantization values for the
IQAHE. (a-d) Temperature dependence of $R_{xx}$and $\delta R_{xy}$ at
different displacement field. (e-h) $\delta R_{xy}$ as a function of $R_{xx}$
with temperature as the implicit varying parameter. Data is taken from the
Source Data of Fig.2d and Extended Fig.3e for device $D(3.9^{\circ})$ in Ref.
3.


# Centrosymmetric and reverse matrices in bivariate orthogonal polynomials

Cleonice F. Bracciali  [C. F. Bracciali] Departamento de Matemática, IBILCE,
UNESP - Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP,
Brazil<EMAIL_ADDRESS>,  Glalco S. Costa  [G. S. Costa]
Departamento de Matemática, Instituto de Ciências Tecnológicas e Exatas \-
ICTE, Universidade Federal do Triângulo Mineiro - UFTM, 38025-180, Uberaba,
MG, Brazil<EMAIL_ADDRESS>and  Teresa E. Pérez  [T. E. Pérez]
Instituto de Matemáticas IMAG & Departamento de Matemática Aplicada, Facultad
de Ciencias. Universidad de Granada. 18071. Granada, Spain<EMAIL_ADDRESS>
###### Abstract.

We introduce the concept of reflexive moment functional in two variables and
the definition of reflexive orthogonal polynomial system. Also reverse
matrices and their interesting algebraic properties are studied. Reverse
matrices and reflexive polynomial systems are directly connected in the
context of bivariate orthogonal polynomials. Centrosymmetric matrices, reverse
matrices and their connections with reflexive orthogonal polynomial systems
are presented. Finally, several particular cases and examples are analysed.

###### Key words and phrases:

Bivariate orthogonal polynomials, centrosymmetric matrices, reverse matrices,
reflexive weight functions, reflexive orthogonal polynomial systems

###### 2020 Mathematics Subject Classification:

Primary: 42C05; 33C50; 15A09; 15B99

##  1\. Introduction

When working with vectorial approach for multivariate orthogonal polynomials,
it is necessary to make choices, such as the monomial order, which does not
occur in the univariate case (see [11]). In addition to the choice of working
with orthogonal, monic orthogonal or orthonormal polynomial vectors, the
choice of a basis can simplify some theoretical objects, such as, coefficient
matrices of three-term relations, structure relations, or explicit expressions
for the polynomial vectors, among others. Within scope of the study of
multivariate polynomials, many difficulties arise naturally with the increase
of the dimension in relation to the univariate case. Then, it is important to
simplify the objects’ treatment as much as possible. The present work is
inserted in this context.

Following [11], we consider the linear space of real polynomials in two
variables given by $\Pi=\mathrm{span}\\{x^{h}\,y^{k}:h,k\geqslant 0\\}$ and
the linear space $\Pi_{n}=\mathrm{span}\\{x^{h}\,y^{k}:h+k\leqslant n\\}$ of
finite dimension $(n+1)(n+2)/2$. A polynomial of total degree $n$ in two
variables is a linear combination of monomials of total degree less than or
equal to $n$. As usual, a two variable polynomial of degree $n$,
$p(x,y)\in\Pi_{n}$, is given by

$p(x,y)=\sum_{i+j\leqslant n}c_{i,j}\,x^{i}\,y^{j},\quad
c_{i,j}\in\mathbb{R}.$

We say that $p(x,y)$ is monic if only has one term of highest degree in the
form

$p(x,y)=x^{n-k}\,y^{k}+\sum_{i+j\leqslant n-1}\,c_{ij}\,x^{i}\,y^{j},\quad
c_{ij}\in\mathbb{R}.$

The vector representation for bivariate polynomials by using the graded
lexicographical order was introduced in [15, 16], developed in [21] and it is
the main representation used in the monography [11]. A polynomial system is a
sequence $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ of vectors of increasing size
$n+1$ whose entries are polynomials, defined as

$\mathbb{P}_{n}=(P_{n,0}^{n}(x,y),P_{n-1,1}^{n}(x,y),\ldots,P_{0,n}^{n}(x,y))^{T},$

such that the bivariate polynomial $P_{n-k,k}^{n}(x,y),$ $k=0,1,\ldots,n,$ has
exactly degree $n$. Also, for $n\geqslant 1$, the set of polynomial entries
$\\{P_{n,0}^{n}(x,y)$, $P_{n-1,1}^{n}(x,y)$, $\ldots,$ $P_{0,n}^{n}(x,y)\\}$
is linearly independent.

We say that a polynomial system is _monic_ if every polynomial entry,
$P_{n-k,k}^{n}(x,y)$, is a monic bivariate polynomial, whose unique highest
degree term is $x^{n-k}\,y^{k}$.

Let $\mathbf{u}$ be a moment linear functional defined on $\Pi$ by

$\langle\mathbf{u},x^{m}y^{n}\rangle=\mu_{m,n},\quad\mbox{for}\quad
m,n\in\mathbb{N}\quad\mbox{and}\quad\mu_{m,n}\in\mathbb{R},$

and extended by linearity.

Let $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ be a polynomial system satisfying

$\displaystyle\langle\mathbf{u},\mathbb{P}_{n}\,\mathbb{P}_{n}^{T}\rangle=H_{n},$
$\displaystyle\langle\mathbf{u},\mathbb{P}_{n}\,\mathbb{P}_{m}^{T}\rangle=\mathtt{0},$

where $H_{n}$ is a non-singular symmetric positive definite matrix of order
$n+1$ and $\mathtt{0}$ is the zero matrix of adequate size. Then,
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is called _an orthogonal polynomial
system_ (OPS) associated with the moment functional $\mathbf{u}$. A moment
functional is called _regular_ or _quasi-definite_ if there exist orthogonal
polynomial systems associated with it. Given a regular moment functional,
there exist a unique monic orthogonal polynomial system ([11]). Conditions
about the existence of OPS associated with a given moment functional can be
found in [11].

In the matrix approach for bivariate orthogonal polynomials, it is known
([11]) that the orthogonal polynomial system satisfies the following three-
term relations

$\displaystyle
x\mathbb{P}_{n}=\Lambda_{n,1}\mathbb{P}_{n+1}+\Gamma_{n,1}\mathbb{P}_{n}+\Upsilon_{n,1}\mathbb{P}_{n-1},$
(1.1) $\displaystyle
y\mathbb{P}_{n}=\Lambda_{n,2}\mathbb{P}_{n+1}+\Gamma_{n,2}\mathbb{P}_{n}+\Upsilon_{n,2}\mathbb{P}_{n-1},$

for $n\geqslant 0$, where $\mathbb{P}_{-1}=0$, $\mathbb{P}_{0}=(1)$,
$\Lambda_{n,i}$, $\Gamma_{n,i}$, and $\Upsilon_{n,i}$, for $i=1,2$ are
matrices of adequate size.

A moment functional $\mathbf{u}$ is called _centrally symmetric_ if
$\mu_{m,n}=0$ when $m+n$ is an odd integer. This condition is equivalent to
the fact that $\Gamma_{n,i}=\mathtt{0}$, for $i=1,2$. Moreover, if $n$ is an
even number, then every polynomial $P_{n-k,k}^{n}(x,y)$, for $0\leqslant
k\leqslant n$, only contains monomials of total even degree, and if $n$ is an
odd number, then only contains monomials of total odd degree.

In this work we deal with a special condition of symmetry on the moment
functional, which we call _reflexive property_. This property can simplify
many objects on the matrix approach, provide additional properties and
facilities for the computational algorithms. The _reflexive property_ for the
moment functional implies a kind of symmetry into the polynomial vectors of
the associated monic orthogonal polynomial systems, and induces a particular
symmetric property (the _reverse property_) for matrices, as we will study in
Section 2.

The _reflexive property_ of a moment functional $\mathbf{u}$ is defined by

$\langle\mathbf{u},x^{m}y^{n}\rangle=\langle\mathbf{u},x^{n}y^{m}\rangle,\,\mbox{
for }m,n\geqslant 0.$

For instance, we will show that the monic orthogonal polynomial system (MOPS)
associated with $\mathbf{u},$ denoted by $\\{\mathbb{Q}_{n}\\}_{n\geqslant
0}$, where

$\mathbb{Q}_{n}=(Q_{n,0}^{n}(x,y),Q_{n-1,1}^{n}(x,y),\ldots,Q_{0,n}^{n}(x,y))^{T},$

satisfies $Q_{n-k,k}^{n}(x,y)=Q_{k,n-k}^{n}(y,x)$, for $k=0,1,\ldots,n$. This
property, that it will be called _reflexive property_ of the MOPS, is useful
since only half of the calculation is needed to build the polynomial vectors,
$\mathbb{Q}_{n},$ ${n\geqslant 0}$. As a consequence, the coefficient matrices
of the three-term relations (1.1) have connections with each other. These
connections provide a _reverse property_ among the associated matrices, and
also a smaller number of calculation is necessary to construct them.

Moreover, the moment functional has the _reflexive property_ if and only if
the matrices involved in the three-term relations of the MOPS satisfy the
_reverse property_. In this work we are interesting to show how _reverse
matrices_ and _centrosymmetric matrices_ are related with bivariate orthogonal
polynomials.

This work is organized into five sections. In the Section 2 the concepts of
reverse matrices and centrosymmetric matrices as well as their most relevant
properties are presented. Section 3 contains the basic theory of bivariate
orthogonal polynomials using matrix approach, it also contains the definition
and some properties of reflexive polynomial vectors. Reflexive moment
functionals and the consequent properties of the associated bivariate
orthogonal polynomial systems are introduced in Section 4. Section 5 brings
the main focus of this work, the connection of _reverse matrices_ ,
_centrosymmetric matrices_ and _reflexive bivariate orthogonal polynomials_.
In Section 6 we present some non-trivial examples to illustrate these
connections.

##  2\. Reverse matrices and centrosymmetric matrices

Centrosymmetric matrices satisfy several properties, including some algebraic
structures. They have been extensively studied and many applications can be
found in the literature, for instance see [1, 3, 7, 8, 12, 19, 20].

We start this section defining the reverse property between two matrices of
size $(m+1)\times(n+1)$.

######  Definition 2.1.

Consider two matrices $X$ and $Y$ of size $(m+1)\times(n+1)$, denoted by
$X=(x_{ij})_{i,j=0}^{m,n}$ and $Y=(y_{ij})_{i,j=0}^{m,n}$, respectively. If

$x_{ij}=y_{m-i,n-j},\quad for\ i=0,1,\ldots,m,\ j=0,1,\ldots,n,$

then $X$ is called reverse matrix of $Y$ (conversely, $Y$ is reverse matrix of
$X$). We will denote $X\leftrightharpoons Y$.

######  Example 2.2.

Consider $X=\left(\begin{matrix}1&2&3\\\ 4&5&6\end{matrix}\right)$ and
$Y=\left(\begin{matrix}6&5&4\\\ 3&2&1\end{matrix}\right)$, then
$X\leftrightharpoons Y$.

One can determinate the matrix $X^{R}$, the reverse matrix of
$X=(x_{ij})_{i,j=0}^{m,n}$, by applying an operation called _reflection_ ,
i.e.,

$X^{R}=(x_{m-i,n-j})_{i,j=0}^{m,n}.$ (2.1)

Hence, $X\leftrightharpoons X^{R}.$

The next result presents the minimum number of permutations between rows and
between columns of a matrix $X$ needed to transform $X$ into its reverse
matrix.

######  Lemma 2.3.

Let $X$ be a matrix of size $n$. Then, there exist
$2\lfloor\frac{n}{2}\rfloor$ permutations between rows and between columns of
$X$ such that transforms $X$ into $X^{R}$.

Now we present some direct properties of matrices that have the reverse
property.

######  Proposition 2.4.

Let $X$ and $Y$ be square matrices such that $X\leftrightharpoons Y$, then
$\det(X)=\det(Y)$.

###### Proof.

Since an even number of permutations of rows or columns of a matrix preserves
the value of its determinant, the result follows from Lemma 2.3. ∎

For the next results, we denote by $X^{ij},\,i=0,1,\ldots,m,$
$j=0,1,\ldots,n,$ the matrix obtained from a matrix $X$ of size
$(m+1)\times(n+1)$ eliminating the row $i$ and the column $j$. The adjugate of
the square matrix $X$, that is, the transposed matrix of cofactors of $X$, is
denoted by $adjX$, see [13, p. 22].

######  Proposition 2.5.

If $X$ and $Y$ are matrices of size $(m+1)\times(n+1)$ satisfying
$X\leftrightharpoons Y$, then

1) $X^{T}\leftrightharpoons Y^{T}$.

2) $XY^{T}\leftrightharpoons YX^{T}$.

3) $X^{ij}\leftrightharpoons Y^{m-i,n-j}$, for $i=0,1,\ldots,m,$
$j=0,1,\ldots,n$.

4) If $m=n$, then $adjX\leftrightharpoons adjY$.

###### Proof.

1) Consider $X=(x_{ij})_{i,j=0}^{m,n}$, $Y=(y_{ij})_{i,j=0}^{m,n}$,
$X^{T}=(\widetilde{x}_{ij})_{i,j=0}^{n,m}$, and
$Y^{T}=(\widetilde{y}_{ij})_{i,j=0}^{n,m}$ such that
$\widetilde{x}_{ji}=x_{ij}$ and $\widetilde{y}_{ji}=y_{ij}$ for
$i=0,1,\ldots,m,$ $j=0,1,\ldots,n$. Since $x_{ij}=y_{m-i,n-j}$, for
$j=0,1,\ldots,n$ and $i=0,1,\ldots,m,$ we have

$\widetilde{x}_{ji}=x_{ij}=y_{m-i,n-j}=\widetilde{y}_{n-j,m-i}.$

Therefore, $X^{T}\leftrightharpoons Y^{T}$.

2) Consider the same notation used in item 1). We know that $XY^{T}$ is a
matrix of size $(m+1)\times(m+1)$. Denoting $XY^{T}=(c_{ij})_{i,j=0}^{m}$ and
$YX^{T}=(\widetilde{c}_{ij})_{i,j=0}^{m}$, since $X\leftrightharpoons Y$, we
see that

$c_{ij}=\sum_{k=0}^{n}x_{ik}\widetilde{y}_{kj}=\sum_{k=0}^{n}y_{m-i,n-k}\widetilde{x}_{n-k,m-j}=\widetilde{c}_{m-i,m-j}.$

Hence, $XY^{T}\leftrightharpoons YX^{T}$.

3) It follows directly.

4) Consider the matrices $X^{*}=(x^{*}_{ij})_{i,j=0}^{m,n}$ and
$Y^{*}=(y^{*}_{ij})_{i,j=0}^{m,n}$ the matrices of cofactors of $X$ and $Y$
respectively. From item 3) we know that $X^{ij}\leftrightharpoons
Y^{n-i,n-j},$ for $i,j=0,1,\ldots,n$. The parity of $(i+j)$ is the same of
$(n-i)+(n-j)$ then, from Proposition 2.4, for $i,j=0,1,\ldots,n,$

$x^{*}_{ij}=(-1)^{i+j}\det(X^{ij})=(-1)^{(n-i)+(n-j)}\det(Y^{n-i,n-j})=y_{n-i,n-j}^{*}.$

Therefore, $X^{*}\leftrightharpoons Y^{*}$. Thus, from item 1),
$adjX\leftrightharpoons adjY$. ∎

There are many approaches for the centrosymmetric property of a matrix, see
[1, 3, 7, 13, 20].

For general matrices of size $(m+1)\times(n+1)$, one approach can be the
following.

######  Definition 2.6.

Consider a matrix $X=(x_{i,j})_{i,j=0}^{m,n}$ of size $(m+1)\times(n+1)$. The
matrix $X$ is centrosymmetric if it satisfies $X\leftrightharpoons X$, it
means $X^{R}=X$ and

$x_{i,j}=x_{m-i,n-j},\quad for\ i=0,1,\ldots,m,\ j=0,1,\ldots,n.$

######  Example 2.7.

The matrices

$A=\begin{pmatrix}1&2&3&4&5\\\ 6&7&8&7&6\\\ 5&4&3&2&1\end{pmatrix},\quad
B=\begin{pmatrix}1&2&3&4\\\ 5&6&7&8\\\ 8&7&6&5\\\ 4&3&2&1\end{pmatrix},\quad
and\quad C=\begin{pmatrix}1&2&3\\\ 4&5&4\\\ 3&2&1\end{pmatrix}$

are centrosymmetric matrices.

In an approach for square matrices of size $(n+1)$, the _reversal matrix_ (or
_exchange matrix_), denoted by $J_{n+1}$, is a matrix that has ones along the
secondary diagonal and zeros elsewhere. Notice that $J_{n+1}=J_{n+1}^{-1}$.

A square matrix $X$ of size $(n+1)$ is a _centrosymmetric matrix_ if
$J_{n+1}X=XJ_{n+1}$. Furthermore, a square matrix $X$ of size $(n+1)$ is
called _skew-centrosymmetric matrix_ if $J_{n+1}X=-XJ_{n+1}$, see [3, 7, 13].

A vector $V=(v_{i})_{i=0}^{n}$ of size $(n+1)$ is called a _symmetric vector_
if $J_{n+1}V=V$, it means $v_{i}=v_{n-i}$, for $i=0,1,\ldots,n$. Moreover, a
vector $V=(v_{i})_{i=0}^{n}$ is called a _skew-symmetric vector_ if
$J_{n+1}V=-V$, it yields $v_{i}=-v_{n-i}$, $i=0,1,\ldots,n,$ ([7]).

The next proposition brings some known direct consequences of the
centrosymmetric property of a matrix.

######  Proposition 2.8.

1) The sum of centrosymmetric matrices is a centrosymmetric matrix.

2) The product of centrosymmetric matrices is a centrosymmetric matrix.

3) The transpose of a centrosymmetric matrix is a centrosymmetric matrix.

4) If $X$ is a square centrosymmetric matrix, then $adjX$ is also
centrosymmetric.

5) If $X$ is an invertible centrosymmetric matrix, then $X^{-1}$ is also
centrosymmetric.

###### Proof.

The proof of items 1), 2) and 3) follow directly from the definition of
centrosymmetric matrix. The proof of item 4) follows from Proposition 2.5 item
4) making $Y=X$.

For completeness we include here a simple proof of item 5), using item 4). We
denote $X^{-1}=(\widehat{x}_{ij})_{i,j=0}^{n}$ and
$adjX=(x^{A}_{ij})_{i,j=0}^{n}$. From item 4), $x^{A}_{ij}=x^{A}_{n-i,n-j}$,
for $i,j=0,1,\ldots,n$. Then,

$\widehat{x}_{ij}=\dfrac{1}{\det X}x^{A}_{ij}=\dfrac{1}{\det
X}x^{A}_{n-i,n-j}=\widehat{x}_{n-i,n-j},$

for $i,j=0,1,\ldots,n.$ ∎

The following result is a straightforward consequence of items 2) and 5) of
Proposition 2.8.

######  Corollary 2.9.

If $XY$ is a centrosymmetric matrix and $X$ is an invertible centrosymmetric
matrix, then $Y$ is also centrosymmetric matrix.

Additionally, some basic properties of symmetric centrosymmetric matrices,
including properties of their eigenvalues and eigenvectors, were studied in
[20]. The next result, a consequence of [20, Th. 11], gives a characteristic
of the eigenvectors of symmetric centrosymmetric positive definite matrices.

######  Proposition 2.10 ([20]).

If $V$ is an eigenvector of a symmetric centrosymmetric positive definite
matrix, then $V$ is either a symmetric vector or a skew-symmetric vector.

Next result shows that one can construct, in different ways, centrosymmetric
matrices from two matrices that have the reverse property.

######  Proposition 2.11.

1) Let $T_{1}$ and $T_{2}$ be matrices of size $(m+1)\times(n+1)$.
$T_{1}\leftrightharpoons T_{2}$ if and only if, the matrix of size
$2(m+1)\times 2(n+1)$

$\widehat{T}=\left(\begin{array}[]{c|c}T_{1}&\mathtt{0}\\\
\hline\cr\mathtt{0}&T_{2}\end{array}\right),$

where $\mathtt{0}$ is a zero matrix of size $(m+1)\times(n+1)$, is
centrosymmetric.
2) If $T_{1}$ and $T_{2}$ are matrices of size $(m+1)\times(n+1)$ such that
$T_{1}\leftrightharpoons T_{2}$, then $T_{1}+T_{2}$ and $T_{1}-T_{2}$ are
centrosymmetric amtrices.

###### Proof.

1) Suppose that $T_{1}\leftrightharpoons T_{2}$, hence, using the reflection
operation (2.1), we know that $T_{1}^{R}=T_{2}$ and $T_{2}^{R}=T_{1}.$
Applying reflection operator in $\widehat{T}$ we get

$\widehat{T}^{R}=\left(\begin{array}[]{c|c}T_{2}^{R}&\mathtt{0}\\\
\hline\cr\mathtt{0}&T_{1}^{R}\end{array}\right)=\left(\begin{array}[]{c|c}T_{1}&\mathtt{0}\\\
\hline\cr\mathtt{0}&T_{2}\end{array}\right)=\widehat{T}.$

Therefore, $\widehat{T}$ is a centrosymmetric matrix.

Reciprocally, supposing that $\widehat{T}^{R}=\widehat{T}$, we see that
$T_{2}^{R}=T_{1}$ and $T_{1}^{R}=T_{2},$ hence $T_{1}\leftrightharpoons
T_{2}$.

2) The proof follows directly from the fact that $T_{1}^{R}=T_{2}$ and
$T_{2}^{R}=T_{1}.$ ∎

##  3\. Bivariate orthogonal polynomials

We start recalling the basic definitions and main tools about bivariate
orthogonal polynomials that we will need in the rest of the paper. We refer
mainly [11].

As we mention on the introduction, a _polynomial system (PS)_ is a sequence of
polynomial vectors $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ of increasing size
$n+1$,

$\mathbb{P}_{n}=(P_{n,0}^{n}(x,y),P_{n-1,1}^{n}(x,y),\ldots,P_{0,n}^{n}(x,y))^{T},$

where the bivariate polynomial $P_{n-k,k}^{n}(x,y),$ $k=0,1,\ldots,n,$ has
exactly degree $n$, and the set of polynomial entries is linearly independent.

The simplest polynomial system is the so-called _canonical basis_
$\\{\mathbb{X}_{n}\\}_{n\geqslant 0}$, defined as

$\mathbb{X}_{n}=(x^{n},x^{n-1}\,y,x^{n-2}\,y^{2},\ldots,x\,y^{n-1},y^{n})^{T}.$

If $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is a polynomial system, then there
exist real matrices $G^{n}_{k}$ of size $(n+1)\times(k+1)$ such that every
polynomial vector $\mathbb{P}_{n}$ can be expressed in terms of the canonical
basis as

$\mathbb{P}_{n}=G_{n}\,\mathbb{X}_{n}+G_{n-1}^{n}\,\mathbb{X}_{n-1}+G_{n-2}^{n}\,\mathbb{X}_{n-2}+\cdots+G_{1}^{n}\,\mathbb{X}_{1}+G_{0}^{n}\,\mathbb{X}_{0}.$
(3.1)

In the particular case when $G_{n}=I_{n+1}$, $\\{\mathbb{P}_{n}\\}_{n\geqslant
0}$ is called _monic polynomial system_.

A polynomial system can satisfy a symmetric property on the entries of the
polynomial vectors.

######  Definition 3.1.

Let
$\mathbb{P}_{n}=(P_{n,0}^{n}(x,y),P_{n-1,1}^{n}(x,y),\ldots,P_{0,n}^{n}(x,y))^{T}$
be a polynomial vector satisfying

$P_{n-k,k}^{n}(x,y)=P_{k,n-k}^{n}(y,x),\quad k=0,1,\ldots,n,$

then $\mathbb{P}_{n}$ is called reflexive polynomial vector. A polynomial
system such that all the polynomial vectors are reflexives will be called
reflexive polynomial system.

We remark that in the univariate case, a reflexive property of a polynomial is
defined according with its coefficients. First consider a complex polynomial
in one variable $r(x)=b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots+b_{0}$,
$b_{i}\in\mathbb{C}$, the polynomial
$r^{*}(x)=x^{n}\overline{r\left(1/\overline{x}\right)}=\overline{b}_{0}x^{n}+\overline{b}_{1}x^{n-1}+\cdots+\overline{b}_{n},$
is called _reciprocal polynomial_ of $r(x)$. If $r^{*}(x)=ur(x)$, with
$|u|=1,$ then $r(x)$ is called _self-inversive polynomial_ , see [18]. These
type of polynomials appear also on the theory of orthogonal polynomials on the
unit circle.

Consider now a real polynomial $r(x)=b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots+b_{0},$
$b_{i}\in\mathbb{R}$, and the polynomial
$x^{n}r\left(1/x\right)=b_{0}x^{n}+b_{1}x^{n-1}+\cdots+b_{n}.$ If
$r(x)=x^{n}r(1/x)$, it means $b_{i}=b_{n-i},$ for $i=0,1,\ldots,n$, then
$r(x)$ is called _self-reciprocal polynomial_ or _palindromic polynomial_. In
several works, for instance in [4, 5, 14, 17], the behaviour of the zeros of
special cases of this type of univariate polynomials was studied.

Next result shows a relation between reflexive polynomial vectors and
centrosymmetric matrices. More precisely, it says that the matrix of change of
basis between two reflexive polynomial vectors is necessarily centrosymmetric.
Reciprocally, the product of centrosymmetric matrix and reflexive polynomial
vector preserves the reflexive property of the polynomial vector.

######  Proposition 3.2.

Let $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ and
$\\{\widetilde{\mathbb{P}}_{n}\\}_{n\geqslant 0}$ be two polynomial systems
such that $\mathbb{P}_{n}$ is a reflexive polynomial and
$\widetilde{\mathbb{P}}_{n}=T_{n}\mathbb{P}_{n}$, for $n\in\mathbb{N}$. Then,
$\widetilde{\mathbb{P}}_{n}$ is reflexive polynomial vector if and only if
$T_{n}$ is centrosymmetric matrix.

###### Proof.

Let $\widetilde{\mathbb{P}}_{n}=T_{n}\mathbb{P}_{n}$ be written as

$\left(\begin{array}[]{c}\widetilde{P}_{n,0}^{n}(x,y)\\\ \vdots\\\
\widetilde{P}_{n-k,k}^{n}(x,y)\\\ \vdots\\\
\widetilde{P}_{0,n}^{n}(x,y)\end{array}\right)=\left(\begin{matrix}t_{00}^{(n)}&t_{01}^{(n)}&\ldots&t_{0n}^{(n)}\\\
\vdots&\vdots&&\vdots\\\ t_{k0}^{(n)}&t_{k1}^{(n)}&\ldots&t_{kn}^{(n)}\\\
\vdots&\vdots&&\vdots\\\
t_{n0}^{(n)}&t_{n1}^{(n)}&\ldots&t_{nn}^{(n)}\end{matrix}\right)\left(\begin{array}[]{c}P_{n,0}^{n}(x,y)\\\
\vdots\\\ P_{n-k,k}^{n}(x,y)\\\ \vdots\\\ P_{0,n}^{n}(x,y)\end{array}\right).$

Since $\mathbb{P}_{n}$ is reflexive polynomial vector,

$\widetilde{P}_{n-k,k}^{n}(x,y)=\sum_{j=0}^{n}t_{kj}^{(n)}P_{n-j,j}^{n}(x,y)=\sum_{j=0}^{n}t_{kj}^{(n)}P_{j,n-j}^{n}(y,x).$
(3.2)

On the other hand, we can write $\widetilde{P}_{k,n-k}^{n}(y,x)$ as

$\widetilde{P}_{k,n-k}^{n}(y,x)=\sum_{j=0}^{n}t_{n-k,j}^{(n)}P_{n-j,j}^{n}(y,x).$
(3.3)

Thus, $\widetilde{P}_{n-k,k}^{n}(x,y)=\widetilde{P}_{k,n-k}^{n}(y,x)$ implies
that

$\sum_{j=0}^{n}t_{kj}^{(n)}P_{j,n-j}^{n}(y,x)=\sum_{j=0}^{n}t_{n-k,j}^{(n)}P_{n-j,j}^{n}(y,x).$

Making $l=n-j$ in the second summation, we have

$\sum_{j=0}^{n}t_{kj}^{(n)}P_{j,n-j}^{n}(y,x)=\sum_{l=0}^{n}t_{n-k,n-l}^{(n)}P_{l,n-l}^{n}(y,x).$

Hence,

$\sum_{j=0}^{n}[t_{k,j}^{(n)}-t_{n-k,n-j}^{(n)}]P_{j,n-j}^{n}(y,x)=0,$

and since $\\{P_{n-j,j}^{n}\\}$ is linearly independent, it follows that
$t_{kj}^{(n)}=t_{n-k,n-j}^{(n)},$ $j=0,1,\ldots,n.$ That is, $T_{n}$ is
centrosymmetric matrix.

Conversely, suppose that $T_{n}$ is centrosymmetric matrix and
$\mathbb{P}_{n}$ is reflexive polynomial vector, from (3.2) we know that

$\widetilde{P}_{n-k,k}^{n}(x,y)=\sum_{j=0}^{n}t_{kj}^{(n)}P_{n-j,j}^{n}(x,y)=\sum_{j=0}^{n}t_{n-k,n-j}^{(n)}P_{j,n-j}^{n}(y,x).$

Making $l=n-j$ on the latter summation, from (3.3), we have

$\widetilde{P}_{n-k,k}^{n}(x,y)=\sum_{j=0}^{n}t_{n-k,l}^{(n)}P_{n-l,l}^{n}(y,x)=\widetilde{P}_{k,n-k}^{n}(y,x).$

Then, $\widetilde{\mathbb{P}}_{n}$ is reflexive polynomial vector. ∎

From now on, we will work with bivariate moment functionals $\mathbf{u}$
defined as

$\langle\mathbf{u},f\rangle=\iint_{\Omega}f(x,y)\,W(x,y)\,dx\,dy,$

where $W(x,y)$ is a weight function defined on a region
$\Omega\subset\mathbb{R}^{2}$ such that its associated moments satisfy

$\mu_{n,m}=\langle\mathbf{u},x^{n}\,y^{m}\rangle=\iint_{\Omega}x^{n}\,y^{m}\,W(x,y)\,dx\,dy<+\infty,$

for $n,m=0,1,\ldots$. Thus, we are considering the inner product

$(f,g):=\langle\mathbf{u},f\,g\rangle=\iint_{\Omega}f(x,y)\,g(x,y)\,W(x,y)\,dx\,dy.$
(3.4)

In terms of the inner product (3.4), if $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$
is a polynomial system satisfying

$\displaystyle(\mathbb{P}_{n},\mathbb{P}_{n}^{T})=\langle\mathbf{u},\mathbb{P}_{n}\,\mathbb{P}_{n}^{T}\rangle=H_{n},$
$\displaystyle(\mathbb{P}_{n},\mathbb{P}_{m}^{T})=\langle\mathbf{u},\mathbb{P}_{n}\,\mathbb{P}_{m}^{T}\rangle=\mathtt{0},$

where $H_{n}$ is non-singular symmetric positive definite matrix of size $n+1$
and $\mathtt{0}$ is zero matrix of adequate size. Then,
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is an _orthogonal polynomial system_
(OPS) associated with the weight function $W(x,y).$ If $H_{n}=I_{n+1}$, the
identity matrix of order $n+1$, then $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is
called _orthonormal polynomial system_ , and they are not unique. As we recall
at the introduction, there is a unique _monic orthogonal polynomial system_
associated with the inner product $(\cdot,\cdot)$. In the sequel, we denote by
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ an orthonormal polynomial system, and by
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ the monic orthogonal polynomial system.

Orthogonal polynomial systems satisfy two three-term relations, one for each
variable, as (1.1). For orthonormal polynomial system,
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$, and monic orthogonal polynomial system,
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$, the three-term relations have,
respectively, the following forms

$x_{i}\mathbb{P}_{n}=A_{n,i}\mathbb{P}_{n+1}+B_{n,i}\mathbb{P}_{n}+A^{T}_{n-1,i}\mathbb{P}_{n-1},\quad
i=1,2,$ (3.5)

for $n\geqslant 0$, $\mathbb{P}_{-1}=0$ and
$\mathbb{P}_{0}=(\mu_{0,0}^{-1/2})$, and

$x_{i}\mathbb{Q}_{n}=L_{n,i}\mathbb{Q}_{n+1}+C_{n,i}\mathbb{Q}_{n}+D_{n,i}\mathbb{Q}_{n-1},\quad
i=1,2,$ (3.6)

for $n\geqslant 0$, $\mathbb{Q}_{-1}=0$ and $\mathbb{Q}_{0}=(1)$. Here
$x_{1}=x$, $x_{2}=y$, in order to simplify the notation. The matrices
$A_{n,i}$ of size $(n+1)\times(n+2)$ are full rank matrices, $B_{n,i}$ and
$C_{n,i}$ are matrices of size $(n+1)\times(n+1)$, $D_{n,i}$ are full rank
matrices of size $(n+1)\times n$, and $L_{n,i}$ are matrices of size
$(n+1)\times(n+2)$ such that $x_{i}\mathbb{X}_{n}=L_{n,i}\mathbb{X}_{n+1},$
$i=1,2$, that are

$L_{n,1}=\left(\begin{array}[]{cccc|c}1&&&\bigcirc&0\\\ &1&&&0\\\
&&\ddots&&\vdots\\\ \bigcirc&&&1&0\\\ \end{array}\right)\quad\mbox{and}\quad
L_{n,2}=\left(\begin{array}[]{c|cccc}0&1&&&\bigcirc\\\ 0&&1&&\\\
\vdots&&&\ddots&\\\ 0&\bigcirc&&&1\\\ \end{array}\right).$

Given $\mathbf{u}$ a moment functional, the _matrix of moments_ $M_{n}$,
$n\in\mathbb{N}$, is a matrix of size $(n+1)(n+2)/2$ defined by

$M_{n}=\left(\begin{matrix}\langle\mathbf{u},\mathbb{X}_{0}\mathbb{X}_{0}^{T}\rangle&\langle\mathbf{u},\mathbb{X}_{0}\mathbb{X}_{1}^{T}\rangle&\cdots&\langle\mathbf{u},\mathbb{X}_{0}\mathbb{X}_{n}^{T}\rangle\\\
\langle\mathbf{u},\mathbb{X}_{1}\mathbb{X}_{0}^{T}\rangle&\langle\mathbf{u},\mathbb{X}_{1}\mathbb{X}_{1}^{T}\rangle&\cdots&\langle\mathbf{u},\mathbb{X}_{1}\mathbb{X}_{n}^{T}\rangle\\\
\vdots&\vdots&&\vdots\\\
\langle\mathbf{u},\mathbb{X}_{n}\mathbb{X}_{0}^{T}\rangle&\langle\mathbf{u},\mathbb{X}_{n}\mathbb{X}_{1}^{T}\rangle&\cdots&\langle\mathbf{u},\mathbb{X}_{n}\mathbb{X}_{n}^{T}\rangle\end{matrix}\right),$
(3.7)

where $\langle\mathbf{u},\mathbb{X}_{m}\mathbb{X}_{n}^{T}\rangle$ is a block
of size $(m+1)\times(n+1)$ given by

$\langle\mathbf{u},\mathbb{X}_{m}\mathbb{X}_{n}^{T}\rangle=\left(\begin{matrix}\langle\mathbf{u},x^{m+n}\rangle&\langle\mathbf{u},x^{m+n-1}y\rangle&\ldots&\langle\mathbf{u},x^{m}y^{n}\rangle\\\
\langle\mathbf{u},x^{m+n-1}y\rangle&\langle\mathbf{u},x^{m+n-2}y^{2}\rangle&\ldots&\langle\mathbf{u},x^{m-1}y^{n+1}\rangle\\\
\vdots&\vdots&&\vdots\\\
\langle\mathbf{u},x^{n}y^{m}\rangle&\langle\mathbf{u},x^{n-1}y^{m+1}\rangle&\ldots&\langle\mathbf{u},y^{m+n}\rangle\end{matrix}\right).$
(3.8)

See [11] for more details.

Let $\alpha=(\alpha_{1},\alpha_{2})$ be a bi-index such that
$|\alpha|=\alpha_{1}+\alpha_{2}=n$, it means that $\alpha=(n-k,k)$, for
$k=0,1,\ldots,n$. Using the notation
$\mathbf{x}^{\alpha}=x^{\alpha_{1}}y^{\alpha_{2}}$, the matrix
$M_{\alpha}(x,y)$ is defined by

$M_{\alpha}(x,y)=\left(\begin{tabular}[]{c|c}&$\langle\mathbf{u},\mathbf{x}^{\alpha}\mathbb{X}_{0}\rangle$\\\
$M_{n-1}$&$\langle\mathbf{u},\mathbf{x}^{\alpha}\mathbb{X}_{1}\rangle$\\\
&$\vdots$\\\
&$\langle\mathbf{u},\mathbf{x}^{\alpha}\mathbb{X}_{n-1}\rangle$\\\
\hline\cr$\mathbb{X}_{0}^{T}\quad\mathbb{X}_{1}^{T}\quad...\quad\mathbb{X}_{n-1}^{T}$&$\textbf{x}^{\alpha}$\end{tabular}\right).$
(3.9)

######  Proposition 3.3 ([11]).

Consider the monic polynomials $Q^{n}_{n-k,k}(x,y)$, for $k=0,1,\ldots,n$, the
entries of
$\mathbb{Q}_{n}=(Q_{n,0}^{n}(x,y),Q_{n-1,1}^{n}(x,y),\ldots,Q_{0,n}^{n}(x,y))^{T},$
given by

$Q^{n}_{n-k,k}(x,y)=\dfrac{\det(M_{(n-k,k)}(x,y))}{\det(M_{n-1})},\quad
k=0,1,\ldots,n,$

then $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is the MOPS with respect to the
moment functional $\mathbf{u}$.

##  4\. Reflexive Bivariate Orthogonal Polynomials

First we need the definition of _reflexive weight function_.

######  Definition 4.1.

Consider a region $\Omega\subseteq\mathbb{R}^{2}$, such that it satisfies
$(x,y)\in\Omega\Leftrightarrow(y,x)\in\Omega.$ A weight function $W(x,y)$
defined in $\Omega$, satisfying

$W(x,y)=W(y,x),\quad(x,y)\in\Omega,$ (4.1)

is called a reflexive weight function.

As we define at the introduction, a _reflexive moment functional_ is a
functional $\mathbf{u}$ such that its associated moments satisfy

$\mu_{m,n}=\mu_{n,m},\quad m,n=0,1,\ldots.$

In particular, if $W(x,y)$ is a _reflexive weight function_ , the moment
functional

$\langle\mathbf{u},f\rangle=\iint_{\Omega}f(x,y)\,W(x,y)\,dx\,dy,$

is a _reflexive moment functional_.

Now we can show that a reflexive weight function yields reflexive MOPS. This
is one of the main results involving reflexive weight function.

######  Theorem 4.2.

The polynomial vectors of the MOPS $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$
associated with a reflexive weight function are reflexives, it means

$Q_{n-k,k}^{n}(x,y)=Q_{k,n-k}^{n}(y,x),\quad k=0,1\ldots,n.$

###### Proof.

From Proposition 3.3, we can write, for $k=0,1,\ldots,n,$

$Q_{n-k,k}^{n}(x,y)=\dfrac{\det(M_{(n-k,k)}(x,y))}{\det(M_{n-1}(x,y))}\quad\mbox{and}\quad
Q_{k,n-k}^{n}(y,x)=\dfrac{\det(M_{(k,n-k)}(y,x))}{\det(M_{n-1}(y,x))},$

where, from (3.7) and (3.8), the matrices $M_{n-1}(x,y)$ and $M_{n-1}(y,x)$
are

$M_{n-1}(x,y)=\left(\begin{matrix}\mu_{00}&\mu_{10}&\mu_{01}&\cdots&\mu_{n-1,0}&\cdots&\mu_{0,n-1}\\\\[4.30554pt]
\mu_{10}&\mu_{20}&\mu_{11}&\cdots&\mu_{n0}&\cdots&\mu_{1,n-1}\\\\[4.30554pt]
\mu_{01}&\mu_{11}&\mu_{02}&\cdots&\mu_{n-1,1}&\cdots&\mu_{0n}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots\\\
\mu_{n-1,0}&\mu_{n,0}&\mu_{n-1,1}&\cdots&\mu_{2n-2,0}&\cdots&\mu_{n-1,n-1}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots\\\
\mu_{0,n-1}&\mu_{1,n-1}&\mu_{0,n}&\cdots&\mu_{n-1,n-1}&\cdots&\mu_{0,2n-2}\\\\[4.30554pt]
\end{matrix}\right)$

and

$M_{n-1}(y,x)=\left(\begin{matrix}\mu_{00}&\mu_{01}&\mu_{10}&\cdots&\mu_{0,n-1}&\cdots&\mu_{n-1,0}\\\\[4.30554pt]
\mu_{01}&\mu_{02}&\mu_{11}&\cdots&\mu_{0n}&\cdots&\mu_{n-1,1}\\\\[4.30554pt]
\mu_{10}&\mu_{11}&\mu_{20}&\cdots&\mu_{1,n-1}&\cdots&\mu_{n,0}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots\\\
\mu_{0,n-1}&\mu_{0,n}&\mu_{1,n-1}&\cdots&\mu_{0,2n-2}&\cdots&\mu_{n-1,n-1}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots\\\
\mu_{n-1,0}&\mu_{n-1,1}&\mu_{n,0}&\cdots&\mu_{n-1,n-1}&\cdots&\mu_{2n-2,0}\\\\[4.30554pt]
\end{matrix}\right).$

Since the weight function is reflexive, i.e., $\mu_{m,n}=\mu_{n,m},$ for
$m,n=0,1,\ldots,$ we observe that $M_{n-1}(x,y)=M_{n-1}(y,x)$ and their
determinants are the same.

Now, from (3.9) the matrices $M_{(n-k,k)}(x,y)$ and $M_{(k,n-k)}(y,x)$ are
given, respectively, by

$\left(\begin{matrix}\mu_{00}&\mu_{10}&\mu_{01}&\cdots&\mu_{n-1,0}&\cdots&\mu_{0,n-1}&\mu_{n-k,k}\\\\[4.30554pt]
\mu_{10}&\mu_{20}&\mu_{11}&\cdots&\mu_{n0}&\cdots&\mu_{1,n-1}&\mu_{n-k+1,k}\\\\[4.30554pt]
\mu_{01}&\mu_{11}&\mu_{02}&\cdots&\mu_{n-1,1}&\cdots&\mu_{0n}&\mu_{n-k,k+1}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots&\vdots\\\
\mu_{n-1,0}&\mu_{n,0}&\mu_{n-1,1}&\cdots&\mu_{2n-2,0}&\cdots&\mu_{n-1,n-1}&\mu_{2n-k-1,k}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots&\vdots\\\
\mu_{0,n-1}&\mu_{1,n-1}&\mu_{0,n}&\cdots&\mu_{n-1,n-1}&\cdots&\mu_{0,2n-2}&\mu_{n-k,k+n-1}\\\\[4.30554pt]
1&x&y&\cdots&x^{n-1}&\cdots&y^{n-1}&x^{n-k}y^{k}\end{matrix}\right)$

and

$\left(\begin{matrix}\mu_{00}&\mu_{01}&\mu_{10}&\cdots&\mu_{0,n-1}&\cdots&\mu_{n-1,0}&\mu_{n-k,k}\\\\[4.30554pt]
\mu_{01}&\mu_{02}&\mu_{11}&\cdots&\mu_{0n}&\cdots&\mu_{n-1,1}&\mu_{n-k,k+1}\\\\[4.30554pt]
\mu_{10}&\mu_{11}&\mu_{20}&\cdots&\mu_{1,n-1}&\cdots&\mu_{n,0}&\mu_{n-k+1,k}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots&\vdots\\\
\mu_{0,n-1}&\mu_{0,n}&\mu_{1,n-1}&\cdots&\mu_{0,2n-2}&\cdots&\mu_{n-1,n-1}&\mu_{n-k,k+n-1}\\\
\vdots&\vdots&\vdots&&\vdots&&\vdots&\vdots\\\
\mu_{n-1,0}&\mu_{n-1,1}&\mu_{n,0}&\cdots&\mu_{n-1,n-1}&\cdots&\mu_{2n-2,0}&\mu_{2n-k-1,k}\\\\[4.30554pt]
1&y&x&\cdots&y^{n-1}&\cdots&x^{n-1}&x^{n-k}y^{k}\end{matrix}\right).$

Once more using the fact that $\mu_{m,n}=\mu_{n,m},$ for $m,n=0,1,\ldots,$ we
see that the number of permutations of rows and permutations of columns that
transforms the matrix $M_{(n-k,k)}(x,y)$ into the matrix $M_{(k,n-k)}(y,x)$ is
even, hence

$\det(M_{n-k,k}(x,y))=\det(M_{k,n-k}(y,x)),\quad for\ k=0,1\ldots,n,$

and the result holds. ∎

When the polynomial vectors of the MOPS $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$
are reflexives for $n\geqslant 0$, then $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$
is called reflexive MOPS.

Next results give some connections of reflexive MOPS and some centrosymmetric
matrices.

######  Corollary 4.3.

Let $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ be a MOPS associated with a
reflexive weight function satisfying (4.1). Then, for $n\in\mathbb{N}$, the
matrix $H_{n}$ of size $n+1$, given by
$H_{n}=\langle\mathbf{u},\mathbb{Q}_{n}\mathbb{Q}_{n}^{T}\rangle,$ is
centrosymmetric, i.e., $H_{n}=(h_{ij})_{i,j=0}^{n},$ satisfies

$h_{ij}=h_{n-i,n-j},\quad\mbox{for}\ i,j=0,1,\ldots,n.$

###### Proof.

Observe that
$h_{ij}=\langle\mathbf{u},Q_{n-i,i}^{n}(x,y)Q_{n-j,j}^{n}(x,y)\rangle$, for
$i,j=0,1,\ldots,n$.

From Theorem 4.2, we can write

$\displaystyle
h_{ij}=\displaystyle\iint_{\Omega}Q_{i,n-i}^{n}(y,x)Q_{j,n-j}^{n}(y,x)W(x,y)dxdy.$

Making a change of variables $x\leftrightarrow y$, we have

$\displaystyle
h_{ij}=\displaystyle\iint_{\Omega}Q_{i,n-i}^{n}(x,y)Q_{j,n-j}^{n}(x,y)W(y,x)dxdy.$

Finally, using the property (4.1), we obtain

$\displaystyle
h_{ij}=\displaystyle\iint_{\Omega}Q_{i,n-i}^{n}(x,y)Q_{j,n-j}^{n}(x,y)W(x,y)dxdy=h_{n-i,n-j}.$

∎

Consider $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ the MOPS associated with a
weight function $W(x,y)$, and $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ another
OPS also associated with $W(x,y)$. Using the explicit expression (3.1) and the
orthogonality, the polynomial vectors $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$
are related by $\mathbb{P}_{n}=G_{n}\mathbb{Q}_{n},$ $n\geqslant 0$. In
particular, as a consequence of Proposition 3.2, we have the following result.

######  Corollary 4.4.

If $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is the MOPS associated with a
reflexive weight function $W(x,y)$ and $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$
is an OPS associated with $W(x,y)$ and given by
$\mathbb{P}_{n}=G_{n}\mathbb{Q}_{n}$, then $\mathbb{P}_{n}$ is reflexive
polynomial vector if and only if $G_{n}$ is a centrosymmetric matrix.

##  5\. Main Results

We now present more connections involving reverse matrices, centrosymmetric
matrices and reflexive bivariate orthogonal polynomial systems.

The first result shows that the coefficient matrices of the three-term
relations for reflexive MOPS given by (3.6) have the reverse property.

######  Theorem 5.1.

Let $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ be a MOPS associated with a
reflexive weight function satisfying (4.1) and let (3.6) be their three-term
relations. Then, for $n\in\mathbb{N}$, $C_{n,1}\leftrightharpoons C_{n,2}$,
and $D_{n,1}\leftrightharpoons D_{n,2}$.

###### Proof.

First we denote the matrices $C_{n,k}=(c^{(k)}_{ij})_{i,j=0}^{n,n}$ and
$D_{n,k}=(d^{(k)}_{ij})_{i,j=0}^{n,n-1}$, for $k=1,2.$ From the three-term
relation (3.6), omitting the variables $(x,y)$, we can write

$x\left(\begin{array}[]{c}Q_{n,0}^{n}\\\ \vdots\\\ Q_{n-k,k}^{n}\\\ \vdots\\\
Q_{0,n}^{n}\end{array}\right)=\left(\begin{array}[]{c}Q_{n+1,0}^{n+1}\\\
\vdots\\\ Q_{n+1-k,k}^{n+1}\\\ \vdots\\\
Q_{1,n}^{n+1}\end{array}\right)+C_{n,1}\left(\begin{array}[]{c}Q_{n,0}^{n}\\\
\vdots\\\ Q_{n-l,l}^{n}\\\ \vdots\\\
Q_{0,n}^{n}\end{array}\right)+D_{n,1}\left(\begin{array}[]{c}Q_{n-1,0}^{n-1}\\\
\vdots\\\ Q_{n-1-s,s}^{n-1}\\\ \vdots\\\ Q_{0,n-1}^{n-1}\end{array}\right)$
(5.1)

and

$y\left(\begin{array}[]{c}Q_{n,0}^{n}\\\ \vdots\\\ Q_{k,n-k}^{n}\\\ \vdots\\\
Q_{0,n}^{n}\end{array}\right)=\left(\begin{array}[]{c}Q_{n,1}^{n+1}\\\
\vdots\\\ Q_{k,n+1-k}^{n+1}\\\ \vdots\\\
Q_{0,n+1}^{n+1}\end{array}\right)+C_{n,2}\left(\begin{array}[]{c}Q_{n,0}^{n}\\\
\vdots\\\ Q_{n-l,l}^{n}\\\ \vdots\\\
Q_{0,n}^{n}\end{array}\right)+D_{n,2}\left(\begin{array}[]{c}Q_{n-1,0}^{n-1}\\\
\vdots\\\ Q_{n-1-s,s}^{n-1}\\\ \vdots\\\ Q_{0,n-1}^{n-1}\end{array}\right).$
(5.2)

From (5.1), for $k=0,1,\ldots,n$, we can write $xQ_{n-k,k}^{n}(x,y)$ as

$\displaystyle xQ_{n-k,k}^{n}(x,y)=Q_{n+1-k,k}^{n+1}(x,y)$
$\displaystyle+c^{(1)}_{k0}Q_{n,0}^{n}(x,y)+\cdots+c^{(1)}_{kl}Q_{n-l,l}^{n}(x,y)+\cdots+c^{(1)}_{kn}Q_{0,n}^{n}(x,y)$
(5.3)
$\displaystyle+d^{(1)}_{k0}Q_{n-1,0}^{n-1}(x,y)+\cdots+d^{(1)}_{ks}Q_{n-1-s,s}^{n-1}(x,y)+\cdots+d^{(1)}_{k,n-1}Q_{0,n-1}^{n-1}(x,y).$

In the same way, working on (5.2), we can write $yQ_{k,n-k}^{n}(x,y),$ for
$k=0,1,\ldots,n$, as

$\displaystyle yQ_{k,n-k}^{n}(x,y)=Q_{k,n+1-k}^{n+1}(x,y)$
$\displaystyle+c^{(2)}_{n-k,0}Q_{n,0}^{n}(x,y)+\cdots+c^{(2)}_{n-k,l}Q_{n-l,l}^{n}(x,y)+\cdots+c^{(2)}_{n-k,n}Q_{0,n}^{n}(x,y)$
(5.4)
$\displaystyle+d^{(2)}_{n-k,0}Q_{n-1,0}^{n-1}(x,y)+\cdots+d^{(2)}_{n-k,s}Q_{n-1-s,s}^{n-1}(x,y)+\cdots+d^{(2)}_{n-k,n-1}Q_{0,n-1}^{n-1}(x,y)$

Making the change of variables $x\leftrightarrow y$ in (5) and using Theorem
4.2, we obtain

$\displaystyle xQ_{n-k,k}^{n}(x,y)\\!\\!\\!$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\\!\\!\\!Q_{n+1-k,k}^{n+1}(x,y)+c^{(2)}_{n-k,0}Q_{0,n}^{n}(x,y)+\cdots+c^{(2)}_{n-k,l}Q_{l,n-l}^{n}(x,y)+\cdots$
$\displaystyle\\!\\!+c^{(2)}_{n-k,n-l}Q_{n-l,l}^{n}(x,y)+\cdots+c^{(2)}_{n-k,n}Q_{n,0}^{n}(x,y)$
$\displaystyle\\!\\!+d^{(2)}_{n-k,0}Q_{0,n-1}^{n-1}(x,y)+\cdots+d^{(2)}_{n-k,s}Q_{s,n-1-s}^{n-1}(x,y)+\cdots$
$\displaystyle\\!\\!+d^{(2)}_{n-k,n-1-s}Q_{n-1-s,s}^{n-1}(x,y)+\cdots+d^{(2)}_{n-k,n-1}Q_{n-1,0}^{n-1}(x,y).$

Since $\\{Q_{n-l,l}^{n}\\}^{n}_{l=0}\cup\\{Q_{n-1-s,s}^{n-1}\\}^{n-1}_{s=0}$
is a linearly independent set, comparing (5) and (5), we obtain

$c^{(1)}_{ij}=c^{(2)}_{n-i,n-j},\quad i,j=0,1,\ldots,n,$

and

$d^{(1)}_{ij}=d^{(2)}_{n-i,n-1-j},\quad i=0,1,\ldots,n,\ \ j=0,1,\ldots,n-1.$

Hence, $C_{n,1}\leftrightharpoons C_{n,2}$ and $D_{n,1}\leftrightharpoons
D_{n,2}$. ∎

We remark that, since $C_{n,1}\leftrightharpoons C_{n,2}$ and
$D_{n,1}\leftrightharpoons D_{n,2}$, it is enough to calculate the coefficient
matrices associated with one variable and the other coefficient matrices are
directly obtained.

The reciprocal is also true.

######  Theorem 5.2.

If the matrices $C_{n,i}$ and $D_{n,i}$, $i=1,2$, in (3.6) satisfy
$C_{n,1}\leftrightharpoons C_{n,2}$ and $D_{n,1}\leftrightharpoons D_{n,2}$,
then the associated MOPS $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is reflexive.

###### Proof.

Let $C_{n,k}=(c^{(k)}_{ij})_{i,j=0}^{n,n}$,
$D_{n,k}=(d^{(k)}_{ij})_{i,j=0}^{n,n-1}$, $k=1,2,$ and $\mathbb{Q}_{n}$ be the
monic orthogonal polynomial vector. The three-term relations (3.6) for $n=0$,
are

$x\mathbb{Q}_{0}=L_{0,1}\mathbb{Q}_{1}+C_{0,1}\mathbb{Q}_{0},$
$y\mathbb{Q}_{0}=L_{0,2}\mathbb{Q}_{1}+C_{0,2}\mathbb{Q}_{0}.$

Denoting $\mathbb{Q}_{0}=(1)$ and
$\mathbb{Q}_{1}=\left(\begin{matrix}x+\alpha\\\ y+\beta\end{matrix}\right)$,
it follows that

$\begin{array}[]{lll}x=(1\quad 0)\left(\begin{matrix}x+\alpha\\\
y+\beta\end{matrix}\right)+c_{00}^{(1)},\\\\[12.91663pt] y=(0\quad
1)\left(\begin{matrix}x+\alpha\\\
y+\beta\end{matrix}\right)+c_{00}^{(2)}.\end{array}$

It is easy to see that $\alpha=\beta$, since by hypothesis,
$c_{00}^{(1)}=c_{00}^{(2)}$. Hence, $Q^{1}_{1,0}(x,y)=Q^{1}_{0,1}(y,x)$, that
is, $\mathbb{Q}_{0}$ and $\mathbb{Q}_{1}$ are reflexive polynomial vectors.

We now prove by mathematical induction that, if $\mathbb{Q}_{n}$ and
$\mathbb{Q}_{n-1}$ are reflexive polynomial vectors, then it holds for
$\mathbb{Q}_{n+1}$. Omitting the variables $(x,y)$, the three-term relations
(3.6) can be written as

$L_{n,1}\mathbb{Q}_{n+1}=x\mathbb{Q}_{n}-C_{n,1}\mathbb{Q}_{n}-D_{n,1}\mathbb{Q}_{n-1},$
(5.6)
$L_{n,2}\mathbb{Q}_{n+1}=y\mathbb{Q}_{n}-C_{n,2}\mathbb{Q}_{n}-D_{n,2}\mathbb{Q}_{n-1}.$
(5.7)

From (5.6), for $k=0,1,\ldots,n,$ we can write $Q_{n+1-k,k}^{n+1}(x,y)$ as

$Q_{n+1-k,k}^{n+1}(x,y)=xQ_{n-k,k}^{n}(x,y)-\sum_{j=0}^{n}c^{(1)}_{kj}Q_{n-j,j}^{n}(x,y)-\sum_{j=0}^{n-1}d^{(1)}_{kj}Q_{n-1-j,j}^{n-1}(x,y).$

On the other hand, from (5.7), we can write $Q_{k,n+1-k}^{n+1}(x,y)$, for
$k=0,1,\ldots,n,$ as

$Q_{k,n+1-k}^{n+1}(x,y)=yQ_{k,n-k}^{n}(x,y)-\sum_{j=0}^{n}c^{(2)}_{n-k,j}Q_{n-j,j}^{n}(x,y)-\sum_{j=0}^{n-1}d^{(2)}_{n-k,j}Q_{n-1-j,j}^{n-1}(x,y),$

Since $\mathbb{Q}_{n}$ and $\mathbb{Q}_{n-1}$ are reflexives, and
$C_{n,1}\leftrightharpoons C_{n,2}$ and $D_{n,1}\leftrightharpoons D_{n,2}$,
we get

$\displaystyle Q_{k,n+1-k}^{n+1}(y,x)=$ $\displaystyle
xQ_{k,n-k}^{n}(y,x)-\sum_{j=0}^{n}c^{(2)}_{n-k,j}Q_{n-j,j}^{n}(y,x)-\sum_{j=0}^{n-1}d^{(2)}_{n-k,j}Q_{n-1-j,j}^{n-1}(y,x)$
$\displaystyle=$ $\displaystyle
xQ_{n-k,k}^{n}(x,y)-\sum_{j=0}^{n}c^{(1)}_{k,n-j}Q_{j,n-j}^{n}(x,y)-\sum_{j=0}^{n-1}d^{(1)}_{k,n-j}Q_{j,n-1-j}^{n-1}(x,y)$
$\displaystyle=$ $\displaystyle
xQ_{n-k,k}^{n}(x,y)-\sum_{j=0}^{n}c^{(1)}_{kj}Q_{n-j,j}^{n}(x,y)-\sum_{j=0}^{n-1}d^{(1)}_{kj}Q_{n-1-j,j}^{n-1}(x,y)$
$\displaystyle=$ $\displaystyle Q_{n+1-k,k}^{n}(x,y).$

Therefore, $\mathbb{Q}_{n+1}$ is reflexive polynomial vector. ∎

In the next result we relate reflexive MOPS with centrosymmetric matrices.

######  Theorem 5.3.

With the same assumptions of the Theorem 5.1, for $n\in\mathbb{N}$, the
following matrices

$\widehat{C}_{n}=\left(\begin{array}[]{c|c}C_{n,1}&\mathtt{0}\\\
\hline\cr\mathtt{0}&C_{n,2}\end{array}\right)\quad\mbox{and}\quad\widehat{D}_{n}=\left(\begin{array}[]{c|c}D_{n,1}&\mathtt{0}\\\
\hline\cr\mathtt{0}&D_{n,2}\end{array}\right)$

are centrosymmetric matrices.

###### Proof.

The proof follows directly from Proposition 2.11 and Theorem 5.1. ∎

We now present the results for orthonormal polynomial systems.

We need the following lemma, about the square root of a symmetric
centrosymmetric positive definite matrix. Following [13, p. 440], if $X$ is a
symmetric positive definite matrix, there exists a unique symmetric positive
definite matrix, $X^{\frac{1}{2}}$, such that
$X^{\frac{1}{2}}X^{\frac{1}{2}}=X$. The matrix $X^{\frac{1}{2}}$ is called the
square root matrix of $X$.

######  Lemma 5.4.

Let $X$ be a symmetric centrosymmetric positive definite matrix. Then, the
square root matrix $X^{\frac{1}{2}}$ is symmetric centrosymmetric positive
definite.

###### Proof.

Considering $X=(x_{ij})_{i,j=0}^{n}$ a symmetric centrosymmetric positive
definite of size $n+1$, there exist an orthogonal matrix
$R=(r_{ij})_{i,j=0}^{n}$ and a diagonal matrix
$D=\operatorname{diag}(\lambda_{0},\lambda_{1},\ldots,\lambda_{n}),$ where
$\lambda_{i}>0,$ for $i=0,1,\ldots,n$, are the eigenvalues of $X$, such that

$X=R\,D\,R^{T},$ (5.8)

see [13]. Hence, the square root matrix of $X$ can be written as

$X^{\frac{1}{2}}=R\,D^{\frac{1}{2}}\,R^{T},$ (5.9)

where
$D^{\frac{1}{2}}=\operatorname{diag}(\sqrt{\lambda_{0}},\sqrt{\lambda_{1}},\ldots,\sqrt{\lambda_{n}})$
and we denote $X^{\frac{1}{2}}=(\tilde{x}_{ij})_{i,j=0}^{n}$.

First we suppose that $\lambda_{0}=\lambda_{1}=\cdots=\lambda_{n}$, then
$X^{\frac{1}{2}}=\sqrt{\lambda}I_{n+1}$ and it is symmetric centrosymmetric
positive definite matrix.

We suppose now that the eigenvalues $\lambda_{i}$, $i=0,1,\ldots,n$, are not
simultaneously equals. From (5.8) we observe that the entries of matrix $X$
are linear combinations of $\\{\lambda_{0},\lambda_{1},\ldots,\lambda_{n}\\}$,
and it is possible to write

$x_{ij}=\sum_{k=0}^{n}\,r_{ik}\,r_{jk}\,\lambda_{k}.=\sum_{k=0}^{n}\,d_{k}^{(i,j)}\,\lambda_{k},$

where $d_{k}^{(i,j)}=r_{ik}\,r_{jk}$, for $k=0,1,\ldots,n.$ Similarly, from
(5.9), the entries of matrix $X^{\frac{1}{2}}$ can be written as

$\tilde{x}_{ij}=\sum_{k=0}^{n}\,r_{ik}\,r_{jk}\,\sqrt{\lambda_{k}}=\sum_{k=0}^{n}\,d_{k}^{(i,j)}\,\sqrt{\lambda_{k}}.$

From Proposition 2.10, we know that the eigenvectors of a symmetric
centrosymmetric positive definite matrix are either symmetric or skew-
symmetric vectors. In the decomposition $X=RDR^{T}$ the columns of matrix $R$
correspondent to the eigenvectors of $X$. Hence the entries of the $k$th
column of $R$ satisfy either

$r_{i,k}=r_{n-i,k},\quad i=0,1,\ldots,n,$

or

$r_{i,k}=-r_{n-i,k},\quad i=0,1,\ldots,n.$

Therefore, for a fixed $k$, one can conclude the following

$d_{k}^{(i,j)}=r_{ik}\,r_{jk}=r_{n-i,k}\,r_{n-j,k}=d_{k}^{(n-i,n-j)}.$

Finally, we observe that, for $i,j=0,1,\ldots,n,$

$\tilde{x}_{ij}=\sum_{k=0}^{n}\,d_{k}^{(i,j)}\,\sqrt{\lambda_{k}}=\sum_{k=0}^{n}\,d_{k}^{(n-i,n-j)}\,\sqrt{\lambda_{k}}=\tilde{x}_{n-i,n-j},$

therefore, $X^{\frac{1}{2}}$ is symmetric centrosymmetric positive definite
matrix. ∎

Next results relate reflexive orthonormal polynomial system with
centrosymmetric matrices.

######  Theorem 5.5.

Consider the orthonormal polynomial system $\\{\mathbb{P}_{n}\\}_{n\geqslant
0}$ associated with a weight function $W(x,y)$ satisfying (4.1) and defined by
$\mathbb{P}_{n}=H_{n}^{-1/2}\mathbb{Q}_{n}$, for $n\geqslant 0$, where
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is the associated MOPS and
$H_{n}=\langle\mathbf{u},\mathbb{Q}_{n}\mathbb{Q}_{n}^{T}\rangle$. Let
$A_{n,i}$ and $B_{n,i}$, $i=1,2$, be the coefficient matrices of the three-
term relations (3.5) for $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$. Then, the
matrices

$\widehat{A}_{n}=\left(\begin{array}[]{c|c}A_{n,1}&\mathtt{0}\\\
\hline\cr\mathtt{0}&A_{n,2}\end{array}\right)\quad\mbox{and}\quad\widehat{B}_{n}=\left(\begin{array}[]{c|c}B_{n,1}&\mathtt{0}\\\
\hline\cr\mathtt{0}&B_{n,2}\end{array}\right)$

are centrosymmetric.

###### Proof.

From [11], we know that, for orthonormal polynomial system
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$, defined by
$\mathbb{P}_{n}=H_{n}^{-1/2}\mathbb{Q}_{n}$, the matrices $A_{n,i}$,
$B_{n,i}$, $C_{n,i}$, and $D_{n,i}$ of the associated three-term relations are
related as

$A_{n,i}=H_{n}^{\frac{1}{2}}D_{n+1,i}^{T}H_{n+1}^{-\frac{1}{2}}\quad and\quad
B_{n,i}=H_{n}^{-\frac{1}{2}}C_{n,i}H_{n}^{\frac{1}{2}},\quad i=1,2,$

where the matrix $H_{n}^{\frac{1}{2}}$ is the symmetric centrosymmetric
positive definite matrix such that
$H_{n}^{\frac{1}{2}}H_{n}^{\frac{1}{2}}=H_{n}$.

Hence, the matrix $\widehat{A}_{n}$ can be written as

$\widehat{A}_{n}=\left(\begin{array}[]{c|c}H_{n}^{\frac{1}{2}}D_{n+1,1}^{T}H_{n+1}^{-\frac{1}{2}}&\mathtt{0}\\\
\hline\cr\mathtt{0}&H_{n}^{\frac{1}{2}}D_{n+1,2}^{T}H_{n+1}^{-\frac{1}{2}}\end{array}\right),$

or

$\widehat{A}_{n}=\left(\begin{array}[]{c|c}H_{n}^{\frac{1}{2}}&\mathtt{0}\\\
\hline\cr\mathtt{0}&H_{n}^{\frac{1}{2}}\end{array}\right)\left(\begin{array}[]{c|c}D_{n+1,1}^{T}&\mathtt{0}\\\
\hline\cr\mathtt{0}&D_{n+1,2}^{T}\end{array}\right)\left(\begin{array}[]{c|c}H_{n+1}^{-\frac{1}{2}}&\mathtt{0}\\\
\hline\cr\mathtt{0}&H_{n+1}^{-\frac{1}{2}}\end{array}\right).$

From Corollary 4.3 and Lemma 5.4, we know that $H_{n}^{\frac{1}{2}}$ is
centrosymmetric matrix, then $H_{n}^{\frac{1}{2}}\leftrightharpoons
H_{n}^{\frac{1}{2}}$. Hence, from Proposition 2.11

$\left(\begin{array}[]{c|c}H_{n}^{\frac{1}{2}}&\mathtt{0}\\\
\hline\cr\mathtt{0}&H_{n}^{\frac{1}{2}}\end{array}\right)\quad\mbox{and}\quad\left(\begin{array}[]{c|c}H_{n+1}^{-\frac{1}{2}}&\mathtt{0}\\\
\hline\cr\mathtt{0}&H_{n+1}^{-\frac{1}{2}}\end{array}\right)$

are centrosymmetric matrices. Furthermore, since $D_{n+1,1}\leftrightharpoons
D_{n+1,2}$, from Theorem 5.3

$\left(\begin{array}[]{c|c}D_{n+1,1}^{T}&\mathtt{0}\\\
\hline\cr\mathtt{0}&D_{n+1,2}^{T}\end{array}\right)$

is also centrosymmetric matrix. Therefore, from item 2) in Proposition 2.8,
the matrix $\widehat{A}_{n}$ is centrosymmetric.

Similarly, since $B_{n,i}=H_{n}^{-\frac{1}{2}}C_{n,i}H_{n}^{\frac{1}{2}},$
$i=1,2,$ and $C_{n,1}\leftrightharpoons C_{n,2}$, it follows that
$\widehat{B}_{n}$ is centrosymmetric matrix. ∎

Finally, we show that the orthonormal polynomial system,
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$, given by
$\mathbb{P}_{n}=H_{n}^{-1/2}\mathbb{Q}_{n}$, associated with a reflexive
weight function also has the reflexive property.

######  Corollary 5.6.

With the same hypotheses of Theorem 5.5, the orthonormal polynomial
$\mathbb{P}_{n}$, defined by $\mathbb{P}_{n}=H_{n}^{-1/2}\mathbb{Q}_{n}$, for
$n\geqslant 0$, is a reflexive polynomial vector. Furthermore, the associated
coefficient matrices of the three-term relations (3.5) satisfy
$A_{n,1}\leftrightharpoons A_{n,2}$ and $B_{n,1}\leftrightharpoons B_{n,2}$.

###### Proof.

From Theorem 4.2, $\mathbb{Q}_{n}$ is a reflexive polynomial vector. From
Corollary 4.3 and Lemma 5.4, the matrix $H_{n}^{\frac{1}{2}}$ is
centrosymmetric. Hence, from Proposition 3.2, it follows that
$\mathbb{P}_{n}=H_{n}^{-1/2}\mathbb{Q}_{n}$ is a reflexive polynomial vector.

The reverse relations $A_{n,1}\leftrightharpoons A_{n,2}$ and
$B_{n,1}\leftrightharpoons B_{n,2}$ follow from Proposition 2.11. ∎

##  6\. Particular Cases

In this section we present several particular cases and examples of reflexive
OPS to illustrate the new concepts.

###  6.1. Tensor product of univariate orthogonal polynomials

The simplest example of bivariate orthogonal polynomials system is associated
with the tensor product of the same weight function in one variable. Consider
$w(x)$ a weight function defined for $x\in(a,b)$, then

$W(x,y)=w(x)w(y)=W(y,x),\quad for\ (x,y)\in(a,b)\times(a,b),$

is a reflexive weight function. The associated OPS,
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$, is given by

$\mathbb{P}_{n}=\left(\begin{matrix}p_{n}(x)p_{0}(y)\\\\[8.61108pt]
p_{n-1}(x)p_{1}(y)\\\\[8.61108pt] \vdots\\\\[8.61108pt]
p_{0}(x)p_{n}(y)\end{matrix}\right),$

where $\\{p_{n}(x)\\}_{n\geqslant 0}$ is the orthogonal polynomial sequence
with respect to $w(x)$. Hence, $\mathbb{P}_{n}$ is a reflexive orthogonal
polynomial vector.

Moreover, consider the three-term recurrence relation for the orthogonal
polynomials $p_{n}(x)$, ${n\geqslant 0},$ given by

$xp_{n}(x)=\lambda_{n}p_{n+1}(x)+\gamma_{n}p_{n}(x)+\upsilon_{n}p_{n-1}(x),\quad
n\geqslant 0,$

with $p_{-1}(x)=0$ and $\lambda_{n},\gamma_{n},\upsilon_{n}\in\mathbb{R}$. It
is known that the tree-term relations for the associated OPS are

$x_{i}\mathbb{P}_{n}=\Lambda_{n,i}\mathbb{P}_{n+1}+\Gamma_{n,i}\mathbb{P}_{n}+\Upsilon_{n,i}\mathbb{P}_{n-1},\quad
i=1,2,$

where $x_{1}=x$, $x_{2}=y$ and the coefficient matrices are given by

$\Lambda_{n,1}=\left(\begin{array}[]{ccccc|c}\lambda_{n}&&&&\bigcirc&0\\\
&\lambda_{n-1}&&&&0\\\ &&&\ddots&&\vdots\\\
\bigcirc&&&&\lambda_{0}&0\end{array}\right),\quad\Lambda_{n,2}=\left(\begin{array}[]{c|ccccc}0&\lambda_{0}&&&&\bigcirc\\\
0&&\ddots&&&\\\ \vdots&&&&\lambda_{n-1}&\\\
0&\bigcirc&&&&\lambda_{n}\end{array}\right),$
$\Gamma_{n,1}=\left(\begin{matrix}\gamma_{n}&&&\bigcirc\\\ &\gamma_{n-1}&&\\\
&&\ddots&\\\
\bigcirc&&&\gamma_{0}\end{matrix}\right),\quad\Gamma_{n,2}=\left(\begin{matrix}\gamma_{0}&&&\bigcirc\\\
&\ddots&&\\\ &&\gamma_{n-1}&\\\ \bigcirc&&&\gamma_{n}\end{matrix}\right),$
$\Upsilon_{n,1}=\left(\begin{matrix}\upsilon_{n}&&&\bigcirc\\\
&\upsilon_{n-1}&&\\\ &&\ddots&\\\ \bigcirc&&&\upsilon_{1}\\\ \hline\cr
0&0&\ldots&0\end{matrix}\right),\quad\mbox{and}\quad\Upsilon_{n,2}=\left(\begin{matrix}0&\ldots&0&0\\\
\hline\cr\upsilon_{1}&&&\bigcirc\\\ &\ddots&&\\\ &&\upsilon_{n-1}&\\\
\bigcirc&&&\upsilon_{n}\end{matrix}\right).$

Clearly, $\Lambda_{n,1}\leftrightharpoons\Lambda_{n,2},$
$\Gamma_{n,1}\leftrightharpoons\Gamma_{n,2}$, and
$\Upsilon_{n,1}\leftrightharpoons\Upsilon_{n,2}$.

###  6.2. Bivariate orthogonal polynomials on the simplex

Let us consider the family of the monic bivariate orthogonal polynomials on
the simplex $\Omega=\\{(x,y)\in\mathbb{R}^{2}\,|\,x,y\geqslant
0,1-x-y\geqslant 0\\}$ associated with the weight function

$W^{(\alpha,\beta,\gamma)}(x,y)=x^{\alpha}y^{\beta}(1-x-y)^{\gamma},\quad\alpha,\beta,\gamma>-1.$

We consider $\alpha=\beta$ and the MOPS associated with the weight function

$W_{\alpha,\alpha,\gamma}(x,y)=(xy)^{\alpha}(1-x-y)^{\gamma},\quad\alpha,\gamma>-1.$
(6.1)

Observe that $W_{\alpha,\alpha,\gamma}(x,y)$ defined in $\Omega$ is a
reflexive weight function. Following [11, p.36], let
$\\{\mathbb{V}^{(\alpha,\gamma)}_{n}\\}_{n\geqslant 0}$ be the monic OPS on
the simplex orthogonal with respect to the weight function
$W_{\alpha,\alpha,\gamma}(x,y)$.

From the explicit formula given in [11, p. 36], we can calculate the first
vectors of the MOPS, that are

$\mathbb{V}^{(\alpha,\gamma)}_{0}=(1),\quad\mathbb{V}^{(\alpha,\gamma)}_{1}=\begin{pmatrix}x-\dfrac{2\alpha+1}{4\alpha+2\gamma+3}\\\\[8.61108pt]
y-\dfrac{2\alpha+1}{4\alpha+2\gamma+3}\end{pmatrix},$

and

$\mathbb{V}^{(\alpha,\gamma)}_{2}=\begin{pmatrix}x^{2}-\dfrac{2(2\alpha+3)}{4\alpha+2\gamma+7}x+\dfrac{(2\alpha+1)(2\alpha+3)}{(4\alpha+2\gamma+5)(4\alpha+2\gamma+7)}\\\\[12.91663pt]
xy-\dfrac{2\alpha+1}{4\alpha+2\gamma+7}(x+y)+\dfrac{(2\alpha+1)^{2}}{(4\alpha+2\gamma+5)(4\alpha+2\gamma+7)}\\\\[12.91663pt]
y^{2}-\dfrac{2(2\alpha+3)}{4\alpha+2\gamma+7}y+\dfrac{(2\alpha+1)(2\alpha+3)}{(4\alpha+2\gamma+5)(4\alpha+2\gamma+7)}\end{pmatrix}.$

These polynomial vectors are reflexives. Now, consider the three-term relation
satisfied by $\\{\mathbb{V}^{(\alpha,\gamma)}_{n}\\}_{n\geqslant 0}$,

$x_{i}\mathbb{V}^{(\alpha,\gamma)}_{n}=L_{n,i}\mathbb{V}^{(\alpha,\gamma)}_{n+1}+C^{(\alpha,\gamma)}_{n,i}\mathbb{V}^{(\alpha,\gamma)}_{n}+D^{(\alpha,\gamma)}_{n,i}\mathbb{V}^{(\alpha,\gamma)}_{n-1},\quad
i=1,2.$

From [2], the shape of the coefficient matrices is

$C^{(\alpha,\gamma)}_{n,1}=\begin{pmatrix}c^{(1)}_{00}&\quad&\quad&\bigcirc\\\
c^{(1)}_{10}&c^{(1)}_{11}&\quad&\quad\\\ \quad&\ddots&\ddots&\quad\\\
\bigcirc&\quad&c^{(1)}_{n,n-1}&c^{(1)}_{nn}\end{pmatrix},C^{(\alpha,\gamma)}_{n,2}=\begin{pmatrix}c^{(2)}_{00}&c^{(2)}_{01}&\quad&\bigcirc\\\
\quad&c^{(2)}_{11}&\ddots&\quad\\\ \quad&\quad&\ddots&c_{n-1,n}^{(2)}\\\
\bigcirc&\quad&\quad&c^{(2)}_{nn}\end{pmatrix},$
$D^{(\alpha,\gamma)}_{n,1}=\begin{pmatrix}d^{(1)}_{00}&\quad&\quad&\quad&\bigcirc\\\\[4.30554pt]
d^{(1)}_{10}&d^{(1)}_{11}&\quad&\quad&\quad\\\\[4.30554pt]
d^{(1)}_{20}&d^{(1)}_{21}&d^{(1)}_{22}&\quad&\quad\\\\[4.30554pt]
\quad&\ddots&\ddots&\ddots&\quad\\\\[4.30554pt]
\quad&\quad&\ddots&\ddots&d^{(1)}_{n-1,n-1}\\\\[4.30554pt]
\bigcirc&\quad&\quad&d^{(1)}_{n,n-2}&d^{(1)}_{n,n-1}\end{pmatrix}$

and

$D^{(\alpha,\gamma)}_{n,2}=\begin{pmatrix}d^{(2)}_{00}&d^{(2)}_{01}&\quad&\quad&\bigcirc\\\\[4.30554pt]
d^{(2)}_{10}&d^{(2)}_{11}&d^{(2)}_{12}&\quad&\quad\\\\[4.30554pt]
\quad&d^{(2)}_{2,1}&d^{(2)}_{22}&\ddots&\quad\\\\[4.30554pt]
\quad&\quad&\ddots&\ddots&d^{(2)}_{n-2,n-1}\\\\[4.30554pt]
\quad&\quad&\quad&d^{(2)}_{n-1,n-2}&d^{(2)}_{n-1,n-1}\\\\[4.30554pt]
\bigcirc&\quad&\quad&\quad&d^{(2)}_{n,n-1}\end{pmatrix}.$

The explicit expressions of the entries are

$\displaystyle c^{(1)}_{ii}$
$\displaystyle=\dfrac{(i-n)(\alpha-i+n)}{2\alpha+\gamma+2n+1}+\dfrac{(-i+n+1)(\alpha-i+n+1)}{2\alpha+\gamma+2n+3}$
$\displaystyle=\dfrac{(n-i+1)(\alpha+n-i+1)}{2\alpha+\gamma+2n+3}-\dfrac{(n-i)(\alpha+n-i)}{2\alpha+\gamma+2n+1}=c^{(2)}_{n-i,n-i},\quad
0\leqslant i\leqslant n,$ $\displaystyle c^{(1)}_{i+1,i}$
$\displaystyle=-\dfrac{2(i+1)(\alpha+i+1)}{(2\alpha+\gamma+2n+1)(2\alpha+\gamma+2n+3)}$
$\displaystyle=\dfrac{2(n-i-1-n)(\alpha-n+i+1+n)}{(2\alpha+\gamma+2n+1)(2\alpha+\gamma+2n+3)}=c^{(2)}_{n-(i+1),n-i},\quad
0\leqslant i\leqslant n-1,$

and $c^{(1)}_{ij}=0$, for $j>i$ or $j<i-1$, $c^{(2)}_{ij}=0$, for $j<i$ or
$j>i+1$, it follows that $C^{(\alpha,\gamma)}_{n,1}\leftrightharpoons
C^{(\alpha,\gamma)}_{n,2}$.

Analogously, working with the explicit expressions of the entries of the
matrices $D^{(\alpha,\gamma)}_{n,i}$, $i=1,2$, given in [2], we get

$\displaystyle d^{(1)}_{ii}=d^{(2)}_{n-i,n-1-i},\quad 0\leqslant i\leqslant
n-1,$ $\displaystyle d^{(1)}_{i+1,i}=d^{(2)}_{n-i-1,n-1-i},\quad 0\leqslant
i\leqslant n-2,$ $\displaystyle d^{(1)}_{i+2,i}=d^{(2)}_{n-i-2,n-1-i},\quad
0\leqslant i\leqslant n-2,$

and, since $d^{(1)}_{ij}=0$, for $j>i$ or $j<i-2$, $d^{(2)}_{ij}=0$, for
$j<i-1$ or $j>i+1$, it follows that
$D^{(\alpha,\gamma)}_{n,1}\leftrightharpoons D^{(\alpha,\gamma)}_{n,2}$.

###  6.3. Bivariate Freud weight function

In [6] we investigate the bivariate Freud weight function given by

$W(x,y)=e^{-q(x,y)},\qquad(x,y)\in\mathbb{R}^{2},$

where

$q(x,y)=a_{4,0}\,x^{4}+a_{2,2}\,x^{2}\,y^{2}+a_{0,4}\,y^{4}+a_{2,0}\,x^{2}+a_{0,2}\,y^{2}$

and $a_{i,j}$ are real parameters. Setting $a_{4,0}=a_{0,4}=a$, $a_{2,2}=b$,
and $a_{2,0}=a_{0,2}=c$, the particular case,

$W(x,y)=e^{-[a(x^{4}+y^{4})+b\,x^{2}\,y^{2}+c(x^{2}+y^{2})]}$

is a reflexive weight function. Since $W(x,y)$ is an even function, the MOPS
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ satisfy the three-term relations

$x_{i}\mathbb{Q}_{n}=L_{n,i}\mathbb{Q}_{n+1}+D_{n,i}\mathbb{Q}_{n-1},\quad
i=1,2,$

for $n\geqslant 0$, $\mathbb{Q}_{-1}=0$ and $\mathbb{Q}_{0}=(1)$, where
$D_{n,1}\leftrightharpoons D_{n,2}$.

###  6.4. Uvarov modification

Let $W(x,y)$ be a weight function defined on a domain
$\Omega\subset\mathbb{R}^{2}$ such that all moments exist

$\iint_{\Omega}x^{k}\,y^{l}\,W(x,y)dxdy<+\infty,$

for $k,l\geqslant 0$. Define the inner products

$(f,g)=\iint_{\Omega}f(x,y)g(x,y)W(x,y)dxdy,$

and

$(f,g)_{U}=(f,g)+\mathbf{f(x)}\,\Lambda\,\mathbf{g(x)}^{T},$

where $\mathbf{x}=((x_{0},y_{0}),(x_{1},y_{1}),\ldots,(x_{n},y_{n}))$,
$(x_{i},y_{i})\in\mathbb{R}^{2}$, $i=0,1,\ldots,n$, are fixed points,
$\Lambda$ is a symmetric positive semi-definite matrix of size
$(n+1)\times(n+1)$, $\mathbf{f(x)}=(f(x_{0},y_{0}),$
$f(x_{1},y_{1}),\ldots,f(x_{n},y_{n}))$, and
$\mathbf{g(x)}=(g(x_{0},y_{0}),g(x_{1},y_{1}),\ldots,g(x_{n},y_{n}))$. This
modification of the original inner product is known as Uvarov modification,
see [2, 9].

Now, we show that if $W(x,y)$ is reflexive weight function, $\Lambda$ is
centrosymmetric matrix, and
$\mathbf{x}=((x_{0},y_{0}),(x_{1},y_{1}),\ldots,(x_{n},y_{n}))$ is such that
$x_{i}=y_{n-i},$ $i=0,1,\ldots,n$, then the inner product $(f,g)_{U}$ is
reflexive in the sense that

$(x^{k},y^{l})_{U}=(x^{l},y^{k})_{U},\quad k,l\geqslant 0.$

In fact, let $\Lambda=(\lambda_{ij})_{i,j=0}^{n,n}$, and

$(x^{l},y^{k})_{U}=\displaystyle\iint_{\Omega}x^{l}y^{k}W(x,y)dxdy+\mathbf{x^{l}}\Lambda(\mathbf{y^{k}})^{T},$

where $\mathbf{x^{l}}=(x_{0}^{l},x_{1}^{l},\ldots,x_{n}^{l})$ and
$\mathbf{y^{k}}=(y_{0}^{k},y_{1}^{k},\ldots,y_{n}^{k})$. Then,

$\displaystyle(x^{l},y^{k})_{U}$
$\displaystyle=\iint_{\Omega}x^{l}y^{k}W(x,y)dxdy+\displaystyle\sum_{i=0}^{n}\sum_{j=0}^{n}\lambda_{ij}x_{i}^{l}y_{j}^{k}$
$\displaystyle=\iint_{\Omega}y^{l}x^{k}W(y,x)dxdy+\displaystyle\sum_{i=0}^{n}\sum_{j=0}^{n}\lambda_{ij}x_{i}^{l}y_{j}^{k}$
$\displaystyle=\iint_{\Omega}x^{k}y^{l}W(x,y)dxdy+\displaystyle\sum_{i=0}^{n}\sum_{j=0}^{n}\lambda_{ij}x_{i}^{l}y_{j}^{k}$
$\displaystyle=\iint_{\Omega}x^{k}y^{l}W(x,y)dxdy+\displaystyle\sum_{i=0}^{n}\sum_{j=0}^{n}\lambda_{n-i,n-j}x_{n-j}^{k}y_{n-i}^{l},$

making the change of variables $x\leftrightarrow y$ in the first above
integral, using the fact that $W$ is reflexive, $\Lambda$ is centrosymmetric
matrix and $x_{i}=y_{n-i},\,i=0,1,\ldots,n$. Hence, making $r=n-i,\,s=n-j$ and
using that $\Lambda$ is symmetric, it becomes

$(x^{l},y^{k})_{U}=\iint_{\Omega}x^{k}y^{l}W(x,y)dxdy+\displaystyle\sum_{s=0}^{n}\sum_{r=0}^{n}\lambda_{s,r}x_{s}^{k}y_{r}^{l}=(x^{k},y^{l})_{U}.$

As a numerical example, we consider again OPS on simplex in $\mathbb{R}^{2}$,
with the weight function $W^{(1,1/2)}(x,y)$ defined in (6.1). Let
$\Lambda=\frac{1}{2}I_{3}$ and $\mathbf{x}=((1,0),(0,0),$ $(0,1))$, hence, the
reflexive Uvarov inner product is given by

$\displaystyle(f,g)_{U}=$
$\displaystyle\iint_{\Omega}\,f(x,y)\,g(x,y)\,x\,y\,\sqrt{1-x-y}\,dx\,dy$
$\displaystyle+\dfrac{1}{2}[f(1,0)g(1,0)+f(0,0)g(0,0)+f(0,1)g(0,1)].$

The first polynomial vectors of the associated MOPS were calculated in [2],
and they are reflexives:

$\mathbb{Q}_{0}=(1),\quad\mathbb{Q}_{1}=\begin{pmatrix}x-\dfrac{10459}{31361}\\\\[8.61108pt]
y-\dfrac{10459}{31361}\end{pmatrix},$

and

$\mathbb{Q}_{2}=\begin{pmatrix}x^{2}-\dfrac{320811709991693}{321113175737485}x+\dfrac{36006461568}{64222635147497}y+\dfrac{51957376}{30832708855}\\\\[8.61108pt]
xy-\dfrac{5355008}{7115240505}(x+y)-\dfrac{69058048}{92498126565}\\\\[8.61108pt]
y^{2}+\dfrac{36006461568}{64222635147497}x-\dfrac{320811709991693}{321113175737485}y+\dfrac{51957376}{30832708855}\end{pmatrix}.$

###  6.5. Christoffel modification

Let $\lambda(x,y)$ be a real bivariate polynomial given as

$\lambda(x,y)=a\,(x^{2}+y^{2})+b\,x\,y+c\,(x+y)+d,$ (6.2)

where $a,b,c,d\in\mathbb{R}$ and $|a|+|b|>0$. We consider the functional
$\mathbf{v}$ obtained from a polynomial modification of a reflexive functional
$\mathbf{u}$, given by

$\langle\mathbf{v},p(x,y)\rangle=\langle\lambda(x,y)\mathbf{u},p(x,y)\rangle=\langle\mathbf{u},\lambda(x,y)\,p(x,y)\rangle,$

for $p(x,y)\in\Pi$. The Christoffel modification on a multivariate functional,
using multiplication by a polynomial of degree 2, was studied in [10].

Let ${v}_{mn}$ be the associated moments of the functional $\mathbf{v}.$
Hence,

$\displaystyle{v}_{m,n}=$ $\displaystyle\ \langle\mathbf{v},x^{m}y^{n}\rangle$
$\displaystyle=$ $\displaystyle\
\langle\mathbf{u},[a(x^{2}+y^{2})+b\,x\,y+c(x+y)+d]x^{m}y^{n}\rangle$
$\displaystyle=$ $\displaystyle\
a\langle\mathbf{u},x^{m+2}y^{n}+x^{m}y^{n+2}\rangle+b\langle\mathbf{u},x^{m+1}y^{n+1}\rangle$
$\displaystyle\
+c\langle\mathbf{u},x^{m+1}y^{n}+x^{m}y^{n+1}\rangle+d\langle\mathbf{u},x^{m}y^{n}\rangle$
$\displaystyle=$ $\displaystyle\
a\langle\mathbf{u},x^{n}y^{m+2}+x^{n+2}y^{m}\rangle+b\langle\mathbf{u},x^{n+1}y^{m+1}\rangle$
$\displaystyle\
+c\langle\mathbf{u},x^{n}y^{m+1}+x^{n+1}y^{m}\rangle+d\langle\mathbf{u},x^{n}y^{m}\rangle$
$\displaystyle=$ $\displaystyle\
\langle\mathbf{u},[a(x^{2}+y^{2})+b\,x\,y+c(x+y)+d]x^{n}y^{m}\rangle={v}_{n,m},$

since $\mathbf{u}$ is reflexive. Therefore, the Christoffel modification of a
reflexive moment functional by a polynomial of type (6.2) preserves the
reflexive property.

Let $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ and
$\\{\widetilde{\mathbb{Q}}_{n}\\}_{n\geqslant 0}$ be the respective MOPS
associated respectively with $\mathbf{u}$ and $\mathbf{v}$. We remark that,
since $\mathbf{u}$ and $\mathbf{v}$ are reflexive moment functional, then
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ and
$\\{\widetilde{\mathbb{Q}}_{n}\\}_{n\geqslant 0}$ are reflexive MOPS.

From [10, Th. 4.1], it is known that, for $n\geqslant 1$, there exist real
matrices $R_{n}$ and $S_{n}$ of respective sizes $(n+1)\times n$ and
$n\times(n-1)$, with $S_{2}\not\equiv\mathtt{0}$, such that

$\mathbb{Q}_{n}=\widetilde{\mathbb{Q}}_{n}+R_{n}\widetilde{\mathbb{Q}}_{n-1}+S_{n}\widetilde{\mathbb{Q}}_{n-2},\quad
n\geqslant 1.$ (6.3)

We now verify that the matrices $R_{n}$ and $S_{n}$ are centrosymmetric. First
we denote $R_{n}=(r^{(n)}_{ij})_{i,j=0}^{n,n-1}$,
$S_{n}=(s^{(n)}_{ij})_{i,j=0}^{n,n-2}$,

$\mathbb{Q}_{n}=(Q_{n,0}^{n}(x,y),Q_{n-1,1}^{n}(x,y),\ldots,Q_{0,n}^{n}(x,y))^{T},$

and

$\widetilde{\mathbb{Q}}_{n}=(\widetilde{Q}_{n,0}^{n}(x,y),\widetilde{Q}_{n-1,1}^{n}(x,y),\ldots,\widetilde{Q}_{0,n}^{n}(x,y))^{T}.$

From relation (6.3) we get

$Q_{n-k,k}^{n}(x,y)=\widetilde{Q}_{n-k,k}^{n}(x,y)+\sum_{j=0}^{n-1}r_{n-k,j}^{(n)}\widetilde{Q}_{n-1-j,j}^{n-1}(x,y)+\sum_{j=0}^{n-2}s^{(n)}_{n-k,j}\widetilde{Q}_{n-2-j,j}^{n-2}(x,y),$
(6.4)

for $k=0,1,\ldots,n,$ and

${Q}_{k,n-k}^{n}(y,x)=\widetilde{Q}_{k,n-k}^{n}(y,x)+\sum_{j=0}^{n-1}r_{kj}^{(n)}\widetilde{Q}_{n-1-j,j}^{n-1}(y,x)+\sum_{j=0}^{n-2}s^{(n)}_{kj}\widetilde{Q}_{n-2-j,j}^{n-2}(y,x).$
(6.5)

Using the fact that $\\{\widetilde{\mathbb{Q}}_{n}\\}_{n\geqslant 0}$ is
reflexive, the representation (6.5) becomes

${Q}_{k,n-k}^{n}(y,x)=\widetilde{Q}_{n-k,k}^{n}(x,y)+\sum_{j=0}^{n-1}r_{kj}^{(n)}\widetilde{Q}_{j,n-1-j}^{n-1}(x,y)+\sum_{j=0}^{n-2}s^{(n)}_{kj}\widetilde{Q}_{j,n-2-j}^{n-2}(x,y).$

Making $l=n-1-j$ in the first summation and $l=n-2-j$ in the second, we get

$Q_{k,n-k}^{n}(y,x)=\widetilde{Q}_{n-k,k}^{n}(x,y)+\sum_{l=0}^{n-1}r_{k,n-1-l}^{(n)}\widetilde{Q}_{n-1-l,l}^{n-1}(x,y)+\sum_{l=0}^{n-2}s^{(n)}_{k,n-2-l}\widetilde{Q}_{n-2-l,l}^{n-2}(x,y)$
(6.6)

for $k=0,1,\ldots,n.$

Comparing expressions (6.4) and (6.6), and using that $\mathbb{Q}_{n}$ is
reflexive polynomial vector, we have $r^{(n)}_{n-k,j}=r^{(n)}_{k,n-1-j}$ and
$s^{(n)}_{n-k,j}=s^{(n)}_{k,n-2-j}$. Finally, setting $i=n-k$, then

$r^{(n)}_{ij}=r^{(n)}_{k,n-1-j}\quad\mbox{and}\quad
s^{(n)}_{ij}=s^{(n)}_{k,n-2-j}.$

Therefore, $R_{n}$ and $S_{n}$, $n\geqslant 1$, are centrosymmetric matrices.

Conversely, from [10, Th. 4.3], consider $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$
a MOPS associated with a moment functional $\mathbf{u}$ and two sequences of
matrices $\\{R_{n}\\}_{n\geqslant 1}$ and $\\{S_{n}\\}_{n\geqslant 2}$ of size
$(n+1)\times n$ and $(n+1)\times(n-1)$ respectively, with
$S_{2}\not\equiv\mathtt{0}$. Then, the monic polynomial system,
$\\{\widetilde{\mathbb{Q}}_{n}\\}_{n\geqslant 0}$, defined by
$\widetilde{\mathbb{Q}}_{0}=\mathbb{Q}_{0},$

$\displaystyle\widetilde{\mathbb{Q}}_{1}$
$\displaystyle=\mathbb{Q}_{1}-R_{1}\mathbb{Q}_{0},$
$\displaystyle\widetilde{\mathbb{Q}}_{n}$
$\displaystyle=\mathbb{Q}_{n}-R_{n}\mathbb{Q}_{n-1}-S_{n}\mathbb{Q}_{n-2},\quad
n\geqslant 2,$

is a MOPS associated with a moment functional
$\mathbf{v}=\lambda(x,y)\mathbf{v}$, where

$\lambda(x,y)=S_{2}^{T}H_{2}^{-1}\mathbb{Q}_{2}+M_{1}^{T}H_{1}^{-1}\mathbb{Q}_{1}+H_{0}^{-1}\mathbb{Q}_{0}$

and $H_{n}=\langle\mathbf{u},\mathbb{Q}_{n}\mathbb{Q}_{n}^{T}\rangle.$

Now we consider the particular case when $\mathbf{u}$ is a reflexive moment
functional and $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is a reflexive MOPS. If
the matrices $R_{n}$, ${n\geqslant 1}$ and $S_{n},$ $n\geqslant 2,$ are
centrosymmetric matrices, from Proposition 3.2 follows that the MOPS
$\\{\widetilde{\mathbb{Q}}_{n}\\}_{n\geqslant 0}$ is also a reflexive MOPS.

Moreover, from Corollary 4.3 and items 3) and 5) of Proposition 2.8, the
polynomial $\lambda(x,y)$ is a reflexive polynomial. Therefore,
$\mathbf{v}=\lambda(x,y)\mathbf{u}$ is a reflexive moment functional.

## Acknowledgements

This work is part of the doctoral thesis of the second author (GSC) at UNESP,
São José do Rio Preto, SP, Brazil, and it began to be developed during a
research visit by GSC to the University of Granada, Spain, supported by the
grant 88887.716711/2022-00 from CAPES, in the scope of the Program CAPES-
PrInt, International Cooperation Project number 88887.310463/2018-00, Brazil.

First author (CFB) thanks for the supported, through the Grant 2022/09575-5,
from FAPESP, São Paulo, Brazil.

Third author (TEP) thanks Grant CEX2020-001105-M funded by MCIN/AEI/
10.13039/501100011033 and Research Group Goya-384, Spain.

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