Multivariate Orthogonal Polynomials and Modified Moment Functionals

Multivariate Orthogonal Polynomials
and Modified Moment Functionals\star\star$\star$This paper is a contribution
to the Special Issue on Orthogonal Polynomials, Special Functions and
Applications. The full collection is available at
http://www.emis.de/journals/SIGMA/OPSFA2015.html

Antonia M. DELGADO, Lidia FERNÁNDEZ, Teresa E. PÉREZ and Miguel A. PIÑAR A.M.
Delgado, L. Fernández, T.E. Pérez and M.A. Piñar IEMath – Math Institute and
Department of Applied Mathematics, University of Granada,
18071, Granada, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>http://www.ugr.es/local/goya/

Received January 28, 2016, in final form September 05, 2016; Published online
September 10, 2016

Multivariate orthogonal polynomials can be introduced by using a moment
functional defined on the linear space of polynomials in several variables
with real coefficients. We study the so-called Uvarov and Christoffel
modifications obtained by adding to the moment functional a finite set of mass
points, or by multiplying it times a polynomial of total degree $2$,
respectively. Orthogonal polynomials associated with modified moment
functionals will be studied, as well as the impact of the modification in
useful properties of the orthogonal polynomials. Finally, some illustrative
examples will be given.

multivariate orthogonal polynomials; moment functionals; Christoffel
modification; Uvarov modification; ball polynomials

33C50; 42C10

##  1 Introduction

Using a moment functional approach as in [20, 21, 22], one interesting problem
in the theory of orthogonal polynomials in one and several variables is the
study of the modifications of a quasi-definite moment functional $u$ defined
on $\Pi$, the linear space of polynomials with real coefficients. In fact,
there are many works devoted to this topic since modifications of moment
functionals are underlying some well known facts, such as quasi-orthogonality,
relations between adjacent families, quadrature and cubature formulas, higher-
order (partial) differential equations, etc.

In the univariate case, given a quasi-definite moment functional $u$ defined
on $\Pi$, the (basic) Uvarov modification is defined by means of the addition
of a Dirac delta on $a\in\mathbb{R}$,

$\displaystyle v=u+\lambda\delta_{a},\qquad\textrm{such that}\quad\langle
v,p(x)\rangle=\langle u,p(x)\rangle+\lambda p(a),\qquad\lambda\neq 0.$

Apparently, this modification was introduced by V.B. Uvarov in 1969 [27], who
studied the case where a finite number of mass points is added to a measure,
and proved connection formulas for orthogonal polynomials with respect to the
modified measure in terms of those with respect to the original one.

In some special cases of classical Laguerre and Jacobi measures, if the
perturbations are given at the end points of the support of the measure, then
the new polynomials are eigenfunctions of higher-order differential operators
with polynomial coefficients and they are called Krall polynomials (see, for
instance, [32] and the references therein).

In the multivariate case, the addition of Dirac masses to a multivariate
measure was studied in [11] and [13]. Moreover, Uvarov modification for disk
polynomials was analysed in [10].

Besides Uvarov modifications by means of Dirac masses at finite discrete set
of points, in the context of several variables it is possible to modify the
moment functional by means of moment functionals defined on lower dimensional
manifolds such as curves, surfaces, etc. Recently, a family of orthogonal
polynomials with respect to such a Uvarov modification of the classical ball
measure by means of a mass uniformly distributed over the sphere was
introduced in [23]. The authors proved that, at least in the Legendre case,
these multivariate orthogonal polynomials satisfy a fourth-order partial
differential equation, which constitutes a natural extension of Krall
orthogonal polynomials [16] to the multivariate case. In [5], a modification
of a moment functional by adding another moment functional defined on a curve
is presented, and a Christoffel formula built up in terms of a Fredholm
integral equation is discussed. As far as we know, a general theory about
Uvarov modifications by means of moment functionals defined on lower
dimensional manifolds remains as an open problem.

Following [31], the univariate (basic) Christoffel modification is given by
the multiplication of a moment functional $u$ times a polynomial of degree
$1$, usually $x-a$, $a\in\mathbb{R}$,

$\displaystyle v=(x-a)u,\quad\textrm{acting as}\quad\langle
v,p(x)\rangle=\langle u,(x-a)p(x)\rangle.$

This type of transformations were first considered by E.B. Christoffel in 1858
[9] within the framework of Gaussian quadrature theory. Nowadays, Christoffel
formulas are classical results in the theory of orthogonal polynomials, and
they are presented in many general references (e.g., [8, 26]).

Christoffel modification is characterized by the linear relations that both
families, modified and non-modified orthogonal polynomials, satisfy. They are
called connection formulas. It is well known that some families of classical
orthogonal polynomials can be expressed as linear combinations of polynomials
of the same family for different values of their parameters, the so-called
_relations between adjacent families_ (e.g., see formulas in Chapter 22 in [1]
for Jacobi polynomials, or (5.1.13) in [26] for Laguerre polynomials). The
study of such type of linear combinations is also related with the concept of
_quasi-orthogonality_ introduced by M. Riesz in 1921 (see [8, p. 64]) as the
basis of his analysis of the moment problem, and it is related with quadrature
formulas based on the zeros of orthogonal polynomials.

The extension of this kind of results to the multivariate case is not always
possible. Gaussian cubature formulas of degree $2n-1$ were characterized by
Mysovskikh [24] in terms of the number of common zeros of the multivariate
orthogonal polynomials. However, these formulas only exist in very special
cases and the case of degree $2n-2$ becomes interesting. Here, linear
combinations of multivariate orthogonal polynomials play an important role, as
it can be seen for instance in [6, 25, 28, 29].

To our best knowledge, one of the first studies about Christoffel
modifications in several variables, multiplying a moment functional times a
polynomial of degree $1$, was done in [2], where the modification naturally
appears in the study of linear relations between two families of multivariate
polynomials. Necessary and sufficient conditions about the existence of
orthogonality properties for one of the families was given in terms of the
three term relations, by using Favard’s theorem in several variables [12].

In [7] the authors show that modifications of univariate moment functionals
are related with the Darboux factorization of the associated Jacobi matrix. In
this direction, in [3, 4, 5] long discussions about several aspects of the
theory of multivariate orthogonal polynomials can be found. In particular,
Darboux transformations for orthogonal polynomials in several variables were
presented in [3], and in [4] we can find an extension of the univariate
Christoffel determinantal formula to the multivariate context. Also, as in the
univariate case, they proved a connection with the Darboux factorization of
the Jacobi block matrix associated with the three term recurrence relations
for multivariate orthogonal polynomials. Similar considerations for
multivariate Geronimus and more general linear spectral transformations of
moment functionals can be found, among other topics, in [5].

In this paper, we study Uvarov and Christoffel modifications of quasi-definite
moment functionals. The study of orthogonal polynomials associated with moment
functionals fits into a quite general frame that includes families of
orthogonal polynomials associated either with positive-definite or non
positive-definite moment functionals such as those generated using Bessel
polynomials, among others. We give necessary and sufficient conditions in
order to obtain the quasi-definiteness of the modified moment functional in
both cases, Uvarov and Christoffel modifications (see Theorems 3.1 and 4.3).
We also investigate properties of the polynomials associated with the modified
functional, relations between original and modified orthogonal polynomials as
well as the impact of the modification in some useful properties of the
orthogonal polynomials.

When dealing with the Christoffel modification, and in the case where both
moment functionals, the original and the modified, are quasi-definite, some of
the results are similar to those obtained in [3, 4, 5] using a different
technique. In particular, the necessary condition in our Theorem 4.3 was
proven there for an arbitrary degree polynomial, but the sufficient condition
was not discussed there.

The main results of this work can be divided in three parts, corresponding
with Sections 3, 4 and 5. Section 2 is devoted to establish the basic concepts
and tools we will need along the paper. For that section, we recall the
standard notations and basic results in the theory of multivariate orthogonal
polynomials following mainly [12].

Uvarov modification of a quasi-definite moment functional is studied in
Section 3. In this case, we modify a quasi-definite moment functional by
adding several Dirac deltas at a finite discrete set of fixed points. First,
we will give a necessary and sufficient condition for the quasi-definiteness
of the moment functional associated with this Uvarov modification, and, in the
affirmative case, we will deduce the connection between both families of
orthogonal polynomials. A similar study was done in [11] in the case when the
original moment functional is defined from a measure, and the modification is
defined by means of a positive semi-definite matrix. In that case, both moment
functionals are positive-definite, and both orthogonal polynomial systems
exist. As a consequence, since it is possible to work with orthonormal
polynomials, in [11] the first family of orthogonal polynomials is considered
orthonormal, and formulas are simpler than in the general case considered
here.

In Section 4 we study the Christoffel modification by means of a second degree
polynomial. This is not a trivial extension of the case when the degree of the
polynomial is $1$ studied in [2], since in several variables not every
polynomial of degree $2$ factorizes as a product of polynomials of degree $1$.
Again, we relate both families of orthogonal polynomials and also we deduce
the orthogonality by using Favard’s theorem in several variables. In fact,
from a recursive expression for the modified polynomials, we give necessary
and sufficient conditions for the existence of a three term relation, and we
show that there exists a polynomial of second degree constructed in terms of
the first connection coefficients. Since three term relations can be
reformulated in terms of Jacobi block matrices, similar connection results can
be found in [4, 5] for arbitrary degree polynomials by using a block matrix
formalism. Moreover, we study this kind of Christoffel modification in the
particular case when the original moment functional is centrally symmetric,
the natural extension of the concept of symmetry for univariate moment
functionals.

Finally, Section 5 is devoted to apply our results to two families of
multivariate orthogonal polynomials: the classical orthogonal polynomials on
the ball in $\mathbb{R}^{d}$, and the Bessel–Laguerre polynomials, a family of
bivariate polynomials orthogonal with respect to a non positive-definite
moment functional, that can be found in [19] as solution of one of the
classical Krall and Sheffer’s second-order partial differential equation [18].

First, we modify the classical moment functional on the ball by adding a Dirac
mass at $\mathbf{0}\in\mathbb{R}^{d}$. This example was introduced in [13],
and here, we complete that study giving, as a new result, the asymptotics of
the modification. Next, we use the Christoffel modification by means of a
second degree polynomial to deduce a relation between adjacent families of
classical ball orthogonal polynomials in several variables. As far as we know,
there is not a relation of this kind for ball polynomials in the literature.
This relation can be seen as an extension of (22.7.23) in [1, p. 782] for
Gegenbauer polynomials in one variable. Last example corresponds to a Uvarov
modification for the non positive-definite classical Bessel–Laguerre bivariate
polynomials defined in [19].

##  2 Basic tools

In this section we collect the definitions and basic tools about orthogonal
polynomials in several variables that will be used later. We follow mainly
[12].

Along this paper, we will denote by $\mathcal{M}_{h\times k}(\mathbb{R})$ the
linear space of matrices of size $h\times k$ with real entries, and the
notation will simplify when $h=k$ as $\mathcal{M}_{h}(\mathbb{R})$. As usual,
we will say that $M\in\mathcal{M}_{h}(\mathbb{R})$ is non-singular (or
invertible) if $\det M\neq 0$, and symmetric if $M^{t}=M$, where $M^{t}$
denotes its transpose. Moreover, $I_{h}$ will denote the identity matrix of
size $h$, and we will omit the subscript when the size is clear from the
context.

Let us consider the $d$-dimensional space $\mathbb{R}^{d}$, with $d\geqslant
1$. Given $\nu=(n_{1},n_{2},\ldots,n_{d})\in\mathbb{N}_{0}^{d}$ a multi-index,
a monomial in $d$ variables is an expression of the form

$\displaystyle\mathbf{x}^{\nu}=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots
x_{d}^{n_{d}},$

where $\mathbf{x}=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$. The total degree of
the monomial is denoted by $n=|\nu|=n_{1}+n_{2}+\cdots+n_{d}$. Then, a
polynomial of total degree $n$ in $d$ variables with real coefficients is a
finite linear combination of monomials of degree at most $n$,

$\displaystyle p(\mathbf{x})=\sum_{|\nu|\leqslant
n}a_{\nu}\mathbf{x}^{\nu},\qquad a_{\nu}\in\mathbb{R}.$

Let us denote by $\Pi^{d}_{n}$ the linear space of polynomials in $d$
variables of total degree less than or equal to $n$, and by $\Pi^{d}$ the
linear space of all polynomials in $d$ variables.

For $d\geqslant 2$ and $n\geqslant 0$, if we denote

$\displaystyle
r^{d}_{n}=\\#\\{\mathbf{x}^{\nu}\colon|\nu|=n\\}=\binom{n+d-1}{n},$

then, unlike the univariate case, $r_{n}^{d}>1$ for $d\geqslant 2$. Moreover,

$\displaystyle\mathbf{r}_{n}^{d}=\dim\Pi^{d}_{n}=\sum_{m=0}^{n}r^{d}_{m}=\binom{n+d}{n}.$
(2.1)

When we deal with more than one variable, the first problem we have to face is
that there is not a natural way to order the monomials. As in [12], we use the
graded lexicographical order, that is, we order the monomial by their total
degree, and then by reverse lexicographical order. For instance, if $d=2$, the
order of the monomials is

$\displaystyle\big{\\{}1;x_{1},x_{2};x_{1}^{2},x_{1}x_{2},x_{2}^{2};x_{1}^{3},x_{1}^{2}x_{2},x_{1}x_{2}^{2},x_{2}^{3};\dots\big{\\}}.$

A useful tool in the theory of orthogonal polynomials in several variables is
the representation of a basis of polynomials as a polynomial system (PS).

######  Definition 2.1.

A _polynomial system $($PS$)$_ is a sequence of column vectors of increasing
size $r_{n}^{d}$, $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$, whose entries are
independent polynomials of total degree $n$

$\displaystyle\mathbb{P}_{n}=\mathbb{P}_{n}(\mathbf{x})=\big{(}P_{\nu}^{n}(\mathbf{x})\big{)}_{|\nu|=n}=\big{(}P_{\nu_{1}}^{n}(\mathbf{x}),P_{\nu_{2}}^{n}(\mathbf{x}),\dots,P_{\nu_{r^{d}_{n}}}^{n}(\mathbf{x})\big{)}^{t},$

where
$\nu_{1},\nu_{2},\dots,\nu_{r^{d}_{n}}\in\big{\\{}\nu\in\mathbb{N}_{0}^{d}\colon|\nu|=n\big{\\}}$
are different multi-indexes arranged in the reverse lexicographical order.

Observe that, for $n\geqslant 0$, the entries of
$\\{\mathbb{P}_{0},\mathbb{P}_{1},\dots,\mathbb{P}_{n}\\}$ form a basis of
$\Pi_{n}^{d}$, and, by extension, we will say that the vector polynomials
$\\{\mathbb{P}_{m}\\}_{m=0}^{n}$ is a basis of $\Pi_{n}^{d}$.

Using this representation, we can define the _canonical polynomial system_ ,
as the sequence of vector polynomials whose entries are the basic monomials
arranged in the reverse lexicographical order,

$\displaystyle\\{\mathbb{X}_{n}\\}_{n\geqslant
0}=\big{\\{}\big{(}\mathbf{x}^{\nu_{1}},\mathbf{x}^{\nu_{2}},\dots,\mathbf{x}^{\nu_{r_{n}^{d}}}\big{)}^{t}\colon|\nu_{i}|=n\big{\\}}_{n\geqslant
0}.$

Thus, each polynomial vector $\mathbb{P}_{n}$, for $n\geqslant 0$, can be
expressed as a unique linear combination of the canonical polynomial system
with matrix coefficients

$\displaystyle\mathbb{P}_{n}=G_{n,n}\mathbb{X}_{n}+G_{n,n-1}\mathbb{X}_{n-1}+\cdots+G_{n,0}\mathbb{X}_{0},$

where $G_{n,k}\in\mathcal{M}_{r_{n}^{d}\times r_{k}^{d}}(\mathbb{R})$ for
$k=0,1,\ldots,n$. Since both vectors $\mathbb{P}_{n}$ and $\mathbb{X}_{n}$
contain a system of independent polynomials, the square matrix $G_{n,n}$ is
non-singular and it is called the leading coefficient of the vector polynomial
$\mathbb{P}_{n}$. Usually, it will be denoted by

$\displaystyle
G_{n}(\mathbb{P}_{n})=G_{n,n}\in\mathcal{M}_{r_{n}^{d}}(\mathbb{R}).$

In the case when the leading coefficient is the identity matrix for all vector
polynomial in a PS

$\displaystyle G_{n}(\mathbb{P}_{n})=I_{r_{n}^{d}},\qquad n\geqslant 0,$

then we will say that $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is a monic PS.
With this notation, we will say that two vector polynomials $\mathbb{P}_{n}$
and $\mathbb{Q}_{n}$, for $n\geqslant 0$, have the same leading coefficient if
$G_{n}(\mathbb{P}_{n})=G_{n}(\mathbb{Q}_{n})$, or, equivalently, if any entry
of the vector $\mathbb{P}_{n}-\mathbb{Q}_{n}$ is a polynomial of degree at
most $n-1$.

The _shift operator_ in several variables, that is, the multiplication of a
polynomial times a variable $x_{i}$ for $i=1,2,\ldots,d$, can be expressed in
terms of the canonical PS as

$\displaystyle x_{i}\mathbb{X}_{n}=L_{n,i}\mathbb{X}_{n+1},\qquad n\geqslant
0,$

where $L_{n,i}$, $i=1,2,\ldots,d$ are $r_{n}^{d}\times r^{d}_{n+1}$ full rank
matrices such that (see [12])

$\displaystyle L_{n,i}L^{t}_{n,i}=I_{r^{d}_{n}}.$

In particular, $L_{n,i}$ is a matrix containing the columns of the identity
matrix of size $r_{n}^{d}$ but including $r^{d}_{n+1}-r_{n}^{d}$ columns of
zeros eventually separating the columns of the identity.

Moreover, for $i,j=1,2,\ldots,d$, the matrix products
$L_{n,i}L_{n+1,j}\in\mathcal{M}_{r_{n}^{d}\times r^{d}_{n+2}}(\mathbb{R})$,
are also full rank matrices and of the same type of $L_{n,i}$. In addition,
since $x_{i}x_{j}\mathbb{X}_{n}=x_{j}x_{i}\mathbb{X}_{n}$, for $n\geqslant 0$,
we get

$\displaystyle L_{n,i}L_{n+1,j}=L_{n,j}L_{n+1,i},\qquad
i,j=1,2,\ldots,d,\qquad n\geqslant 0.$ (2.2)

For instance, if $d=2$,

$\displaystyle L_{n,1}=\left(\begin{array}[]{@{}ccc|c@{}}1&&\bigcirc&0\\\
&\ddots&&\vdots\\\ \bigcirc&&1&0\\\ \end{array}\right),\qquad
L_{n,2}=\left(\begin{array}[]{@{}c|ccc@{}}0&1&&\bigcirc\\\ \vdots&&\ddots&\\\
0&\bigcirc&&1\\\ \end{array}\right),$

and

$\displaystyle
L_{n,1}L_{n+1,1}=\left(\begin{array}[]{@{}ccc|cc@{}}1&&\bigcirc&0&0\\\
&\ddots&&\vdots&\vdots\\\ \bigcirc&&1&0&0\\\ \end{array}\right),\qquad
L_{n,2}L_{n+1,2}=\left(\begin{array}[]{@{}cc|ccc@{}}0&0&1&&\bigcirc\\\
\vdots&\vdots&&\ddots&\\\ 0&0&\bigcirc&&1\\\ \end{array}\right),$
$\displaystyle
L_{n,1}L_{n+1,2}=\left(\begin{array}[]{@{}c|ccc|c@{}}0&1&&\bigcirc&0\\\
\vdots&&\ddots&&\vdots\\\ 0&\bigcirc&&1&0\\\
\end{array}\right)=L_{n,2}L_{n+1,1}.$

Let us now turn to deal with moment functionals and orthogonality in several
variables. Given a sequence of real numbers $\\{\mu_{\nu}\\}_{|\nu|=n\geqslant
0}$, the moment functional $u$ is defined by means of its moments

$\displaystyle u\colon\ \Pi^{d}$ $\displaystyle\longrightarrow\mathbb{R},$
$\displaystyle\mathbf{x}^{\nu}$ $\displaystyle\longmapsto\langle
u,\mathbf{x}^{\nu}\rangle=\mu_{\nu},$

and extended to polynomials by linearity, i.e., if
$p(\mathbf{x})=\sum\limits_{|\nu|\leqslant n}a_{\nu}\mathbf{x}^{\nu}$, then

$\displaystyle\langle u,p(\mathbf{x})\rangle=\sum_{|\nu|\leqslant
n}a_{\nu}\mu_{\nu}.$

Now, we recall some basic operations acting over a moment functional $u$. The
action of $u$ over a polynomial matrix is defined by

$\displaystyle\langle u,M\rangle=\left(\langle
u,m_{i,j}(\mathbf{x})\rangle\right)_{i,j=1}^{h,k}\in{\mathcal{M}}_{h\times
k}(\mathbb{R}),$

where $M=(m_{i,j}(\mathbf{x}))_{i,j=1}^{h,k}\in{\mathcal{M}}_{h\times
k}(\Pi^{d})$, and the left product of a polynomial $p\in\Pi^{d}$ times $u$ by

$\displaystyle\langle pu,q\rangle=\langle
u,pq\rangle,\qquad\forall\,q\in\Pi^{d}.$

Using the canonical polynomial system, for $h,k\geqslant 0$, we define the
$r_{h}^{d}\times r_{k}^{d}$ block of moments

$\displaystyle\textbf{m}_{h,k}=\langle
u,\mathbb{X}_{h}\mathbb{X}^{t}_{k}\rangle=\textbf{m}_{k,h}^{t}.$

We must remark that $\textbf{m}_{h,k}$ contains all of moments of order
$h\times k$. Then, we can see the moment matrix as a block matrix in the form

$\displaystyle\textbf{M}_{n}=\left(\begin{array}[]{@{}c|c|c|c|c@{}}\textbf{m}_{0,0}&\textbf{m}_{0,1}&\textbf{m}_{0,2}&\cdots&\textbf{m}_{0,n}\\\
\hline\cr\textbf{m}_{1,0}&\textbf{m}_{1,1}&\textbf{m}_{1,2}&\cdots&\textbf{m}_{1,n}\\\
\hline\cr\textbf{m}_{2,0}&\textbf{m}_{2,1}&\textbf{m}_{2,2}&\cdots&\textbf{m}_{2,n}\\\
\hline\cr\vdots&\vdots&\vdots&&\vdots\\\
\hline\cr\textbf{m}_{n,0}&\textbf{m}_{n,1}&\textbf{m}_{n,2}&\cdots&\textbf{m}_{n,n}\end{array}\right)$

of dimension $\mathbf{r}_{n}^{d}$ defined in (2.1).

######  Definition 2.2.

A moment functional $u$ is called quasi-definite or regular if and only if

$\displaystyle\Delta_{n}^{d}=\det\textbf{M}_{n}\neq 0,\qquad n\geqslant 0.$

Now, we are ready to introduce the orthogonality. Two polynomials
$p,q\in\Pi^{d}$ are said to be orthogonal with respect to $u$ if $\langle
u,pq\rangle=0$. A given polynomial $p\in\Pi^{d}_{n}$ of exact degree $n$ is an
orthogonal polynomial if it is orthogonal to any polynomial of lower degree.

We can also introduce the orthogonality in terms of a polynomial system. We
will say that a PS $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is orthogonal (OPS)
with respect to $u$ if

$\displaystyle\langle u,\mathbb{X}_{n}\mathbb{P}_{n}^{t}\rangle=S_{n},\quad
n\geqslant 0,\qquad\langle u,\mathbb{X}_{m}\mathbb{P}_{n}^{t}\rangle=0,\quad
m<n,$

where $S_{n}$ is a non-singular matrix of size $r_{n}^{d}\times r_{n}^{d}$. As
a consequence, it is clear that

$\displaystyle\langle u,\mathbb{P}_{n}\mathbb{P}_{n}^{t}\rangle=H_{n},\quad
n\geqslant 0,\qquad\langle u,\mathbb{P}_{m}\mathbb{P}_{n}^{t}\rangle=0,\quad
m\neq n,$

where the matrix $H_{n}$ is symmetric and non-singular. At this point we have
to notice that, with this definition, the orthogonality between the
polynomials of the same total degree may not hold. In the case when the matrix
$H_{n}$ is diagonal for all $n\geqslant 0$ we say that the OPS is mutually
orthogonal.

A moment functional $u$ is _quasi-definite or regular_ if and only if there
exists an OPS. If $u$ is quasi-definite then there exists a unique _monic
OPS_. As usual $u$ is said _positive definite_ if $\langle u,p^{2}\rangle>0$,
for all polynomial $p\neq 0$, and a positive definite moment functional $u$ is
quasi-definite. In this case, there exist an _orthonormal basis_ satisfying

$\displaystyle\langle
u,\mathbb{P}_{n}\mathbb{P}_{n}^{t}\rangle=H_{n}=I_{r_{n}^{d}}.$

For $n\geqslant 0$, the kernel functions in several variables are the
symmetric functions defined by

$\displaystyle
P_{m}(u;\mathbf{x},\mathbf{y})=\mathbb{P}_{m}(\mathbf{x})^{t}H_{m}^{-1}\mathbb{P}_{m}(\mathbf{y}),\qquad
m\geqslant 0,$ $\displaystyle
K_{n}(u;\mathbf{x},\mathbf{y})=\sum_{m=0}^{n}\mathbb{P}_{m}(\mathbf{x})^{t}H_{m}^{-1}\mathbb{P}_{m}(\mathbf{y})=\sum_{m=0}^{n}P_{m}(u;\mathbf{x},\mathbf{y}).$
(2.3)

Both kernels satisfy the usual reproducing property, and they are independent
of the particularly chosen orthogonal polynomial system. For $n=0$, we assume
$P_{-1}(u;\mathbf{x},\mathbf{y})=K_{-1}(u;\mathbf{x},\mathbf{y})=0$.

To finish this section, we need to recall the _three term relations_ satisfied
by orthogonal polynomials in several variables. As in the univariate case, the
orthogonality can be characterized by means of the _three term relations_ [12,
p. 74].

######  Theorem 2.3.

Let $\\{\mathbb{P}_{n}\\}_{n\geqslant
0}=\\{P^{n}_{\nu}(\mathbf{x})\colon|\nu|=n,n\geqslant 0\\}$,
$\mathbb{P}_{0}=1$, be an arbitrary sequence in $\Pi^{d}$. Then the following
statements are equivalent.

  1. $(1)$

There exists a linear functional $u$ which defines a quasi-definite moment
functional on $\Pi^{d}$ and which makes $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$
an orthogonal basis in $\Pi^{d}$.

  2. $(2)$

For $n\geqslant 0$, $1\leqslant i\leqslant d$, there exist matrices $A_{n,i}$,
$B_{n,i}$ and $C_{n,i}$ of respective sizes $r_{n}^{d}\times r_{n+1}^{d}$,
$r_{n}^{d}\times r_{n}^{d}$ and $r_{n}^{d}\times r_{n-1}^{d}$, such that

    1. $(a)$

the polynomials $\mathbb{P}_{n}$ satisfy the three term relations

$\displaystyle
x_{i}\mathbb{P}_{n}=A_{n,i}\mathbb{P}_{n+1}+B_{n,i}\mathbb{P}_{n}+C_{n,i}\mathbb{P}_{n-1},\qquad
1\leqslant i\leqslant d,$ (2.4)

with $\mathbb{P}_{-1}=0$ and $C_{-1,i}=0$,

    2. $(b)$

for $n\geqslant 0$ and $1\leqslant i\leqslant d$, the matrices $A_{n,i}$ and
$C_{n+1,i}$ satisfy the rank conditions

$\displaystyle\operatorname{rank}A_{n,i}=\operatorname{rank}C_{n+1,i}=r_{n}^{d},$
(2.5)

and,

$\displaystyle\operatorname{rank}A_{n}=\operatorname{rank}C_{n+1}^{t}=r_{n+1}^{d},$
(2.6)

where $A_{n}$ is the joint matrix of $A_{n,i}$, defined as

$\displaystyle
A_{n}=\big{(}A_{n,1}^{t},A_{n,2}^{t},\ldots,A_{n,d}^{t}\big{)}^{t}\in\mathcal{M}_{dr_{n}^{d}\times
r_{n+1}^{d}}(\mathbb{R}),$

and $C_{n+1}^{t}$ is the joint matrix of $C_{n+1,i}^{t}$. Moreover,

$\displaystyle A_{n,i}H_{n+1}=\langle
u,x_{i}\mathbb{P}_{n}\mathbb{P}_{n+1}^{t}\rangle,\qquad B_{n,i}H_{n}=\langle
u,x_{i}\mathbb{P}_{n}\mathbb{P}_{n}^{t}\rangle,$ $\displaystyle
C_{n,i}H_{n-1}=\langle
u,x_{i}\mathbb{P}_{n}\mathbb{P}_{n-1}^{t}\rangle=H_{n}A^{t}_{n-1,i}.$ (2.7)

In the case when the orthogonal polynomial system is monic, it follows that
$A_{n,i}=L_{n,i}$, $n\geqslant 0$ for $1\leqslant i\leqslant d$. In this case,
the rank conditions for the matrices $A_{n,i}=L_{n,i}$ and $A_{n}=L_{n}$
obviously hold.

##  3 Uvarov modification

Let $u$ be a quasi-definite moment functional defined on $\Pi^{d}$, let
$\\{\xi_{1},\xi_{2},\ldots,\xi_{N}\\}$ be a fixed set of distinct points in
$\mathbb{R}^{d}$, and let $\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\\}$
be a finite set of non zero real numbers. Then, for $p\in\Pi^{d}$, the
expression

$\displaystyle\langle v,p\rangle=\langle
u,p\rangle+\lambda_{1}p(\xi_{1})+\lambda_{2}p(\xi_{2})+\cdots+\lambda_{N}p(\xi_{N}),$
(3.1)

defines a moment functional on $\Pi^{d}$ which is known as a Uvarov
modification of $u$.

If we define the diagonal matrix
$\Lambda=\operatorname{diag}\\{\lambda_{1},\ldots,\lambda_{N}\\}$, then, for
$p,q\in\Pi^{d}$, we can write

$\displaystyle\langle v,pq\rangle=\langle
u,pq\rangle+(p(\xi_{1}),p(\xi_{2}),\ldots,p(\xi_{N}))\Lambda\begin{pmatrix}q(\xi_{1})\\\
q(\xi_{2})\\\ \vdots\\\ q(\xi_{N})\end{pmatrix}=\langle
u,pq\rangle+\mathsf{p}(\xi)^{t}\Lambda\mathsf{q}(\xi),$ (3.2)

where $\mathsf{p}(\xi)=(p(\xi_{1}),p(\xi_{2}),\ldots,p(\xi_{N}))^{t}$ and
$\mathsf{q}(\xi)=(q(\xi_{1}),q(\xi_{2}),\ldots,q(\xi_{N}))^{t}$, for all
$p,q\in\Pi^{d}$.

If we generalize equation (3.2) using a non-diagonal matrix $\Lambda$, then
$v$ is not a moment functional but it can be seen as a bilinear form.

If the moment functional $v$ is quasi-definite, our first result shows that
orthogonal polynomials with respect to $v$ can be derived in terms of those
with respect to $u$. To simplify the proof of this result we will make use of
a vector-matrix notation which we will introduce next.

Throughout this section, we shall fix $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ as
an orthogonal polynomial system associated with $u$. We denote by
$\mathsf{P}_{n}(\xi)$ the matrix whose columns are $\mathbb{P}_{n}(\xi_{i})$

$\displaystyle\mathsf{P}_{n}(\xi)=\left(\mathbb{P}_{n}(\xi_{1})|\mathbb{P}_{n}(\xi_{2})|\dots|\mathbb{P}_{n}(\xi_{N})\right)\in\mathcal{M}_{r_{n}^{d}\times
N}(\mathbb{R}),$ (3.3)

denote by $\mathcal{K}_{n}$ the matrix whose entries are the kernels
$K_{n}(u;\xi_{i},\xi_{j})$ defined in (2.3),

$\displaystyle\mathcal{K}_{n}=\big{(}K_{n}(u;\xi_{i},\xi_{j})\big{)}_{i,j=1}^{N}\in\mathcal{M}_{N\times
N}(\mathbb{R}),$ (3.4)

and, finally, denote by $\mathsf{K}_{n}(\xi,\mathbf{x})$ the vector of
polynomials

$\displaystyle{\mathsf{K}_{n}(\xi,\mathbf{x})}=\big{(}K_{n}(u;\xi_{1},\mathbf{x}),K_{n}(u;\xi_{2},\mathbf{x}),\ldots,K_{n}(u;\xi_{N},\mathbf{x})\big{)}^{t}.$
(3.5)

From the fact that
$K_{n}(u;\mathbf{x},\mathbf{y})-K_{n-1}(u;\mathbf{x},\mathbf{y})=P_{n}(u;\mathbf{x},\mathbf{y})=\mathbb{P}_{n}(\mathbf{x})^{t}H_{n}^{-1}\mathbb{P}_{n}(\mathbf{y})$,
we have immediately the following relations

$\displaystyle\mathsf{P}_{n}^{t}(\xi)H_{n}^{-1}\mathbb{P}_{n}(\mathbf{x})=\mathsf{K}_{n}(\xi,\mathbf{x})-\mathsf{K}_{n-1}(\xi,\mathbf{x}),$
(3.6)
$\displaystyle\mathsf{P}_{n}^{t}(\xi)H_{n}^{-1}\mathsf{P}_{n}(\xi)=\mathcal{K}_{n}-\mathcal{K}_{n-1},$
(3.7)

which will be used below.

Now we are ready to state and prove the main result in this section. In fact,
we give a necessary and sufficient condition in order to ensure the quasi-
definiteness of the modified moment functional in terms of the non-singularity
of a matrix.

######  Theorem 3.1.

Let $u$ be a quasi-definite moment functional and assume that the moment
functional $v$ defined in (3.1) is quasi-definite. Then the matrices

$\displaystyle I_{N}+\Lambda{\mathcal{K}_{n-1}}$ (3.8)

are invertible for $n=1,2,\ldots$, and any polynomial system
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ orthogonal with respect to $v$ can be
written in the form

$\displaystyle\mathbb{Q}_{0}(\mathbf{x})=\mathbb{P}_{0}(\mathbf{x}),$
$\displaystyle\mathbb{Q}_{n}(\mathbf{x})=\mathbb{P}_{n}(\mathbf{x})-{\mathsf{P}_{n}(\xi)}(I_{N}+\Lambda{\mathcal{K}_{n-1}})^{-1}\Lambda{\mathsf{K}_{n-1}(\xi,\mathbf{x})},\qquad
n\geqslant 1,$ (3.9)

where $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is a polynomial system orthogonal
with respect to $u$. Moreover, the invertible matrices $\hat{H}_{n}=\langle
v,\mathbb{Q}_{n}\mathbb{Q}^{t}_{n}\rangle$ satisfy

$\displaystyle\hat{H}_{n}=H_{n}+\mathsf{P}_{n}(\xi)(I_{N}+\Lambda\mathcal{K}_{n-1})^{-1}\Lambda\mathsf{P}_{n}^{t}(\xi).$
(3.10)

Conversely, if the matrices defined in (3.8) and (3.10) are invertible then
the polynomial system $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ defined by (3.9)
constitutes an orthogonal polynomial system with respect to $v$, and therefore
$v$ is quasi-definite.

###### Proof.

Let us assume that $v$ is a quasi-definite moment functional and let
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ be an OPS with respect to $v$. We can
select an OPS $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ with respect to $u$ such
that $\mathbb{Q}_{n}$ has the same leading coefficient as $\mathbb{P}_{n}$,
for $n\geqslant 0$, in particular we have $\mathbb{Q}_{0}=\mathbb{P}_{0}$.

From this assumption, the components of $\mathbb{Q}_{n}-\mathbb{P}_{n}$ are
polynomials in $\Pi_{n-1}^{d}$ for $n\geqslant 1$, then, we can express them
as linear combinations of orthogonal polynomials
$\mathbb{P}_{0},\mathbb{P}_{1},\ldots,\mathbb{P}_{n-1}$. In vector-matrix
notation, this means that

$\displaystyle\mathbb{Q}_{n}(\mathbf{x})=\mathbb{P}_{n}(\mathbf{x})+\sum_{j=0}^{n-1}M_{j}^{n}\mathbb{P}_{j}(\mathbf{x}),$

where $M_{j}^{n}$ are matrices of size $r_{n}^{d}\times r_{j}^{d}$. These
coefficient matrices can be determined from the orthogonality of
$\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$. Indeed, $\langle
v,\mathbb{Q}_{n}\mathbb{P}^{t}_{j}\rangle=0$ for $0\leqslant j\leqslant n-1$,
which shows, by definition of $v$ and the fact that $\mathbb{P}_{j}$ is
orthogonal,

$\displaystyle M_{j}^{n}=\langle u,\mathbb{Q}_{n}\mathbb{P}^{t}_{j}\rangle
H_{j}^{-1}=-\mathsf{Q}_{n}(\xi)\Lambda\mathsf{P}^{t}_{j}(\xi)H_{j}^{-1},$

where $\mathsf{P}_{j}(\xi)$ is defined as in (3.3) and
$\mathsf{Q}_{n}(\xi)=\big{(}\mathbb{Q}_{n}(\xi_{1})|\mathbb{Q}_{n}(\xi_{2})|\dots|\mathbb{Q}_{n}(\xi_{N})\big{)}$
is the analogous matrix with $\mathbb{Q}_{n}(\xi_{i})$ as its column vectors.
Consequently, we obtain

$\displaystyle\mathbb{Q}_{n}(\mathbf{x})=\mathbb{P}_{n}(\mathbf{x})-\sum_{j=0}^{n-1}\mathsf{Q}_{n}(\xi)\Lambda\mathsf{P}^{t}_{j}(\xi)H_{j}^{-1}\mathbb{P}_{j}(\mathbf{x})=\mathbb{P}_{n}(\mathbf{x})-\mathsf{Q}_{n}(\xi)\Lambda\mathsf{K}_{n-1}(\xi,\mathbf{x}),$
(3.11)

where the second equation follows from relation (3.6), which leads to a
telescopic sum that sums up to $\mathsf{K}_{n-1}(\xi,\mathbf{x})$. Setting
$\mathbf{x}=\xi_{i}$, we obtain

$\displaystyle\mathbb{Q}_{n}(\xi_{i})=\mathbb{P}_{n}(\xi_{i})-\mathsf{Q}_{n}(\xi)\Lambda\mathsf{K}_{n-1}(\xi,\xi_{i}),\qquad
1\leqslant i\leqslant N,$

which by the definition of $\mathcal{K}_{n-1}$ at (3.4) leads to

$\displaystyle\mathsf{Q}_{n}(\xi)=\mathsf{P}_{n}(\xi)-\mathsf{Q}_{n}(\xi)\Lambda\mathcal{K}_{n-1},$

and therefore

$\displaystyle\mathsf{Q}_{n}(\xi)(I_{N}+\Lambda\mathcal{K}_{n-1})=\mathsf{P}_{n}(\xi).$
(3.12)

Next, we are going to show that the matrices $I_{N}+\Lambda\mathcal{K}_{n-1}$
are invertible. Assume that there exists an index $k$ such that
$I_{N}+\Lambda\mathcal{K}_{k-1}$ is non regular, then there exists a vector
$C=(c_{1},c_{2},\ldots,c_{N})^{t}$ satisfying
$(I_{N}+\Lambda\mathcal{K}_{k-1})C=0$. From (3.12) we deduce
$\mathsf{P}_{k}(\xi)C=0$, which implies $(I_{N}+\Lambda\mathcal{K}_{k})C$ $=0$
using (3.7), and therefore we conclude

$\displaystyle\mathsf{P}_{n}(\xi)C=0,\qquad n=k,k+1,\ldots.$ (3.13)

Let us consider the discrete moment functional
$\mathcal{L}=c_{1}\delta_{\xi_{1}}+\dots+c_{N}\delta_{\xi_{N}}$, then (3.13)
implies that $\mathcal{L}$ vanishes on every orthogonal polynomial of total
degree $n\geqslant k$. Using duality we can deduce the existence of a
polynomial $q(\mathbf{x})$ of total degree $k-1$ such that
$\mathcal{L}=q(\mathbf{x})u$. Let $p(\mathbf{x})$ be a non zero polynomial
vanishing at $\xi_{1},\xi_{2},\ldots,\xi_{N}$, then we get
$p(\mathbf{x})q(\mathbf{x})u=p(\mathbf{x})\mathcal{L}=0$, which contradicts
the quasi-definite character of $u$.

Now, solving for $\mathsf{Q}_{n}(\xi)$ in equation (3.12) we get

$\displaystyle\mathsf{Q}_{n}(\xi)=\mathsf{P}_{n}(\xi)(I_{N}+\Lambda\mathcal{K}_{n-1})^{-1},$
(3.14)

and substituting this expression into (3.11) establishes (3.9).

Finally, from (3.9) and (3.14) we obtain

$\displaystyle\hat{H}_{n}=\langle
v,\mathbb{Q}_{n}\mathbb{Q}_{n}^{t}\rangle=\langle
v,\mathbb{Q}_{n}\mathbb{P}_{n}^{t}\rangle=\langle
u,\mathbb{Q}_{n}\mathbb{P}_{n}^{t}\rangle+\mathsf{Q}_{n}(\xi)\Lambda\mathsf{P}_{n}^{t}(\xi)$
$\displaystyle\hphantom{\hat{H}_{n}}{}=H_{n}+\mathsf{P}_{n}(\xi)(I_{N}+\Lambda\mathcal{K}_{n-1})^{-1}\Lambda\mathsf{P}_{n}^{t}(\xi),$

which proves (3.10).

Conversely, if we define polynomials $\mathbb{Q}_{n}$ as in (3.9), then above
proof shows that $\mathbb{Q}_{n}$ is orthogonal with respect to $v$. Since
$\mathbb{Q}_{n}$ and $\mathbb{P}_{n}$ have the same leading coefficient, it is
evident that $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is an OPS in $\Pi^{d}$. ∎

From now on, let us assume that $v$ is a quasi-definite moment functional and
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is an OPS with respect to $v$ as given
in (3.9). Then, the invertible matrix $\hat{H}_{n}=\langle
v,\mathbb{Q}_{n}\mathbb{Q}^{t}_{n}\rangle$ can be expressed in terms of
matrices involving only $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$, as we have
shown in Theorem 3.1. It turns out that this happens also for
$\hat{H}_{n}^{-1}$.

######  Proposition 3.2.

In the conditions of Theorem 3.1, for $n\geqslant 0$, the following identity
holds,

$\displaystyle\hat{H}^{-1}_{n}=H^{-1}_{n}-H^{-1}_{n}\mathsf{P}_{n}(\xi)(I_{N}+\Lambda\mathcal{K}_{n})^{-1}\Lambda\mathsf{P}_{n}^{t}(\xi)H^{-1}_{n}.$
(3.15)

###### Proof.

Formula (3.15) is a direct consequence of the Sherman–Morrison–Woodbury
identity for the inverse of the perturbation of a non singular matrix (see
[14, p. 51]). ∎

Our next result gives explicit formulas for the reproducing kernels associated
with $v$, which we denote by

$\displaystyle
P_{m}(v;\mathbf{x},\mathbf{y})=\mathbb{Q}^{t}_{m}(\mathbf{x})\hat{H}^{-1}_{m}\mathbb{Q}_{m}(\mathbf{y}),$
$\displaystyle
K_{n}(v;\mathbf{x},\mathbf{y})=\sum_{m=0}^{n}P_{m}(v;\mathbf{x},\mathbf{y})=\sum_{m=0}^{n}\mathbb{Q}^{t}_{m}(\mathbf{x})\hat{H}^{-1}_{m}\mathbb{Q}_{m}(\mathbf{y}).$

######  Lemma 3.3.

Let $u$ be a quasi-definite moment functional, $\mathcal{K}_{m}$ defined by
(3.4), and let
$\Lambda=\operatorname{diag}\\{\lambda_{1},\ldots,\lambda_{N}\\}$, with
$\lambda_{i}\neq 0$, $i=1,2,\ldots,N$. Then, for $m\geqslant 0$,
$(I_{N}+\Lambda\mathcal{K}_{m})^{-1}\Lambda$ is a symmetric matrix.

###### Proof.

$(I_{N}+\Lambda\mathcal{K}_{m})^{-1}\Lambda$ is a symmetric matrix as it is
the inverse of the symmetric matrix
$\Lambda^{-1}(I_{N}+\Lambda\mathcal{K}_{m})=\Lambda^{-1}+\mathcal{K}_{m}$. ∎

Next theorem establishes a relation between the kernels of both families.
Similar tools as those used in the proof of Theorem 2.5 in [11] can be applied
to obtain this result.

######  Theorem 3.4.

Suppose that we are in the conditions of Theorem 3.1. Then, for $m\geqslant
0$, we get

$\displaystyle
P_{m}(v;\mathbf{x},\mathbf{y})=P_{m}(u;\mathbf{x},\mathbf{y})-\mathsf{K}_{m}^{t}(\xi,\mathbf{x})(I_{N}+\Lambda\mathcal{K}_{m})^{-1}\Lambda\mathsf{K}_{m}(\xi,\mathbf{y})$
$\displaystyle\hphantom{P_{m}(v;\mathbf{x},\mathbf{y})=}{}+\mathsf{K}_{m-1}^{t}(\xi,\mathbf{x})(I_{N}+\Lambda\mathcal{K}_{m-1})^{-1}\Lambda\mathsf{K}_{m-1}(\xi,\mathbf{y}),$

where we assume $\mathsf{K}_{-1}(\mathbf{x},\mathbf{y})\equiv 0$. Furthermore,
for $n\geqslant 0$,

$\displaystyle
K_{n}(v;\mathbf{x},\mathbf{y})=K_{n}(u;\mathbf{x},\mathbf{y})-\mathsf{K}_{n}^{t}(\xi,\mathbf{x})(I_{N}+\Lambda\mathcal{K}_{n})^{-1}\Lambda\mathsf{K}_{n}(\xi,\mathbf{y}).$
(3.16)

##  4 Christoffel modification

Christoffel modification of a quasi-definite moment functional $u$ will be
studied in this section. We define the Christoffel modification of $u$ as the
moment functional

$\displaystyle v=\lambda(\mathbf{x})u,$

acting as follows

$\displaystyle\langle
v,p(\mathbf{x})\rangle=\langle\lambda(\mathbf{x})u,p(\mathbf{x})\rangle=\langle
u,\lambda(\mathbf{x})p(\mathbf{x})\rangle,\qquad\forall\,p(\mathbf{x})\in\Pi^{d}.$

We will work with the particular case when the polynomial
$\lambda(\mathbf{x})$ has total degree $2$. We must remark that in several
variables there exist polynomials of second degree that they can not be
factorized as a product of two polynomials of degree $1$, and then this case
is not a trivial extension of the case considered in [2]. Using a block matrix
formalism for the three term relations, this case have been also considered in
[4] and [5] for arbitrary degree polynomials.

Let $u$ be a quasi-definite moment functional, and let $\lambda(\mathbf{x})$
be a polynomial in $d$ variables of total degree $2$. In terms of the
canonical basis, this polynomial can be written as

$\displaystyle\lambda(\mathbf{x})=\textbf{a}_{2}\mathbb{X}_{2}+\textbf{a}_{1}\mathbb{X}_{1}+\textbf{a}_{0}\mathbb{X}_{0}=\sum_{i=1}^{d}\sum_{j=i}^{d}a^{(2)}_{ij}x_{i}x_{j}+\sum_{i=1}^{d}a^{(1)}_{i}x_{i}+a^{(0)},$
(4.1)

where $\textbf{a}_{k}\in\mathcal{M}_{1\times r_{k}^{d}}(\mathbb{R})$, for
$k=0,1,2$, whose explicit expressions are

$\displaystyle\textbf{a}_{2}=\big{(}a^{(2)}_{11},a^{(2)}_{12},\ldots,a^{(2)}_{1d},a^{(2)}_{22},\ldots,a^{(2)}_{2d},\ldots,a^{(2)}_{dd}\big{)},$
$\displaystyle\textbf{a}_{1}=\big{(}a^{(1)}_{1},a^{(1)}_{2},\ldots,a^{(1)}_{d}\big{)},$
$\displaystyle\textbf{a}_{0}=(a^{(0)}),$

with the conditions
$|\textbf{a}_{2}|=\sum\limits_{i=1}^{d}\sum\limits_{j=i}^{d}|a^{(2)}_{ij}|\neq
0$, and

$\displaystyle\langle u,\lambda(\mathbf{x})\rangle\neq 0.$ (4.2)

Observe that the first moment of $v$ is given by

$\displaystyle\hat{\mu}_{0}=\langle v,1\rangle=\langle
u,\lambda(\mathbf{x})\rangle,$

and using (4.2), we get $\hat{\mu}_{0}\neq 0$.

First we are going to describe the relations between the moment matrices of
both functionals $u$ and $v$. Taking into account that, for $h\geqslant 0$, we
get

$\displaystyle x_{j}\mathbb{X}_{h}=L_{h,j}\mathbb{X}_{h+1},\qquad
x_{i}x_{j}\mathbb{X}_{h}=L_{h,j}L_{h+1,i}\mathbb{X}_{h+2},$

the $r^{d}_{h}\times r^{d}_{k}$ block of moments for the functional $v$,
$\hat{\textbf{m}}_{h,k}$, can be expressed in terms of the block of moments
for the functional $u$, $\textbf{m}_{h,k}$, in the following way

$\displaystyle\hat{\textbf{m}}_{h,k}=\langle
v,\mathbb{X}_{h}\mathbb{X}_{k}^{t}\rangle=\langle\lambda(\mathbf{x})u,\mathbb{X}_{h}\mathbb{X}_{k}^{t}\rangle=\langle
u,\lambda(\mathbf{x})\mathbb{X}_{h}\mathbb{X}_{k}^{t}\rangle$
$\displaystyle\hphantom{\hat{\textbf{m}}_{h,k}}{}=a^{(0)}\textbf{m}_{h,k}+A_{h,1}\textbf{m}_{h+1,l}+A_{h,2}\textbf{m}_{h+2,k},$

where

$\displaystyle A_{h,1}=\sum_{i=1}^{d}a^{(1)}_{i}L_{h,i},\qquad
A_{h,2}=\sum_{i=1}^{d}\sum_{j=i}^{d}a^{(2)}_{ij}L_{h,j}L_{h+1,i}$ (4.3)

are matrices of orders $r_{h}^{d}\times r_{h+1}^{d}$ and $r_{h}^{d}\times
r_{h+2}^{d}$, respectively, and $A_{h,2}$ has full rank. If we define the
block matrices

$\displaystyle\textbf{A}_{n,1}=\left(\begin{array}[]{@{}cccc|c@{}}0&A_{0,1}&&&0\\\
&0&A_{1,1}&&0\\\ &&\ddots&\ddots&\vdots\\\
&&&0&A_{n,1}\end{array}\right),\qquad\textbf{L}_{n}=\left(\begin{array}[]{@{}cccc|c@{}}I_{r^{d}_{0}}&&&&0\\\
&I_{r^{d}_{1}}&&&0\\\ &&\ddots&&\vdots\\\
&&&I_{r^{d}_{n}}&0\end{array}\right),$

both of dimension $\textbf{r}^{d}_{n}\times\textbf{r}^{d}_{n+1}$, and

$\displaystyle\textbf{A}_{n,2}=\left(\begin{array}[]{@{}cccccc|c@{}}0&0&A_{0,2}&&&&0\\\
&0&0&A_{1,2}&&&0\\\ &&0&0&A_{2,2}&&0\\\ &&&\ddots&\ddots&\ddots&\vdots\\\
&&&&0&0&A_{n,2}\end{array}\right)$

of dimension $\textbf{r}^{d}_{n}\times\textbf{r}^{d}_{n+2}$, then we can write
the moment matrix for the functional $v$ as the following perturbation of the
original moment matrix

$\displaystyle\hat{\textbf{M}}_{n}=\textbf{a}_{0}\textbf{M}_{n}+\textbf{A}_{n,1}\textbf{M}_{n+1}\textbf{L}_{n}^{t}+\textbf{A}_{n,2}\textbf{M}_{n+2}\textbf{L}_{n+1}^{t}\textbf{L}_{n}^{t}.$

If $u$ and $v$ are quasi-definite, we want to relate both orthogonal
polynomial systems $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ and
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ associated with $u$ and $v$,
respectively. As usual in this paper, for $n\geqslant 0$, we denote
$H_{n}=\langle u,\mathbb{P}_{n}\mathbb{P}_{n}^{t}\rangle$, and
$\hat{H}_{n}=\langle u,\mathbb{Q}_{n}\mathbb{Q}_{n}^{t}\rangle$, both
symmetric and invertible matrices.

######  Theorem 4.1.

Let $u$ and $v$ be two quasi-definite moment functionals, and let
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ and $\\{\mathbb{Q}_{n}\\}_{n\geqslant
0}$ be monic OPS associated with $u$ and $v$, respectively. The following
statements are equivalent:

  1. $(1)$

There exists a polynomial $\lambda(\mathbf{x})$ of exact degree two such that

$\displaystyle v=\lambda(\mathbf{x})u.$

  2. $(2)$

For $n\geqslant 1$, there exist matrices
$M_{n}\in{\mathcal{M}}_{r^{d}_{n}\times r^{d}_{n-1}}(\mathbb{R})$,
$N_{n}\in{\mathcal{M}}_{r^{d}_{n}\times r^{d}_{n-2}}(\mathbb{R})$, with
$N_{2}\not\equiv 0$, such that

$\displaystyle\mathbb{P}_{n}=\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2}.$
(4.4)

###### Proof.

First, we prove (1) $\Rightarrow$ (2). Let us assmue that
$v=\lambda(\mathbf{x})u$ where
$\lambda(\mathbf{x})=\textbf{a}_{2}\mathbb{X}_{2}+\textbf{a}_{1}\mathbb{X}_{1}+\textbf{a}_{0}\mathbb{X}_{0}$,
and $|\textbf{a}_{2}|\neq 0$.

Since $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is a basis of the space of
polynomials, and $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$ are monic, then

$\displaystyle\mathbb{P}_{n}=\mathbb{Q}_{n}+\sum_{j=0}^{n-1}M^{n}_{j}\mathbb{Q}_{j},$

where $M^{n}_{j}\in\mathcal{M}_{r^{d}_{n}\times r^{d}_{j}}(\mathbb{R})$, and

$\displaystyle M_{j}^{n}=\langle
v,\mathbb{P}_{n}\mathbb{Q}_{j}^{t}\rangle\hat{H}_{j}^{-1},\qquad 0\leqslant
j\leqslant n-1.$

Given that the degree of $\lambda(\mathbf{x})$ is $2$, from the orthogonality
of $\mathbb{P}_{n}$ we get

$\displaystyle\langle
v,\mathbb{P}_{n}\mathbb{Q}_{j}^{t}\rangle=\langle\lambda(\mathbf{x})u,\mathbb{P}_{n}\mathbb{Q}_{j}^{t}\rangle=\langle
u,\mathbb{P}_{n}\lambda(\mathbf{x})\mathbb{Q}_{j}^{t}\rangle=0,\qquad\text{for
}j<n-2,$

and then $M_{j}^{n}=0$ for $j<n-2$. Therefore, (4.4) holds with
$M^{n}_{n-1}=M_{n}$ and $M^{n}_{n-2}=N_{n}$.

To compute $N_{2}$, we use
$\mathbb{P}_{2}=\mathbb{Q}_{2}+M_{2}\mathbb{Q}_{1}+N_{2}\mathbb{Q}_{0}$, and
thus

$\displaystyle\langle v,\mathbb{P}_{2}\mathbb{Q}_{0}^{t}\rangle=\langle
v,(\mathbb{Q}_{2}+M_{2}\mathbb{Q}_{1}+N_{2}\mathbb{Q}_{0})\mathbb{Q}_{0}^{t}\rangle=N_{2}\hat{H}_{0}.$

On the other hand,

$\displaystyle\langle v,\mathbb{P}_{2}\mathbb{Q}_{0}^{t}\rangle=\langle
u,\lambda(\mathbf{x})\mathbb{P}_{2}\rangle=\langle
u,\mathbb{P}_{2}(\mathbb{X}_{2}^{t}\textbf{a}^{t}_{2}+\mathbb{X}_{1}^{t}\textbf{a}^{t}_{1}+\mathbb{X}_{0}^{t}\textbf{a}^{t}_{0})\rangle=H_{2}\textbf{a}^{t}_{2},$

and therefore

$\displaystyle
N_{2}=H_{2}\textbf{a}^{t}_{2}\hat{H}_{0}^{-1}\in\mathcal{M}_{r_{2}^{d}\times
1}(\mathbb{R}),$

has full rank since $|\textbf{a}_{2}|\neq 0$. So, $N_{2}\not\equiv 0$ since
$N_{2}$ is a column matrix.

Conversely, we see (2) $\Rightarrow$ (1). Using the dual basis, and the same
reasoning as in the proof of Lemma 1 in [2], we obtain

$\displaystyle v=\sum_{n=0}^{+\infty}\mathbb{P}_{n}^{t}H_{n}^{-1}E_{n}u,$

where $E_{n}^{t}=\langle v,\mathbb{P}_{n}^{t}\rangle$, $n\geqslant 0$. By
(4.4), we get

$\displaystyle E_{0}^{t}=\langle v,\mathbb{Q}_{0}^{t}\rangle=\hat{H}_{0}\neq
0,$ $\displaystyle E_{1}^{t}=\langle
v,\mathbb{Q}_{1}^{t}+\mathbb{Q}_{0}^{t}M_{1}^{t}\rangle=\hat{H}_{0}M_{1}^{t},$
$\displaystyle E_{2}^{t}=\langle
v,\mathbb{Q}_{2}^{t}+\mathbb{Q}_{1}^{t}M_{2}^{t}+\mathbb{Q}_{0}^{t}N_{2}^{t}\rangle=\hat{H}_{0}N_{2}^{t},$
$\displaystyle E_{n}^{t}=\langle
v,\mathbb{Q}_{n}^{t}+\mathbb{Q}_{n-1}^{t}M_{n}^{t}+\mathbb{Q}_{n-2}^{t}N_{n}^{t}\rangle=0,\qquad
n\geqslant 3.$

Then,

$\displaystyle
v=\big{(}\mathbb{P}_{2}^{t}H_{2}^{-1}N_{2}\hat{H}_{0}+\mathbb{P}_{1}^{t}H_{1}^{-1}M_{1}\hat{H}_{0}+\mathbb{P}_{0}^{t}H_{0}^{-1}\hat{H}_{0}\big{)}u,$

or equivalently, there exists a polynomial

$\displaystyle\lambda(\mathbf{x})=\hat{H}_{0}\big{(}N_{2}^{t}H_{2}^{-1}\mathbb{P}_{2}+M_{1}^{t}H_{1}^{-1}\mathbb{P}_{1}+H_{0}^{-1}\mathbb{P}_{0}\big{)},$

such that $v=\lambda(\mathbf{x})u$. Since $N_{2}\not\equiv 0$, then
$\lambda(\mathbf{x})$ has exact degree 2.

Moreover, we will prove that $N_{n}$ has full rank for $n\geqslant 2$. In
fact, using (4.4), we get

$\displaystyle\langle v,\mathbb{P}_{n}\mathbb{Q}_{n-2}^{t}\rangle=\langle
v,[\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2}]\mathbb{Q}_{n-2}^{t}\rangle=N_{n}\hat{H}_{n-2}.$

On the other hand,

$\displaystyle\langle v,\mathbb{P}_{n}\mathbb{Q}_{n-2}^{t}\rangle=\langle
u,\mathbb{P}_{n}\lambda(\mathbf{x})\mathbb{Q}_{n-2}^{t}\rangle=\sum_{i=1}^{d}\sum_{j=i}^{d}a^{(2)}_{ij}\langle
u,\mathbb{P}_{n}x_{i}x_{j}\mathbb{Q}_{n-2}^{t}\rangle$
$\displaystyle\hphantom{\langle
v,\mathbb{P}_{n}\mathbb{Q}_{n-2}^{t}\rangle}{}=\sum_{i=1}^{d}\sum_{j=i}^{d}a^{(2)}_{ij}\langle
u,\mathbb{P}_{n}x_{i}x_{j}\mathbb{X}_{n-2}^{t}\rangle=\sum_{i=1}^{d}\sum_{j=i}^{d}a^{(2)}_{ij}\langle
u,\mathbb{P}_{n}\mathbb{X}_{n}^{t}\rangle L_{n-1,i}^{t}L_{n-2,j}^{t}$
$\displaystyle\hphantom{\langle
v,\mathbb{P}_{n}\mathbb{Q}_{n-2}^{t}\rangle}{}=\sum_{i=1}^{d}\sum_{j=i}^{d}a^{(2)}_{ij}H_{n}L_{n-1,i}^{t}L_{n-2,j}^{t}=H_{n}A_{n-2,2}^{t},$

where $A_{n-2,2}$ was defined in (4.3). Then,

$\displaystyle N_{n}\hat{H}_{n-2}=H_{n}A_{n-2,2}^{t},\qquad n\geqslant 2.$
(4.5)

Therefore $N_{n}$ is full rank for $n\geqslant 2$ since $H_{n}$ and
$\hat{H}_{n-2}$ are invertible matrices, and the rank of a matrix is invariant
by multiplication times non-singular matrices [15, p. 13]. ∎

######  Remark 4.2.

When both moment functionals are quasi-definite, that is, when both OPS
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ and $\\{\mathbb{Q}_{n}\\}_{n\geqslant
0}$ exist, the orthogonality condition of the second family with respect to
$v=\lambda(\mathbf{x})u$ trivially implies that the polynomial entries in
$\lambda(\mathbf{x})\mathbb{Q}_{n}$ are quasi-orthogonal with respect to the
first moment functional $u$. In fact, there exist matrices of adequate size
such that

$\displaystyle\lambda(\mathbf{x})\mathbb{Q}_{n}=\sum_{k=0}^{n+\deg\lambda(\mathbf{x})}A_{k}^{n}\mathbb{P}_{k},$

where

$\displaystyle A_{k}^{n}H_{k}=\langle
u,\lambda(\mathbf{x})\mathbb{Q}_{n}\mathbb{P}_{k}\rangle=\langle\lambda(\mathbf{x})u,\mathbb{Q}_{n}\mathbb{P}_{k}\rangle=\langle
v,\mathbb{Q}_{n}\mathbb{P}_{k}\rangle.$

Then, $A_{k}^{n}=0$, for $0\leqslant k\leqslant n-1$, and therefore

$\displaystyle\lambda(\mathbf{x})\mathbb{Q}_{n}=\sum_{k=n}^{n+\deg\lambda(\mathbf{x})}A_{k}^{n}\mathbb{P}_{k}.$

The matrix version of this relation is the first identity in Proposition 2.7
of [4].

Now, we assume that $u$ is a quasi-definite moment functional, and
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ is the monic OPS associated with $u$.
Then $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ satisfy the three term relations
(2.4) with the rank conditions (2.5), (2.6). Defining recursively the monic
polynomial system $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ by means of (4.4), we
want to deduce its relation with $u$ as well as conditions for its quasi-
definiteness.

######  Theorem 4.3.

Let $\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$ be a monic OPS associated with the
quasi-definite moment functional $u$, and let $\\{M_{n}\\}_{n\geqslant 1}$ and
$\\{N_{n}\\}_{n\geqslant 2}$ be two sequences of matrices of orders
$r_{n}^{d}\times r_{n-1}^{d}$ and $r_{n}^{d}\times r_{n-2}^{d}$ respectively,
such that $N_{2}\not\equiv 0$. Define recursively the monic polynomial system

$\displaystyle\mathbb{Q}_{0}=\mathbb{P}_{0},$
$\displaystyle\mathbb{Q}_{1}=\mathbb{P}_{1}-M_{1}\mathbb{P}_{0},$
$\displaystyle\mathbb{Q}_{n}=\mathbb{P}_{n}-M_{n}\mathbb{Q}_{n-1}-N_{n}\mathbb{Q}_{n-2},\qquad
n\geqslant 2.$

Then $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is a monic OPS associated with a
quasi-definite moment functional $v$, satisfying the three term relation

$\displaystyle
x_{i}\mathbb{Q}_{n}(\mathbf{x})=L_{n,i}\mathbb{Q}_{n+1}(\mathbf{x})+\hat{B}_{n,i}\mathbb{Q}_{n}(\mathbf{x})+\hat{C}_{n,i}\mathbb{Q}_{n-1}(\mathbf{x}),\qquad
1\leqslant i\leqslant d,$ (4.6)

with initial conditions $\mathbb{Q}_{-1}(\mathbf{x})=0$, $\hat{C}_{-1,i}=0$,
if and only if

$\displaystyle\hat{B}_{n,i}=B_{n,i}-M_{n}L_{n-1,i}+L_{n,i}M_{n+1},$ (4.7)
$\displaystyle\hat{C}_{n,i}=C_{n,i}-M_{n}\hat{B}_{n-1,i}+B_{n,i}M_{n}-N_{n}L_{n-2,i}+L_{n,i}N_{n+1},$
(4.8) $\displaystyle
M_{n}\hat{C}_{n-1,i}+N_{n}\hat{B}_{n-2,i}=C_{n,i}M_{n-1}+B_{n,i}N_{n},$ (4.9)
$\displaystyle C_{n,i}N_{n-1}=N_{n}\hat{C}_{n-2,i}.$ (4.10)

In such a case, there exists a polynomial of exact degree two given by

$\displaystyle\lambda(\mathbf{x})=N_{2}^{t}H_{2}^{-1}\mathbb{P}_{2}+M_{1}^{t}H_{1}^{-1}\mathbb{P}_{1}+H_{0}^{-1}\mathbb{P}_{0},$

satisfying

$\displaystyle v=\lambda(\mathbf{x})u.$

###### Proof.

Replacing (4.4) in (2.4) we get

$\displaystyle
x_{i}(\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2})=L_{n,i}\mathbb{Q}_{n+1}+(L_{n,i}M_{n+1}+B_{n,i})\mathbb{Q}_{n}$
$\displaystyle\hphantom{x_{i}(\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2})=}{}+(L_{n,i}N_{n+1}+B_{n,i}M_{n}+C_{n,i})\mathbb{Q}_{n-1}$
$\displaystyle\hphantom{x_{i}(\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2})=}{}+(B_{n,i}N_{n}+C_{n,i}M_{n-1})\mathbb{Q}_{n-2}+C_{n,i}N_{n-1}\mathbb{Q}_{n-3}.$

On the other hand, if $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ satisfy (4.6) then

$\displaystyle
x_{i}(\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2})=L_{n,i}\mathbb{Q}_{n+1}+(M_{n}L_{n-1,i}+\hat{B}_{n,i})\mathbb{Q}_{n}$
$\displaystyle\hphantom{x_{i}(\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2})=}{}+(\hat{C}_{n,i}+M_{n}\hat{B}_{n-1,i}+N_{n}L_{n-2,i})\mathbb{Q}_{n-1}$
$\displaystyle\hphantom{x_{i}(\mathbb{Q}_{n}+M_{n}\mathbb{Q}_{n-1}+N_{n}\mathbb{Q}_{n-2})=}{}+(M_{n}\hat{C}_{n-1,i}+N_{n}\hat{B}_{n-2,i})\mathbb{Q}_{n-2}+N_{n}\hat{C}_{n-2,i}\mathbb{Q}_{n-3}.$

Subtracting both expressions we get (4.7)–(4.10).

Conversely, we will prove the three term relation for
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ using induction. In fact, for $n=1$,
multiplying relation (4.4) times $L_{0,i}$, we get

$\displaystyle
x_{i}\mathbb{Q}_{0}=L_{0,i}\mathbb{Q}_{1}+(B_{0,i}+L_{0,i}M_{1})\mathbb{Q}_{0}.$

Let us suppose that (4.6) is satisfied for $n-1$. Multiplying (4.4) for $n+1$
times $L_{n,i}$ and applying the three term relation for
$\\{\mathbb{P}_{n}\\}_{n\geqslant 0}$, we get

$\displaystyle
x_{i}\mathbb{P}_{n}-B_{n,i}\mathbb{P}_{n}-C_{n,i}\mathbb{P}_{n-1}=L_{n,i}\mathbb{Q}_{n+1}+L_{n,i}M_{n+1}\mathbb{Q}_{n}+L_{n,i}N_{n+1}\mathbb{Q}_{n-1}.$

Replacing again (4.4) in the left hand side, using induction hypotheses and
relations (4.7)–(4.10) we get the announced three term relations for
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$.

Now, we define the moment functional $v$ as

$\displaystyle\langle v,1\rangle=1,\qquad\langle
v,\mathbb{Q}_{n}\rangle=0,\qquad n\geqslant 1.$

Since $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is a basis of $\Pi^{d}$, then $v$
is well defined. Following [12, p. 74], since $L_{n,i}$ and $L_{n}$ have full
rank, then

$\displaystyle\langle v,\mathbb{Q}_{n}\mathbb{Q}^{t}_{m}\rangle=0,\qquad n\neq
m.$

Finally, we need to prove that $\hat{H}_{n}=\langle
v,\mathbb{Q}_{n}\mathbb{Q}^{t}_{n}\rangle$, is an invertible matrix, for
$n\geqslant 0$.

From the definition of $v$, $\hat{H}_{0}=\langle
v,\mathbb{Q}_{0}\mathbb{Q}^{t}_{0}\rangle=\langle v,1\rangle=1$. On the other
hand, since expression (4.5) still holds in this case,

$\displaystyle N_{n}\hat{H}_{n-2}=H_{n}A_{n-2,2}^{t},\qquad n\geqslant 2,$

using the properties of the rank of a product of matrices, we get

$\displaystyle\operatorname{rank}\big{(}N_{n}\hat{H}_{n-2}\big{)}=\operatorname{rank}\big{(}H_{n}A_{n-2,2}^{t}\big{)}=\operatorname{rank}A_{n-2,2}^{t}=r_{n-2}^{d}.$

Therefore

$\displaystyle
r_{n-2}^{d}=\operatorname{rank}\big{(}N_{n}\hat{H}_{n-2}\big{)}\leqslant\min\big{\\{}\operatorname{rank}N_{n},\operatorname{rank}\hat{H}_{n-2}\big{\\}}\leqslant\operatorname{rank}\hat{H}_{n-2}\leqslant
r_{n-2}^{d},$

and then $\operatorname{rank}\hat{H}_{n-2}=r^{d}_{n-2}$. In this way, the
moment functional $v$ is quasi-definite and $\\{\mathbb{Q}_{n}\\}_{n\geqslant
0}$ is a monic OPS associated with $v$. Using Theorem 4.1, both moment
functionals are related by means of a Christoffel modification

$\displaystyle v=\lambda(\mathbf{x})u,$

where
$\lambda(\mathbf{x})=N_{2}^{t}H_{2}^{-1}\mathbb{P}_{2}+M_{1}^{t}H_{1}^{-1}\mathbb{P}_{1}+H_{0}^{-1}\mathbb{P}_{0}$.
∎

######  Remark 4.4.

Observe that relation (4.10) always holds when
$\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is an OPS. In fact, using (2.7), we get

$\displaystyle
C_{n,i}H_{n-1}=H_{n}L^{t}_{n-1,i}\qquad\text{and}\qquad\hat{C}_{n,i}\hat{H}_{n-1}=\hat{H}_{n}L^{t}_{n-1,i},$

and jointly with (4.5), it follows

$\displaystyle
C_{n,i}N_{n-1}-N_{n}\hat{C}_{n-2,i}=H_{n}\big{[}L^{t}_{n-1,i}A^{t}_{n-3,2}-A^{t}_{n-2,2}L^{t}_{n-3,i}\big{]}\hat{H}^{-1}_{n-3}.$

On the other hand, from (4.3),

$\displaystyle
L_{n-3,i}A_{n-2,2}-A_{n-3,2}L_{n-1,i}=L_{n-3,i}\left(\sum_{k=1}^{d}\sum_{j=k}^{d}a^{(2)}_{kj}L_{n-2,j}L_{n-1,k}\right)$
$\displaystyle\hphantom{L_{n-3,i}A_{n-2,2}-A_{n-3,2}L_{n-1,i}=}{}-\left(\sum_{k=1}^{d}\sum_{j=k}^{d}a^{(2)}_{kj}L_{n-3,j}L_{n-2,k}\right)L_{n-1,i}=0,$

from property (2.2). However, if $\\{\mathbb{Q}_{n}\\}_{n\geqslant 0}$ is not
orthogonal, then we can not assume a priori that $\hat{H}_{n-3}$ is non
singular, and so (4.10) does not necessarily hold.

######  Remark 4.5.

In the case when both functionals $u$ and $v$ are quasi-definite, Theorem 4.3
can be rewritten by using a matrix formalism, as it is done in [4, 5]. For
$1\leqslant i\leqslant d$, we denote by

$\displaystyle\mathcal{J}_{i}=\begin{pmatrix}B_{0,i}&L_{0,i}&&&\bigcirc\\\
C_{1,i}&B_{1,i}&L_{1,i}&&\\\ &C_{2,i}&B_{2,i}&\ddots&\\\ &&\ddots&\ddots&\\\
\bigcirc&&&&\end{pmatrix},\qquad\hat{\mathcal{J}}_{i}=\begin{pmatrix}\hat{B}_{0,i}&L_{0,i}&&&\bigcirc\\\
\hat{C}_{1,i}&\hat{B}_{1,i}&L_{1,i}&&\\\
&\hat{C}_{2,i}&\hat{B}_{2,i}&\ddots&\\\ &&\ddots&\ddots&\\\
\bigcirc&&&&\end{pmatrix},$

the respective _block Jacobi matrices_ associated with the three term
relations [12, p. 82]. Also we define the _lower triangular block matrix_ with
identity matrices as diagonal blocks

$\displaystyle\mathcal{M}=\begin{pmatrix}I&&&&&\bigcirc\\\ M_{1}&I&&&&\\\
N_{2}&M_{2}&I&&&\\\ &N_{3}&M_{3}&I&&\\\ &&\ddots&\ddots&\ddots&\\\
\bigcirc&&&&&\end{pmatrix},$

where $M_{n}$, $N_{n}$ are defined in Theorem 4.1. Then, formulas (4.7)–(4.10)
can be expressed as the matrix product

$\displaystyle\mathcal{J}_{i}\mathcal{M}=\mathcal{M}\hat{\mathcal{J}}_{i},\qquad
i=1,2,\ldots,d.$

The explicit expressions of the matrices $M_{n}$ and $N_{n}$ given in Theorem
4.1 lead to matrix relations of Proposition 2.4 in [4].

###  4.1 Centrally symmetric functionals

Following [12, p. 76], a moment functional $u$ is called _centrally symmetric_
if it satisfies

$\displaystyle\langle
u,x^{\nu}\rangle=0,\qquad\nu\in\mathbb{N}^{d},\qquad|\nu|\quad\textrm{is an
odd integer}.$

This definition constitutes the multivariate extension of the symmetry for a
moment functional.

Quasi-definite centrally symmetric moment functionals can be characterized in
terms of the matrix coefficients of the three term relations (2.4). In fact,
$u$ is centrally symmetric if and only if $B_{n,i}=0$ for all $n\geqslant 0$
and $1\leqslant i\leqslant d$.

As a consequence, an orthogonal polynomial of degree $n$ with respect to $u$
is a sum of monomials of even degree if $n$ is even and a sum of monomials of
odd degree if $n$ is odd.

Let us suppose that $u$ is a quasi-definite centrally symmetric moment
functional, and we define its Christoffel modification by

$\displaystyle v=\lambda(\mathbf{x})u,$

where $\lambda(\mathbf{x})$ is a polynomial of second degree as (4.1). Then

######  Proposition 4.6.

$v$ is centrally symmetric if and only if $\textbf{a}_{1}=0$, that is,

$\displaystyle\lambda(\mathbf{x})=\textbf{a}_{2}\mathbb{X}_{2}+\textbf{a}_{0}\mathbb{X}_{0}=\sum_{i=1}^{d}\sum_{j=i}^{d}a^{(2)}_{ij}x_{i}x_{j}+a^{(0)}.$

If $v$ is quasi-definite and centrally symmetric, then relation (4.4) is given
by

$\displaystyle\mathbb{P}_{0}=\mathbb{Q}_{0},\qquad\mathbb{P}_{1}=\mathbb{Q}_{1},\qquad\mathbb{P}_{n}=\mathbb{Q}_{n}+N_{n}\mathbb{Q}_{n-2},\qquad
n\geq 2,$ (4.11)

where $N_{n}=H_{n}A_{n-2,2}^{t}\hat{H}_{n-2}^{-1}$, $n\geqslant 2$.

##  5 Examples

###  5.1 Two modifications on the ball

Let us denote by

$\displaystyle\mathbb{B}^{d}=\big{\\{}\mathbf{x}\in\mathbb{R}^{d}\colon\|\mathbf{x}\|\leqslant
1\big{\\}}\qquad\textrm{and}\qquad\mathbb{S}^{d-1}=\big{\\{}\xi\in\mathbb{R}^{d}\colon\|\xi\|=1\big{\\}},$

the unit ball and the unit sphere on $\mathbb{R}^{d}$, respectively, where

$\displaystyle\|\mathbf{x}\|=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{d}^{2}},$

denotes the usual Euclidean norm. Consider the weight function

$\displaystyle
W_{\mu}(\mathbf{x})=\big{(}1-\|\mathbf{x}\|^{2}\big{)}^{\mu-1/2},\qquad\mu>-1/2,\qquad\mathbf{x}\in\mathbb{B}^{d}.$

Associated with $W_{\mu}(\mathbf{x})$, we define the usual inner product on
the unit ball

$\displaystyle\langle
f,g\rangle_{\mu}=\omega_{\mu}\int_{\mathbb{B}^{d}}f(\mathbf{x})g(\mathbf{x})W_{\mu}(\mathbf{x})d\mathbf{x},$

where the normalizing constant

$\displaystyle\omega_{\mu}=\left[\int_{\mathbb{B}^{d}}W_{\mu}(\mathbf{x})d\mathbf{x}\right]^{-1}=\frac{\Gamma(\mu+(d+1)/2)}{\pi^{d/2}\Gamma(\mu+1/2)}$

is chosen in order to have $\langle 1,1\rangle_{\mu}=1$.

The associated moment functional is defined by means of its moments

$\displaystyle\langle
u_{\mu},\mathbf{x}^{\nu}\rangle=\omega_{\mu}\int_{\mathbb{B}^{d}}\mathbf{x}^{\nu}W_{\mu}(\mathbf{x})d\mathbf{x}=\omega_{\mu}\int_{\mathbb{B}^{d}}\mathbf{x}^{\nu}\big{(}1-\|\mathbf{x}\|^{2}\big{)}^{\mu-1/2}d\mathbf{x}.$

Observe that $u_{\mu}$ is a centrally symmetric positive-definite moment
functional, that is,

$\displaystyle\langle
u_{\mu},\mathbf{x}^{\nu}\rangle=0,\qquad\text{whenever}\quad|\nu|\quad\text{is
an odd integer}.$

Let us denote by $\\{\mathbb{P}^{(\mu)}_{n}\\}_{n\geqslant 0}$ a ball OPS. In
this case, several explicit bases are known, and we will describe one of them.
An orthogonal basis in terms of classical Jacobi polynomials and spherical
harmonics is presented in [12].

Harmonic polynomials (see [12, p. 114]) are homogeneous polynomials in $d$
variables $Y(\mathbf{x})$ satisfying the Laplace equation

$\displaystyle\Delta Y=0.$

Let $\mathcal{H}_{n}^{d}$ denote the space of harmonic polynomials of degree
$n$. It is well known that

$\displaystyle\sigma_{n}=\dim\mathcal{H}_{n}^{d}=\binom{n+d-1}{n}-\binom{n+d-3}{n},\qquad
n\geqslant 3,$

and $\sigma_{0}=\dim\mathcal{H}_{0}^{d}=1$,
$\sigma_{1}=\dim\mathcal{H}_{1}^{d}=d$.

The restriction of $Y(\mathbf{x})\in\mathcal{H}_{n}^{d}$ to $\mathbb{S}^{d-1}$
is called a spherical harmonic. We will use spherical polar coordinates
$\mathbf{x}=r\xi$, for $\mathbf{x}\in\mathbb{R}^{d}$, $r\geqslant 0$, and
$\xi\in\mathbb{S}^{d-1}$. If $Y\in\mathcal{H}_{n}^{d}$, we use the notation
$Y(\mathbf{x})$ to denote the harmonic polynomial, and $Y(\xi)$ to denote the
spherical harmonic. This notation is coherent with the fact that if
$\mathbf{x}=r\xi$ then $Y(\mathbf{x})=r^{n}Y(\xi)$. Moreover, spherical
harmonics of different degree are orthogonal with respect to the surface
measure on $\mathbb{S}^{d-1}$, and we can choose an orthonormal basis.

Then, an orthonormal basis for the classical ball inner product (see [12, p.
142]) is given by the polynomials

$\displaystyle
P_{j,k}^{n}(\mathbf{x};\mu)=h_{j,n}^{-1}P_{j}^{(\mu-\frac{1}{2},n-2j+\frac{d-2}{2})}\big{(}2\|\mathbf{x}\|^{2}-1\big{)}Y_{k}^{n-2j}(\mathbf{x})$
(5.1)

for $0\leqslant j\leqslant n/2$ and $1\leqslant k\leqslant\sigma_{n-2j}$. Here
$P_{j}^{(\alpha,\beta)}(t)$ denotes the $j$-th classical Jacobi polynomial,
$\\{Y_{k}^{n-2j}\colon 1\leqslant k\leqslant\sigma_{n-2j}\\}$ is an
orthonormal basis for $\mathcal{H}_{n-2j}^{d}$ and the constants $h_{j,n}$ are
defined by

$\displaystyle[h_{j,n}]^{2}=[h_{j,n}(\mu)]^{2}=\frac{\big{(}\mu+\frac{1}{2}\big{)}_{j}\big{(}\frac{d}{2}\big{)}_{n-j}\big{(}n-j+\mu+\frac{d-1}{2}\big{)}}{j!\big{(}\mu+\frac{d+1}{2}\big{)}_{n-j}\big{(}n+\mu+\frac{d-1}{2}\big{)}},$

where $(a)_{m}=a(a+1)\cdots(a+m-1)$ denotes the Pochhammer symbol.

This basis can be written as an orthonormal polynomial system in the form

$\displaystyle\mathbb{P}_{n}^{(\mu)}=\big{(}P^{n}_{[\frac{n}{2}],\sigma_{n-2[\frac{n}{2}]}},\ldots,P^{n}_{[\frac{n}{2}],1};\dots;P^{n}_{1,\sigma_{n-2}},\ldots,P^{n}_{1,2},P^{n}_{1,1};P^{n}_{0,\sigma_{n}},\ldots,P^{n}_{0,2},P^{n}_{0,1}\big{)}^{t},$

where we order the entries by reverse lexicographical order of their indexes.
Observe that for $n$ odd, then $n-2[\frac{n}{2}]=1$, and $\sigma_{1}=d$, while
for $n$ even, $n-2[\frac{n}{2}]=0$, and $\sigma_{0}=1$.

####  5.1.1 Connection properties for adjacent families of ball polynomials

Christoffel modification allows us to relate two families of ball polynomials
for two values of the parameter $\mu$ differing by one unity. In fact,

$\displaystyle
W_{\mu+1}(\mathbf{x})=\big{(}1-\|\mathbf{x}\|^{2}\big{)}^{\mu+1}=\big{(}1-\|\mathbf{x}\|^{2}\big{)}W_{\mu}(\mathbf{x}).$

Then, if we define
$\lambda(\mathbf{x})=1-\|\mathbf{x}\|^{2}=1-x_{1}^{2}-x_{2}^{2}-\cdots-
x_{d}^{2}$, using the matrix formalism (4.11), we get the following relation

$\displaystyle\mathbb{P}_{n}^{(\mu)}(\mathbf{x})=F_{n}\mathbb{P}_{n}^{(\mu+1)}(\mathbf{x})+N_{n}\mathbb{P}_{n-2}^{(\mu+1)}(\mathbf{x}),$

where $F_{n}$ is the non-singular matrix needed to change the leading
coefficients, and $N_{n}$ has full rank. However, in this case, we can
explicitly give both matrices using the previously described basis.

Here we use the relation for Jacobi polynomials (formula (22.7.18) in [1])

$\displaystyle
P_{n}^{(\alpha,\beta)}(t)=\frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1}P_{n}^{(\alpha+1,\beta)}(t)-\frac{n+\beta}{2n+\alpha+\beta+1}P_{n-1}^{(\alpha+1,\beta)}(t),$

for $n\geqslant 0$, in (5.1) and we can relate ball polynomials corresponding
to adjacent families

$\displaystyle
P_{j,k}^{n}(\mathbf{x};\mu)=a^{n}_{j}P_{j,k}^{n}(\mathbf{x};\mu+1)-b^{n}_{j}P_{j-1,k}^{n-2}(\mathbf{x};\mu+1),$

where

$\displaystyle
a^{n}_{j}=\frac{h_{j,n}(\mu+1)}{h_{j,n}(\mu)}\frac{n-j+\mu+(d-1)/2}{n+\mu+(d-1)/2},\qquad
b^{n}_{j}=\frac{h_{j-1,n-2}(\mu+1)}{h_{j,n}(\mu)}\frac{n-j-1+d/2}{n+\mu+(d-1)/2},$

for $n\geqslant 2$ and $1\leqslant j\leqslant n/2$.

Then, the matrix $F_{n}$ is a diagonal and invertible matrix given by

$\displaystyle
F_{n}=\operatorname{diag}\big{(}a^{n}_{[\frac{n}{2}]},\ldots,a^{n}_{[\frac{n}{2}]};\dots;a^{n}_{1},\ldots,a^{n}_{1};a^{n}_{0},\ldots,a^{n}_{0}\big{)},$

that is, we repeat every $a^{n}_{i}$ in the diagonal $\sigma_{n-2i}$ times,
for $0\leqslant i\leqslant[\frac{n}{2}]$.

Moreover, the $r^{d}_{n}\times r^{d}_{n-2}$ block matrix $N_{n}$ has full
rank, and

$\displaystyle
N_{n}=-L^{t}_{n-1,1}L^{t}_{n-2,1}\operatorname{diag}\big{(}b^{n}_{[\frac{n}{2}]},\ldots,b^{n}_{[\frac{n}{2}]};\dots;b^{n}_{1},\ldots,b^{n}_{1}\big{)}.$

####  5.1.2 Ball–Uvarov polynomials

Let $u_{\mu}$ be the classical ball moment functional defined as above, and we
add a mass point at the origin and consider the moment functional

$\displaystyle\langle v,p(\mathbf{x})\rangle=\langle
u_{\mu},p(\mathbf{x})\rangle+\lambda p(\mathbf{0}).$

Using Theorem 3.1 we deduce the quasi-definite character of the moment
functional $v$ except for a denumerable set of negative values of $\lambda$.
Of course, $v$ is positive definite for every positive value of $\lambda$.

Let $\\{P_{j,k}^{n}(\mathbf{x})\\}$ be the mutually orthogonal basis for the
ball polynomials given in (5.1). Since spherical harmonics are homogeneous
polynomials, we get $Y_{k}^{n-2j}(\mathbf{0})=0$ whenever $n-2j>0$. Hence, it
follows

$\displaystyle
P_{j,k}^{n}(\mathbf{0})=\begin{cases}h^{-1}_{{\lfloor\frac{n}{2}\rfloor},n}P_{{\lfloor\frac{n}{2}\rfloor}}^{(\mu-\frac{1}{2},\frac{d-2}{2})}(-1)&\text{if
$n$ is even, $j={\lfloor\frac{n}{2}\rfloor}$ and $k=1$,}\\\ 0&\text{in any
other case.}\end{cases}$ (5.2)

As a consequence we get

######  Lemma 5.1.

For $n\geqslant 0$

$\displaystyle
K_{n}(u_{\mu};\mathbf{x},\mathbf{0})=\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}P_{{\lfloor\frac{n}{2}\rfloor}}^{(\frac{d-2}{2},\mu-\frac{1}{2})}\big{(}1-2\|\mathbf{x}\|^{2}\big{)},$
$\displaystyle
K_{n}(u_{\mu};\mathbf{0},\mathbf{0})=\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}\binom{{\lfloor\frac{n}{2}\rfloor}+\frac{d}{2}}{{\lfloor\frac{n}{2}\rfloor}}.$
(5.3)

###### Proof.

Let $h_{n}^{(\alpha,\beta)}$ be the norm of the Jacobi polynomial
$P_{n}^{(\alpha,\beta)}(t)$ as given in [26, (4.3.3)]

$\displaystyle
h_{n}^{(\alpha,\beta)}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{n!\Gamma(n+\alpha+\beta+1)}.$

Then we have
$h^{2}_{{\lfloor\frac{n}{2}\rfloor},n}=d_{\mu}h_{\lfloor\frac{n}{2}\rfloor}^{(\mu-\frac{1}{2},\frac{d-2}{2})}$,
where

$\displaystyle
d_{\mu}=\frac{1}{2^{\mu+\frac{d-1}{2}}}\frac{\Gamma\big{(}\mu+\frac{d+1}{2}\big{)}}{\Gamma\big{(}\mu+\frac{1}{2}\big{)}\Gamma\big{(}\frac{d}{2}\big{)}}.$

From (5.2) we get

$\displaystyle
K_{n}(u_{\mu};\mathbf{x},\mathbf{0})=d_{\mu}^{-1}\sum_{j=0}^{{\lfloor\frac{n}{2}\rfloor}}\big{[}h_{j}^{(\mu-\frac{1}{2},\frac{d-2}{2})}\big{]}^{-1}P_{j}^{(\mu-\frac{1}{2},\frac{d-2}{2})}\big{(}2\|\mathbf{x}\|^{2}-1\big{)}P_{j}^{(\mu-\frac{1}{2},\frac{d-2}{2})}(-1).$

Here, we can recognize the ${\lfloor\frac{n}{2}\rfloor}$-th kernel of the
Jacobi polynomials with parameters $(\mu-\frac{1}{2},\frac{d-2}{2})$, using
$P_{j}^{(\alpha,\beta)}(t)=(-1)^{j}P_{j}^{(\beta,\alpha)}(-t)$ and relation
(4.5.3) in [26], we get

$\displaystyle
K_{n}(u_{\mu};\mathbf{x},\mathbf{0})=\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}P_{{\lfloor\frac{n}{2}\rfloor}}^{(\frac{d}{2},\mu-\frac{1}{2})}\big{(}1-2\|\mathbf{x}\|^{2}\big{)}.$

In particular, setting $\mathbf{x}=\mathbf{0}$ we obtain

$\displaystyle
K_{n}(u_{\mu};\mathbf{0},\mathbf{0})=\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}P_{{\lfloor\frac{n}{2}\rfloor}}^{(\frac{d}{2},\mu-\frac{1}{2})}(1)=\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}\binom{{\lfloor\frac{n}{2}\rfloor}+\frac{d}{2}}{{\lfloor\frac{n}{2}\rfloor}}.$
∎

Summarizing the previous results, in next theorem we get the connection
formulas for the orthogonal polynomials and kernels corresponding to the
moment functionals $u_{\mu}$ and $v$.

######  Theorem 5.2.

Let us assume $v$ is quasi-definite and define the polynomials

$\displaystyle
Q_{j,k}^{n}(\mathbf{x})=\begin{cases}P_{{\lfloor\frac{n}{2}\rfloor},1}^{n}(\mathbf{x})-a_{n}K_{n-1}(u_{\mu};\mathbf{x},\mathbf{0})&\text{if
$n$ is even, $j={\lfloor\frac{n}{2}\rfloor}$ and $k=1$,}\\\
P_{j,k}^{n}(\mathbf{x})&\text{in any other case,}\end{cases}$

where

$\displaystyle a_{n}=\frac{\lambda
P_{{\lfloor\frac{n}{2}\rfloor},1}^{n}(\mathbf{0})}{1+\lambda
K_{n-1}(u_{\mu};\mathbf{0},\mathbf{0})}.$

Then according to Theorem 3.1, $\\{Q_{j,k}^{n}(\mathbf{x})\\}$ constitutes an
orthogonal basis with respect to the moment functional $v$. Moreover, the
corresponding kernels are related by the expression

$\displaystyle
K_{n}(v;\mathbf{x},\mathbf{y})=K_{n}(u_{\mu};\mathbf{x},\mathbf{y})-b_{n}P_{{\lfloor\frac{n}{2}\rfloor}}^{(\frac{d}{2},\mu-\frac{1}{2})}\big{(}1-2\|\mathbf{x}\|^{2}\big{)}P_{{\lfloor\frac{n}{2}\rfloor}}^{(\frac{d}{2},\mu-\frac{1}{2})}\big{(}1-2\|\mathbf{y}\|^{2}\big{)},$
(5.4)

where

$\displaystyle b_{n}=\frac{\lambda}{1+\lambda
K_{n}(u_{\mu};\mathbf{0},\mathbf{0})}\left[\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}\right]^{2}.$

As a consequence of (5.4) we can easily deduce the asymptotic behaviour of the
Christoffel functions. From now on we will assume $\lambda>0$, also we have to
impose the restriction $\mu\geqslant 0$ because existing asymptotics for
Christoffel functions in the classical case have only been established for
this range of $\mu$. The symbol $C$ will represent a constant but it does not
have always the same value.

First, we get the asymptotics for the interior of the ball. For this purpose,
we will need the following classical estimate for Jacobi polynomials (see [26,
equations (4.1.3) and (7.32.5)]):

######  Lemma 5.3.

For arbitrary real numbers $\alpha>-1$, $\beta>-1$ and $t\in[0,1]$,

$\displaystyle\big{|}P_{n}^{(\alpha,\beta)}(t)\big{|}\leqslant
Cn^{-\frac{1}{2}}\big{(}1-t+n^{-2}\big{)}^{-(\alpha+1/2)/2}.$

And an analogous estimate on $[-1,0]$ follows from
$P_{n}^{(\alpha,\beta)}(t)=(-1)^{n}P_{n}^{(\beta,\alpha)}(-t)$.

######  Theorem 5.4.

Let $r=\|\mathbf{x}\|$. For $0<r\leqslant 1/2$ we have

$\displaystyle
0<K_{n}(u_{\mu};\mathbf{x},\mathbf{x})-K_{n}(v;\mathbf{x},\mathbf{x})\leqslant
Cn^{-1}\left(2r^{2}+\frac{4}{n^{2}}\right)^{-\frac{d+1}{2}}.$ (5.5)

For $1/2\leqslant r<1$ we have

$\displaystyle
0<K_{n}(u_{\mu};\mathbf{x},\mathbf{x})-K_{n}(v;\mathbf{x},\mathbf{x})\leqslant
Cn^{-1}\left(2\big{(}1-r^{2}\big{)}+\frac{4}{n^{2}}\right)^{-\mu}.$ (5.6)

Here $C$ is independent of $n$ and $x$. Consequently if $\mu\geqslant 0$,
uniformly for $\mathbf{x}$ in compact subsets of $\\{\mathbf{x}\colon
0<\|\mathbf{x}\|<1\\}$,

$\displaystyle\lim_{n\rightarrow\infty}K_{n}(v;\mathbf{x},\mathbf{x})/\binom{n+d}{d}=\frac{1}{\sqrt{\pi}}\frac{\Gamma\big{(}\mu+\frac{1}{2}\big{)}\Gamma\big{(}\frac{d+1}{2}\big{)}}{\Gamma\big{(}\mu+\frac{d+1}{2}\big{)}}\big{(}1-\|\mathbf{x}\|^{2}\big{)}^{-\mu}.$
(5.7)

###### Proof.

From (5.4) we get

$\displaystyle
0<K_{n}(u_{\mu};\mathbf{x},\mathbf{x})-K_{n}(v;\mathbf{x},\mathbf{x})=b_{n}\left(P_{{\lfloor\frac{n}{2}\rfloor}}^{(\frac{d}{2},\mu-\frac{1}{2})}\big{(}1-2\|\mathbf{x}\|^{2}\big{)}\right)^{2}$

with

$\displaystyle
0<b_{n}=\frac{\lambda}{1+\lambda\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}\left(\begin{matrix}\lfloor\frac{n}{2}\rfloor+\frac{d}{2}\\\
\lfloor\frac{n}{2}\rfloor\end{matrix}\right)}\left[\frac{\big{(}\mu+\frac{d+1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}{\big{(}\mu+\frac{1}{2}\big{)}_{{\lfloor\frac{n}{2}\rfloor}}}\right]^{2}.$

Using Stirling’s formula we can easily deduce the convergence of the sequence
$\\{b_{n}\\}$ to a positive value. Therefore we can find a constant $C$ such
that $0<b_{n}<C$, $n=0,1,2,\dots$, that is

$\displaystyle
0<K_{n}(u_{\mu};\mathbf{x},\mathbf{x})-K_{n}(v;\mathbf{x},\mathbf{x})<C\left(P_{{\lfloor\frac{n}{2}\rfloor}}^{(\frac{d}{2},\mu-\frac{1}{2})}\big{(}1-2\|\mathbf{x}\|^{2}\big{)}\right)^{2},$

and Lemma 5.3 gives (5.5) and (5.6). Finally, (5.7) follows from the
corresponding asymptotic result for Christoffel functions on the ball obtained
by Y. Xu in [30]. ∎

At the origin the asymptotic is completely different, in fact, we recover the
value of the mass $\lambda$ as the limit of the Christoffel functions
evaluated at $\mathbf{x}=\mathbf{0}$, as we show in our next result.

######  Theorem 5.5.

$\displaystyle\lim_{n\rightarrow\infty}K_{n}(v;\mathbf{0},\mathbf{0})=\frac{1}{\lambda}.$

###### Proof.

From (3.16) we get

$\displaystyle K_{n}(v;\mathbf{0},\mathbf{0})$
$\displaystyle=K_{n}(u_{\mu};\mathbf{0},\mathbf{0})-\frac{\lambda\left[K_{n}(u_{\mu};\mathbf{0},\mathbf{0})\right]^{2}}{1+\lambda
K_{n}(u_{\mu};\mathbf{0},\mathbf{0})}=\frac{K_{n}(u_{\mu};\mathbf{0},\mathbf{0})}{1+\lambda
K_{n}(u_{\mu};\mathbf{0},\mathbf{0})},$

and the result follows from (5.3) and Stirling’s formula. ∎

###  5.2 Uvarov modification of bivariate Bessel–Laguerre orthogonal
polynomials

As usual, let $\\{L_{n}^{(\alpha)}(t)\\}_{n\geqslant 0}$ denote the classical
Laguerre polynomials orthogonal with respect to the moment functional

$\displaystyle\big{\langle}\ell^{(\alpha)},f\big{\rangle}=\int_{0}^{+\infty}f(t)t^{\alpha}e^{-t}dt,\qquad\alpha>-1,$

normalized by the condition (equation (22.2.12) in [1])

$\displaystyle
h_{n}^{(\alpha)}=\big{\langle}\ell^{(\alpha)},(L_{n}^{(\alpha)}(t))^{2}\big{\rangle}=\frac{\Gamma(\alpha+n+1)}{n!}.$

Following [17], let $\\{B_{n}^{(a,b)}(z)\\}_{n\geqslant 0}$ denote the
univariate classical Bessel polynomials orthogonal with respect to the non
positive-definite moment functional

$\displaystyle\langle b^{(a,b)},f\rangle=\int_{c}f(z)(2\pi
i)^{-1}z^{a-2}e^{-b/z}dz,\qquad a\neq 0,-1,-2,\ldots,\qquad b\neq 0,$

where $c$ is the unit circle oriented in the counter-clockwise direction,
normalized by the condition

$\displaystyle B_{n}^{(a,b)}(0)=1,$

and [17, equation (58), p. 113]

$\displaystyle
h_{n}^{(a,b)}=\big{\langle}b^{(a,b)},(B_{n}^{(a,b)}(z))^{2}\big{\rangle}=\frac{(-1)^{n+1}n!b}{(2n+a-1)(a)_{n-1}}.$

In [19], the Krall and Sheffer’s partial differential equation (5.55)

$\displaystyle
x^{2}w_{xx}+2xyw_{xy}+\big{(}y^{2}-y\big{)}w_{yy}+g(x-1)w_{x}+g(y-\gamma)w_{y}=n(n+g-1)w,$

was considered and the authors proved that it has an OPS
$\big{\\{}P^{(g,\gamma)}_{n,m}(x,y)\colon 0\leqslant m\leqslant
n\big{\\}}_{n\geqslant 0}$ for $g\gamma+n\neq 0$, for $n\geqslant 0$. The
corresponding moment functional is not positive-definite.

Moreover, they proved that an explicit expression for the polynomial solutions
is given by

$\displaystyle
P^{(g,\gamma)}_{n,m}(x,y)=B_{n-m}^{(g+2m,-g)}(x)x^{m}L_{m}^{(g\gamma-1)}\left(\frac{gy}{x}\right),$
(5.8)

for $g\neq 0$, $g\gamma>-2$, $g+n\neq 0$ and $g\gamma+n\neq 0$, when
$n\geqslant 0$. These polynomials are orthogonal with respect to the moment
functional $u^{(g,\gamma)}$ defined as (see Theorem 3.4 in [19])

$\displaystyle\big{\langle}u^{(g,\gamma)},x^{h}y^{k}\big{\rangle}=\big{\langle}x^{k}\ell_{x}^{(g,-g)},x^{h}\big{\rangle}\big{\langle}b_{y}^{(g\gamma-1)},y^{k}\big{\rangle},\qquad
h,k\geqslant 0.$

For the bivariate polynomials (5.8), we get

$\displaystyle
h^{(g,\gamma)}_{n,m}=\big{\langle}u^{(g,\gamma)},\big{(}P^{(g,\gamma)}_{n,m}(x,y)\big{)}^{2}\big{\rangle}$
$\displaystyle\hphantom{h^{(g,\gamma)}_{n,m}}{}=\big{\langle}b_{x}^{(g+2m,-g)},\big{(}B_{n-m}^{(g+2m,-g)}(x)\big{)}^{2}\big{\rangle}\big{\langle}\ell^{(g\gamma-1)},\big{(}L_{m}^{(g\gamma-1)}(y)\big{)}^{2}\big{\rangle}$
$\displaystyle\hphantom{h^{(g,\gamma)}_{n,m}}{}=h_{n-m}^{(g+2m,-g)}h_{m}^{(g\gamma-1)}.$

Now, we will modify the above moment functional adding a mass at the point
$\mathbf{0}=(0,0)$. Thus, we define

$\displaystyle\big{\langle}v^{(g,\gamma)},p(x,y)\big{\rangle}=\big{\langle}u^{(g,\gamma)},p(x,y)\big{\rangle}+\lambda
p(0,0),\qquad\forall\,p\in\Pi^{2}.$

Following Section 3, since the polynomial
$x^{m}L_{m}^{(g\gamma-1)}\left(\frac{gy}{x}\right)$ vanishes at $(0,0)$ except
for $m=0$, we get

$\displaystyle P^{(g,\gamma)}_{n,0}(0,0)=B_{n}^{(g,-g)}(0)=1,\qquad
P^{(g,\gamma)}_{n,m}(0,0)=0,\qquad 0<m\leqslant n.$

Therefore, (3.3) becomes

$\displaystyle\mathsf{P}_{n}((0,0))=(1,0,\dots,0)^{t},$

and (3.5) can be explicitly computed as follows

$\displaystyle\mathsf{K}_{n}((0,0),(x,y))=K_{n}(u^{(g,\gamma)};(0,0),(x,y))$
$\displaystyle\hphantom{\mathsf{K}_{n}((0,0),(x,y))}{}=\sum_{m=0}^{n}\sum_{k=0}^{m}\frac{P^{(g,\gamma)}_{m,k}(0,0)P^{(g,\gamma)}_{m,k}(x,y)}{h^{(g,\gamma)}_{m,k}}=\sum_{m=0}^{n}\frac{B_{m}^{(g,-g)}(0)B_{m}^{(g,-g)}(x)}{h^{(g,-g)}_{m}h_{0}^{(g\gamma-1)}}$
$\displaystyle\hphantom{\mathsf{K}_{n}((0,0),(x,y))}{}=\frac{1}{h_{0}^{(g\gamma-1)}}K_{n}(b^{(g,-g)};0,x)=\frac{1}{g\Gamma(g\gamma)}\sum_{m=0}^{n}\frac{(-1)^{m}(2m+g-1)(g)_{m-1}}{m!}$
$\displaystyle\hphantom{\mathsf{K}_{n}((0,0),(x,y))}{}=\frac{1}{g\Gamma(g\gamma)}\frac{(-1)^{n}(g)_{n}}{n!}=\frac{(-1)^{n}}{g\Gamma(g\gamma)}\binom{g+n-1}{n}.$

Then, using Theorem 3.1, $v^{(g,\gamma)}$ is quasi-definite if and only if we
choose $\lambda\in\mathbb{R}$ such that the value

$\displaystyle\lambda_{n}=1+\lambda\mathsf{K}_{n}((0,0),(x,y))=1+\lambda\frac{(-1)^{n}}{g\Gamma(g\gamma)}\binom{g+n-1}{n}$

is different from zero for $n\geqslant 0$. Using the above computations, there
exists only a numerable set of values of $\lambda\in\mathbb{R}$ such that the
modified moment functional $v^{(g,\gamma)}$ is not quasi-definite.

Therefore, let $\lambda\in\mathbb{R}$ such that $\lambda_{n}\neq 0$,
$n\geqslant 0$ and consider the quasi-definite moment functional
$v^{(g,\gamma)}$. The modified polynomials are given by

$\displaystyle Q^{(g,\gamma)}_{n,m}(x,y)=P^{(g,\gamma)}_{n,m}(x,y),\qquad
0<m\leqslant n,$ $\displaystyle
Q^{(g,\gamma)}_{n,0}(x,y)=P^{(g,\gamma)}_{n,0}(x,y)-\frac{\lambda}{\lambda_{n-1}}\mathsf{K}_{n-1}((0,0),(x,y)).$

We can compute explicitly $Q^{(g,\gamma)}_{n,0}(x,y)$. If
$\tilde{\lambda}=\lambda/h_{0}^{(g\gamma-1)}$, then

$\displaystyle
Q^{(g,\gamma)}_{n,0}(x,y)=B_{n}^{(g,-g)}(x)-\frac{\tilde{\lambda}}{1+\tilde{\lambda}K_{n}(b^{(g,-g)};0,0)}K_{n-1}\big{(}b^{(g,-g)};0,x\big{)},$

that is, $Q^{(g,\gamma)}_{n,0}(x,y)$ coincides with the (univariate)
orthogonal polynomial associated with the univariate modification of the
Bessel moment functional given by

$\displaystyle\big{\langle}\tilde{b}^{(g,-g)},p(x)\big{\rangle}=\big{\langle}b^{(g,-g)},p(x)\big{\rangle}+\tilde{\lambda}p(0).$

As a conclusion, the perturbation $v^{(g,\gamma)}$ of the bivariate moment
functional $u^{(g,\gamma)}$ only affects the bivariate polynomials
$Q^{(g,\gamma)}_{n,0}(x,y)$, $n\geqslant 0$.

### Acknowledgements

The authors are really grateful to the anonymous referees for their valuable
suggestions and comments which led us to improve this paper. This work has
been partially supported by MINECO of Spain and the European Regional
Development Fund (ERDF) through grant MTM2014–53171–P, and Junta de Andalucía
grant P11–FQM–7276 and research group FQM–384.

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