# Christoffel transform and multiple orthogonal polynomials

Rostyslav Kozhan1 and  Marcus Vaktnäs2 1Department of Mathematics, Uppsala
University, S-751 06 Uppsala, Sweden<EMAIL_ADDRESS>2Department of
Mathematics, Uppsala University, S-751 06 Uppsala, Sweden,
<EMAIL_ADDRESS>
###### Abstract.

We identify a connection between the Christoffel transform of orthogonal
polynomials and multiple orthogonality systems containing a finitely supported
measure. In consequence, the compatibility relations for the nearest neighbour
recurrence coefficients provide a new algorithm for the computation of the
Jacobi coefficients of the one-step or multi-step Christoffel transforms.

More generally, we investigate multiple orthogonal polynomials associated with
the system of measures obtained by applying a Christoffel transform to each of
the orthogonality measures. We present an algorithm for computing the
transformed recurrence coefficients and determinantal formulas for the
transformed multiple orthogonal polynomials of type I and type II.

Finally, we show that zeros of multiple orthogonal polynomials of an Angelesco
or an AT system interlace with the zeros of the polynomial corresponding to
the one-step Christoffel transform. This allows us to prove a number of
interlacing properties satisfied by the multiple orthogonality analogues of
classical orthogonal polynomials. For the discrete polynomials, this also
produces an estimate on the smallest distance between consecutive zeros.

##  1\. Introduction

Let $\mu$ be a positive Borel measure on the real line with all the moments
$c_{j}:=\int x^{j}\,\mathrm{d}\mu(x)$ finite. Denote $P_{n}(x)$,
$n\in\mathbb{N}:=\\{k\in\mathbb{Z}:k\geq 0\\}$, to be the monic orthogonal
polynomial of degree $n$ with respect to the inner product

$\left\langle
f(x),g(x)\right\rangle=\int_{\mathbb{R}}f(x)g(x)\,\mathrm{d}\mu(x).$ (1)

These polynomials satisfy the famous three-term recurrence relation

$xP_{n}(x)=P_{n+1}(x)+b_{n}P_{n}(x)+a_{n}P_{n-1}(x)$ (2)

for some $a_{n}>0$ and $b_{n}\in\mathbb{R}$, called the Jacobi coefficients of
$\mu$.

Given a point $z_{0}\in\mathbb{R}$, a one-step Christoffel transform of $\mu$
is a new measure $\widehat{\mu}$ defined by

$\int f(x)\,\mathrm{d}\widehat{\mu}(x)=\int
f(x)({x-z_{0}})\,\mathrm{d}\mu(x).$ (3)

More generally, given a polynomial $\Phi(x)=\prod_{j=1}^{m}(x-z_{j})$ we
define the multi-step Christoffel transform to be $\widehat{\mu}$ given by

$\int f(x)\,\mathrm{d}\widehat{\mu}(x)=\int f(x){\Phi(x)}\,\mathrm{d}\mu(x),$
(4)

which, of course, can be viewed as the one-step transform repeated $m$ times
for each of the roots of $\Phi(x)$.

It is a natural question to understand the relationship between the orthogonal
polynomials $(P_{n}(x))_{n=0}^{\infty}$ of $\mu$ and
$(\widehat{P}_{n}(x))_{n=0}^{\infty}$ of $\widehat{\mu}$. In the one-step case
(3) the following simple relation holds true [Christoffel]:

$\begin{split}\widehat{P}_{n}(x)=\frac{1}{x-z_{0}}\left({P}_{n+1}(x)-\frac{{P}_{n+1}(z_{0})}{{P}_{n}(z_{0})}{P}_{n}(x)\right).\end{split}$
(5)

More generally, if $\deg\Phi=m$, then $\widehat{P}_{n}$ can be expressed in
terms of $P_{n},P_{n+1},\ldots,P_{n+m}$ using the Christoffel determinantal
formula, see (26) below.

Another matter of interest here is the relationship between the Jacobi
coefficients of $\widehat{\mu}$ and of $\mu$. For the one-step case
$\deg\Phi=1$ there are a number of closely related algorithms (among them are
the $qd$ algorithm of Rutishauser [QD], Galant’s [Galant], and Gautschi’s
[Gautschibook]) that allow to compute $\widehat{a}_{n}$’s and
$\widehat{b}_{n}$’s recursively from $a_{n}$’s and $b_{n}$’s. We present one
of them below (31). There exist explicit algorithms for the quadratic factors,
$\deg\Phi=2$, see, e.g., [Gautschibook, Sect 2.4.3]. For the general case
$\deg\Phi=m$, one typically applies the one-step or two-step algorithms
repeatedly.

Now let us introduce multiple orthogonal polynomials with respect to a system
of two measures $(\mu_{1},\mu_{2})$ on $\mathbb{R}$ (we work in the more
general setting of $r$ measures from Section 2 onwards). For
$(n_{1},n_{2})\in\mathbb{N}^{2}$, let $P_{n_{1},n_{2}}(x)$ be a non-zero monic
polynomial of degree $n_{1}+n_{2}$ satisfying

$\displaystyle\int_{\mathbb{R}}P_{n_{1},n_{2}}(x)x^{p}\,\mathrm{d}\mu_{1}(x)$
$\displaystyle=0,\qquad p=0,1,\dots,n_{1}-1,$ (6)
$\displaystyle\int_{\mathbb{R}}P_{n_{1},n_{2}}(x)x^{p}\,\mathrm{d}\mu_{2}(x)$
$\displaystyle=0,\qquad p=0,1,\dots,n_{2}-1.$ (7)

$P_{n_{1},n_{2}}(x)$ is then called a multiple orthogonal polynomial at the
multi-index $(n_{1},n_{2})$. We say that $(n_{1},n_{2})$ is normal for
$(\mu_{1},\mu_{2})$ if such $P_{n_{1},n_{2}}$ exists and is unique.

Assuming sufficiently many indices are normal, these polynomials satisfy [NNR,
Ismail] the nearest neighbour recurrence relations (compare with (2))

$\displaystyle
xP_{n_{1},n_{2}}(x)=P_{n_{1}+1,n_{2}}(x)+b_{n_{1},n_{2};1}P_{n_{1},n_{2}}(x)+a_{n_{1},n_{2};1}P_{n_{1}-1,n_{2}}(x)+a_{n_{1},n_{2};2}P_{n_{1},n_{2}-1}(x),$
(8) $\displaystyle
xP_{n_{1},n_{2}}(x)=P_{n_{1},n_{2}+1}(x)+b_{n_{1},n_{2};2}P_{n_{1},n_{2}}(x)+a_{n_{1},n_{2};1}P_{n_{1}-1,n_{2}}(x)+a_{n_{1},n_{2};2}P_{n_{1},n_{2}-1}(x).$
(9)

Subtracting (8) and (9) we can also obtain

$P_{n_{1}+1,n_{2}}(x)-P_{n_{1},n_{2}+1}(x)=(b_{n_{1},n_{2};2}-b_{n_{1},n_{2};1})P_{n_{1},n_{2}}(x).$
(10)

The coefficients
$a_{n_{1},n_{2};1},a_{n_{1},n_{2};2},b_{n_{1},n_{2};1},b_{n_{1},n_{2};2}$ in
(8) and (9) are called the nearest neighbour coefficients. These coefficients
satisfy [NNR] a set of nearest neighbour compatibility conditions (CC, for
short) explicitly written below in (41)–(43).

[ComputingNNR] showed that these compatibility conditions provide an algorithm
that allows to recursively compute all the nearest neighbour recurrence
coefficients
$a_{n_{1},n_{2};1},a_{n_{1},n_{2};2},b_{n_{1},n_{2};1},b_{n_{1},n_{2};2}$ from
the Jacobi coefficients of $\mu_{1}$ and $\mu_{2}$ (that is, from
$a_{n_{1},0;1},a_{0,n_{2};2},b_{n_{1},0;1},b_{0,n_{2};2}$), see Section 3.2
below for the details.

The central idea of our paper is the simple observation that if $\mu_{2}$ is
supported on $N$ distinct points $\\{z_{j}\\}_{j=1}^{N}$, then the multiple
orthogonal polynomial $P_{n,N}(x)$ of the system $(\mu_{1},\mu_{2})$ coincides
with $\widehat{P}_{n}(x)\Phi(x)$, where $\Phi(x)=\prod_{j=1}^{N}(x-z_{j})$ and
$\widehat{P}_{n}(x)$ is the $n$-th orthogonal polynomial of the Christoffel
transform $\widehat{\mu}_{1}$ (4) of $\mu_{1}$ corresponding to the polynomial
$\Phi(x)$. Indeed, $\widehat{P}_{n}(x)\Phi(x)$ is monic, has the right degree,
trivially satisfies (6)–(7), and it only remains to resolve the issue of
uniqueness which we do in Theorem 3.1.

In particular, the nearest neighbour recurrence relation (8) along locations
$\big{\\{}(n,N):n\in\mathbb{N}\big{\\}}$ reduces to the three-term recurrence
relation for $\widehat{\mu}_{1}$ (one should observe that $a_{n,N;2}=0$ for
all $n$, see Theorem 3.5), and the nearest neighbour coefficients along these
locations coincide with the Jacobi coefficients of $\widehat{\mu}_{1}$.

Taking the simplest case $N=1$, one realizes that Christoffel’s formula (5) is
just (10), while Gautschi’s algorithm (31) for computing the Jacobi
coefficients of the Christoffel transform is effectively the CC algorithm of
[ComputingNNR] (after minor modifications related to restricting the
coefficients to the strip $\mathbb{N}\times\\{0,1\\}$, see Section 3.2).
Furthermore, one can show that the well-known Gauss–Radau quadrature rule for
$\mu_{1}$ is just the multiple Gauss quadrature rule for $(\mu_{1},\mu_{2})$
with $N=|\operatorname{supp}\mu_{2}|=1$ (the Gauss–Lobatto rule corresponds to
$N=|\operatorname{supp}\mu_{2}|=2$).

For any $N$ the modified CC algorithm (see Section 3.2) therefore provides an
algorithm for computation of the Jacobi coefficients of the multi-step
Christoffel transform. It would be interesting to find out if there is any
computational benefit of this algorithm compared to the repeated use of the
one-step/two-step Gautschi/Galant algorithm. Such questions are important in
numerical mathematics, see, e.g., [BuenoandDopico, Galant, Gautschi,
Gautschibook] and references therein. We will not pursue this in this paper.

Our main focus is the study of the multiple Christoffel transform
$(\widehat{\mu}_{1},\ldots,\widehat{\mu}_{r})$ of the multiple orthogonality
system $(\mu_{1},\ldots,\mu_{r})$ for $r\geq 2$. Here $\widehat{\mu}_{j}$ are
defined as in (4) with the same polynomial $\Phi(x)$ for each $j$. Such
transform appears naturally when one studies the multiple Gauss quadrature
with fixed nodes at the zeros of $\Phi$.

We show how one can use the CC algorithm to compute the nearest neighbour
recurrence coefficients of $(\widehat{\mu}_{1},\ldots,\widehat{\mu}_{r})$
(Section 3.2) and establish the determinantal formula for the multiple
orthogonal polynomials for $(\widehat{\mu}_{1},\ldots,\widehat{\mu}_{r})$ (see
Section 3.1 for type II, and Section 3.4 for type I).

For the special case $N=1$ the determinantal formulas reduce to (75) for type
I and to (61) for type II. The relation (61) was known earlier [ADMVA, Prop
3.2], while (75) has appeared in [BFM22] for the case of two measures and
multi-indices along the step-line. During the preparation of the manuscript
there appeared [ManasRojas] studying multiple Christoffel transforms using
another approach (the Gauss–Borel factorization) for the step-line multi-
indices.

In Section 3.6 we demonstrate that CC algorithm can be used to compute
repeated Christoffel transforms, which is the natural setting for the $qd$
algorithm of Rutishauser [QD] and the discrete-time Toda lattices in one (see,
e.g., [SpiZhe]) and multiple dimensions, see [ADMVA, Dol] and references
therein.

In Section 3.7 we classify all possible nearest neighbour recurrence
coefficients $\\{a_{\bm{n},j},b_{\bm{n},j}\\}$ that can occur for maximally-
normal systems (such systems are called perfect). This was proved for systems
with two positive infinitely-supported measures in [IntegrableSystems]. We
provide an alternative simple proof that allows either measures or linear
moment functionals which may be finitely or infinitely supported. The main
difference is that for finitely supported $\mu_{j}$’s the
$a_{\bm{n},j}$-coefficients must be zero not only on the initial marginal
indices (that is, with $n_{j}=0$) but also on the final ones (with
$n_{j}=|\operatorname{supp}\,\mu_{j}|$), see Theorem 3.17 for the details.

The next portion of the results (Section 3.8) concerns interlacing of the
zeros of multiple orthogonal polynomials $P_{\bm{n}}$ for
$(\mu_{1},\ldots,\mu_{r})$ and $\widehat{P}_{\bm{n}}$ for
$(\widehat{\mu}_{1},\ldots,\widehat{\mu}_{r})$ when $\deg\Phi=1$. We show that
the zeros of $P_{\bm{n}}$ and $\widehat{P}_{\bm{n}}$ interlace for a wide
class of systems including all Angelesco and AT systems, and the same result
holds for type I polynomials for a class of measures containing all Angelesco
systems. In particular, this applies to multiple Laguerre of the first and
second kind, Jacobi–Piñeiro, Angelesco–Jacobi, Jacobi–Laguerre,
Jacobi–Hermite, Charlier, Meixner of the first and second kind, Krawtchouk,
and Hahn (Sections 3.9 and 3.9). This type of interlacing was shown very
recently in [MarMor24, dosSan] for Angelesco–Jacobi, Jacobi–Laguerre,
Jacobi–Hermite systems for type II polynomials along the step-line multi-
indices using much more involved arguments, see also [MarMorPer] for related
results which use the notion of free convolution.

The interlacing results for the discrete systems then produces the lower bound
1 for the distance between two consecutive zeros (see Section 3.11), a result
that is well-known for the $r=1$ case, see [ChiSta, Lev67, KraZar].

Note that the result of one-step or multi-step Christoffel transform is always
a positive measure if all of the zeros of $\Phi(x)$ fall outside of the
interior of the convex hull $\operatorname{supp}\,\mu$ (or if there are even
number of them at each gap of the support). Otherwise however, $\widehat{\mu}$
is a signed/complex measure, which is convenient to view as a linear moment
functional. It does not take too much extra effort to allow $\mu$ to be a
linear moment functional from the beginning, which is what we do starting from
Section 2 onwards.

Finally, in Section 2.6 we collect several criteria that allow to establish
the normality of indices for multiple orthogonality systems. While basic,
these results may be of interest on their own. For example, we apply them to
find a necessary and sufficient condition for
$(\widehat{\mu}_{1},\ldots,\widehat{\mu}_{r})$ to be perfect. It is given in
terms of certain determinants involving $P_{\bm{n}}$’s (see Theorem 3.10).

In a companion paper [KVGeronimus] we obtain determinantal formulas for type I
and type II multiple orthogonal polynomials for rational perturbations of
measures, which includes the Geronimus transforms as well as the Christoffel
transforms with different polynomials $\Phi_{j}$ for each $\mu_{j}$.

#### Acknowledgements

Most of the results that appear in the current paper were part of M.V.’s 2021
Master Thesis in Uppsala University as reported in [MVthesis].

##  2\. Preliminaries

###  2.1. Orthogonal polynomials with respect to moment functionals

We use the notation $\mathbb{N}:=\\{k\in\mathbb{Z}:k\geq 0\\}$ (notice that we
include $0$) and $\mathbb{Z}_{>0}:=\\{k\in\mathbb{Z}:k>0\\}$. Let us assume
that we are given an arbitrary sequence $\\{c_{n}\\}_{n=0}^{\infty}$ of
complex numbers which will be referred to as the moment sequence. Define the
corresponding moment functional $\mu$ to be the linear map on the space of all
polynomials such that

$\mu[x^{n}]=c_{n},\qquad n\in\mathbb{N}.$ (11)

Associated to $\mu$ we have the bilinear form

${\left\langle P(x),Q(x)\right\rangle=\mu[P(x)Q(x)].}$ (12)

In particular, note that

${\left\langle P(x)R(x),Q(x)\right\rangle=\left\langle
P(x),R(x)Q(x)\right\rangle}$ (13)

for any choice of polynomials $P$, $Q$, and $R$. Orthogonal polynomials with
respect to $\mu$ are non-zero polynomials $P_{n}(x)$ such that
$\deg{P_{n}}\leq n$ and

${\left\langle P_{n}(x),x^{p}\right\rangle=0,\qquad p=0,1,\dots,n-1.}$ (14)

Such polynomials always exist, since solving (14) for the first $n+1$
Maclaurin coefficients results in a homogeneous system of linear equations
with more columns than rows. If we fix the $x^{n}$-coefficient we get a linear
system with coefficient matrix

${M_{n}=\begin{pmatrix}c_{0}&c_{1}&\cdots&c_{n-1}\\\
c_{1}&c_{2}&\cdots&c_{n}\\\ \vdots&\vdots&\ddots&\vdots\\\
c_{n-1}&c_{n}&\cdots&c_{2n-2}\\\ \end{pmatrix}.}$ (15)

Then $\Delta_{n}=\det{M_{n}}\neq 0$ if and only if $P_{n}$ is unique up to
multiplication by a constant and $\deg{P_{n}}=n$. In this case we always take
$P_{n}$ to be monic. If $\Delta_{n}=0$ then there is always at least one non-
zero solution $P_{n}$ of (14) with $\deg{P_{n}}<n$. Indeed, there would be at
least two different monic $P_{n}$ with degree $n$ solving (14), and then their
difference would be a non-zero solution of smaller degree.

Denote $\mathcal{L}_{\infty}$ to be the set of all quasi-definite moment
functionals, which are those $\mu$ for which $\Delta_{n}\neq 0$ for all
$n\in\mathbb{N}$. For such $\mu$ the monic orthogonal polynomial $P_{n}$ is
unique for each $n\in\mathbb{N}$. The polynomials satisfy the three-term
recurrence relation

$\begin{split}xP_{n}(x)=P_{n+1}(x)+b_{n}P_{n}(x)+a_{n}P_{n-1}(x),\qquad
n\in\mathbb{N}\end{split}$ (16)

for some complex numbers $a_{n}$ and $b_{n}$, called the Jacobi coefficients
of $\mu$ (in the case $n=0$ we formally take $a_{0}=0$ and $P_{-1}=0$).

It can be shown that $a_{n}\neq 0$ for all $n>0$ (and a simple proof is
included here in Lemma 2.18). Conversely, Favard’s Theorem states that any set
of $a_{n}$ and $b_{n}$, with $a_{0}=0$ and $a_{n}\neq 0$ for $n>0$, generates
a sequence of polynomials from (16) that are the orthogonal polynomials with
respect to some moment functional $\mu$.

The tridiagonal matrix

$J=\left(\begin{array}[]{cccc}b_{0}&1&{0}&\\\ a_{1}&b_{1}&1&\ddots\\\
{0}&a_{2}&b_{3}&\ddots\\\ &\ddots&\ddots&\ddots\end{array}\right)$ (17)

will be called the (“monic”) Jacobi matrix associated with $\mu$. It is the
matrix of the map $P(x)\mapsto xP(x)$ in the basis
$\left\\{P_{n}\right\\}_{n=0}^{\infty}$, by (16).

Finally, define the Christoffel–Darboux kernel via

$K_{n}(x,y)=\sum_{j=0}^{n-1}\frac{{P_{j}(x)}P_{j}(y)}{\left\langle
P_{j},P_{j}\right\rangle}.$ (18)

Then the Christoffel–Darboux identity takes place:

$K_{n}(x,y)=\frac{1}{\left\langle
P_{n-1},P_{n-1}\right\rangle}\frac{P_{n}(x)P_{n-1}(y)-P_{n-1}(x)P_{n}(y)}{x-y}.$
(19)

For more on the basics of orthogonal polynomials with respect to moment
functionals, see for example [Chihara].

Let $\mathcal{M}_{\infty}$ be the set of measures $\mu$ on $\mathbb{R}$ with
infinite support and all the moments

$c_{n}=\int x^{n}d\mu(x),\quad n\in\mathbb{N},$ (20)

finite. Such a measure generates a quasi-definite moment functional (11) which
with a mild abuse of notation we also denote by $\mu$. In particular, (12)
becomes the usual inner product in $L^{2}(\mu)$. With this convention, we can
view $\mathcal{M}_{\infty}$ as a subset of $\mathcal{L}_{\infty}$. The setting
$\mu\in\mathcal{M}_{\infty}$ corresponds to the standard theory of orthogonal
polynomials with $a_{n}>0$ for all $n\geq 1$ and $b_{n}\in\mathbb{R}$ for all
$n\geq 0$.

###  2.2. $\mu$ of finite support

If one takes a measure $\mu$ on $\mathbb{R}$ supported on exactly $N$ distinct
points $\\{z_{j}\\}_{j=1}^{N}\subset\mathbb{R}$, $N\in\mathbb{Z}_{>0}$, then
it is known that $\Delta_{n}\neq 0$ for $0\leq n\leq N$ and $\Delta_{n}=0$ for
$n>N$. Consequently, only $\\{P_{n}(x)\\}_{n=0}^{N}$ are uniquely defined,
with $P_{N}(x)=\prod_{j=1}^{N}(x-z_{j})$. The three-term recurrence (16) holds
for $0\leq n\leq N-1$ with $a_{n}>0$ for $1\leq n\leq N-1$. The corresponding
Jacobi matrix $J$ in (17) is finite of size $N\times N$ with
$\\{b_{n}\\}_{n=0}^{N-1}$ on the diagonal and $\\{a_{n}\\}_{n=1}^{N-1}$ on the
subdiagonal. Denote such $N$-finitely supported measures (or their associated
moment functionals) by $\mathcal{M}_{N}$.

In what follows we want to allow complex finite Jacobi matrices and the
associated linear functionals. Therefore we define $\mathcal{L}_{N}$, for each
$N\in\mathbb{Z}_{>0}$, to be the set of all the moment functionals $\mu$ for
which $\Delta_{n}\neq 0$ for $1\leq n\leq N$ and $\Delta_{n}=0$ for $n>N$. We
will refer to $\bigcup_{N=1}^{\infty}\mathcal{L}_{N}$ as the moment
functionals of finite support.

######  Lemma 2.1.

Suppose $\mu\in\mathcal{L}_{N}$ for some $N\in\mathbb{Z}_{>0}$. Then

$\left\langle P_{N}(x),x^{p}\right\rangle=0,\qquad p\in\mathbb{N}.$ (21)

Moreover, we have

$\left\langle P(x),x^{p}\right\rangle=0,\qquad p=0,1,\ldots,N-1,$ (22)

if and only if $P(x)$ is divisible by $P_{N}(x)$.

###### Proof.

Since $\Delta_{N+1}=0$, there is some (non-zero) monic $P_{N+1}$ solving (14)
for $n=N+1$ with degree $\deg{P_{N+1}}<N+1$. Now by $\Delta_{N}\neq 0$, we
must have $P_{N+1}=P_{N}$. By the orthogonality relations of $P_{N+1}$ we then
must have $\left\langle P_{N}(x),x^{N}\right\rangle=0$. We proceed to prove by
induction that $\left\langle P_{N}(x),x^{p}\right\rangle=0$ for all $p\geq N$.
By $\Delta_{n+1}=0$ (assuming $n\geq N$), there is some monic $P_{n}$ solving
(14) with $\deg{P_{n}}=n-k$, $0\leq k\leq n-N$, such that $\left\langle
P_{n}(x),x^{n}\right\rangle=0$ (since there is some monic $P_{n+1}$ with
$\deg{P_{n+1}}\leq n$). In particular, $\left\langle
P_{n}(x),Q(x)\right\rangle=0$ for every polynomial $Q$ with $\deg{Q}\leq n$,
so we must have $\left\langle P_{N}(x),x^{k}P_{n}(x)\right\rangle=\left\langle
P_{n}(x),x^{k}P_{N}(x)\right\rangle=0$, by (13). Assuming $\left\langle
P_{N}(x),x^{p}\right\rangle=0$ is true for all $p<n$ (and here $n>N$), we then
end up with $\left\langle P_{N}(x),x^{n}\right\rangle=\left\langle
P_{N}(x),x^{k}P_{n}(x)\right\rangle=0$, since $x^{k}P_{n}(x)=x^{n}+o(x^{n})$,
and $\left\langle P_{N}(x),o(x^{n})\right\rangle=0$. This proves (21).

Now for the second part, we have $\left\langle
P_{N}(x)Q(x),x^{p}\right\rangle=\left\langle
P_{N}(x),x^{p}Q(x)\right\rangle=0$ by (21). Conversely, suppose that $P$
satisfies (22) and write $P=P_{N}q+r$ with $\deg r<N$. Since $P_{N}$ is
orthogonal to any polynomial, we get that $r$ also satisfies (22). This is
only possible if $r\equiv 0$. ∎

Now we are ready to prove the analogue of the Favard theorem for the finite
support functionals.

######  Theorem 2.2.

If $\mu\in\mathcal{L}_{N}$, $N\leq\infty$, then $\\{P_{n}(x)\\}_{n=0}^{N}$
satisfy the three-term recurrence relation (16) for $0\leq n\leq N-1$ with
$a_{n}\in\mathbb{C}\setminus\\{0\\}$ for $1\leq n\leq N-1$ and
$b_{n}\in\mathbb{C}$ for $0\leq n\leq N-1$. Conversely, for any non-zero
$\left\\{a_{n}\right\\}_{n=1}^{N-1}$ and any
$\left\\{b_{n}\right\\}_{n=0}^{N-1}$ there exists a functional
$\mu\in\mathcal{L}_{N}$, unique up to multiplication by a non-zero constant,
with exactly these Jacobi coefficients.

######  Remark 2.3.

See [Gus, Thm 4] or a more detailed description of $\mu$ from
$\mathcal{L}_{N}$: if the $N$ roots of $P_{N}$ are all distinct then $\mu$ is
a (complex) measure supported on these roots. If some of the roots overlap
then $\mu$ contains derivative(s) of the Dirac delta function at the
corresponding root.

###### Proof.

The first part of the statement follows from the same arguments as in
[Chihara, Ch. 1, Sect. 3–4]. The second part follows very similarly to the
proof of [Chihara, Ch.1, Thm 4.4]. First define $\mu[1]=c\neq 0$ and
$\mu[P_{n}]=0$ for $n=1,\dots,N-1$, where $P_{n}$ are generated by the
recurrence coefficients through (16). If we also define $\mu[x^{p}P_{N}(x)]=0$
for all $p\in\mathbb{N}$ (see Lemma 2.1) we can then extend $\mu$ uniquely to
all polynomials. From the recurrence relation one can prove that $P_{n}$ are
orthogonal polynomials with respect to $\mu$, for $n=0,\dots,N-1$, and also
$\left\langle P_{n},x^{n}\right\rangle=a_{n}\dots a_{1}\neq 0$, which implies
$\Delta_{n}\neq 0$ for $n=1,\dots,N$. From $\langle P_{N}(x),x^{p+N}\rangle=0$
we get $\Delta_{n}=0$ for each $n>N$, so $\mu$ belongs to $\mathcal{L}_{N}$.
Since these assumptions on $\mu$ were necessary for
$\left\\{P_{n}\right\\}_{n=0}^{N}$ to be orthogonal with respect to some
$\mu\in\mathcal{L}_{N}$, and the recurrence coefficients are uniquely
determined by the recurrence, we get the full result. ∎

Finally, we define

$\mathcal{L}=\Big{(}\bigcup_{N=1}^{\infty}\mathcal{L}_{N}\Big{)}\cup\mathcal{L}_{\infty}.$
(23)

This is the set of moment functionals that we are working with throughout this
paper. Note that $\mathcal{L}$ includes
$\mathcal{M}=\Big{(}\bigcup_{N=1}^{\infty}\mathcal{M}_{N}\Big{)}\cup\mathcal{M}_{\infty}$,
the set of all positive measures of finite or infinite support on $\mathbb{R}$
with finite moments.

###  2.3. Christoffel transform

For $\mu\in\mathcal{L}$, a one-step Christoffel transform is a functional
$\widehat{\mu}$ given by

$\widehat{\mu}[x^{n}]=\mu[x^{n}(x-z_{0})],\qquad n\in\mathbb{N}.$ (24)

If $\mu\in\mathcal{M}$, then $\widehat{\mu}$ becomes (3).

More generally, if $\mu\in\mathcal{L}$ and $\Phi(x)=\prod_{j=1}^{m}(x-z_{j})$
is any polynomial, then we define the corresponding Christoffel transform

$\widehat{\mu}[x^{n}]=\mu[x^{n}\Phi(x)],\qquad n\in\mathbb{N}.$ (25)

We will also occasionally employ the notation $\Phi\mu$ for $\widehat{\mu}$ to
stress the dependence on $\Phi$. If $\mu\in\mathcal{M}$ and $\Phi$ has real
coefficients and does not change sign on the convex hull of
$\operatorname{supp}(\mu)$, then $\Phi\mu$ is also in $\mathcal{M}$ and is
given as in (4).

Some authors choose to work with the normalized version of $\Phi{\mu}$ given
by $\mu[\Phi(x)]^{-1}\mu[x^{n}\Phi(x)]$, under the assumption
$\mu[\Phi(x)]\neq 0$. Note that the monic orthogonal polynomials and their
recurrence coefficients for both versions of the Christoffel transform are the
same, so this makes no significant difference.

Given any $\mu\in\mathcal{L}$ and its Christoffel transform
$\widehat{\mu}=\Phi\mu$, we write $\widehat{P}_{n}$ for the orthogonal
polynomials with respect to $\widehat{\mu}$, and $\widehat{a}_{n}$ and
$\widehat{b}_{n}$ for the Jacobi coefficients with respect to $\widehat{\mu}$.
If $\mu,\widehat{\mu}\in\mathcal{L}_{\infty}$ then the following Christoffel
determinantal formula [Christoffel] holds:

$\begin{split}\widehat{P}_{n}(x)={\Phi(x)}^{-1}D_{n}^{-1}\det\begin{pmatrix}P_{n+m}(x)&P_{n+m-1}(x)&\cdots&P_{n}(x)\\\
P_{n+m}(z_{1})&P_{n+m-1}(z_{1})&\cdots&P_{n}(z_{1})\\\
\vdots&\vdots&\ddots&\vdots\\\
P_{n+m}(z_{m})&P_{n+m-1}(z_{m})&\cdots&P_{n}(z_{m})\end{pmatrix},\end{split}$
(26)

where $D_{n}$ is the normalizing constant

$\begin{split}D_{n}=\det\begin{pmatrix}P_{n+m-1}(z_{1})&P_{n+m-2}(z_{1})&\cdots&P_{n}(z_{1})\\\
P_{n+m-1}(z_{2})&P_{n+m-2}(z_{2})&\cdots&P_{n}(z_{2})\\\
\vdots&\vdots&\ddots&\vdots\\\
P_{n+m-1}(z_{m})&P_{n+m-2}(z_{m})&\cdots&P_{n}(z_{m})\end{pmatrix}.\end{split}$
(27)

In the one-step case this becomes

$\begin{split}\widehat{P}_{n}(x)=\frac{1}{x-z_{0}}\left({P}_{n+1}(x)-\frac{{P}_{n+1}(z_{0})}{{P}_{n}(z_{0})}{P}_{n}(x)\right).\end{split}$
(28)

By the Christoffel–Darboux formula (19) we get

$\begin{split}\widehat{P}_{n}(x)=\frac{\left\langle
P_{n},P_{n}\right\rangle}{P_{n}(z_{0})}K_{n+1}(z_{0},x).\end{split}$ (29)

Polynomials $K_{n}(z_{0},z)$ are sometimes referred to as the kernel
polynomials of $\mu$.

(28) together with the three-term recurrence (16) can be used to generate the
recurrence equations (see [Gautschibook])

$\begin{split}\widehat{b}_{n}-\delta_{n+1}&=b_{n+1}-\delta_{n},\\\
\widehat{a}_{n}-\delta_{n}\widehat{b}_{n}&=a_{n+1}-\delta_{n}b_{n},\\\
\delta_{n-1}\widehat{a}_{n}&=\delta_{n}a_{n},\end{split}$ (30)

with initial conditions $\widehat{a}_{0}=0$ and $\delta_{0}=z_{0}-b_{0}$. From
this it is possible to compute the recurrence coefficients of $\widehat{\mu}$
from the recurrence coefficients of $\mu$, through the following algorithm,

$\displaystyle\begin{split}&\delta_{0}:=z_{0}-b_{0};\\\ &\widehat{a}_{0}=0;\\\
&\widehat{b}_{0}=b_{0}-\frac{a_{1}}{\delta_{0}}\\\ &\mbox{for all
}n\in\mathbb{N}:\\\
&\qquad\delta_{n}:=\widehat{b}_{n-1}-b_{n}-\delta_{n-1};\\\
&\qquad\widehat{a}_{n}=a_{n}\frac{\delta_{n}}{\delta_{n-1}};\\\
&\qquad\widehat{b}_{n}=b_{n}+\frac{\widehat{a}_{n}-a_{n+1}}{\delta_{n}}.\end{split}$
(31)

In Section 3.1 we show how this can be generalized to the multi-step
Christoffel transform using multiple orthogonal polynomials.

###  2.4. Basics of MOPRL

Let $r\geq 1$ and consider a system of functionals
$\bm{\mu}=(\mu_{1},\dots,\mu_{r})\in\mathcal{L}^{r}$. Let us write
$\left\langle P(x),Q(x)\right\rangle_{j}$ for $\mu_{j}[P(x)Q(x)]$.

######  Definition 2.4.

Given a multi-index $\bm{n}\in\mathbb{N}^{r}$, a type II multiple orthogonal
polynomial is a non-zero polynomial $P_{\bm{n}}(x)$ such that
$\deg{P_{\bm{n}}}\leq\left|\bm{n}\right|:=n_{1}+\ldots+n_{r}$, and

$\begin{split}\left\langle P_{\bm{n}}(x),x^{p}\right\rangle_{j}=0,\qquad
p=0,1,\dots,n_{j}-1,\qquad j=1,\dots,r.\end{split}$ (32)

######  Definition 2.5.

A type I multiple orthogonal polynomial is a non-zero vector of polynomials
$\bm{A_{\bm{n}}}=(A_{\bm{n}}^{(1)},\dots,A_{\bm{n}}^{(r)})$ such that
$\deg{A_{\bm{n}}^{(j)}}\leq n_{j}-1$, $j=1,\dots,r$, and

$\begin{split}\sum_{j=1}^{r}\left\langle
A_{\bm{n}}^{(j)}(x),x^{p}\right\rangle_{j}=0,\qquad
p=0,1,\dots,\left|\bm{n}\right|-2.\end{split}$ (33)

Note that $A_{\bm{n}}^{(j)}=0$ when $n_{j}=0$ (we take the degree of $0$ to be
$-\infty$). Hence for $\bm{n}=\bm{0}$ there would be no non-zero solutions to
(33). In this case we take $\bm{A}_{\bm{0}}=\bm{0}$ (the $r$-vector of zero
polynomials) as the only type I polynomial.

It is easy to show that for any multi-index
$\bm{n}\in\mathbb{N}^{r}\setminus\left\\{\bm{0}\right\\}$ the following
statements are equivalent:

  * (i)

There is a unique monic type II multiple orthogonal polynomial $P_{\bm{n}}$
such that $\deg{P_{\bm{n}}}=\left|\bm{n}\right|$;

  * (ii)

There is a unique type I multiple orthogonal polynomial
$(A_{\bm{n}}^{(1)},\dots,A_{\bm{n}}^{(r)})$ such that

$\sum_{j=1}^{r}\left\langle
A_{\bm{n}}^{(j)}(x),x^{\left|\bm{n}\right|-1}\right\rangle_{j}=1.$ (34)

  * (iii)

$\deg P_{\bm{n}}=\left|\bm{n}\right|$ for every non-zero solution of (32);

  * (iv)

$\sum_{j=1}^{r}\left\langle
A_{\bm{n}}^{(j)}(x),x^{\left|\bm{n}\right|-1}\right\rangle_{j}\neq 0$ for
every non-zero solution of (33);

  * (v)

$\det M_{\bm{n}}\neq 0$, where

$M_{\bm{n}}=\begin{pmatrix}c_{0}^{1}&c_{1}^{1}&\cdots&c_{\left|\bm{n}\right|-1}^{1}\\\
c_{1}^{1}&c_{2}^{1}&\cdots&c_{\left|\bm{n}\right|}^{1}\\\
\vdots&\vdots&\ddots&\vdots\\\
c_{n_{1}-1}^{1}&c_{n_{1}}^{1}&\cdots&c_{\left|\bm{n}\right|+n_{1}-2}^{1}\\\
\hline\cr&&\vdots\\\ \hline\cr
c_{0}^{r}&c_{1}^{r}&\cdots&c_{\left|\bm{n}\right|-1}^{r}\\\
c_{1}^{r}&c_{2}^{r}&\cdots&c_{\left|\bm{n}\right|}^{r}\\\
\vdots&\vdots&\ddots&\vdots\\\
c_{n_{r}-1}^{r}&c_{n_{r}}^{r}&\cdots&c_{\left|\bm{n}\right|+n_{r}-2}^{r}\\\
\end{pmatrix},$ (35)

where $c_{n}^{j}$ are the moments

$\begin{split}c_{n}^{j}=\mu_{j}[x^{n}].\end{split}$

######  Definition 2.6.

If any and therefore each of the above conditions (i)–(iv) are satisfied for
an index $\bm{n}\neq\bm{0}$ then we say that $\bm{n}$ is normal. We always
take the index $\bm{0}$ to be normal.

######  Proposition 2.7.

If $\mu_{j}\in\mathcal{L}_{N_{j}}$ with $N_{j}<\infty$, then indices $\bm{n}$
with $n_{j}>N_{j}$ can never be normal.

###### Proof.

In the case $r=1$ this follows from Lemma 2.1. In general, if $n_{j}>N_{j}$,
we have

$\begin{split}\left\langle
P_{\bm{n}-\bm{e}_{j}}(x),x^{n_{j}-1}\right\rangle_{j}=\left\langle
P_{\bm{n}-\bm{e}_{j}}(x),Q(x)\right\rangle_{j}\end{split}$

for any degree $n_{j}-1$ monic polynomial $Q$, given any choice of type II
polynomial $P_{\bm{n}-\bm{e}_{j}}$ (for the index $\bm{n}-\bm{e}_{j}$). If we
choose $Q(x)=x^{n_{j}-1-N_{j}}P_{N_{j}}(x)$, where $P_{N_{j}}$ is the degree
$N_{j}$ orthogonal polynomials with respect to $\mu_{j}$, then we get

$\begin{split}\left\langle
P_{\bm{n}-\bm{e}_{j}}(x),x^{n_{j}-1}\right\rangle_{j}=\left\langle
P_{N_{j}}(x),P_{\bm{n}-\bm{e}_{j}}(x)x^{n_{j}-1-N_{j}}\right\rangle_{j}=0\end{split}$

by Lemma 2.1. This means that $P_{\bm{n}-\bm{e}_{j}}$ satisfies all the
orthogonality relations (32) for the index $\bm{n}$, but
$\deg{P_{\bm{n}-\bm{e}_{j}}}<\left|\bm{n}\right|$, so $\bm{n}$ cannot be
normal. ∎

A system $\bm{\mu}=(\mu_{1},\dots,\mu_{r})$ in $\mathcal{L}^{r}$ or in
$\mathcal{M}^{r}$ is usually said to be perfect if every
$\mathbb{N}^{r}$-index is normal. We modify this notion to the case when
$\mu_{j}\in\mathcal{L}_{N_{j}}$ where $N_{j}$ may be finite. We write
$\bm{N}=(N_{1},\ldots,N_{r})\in(\mathbb{Z}_{>0}\cup\\{+\infty\\})^{r}$ and
denote

$\mathbb{N}_{\bm{N}}^{r}:=\left\\{\bm{n}\in\mathbb{N}^{r}\mid 0\leq n_{j}\leq
N_{j}\right\\}.$

######  Definition 2.8.

We say that $(\mu_{1},\dots,\mu_{r})$ is a perfect system if all indices in
$\mathbb{N}_{\bm{N}}^{r}$ are normal.

One important class of perfect systems is Angelesco systems. Usually they are
defined for measures with infinite support, but we extend the notion of
Angelesco system to allow measures of finite support. If
$\bm{\mu}\in\mathcal{M}^{r}$ we write $\Delta_{j}$ for the convex hull of
$\operatorname{supp}(\mu_{j})$ and $\Delta_{j}^{\mathrm{o}}$ for the interior
of $\Delta_{j}$.

######  Definition 2.9.

An Angelesco system is a $\mathcal{M}^{r}$-system such that
$\Delta_{j}^{\mathrm{o}}\cap\Delta_{k}^{\mathrm{o}}=\emptyset$ and if
$\mu_{k}$ and $\mu_{j}$ are both finitely supported then we also require
$\Delta_{k}\cap\Delta_{j}=\emptyset$ ($k\neq j$).

With this definition one can show that any Angelesco system is perfect in the
sense of Definition 2.8. The proof goes along the same lines (counting real
zeros of odd multiplicity of $P_{\bm{n}}$) as the standard argument for the
case $\bm{N}=(\infty,\ldots,\infty)$, see e.g. [Ismail, Thm 23.1.3].

###  2.5. MOPRL: nearest neighbour recurrence relations

It was shown by Van Assche [NNR, Ismail] that multiple orthogonal polynomials
of type II and type I satisfy the following set of equations, called the
nearest neighbour recurrence relations (NNRR for short). Although it was
orinigally stated in $\mathcal{M}_{\infty}^{r}$, the proof is no different in
$\mathcal{L}^{r}$.

######  Theorem 2.10.

[Van Assche, [NNR]] Let $\bm{\mu}\in\mathcal{L}^{r}$. If $\bm{n}$ and
$\bm{n}+\bm{e}_{j}$ are normal, then

$\begin{split}xP_{\bm{n}}(x)=P_{\bm{n}+\bm{e}_{j}}(x)+b_{\bm{n},j}P_{\bm{n}}(x)+\sum_{i=1}^{r}a_{\bm{n},i}P_{\bm{n}-\bm{e}_{i}}(x)\end{split}$
(36)

for some constants $a_{\bm{n},1},\dots,a_{\bm{n},r}$ and $b_{\bm{n},j}$. Here
$a_{\bm{n},l}:=0$ whenever $n_{l}=0$.

######  Remark 2.11.

The recurrence coefficient $b_{\bm{n},j}$ is defined when both $\bm{n}$ and
$\bm{n}+\bm{e}_{j}$ are normal, and it is always unique. The coefficient
$a_{\bm{n},i}$ is defined when $\bm{n}$ is normal and $\bm{n}+\bm{e}_{l}$ is
normal for some $l$. $a_{\bm{n},i}$ is unique if $\bm{n}-\bm{e}_{i}$ is also
normal. Indeed, (36) implies

$a_{\bm{n},i}=\frac{\left\langle
P_{\bm{n}}(x),x^{n_{i}}\right\rangle_{i}}{\left\langle
P_{\bm{n}-\bm{e}_{i}}(x),x^{n_{i}-1}\right\rangle_{i}}$ (37)

if $n_{i}>0$. Therefore the constants $a_{\bm{n},i}$ are independent of the
choice of $j$ in (36) (i.e., if $\bm{n}+\bm{e}_{j}$ are normal for several
choices of $j$), but may depend on the choice of
$P_{\bm{n}-\bm{e}_{1}},\dots,P_{\bm{n}-\bm{e}_{r}}$ if any of the indices
$\bm{n}-\bm{e}_{1},\ldots,\bm{n}-\bm{e}_{r}$ are not normal.

######  Remark 2.12.

In particular, for perfect systems $\bm{\mu}\in\mathcal{L}^{r}$ with
$N_{j}\leq\infty$, we have that $a_{\bm{n},j}$ is defined for all $1\leq j\leq
r$ and all $\bm{n}\in\mathbb{N}^{r}_{\bm{N}}$ except for $\bm{n}=\bm{N}$. This
exception is only relevant if $N_{j}<\infty$ for all $j$, of course. Let us
adopt the convention $a_{\bm{N},j}:=0$ for that case, for reasons that will
become more clear later on in the paper. Assuming this, all perfect systems
then have well-defined NNRR coefficiens
$\\{a_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}}}$ and
$\\{b_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}-\bm{e}_{j}}}$ for each
$j$.

It is well-known that the type I polynomials satisfy similar recurrence
relations. Since the explicit proof is hard to locate we provide a proof in
Section 2.7, using minimal normality assumptions.

######  Theorem 2.13.

Let $\bm{\mu}\in\mathcal{L}^{r}$. If $\bm{n}$ and $\bm{n}-\bm{e}_{j}$ are
normal, then

$\begin{split}x\bm{A}_{\bm{n}}(x)=\bm{A}_{\bm{n}-\bm{e}_{j}}(x)+b_{\bm{n}-\bm{e}_{j},j}\bm{A}_{\bm{n}}(x)+\sum_{i=1}^{r}c_{\bm{n},i}\bm{A}_{\bm{n}+\bm{e}_{i}}(x),\end{split}$
(38)

for some constants $c_{\bm{n},1},\dots,c_{\bm{n},r}$. If $\bm{n}+\bm{e}_{i}$
and $\bm{n}-\bm{e}_{i}$ are normal then $c_{\bm{n},i}=a_{\bm{n},i}$.

######  Remark 2.14.

As above, $c_{\bm{n},i}$ are independent of the choice of $j$, but may depend
on the choice of the vectors
$(A_{\bm{n}+\bm{e}_{i},1},\dots,A_{\bm{n}+\bm{e}_{i},r})$ when
$\bm{n}+\bm{e}_{i}$ is not normal (for the details, see the Appendix).

By comparing (36) for two different $j$, and doing the same for (38), we
obtain the following.

######  Corollary 2.15.

Assume $\bm{n}$, $\bm{n}+\bm{e}_{j}$ and $\bm{n}+\bm{e}_{l}$ are normal. Then

$P_{\bm{n}+\bm{e}_{l}}-P_{\bm{n}+\bm{e}_{j}}=(b_{\bm{n},j}-b_{\bm{n},l})P_{\bm{n}}.$
(39)

Similarly, if $\bm{n}$, $\bm{n}-\bm{e}_{j}$ and $\bm{n}-\bm{e}_{l}$ are
normal, then

$\bm{A}_{\bm{n}-\bm{e}_{l}}-\bm{A}_{\bm{n}-\bm{e}_{j}}=(b_{\bm{n}-\bm{e}_{j},j}-b_{\bm{n}-\bm{e}_{l},l})\bm{A}_{\bm{n}}$
(40)

Van Assche showed in [NNR] that (assuming sufficient normality conditions) the
recurrence coefficients satisfy the partial difference equations in the
theorem below. Their proof works for $\bm{\mu}\in\mathcal{L}^{r}$ without any
changes. When we write $\bm{m}\leq\bm{n}$ we mean $m_{j}\leq n_{j}$ for each
$j=1,\dots,r$.

######  Theorem 2.16.

[Van Assche, [NNR]] Let $\bm{\mu}\in\mathcal{L}^{r}$. Assume all indices
$\bm{m}\leq\bm{n}+\bm{e}_{j}+\bm{e}_{l}$ is normal. If $j\neq l$ then

$\displaystyle
b_{\bm{n}+\bm{e}_{l},j}-b_{\bm{n}+\bm{e}_{j},l}=b_{\bm{n},j}-b_{\bm{n},l},$
(41) $\displaystyle
b_{\bm{n},l}b_{\bm{n}+\bm{e}_{l},j}-b_{\bm{n},j}b_{\bm{n}+\bm{e}_{j},l}=\sum_{i=1}^{r}a_{\bm{n}+\bm{e}_{l},i}-\sum_{i=1}^{r}a_{\bm{n}+\bm{e}_{j},i},$
(42) $\displaystyle
a_{\bm{n}+\bm{e}_{l},j}\left(b_{\bm{n}-\bm{e}_{j},j}-b_{\bm{n}-\bm{e}_{j},l}\right)=a_{\bm{n},j}\left(b_{\bm{n},j}-b_{\bm{n},l}\right).$
(43)

If $n_{j}=0$ then we only get the first two equations.

We can use (41) to rewrite the left hand side of (42) as

$\begin{split}b_{\bm{n},l}b_{\bm{n}+\bm{e}_{l},j}-b_{\bm{n},j}b_{\bm{n}+\bm{e}_{j},l}&=b_{\bm{n},l}b_{\bm{n}+\bm{e}_{l},j}-b_{\bm{n},j}(b_{\bm{n}+\bm{e}_{l},j}+b_{\bm{n},l}-b_{\bm{n},j})\\\
&=(b_{\bm{n},l}-b_{\bm{n},j})b_{\bm{n}+\bm{e}_{l},j}-(b_{\bm{n},l}-b_{\bm{n},j})b_{\bm{n},j}.\end{split}$

We write $\delta_{\bm{n},j,l}$ for $b_{\bm{n},l}-b_{\bm{n},j}$. Now we can
rewrite (41)–(43) in the alternative form,

$\displaystyle
b_{\bm{n}+\bm{e}_{l},j}-\delta_{\bm{n}+\bm{e}_{j},j,l}=b_{\bm{n}+\bm{e}_{j},j}-\delta_{\bm{n},j,l},$
(44)
$\displaystyle\sum_{i=1}^{r}a_{\bm{n}+\bm{e}_{l},i}-\delta_{\bm{n},j,l}b_{\bm{n}+\bm{e}_{l},j}=\sum_{i=1}^{r}a_{\bm{n}+\bm{e}_{j},i}-\delta_{\bm{n},j,l}b_{\bm{n},j},$
(45)
$\displaystyle\delta_{\bm{n}-\bm{e}_{j},j,l}a_{\bm{n}+\bm{e}_{l},j}=\delta_{\bm{n},j,l}a_{\bm{n},j},$
(46)

which will be better adapted for our purposes.

Consider a sequence of normal indices
$\left\\{\bm{n}_{k}\right\\}_{k=0}^{\infty}$ that forms a path starting at the
origin, i.e., $\bm{n}_{0}=\bm{0}$ and $\bm{n}_{k+1}=\bm{n}_{k}+\bm{e}_{j_{k}}$
for some $j_{k}\in\left\\{1,\dots,r\right\\}$, $k\in\mathbb{N}$. Define the
generalized Christoffel–Darboux kernel
$\bm{K}_{\bm{n}}(x,y)=(K_{\bm{n}}^{(1)}(x,y),\dots,K_{\bm{n}}^{(r)}(x,y))$ via

$\bm{K}_{\bm{n}}(x,y)=\sum_{k=0}^{N-1}P_{\bm{n}_{k}}(x)\bm{A}_{\bm{n}_{k+1}}(y).$
(47)

Using the NNRR one can show [NNR] that the following Christoffel–Darboux
formula holds

$(x-y)\bm{K}_{\bm{n}}(x,y)=P_{\bm{n}_{N}}(x)\bm{A}_{\bm{n}_{N}}(y)-\sum_{j=1}^{r}a_{\bm{n},j}P_{\bm{n}_{N}-\bm{e}_{j}}(x)\bm{A}_{\bm{n}_{N}+\bm{e}_{j_{k}}}(y),$
(48)

originally due to [DK04].

###  2.6. Normality criteria

Here we collect a series of results that allow to establish normality of
indices for certain multiple orthogonality systems
$\bm{\mu}\in\mathcal{L}^{r}$. We start with the following simple lemma. One
direction is standard, see, e.g., [Ismail, Cor 23.1.1–23.1.2], while the other
has appeared in [mopucrecurrence] (for the unit circle), see also [KVMOPUC].

######  Lemma 2.17.

$\bm{n}+\bm{e}_{j}$ is normal if and only if

$\begin{split}\left\langle P_{\bm{n}}(x),x^{n_{j}}\right\rangle_{j}\neq
0\end{split}$ (49)

for every type II multiple orthogonal polynomial $P_{\bm{n}}$ corresponding to
the index $\bm{n}$.

Similarly, $\bm{n}-\bm{e}_{j}$ is normal if and only if $A_{\bm{n}}^{(j)}$ has
degree $n_{j}-1$ for every vector of type I multiple orthogonal polynomials
$(A_{\bm{n}}^{(1)},\dots,A_{\bm{n}}^{(r)})$.

###### Proof.

If $\left\langle P_{\bm{n}}(x),x^{n_{j}}\right\rangle_{j}$ is zero for some
multiple orthogonal polynomial $P_{\bm{n}}$, then $P_{\bm{n}}$ satisfies every
orthogonality condition for the index $\bm{n}+\bm{e}_{j}$, but $P_{\bm{n}}$
has degree $\leq\left|\bm{n}\right|$, so $\bm{n}+\bm{e}_{j}$ cannot be normal.
Conversely, if $\bm{n}+\bm{e}_{j}$ is not normal then (32) has a solution
$P_{\bm{n}+\bm{e}_{j}}$ of degree $\leq\left|\bm{n}\right|$.
$P_{\bm{n}+\bm{e}_{j}}$ satisfies the orthogonality conditions (32) for the
index $\bm{n}$, so it is a multiple orthogonal polynomial for the index
$\bm{n}$. In other words, $P_{\bm{n}+\bm{e}_{j}}=P_{\bm{n}}$ for some choice
of $P_{\bm{n}+\bm{e}_{j}}$ and $P_{\bm{n}}$, but then (32) for the index
$\bm{n}+\bm{e}_{j}$ implies $\left\langle
P_{\bm{n}}(x),x^{n_{j}}\right\rangle_{j}=0$.

If there is some $A_{\bm{n}}^{(j)}$ of degree $<n_{j}-1$ then we have a non-
zero solution to the system (33) for the index $\bm{n}-\bm{e}_{j}$, so
$\bm{n}-\bm{e}_{j}$ is not normal. Conversely, if $\bm{n}-\bm{e}_{j}$ is not
normal then there is a non-zero solution
$(A_{\bm{n}-\bm{e}_{j}}^{(1)},\dots,A_{\bm{n}-\bm{e}_{j}}^{(r)})$ to

$\begin{split}\sum_{i=1}^{r}\left\langle
A_{\bm{n}-\bm{e}_{j}}^{(i)}(x),x^{p}\right\rangle_{i}=0,\qquad
p=0,1,\dots,\left|\bm{n}\right|-2,\end{split}$

but then this is also a solution for the index $\bm{n}$, which means that
there is an $A_{\bm{n},j}$ of degree $<n_{j}-1$. ∎

In order to define the recurrence coefficients we required some indices to be
normal. In turn, the recurrence coefficients can give us information about the
normality of neighbouring indices, as we show next.

######  Lemma 2.18.

The following holds true for the recurrence coefficients of a system
$\mu\in\mathcal{L}^{r}$.

  1. a)

Let $\bm{n}$ and $\bm{n}+\bm{e}_{i}$ be normal for some $i$ and $n_{j}>0$.
Then $a_{\bm{n},j}\neq 0$ if and only if $\bm{n}+\bm{e}_{j}$ is normal.

  2. b)

Let $\bm{n}$, $\bm{n}+\bm{e}_{j}$ and $\bm{n}+\bm{e}_{l}$ be normal for some
$j\neq l$. Then $b_{\bm{n},j}\neq b_{\bm{n},l}$ if and only if
$\bm{n}+\bm{e}_{j}+\bm{e}_{l}$ is normal.

###### Proof.

a) follows by Lemma 2.17 and (37). To show b), take the $j$-th inner product
with respect to $x^{n_{j}}$ in (39) to get

$\begin{split}b_{\bm{n},j}-b_{\bm{n},l}=\frac{\left\langle
P_{\bm{n}+\bm{e}_{l}}(x),x^{n_{j}}\right\rangle_{j}}{\left\langle
P_{\bm{n}}(x),x^{n_{j}}\right\rangle{{}_{j}}}.\end{split}$ (50)

Then b) follows from Lemma 2.17. ∎

To check perfectness of certain systems the following two results may be
useful. Note that direct application of Lemma 2.17 would require to check the
condition (49) for every orthogonal polynomial at every index $\bm{n}$. We
show that it is enough to check it for only one choice of orthogonal
polynomial at every index $\bm{n}$. This is useful since occasionally finding
one such choice is easy from a Rodrigues-type formula or other arguments. One
such application can be found Section 3.1 below.

######  Theorem 2.19.

$\bm{\mu}\in\mathcal{L}^{r}$ is a perfect system if and only if there is a
choice of type II multiple orthogonal polynomials $P_{\bm{n}}$ for each
$\bm{n}\in\mathbb{N}^{r}_{\bm{N}}$, such that

${\left\langle P_{\bm{n}}(x),x^{n_{j}}\right\rangle_{j}\neq 0}$ (51)

whenever $\bm{n}\in\mathbb{N}^{r}_{\bm{N}-\bm{e}_{j}}$.

###### Proof.

Necessity of (51) is trivial from Lemma 2.17. Let us now show sufficiency.
Note that (51) is only imposed on one choice of $P_{\bm{n}}$ so this direction
is not immediate from Lemma 2.17. We use induction on $N=\left|\bm{n}\right|$
to prove that $\bm{n}$ is normal whenever $0\leq n_{j}\leq N_{j}$. The case
$N=0$ is obvious. If $N>0$, assume $\bm{n}$ is normal whenever
$\left|\bm{n}\right|<N$ and take $\bm{m}=(m_{1},\dots,m_{r})$ such that
$\left|\bm{m}\right|=N$. There is some $j$ such that
$\bm{m}=\bm{n}+\bm{e}_{j}$. $\bm{n}$ is normal by assumption, and

$\begin{split}\left\langle P_{\bm{n}}(x),x^{n_{j}}\right\rangle_{j}\neq
0,\end{split}$

so $\bm{n}+\bm{e}_{j}$ is normal by Lemma 2.17. ∎

######  Remark 2.20.

In fact the same proof shows that all indices along any path
$\\{\bm{n}_{l}\\}_{l=0}^{m}$ (where $m$ may be infinite) with
$\bm{n}_{0}=\bm{0}$ and $\bm{n}_{l+1}=\bm{n}_{l}+\bm{e}_{j_{l}}$, where $1\leq
j_{l}\leq r$, are normal if and only if

$\left\langle
P_{\bm{n}_{l}}(x),x^{(\bm{n}_{l})_{j_{l}}}\right\rangle_{j_{l}}\neq 0,\qquad
l=0,\dots,m-1,$ (52)

for a choice of type II multiple orthogonal polynomials $P_{\bm{n}_{l}}$.

######  Theorem 2.21.

$\bm{\mu}\in\mathcal{L}^{r}$ is a perfect system if and only if there is a
choice of type II multiple orthogonal polynomials $P_{\bm{n}}$ for each
$\bm{n}\in\mathbb{N}^{r}_{\bm{N}}$, such that for each
$\bm{n}\in\mathbb{N}^{r}_{\bm{N}-\bm{e}_{j}-\bm{e}_{l}}$

$\begin{split}P_{\bm{n}+\bm{e}_{l}}-P_{\bm{n}+\bm{e}_{j}}=c_{\bm{n},j,l}P_{\bm{n}}\end{split}$
(53)

for some constants $c_{\bm{n},j,l}\neq 0$ ($j\neq l$).

###### Proof.

We prove that $\bm{n}$ is normal by induction on $M=\left|\bm{n}\right|$. The
cases $M=0$ and $M=1$ are obvious. For $M>1$, assume $\bm{n}$ is normal
whenever $\left|\bm{n}\right|<M$ and take $\bm{m}=(m_{1},\dots,m_{r})$ such
that $\left|\bm{m}\right|=M$. Note that we know the normality in the case
$\bm{m}=m\bm{e}_{i}$, so assume $m_{j}>0$ and $m_{l}>0$ for some $j$ and $k$
with $j\neq l$. Then $\bm{n}=\bm{m}-\bm{e}_{j}-\bm{e}_{l}$ is normal, as well
as $\bm{n}+\bm{e}_{j}=\bm{m}-\bm{e}_{l}$ and
$\bm{n}+\bm{e}_{l}=\bm{m}-\bm{e}_{j}$. Hence by Corollary 2.15 we get

$\begin{split}(b_{\bm{n},j}-b_{\bm{n},l})P_{\bm{n}}=P_{\bm{n}+\bm{e}_{l}}-P_{\bm{n}+\bm{e}_{j}}=c_{\bm{n},j,l}P_{\bm{n}},\end{split}$

so we must have $b_{\bm{n},j}-b_{\bm{n},l}=c_{\bm{n},j,l}\neq 0$, which
implies that $\bm{m}=\bm{n}+\bm{e}_{j}+\bm{e}_{l}$ is normal, by Lemma 2.18
b). ∎

###  2.7. Appendix: Proof of Theorem 2.13

######  Lemma 2.22.

Suppose $\bm{n}$ is normal. Then the vectors
$\left\\{(A_{\bm{n}+\bm{e}_{i},1},\dots,A_{\bm{n}+\bm{e}_{i},r})\right\\}_{i=1}^{r}$
are linearly independent.

###### Proof.

Consider the equation

$\begin{split}\sum_{i=1}^{r}c_{i}(A_{\bm{n}+\bm{e}_{i},1},\dots,A_{\bm{n}+\bm{e}_{i},r})=0.\end{split}$

Since $\bm{n}$ is normal $A_{\bm{n}+\bm{e}_{j},j}$ has degree $n_{j}$, by
Lemma 2.17, but $A_{\bm{n}+\bm{e}_{i},j}$ has degree $<n_{j}$ when $i\neq j$,
so $c_{j}=0$, which proves linear independence. ∎

######  Remark 2.23.

The type II equivalent of Lemma 2.22 is the linear independence of
$P_{\bm{n}-\bm{e_{1}}},\dots,P_{\bm{n}-\bm{e}_{r}}$, which also follows from
Lemma 2.17 (see [DK04, Ismail]).

###### Proof of Theorem 2.13.

We have

$\begin{split}\sum_{k=1}^{r}\left\langle
xA_{\bm{n},k}(x)-A_{\bm{n}-\bm{e}_{j},k},x^{p}\right\rangle=0,\qquad
p=0,1,\dots,\left|\bm{n}\right|-2.\end{split}$

Now choose $d_{\bm{n},j}$ such that

$\begin{split}\sum_{k=1}^{r}\left\langle
xA_{\bm{n},k}(x)-A_{\bm{n}-\bm{e}_{j},k}-d_{\bm{n},j}A_{\bm{n},k},x^{p}\right\rangle=0,\qquad
p=0,1,\dots,\left|\bm{n}\right|-1.\end{split}$

The polynomials
$B_{\bm{n},k}=xA_{\bm{n},k}(x)-A_{\bm{n}-\bm{e}_{j},k}-d_{\bm{n},j}A_{\bm{n},k}$
have degree $n_{k}$ the above system of orthogonality relations. This system
is homogeneous with matrix $M_{\bm{n}}^{t}$ with $r$ columns added, where
$M_{\bm{n}}$ is given by (15). This matrix has nullity $r$, since
$M_{\bm{n}}^{t}$ has nullity $0$, by normality. Hence the solution space is of
dimension $r$, and
$(B_{\bm{n},1},\dots,B_{\bm{n},r})=(A_{\bm{n}+\bm{e}_{i},1},\dots,A_{\bm{n}+\bm{e}_{i},r})$
for each $i=1,\dots,r$ are solutions. By Lemma 2.22 every solution is thus on
the form

$\begin{split}B_{\bm{n},k}=\sum_{i=1}^{r}c_{\bm{n},k}A_{\bm{n}+\bm{e}_{i},k}.\end{split}$

In other words we have the recurrence relation

$\begin{split}xA_{\bm{n},k}(x)=A_{\bm{n}-\bm{e}_{j},k}+d_{\bm{n},j}A_{\bm{n},k}+\sum_{i=1}^{r}c_{\bm{n},k}A_{\bm{n}+\bm{e}_{i},k}.\end{split}$

To show that $d_{\bm{n},j}=b_{\bm{n}-\bm{e}_{j},j}$, we compare with the
recurrence relation

$\begin{split}xP_{\bm{n}-\bm{e}_{j}}(x)=P_{\bm{n}}(x)+b_{\bm{n}-\bm{e}_{j},j}P_{\bm{n}-\bm{e}_{j}}(x)+\sum_{i=1}^{r}a_{\bm{n}-\bm{e}_{j},i}P_{\bm{n}-\bm{e}_{j}-\bm{e}_{i}}(x).\end{split}$

From here we can write

$\begin{split}\sum_{k=1}^{r}\left\langle
xA_{\bm{n},k}(x),P_{\bm{n}-\bm{e}_{j}}(x)\right\rangle&=\sum_{k=1}^{r}\left\langle
A_{\bm{n},k}(x),P_{\bm{n}}(x)\right\rangle\\\
&+b_{\bm{n}-\bm{e}_{j},j}\sum_{k=1}^{r}\left\langle
A_{\bm{n},k}(x),P_{\bm{n}-\bm{e}_{j}}(x)\right\rangle\\\
&\sum_{i=1}^{r}a_{\bm{n}-\bm{e}_{j},i}\sum_{k=1}^{r}\left\langle
A_{\bm{n},k}(x),P_{\bm{n}-\bm{e}_{j}-\bm{e}_{i}}(x)\right\rangle.\end{split}$

The first term vanishes by the orthogonality relations of $P_{\bm{n}}$, and
the last term vanishes by the orthogonality relations of $\bm{A}_{\bm{n}}$.
For the second term, we have

$\begin{split}\sum_{k=1}^{r}\left\langle
xA_{\bm{n},k}(x),P_{\bm{n}-\bm{e}_{j}}(x)\right\rangle=b_{\bm{n}-\bm{e}_{j},j}.\end{split}$

On the other hand, if we instead apply a similar argument using the type I
recurrence relation we end up with

$\begin{split}\sum_{k=1}^{r}\left\langle
xA_{\bm{n},k}(x),P_{\bm{n}-\bm{e}_{j}}(x)\right\rangle=d_{\bm{n},j},\end{split}$

so $d_{\bm{n},j}=b_{\bm{n}-\bm{e}_{j},j}$.

Write $\kappa_{\bm{n},j}$ for the degree $n_{j}-1$ coefficient of
$A_{\bm{n},j}$, so that
$A_{\bm{n},j}(x)=\kappa_{\bm{n},j}x^{n_{j}-1}+o(x^{n_{j}-1})$. By comparing
the degree $n_{i}$ coefficients in (38) we see that
$\kappa_{\bm{n},i}=c_{\bm{n},i}\kappa_{\bm{n}+\bm{e}_{i},i}$. By Lemma 2.17 we
know that $\kappa_{\bm{n}+\bm{e}_{i},i}\neq 0$, so we get

$\begin{split}c_{\bm{n},i}=\frac{\kappa_{\bm{n},i}}{\kappa_{\bm{n}+\bm{e}_{i},i}},\end{split}$

which shows independence of $j$.

What remains is to show that $c_{\bm{n},i}=a_{\bm{n},i}$. Note that

$\begin{split}\sum_{k=1}^{r}\left\langle
A_{\bm{n},k}(x),P_{\bm{n}-\bm{e}_{i}}(x)\right\rangle=1,\end{split}$

by the orthogonality relations of $\bm{A}_{\bm{n}}$, assuming
$\bm{n}-\bm{e}_{i}$ is normal. By the orthogonality relations of
$P_{\bm{n}-\bm{e}_{i}}$ we instead get

$\begin{split}\sum_{k=1}^{r}\left\langle
A_{\bm{n},k}(x),P_{\bm{n}-\bm{e}_{i}}(x)\right\rangle&=\left\langle
A_{\bm{n},i}(x),P_{\bm{n}-\bm{e}_{i}}(x)\right\rangle\\\
&=\kappa_{\bm{n},i}\left\langle
x^{n_{i}-1},P_{\bm{n}-\bm{e}_{i}}(x)\right\rangle.\end{split}$

Similarly, if $\bm{n}+\bm{e}_{i}$ is normal then

$\begin{split}\kappa_{\bm{n}+\bm{e}_{i},i}\left\langle
x^{n_{i}},P_{\bm{n}}(x)\right\rangle=1,\end{split}$

so we get

$\begin{split}a_{\bm{n},i}=\frac{\left\langle
x^{n_{i}},P_{\bm{n}}(x)\right\rangle}{\left\langle
x^{n_{i}-1},P_{\bm{n}-\bm{e}_{i}}(x)\right\rangle}=\frac{\kappa_{\bm{n},i}}{\kappa_{\bm{n}+\bm{e}_{i},i}}=c_{\bm{n},i},\end{split}$

which completes the proof. ∎

##  3\. Main results

Let us setup the following notation for the remainder of this paper. We write

$\displaystyle\bm{\mu}$
$\displaystyle=(\mu_{1},\dots,\mu_{r})\in\mathcal{L}^{r},$
$\displaystyle\bm{\nu}$
$\displaystyle=(\mu_{1},\ldots,\mu_{r-1})\in\mathcal{L}^{r-1},$

that is, $\bm{\nu}$ is $\bm{\mu}$ with $\mu_{r}$ removed, where $r\geq 2$.

As usual we allow $\mu_{j}\in\mathcal{L}_{N_{j}}$ with $N_{j}\leq\infty$, and
denote $\bm{N}=(N_{1},\ldots,N_{r})$. We use $\bm{n}$ for a multi-index in
$\mathbb{N}^{r}_{\bm{N}}$.

We use $\bm{k}=(k_{1},\ldots,k_{r-1})$ for a multi-index in
$\mathbb{N}^{r-1}_{\bm{K}}$, where we denote $\bm{K}=(N_{1},\ldots,N_{r-1})$.

Finally, it is clear that the type II multiple orthogonal polynomials for
$\bm{\nu}$ at a location $\bm{k}$ coincide with type II multiple orthogonal
polynomials for $\bm{\mu}$ at the location $(\bm{k},0)=:\bm{n}$. So we use the
notation $P_{\bm{n}}=P_{(\bm{k},0)}=P_{\bm{k}}$ interchangeably, as it cannot
lead to any ambiguity.

###  3.1. Christoffel Transforms of Type II Polynomials

Let $\mu_{r}\in\mathcal{L}_{m}$ with $m=N_{r}<\infty$, and write
$\Phi(x)=\prod_{j=1}^{m}(x-z_{j})$ for the unique monic orthogonal polynomial
of degree $m$ with respect to $\mu_{r}$. For example, if all $z_{j}$’s are
distinct (which happens if $\mu_{r}\in\mathcal{M}_{m}\subset\mathcal{L}_{m}$),
then $\mu_{r}$ is of the form

$\sum_{j=1}^{m}w_{j}\delta_{z_{j}}$ (54)

where $w_{j}\in\mathbb{C}\setminus\left\\{0\right\\}$ for each $j=1,\dots,m$.
But in general $z_{j}$’s may overlap leading to more complicated functionals,
see Remark 2.3.

Consider the Christoffel transforms $\widehat{\mu}_{j}=\Phi\mu_{j}$ for $1\leq
j\leq r-1$, as in (25). We refer to
$\widehat{\bm{\nu}}=\Phi\bm{\nu}=(\widehat{\mu}_{1},\ldots,\widehat{\mu}_{r-1})$
as the Christoffel transform of $\bm{\nu}=(\mu_{1},\ldots,\mu_{r-1})$. We
write $P_{\bm{k}}$ for the type II polynomials with respect to $\bm{\nu}$ and
$\widehat{P}_{\bm{k}}$ for the type II polynomials with respect to
$\widehat{\bm{\nu}}$, $\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}}$. Let
$\\{a_{\bm{k},j},b_{\bm{k},j}\\}$ denote the recurrence coefficients of
$\bm{\nu}$, and $\\{\widehat{a}_{\bm{k},j},\widehat{b}_{\bm{k},j}\\}$ denote
the recurrence coefficients of $\widehat{\bm{\nu}}$. The connection between
the systems $\widehat{\bm{\nu}}$ and $\bm{\mu}=(\bm{\nu},\mu_{r})$ is
summarized by the following result.

######  Theorem 3.1.

An index $(\bm{k},m)$ is normal for the system $\bm{\mu}=(\bm{\nu},\mu_{r})$
if and only if $\bm{k}$ is normal for the system $\widehat{\bm{\nu}}$. In that
case we have $P_{(\bm{k},m)}(x)=\Phi(x)\widehat{P}_{\bm{k}}(x)$.

###### Proof.

Suppose $(\bm{k},m)$ is normal. By the orthogonality relations

$\begin{split}\left\langle P_{(\bm{k},m)}(x),x^{p}\right\rangle_{r}=0,\qquad
p=0,1,\dots,m-1,\end{split}$ (55)

we must have $P_{(\bm{k},m)}(x)=\Phi(x)Q(x)$ for some polynomial $Q$ with
$\deg{Q}\leq\left|\bm{k}\right|$, by Lemma 2.1. For the other inner products
we then have

$\begin{split}\left\langle\Phi(x)Q(x),x^{p}\right\rangle_{j}=0,\qquad
p=0,1,\dots,k_{j}-1,\qquad j=1,\dots,r-1.\end{split}$ (56)

Hence $Q$ must be a multiple orthogonal polynomial with respect to
$\widehat{\bm{\nu}}$. Conversely, any multiple orthogonal polynomial
$Q=\widehat{P}_{\bm{k}}$ with respect to $\widehat{\bm{\nu}}$ satisfies (56)
by definition. (55) is then satisfied with $P_{(\bm{k},m)}(x)$ replaced by
$\Phi(x)\widehat{P}_{\bm{k}}(x)$, by Lemma 2.1. Now, $(\bm{k},m)$ is normal
for $(\bm{\nu},\mu_{r})$ if and only if every $P_{(\bm{k},m)}$ has degree
$\left|\bm{k}\right|+m$, if and only if every $\widehat{P}_{\bm{k}}$ has
degree $\left|\bm{k}\right|$, if and only if $\bm{k}$ is normal for
$\widehat{\bm{\nu}}$. ∎

Now we find the determinantal formula, together with a necessary and
sufficient condition on the perfectness of $\widehat{\bm{\nu}}$.

######  Theorem 3.2.

Suppose that $\bm{\nu}\in\mathcal{L}_{\infty}^{r-1}$ is a perfect system, and
all $\\{z_{j}\\}_{j=1}^{m}$ are distinct.

  * i)

If $(\bm{k},m)$ is normal, then for any sequence of $\mathbb{N}^{r-1}$-indices
$\left\\{\bm{s}_{j}\right\\}_{j=0}^{m}$ with $\bm{s}_{0}=0$ and
$\left|\bm{s}_{j}\right|=j$, we have
$D_{\bm{k}}=\det(P_{\bm{k}+\bm{s}_{m-i}}(z_{j}))_{i,j=1}^{m}\neq 0$ and the
determinantal formula

$\begin{split}\widehat{P}_{\bm{k}}(x)=\Phi(x)^{-1}D_{\bm{k}}^{-1}\det\begin{pmatrix}P_{\bm{k}+\bm{s}_{m}}(x)&P_{\bm{k}+\bm{s}_{m-1}}(x)&\cdots&P_{\bm{k}}(x)\\\
P_{\bm{k}+\bm{s}_{m}}(z_{1})&P_{\bm{k}+\bm{s}_{m-1}}(z_{1})&\cdots&P_{\bm{k}}(z_{1})\\\
\vdots&\vdots&\ddots&\vdots\\\
P_{\bm{k}+\bm{s}_{m}}(z_{m})&P_{\bm{k}+\bm{s}_{m-1}}(z_{m})&\cdots&P_{\bm{k}}(z_{m})\end{pmatrix},\\\
\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}}.\end{split}$ (57)

  * ii)

If $D_{\bm{k}}=\det(P_{\bm{k}+\bm{s}_{m-i}}(z_{j}))_{i,j=1}^{m}\neq 0$ for any
sequence of $\mathbb{N}^{r-1}$-indices $\left\\{\bm{s}\right\\}_{j=0}^{m}$
with $\bm{s}_{0}=0$ and $\left|\bm{s}_{j}\right|=j$, then $\widehat{\bm{\nu}}$
is perfect.

###### Proof.

i) If $D_{\bm{n}}=0$ then the columns of
$(P_{\bm{k}+\bm{s}_{m-i}}(z_{j}))_{i,j=1}^{m}$ are linearly dependent. This
would imply

$\begin{split}P(x)=\sum_{j=0}^{m-1}c_{j}P_{\bm{k}+\bm{s}_{j}}(x)=0,\qquad
x=z_{1},\dots,z_{r},\end{split}$

for some $(c_{1},\dots,c_{r})\neq(0,\dots,0)$. However, since we have a linear
combination of polynomials with different degrees, $P$ cannot be identically
$0$. $P$ satisfies the orthogonality conditions at the index $(\bm{k},m)$ with
respect to $\mu_{j}$ with $1\leq j\leq r-1$ since
$\bm{k}\leq\bm{k}+\bm{s}_{i}$ for every $i=0,\dots,m-1$, and with respect to
$\mu_{r}$ by Lemma 2.1 since $P$ is divisible by $\Phi$. Since
$\deg{P}\leq\left|\bm{k}\right|+m-1$ this would contradict the normality of
$(\bm{k},m)$.

The determinant in (57) is a polynomial of degree $\left|\bm{k}\right|+m$. It
is a linear combination of $P_{\bm{k}},\dots,P_{\bm{k}+\bm{s}_{m}}$, and
$\bm{k}\leq\bm{k}+\bm{s}_{i}$, so it is orthogonal to $x^{p}$ with respect to
$\mu_{j}$ for $p=0,\dots,n_{j}-1$, $j=1,\dots,r-1$. Since it vanishes at
$z_{1},\dots,z_{m}$ it is orthogonal to everything with respect to $\mu_{r}$,
by Lemma 2.1. Hence the normality of $(\bm{k},m)$ implies that
$P_{(\bm{k},m)}$ coincides with the determinant in (57) divided by the
normalization constant $D_{\bm{n}}$. Finally, Theorem 3.1 implies (57).

ii) Conversely, assume $\bm{\nu}$ is perfect and $D_{\bm{k}}\neq 0$ for any
choice of path and polynomials for all $\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}}$.
Fix any path $\left\\{\bm{k}_{l}\right\\}_{k=0}^{M}$ of multi-indices in
$\mathbb{N}^{r-1}_{\bm{K}}$ with $\bm{k}_{0}=\bm{0}$ and
$\bm{k}_{l+1}=\bm{k}_{l}+\bm{e}_{j_{l}}$. For each
$\bm{k}_{l}\in\mathbb{N}_{\bm{K}}^{r-1}$, denote $Q_{\bm{k}_{l}}$ to be the
polynomial on the right-hand side of (57), where we choose $\bm{s}_{j}$ in
such a way that $\bm{k}+\bm{s}_{j}$ for each $j$ belongs to the chosen path
$\left\\{\bm{k}_{l}\right\\}_{l=0}^{M}$. It is clear that $Q_{\bm{k}_{l}}$ is
monic of degree $\left|\bm{k}_{l}\right|$ and satisfies all the type II
orthogonality conditions for $\widehat{\bm{\nu}}$. Let us show that
$Q_{\bm{k}_{l}}$ satisfy the conditions of Remark 2.20 with respect to the
system $\widehat{\nu}$. Indeed, by (57),

$\left\langle\Phi(x)Q_{\bm{k}_{l}}(x),x^{(\bm{k}_{l})_{j_{l}}}_{j_{l}}\right\rangle=\frac{(-1)^{m}D_{\bm{k}_{l+1}}}{D_{\bm{k}_{l}}}\left\langle
P_{\bm{k}_{l}}(x),x^{(\bm{k}_{l})_{j_{l}}}\right\rangle_{j_{l}},$ (58)

which is non-zero by the assumptions and Lemma 2.17 applied to $\bm{\nu}$.
Hence by Remark 2.20 every index along the path
$\left\\{\bm{k}_{l}\right\\}_{l=0}^{M}$ is normal for $\widehat{\bm{\nu}}$,
and by taking different paths that cover all
$\mathbb{N}_{\bm{K}}^{r-1}$-indices, we see that $\widehat{\bm{\nu}}$ is
perfect.

∎

$\mu_{2}$$\mu_{1}$$\widehat{\mu}_{1}$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$
Figure 1. The recurrence coefficients of $(\mu_{1},\mu_{2})$ can be computed
via the CC algorithm (Theorem 3.6). On the upper boundary we find the
recurrence coefficients of ${\widehat{\mu}}_{1}$.

######  Remark 3.3.

If we remove the assumption $\bm{\nu}\in\mathcal{L}_{\infty}^{r-1}$ then we
get complications when $N_{j}<\infty$ for all $j=1,\dots,r-1$, since a
sequence of normal indices $\left\\{\bm{k}+\bm{s}_{j}\right\\}_{j=0}^{m}$ will
not exist every $\bm{k}\in\mathbb{N}_{\bm{K}}^{r-1}$. Indeed, if
$|\bm{k}|>|\bm{K}|-m$, then $\bm{k}+\bm{s}_{j}$ will have to be outside of
$\mathbb{N}_{\bm{K}}^{r-1}$ for $j>|\bm{K}|-|\bm{k}|$ and therefore will not
be normal. Theorem 3.2 still works if one replaces $P_{\bm{k}+\bm{s}_{j}}(x)$
for those $j$’s with, for example,
$x^{\left|\bm{k}\right|-\left|\bm{K}\right|+j}P_{\bm{K}}(x)$. The above proof
works without any significant change. More generally, as long as
$\deg{P_{\bm{k}+\bm{s}_{j}}}=\left|\bm{k}\right|+j$ for each $j=1,\dots,m$,
the proof still holds even if these indices are not normal.

######  Remark 3.4.

If we remove the assumption that $\\{z_{j}\\}_{j=1}^{m}$ are distinct, then
the theorem still holds (the second proof works) but the matrix in (57) should
be modified as follows. If $z_{j}$ is a root of $\Phi$ of multiplicity $l$,
then instead of $l$ copies of the row
$\begin{pmatrix}P_{\bm{k}+\bm{s}_{m}}(z_{j})&P_{\bm{k}+\bm{s}_{m-1}}(z_{j})&\cdots&P_{\bm{k}}(z_{j})\end{pmatrix}$
we have the rows

$\begin{split}\begin{pmatrix}P_{\bm{k}+\bm{s}_{m}}(z_{j})&P_{\bm{k}+\bm{s}_{m-1}}(z_{j})&\cdots&P_{\bm{k}}(z_{j})\\\
P_{\bm{k}+\bm{s}_{m}}^{\prime}(z_{j})&P_{\bm{k}+\bm{s}_{m-1}}^{\prime}(z_{j})&\cdots&P_{\bm{k}}^{\prime}(z_{j})\\\
\vdots&\vdots&\ddots&\vdots\\\
P_{\bm{k}+\bm{s}_{m}}^{(l-1)}(z_{j})&P_{\bm{k}+\bm{s}_{m-1}}^{(l-1)}(z_{j})&\cdots&P_{\bm{k}}^{(l-1)}(z_{j})\end{pmatrix},\end{split}$

and the corresponding modifications should be done in the determinant
$D_{\bm{k}}$ to make $\widehat{P}_{\bm{k}}$ monic. This change ensures that
$z_{j}$ is a root of multiplicity $l$ of the right-hand side of (57). The
previous remark still holds.

###  3.2. Nearest Neighbour Recurrence Coefficients for the Christoffel
Transforms

In Theorem 3.1 we stated the relationship between the type II polynomials of
$\widehat{\bm{\nu}}$ and $\bm{\mu}$ (for type I polynomials, see Theorem 3.12
below). It will come as no surprise that the nearest neighbour recurrence
coefficients of $\widehat{\bm{\nu}}$ and $\bm{\mu}$ are also related.

######  Theorem 3.5.

Suppose $\bm{\mu}\in\mathcal{L}^{r}$ with $\mu_{r}\in\mathcal{L}_{m}$, where
$m<\infty$. Then:

  * i)

$a_{(\bm{k},m),r}=0$ for all $\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}}$.

  * ii)

Additionally, suppose $\bm{\nu}$ and $\widehat{\bm{\nu}}$ are perfect. Then

$\displaystyle a_{(\bm{k},m),j}=\widehat{a}_{\bm{k},j},$
$\displaystyle\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}},\qquad$ $\displaystyle
j=1,\dots,r-1,$ (59) $\displaystyle b_{(\bm{k},m),j}=\widehat{b}_{\bm{k},j},$
$\displaystyle\bm{k}\in\mathbb{N}^{r}_{\bm{K}-\bm{e}_{j}},\qquad$
$\displaystyle j=1,\dots,r-1.$ (60)

###### Proof.

i) This follows immediately from Lemma 2.18 a) along with Proposition 2.7
(except in the case $\bm{k}=\bm{K}$, in which case it follows by definition
through Remark 2.12, for the sake of convenience).

ii)  This is clear from Theorem 3.1 and i). ∎

Now let $\bm{\mu}\in\mathcal{L}^{r}$ be perfect in the extended sense of
Definition 2.8. Suppose that the Jacobi coefficients of each
$\mu_{j}\in\mathcal{L}$ is given, i.e.,
$\\{a_{n\bm{e}_{j},j}\\}_{n=1}^{N_{j}}$ and
$\\{b_{n\bm{e}_{j},j}\\}_{n=0}^{N_{j}-1}$ for each $1\leq j\leq r$ (here
$a_{n\bm{e}_{j},N_{j}}=0$ if $N_{j}<\infty$). It is clear that this
information is sufficient to determine all the NNRR coefficients
$\\{a_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}}}$ and
$\\{b_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}-\bm{e}_{j}}}$ for each
$1\leq j\leq r$. Filipuk, Haneczok, and Van Assche in [ComputingNNR] put
forward a recursive algorithm for this based on the compatibility conditions
(41)–(43). In Theorem 3.6 we state their algorithm in the more general setting
$\bm{\mu}\in\mathcal{L}^{r}$, where we allow $N_{j}$’s to be finite. The
proof, which is a straightforward application of Theorem 2.16, is following
the same lines as the proof of [ComputingNNR, Thm 3.3] except that extra care
needs to be taken with the indices $\bm{n}$ with $n_{j}=N_{j}$ if
$N_{j}<\infty$. Indeed, the $a_{\bm{n},j}$ coefficients vanish there, as we
just showed in Theorem 3.5i).

######  Theorem 3.6 (The CC Algorithm, [ComputingNNR]).

Suppose $\bm{\mu}=(\mu_{1},\dots,\mu_{r})$ is perfect. Given the Jacobi
coefficients of each $\mu_{j}$, $j=1,\dots,r$, the following algorithm
produces all the NNRR coefficients:

$\displaystyle\mbox{for all }1\leq j,k\leq r\mbox{ with }j\neq k:$
$\displaystyle\qquad a_{\bm{n},j}:=0\mbox{ for all
}\bm{n}\in\mathbb{N}^{r}_{\bm{N}}\mbox{ with }n_{j}=0;$
$\displaystyle\qquad\delta_{\bm{0},j,k}:=b_{\bm{0},k}-b_{\bm{0},j};$
$\displaystyle\mbox{for all }d\in\mathbb{N}:$ $\displaystyle\qquad\mbox{for
all }1\leq j,k\leq r\mbox{ with }j\neq k:$ $\displaystyle\qquad\qquad\mbox{for
all }\bm{n}\in\mathbb{N}_{\bm{N}}^{r}\mbox{ with }n_{k}>0,n_{j}>0,|\bm{n}|=d:$
$\displaystyle\qquad\qquad\qquad a_{\bm{n},j}=\begin{cases}0&\mbox{if
}a_{\bm{n}-\bm{e}_{k},j}=0,\\\
a_{\bm{n}-\bm{e}_{k},j}\frac{\delta_{\bm{n}-\bm{e}_{k},k,j}}{\delta_{\bm{n}-\bm{e}_{k}-\bm{e}_{j},k,j}}&\mbox{otherwise};\end{cases}$
$\displaystyle\qquad\mbox{for all }1\leq j,k\leq r\mbox{ with }j\neq k:$
$\displaystyle\qquad\qquad\mbox{for all
}\bm{n}\in\mathbb{N}_{\bm{N}-\bm{e}_{j}}^{r}\mbox{ with }n_{k}>0,|\bm{n}|=d:$
$\displaystyle\qquad\qquad\qquad
b_{\bm{n},j}=b_{\bm{n}-\bm{e}_{k},j}+\frac{\sum_{i=1}^{r}a_{\bm{n},i}-\sum_{i=1}^{r}a_{\bm{n}+\bm{e}_{j}-\bm{e}_{k},i}}{\delta_{\bm{n}-\bm{e}_{k},k,j}};$
$\displaystyle\qquad\mbox{for all }1\leq j,k\leq r\mbox{ with }j\neq k:$
$\displaystyle\qquad\qquad\mbox{ for all
}\bm{n}\in\mathbb{N}_{\bm{N}-\bm{e}_{j}-\bm{e}_{k}}^{r},|\bm{n}|=d:$
$\displaystyle\qquad\qquad\qquad\delta_{\bm{n},j,k}:=b_{\bm{n},k}-b_{\bm{n},j}.$

###### Proof.

Assume all coefficients of indices with size $<d$ are computed and consider
indices $\bm{n}$ with $\left|\bm{n}\right|=d$. Using (46) we can compute the
coefficients $a_{\bm{n},j}$ when $0<n_{j}<N_{j}$ by

$\begin{split}a_{\bm{n},j}=a_{\bm{n}-\bm{e}_{k},j}\frac{\delta_{\bm{n}-\bm{e}_{k},k,j}}{\delta_{\bm{n}-\bm{e}_{k}-\bm{e}_{j},k,j}},\end{split}$

where $k\neq j$ and $n_{k}>0$ (if such a $k$ does not exist then
$\bm{n}=d\bm{e}_{j}$, in which case $a_{\bm{n},j}$ is already given). If
$n_{j}=0$ then we have $a_{\bm{n},j}=0$ by definition, and if $n_{j}=N_{j}$
then we have $a_{\bm{n},j}=0$ by Theorem 3.5, which agrees with the algorithm.

With every $a_{\bm{n},j}$ computed, we can then use (45) to compute the
coefficients $b_{\bm{n},j}$ by

$\begin{split}b_{\bm{n},j}=b_{\bm{n}-\bm{e}_{k},j}+\frac{\sum_{i=1}^{r}a_{\bm{n},i}-\sum_{i=1}^{r}a_{\bm{n}+\bm{e}_{j}-\bm{e}_{k},i}}{\delta_{\bm{n}-\bm{e}_{k},k,j}},\end{split}$

with $k$ chosen such that $k\neq j$ and $n_{k}>0$. Hence all the NNRR
coefficients can be computed, by induction on $\left|\bm{n}\right|$ (in the
case $\left|\bm{n}\right|=0$ there is nothing to compute).

∎

######  Remark 3.7.

Note that the algorithm never breaks down for perfect systems since
$\delta_{\bm{n},j,k}\neq 0$ for all
$\bm{n}\in\mathbb{N}_{\bm{N}-\bm{e}_{j}-\bm{e}_{k}}^{r}$, by Lemma 2.18 b).

######  Remark 3.8.

The algorithm in the way that is written here is not optimized since most of
the coefficients are computed more than once (generically $r$ times, of course
all leading to the same answer); if one cares about the computational
efficiency then one would need to be more careful with the loops to avoid
repetitions.

######  Corollary 3.9.

Suppose $\bm{\mu}$ is perfect and $\mu_{r}\in\mathcal{L}_{m}$. Then the
recurrence coefficients of $\widehat{\bm{\nu}}$ can be computed from the
recurrence coefficients of $\bm{\nu}$ using the CC algorithm (see Theorem 3.6)
applied to ${\bm{\mu}}$.

###### Proof.

This is immediate from Theorem 3.5 and Theorem 3.6. ∎

###  3.3. Two examples

When $m=1$ in Theorem 3.2 we get the one-step Christoffel transform
$\widehat{\bm{\nu}}$ of a multiple orthogonal system $\bm{\nu}$. This
corresponds to taking $\mu_{r}$ to be the Dirac delta measure $\delta_{z_{0}}$
at $z_{0}$ for some $z_{0}\in\mathbb{C}$ (i.e.,
$\delta_{z_{0}}[x^{n}]:=z_{0}^{n}$) and $\Phi(x)=x-z_{0}$. Here, the
determinantal condition is particularly interesting. The formula (61) has
appeared earlier in [ADMVA].

######  Theorem 3.10.

Suppose $\bm{\nu}\in\mathcal{L}^{r-1}$ is perfect. Then $\widehat{\bm{\nu}}$
is perfect if and only if $z_{0}$ is not a root of any $P_{\bm{k}}(x)$,
$\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}}$, and then the type II polynomials with
respect to $\bm{\nu}$ are given by

$\widehat{P}_{\bm{k}}(x)=\frac{1}{x-z_{0}}\left({P}_{\bm{k}+\bm{e}_{j}}(x)-\frac{{P}_{\bm{k}+\bm{e}_{j}}(z_{0})}{{P}_{\bm{k}}(z_{0})}{P}_{\bm{k}}(x)\right),\qquad\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}-\bm{e}_{j}},\qquad
j=1,\dots,r-1.$ (61)

###### Proof.

$D_{\bm{k}}=P_{\bm{k}}(z_{0})$. ∎

The compatibility conditions/CC algorithm now starts to resemble the original
computation algorithm (30). If we write $\delta_{\bm{k},j}$ for
$\delta_{(\bm{k},0),j,r}$, $j=1,\dots,r-1$, the equations (44)–(46) for the
system $\bm{\mu}=(\bm{\nu},\delta_{z_{0}})$ with $k=r$ then turn into

$\displaystyle\widehat{b}_{\bm{k},j}-\delta_{\bm{k}+\bm{e}_{j},j}=b_{\bm{k}+\bm{e}_{j},j}-\delta_{\bm{k},j},$
(62)
$\displaystyle\sum_{i=1}^{r-1}\widehat{a}_{\bm{k},i}-\delta_{\bm{k},j}\widehat{b}_{\bm{k},j}=\sum_{i=1}^{r-1}a_{\bm{k}+\bm{e}_{j},i}-\delta_{\bm{k},j}b_{\bm{k},j},$
(63)
$\displaystyle\delta_{\bm{k}-\bm{e}_{j},j}\widehat{a}_{\bm{k},j}=\delta_{\bm{k},j}a_{\bm{k},j},$
(64)

with the initial conditions $\delta_{\bm{0},j}=z_{0}-b_{\bm{0},j}$ and
$\widehat{a}_{\bm{k},j}=0$ when $k_{j}=0$. In the case $r=2$ we end up with
exactly (30). We can now generalize the computation algorithm to one-step
Christoffel transforms for multiple orthogonal polynomials.

######  Theorem 3.11.

Suppose $\bm{\nu}$ is perfect and $z_{0}$ is not a root of any $P_{\bm{k}}$,
$\bm{k}\in\mathbb{N}_{\bm{K}}^{r-1}$. Given the NNRR coefficients of
$\bm{\nu}$, the following algorithm produces all the NNRR coefficients of
$\widehat{\bm{\nu}}:=(z-z_{0})\nu$,

$\displaystyle\mbox{for all }1\leq j\leq r-1:$
$\displaystyle\qquad\delta_{\bm{0},j}:=z_{0}-b_{\bm{0},j};$
$\displaystyle\qquad\widehat{a}_{\bm{k},j}:=0\mbox{ for all
}\bm{k}\in\mathbb{Z}^{r-1}_{+,\bm{K}}\mbox{ with }k_{j}=0;$
$\displaystyle\qquad\widehat{b}_{\bm{0},j}:=b_{\bm{0},j}-\frac{a_{\bm{e_{j}},j}}{\delta_{\bm{0},j}}$
$\displaystyle\mbox{for all }d\in\mathbb{N}:$ $\displaystyle\qquad\mbox{for
all }1\leq j\leq r-1:$ $\displaystyle\qquad\qquad\mbox{for all
}\bm{k}\in\mathbb{N}_{\bm{K}-\bm{e}_{j}}^{r-1}\mbox{ with }|\bm{k}|=d:$
$\displaystyle\qquad\qquad\qquad\delta_{\bm{k},j}:=\widehat{b}_{\bm{k}-\bm{e}_{j},j}-b_{\bm{k},j}-\delta_{\bm{k}-\bm{e}_{j},j};$
$\displaystyle\qquad\mbox{for all }1\leq j\leq r-1:$
$\displaystyle\qquad\qquad\mbox{for all
}\bm{k}\in\mathbb{N}_{\bm{K}}^{r-1}\mbox{ with }k_{j}>0,|\bm{k}|=d:$
$\displaystyle\qquad\qquad\qquad\widehat{a}_{\bm{k},j}=\begin{cases}a_{\bm{k},j}\frac{\delta_{\bm{k},j}}{\delta_{\bm{k}-\bm{e}_{j},j}}&\mbox{if
}k_{j}<N_{j},\\\ 0&\mbox{if }k_{j}=N_{j};\end{cases}$
$\displaystyle\qquad\mbox{for all }1\leq j\leq r-1:$
$\displaystyle\qquad\qquad\mbox{ for all
}\bm{k}\in\mathbb{N}_{\bm{K}-\bm{e}_{j}}^{r-1}\mbox{ with }|\bm{k}|=d:$
$\displaystyle\qquad\qquad\qquad\widehat{b}_{\bm{k},j}=b_{\bm{k},j}+\frac{\sum_{i=1}^{r-1}\widehat{a}_{\bm{k},i}-\sum_{i=1}^{r-1}a_{\bm{k}+\bm{e}_{j},i}}{\delta_{\bm{k},j}}.$

Another example one often wants to consider is the two-step Christoffel
transform of $\nu\in\mathcal{M}$ with $\Phi(z)=(x-z_{0})(x-\bar{z}_{0})$ where
$z_{0}\in\mathbb{C}\setminus\mathbb{R}$. The most obvious choice of the
multiple orthogonality system system
$(\nu,\tfrac{1}{2}\delta_{z_{0}}+\tfrac{1}{2}\delta_{\bar{z}_{0}})$ might not
be perfect however: for example, if $\operatorname{Re}z_{0}$ happens to be a
zero of some $P_{n}(x)$ then the multi-index $(n,1)$ can be shown to be not
normal. Instead, let us consider the multiple orthogonality system system
$(\nu,\omega)$, where $\omega\in\mathcal{L}_{2}$ given by

$\omega=w_{0}\delta_{z_{0}}+(1-w_{0})\delta_{\bar{z}_{0}}$ (65)

with $w_{0}\in\mathbb{R}\setminus\\{0,\tfrac{1}{2},1\\}$. Indices $(n,0)$ and
$(n,2)$ are normal for all $n$ since $\nu$ and its transform are both in
$\mathcal{M}$. Suppose that some index $(n,1)$ is not normal. Then $P_{n}(x)$
must be one of the type II polynomials at index $(n,1)$, implying

$0=\omega[P_{n}(x)]=w_{0}P_{n}(z_{0})+(1-w_{0})\overline{P_{n}(z_{0})}.$ (66)

The latter equality implies $\operatorname{Re}P_{n}(z_{0})=0$ and
$(2w_{0}-1)\operatorname{Im}P_{n}(z_{0})=0$. Hence
$P_{n}(z_{0})=P_{n}(\bar{z}_{0})=0$, but then $P_{n}$ solves all orthogonality
relations with respect to the index $(n,2)$, which contradicts the normality
of $(n,2)$. This proves that $(n,1)$ is normal and therefore $(\nu,\omega)$ is
perfect for any $w_{0}\in\mathbb{R}\setminus\\{0,\tfrac{1}{2},1\\}$. In order
to run the CC algorithm with $(\nu,\omega)$ for computing $\widehat{\nu}$ one
needs to know the Jacobi coefficients (17) of $\omega$. Elementary
calculations show that these are

$\displaystyle b_{0}$ $\displaystyle=x_{0}-iy_{0}(1-2w_{0}),$ (67)
$\displaystyle b_{1}$ $\displaystyle=x_{0}+iy_{0}(1-2w_{0}),$ (68)
$\displaystyle a_{1}$ $\displaystyle=4w_{0}(1-w_{0})y_{0}^{2},$ (69)

where $x_{0}=\operatorname{Re}z_{0}$, $y_{0}=\operatorname{Im}z_{0}$. Note
that all the arguments used here work just as well in the more general case
$\bm{\nu}\in\mathcal{M}^{r-1}$ with $r-1>2$ except that the perfectness of
$\nu$ and of $\widehat{\nu}$ is no longer given for free (but this easily
holds if we assume that $\bm{\nu}$ is an Angelesco or an AT system, for
example).

###  3.4. Christoffel Transforms of Type I polynomials

We write $\bm{A}_{\bm{k}}=(A_{\bm{k}}^{(1)},\dots,A_{\bm{k}}^{(r-1)})$ and
$\widehat{\bm{A}}_{\bm{k}}=(\widehat{A}_{\bm{k}}^{(1)},\dots,\widehat{A}_{\bm{k}}^{(r-1)})$
for the type I polynomials with respect to $\bm{\nu}$ and
$\widehat{\bm{\nu}}$, respectively. We get similar results to the previous two
sections here. However, in this case the matrix is larger and the sequence of
indices requires the extra constraints $\bm{s}_{j}\leq m\bm{1}$ where
$m\bm{1}=(m,\dots,m)\in\mathbb{N}^{r-1}$. For simplicity we assume that
$\bm{\nu}\in\mathcal{L}_{\infty}^{r-1}$ and all zeros of $\Phi$ are simple,
ignoring the details of the special cases discussed in Remark 3.3-3.4.

######  Theorem 3.12.

Let $\bm{\nu}$ be a perfect system and $\bm{k}\in\mathbb{N}_{\bm{K}}^{r-1}$.
Let $\left\\{\bm{k}+\bm{s}_{j}\right\\}_{j=0}^{(r-1)m}$ be a sequence of
$\mathbb{N}^{r-1}$-indices where $\left|\bm{s}_{j}\right|=j$ and
$\bm{s}_{j}\in\mathbb{N}_{m\bm{1}}^{r-1}$ for each $j=0,1,\dots,(r-1)m$.
Consider the determinant

$\begin{split}D_{\bm{k}}=\det{\begin{pmatrix}\bm{A}_{\bm{k}+\bm{s}_{1}}(z_{1})&\bm{A}_{\bm{k}+\bm{s}_{2}}(z_{1})&\cdots&\bm{A}_{\bm{k}+\bm{s}_{(r-1)m}}(z_{1})\\\
\bm{A}_{\bm{k}+\bm{s}_{1}}(z_{2})&\bm{A}_{\bm{k}+\bm{s}_{2}}(z_{2})&\cdots&\bm{A}_{\bm{k}+\bm{s}_{(r-1)m}}(z_{2})\\\
\vdots&\vdots&\ddots&\vdots\\\
\bm{A}_{\bm{k}+\bm{s}_{1}}(z_{m})&\bm{A}_{\bm{k}+\bm{s}_{2}}(z_{m})&\cdots&\bm{A}_{\bm{k}+\bm{s}_{(r-1)m}}(z_{m})\end{pmatrix},}\end{split}$
(70)

where $\bm{A}_{\bm{k}+\bm{s}_{j}}(z_{k})$ denotes the column vector with
elements
$A_{\bm{k}+\bm{s}_{j}}^{(1)}(z_{k}),\dots,A_{\bm{k}+\bm{s}_{j}}^{(r-1)}(z_{k})$.

If $(\bm{k},m)$ is normal then $D_{\bm{k}}\neq 0$ and the following
determinantal formula holds

$\begin{split}A_{{(}\bm{k},m{)}}^{(j)}(x)=\widehat{A}_{\bm{k}}^{(j)}(x)={\Phi(x)}^{-1}D_{\bm{k}}^{-1}\det{\begin{pmatrix}A_{\bm{k}}^{(j)}(x)&A_{\bm{k}+\bm{s}_{1}}^{(j)}(x)&\cdots&A_{\bm{k}+\bm{s}_{(r-1)m}}^{(j)}(x)\\\
\bm{A}_{\bm{k}}(z_{1})&\bm{A}_{\bm{k}+\bm{s}_{1}}(z_{1})&\cdots&\bm{A}_{\bm{k}+\bm{s}_{(r-1)m}}(z_{1})\\\
\vdots&\vdots&\ddots&\vdots\\\
\bm{A}_{\bm{k}}(z_{m})&\bm{A}_{\bm{k}+\bm{s}_{1}}(z_{m})&\cdots&\bm{A}_{\bm{k}+\bm{s}_{(r-1)m}}(z_{m})\end{pmatrix}},\\\
1\leq j\leq r-1.\end{split}$ (71)

###### Proof.

In the proof we use the notation $\left\langle\bm{A}(x),P(x)\right\rangle$,
for polynomials $P(x)$ and vectors of polynomials
$(A^{(1)}(x),\dots,A^{(r-1)}(x))$, to represent $\sum_{j=1}^{r-1}\left\langle
A^{(j)}(x),P(x)\right\rangle_{j}$. Note that this is clearly bilinear. Then
the orthogonality relations for $\bm{A}_{\bm{k}}(x)$ can be rephrased as

$\left\langle\bm{A}_{\bm{k}}(x),x^{p}\right\rangle=0,\qquad
p=0,\dots,\left|\bm{k}\right|-2.$ (72)

We first prove that $D_{\bm{k}}\neq 0$. $D_{\bm{k}}=0$ would imply that the
columns are linearly dependent. In particular this means that

$\begin{split}\sum_{j=1}^{(r-1)m}c_{j}\bm{A}_{\bm{k}+\bm{s}_{j}}(x)=\bm{0},\qquad
x=z_{1},\dots,z_{r},\end{split}$

for some $(c_{1},\dots,c_{(r-1)m})\neq(0,\dots,0)$. On the other hand, we note
that the sum cannot vanish for every $x$, since in that case the smallest $l$
such that $c_{l}\neq 0$ would satisfy

$\begin{split}\left\langle\sum_{j=1}^{(r-1)m}c_{j}\bm{A}_{\bm{k}+\bm{s}_{j}}(x),x^{{\left|n\right|}-1+l}\right\rangle=c_{l}\left\langle\bm{A}_{\bm{k}+\bm{s}_{l}}(x),x^{{\left|n\right|}-1+l}\right\rangle=0.\end{split}$

Normality of $\bm{k}+\bm{s}_{l}$ would then imply $c_{l}=0$. Hence we have a
non-zero vector $\sum_{j=1}^{(r-1)m}c_{j}\bm{A}_{\bm{k}+\bm{s}_{j}}$ which is
also divisible by $\Phi$. Write $\bm{A}=(A^{(1)},\dots,A^{(r-1)})$ for
$\Phi(x)^{-1}\sum_{j=1}^{(r-1)m}c_{j}\bm{A}_{\bm{k}+\bm{s}_{j}}$. Since
$\bm{s}_{j}\in\mathbb{N}_{m\bm{1}}$ for each $j=0,1,\dots,(r-1)m$ we have
$\deg{A^{(i)}}\leq k_{i}-1$. This vector also satisfies

$\begin{split}\left\langle\Phi(x)\bm{A}(x),x^{p}\right\rangle=0,\qquad
p=0,\dots,\left|\bm{k}\right|-1,\end{split}$

which implies that $\bm{k}$ is not $\widehat{\nu}$-normal. This contradicts
the normality of $(\bm{k},m)$ by Theorem 3.1. Hence we must have
$D_{\bm{k}}\neq 0$.

Next, we note that the determinant in (71) vanishes at $z_{1}\dots,z_{r}$, so
it is divisible by $\Phi(x)$ (for each $j=0,\dots,r-1$). Denote the rightmost
expression of (71) by $B_{\bm{k}}^{(j)}$ and
$\bm{B}_{\bm{k}}=(B^{(1)}_{\bm{k}},\dots,B^{(r-1)}_{\bm{k}})$. Clearly
$\bm{B}_{\bm{k}}$ is then orthogonal to $x^{p}$ for
$p=0,\dots,\left|\bm{k}\right|-2$ with respect to $\widehat{\bm{\nu}}$ in the
sense of (72). Now note that

${\left\langle\bm{B}_{\bm{k}}(x),x^{\left|\bm{k}\right|-1}\right\rangle_{\widehat{\bm{\nu}}}=\left\langle
D_{\bm{k}}^{-1}D_{\bm{k}}\bm{A}_{\bm{k}}(x),x^{\left|\bm{k}\right|-1}\right\rangle_{{\bm{\nu}}}={1}.}$
(73)

This shows that $\widehat{\bm{A}}_{\bm{k}}=\bm{B}_{\bm{k}}$. Finally,

$\begin{split}\left\langle\bm{A}_{(\bm{k},m)},x^{p+m}\right\rangle_{\bm{\mu}}=\left\langle\bm{A}_{(\bm{k},m)},x^{p}\Phi(x)\right\rangle_{\bm{\mu}}=\sum_{j=1}^{r-1}\left\langle\Phi(x)A_{(\bm{k},m)}^{(j)},x^{p}\right\rangle_{j}=\begin{cases}0,\qquad
p=0,\dots,\left|\bm{k}\right|-2,\\\ {1,\qquad
p=|\bm{k}|-1,}\end{cases}\end{split}$

by Lemma 2.1. Also $\deg{A_{\bm{k},m}}\leq k_{j}-1$, $j=1,\dots,r-1$. Since
$\bm{k}$ is normal for $\widehat{\bm{\nu}}$, we conclude with
$\widehat{A}_{\bm{k}}^{(j)}=A_{(\bm{k},m)}^{(j)}$. ∎

###  3.5. Determinantal Formula for Type I Polynomials: one-step case

For the case $m=1$, the determinantal formula in Theorem 3.12 is not as simple
as in Theorem 3.10. Rather, we get a formula in terms of
$\bm{A}_{\bm{k}},\bm{A}_{\bm{k}+\bm{s}_{1}},\dots,\bm{A}_{\bm{k}+\bm{s}_{r-1}}$.
However, we can get new formulas by deforming the sequence
$\left\\{\bm{k}+\bm{s}_{j}\right\\}_{j=0}^{r-1}$ using (40). Perhaps the
nicest choice is writing the columns in terms of the nearest neighbours, which
the following theorem shows is possible, even with very relaxed normality
assumptions.

######  Theorem 3.13.

Suppose $\bm{k}$ is normal for both $\bm{\nu}$ and
$\widehat{\bm{\nu}}=(x-z_{0})\bm{\nu}$. Consider the matrix

$\begin{split}D_{\bm{k}}=\det\begin{pmatrix}A_{\bm{k}+\bm{e}_{1}}^{(1)}(z_{0})&A_{\bm{k}+\bm{e}_{2}}^{(1)}(z_{0})&\cdots&A_{\bm{k}+\bm{e}_{r-1}}^{(1)}(z_{0})\\\
A_{\bm{k}+\bm{e}_{1}}^{(2)}(z_{0})&A_{\bm{k}+\bm{e}_{2}}^{(2)}(z_{0})&\cdots&A_{\bm{k}+\bm{e}_{r-1}}^{(2)}(z_{0})\\\
\vdots&\vdots&\ddots&\vdots\\\
A_{\bm{k}+\bm{e}_{1}}^{(r-1)}(z_{0})&A_{\bm{k}+\bm{e}_{2}}^{(r-1)}(z_{0})&\cdots&A_{\bm{k}+\bm{e}_{r-1}}^{(r-1)}(z_{0})\end{pmatrix}.\end{split}$
(74)

Then $\det{D_{\bm{k}}}\neq 0$ and the determinantal formula holds

$\begin{split}\widehat{A}_{\bm{k}}^{(j)}(x)=(x-z_{0})^{-1}D_{\bm{k}}^{-1}\det{\begin{pmatrix}A_{\bm{k}}^{(j)}(x)&A_{\bm{k}+\bm{e}_{1}}^{(j)}(x)&\cdots&A_{\bm{k}+\bm{e}_{r-1}}^{(j)}(x)\\\
A_{\bm{k}}^{(1)}(z_{0})&A_{\bm{k}+\bm{e}_{1}}^{(1)}(z_{0})&\cdots&A_{\bm{k}+\bm{e}_{r-1}}^{(1)}(z_{0})\\\
\vdots&\vdots&\ddots&\vdots\\\
A_{\bm{k}}^{(r-1)}(z_{0})&A_{\bm{k}+\bm{e}_{1}}^{(r-1)}(z_{0})&\cdots&A_{\bm{k}+\bm{e}_{r-1}}^{(r-1)}(z_{0})\end{pmatrix}},\qquad
1\leq j\leq r-1.\end{split}$ (75)

###### Proof.

If $D_{\bm{k}}=0$ then for some non-trivial linear combination,

$\begin{split}\sum_{j=1}^{r-1}c_{k}\bm{A}_{\bm{k}+\bm{e}_{j}}(z_{0})=0,\qquad
j=1,\dots,r-1.\end{split}$

Then define the vector $\bm{A}=(A^{(1)},\dots,A^{(r-1)})$ by

$\begin{split}\bm{A}(x)=(x-z_{0})^{-1}\sum_{j=1}^{r-1}c_{j}\bm{A}_{\bm{k}+\bm{e}_{j}}(x).\end{split}$

Clearly $\deg A^{(i)}\leq k_{i}-1$, $i=1,\dots,r-1$, and $\bm{A}$ satisfies
the orthogonality conditions

$\begin{split}\sum_{i=1}^{r-1}\left\langle(x-z_{0})A^{(i)}(x),x^{p}\right\rangle_{i}=0,\qquad
p=0,\dots,\left|\bm{k}\right|-1.\end{split}$

Since $\bm{A}\neq\bm{0}$ by Lemma 2.22 we get a contradiction with the
normality of the index $\bm{k}$ for the system $\widehat{\bm{\nu}}$. We
conclude that $D_{\bm{k}}\neq 0$, and (75) now follows from similar arguments
as in the proof of Theorem 3.12. ∎

######  Remark 3.14.

We can get a generalized version of Theorem 3.13 for arbitrary $m$, by
considering a determinantal formula in terms of $A_{\bm{k}}^{(j)}$ and
$A_{\bm{k}+l\bm{e}_{i}}^{(j)}$ for each $l=1,\dots,m$ and $i=1,\dots,r-1$
(although here we will need to assume more indices than just $\bm{k}$ to be
normal). This can also be seen through repeated elementary row operations in
the determinant in (71) combined with the relations in (40).

We can also relate the one-step Christoffel transform to the kernel
polynomials (47), similarly to (29), by the following result.

######  Theorem 3.15.

Suppose $\bm{k}$ is normal for both $\bm{\nu}$ and
$\widehat{\bm{\nu}}=(x-z_{0})\bm{\nu}$. Then $P_{\bm{k}}(z_{0})\neq 0$ and

$\begin{split}\bm{K}_{\bm{k}}(z_{0},x)=-P_{\bm{k}}(z_{0})\widehat{\bm{A}}_{\bm{k}}(x).\end{split}$
(76)

###### Proof.

If $P_{\bm{k}}(z_{0})=0$ then $P_{\bm{k}}$ satisfies all the orthogonality
conditions for the index $(\bm{k},1)$ with respect to the system
$(\bm{\nu},\delta_{z_{0}})$, which is impossible by Theorem 3.1, so
$P_{\bm{k}}(z_{0})\neq 0$. By (47) we have

$\begin{split}\sum_{j=1}^{r}\left\langle(x-z_{0})K_{\bm{k}}^{(j)}(z_{0},x),x^{p}\right\rangle_{j}=-P_{\bm{k}}(z_{0})\sum_{j=1}^{r}\left\langle
A_{\bm{k}}^{(j)}(x),x^{p}\right\rangle_{j},\qquad
p=0,\dots,\left|\bm{k}\right|-1,\end{split}$

so $\bm{K}_{\bm{k}}(z_{0},x)$ satisfies the same orthogonality relations as
$-P_{\bm{k}}(z_{0})\widehat{\bm{A}}_{\bm{k}}(x)$. Clearly we also have
$\deg{K_{\bm{k}}^{(j)}(z_{0},x)}\leq k_{j}-1$, so we get (76). ∎

######  Remark 3.16.

For the type II polynomials we see that (61) does not resemble
$\bm{K}_{\bm{k}}(x,z_{0})$. We can however get a similar result to (76) by
considering linear combinations of
$\widehat{P}_{\bm{k}-\bm{e}_{1}},\dots,\widehat{P}_{\bm{k}-\bm{e}_{r}}$. A
quick check using (61) and $\eqref{eq:cd formula moprl}$ shows that

$\begin{split}P_{\bm{k}}(z_{0})K_{\bm{k}}^{(i)}(z_{0},x)=\sum_{j=1}^{r}a_{\bm{k},j}P_{\bm{k}-\bm{e}_{j}}(z_{0})A_{\bm{k}+\bm{e}_{j}}^{(i)}(z_{0})\widehat{P}_{\bm{k}-\bm{e}_{j}}(x).\end{split}$
(77)

###  3.6. Repeated Christoffel transform

In the MOPRL construction, even for the discrete case
$\mu_{l}\in\mathcal{L}_{N_{l}}$, we always have $a_{j\bm{e}_{l},l}\neq 0$ for
$1\leq j\leq N_{l}-1$ and we terminate $a$’s as soon as we reach
$a_{N_{l}\bm{e}_{l},l}=0$. If one thinks in terms of the Jacobi matrices
rather than in terms of the orthogonality measures then it is natural to allow
some of the $a_{j\bm{e}_{l},l}$’s to be zero. Let us do so for the last
measure $\mu_{r}$ only.

To this end, for each $l=1,\ldots,r$, let $J_{l}$ be an $N_{l}\times N_{l}$
Jacobi matrix (with $N_{l}$ finite or infinite) with Jacobi coefficients
$\\{a_{j\bm{e}_{l},l}\\}_{j=1}^{N_{l}-1}$ and
$\\{b_{j\bm{e}_{l},l}\\}_{j=0}^{N_{l}-1}$. We assume that
$J_{1},\ldots,J_{r-1}$ are proper, that is, $a_{j\bm{e}_{l},l}\neq 0$ for each
$1\leq j\leq N_{l}-1$ and $1\leq l\leq r-1$. Suppose $J_{r}$ is the direct sum
$J_{r}=\bigoplus_{j=1}^{\infty}J^{(j)}_{r}$ (which is improper) where
$J^{(j)}_{r}$ is a proper Jacobi matrix of size $m_{j}\times m_{j}$, where
$m_{j}<\infty$. Denote $\det(z-J^{(j)}_{r})=\Phi_{j}(z)$. We then have
$\deg\Phi_{j}=m_{j}$.

Now observe that we can still run the CC algorithm without changes. We then
treat $N_{r}$ as $\infty$, and not as $m_{1}$, as we would normally. The
algorithm will be able to compute all the recurrence coefficients
$\\{a_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{(\bm{K},\infty)}}$ and
$\\{b_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{(\bm{K},\infty)-\bm{e}_{j}}}$
for each $1\leq j\leq r$, as long as all the $\delta_{\bm{n},j,i}$
coefficients are nonzero. All of these coefficients satisfy the compatibility
equations (41)–(43) on the extended lattice $\mathbb{N}^{r}_{(\bm{K},\infty)}$
(here $\bm{K}=(N_{1},\ldots,N_{r-1})$ as usual), and therefore one can
uniquely define the polynomials $P_{\bm{n}}$ and $\bm{A}_{\bm{n}}$ using (36)
and (38) for all $\bm{n}\in\mathbb{N}^{r}_{(\bm{K},\infty)}$.

By the results of the previous section it is clear that if
$\bm{n}\in\mathbb{N}^{r}_{(\bm{K},\infty)}$ has the form
$\bm{n}=(\bm{k},m_{1}+\ldots+m_{s})$ with $\bm{k}\in\mathbb{N}^{r-1}_{\bm{K}}$
for some $s\in\mathbb{N}$, then:

  * i)

$a_{\bm{n},r}=0$;

  * ii)

$a_{\bm{n},j}$ and $b_{\bm{n},j}$ for $1\leq j\leq r-1$ are the NNRR
coefficients of $\Phi_{1}\ldots\Phi_{s}\bm{\nu}$, the Christoffel transform of
the system $\bm{\nu}=(\mu_{1},\ldots,\mu_{r-1})$ corresponding to the
polynomial $\Phi_{1}(x)\ldots\Phi_{s}(x)$;

  * iii)

$P_{\bm{n}}(x)$ is equal to $\Phi_{1}(x)\ldots\Phi_{s}(x)$ times the type II
multiple orthogonal polynomial with respect to the system
$\Phi_{1}\ldots\Phi_{s}\bm{\nu}$.

  * iv)

$A_{\bm{n}}^{(j)}(x)$ for $1\leq j\leq r-1$ is the type I multiple orthogonal
polynomial with respect to the system $\Phi_{1}\ldots\Phi_{s}\bm{\nu}$.

Furthermore, in the region $\bm{n}\in\mathbb{N}^{r}_{(\bm{K},\infty)}$ with
$m_{1}+\ldots+m_{s}\leq n_{r}<m_{1}+\ldots+m_{s+1}$ for some $s\in\mathbb{N}$,
we have:

  * v)

$a_{\bm{n},j}$ and $b_{\bm{n},j}$ for $1\leq j\leq r$ are the NNRR
coefficients of the system $(\Phi_{1}\ldots\Phi_{s}\bm{\nu},\omega_{s})$,
where $\omega_{s}$ is the spectral measure/moment functional corresponding to
the Jacobi submatrix $J_{r}^{(s)}$.

  * vi)

$P_{\bm{n}}(x)$ is equal to $\Phi_{1}(x)\ldots\Phi_{s-1}(x)$ times the type II
multiple orthogonal polynomial with respect to the system
$(\Phi_{1}\ldots\Phi_{s}\bm{\nu},\omega_{s})$.

  * vii)

$A_{\bm{n}}^{(j)}(x)$ is equal to the type I multiple orthogonal polynomial
with respect to the system $(\Phi_{1}\ldots\Phi_{s}\bm{\nu},\omega_{s})$.

The condition $\delta_{\bm{n},j,l}\neq 0$ that ensures that the CC algorithm
in Theorem 3.6 does not break down is then equivalent to the perfectness of
all the systems $(\Phi_{1}\ldots\Phi_{s}\bm{\nu},\omega_{s})$ for all $s\geq
1$. For example, this is easily seen to be true if
$(\mu_{1},\ldots,\mu_{r-1})\in\mathcal{M}^{r-1}$ is Angelesco and
$\sigma(J_{r})$ is disjoint from each $\Delta_{j}$ (the convex hull of
$\operatorname{supp}\,(\mu_{j})$). The same holds true if
$(\mu_{1},\ldots,\mu_{r-1})\in\mathcal{M}^{r-1}$ is an AT system (see, e.g.,
[Ismail, Sect 23.1.2] for the definition) on an interval $I$, and
$\sigma(J_{r})$ is disjoint from $I$.

$\mu$$\Phi_{1}\mu$$\Phi_{1}\Phi_{2}\mu$$\Phi_{1}\Phi_{2}\Phi_{3}\mu$$\mu$$J^{(1)}$$J^{(2)}$$J^{(3)}$$J^{(4)}\dots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\ddots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$
Figure 2. Along the blue vertical arrows we find the Christoffel transforms
$\Phi_{1}\mu,\Phi_{1}\Phi_{2}\mu,\Phi_{1}\Phi_{2}\Phi_{3}\mu,\dots$. The
Jacobi coefficients of each transform can be computed via the NNCC algorithm.

The simplest example of this construction is to take $r=2$ with an arbitrary
$J_{1}$, the Jacobi matrix of some $\mu_{1}\in\mathcal{M}$ with
$\operatorname{supp}\,(\mu_{1})\subseteq\mathbb{R}_{+}$, and $J_{2}$ to be the
zero matrix, i.e., $m_{j}=1$ with $J_{r}^{(j)}$ being the $1\times 1$ zero
matrices. Then for each $k\in\mathbb{N}$, the
$\\{a_{(k,s),1}\\}_{k=1}^{\infty}$ and $\\{b_{(k,s),1}\\}_{k=0}^{\infty}$ are
then the Jacobi coefficients of the measure $x^{s}d\mu_{1}(x)$. The
corresponding polynomials are often called the associated polynomials. The CC
algorithm from above has a close connection to the $qd$-algorithm of
Rutishauser [QD] and the Toda lattice with discrete time.

Similarly, the special case of the given construction with $r>2$ and $m_{j}=1$
for all $j$ (so that $J_{r}^{(j)}$ is the diagonal matrix with
$(\lambda_{1},\lambda_{2},\ldots)$ on the diagonal) leads to the multi-
dimensional Toda lattice with discrete time (“dm-Toda lattice”), explored in
[ADMVA]. We would like to comment that in their Remark 3.5 they posed the
question of finding examples beyond the Laguerre polynomials of the second
kind with $\lambda_{j}\equiv 0$ for all $j$ when this integrable system is
well-defined, or equivalently, when each of the MOPRL systems
$\bm{\nu},(x-\lambda_{1})\bm{\nu},(x-\lambda_{1})(x-\lambda_{2})\bm{\nu},\ldots$
is perfect. As was explained in this section, such perfectness happens quite
generally, for example if the initial multiple orthogonality system $\bm{\nu}$
is Angelesco or AT and each $\lambda_{j}$ ($j\geq 1$) is outside of the convex
hull of $\operatorname{supp}\,\mu_{l}$ for any $1\leq l\leq r-1$.

###  3.7. Recurrence Coefficients and Normality

As discussed in Section 2.5, every perfect system (in the extended sense of
Definition 2.8) $\bm{\mu}\in\mathcal{L}^{r}$ produces the nearest neighbour
recurrence coefficients
$\\{a_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}}}$ and
$\\{b_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}-\bm{e}_{j}}}$ for each
$1\leq j\leq r$ that satisfy the compatibility conditions (41), (42), (43). In
this section we identify what conditions are necessary and sufficient in order
for coefficients $\\{a_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}}}$ and
$\\{b_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}-\bm{e}_{j}}}$ to be the
nearest neighbour recurrence coefficients of some perfect system.

This corresponds to a result central to the paper [IntegrableSystems], solved
for $\bm{\mu}\in\mathcal{L}_{\infty}^{r}$. Here we give an alternative simple
proof, based on the lemmas in Section 2.6, along with the CC algorithm. The
special feature that appears when some of the measures/functionals are
finitely supported is property b) below.

######  Theorem 3.17.

Let $\bm{N}=(N_{j})_{j=1}^{r}$, where $N_{j}\leq\infty$. Suppose we are given
the sets of complex coefficients
$\\{a_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}}}$ and
$\\{b_{\bm{n},j}\\}_{\bm{n}\in\mathbb{N}^{r}_{\bm{N}-\bm{e}_{j}}}$ for each
$1\leq j\leq r$. Assume they satisfy the partial difference equations (41),
(42), (43) along with the boundary conditions

  1. a)

$a_{\bm{n},j}=0$ when $n_{j}=0$,

  2. b)

$a_{\bm{n},j}=0$ when $n_{j}=N_{j}$.

and the normality conditions

  1. c)

$a_{\bm{n},j}\neq 0$ when $0<n_{j}<N_{j}$,

  2. d)

$b_{\bm{n},j}-b_{\bm{n},k}\neq 0$ when $j\neq k$.

Then there is a unique perfect system
$\bm{\mu}=(\mu_{1},\dots,\mu_{r})\in\mathcal{L}^{r}$ such that $a_{\bm{n},i}$
and $b_{\bm{n},i}$ are the nearest neighbour recurrence coefficients of
$\bm{\mu}$ with $\mu_{j}\in\mathcal{L}_{N_{j}}$ for each $j=1,\dots,r$.

######  Remark 3.18.

By induction using (41), we see that d) can be replaced with
$b_{\bm{0},j}-b_{\bm{0},k}\neq 0$ when $j\neq k$. Alternatively, c) can be
replaced with $a_{n\bm{e}_{j},j}\neq 0$ for each $0<n<N_{j}$ when $1\leq j\leq
r$.

######  Remark 3.19.

If we also assume $a_{n\bm{e}_{j},j}>0$ for each $0<n<N_{j}$ and some $1\leq
j\leq r$, then additionally $\mu_{j}\in\mathcal{M}_{N_{j}}$.

###### Proof.

By the Favard Theorem 2.2, we can get functionals $\mu_{1},\dots,\mu_{r}$ such
that $a_{n\bm{e}_{j},j}$ and $b_{n\bm{e}_{j},j}$ are the Jacobi coefficients
of $\mu_{j}\in\mathcal{L}_{N_{j}}$ for each $j=1,\dots,r$, by c). We now want
to prove that $\bm{\mu}=(\mu_{1},\dots,\mu_{r})$ is perfect and that
$a_{\bm{n},j}$ and $b_{\bm{n},j}$ are the recurrence coefficients of
$\bm{\mu}$. We will use induction, to prove that for each $M\in\mathbb{N}$,

  1. 1)

Every index $\bm{n}\leq\bm{N}$ such that $\left|\bm{n}\right|\leq M$ is
normal.

  2. 2)

The nearest neighbour recurrence coefficients of $\bm{\mu}$, for every index
$\bm{n}\leq\bm{N}$ such that $\left|\bm{n}\right|\leq M-1$ are exactly the
recurrence coefficients
$\left\\{a_{\bm{n},i},b_{\bm{n},i}\right\\}_{i=1}^{r}$.

This is true when $M=0$ and $M=1$, since then we already know which indices of
the form $\bm{n}=n_{j}\bm{e}_{j}$ are normal, and the nearest neighbour
recurrence coefficients of $\bm{n}=\bm{0}$ are $a_{\bm{0},j}$ and
$b_{\bm{0},j}$. Now assume 1) and 2) hold for $1,\dots,M$ and suppose
$\bm{n}\leq\bm{N}$ and $\left|\bm{n}\right|=M+1$. If $\bm{n}=n_{j}\bm{e}_{j}$
then we already know that 1) holds. Otherwise we can write
$\bm{n}=\bm{m}+\bm{e}_{j}+\bm{e}_{k}$ for some index $\bm{m}$ and $k\neq j$.
Since $\left|\bm{m}\right|=M-1$ we know that the nearest neighbour recurrence
coefficients for $\bm{m}$ are $a_{\bm{m},i}$ and $b_{\bm{m},i}$. In particular
$b_{\bm{m},j}\neq b_{\bm{m},k}$, so that $\bm{n}=\bm{m}+\bm{e}_{j}+\bm{e}_{k}$
is normal by Corollary 2.15. Now the conditions of Theorem 2.16 are satisfied
and we can apply CC algorithm to compute the nearest neighbour recurrence
coefficients of the system $\bm{\mu}$ for every $\bm{n}$ with
$\left|\bm{n}\right|=M$. Since our recurrence coefficients given in the
Theorem are computed we get 2), and the perfectness of $\bm{\mu}$ follows. ∎

###  3.8. Zero interlacing for Christoffel transforms: general results

In this section we prove some new interlacing results for multiple orthogonal
polynomials of type I and type II. Given two real polynomials $p(x)$ and
$q(x)$ we write $p(x)\sim q(x)$ and say that the zeros of polynomials $p(x)$
and $q(x)$ strictly interlace if all the zeros of $p(x)$ and $q(x)$ are real,
simple, and between every two consecutive zeros of one of the polynomials
there lies exactly one zero of the other polynomial. We also employ the
notation $p\,\bm{\mathcal{\prec}}\,q$ or $q\,\bm{\mathcal{\succ}}\,p$ if zeros
of $p(x)$ and $q(x)$ strictly interlace and

$z_{1}(p)<z_{1}(q)<z_{2}(p)<\ldots,$

where notation $z_{j}(p)$, $z_{j}(q)$ means the $j$-th lowest zero of $p$ and
$q$, respectively. Note that this means that either $\deg p=\deg q=n$ with
$z_{n}(p)<z_{n}(q)$ or $\deg p=\deg q-1=n$ with $z_{n-1}(q)<z_{n}(p)$.

Let us call a polynomial real-rooted if all of its zeros are real. Recall
(see, e.g., [Ismail]) that type II and type I polynomials for any Angelesco
system are real-rooted. Type II polynomials for any AT systems are also real-
rooted.

The following result is well-known (this is part of the
Hermite–Kakeya–Obreschkoff theorem).

######  Lemma 3.20.

Suppose $p$ and $q$ are two polynomials with positive leading coefficients
satisfying $p(x)\,\bm{\mathcal{\prec}}\,q(x)$. Then

$\displaystyle p(x)+\alpha q(x)\,\bm{\mathcal{\prec}}\,q(x),\mbox{ for any
}\alpha\in\mathbb{R},$ (78) $\displaystyle p(x)+\alpha
q(x)\,\bm{\mathcal{\succ}}\,p(x),\mbox{ for any }\alpha>0,$ (79)
$\displaystyle p(x)+\alpha q(x)\,\bm{\mathcal{\prec}}\,p(x),\mbox{ for any
}\alpha<0.$ (80)

######  Corollary 3.21.

Suppose $\bm{\nu}$ and $\widehat{\bm{\nu}}=(x-z_{0})\bm{\nu}$ are perfect. If
${P}_{\bm{k}+\bm{e}_{j}}(x)\sim{P}_{\bm{k}}(x)$, then $\widehat{P}_{\bm{k}}$
is real-rooted. Moreover, $(x-x_{0})\widehat{P}_{\bm{k}}(x)\sim
P_{{\bm{k}}}(x)$ and $(x-x_{0})\widehat{P}_{\bm{k}}(x)\sim
P_{{\bm{k}}+\bm{e}_{j}}(x)$.

###### Proof.

Immediate from (61) and Lemma 3.20 ∎

######  Remark 3.22.

Let $\bm{\nu}$ be any Angelesco or AT system and suppose $z_{0}$ is not a zero
of any $P_{\bm{k}}(x)$. Since interlacing for $\bm{\nu}$ holds at every multi-
index [Jacobioperatorontrees, Ismail, Interlacing], we obtain that
$\widehat{\bm{\nu}}$ is also perfect, and every $\widehat{P}_{\bm{k}}$ is
real-rooted. Notice that if $z_{0}$ belongs to the support of one of the
measures, then $\widehat{\bm{\nu}}$ is no longer a system of positive
measures, so neither perfectness nor interlacing are trivial in that case.

######  Corollary 3.23.

Suppose $\bm{\nu}$ and $\widehat{\bm{\nu}}=(x-z_{0})\bm{\nu}$ are perfect. If
$\widehat{A}^{(j)}_{\bm{k}-\bm{e}_{j}}(x)\sim\widehat{A}^{(j)}_{\bm{k}}(x)$,
then ${A}^{(j)}_{\bm{k}}(x)$ is real-rooted. Moreover,
${A}^{(j)}_{\bm{k}}(x)\sim\widehat{A}^{(j)}_{\bm{k}}(x)$ and
${A}^{(j)}_{\bm{k}}(x)\sim\widehat{A}^{(j)}_{{\bm{k}-\bm{e}_{j}}}(x)$.

###### Proof.

Use (40) and the leftmost equality in (71) to get

$\bm{A}_{\bm{k}}-\widehat{\bm{A}}_{\bm{k}-\bm{e}_{j}}=\delta\widehat{\bm{A}}_{\bm{k}},$
(81)

where $\delta$ is non-zero by Lemma 2.18 b), and (41). Then Lemma 3.20
completes the proof. ∎

######  Remark 3.24.

Any Angelesco system $\widehat{\bm{\nu}}$ satisfies the property that zeros of
$\widehat{A}^{(j)}_{\bm{k}-\bm{e}_{j}}(x)$ and $\widehat{A}^{(j)}_{\bm{k}}(x)$
interlace for any index $\bm{k}$ and any $j$ (see [DenYat, FMM]), so Corollary
3.23 applies, assuming $\bm{\nu}$ is perfect.

###  3.9. Zero interlacing: continuous examples

While Corollaries 3.21 and 3.23 might seem tame and almost obvious, they allow
to obtain interesting interlacing results when applied to the multiple
analogues of the classical orthogonal polynomials. We collect them in Sections
3.9 and 3.10.

We remark that recurrences (61) and (81) (as well as the NNRR for type II and
type I polynomials) provide not only interlacing properties but also various
relations that the classical MOPRL must satisfy. It is highly likely that
these have appeared in the literature in various forms. We focus on
interlacing properties only.

####  3.9.1. The multiple Laguerre polynomials of the first kind

Let $P_{\bm{n}}^{\bm{\alpha}}$ be the type II multiple orthogonal polynomials
corresponding to the system $\bm{\mu}^{\bm{\alpha}}\in\mathcal{M}^{r}$ given
by

$d\mu_{j}^{\bm{\alpha}}=x^{\alpha_{j}}e^{-x}\chi_{[0,\infty)}dx$

where $\alpha_{j}>-1$ and $\alpha_{j}-\alpha_{k}\notin\mathbb{Z}$ for $j\neq
k$.

Since $\bm{\mu}^{\bm{\alpha}}$ is an AT system and
$x\bm{\mu}^{\bm{\alpha}}=\bm{\mu}^{\bm{\alpha}+\bm{1}}$ (where $\bm{1}$ is the
vector of $1$’s), Corollary 3.21 gives

$P_{\bm{n}}^{\bm{\alpha}+\bm{1}}(x)\sim
P_{\bm{n}}^{\bm{\alpha}}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{\bm{\alpha}+\bm{1}}(x)\sim P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha}}(x)$

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ and any $1\leq j\leq r$.

####  3.9.2. The multiple Laguerre polynomials of the second kind

Let $P_{\bm{n}}^{\bm{c};\alpha}$ be the type II multiple orthogonal
polynomials corresponding to the system
$\bm{\mu}^{\bm{c};\alpha}\in\mathcal{M}^{r}$ given by

$d\mu_{j}^{\bm{c};\alpha}=x^{\alpha}e^{-c_{j}x}\chi_{[0,\infty)}dx$

where $\alpha>-1$, $c_{j}>0$, and $c_{j}\neq c_{k}$ for $j\neq k$.

Since $\bm{\mu}^{\bm{c};\alpha}$ is an AT system and
$x\bm{\mu}^{\bm{c};\alpha}=\bm{\mu}^{\bm{c};\alpha+1}$, Corollary 3.21 gives

$P_{\bm{n}}^{\bm{c};\alpha+1}(x)\sim
P_{\bm{n}}^{\bm{c};\alpha}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{\bm{c};\alpha+1}(x)\sim P_{\bm{n}+\bm{e}_{j}}^{\bm{c};\alpha}(x)$

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ and any $1\leq j\leq r$.

####  3.9.3. The Jacobi–Piñeiro polynomials

Let $P_{\bm{n}}^{\bm{\alpha};\beta}$ be the type II multiple orthogonal
polynomials corresponding to the system
$\bm{\mu}^{\bm{\alpha};\beta}\in\mathcal{M}^{r}$ given by

$d\mu_{j}^{\bm{\alpha};\beta}=x^{\alpha_{j}}(1-x)^{\beta}\chi_{[0,1]}dx$

where $\beta>-1$, $\alpha_{j}>-1$, and $\alpha_{j}-\alpha_{k}\notin\mathbb{Z}$
for $j\neq k$.

Since $\bm{\mu}^{\bm{\alpha};\beta}$ is an AT system and
$x\bm{\mu}^{\bm{\alpha};\beta}=\bm{\mu}^{\bm{\alpha}+\bm{1};\beta}$ and
$(1-x)\bm{\mu}^{\bm{\alpha};\beta}=\bm{\mu}^{\bm{\alpha};\beta+1}$ Corollary
3.21 gives

$\displaystyle P_{\bm{n}}^{\bm{\alpha}+\bm{1};\beta}(x)\sim
P_{\bm{n}}^{\bm{\alpha};\beta}(x)$ and $\displaystyle
P_{\bm{n}}^{\bm{\alpha}+\bm{1};\beta}(x)\sim
P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta}(x),$ $\displaystyle
P_{\bm{n}}^{\bm{\alpha};\beta+1}(x)\sim P_{\bm{n}}^{\bm{\alpha};\beta}(x)$ and
$\displaystyle P_{\bm{n}}^{\bm{\alpha};\beta+1}(x)\sim
P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta}(x)$

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ and any $1\leq j\leq r$.

####  3.9.4. The Angelesco–Jacobi polynomials

These are the type II multiple orthogonal polynomials
$P_{\bm{n}}^{\alpha,\beta,\gamma}$ corresponding to the system
$\bm{\mu}^{\alpha,\beta,\gamma}\in\mathcal{M}^{2}$ where

$\displaystyle d\mu^{\alpha,\beta,\gamma}_{1}$
$\displaystyle=(1-x)^{\alpha}(x-a)^{\beta}|x|^{\gamma}\chi_{[a,0]}\,dx,$ (82)
$\displaystyle d\mu^{\alpha,\beta,\gamma}_{2}$
$\displaystyle=(1-x)^{\alpha}(x-a)^{\beta}|x|^{\gamma}\chi_{[0,1]}\,dx,$ (83)

where $\alpha,\beta,\gamma>-1$, $a<0$. Denote
$\bm{A}_{\bm{n}}^{\alpha,\beta,\gamma}=({A}_{\bm{n},1}^{\alpha,\beta,\gamma},{A}_{\bm{n},2}^{\alpha,\beta,\gamma})$
to be the corresponding type I polynomials.

Since $\bm{\mu}^{\alpha,\beta,\gamma}\in\mathcal{M}^{2}$ is an Angelesco
system, Corollary 3.21 and Corollary 3.23 can be applied with any choice of
$z_{0}$ away from the zeros of type II multiple orthogonal polynomials which
are known to all belong to $(a,0)\cup(0,1)$. Note that in our notation,
$(x-1)\bm{\mu}^{\alpha,\beta,\gamma}=\bm{\mu}^{\alpha+1,\beta,\gamma}$,
$(x-a)\bm{\mu}^{\alpha,\beta,\gamma}=\bm{\mu}^{\alpha,\beta+1,\gamma}$, and
$x\bm{\mu}^{\alpha,\beta,\gamma}=\bm{\mu}^{\alpha,\beta,\gamma+1}$ up to a
trivial multiplicative normalization. Then

  * i)

$P_{\bm{n}}^{\alpha+1,\beta,\gamma}(x)\sim
P_{\bm{n}}^{\alpha,\beta,\gamma}(x)$ and
$P_{\bm{n}}^{\alpha+1,\beta,\gamma}(x)\sim
P^{\alpha,\beta,\gamma}_{{\bm{n}}+\bm{e}_{j}}(x)$;

  * ii)

$P_{\bm{n}}^{\alpha,\beta+1,\gamma}(x)\sim
P_{\bm{n}}^{\alpha,\beta,\gamma}(x)$ and
$P_{\bm{n}}^{\alpha,\beta+1,\gamma}(x)\sim
P^{\alpha,\beta,\gamma}_{{\bm{n}}+\bm{e}_{j}}(x)$;

  * iii)

$xP_{\bm{n}}^{\alpha,\beta,\gamma+1}(x)\sim
P_{\bm{n}}^{\alpha,\beta,\gamma}(x)$ and
$xP_{\bm{n}}^{\alpha,\beta,\gamma+1}(x)\sim
P^{\alpha,\beta,\gamma}_{{\bm{n}}+\bm{e}_{j}}(x)$;

  * iv)

$A_{\bm{n},j}^{\alpha+1,\beta,\gamma}(x)\sim
A_{\bm{n},j}^{\alpha,\beta,\gamma}(x)$ and
$A_{\bm{n},j}^{\alpha+1,\beta,\gamma}(x)\sim
A^{\alpha,\beta,\gamma}_{{\bm{n}}+\bm{e}_{j},j}(x)$;

  * v)

$A_{\bm{n},j}^{\alpha,\beta+1,\gamma}(x)\sim
A_{\bm{n},j}^{\alpha,\beta,\gamma}(x)$ and
$A_{\bm{n},j}^{\alpha,\beta+1,\gamma}(x)\sim
A^{\alpha,\beta,\gamma}_{{\bm{n}}+\bm{e}_{j},j}(x)$;

  * vi)

$A_{\bm{n},j}^{\alpha,\beta,\gamma+1}(x)\sim
A_{\bm{n},j}^{\alpha,\beta,\gamma}(x)$ and
$A_{\bm{n},j}^{\alpha,\beta,\gamma+1}(x)\sim
A^{\alpha,\beta,\gamma}_{{\bm{n}}+\bm{e}_{j},j}(x)$,

for any $\bm{n}\in\mathbb{Z}^{2}_{+}$ and any $j=1,2$.

Note that i)–iii) follows from Corollary 3.21. For i) and ii) the extra zero
at $a$ or at $1$ does not matter since all the zeros of type II polynomials
belong to $(a,1)$. iv)–vi) is immediate from Corollary 3.23.

####  3.9.5. The Jacobi–Laguerre polynomials

These are the type II multiple orthogonal polynomials
$P_{\bm{n}}^{\beta,\gamma}$ corresponding to the system
$\bm{\mu}^{\beta,\gamma}\in\mathcal{M}^{2}$ where

$\displaystyle d\mu^{\beta,\gamma}_{1}$
$\displaystyle=(x-a)^{\beta}|x|^{\gamma}e^{-x}\chi_{[a,0]}\,dx,$ (84)
$\displaystyle d\mu^{\beta,\gamma}_{2}$
$\displaystyle=(x-a)^{\beta}|x|^{\gamma}e^{-x}\chi_{[0,\infty)}\,dx,$ (85)

where $\beta,\gamma>-1$, $a<0$. Denote
$\bm{A}_{\bm{n}}^{\beta,\gamma}=({A}_{\bm{n},1}^{\beta,\gamma},{A}_{\bm{n},2}^{\beta,\gamma})$
to be the corresponding type I polynomials.

Since $\bm{\mu}^{\beta,\gamma}\in\mathcal{M}^{2}$ is an Angelesco system, and
$(x-a)\bm{\mu}^{\beta,\gamma}=\bm{\mu}^{\beta+1,\gamma}$,
$x\bm{\mu}^{\beta,\gamma}=\bm{\mu}^{\beta,\gamma+1}$ up to a trivial
multiplicative normalization, we get as in the previous section:

  * i)

$P_{\bm{n}}^{\beta+1,\gamma}(x)\sim P_{\bm{n}}^{\beta,\gamma}(x)$ and
$P_{\bm{n}}^{\beta+1,\gamma}(x)\sim
P^{\beta,\gamma}_{{\bm{n}}+\bm{e}_{j}}(x)$;

  * ii)

$xP_{\bm{n}}^{\beta,\gamma+1}(x)\sim P_{\bm{n}}^{\beta,\gamma}(x)$ and
$xP_{\bm{n}}^{\beta,\gamma+1}(x)\sim
P^{\beta,\gamma}_{{\bm{n}}+\bm{e}_{j}}(x)$;

  * iii)

$A_{\bm{n},j}^{\beta+1,\gamma}(x)\sim A_{\bm{n},j}^{\beta,\gamma}(x)$ and
$A_{\bm{n},j}^{\beta+1,\gamma}(x)\sim
A^{\beta,\gamma}_{{\bm{n}}+\bm{e}_{j},j}(x)$;

  * iv)

$A_{\bm{n},j}^{\beta,\gamma+1}(x)\sim A_{\bm{n},j}^{\beta,\gamma}(x)$ and
$A_{\bm{n},j}^{\beta,\gamma+1}(x)\sim
A^{\beta,\gamma}_{{\bm{n}}+\bm{e}_{j},j}(x)$.

for any $\bm{n}\in\mathbb{Z}^{2}_{+}$ and any $j=1,2$.

####  3.9.6. The Jacobi–Hermite polynomials

These are the type II multiple orthogonal polynomials $P_{\bm{n}}^{\gamma}$
corresponding to the system $\bm{\mu}^{\gamma}\in\mathcal{M}^{2}$ given by

$\displaystyle d\mu^{\gamma}_{1}$
$\displaystyle=|x|^{\gamma}e^{-x^{2}}\chi_{(-\infty,0]}\,dx,$ (86)
$\displaystyle d\mu^{\gamma}_{2}$
$\displaystyle=x^{\gamma}e^{-x^{2}/2}\chi_{[0,\infty)}\,dx,$ (87)

where $\gamma>-1$. Since $\bm{\mu}^{\gamma}\in\mathcal{M}^{2}$ is an Angelesco
system, and $x\bm{\mu}^{\gamma}=\bm{\mu}^{\gamma+1}$ up to a trivial
multiplicative normalization, we get:

  * i)

$xP_{\bm{n}}^{\gamma+1}(x)\sim P_{\bm{n}}^{\gamma}(x)$ and
$xP_{\bm{n}}^{\gamma+1}(x)\sim P^{\gamma}_{{\bm{n}}+\bm{e}_{j}}(x)$;

  * ii)

$A_{\bm{n},j}^{\gamma+1}(x)\sim A_{\bm{n},j}^{\gamma}(x)$ and
$A_{\bm{n},j}^{\gamma+1}(x)\sim A^{\gamma}_{{\bm{n}}+\bm{e}_{j},j}(x)$.

for any $\bm{n}\in\mathbb{Z}^{2}_{+}$ and any $j=1,2$.

Interlacings for the type II Angelesco–Jacobi, Jacobi–Laguerre, and
Jacobi–Hermite polynomials were proved by Martínez-Finkelshtein–Morales in
their recent [MarMor24, Thm 2.2] (see also [dosSan, Lem 2.1]) with more
involved arguments and for $\bm{n}$ along the stepline.

###  3.10. Zero interlacing: discrete examples

####  3.10.1. The multiple Charlier polynomials

These are the type II multiple orthogonal polynomials $P_{\bm{n}}^{\bm{a}}$
corresponding to the discrete system $\bm{\mu}^{\bm{a}}\in\mathcal{M}^{r}$
given by

$\mu_{j}^{\bm{a}}=\sum_{k=0}^{\infty}\frac{a_{j}^{k}}{k!}\delta_{k}$

where $a_{j}>0$, and $\alpha_{j}\neq\alpha_{s}$ for $j\neq s$.

Recall that $\bm{\mu}^{\bm{\alpha}}$ is an AT system. Now note that
$x\bm{\mu}^{\bm{a}}$ is supported on $\\{k\in\mathbb{Z}:k\geq 1\\}$ and its
weight at $\\{k\\}$ is
$\frac{a_{j}^{k}}{k!}k=a_{j}\frac{a_{j}^{k-1}}{(k-1)!}$. This means that up to
an inconsequential multiplicative normalization, $x\bm{\mu}^{\bm{a}}$ shifted
by $1$ to the left coincides with $\bm{\mu}^{\bm{a}}$ itself. Corollary 3.21
then gives

$P_{\bm{n}}^{\bm{a}}(x-1)\sim P_{\bm{n}}^{\bm{a}}(x)\quad\mbox{and}\quad
xP_{\bm{n}}^{\bm{a}}(x-1)\sim P_{\bm{n}+\bm{e}_{j}}^{\bm{a}}(x)$ (88)

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ and any $1\leq j\leq r$.

In particular, the self-interlacing property $xP_{\bm{n}}^{\bm{a}}(x-1)\sim
P_{\bm{n}}^{\bm{a}}(x)$ implies a number of interesting properties, see [Fisk,
Sect 8.7]. In particular, it is easy to see that (88) implies that the
distance between any two roots of $P_{\bm{n}}^{\bm{a}}$ is at least 1. See
Theorem 3.25 for a more general statement.

####  3.10.2. The multiple Meixner polynomials of the first kind

These are the type II multiple orthogonal polynomials
$P_{\bm{n}}^{\bm{c};\beta}$ corresponding to the discrete system
$\bm{\mu}^{\bm{c};\beta}\in\mathcal{M}^{r}$ given by

$\mu_{j}^{\bm{c};\beta}=\sum_{k=0}^{\infty}\frac{(\beta)_{k}c_{j}^{k}}{k!}\delta_{k}$

where $\beta>0$, $0<c_{j}<1$, and $c_{j}\neq c_{s}$ for $j\neq s$. Here
$(x)_{0}=1$, $(x)_{k}=x(x+1)\ldots(x+k-1)$ is the Pochhammer symbol.

Again, $\bm{\mu}^{\bm{c};\beta}$ is an AT system, and
$x\bm{\mu}^{\bm{c};\beta}$ is supported on $\\{k\in\mathbb{Z}:k\geq 1\\}$ with
the weight at $\\{k\\}$ ($k\geq 1$) being
$\frac{(\beta)_{k}c_{j}^{k}}{k!}k=\beta
c_{j}\frac{(\beta+1)_{k-1}c_{j}^{k-1}}{(k-1)!}$. This means that up to an
inconsequential multiplicative normalization, $x\bm{\mu}^{\bm{c};\beta}$
shifted by $1$ to the left coincides with $\bm{\mu}^{\bm{c};\beta+1}$.
Corollary 3.21 then gives

$P_{\bm{n}}^{\bm{c};\beta+1}(x-1)\sim
P_{\bm{n}}^{\bm{c};\beta}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{\bm{c};\beta+1}(x-1)\sim P_{\bm{n}+\bm{e}_{j}}^{\bm{c};\beta}(x)$

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ and any $1\leq j\leq r$.

####  3.10.3. The multiple Meixner polynomials of the second kind

These are the type II multiple orthogonal polynomials
$P_{\bm{n}}^{c;\bm{\beta}}$ corresponding to the discrete system
$\bm{\mu}^{c;\bm{\beta}}\in\mathcal{M}^{r}$ given by

$\mu_{j}^{c;\bm{\beta}}=\sum_{k=0}^{\infty}\frac{(\beta_{j})_{k}c^{k}}{k!}\delta_{k}$

where $\beta_{j}>0$, $0<c<1$, and $\beta_{j}-\beta_{s}\notin\mathbb{Z}$ for
$j\neq s$.

Again, $\bm{\mu}^{c;\bm{\beta}}$ is an AT system, and up to an inconsequential
multiplicative normalization, $x\bm{\mu}^{c;\bm{\beta}}$ shifted by $1$ to the
left coincides with $\bm{\mu}^{c;\bm{\beta}+\bm{1}}$. Corollary 3.21 then
gives

$P_{\bm{n}}^{c;\bm{\beta}+\bm{1}}(x-1)\sim
P_{\bm{n}}^{c;\bm{\beta}}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{c;\bm{\beta}+\bm{1}}(x-1)\sim
P_{\bm{n}+\bm{e}_{j}}^{c;\bm{\beta}}(x)$

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ and any $1\leq j\leq r$.

####  3.10.4. The multiple Krawtchouk polynomials

These are the type II multiple orthogonal polynomials $P_{\bm{n}}^{N;\bm{p}}$
corresponding to the discrete finite system
$\bm{\mu}^{N;\bm{p}}\in\mathcal{M}^{r}$ given by

$\mu_{j}^{N;\bm{p}}=\sum_{k=0}^{N}\binom{N}{k}p_{j}^{k}(1-p_{j})^{N-k}\delta_{k}$

where $N\in\mathbb{Z}_{>0}$, $0<p_{j}<1$, and $p_{j}-p_{s}\neq 0$ for $j\neq
s$.

Again, $\bm{\mu}^{N;\bm{p}}$ is a discrete AT system (see [Discretemoprl]),
and up to an inconsequential multiplicative normalization,
$x\bm{\mu}^{N;\bm{p}}$ shifted by $1$ to the left coincides with
$\bm{\mu}^{N-1;\bm{p}}$. Corollary 3.21 then gives

$P_{\bm{n}}^{N-1;\bm{p}}(x-1)\sim P_{\bm{n}}^{N;\bm{p}}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{N-1;\bm{p}}(x-1)\sim P_{\bm{n}+\bm{e}_{j}}^{N;\bm{p}}(x)$

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ with $|\bm{n}|<N$ and any $1\leq j\leq
r$.

Similarly, up to an inconsequential multiplicative normalization,
$(N-x)\bm{\mu}^{N;\bm{p}}$ coincides with $\bm{\mu}^{N-1;\bm{p}}$. Corollary
3.21 then gives

$P_{\bm{n}}^{N-1;\bm{p}}(x)\sim P_{\bm{n}}^{N;\bm{p}}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{N-1;\bm{p}}(x)\sim P_{\bm{n}+\bm{e}_{j}}^{N;\bm{p}}(x)$

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ with $|\bm{n}|<N$ and any $1\leq j\leq
r$.

####  3.10.5. The multiple Hahn polynomials

These are the type II multiple orthogonal polynomials
$P_{\bm{n}}^{\bm{\alpha};\beta;N}$ corresponding to the discrete finite system
$\bm{\mu}^{\bm{\alpha};\beta;N}\in\mathcal{M}^{r}$ given by

$\mu_{j}^{\bm{\alpha};\beta;N}=\sum_{k=0}^{N}\frac{(\alpha_{j}+1)_{k}(\beta+1)_{N-k}}{k!(N-k)!}\delta_{k}$

where $N\in\mathbb{Z}_{>0}$, $\beta>-1$, $\alpha_{j}>-1$, and
$\alpha_{j}-\alpha_{s}\neq 0$ for $j\neq s$.

Again, $\bm{\mu}^{\bm{\alpha};\beta;N}$ is a discrete AT system (see
[Discretemoprl]), and up to an inconsequential multiplicative normalization,
$x\bm{\mu}^{\bm{\alpha};\beta;N}$ shifted by $1$ to the left coincides with
$\bm{\mu}^{\bm{\alpha};\beta;N-1}$, and $(N-x)\bm{\mu}^{\bm{\alpha};\beta;N}$
coincides with $\bm{\mu}^{\bm{\alpha};\beta;N-1}$. Corollary 3.21 then gives

$P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x-1)\sim
P_{\bm{n}}^{\bm{\alpha};\beta;N}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x-1)\sim
P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta;N}(x)$ (89)

and

$P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x)\sim
P_{\bm{n}}^{\bm{\alpha};\beta;N}(x)\quad\mbox{and}\quad
P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x)\sim
P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta;N}(x)$ (90)

for any $\bm{n}\in\mathbb{Z}^{r}_{+}$ with $|\bm{n}|<N$ and any $1\leq j\leq
r$.

###  3.11. Minimal distance between roots

In Section 3.10.1 we observed that the minimal distance between consecutive
roots (sometimes called the mesh) of any type II multiple Charlier polynomial
is larger than 1. In the next theorem we generalize this to all the other
discrete multiple orthogonal polynomials discussed here. This also serves as a
new elementary proof for the case of one measure $r=1$, for which this
property has been well known, see [ChiSta, Lev67, KraZar].

######  Theorem 3.25.

The minimal distance between roots of any type II multiple Hahn, multiple
Krawtchouk, or multiple Charlier polynomial is larger than 1. The minimal
distance between roots of any type II multiple Meixner polynomial of the first
or second kind is not smaller than 1.

###### Proof.

For the Charlier case this is clear from (88).

Consider the multiple Hahn case. Since
${P^{\bm{\alpha;\beta;N}}_{\bm{n}+\bm{e}_{j}}(0)}/{P^{\bm{\alpha;\beta;N}}_{\bm{n}}(0)}>0$
and
${P^{\bm{\alpha;\beta;N}}_{\bm{n}+\bm{e}_{j}}(N)}/{P^{\bm{\alpha;\beta;N}}_{\bm{n}}(N)}<0$
for any $\bm{n}$, then (61) together with Lemma 3.20 allows to improve (89)
and (90) to a stronger statement
$xP_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x-1)\,\bm{\mathcal{\prec}}\,P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta;N}(x)$
and
$(N-x)P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x-1)\,\bm{\mathcal{\succ}}\,P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta;N}(x)$.

That implies that below the first zero of
$P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta;N}(x)$ there is no zero of either
$P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x-1)$ nor
$P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x)$. Then between the first and the second
zero of $P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta;N}(x)$ there is exactly one
zero of $P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x-1)$ and of
$P_{\bm{n}}^{\bm{\alpha};\beta;N-1}(x)$. This proves that the distance between
the first two zeros of $P_{\bm{n}+\bm{e}_{j}}^{\bm{\alpha};\beta;N}(x)$ is
larger than one. An induction completes the proof.

The proof for the multiple Krawtchouk polynomials is identical.

For the Meixner of the first and second kind, recall that they can be obtained
from the Hahn polynomials by taking a certain limit with respect to the
coefficients $\bm{\alpha},\beta,N$, see [BCVA]. Since the minimal distance for
the Hahn polynomials is $>1$, we get $\geq 1$ in the limit. ∎

  * []
AptekarevA.I.Multiple orthogonal polynomialsJournal of Computational and
Applied Mathematics991998423–447@article{Aptekarev, author = {Aptekarev,
A.I.}, title = {Multiple orthogonal polynomials}, journal = {Journal of
Computational and Applied Mathematics}, volume = {99}, year = {1998}, pages =
{423-447}} AptekarevA.I.DerevyaginM.MikiH.Van AsscheW.Multidimensional Toda
lattices: continuous and discrete timeSIGMA Symmetry Integrability Geom.
Methods Appl.122016Paper No. 054, 30@article{ADMVA, author = {Aptekarev,
A.I.}, author = {Derevyagin, M.}, author = {Miki, H.}, author = {Van Assche,
W.}, title = {Multidimensional {T}oda lattices: continuous and discrete time},
journal = {SIGMA Symmetry Integrability Geom. Methods Appl.}, volume = {12},
year = {2016}, pages = {Paper No. 054, 30}} AptekarevA.I.DerevyaginM.Van
AsscheW.Discrete integrable systems generated by hermite-padé
approximantsNonlinearity29520161487–1506@article{IntegrableSystems, author =
{Aptekarev, A.I.}, author = {Derevyagin, M.}, author = {Van Assche, W.}, title
= {Discrete integrable systems generated by Hermite-Padé approximants},
journal = {Nonlinearity}, volume = {29}, number = {5}, year = {2016}, pages =
{1487-1506}} AptekarevA.I.DenisovS.A.YattselevM.L.Self-adjoint jacobi matrices
on trees and multiple orthogonal polynomialsTransactions of the American
Mathematical Society37322020875–917@article{Jacobioperatorontrees, author =
{Aptekarev, A.I.}, author = {Denisov, S.A.}, author = {Yattselev, M.L.}, title
= {Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials},
journal = {Transactions of the American Mathematical Society}, volume = {373},
number = {2}, year = {2020}, pages = {875-917}} ArvesúJ.CoussementJ.Van
AsscheW.Some discrete multiple orthogonal polynomialsJournal of Computational
and Applied Mathematics153200119–45@article{Discretemoprl, author = {Arvesú,
J.}, author = {Coussement, J.}, author = {Van Assche, W.}, title = {Some
Discrete Multiple Orthogonal Polynomials}, journal = {Journal of Computational
and Applied Mathematics}, volume = {153}, year = {2001}, pages = {19-45}}
BeckermannB.CoussementJ.Van AsscheW.Multiple wilson and jacobi-piñeiro
polynomialsJ. Approx. Theory13220052155–181@article{BCVA, author =
{Beckermann, B.}, author = {Coussement, J.}, author = {Van Assche, W.}, title
= {Multiple Wilson and Jacobi-Pi\~neiro polynomials}, journal = {J. Approx.
Theory}, volume = {132}, date = {2005}, number = {2}, pages = {155–181}}
BranquinhoA.Foulquié-MorenoA.MañasM.Multiple orthogonal polynomials: pearson
equations and christoffel formulasAnal. Math. Phys.1220226Paper No. 129,
59@article{BFM22, author = {Branquinho, A.}, author = {Foulqui\'{e}-Moreno,
A.}, author = {Ma\~{n}as, M.}, title = {Multiple orthogonal polynomials:
Pearson equations and Christoffel formulas}, journal = {Anal. Math. Phys.},
volume = {12}, date = {2022}, number = {6}, pages = {Paper No. 129, 59}}
BuenoM.I.DopicoF.M.A more accurate algorithm for computing the christoffel
transformationsJournal of Computational and Applied
Mathematics2052007567–582@article{BuenoandDopico, author = {Bueno, M.I.},
author = {Dopico, F.M.}, title = {A more accurate algorithm for computing the
Christoffel transformations}, journal = {Journal of Computational and Applied
Mathematics}, volume = {205}, year = {2007}, pages = {567-582}}
ChiharaL.StantonD.Zeros of generalized krawtchouk polynomialsJ. Approx.
Theory601990143–57ISSN 0021-9045@article{ChiSta, author = {Chihara, L.},
author = {Stanton, D.}, title = {Zeros of generalized Krawtchouk polynomials},
journal = {J. Approx. Theory}, volume = {60}, date = {1990}, number = {1},
pages = {43–57}, issn = {0021-9045}} ChiharaT.S.An introduction to orthogonal
polynomialsISBN 9780486479293Mathematics and Its Applications13Gordon and
Breach Science Publishers, Inc.1978@book{Chihara, author = {Chihara, T.S.},
title = {An Introduction to Orthogonal Polynomials}, isbn = {9780486479293},
series = {Mathematics and Its Applications}, volume = {13}, publisher =
{Gordon and Breach Science Publishers, Inc.}, year = {1978}} Über die
gaußische quadratur und eine verallgemeinerung
derselben.ChristoffelE.B.Journal für die Reine und Angewandte Mathematik
(Crelles Journal)55185861–82@article{Christoffel, title = {{\"U}ber die
Gau{\ss}ische Quadratur und eine Verallgemeinerung derselben.}, author =
{Christoffel, E.B.}, journal = {Journal f{\"u}r die Reine und Angewandte
Mathematik (Crelles Journal)}, volume = {55}, year = {1858}, pages = {61-82}}
Cruz-BarrosoR.Díaz MendozaC.OriveR.Multiple orthogonal polynomials on the unit
circle. normality and recurrence relationsJ. Comput. Appl.
Math.2842015115–132@article{mopucrecurrence, author = {Cruz-Barroso, R.},
author = {Díaz Mendoza, C.}, author = {Orive, R.}, title = {Multiple
orthogonal polynomials on the unit circle. Normality and recurrence
relations}, journal = {J. Comput. Appl. Math.}, volume = {284}, year = {2015},
pages = {115–132}} DaemsE.KuijlaarsA. B. J.A christoffel-darboux formula for
multiple orthogonal polynomialsJ. Approx. Theory13020042190–202@article{DK04,
author = {Daems, E.}, author = {Kuijlaars, A. B. J.}, title = {A Christoffel-
Darboux formula for multiple orthogonal polynomials}, journal = {J. Approx.
Theory}, volume = {130}, date = {2004}, number = {2}, pages = {190–202}}
DenisovS. A.YattselevM. L.Spectral theory of jacobi matrices on trees whose
coefficients are generated by multiple orthogonalityAdv. Math.3962022Paper No.
108114, 79@article{DenYat, author = {Denisov, S. A.}, author = {Yattselev, M.
L.}, title = {Spectral theory of Jacobi matrices on trees whose coefficients
are generated by multiple orthogonality}, journal = {Adv. Math.}, volume =
{396}, date = {2022}, pages = {Paper No. 108114, 79}} dos SantosE. J.
C.Monotonicity of zeros of jacobi-angelesco polynomialsProc. Amer. Math.
Soc.1452017114741–4750@article{dosSan, author = {dos Santos, E. J. C.}, title
= {Monotonicity of zeros of Jacobi-Angelesco polynomials}, journal = {Proc.
Amer. Math. Soc.}, volume = {145}, date = {2017}, number = {11}, pages =
{4741–4750}} DoliwaA.Hermite-Padé approximation, multiple orthogonal
polynomials, and multidimensional Toda equationsarXiv:2310.15116@article{Dol,
author = {Doliwa, A.}, title = {Hermite-{P}ad\'{e} approximation, multiple
orthogonal polynomials, and multidimensional {T}oda equations}, journal =
{{}}, volume = {{}}, date = {{}}, number = {{}}, pages = {arXiv:2310.15116}}
DyachenkoA.LysovV.Discrete multiple orthogonal polynomials on shifted
lattices2019@article{DyachenkoandLysov, author = {Dyachenko, A.}, author =
{Lysov, V.}, title = {Discrete multiple orthogonal polynomials on shifted
lattices}, year = {2019}} FilipukG.HaneczokM.Van AsscheW.Computing recurrence
coefficients of multiple orthogonal polynomialsNumerical
Algorithms7020153519–543@article{ComputingNNR, author = {Filipuk, G.}, author
= {Haneczok, M.}, author = {Van Assche, W.}, title = {Computing recurrence
coefficients of multiple orthogonal polynomials}, journal = {Numerical
Algorithms}, volume = {70}, year = {2015}, number = {3}, pages = {519-543}}
Fidalgo PrietaU.Medina PeraltaS.Mínguez CenicerosJ.Mixed type multiple
orthogonal polynomials: perfectness and interlacing properties of zerosLinear
Algebra Appl.438201331229–1239@article{FMM, author = {Fidalgo Prieta, U.},
author = {Medina Peralta, S.}, author = {M\'inguez Ceniceros, J.}, title =
{Mixed type multiple orthogonal polynomials: perfectness and interlacing
properties of zeros}, journal = {Linear Algebra Appl.}, volume = {438}, date =
{2013}, number = {3}, pages = {1229–1239}} FiskS.Polynomials, roots, and
interlacing, v.2https://arxiv.org/abs/math/0612833v2@article{Fisk, author =
{Fisk, S.}, title = {Polynomials, roots, and interlacing, v.2}, journal =
{{}}, volume = {{}}, date = {}, number = {{}}, pages =
{https://arxiv.org/abs/math/0612833v2}} GalantD.An implementation of
christoffel’s theorem in the theory of orthogonal polynomialsMathematics of
Computation251971111–113@article{Galant, author = {Galant, D.}, title = {An
Implementation of Christoffel's Theorem in the Theory of Orthogonal
Polynomials}, journal = {Mathematics of Computation}, volume = {25}, year =
{1971}, pages = {111-113}} GautschiW.An algorithmic implementation of the
generalized christoffel theoremNumerical Integration, International Series of
Numerical Mathematics57198289–106@article{Gautschi, author = {Gautschi, W.},
title = {An Algorithmic Implementation of the Generalized Christoffel
Theorem}, journal = {Numerical Integration, International Series of Numerical
Mathematics}, volume = {57}, year = {1982}, pages = {89-106}}
GautschiW.Orthogonal polynomials, computation and approximationISBN
9780198506720Numerical Mathematics and Scientific ComputationOxford University
Press2004@book{Gautschibook, author = {Gautschi, W.}, title = {Orthogonal
Polynomials, Computation and Approximation}, isbn = {9780198506720}, series =
{Numerical Mathematics and Scientific Computation}, publisher = {Oxford
University Press}, year = {2004}} GuseinovG. Sh.Inverse spectral problems for
tridiagonal $N$ by $N$ complex hamiltoniansSIGMA Symmetry Integrability Geom.
Methods Appl.52009Paper 018, 28@article{Gus, author = {Guseinov, G. Sh.},
title = {Inverse spectral problems for tridiagonal $N$ by $N$ complex
Hamiltonians}, journal = {SIGMA Symmetry Integrability Geom. Methods Appl.},
volume = {5}, date = {2009}, pages = {Paper 018, 28}} HaneczokM.Van
AsscheW.Interlacing properties of zeros of multiple orthogonal
polynomialsJournal of Mathematical Analysis and
Applications2012389429–438@article{Interlacing, author = {Haneczok, M.},
author = {Van Assche, W.}, title = {Interlacing properties of zeros of
multiple orthogonal polynomials}, journal = {Journal of Mathematical Analysis
and Applications}, year = {2012}, volume = {389}, pages = {429-438}}
IsmailM.E.H.Classical and quantum orthogonal polynomials in one variableISBN
9780521782012Encyclopedia of Mathematics and its Applications98Cambridge
University Press2005@book{Ismail, author = {Ismail, M.E.H.}, title =
{Classical and Quantum Orthogonal Polynomials in One Variable}, isbn =
{9780521782012}, series = {Encyclopedia of Mathematics and its Applications},
volume = {98}, publisher = {Cambridge University Press}, year = {2005}}
KozhanR.VaktnäsM.Determinantal formulas for rational perturbations of multiple
orthogonality measuresin preparation@article{KVGeronimus, author = {Kozhan,
R.}, author = {Vaktn\"{a}s, M.}, title = {Determinantal Formulas for Rational
Perturbations of Multiple Orthogonality Measures}, journal = {{}}, volume =
{{}}, date = {{}}, number = {{}}, pages = {in preparation}}
KozhanR.VaktnäsM.Szegő recurrence for multiple orthogonal polynomials on the
unit circleProc. Amer. Math. Soc.152202472983–2997@article{KVMOPUC, author =
{Kozhan, R.}, author = {Vaktn\"as, M.}, title = {Szeg\H o{} recurrence for
multiple orthogonal polynomials on the unit circle}, journal = {Proc. Amer.
Math. Soc.}, volume = {152}, date = {2024}, number = {7}, pages = {2983–2997}}
KrasikovI.ZarkhA.On zeros of discrete orthogonal polynomialsJ. Approx.
Theory15620092121–141@article{KraZar, author = {Krasikov, I.}, author =
{Zarkh, A.}, title = {On zeros of discrete orthogonal polynomials}, journal =
{J. Approx. Theory}, volume = {156}, date = {2009}, number = {2}, pages =
{121–141}} LevitR. J.The zeros of the hahn polynomialsSIAM
Rev.91967191–203@article{Lev67, author = {Levit, R. J.}, title = {The zeros of
the Hahn polynomials}, journal = {SIAM Rev.}, volume = {9}, date = {1967},
pages = {191–203}} MañasM.RojasM.General christoffel perturbations for mixed
multiple orthogonal polynomialsarXiv:2405.11630@article{ManasRojas, author =
{Ma\~{n}as, M.}, author = {Rojas, M.}, title = {General Christoffel
Perturbations for Mixed Multiple Orthogonal Polynomials}, pages =
{arXiv:2405.11630}} Martínez-FinkelshteinA.MoralesR.Interlacing and
monotonicity of zeros of angelesco–jacobi polynomialsPure and Applied
Functional Analysisto appear2024@article{MarMor24, author = {Mart\'{i}nez-
Finkelshtein, A.}, author = {Morales, R.}, title = {Interlacing and
monotonicity of zeros of Angelesco–Jacobi polynomials}, journal = {Pure and
Applied Functional Analysis}, pages = {to appear}, year = {2024}} Martínez-
FinkelshteinA.MoralesR.PeralesD.Zeros of generalized hypergeometric
polynomials via finite free convolution. applications to multiple
orthogonalityhttps://arxiv.org/abs/2404.11479@article{MarMorPer, author =
{Mart\'{i}nez-Finkelshtein, A.}, author = {Morales, R.}, author = {Perales,
D.}, title = {Zeros of generalized hypergeometric polynomials via finite free
convolution. Applications to multiple orthogonality}, journal = {{}}, volume =
{{}}, date = {{}}, number = {{}}, pages = {https://arxiv.org/abs/2404.11479}}
Martínez-FinkelshteinA.Van AsscheWWHAT is…a multiple orthogonal
polynomial?Notices of the American Mathematical
Society6320161029–1031@article{Applications, author = {Martínez-Finkelshtein,
A.}, author = {Van Assche, W}, title = {WHAT IS…A Multiple Orthogonal
Polynomial?}, journal = {Notices of the American Mathematical Society}, volume
= {63}, year = {2016}, pages = {1029-1031}} NikishinE.M.SorokinV.N.Rational
approximations and orthogonalityISBN 9780821845455Translations of Mathematical
Monographs92American Mathematical Society1991@book{Nikishin, author =
{Nikishin, E.M.}, author = {Sorokin, V.N.}, title = {Rational Approximations
and Orthogonality}, isbn = {9780821845455}, series = {Translations of
Mathematical Monographs}, volume = {92}, publisher = {American Mathematical
Society}, year = {1991}} SpiridonovV.ZhedanovA.Discrete darboux
transformations, the discrete-time toda lattice, and the askey-wilson
polynomialsMethods Appl. Anal.219954369–398@article{SpiZhe, author =
{Spiridonov, V.}, author = {Zhedanov, A.}, title = {Discrete Darboux
transformations, the discrete-time Toda lattice, and the Askey-Wilson
polynomials}, journal = {Methods Appl. Anal.}, volume = {2}, date = {1995},
number = {4}, pages = {369–398}} RutishauserH.Der quotienten-differenzen-
algorithmusGermanZ. Angew. Math. Phys.51954233–251ISSN 0044-2275@article{QD,
author = {Rutishauser, H.}, title = {Der Quotienten-Differenzen-Algorithmus},
language = {German}, journal = {Z. Angew. Math. Phys.}, volume = {5}, date =
{1954}, pages = {233–251}, issn = {0044-2275}} SimonB.Szegő’s theorem and its
descendants: spectral theory for $L^{2}$ perturbations of orthogonal
polynomialsISBN 9780691147048Porter LecturesPrinceton University
Press2011@book{Simon, author = {Simon, B.}, title = {Szegő's Theorem and Its
Descendants: Spectral Theory for $L^2$ Perturbations of Orthogonal