# Zeros of generalized hypergeometric polynomials via finite free convolution.
Applications to multiple orthogonality

Andrei Martínez-Finkelshtein  Department of Mathematics, Baylor University,
TX, USA, and Department of Mathematics, University of Almería, Spain
<EMAIL_ADDRESS>,  Rafael Morales  Department of
Mathematics, Baylor University, TX, USA<EMAIL_ADDRESS>and
Daniel Perales  Department of Mathematics, Texas A&M University, TX, USA
<EMAIL_ADDRESS>
###### Abstract.

We address the problem of the weak asymptotic behavior of zeros of families of
generalized hypergeometric polynomials as their degree tends to infinity. The
main tool is the representation of such polynomials as a finite free
convolution of simpler elements; this representation is preserved in the
asymptotic regime, so we can formally write the limit zero distribution of
these polynomials as a free convolution of explicitly computable measures. We
derive a simple expression for the $S$-transform of the limit distribution,
which turns out to be a rational function, and a representation of the Kampé
de Fériet polynomials in terms of finite free convolutions.

We apply these tools, as well as those from [38], to the study of some well-
known families of multiple orthogonal polynomials (Jacobi-Piñeiro and multiple
Laguerre of the first and second kinds), obtaining results on their zeros,
such as interlacing, monotonicity, and asymptotics.

###### Key words and phrases:

Hypergeometric polynomials; Finite free convolution; Free probability;
Multiple ortogonal polynomials; Zero asymptotics

###### 2020 Mathematics Subject Classification:

Primary: 33C45; Secondary: 33C20, 42C05, 46L54

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Hypergeometric polynomials
    3. 2.3 Finite free convolutions
    4. 2.4 Real roots, interlacing, and free finite convolution
    5. 2.5 Integral transforms and free convolution of measures
  3. 3 General results
    1. 3.1 Kampé de Fériet polynomials and finite free convolution
    2. 3.2 Finite free convolutions in the asymptotic regime
    3. 3.3 S-transform for a hypergeometric polynomial
  4. 4 Type I Multiple orthogonal polynomials
    1. 4.1 Type I Jacobi-Piñeiro polynomials
      1. 4.1.1 Real zeros: monotonicity and interlacing
      2. 4.1.2 Zero asymptotics of Jacobi-Piñeiro polynomials of Type I
    2. 4.2 Type I Multiple Laguerre polynomials of the first kind
      1. 4.2.1 Real zeros: monotonicity and interlacing
      2. 4.2.2 Zero asymptotics of Type I Multiple Laguerre polynomials of first kind
    3. 4.3 Type I Multiple Laguerre of the second kind
      1. 4.3.1 Zero asymptotics of Type I Multiple Laguerre polynomials of the second kind
  5. 5 Type II Multiple orthogonal polynomials
    1. 5.1 Type II Jacobi-Piñeiro polynomials
      1. 5.1.1 Real zeros: monotonicity and interlacing
      2. 5.1.2 Zero asymptotics of Type II Jacobi-Piñeiro polynomials
    2. 5.2 Type II Multiple Laguerre polynomials of the first kind
      1. 5.2.1 Real zeros: monotonicity and interlacing
      2. 5.2.2 Zero asymptotics of Type II Multiple Laguerre polynomials of the first kind
    3. 5.3 Type II Multiple Laguerre of the second kind
      1. 5.3.1 Zero asymptotics of Type II Multiple Laguerre polynomials of the second kind
  6. 6 Further examples
  7. A Outline of the proof of Theorem 3.3

##  1\. Introduction

The definition of the generalized hypergeometric function ${}_{i}F_{j}$ with
$i$ numerator and $j$ denominator parameters is well known. If one of the
numerator parameters is equal to a negative integer, say $-n$, with
$n\in\mathbb{N}$, then the series terminates and is a polynomial of degree
$\leq n$. If all its zeros are real, we usually want to know their properties,
such as positivity/negativity, interlacing, and monotonicity with respect to
the parameters. In the case of a sequence of such polynomials, enumerated by
their degree $n$, we also want to investigate their asymptotic behavior as
$n\to\infty$.

For small values of $i$ and $j$, the answer to these questions can usually be
obtained by exploiting their connection to some classical families of
polynomials, in many cases orthogonal, or using their other properties, such
as the differential equation or integral representation. But when $i,j\geq 2$,
the problem becomes more difficult due to the limited number of tools that
allow us to investigate their behavior.

The finite free convolutions (that in this paper come in two flavors,
multiplicative $\boxtimes_{n}$ and additive $\boxplus_{n}$), are binary
operations on polynomials, studied already by Szegő, Schur, Walsh, and others,
although under different names. They behave especially well when applied to
real-rooted polynomials, preserving zero interlacing and monotonicity.
Recently, such convolutions have been rediscovered as expected characteristic
polynomials of a multiplication (or addition) of random matrices [36], and
were also interpreted as finite analogs of the free probability [35] (thus,
named generically as finite free convolution of polynomials).

The connection between these polynomial convolutions and free probability is
revealed in the asymptotic regime, when we consider the zero-counting measure
(also known in this context as the empirical root distribution) of a sequence
of polynomials whose degree tends to infinity. Then the finite free
convolution of polynomials turns into a free convolution of limiting
distributions of their zeros [4, 5].

In a recent paper [38], the authors illustrated the power of finite free
convolution of polynomials to prove the real-rootedness, interlacing, or
monotonicity of zeros with respect to the parameters. The key tool was the
representation of generalized hypergeometric polynomials as finite free
convolutions of simpler “building blocks”, much easier to study. We also
briefly explained the potential of this approach in the study of asymptotic
behavior.

This work is, in a certain sense, a natural continuation of [38]. Here, the
main focus is precisely on the weak asymptotic behavior of zeros of families
of generalized hypergeometric polynomials as their degree tends to infinity.
Using the representation of a generalized hypergeometric polynomial as a
finite free convolution of some simpler elements, we can formally write the
limit zero distribution of these polynomials as a free convolution of simpler
measures. In order to convert this observation into an effective computational
tool, we derive in Section 3.3 an expression for the so-called $S$-transform
of the limit zero-counting measures of the original polynomials. Unlike the
case of the Cauchy transform of such a measure, which is normally an algebraic
function, the $S$-transform turns out to be a rational function, easily
expressible in terms of the main parameters of the problem.

In the second part of the article, we apply these tools, as well as those from
[38], to the study of some well-known families of multiple orthogonal
polynomials. While writing the paper, we became aware that the recent
contribution [55] mentions some connections of multiple orthogonal polynomials
and finite free convolution using the results from [38].

Multiple, also known as Hermite-Padé, orthogonal polynomials (MOPs), are
polynomials of one variable that satisfy the orthogonality conditions with
respect to several measures. They are a very useful extension of orthogonal
polynomials and have recently received renewed interest because tools have
become available to investigate their asymptotic behavior, and they do appear
in a number of fascinating applications, see [40].

In this paper, we consider only the case of absolutely continuous
orthogonality measures on the real line. Thus, let $r$ be a positive integer
and $w_{1},w_{2},\ldots,w_{r}$ be non-negative integrable functions
(“weights”) on $\mathbb{R}$ for which all the moments are finite. Let
$\bm{n}=(n_{1},n_{2},\ldots,n_{r})\in\mathbb{N}^{r}$ be a multi-index of size
$|\bm{n}|=n_{1}+n_{2}+\cdots+n_{r}$. There are two types of multiple
orthogonal polynomials. Type I multiple orthogonal polynomials are given as a
vector $(A_{\bm{n},1},A_{\bm{n},2},\ldots,A_{\bm{n},r})$ of $r$ polynomials,
where $A_{\bm{n},j}$ has degree $\leq n_{j}-1$, for which the function

$Q_{\bm{n}}(x)\overset{\mathrm{def}}{=\joinrel=}\sum_{j=1}^{r}A_{\bm{n},j}(x)w_{j}(x)$
(1)

is orthogonal to all polynomials of degree $\leq|\bm{n}|-2$:

$\int x^{k}Q_{\bm{n}}(x)\,dx=\sum_{j=1}^{r}\int
x^{k}A_{\bm{n},j}(x)w_{j}(x)\,dx=0,\qquad 0\leq k\leq|\bm{n}|-2.$ (2)

One usually adds the normalization

$\int x^{|\bm{n}|-1}Q_{\bm{n}}(x)\,dx=1.$ (3)

The Type II multiple orthogonal polynomial $P_{\bm{n}}$ is the monic
polynomial of degree $|\bm{n}|$ that satisfies the orthogonality conditions

$\int P_{\bm{n}}(x)x^{k}w_{j}(x)\,dx=0,\qquad 0\leq k\leq n_{j}-1,$ (4)

for $1\leq j\leq r$. Both conditions (2)–(3) and condition (4) yield a
corresponding linear system of $|\bm{n}|$ equations in the $|\bm{n}|$
unknowns, either coefficients of polynomials
$(A_{\bm{n},1},\dots,A_{\bm{n},r})$ or coefficients of the monic polynomial
$P_{\bm{n}}$. The matrices of these linear systems are each other’s transpose
and contain moments of the $r$ weights $(w_{1},\ldots,w_{r})$. A solution of
these linear systems may not exist or may not be unique. One needs extra
assumptions on the weights $(w_{1},\ldots,w_{r})$ for a solution to exist and
to be unique. If a unique solution exists for a multi-index $\bm{n}$ then the
multi-index is said to be normal. If all multi-indices are normal, then the
system $(w_{1},\ldots,w_{r})$ is said to be a perfect system.

The weights $(w_{1},\ldots,w_{r})$ form an AT-system on an interval
$\Delta\subset\mathbb{R}$ for a multi-index $\bm{n}\in\mathbb{N}^{r}$ if for
any polynomials $A_{\bm{n},j}$, $j=1,\ldots,r$, in (1), satisfying the
mentioned degree constraints and not all equal to $0$, the function
$Q_{\bm{n}}$ has at most $|\bm{n}|-1$ zeros on $\Delta$, see, e.g. [21,
Chapter 23]. It is known that if $(w_{1},\ldots,w_{r})$ is an AT-system for
every multi-index $\bm{n}\in\mathbb{N}^{r}$, the system is perfect. An example
of such systems are the Nikishin systems, fact proved in [13, 14]. Since this
is not central to our discussion, for the definition of such systems we refer
the reader to [34, 32, 31].

For every AT-system and for any $\bm{n}\in\mathbb{N}^{r}$, the Type I function
for $Q_{\bm{n}}$ defined by (1) and satisfying (2), has exactly $|\bm{n}|-1$
sign changes in $\Delta$, while the Type II multiple orthogonal polynomial
$P_{\bm{n}}$, satisfying (4), has $|\bm{n}|$ simple zeros on $\Delta$. These
zeros exhibit an interlacing property, meaning that there is always a zero of
$P_{\bm{n}}$ between two consecutive zeros of $P_{\bm{n}+\bm{e}_{k}}$, for
each $0\leq k\leq r-1$, where $\bm{e}_{k}\in\mathbb{N}^{r}$ is the multi-index
that has all entries equal to 0 except the entry of index $k$ which is equal
to 1.

Various families of special MOPs have been found, extending the classical
orthogonal polynomials, but also related to completely new special functions
[1], [21, Ch. 23].

One of the well-known AT-systems is given by the weights

$w_{j}(x)=x^{\alpha_{j}}(1-x)^{\beta},\quad j=1,2,\dots,r,$ (5)

on the interval $\Delta=[0,1]$. Here $\alpha_{1},\ldots,\alpha_{r},\beta>-1$
and $\alpha_{i}-\alpha_{j}\notin\mathbb{Z}$ for $i\neq j$. The corresponding
polynomials are known as the Jacobi-Piñeiro polynomials, and are studied in
Sections 4.1 and 5.1.

Two different AT-systems correspond to multiple Laguerre weights on
$[0,+\infty)$, see, e.g. [21, §23.4]. First, we can take

$w_{j}(x)=x^{\alpha_{j}}e^{-x},\quad j=1,2,\dots,r,$ (6)

where again $\alpha_{1},\ldots,\alpha_{r}>-1$ and
$\alpha_{i}-\alpha_{j}\notin\mathbb{Z}$ for $i\neq j$. The corresponding
polynomials are known as  multiple Laguerre polynomials of the first kind, and
their behavior is discussed in Sections 4.2 and 5.2.

Another option is to define the weights

$w_{j}(x)=x^{\alpha}e^{-c_{j}x},\quad j=1,2,\dots,r,$ (7)

where $\alpha>-1$, with all $c_{i}>0$, and such that $c_{i}\neq c_{j}$ for
$i\neq j$. The corresponding polynomials are known as multiple Laguerre
polynomials of the second kind, and their properties are discussed in Sections
4.3 and 5.3. For Type I, they are a special case of the Kampé de Fériet
polynomials, so, in Section 3.1, we prove the more general fact that they can
be represented as a finite free additive convolution of hypergeometric
polynomials.

The asymptotic behavior of MOPs is a highly non-trivial subject. Although the
study of the Hermite-Padé polynomials goes back to the original works of
Hermite, see [19, 20], as well as [45], the first important asymptotic result
appeared in the work of Kalyagin [22]. In the 1980s, the ground-breaking works
of Aptekarev, Gonchar, Rakhmanov, and Stahl, made clear that the asymptotics
of the Type I form (1) (but not of the individual entries $A_{\bm{n},i}$) and
of the Type II polynomials $P_{\bm{n}}$ can be described in terms of a vector
equilibrium problem for logarithmic potentials, [3, 15, 16]. However, solving
such a problem is usually a formidable task.

Another approach is via the higher-order nearest-neighbor recurrence relations
by MOPs, established in the work of Van Assche [51] (see also [21, Ch. 23])
that allow to derive an algebraic equation on a weighted distribution of
zeros. An important ingredient of this method is the expression (or, at least,
the behavior) of the recurrence coefficients. A systematic study of a large
number of classical and semi-classical families of orthogonality weights for
MOP, and in particular, of their recurrence coefficients, was carried out in a
number of contributions by Van Assche and his collaborators in recent years;
see, e.g. [2]. A limitation of this approach is that it allows one to address
only the quasi-diagonal case (step line) for the multi-indices $\bm{n}$.

Finally, another recent and formidable tool for asymptotic analysis is the
non-linear steepest descent method of Deift and Zhou, applied to the Riemann-
Hilbert characterization of MOPs [52]. This technique renders extremely
precise asymptotic information (see, e.g. [6, 9, 40, 39], to mention a few),
but at a very high cost of being technically challenging.

In this paper, we address the problem of the asymptotic zero distribution of
the three families of MOP mentioned above, when the degree
$|\bm{n}|\to\infty$, also allowing a linear dependence of the parameters
$\alpha$’s and $\beta$’s on $\bm{n}$. As recent investigations show, these
polynomials are hypergeometric, so we can use the free convolution approach at
a relatively low cost. We stress that one of the appeals of this technique is
its simplicity. Alternatives such as the general methods described above, or
using the differential equation or the integral representation [56] of these
MOPs are usually much more involved.

Since a representation of a real-rooted polynomial in terms of free
convolution can involve polynomials with complex zeros, the results on the
asymptotic regime from [4, 5] cannot be directly applied. Taking advantage of
the fact the proofs in those references are essentially algebraic, based on
the behavior of moments and cumulants, we adapt these arguments to analyze the
finite free convolutions in the asymptotic regime for measures compactly
supported on the complex plane and not necessarily on the real line; see
Section 3.2 and Appendix A.

As a by-product of the representation of the hypergeometric polynomials in
terms of finite free convolutions, we also derive some zero monotonicity and
interlacing properties for the mentioned families of MOP. They appear to be
new in the majority of cases; they are especially interesting for the
polynomials $A_{\bm{n},i}$ in (1), where not much is known.

##  2\. Preliminaries

###  2.1. Notation

In what follows, $\mathbb{P}_{n}$ stands for all algebraic polynomials of
degree $\leq n$, and $\mathbb{P}\overset{\mathrm{def}}{=\joinrel=}\cup_{n\geq
0}\mathbb{P}_{n}$. Also, for $K\subset\mathbb{C}$, we denote by
$\mathbb{P}_{n}(K)$ the subset of polynomials of degree $\leq n$ with all
zeros in $K$. In particular, $\mathbb{P}_{n}(\mathbb{R})$ denotes the family
of real-rooted polynomials of degree $\leq n$, and
$\mathbb{P}_{n}(\mathbb{R}_{\geq 0})$ is the subset of
$\mathbb{P}_{n}(\mathbb{R})$ of polynomials having only roots in
$\mathbb{R}_{\geq 0}\overset{\mathrm{def}}{=\joinrel=}[0,+\infty)$, e.g real
and non-negative.

The rising factorial (also, Pochhammer’s symbol111 Another standard notation
for the raising factorial is $(a)_{j}$. We prefer to use the notation defined
here.) for $a\neq 0$ and $j\in\mathbb{Z}_{\geq
0}\overset{\mathrm{def}}{=\joinrel=}\mathbb{N}\cup\\{0\\}$ is

$\left(a\right)^{\overline{j}}\overset{\mathrm{def}}{=\joinrel=}a(a+1)\dots(a+j-1)=\frac{\Gamma(a+j)}{\Gamma(a)},\quad\left(a\right)^{\overline{0}}\overset{\mathrm{def}}{=\joinrel=}1,$

while the falling factorial is defined as

$\left(a\right)^{\underline{j}}\overset{\mathrm{def}}{=\joinrel=}a(a-1)\dots(a-j+1)=\left(a-j+1\right)^{\overline{j}},\quad\left(a\right)^{\underline{0}}\overset{\mathrm{def}}{=\joinrel=}1.$

If $\bm{a}=(a_{1},\dots,a_{i})\in\mathbb{R}^{i}$ is a vector (tuple), we
understand by

$\left(\bm{a}\right)^{\overline{k}}=\prod_{s=1}^{i}\left(a_{s}\right)^{\overline{k}}.$

For $n\in\mathbb{N}$, we will use the notation

$\mathbb{Z}_{n}\overset{\mathrm{def}}{=\joinrel=}\\{0,1,2,\dots,n-1\\},\quad(-\mathbb{Z}_{n})\overset{\mathrm{def}}{=\joinrel=}\\{0,-1,-2,\dots,-n+1\\}.$
(8)

Finally, we will write

$p\simeq q$

to indicate that polynomials $p$ and $q$ coincide up to a non-zero
multiplicative constant. We will also use the standard notation of the theory
of orthogonal polynomials,

$p^{*}(z)\overset{\mathrm{def}}{=\joinrel=}z^{n}\,\overline{p(1/\overline{z})}$

for the reversed polynomial of $p$. Clearly, for polynomials $p$ with real
coefficients, $p^{*}(x)=x^{n}\,p(1/x)$, in which case $p^{*}$ is usually
called the reciprocal of $p$.

###  2.2. Hypergeometric polynomials

For $\bm{a}=(a_{1},\dots,a_{i})\in\mathbb{R}^{i}$ and
$\bm{b}=(b_{1},\dots,b_{j})\in\mathbb{R}^{j}$, a generalized hypergeometric
series [23, 46] is an expression

${\ }_{i+1}F_{j}{\left(\genfrac{.}{.}{0.0pt}{}{a_{0},\
\bm{a}}{\bm{b}};x\right)}=\sum_{k=0}^{\infty}\frac{\left(a_{0}\right)^{\overline{k}}\left(\bm{a}\right)^{\overline{k}}}{\left(\bm{b}\right)^{\overline{k}}}\frac{x^{k}}{k!}.$

In the particular case when $a_{0}$ is a negative integer, the series is
terminating:

${\ }_{i+1}F_{j}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}}{\bm{b}};x\right)}=\sum_{k=0}^{n}\frac{\left(-n\right)^{\overline{k}}\left(\bm{a}\right)^{\overline{k}}}{\left(\bm{b}\right)^{\overline{k}}}\frac{x^{k}}{k!}$
(9)

is a (generalized) hypergeometric polynomial of degree $\leq n$, as long as

$b_{1},\dots b_{j}\notin\left(-\mathbb{Z}_{n+1}\right).$ (10)

Additionally, the polynomial in (9) is of degree exactly $n$ if and only if

$a_{1},\dots,a_{i}\notin\left(-\mathbb{Z}_{n}\right).$ (11)

In what follows, we always assume that conditions (10)–(11) hold. From a
direct computation, we get the following equation for the derivative of a
hypergeometric polynomial,

$\frac{d}{dx}\left({\ }_{i+1}F_{j}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}}{\bm{b}};x\right)}\right)\simeq{\
}_{i+1}F_{j}{\left(\genfrac{.}{.}{0.0pt}{}{-n+1,\
\bm{a}+1}{\bm{b}+1};x\right)}.$ (12)

We will also make use of hypergeometric series of two variables (or Kampé de
Fériet series, see, e.g. [47, Section 1.3]): with $\bm{a}\in\mathbb{R}^{p}$,
$\bm{b}\in\mathbb{R}^{r}$, $\bm{c}\in\mathbb{R}^{s}$,
$\bm{\alpha}\in\mathbb{R}^{q}$, $\bm{\beta}\in\mathbb{R}^{n}$, and
$\bm{\gamma}\in\mathbb{R}^{k}$,

$F_{q\mathrel{\mathop{\ordinarycolon}}n;k}^{p\mathrel{\mathop{\ordinarycolon}}r;s}\left[\left.\begin{array}[]{c}\bm{a}\mathrel{\mathop{\ordinarycolon}}\bm{b}\
;\ \bm{c}\\\ \bm{\alpha}\mathrel{\mathop{\ordinarycolon}}\bm{\beta}\ ;\
\bm{\gamma}\end{array}\right\rvert\
x,y\right]\overset{\mathrm{def}}{=\joinrel=}\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{\left(\bm{a}\right)^{\overline{i+j}}\left(\bm{b}\right)^{\overline{i}}\left(\bm{c}\right)^{\overline{j}}}{\left(\bm{\alpha}\right)^{\overline{i+j}}\left(\bm{\beta}\right)^{\overline{i}}\left(\bm{\gamma}\right)^{\overline{j}}}\frac{x^{i}}{i!}\frac{y^{j}}{j!},$

along with the Kampé de Fériet polynomials: for $n\in\mathbb{Z}_{\geq 0}$,

$F_{q\mathrel{\mathop{\ordinarycolon}}n;k}^{p+1\mathrel{\mathop{\ordinarycolon}}r;s}\left[\left.\begin{array}[]{c}-n,\
\bm{a}\mathrel{\mathop{\ordinarycolon}}\bm{b}\ ;\ \bm{c}\\\
\bm{\alpha}\mathrel{\mathop{\ordinarycolon}}\bm{\beta}\ ;\
\bm{\gamma}\end{array}\right\rvert\
x,y\right]\overset{\mathrm{def}}{=\joinrel=}\sum_{i=0}^{n}\sum_{j=0}^{n-i}\frac{\left(-n\right)^{\overline{i+j}}\left(\bm{a}\right)^{\overline{i+j}}\left(\bm{b}\right)^{\overline{i}}\left(\bm{c}\right)^{\overline{j}}}{\left(\bm{\alpha}\right)^{\overline{i+j}}\left(\bm{\beta}\right)^{\overline{i}}\left(\bm{\gamma}\right)^{\overline{j}}}\frac{x^{i}}{i!}\frac{y^{j}}{j!}.$
(13)

###  2.3. Finite free convolutions

######  Definition 2.1 ([36]).

Given two polynomials, $p$ and $q$, of degree at most $n$, the $n$-th
multiplicative finite free convolution of $p$ and $q$, denoted as
$p\boxtimes_{n}q$, is a polynomial of degree at most $n$, which can be defined
in terms of the coefficients of polynomials written in the form

$p(x)=\sum_{j=0}^{n}x^{n-j}(-1)^{j}e_{j}(p)\quad\text{ and }\quad
q(x)=\sum_{j=0}^{n}x^{n-j}(-1)^{j}e_{j}(q).$ (14)

Namely,

$[p\boxtimes_{n}q](x)\overset{\mathrm{def}}{=\joinrel=}\sum_{k=0}^{n}x^{n-k}(-1)^{k}e_{k}(p\boxtimes_{n}q),$

with

$e_{k}(p\boxtimes_{n}q)\overset{\mathrm{def}}{=\joinrel=}\binom{n}{k}^{-1}e_{j}(p)e_{j}(q).$
(15)

In particular, if $p,q$ are of degree $n$ and monic, then also
$p\boxtimes_{n}q$ has the same property.

The multiplicative finite free convolution is a bi-linear operator from
$\mathbb{P}_{n}\times\mathbb{P}_{n}$ to $\mathbb{P}_{n}$: if
$p,q,r\in\mathbb{P}_{n}$, and $\alpha\in\mathbb{R}$, then

$(\alpha p+q)\boxtimes_{n}r=\alpha(p\boxtimes_{n}r)+q\boxtimes_{n}r.$

From Definition 2.1 it easily follows that multiplicative convolution with the
polynomial $(x-\alpha)^{n}$ is equivalent to a _dilation_ of the roots by
$\alpha$:

$\text{Dil}_{\alpha}(p)\overset{\mathrm{def}}{=\joinrel=}\alpha^{n}p\left(\frac{x}{\alpha}\right)=p(x)\boxtimes_{n}(x-\alpha)^{n},\quad\alpha\neq
0.$ (16)

In consequence,

$(\text{Dil}_{\alpha}p)\boxtimes_{n}q=p\boxtimes_{n}(\text{Dil}_{\alpha}q)=\text{Dil}_{\alpha}\left[p\boxtimes_{n}q\right],\quad\alpha\neq
0,$ (17)

and

$\text{Dil}_{\alpha}\text{Dil}_{\beta}(p)=\text{Dil}_{\alpha\beta}(p),\quad\alpha,\beta\neq
0.$ (18)

The following result was proved in [38, Theorem 3.1]:

######  Theorem A.

If $n\in\mathbb{Z}_{\geq 0}$, and

$p(x)={\
}_{i_{1}+1}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{1}}{\bm{b}_{1}};x\right)},\qquad
q(x)={\
}_{i_{2}+1}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{2}}{\bm{b}_{2}};x\right)},$

where the parameters $\bm{a}_{1},\bm{a}_{2},\bm{b}_{1},\bm{b}_{2}$ are tuples
(of sizes $i_{1},i_{2},j_{i},j_{2}$, respectively), then their $n$-th free
multiplicative convolution is given by

$[p\boxtimes_{n}q](x)={\
}_{i_{1}+i_{2}+1}F_{j_{1}+j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{1},\bm{a}_{2}}{\bm{b}_{1},\bm{b}_{2}};x\right)}.$

######  Definition 2.2 ([36]).

Given two polynomials, $p$ and $q$, of degree at most $n$, the $n$-th additive
finite free convolution of $p$ and $q$, denoted as $p\boxplus_{n}q$, is a
polynomial of degree at most $n$, defined in terms of the coefficients of
polynomials written in the form

$p(x)=\sum_{j=0}^{n}x^{n-j}(-1)^{j}e_{j}(p)\quad\text{ and }\quad
q(x)=\sum_{j=0}^{n}x^{n-j}(-1)^{j}e_{j}(q).$ (19)

Namely,

$[p\boxplus_{n}q](x)\overset{\mathrm{def}}{=\joinrel=}\sum_{k=0}^{n}x^{n-k}(-1)^{k}e_{k}(p\boxplus_{n}q),$

with

$e_{k}(p\boxplus_{n}q)\overset{\mathrm{def}}{=\joinrel=}\left(n\right)^{\underline{k}}\sum_{i+j=k}\frac{e_{i}(p)}{\left(n\right)^{\underline{i}}}\frac{e_{j}(q)}{\left(n\right)^{\underline{j}}}$
(20)

(and thus, $e_{0}(p\boxplus_{n}q)=e_{0}(p)e_{0}(q)$).

The additive finite free convolution is a bi-linear operator from
$\mathbb{P}_{n}\times\mathbb{P}_{n}$ to $\mathbb{P}_{n}$: if
$p,q,r\in\mathbb{P}_{n}$, and $\alpha\in\mathbb{R}$, then

$(\alpha p+q)\boxplus_{n}r=\alpha(p\boxplus_{n}r)+q\boxplus_{n}r.$

Moreover,

$p(x)\boxplus_{n}(x-\alpha)^{n}=p(x-\alpha),\quad p\in\mathbb{P}_{n}.$ (21)

and $p\boxplus_{n}q=0$ if and only if $\deg(p)+\deg(q)<n$, or if $\deg p=n$
then $q\equiv 0$. This also shows that the inverse of any $p\in\mathbb{P}_{n}$
under the additive (finite free) convolution $\boxplus_{n}$ is unique.

In analogy to (17), we have

$(\text{Dil}_{\alpha}p)\boxplus_{n}(\text{Dil}_{\alpha}q)\simeq\text{Dil}_{\alpha}\left(p\boxplus_{n}q\right),\quad\alpha\neq
0.$ (22)

The following result appears, although in a slightly different form, in [38,
Theorem 3.4]:

######  Theorem B.

Let $p$ and $q$ be hypergeometric polynomials of the following form:

$p(x)={\ }_{i_{1}+1}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{1}}{\bm{b}_{1}};(-1)^{l_{1}}x\right)},\qquad q(x)={\
}_{i_{2}+1}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{2}}{\bm{b}_{2}};(-1)^{l_{2}}x\right)},$

where $l_{1},l_{2}\in\\{0,1\\}$ the parameters
$\bm{a}_{1},\bm{a}_{2},\bm{b}_{1},\bm{b}_{2}$ are tuples (of sizes
$i_{1},i_{2},j_{i},j_{2}$, respectively). Then, with the notation
$\mathfrak{d}=d/dx$,

$\begin{split}[p\boxplus_{n}q](x)\simeq{\
}_{j_{1}}F_{i_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-\bm{b}_{1}-n+1}{-\bm{a}_{1}-n+1};(-1)^{i_{1}+j_{1}+l_{1}+1}\mathfrak{d}\right)}{\
}_{j_{2}}F_{i_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-\bm{b}_{2}-n+1}{-\bm{a}_{2}-n+1};(-1)^{i_{2}+j_{2}+l_{2}+1}\mathfrak{d}\right)}[x^{n}].\end{split}$

From here, an immediate consequence is the following auxiliary result:

######  Lemma 2.3.

Assume that a polynomial $p$ of degree $n$ can be represented as a product of
hypergeometric functions,

$p(x)={\
}_{i_{1}}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-\bm{b}_{1}-n+1}{-\bm{a}_{1}-n+1};(-1)^{i_{1}+j_{1}+l_{1}+1}x\right)}{\
}_{i_{2}}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-\bm{b}_{2}-n+1}{-\bm{a}_{2}-n+1};(-1)^{i_{2}+j_{2}+l_{2}+1}x\right)},$

where $l_{1},l_{2}\in\\{0,1\\}$. Then, for the reciprocal polynomial $p^{*}$,

$p^{*}(x)\simeq{\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n,1}{\cdot};x\right)}\boxtimes_{n}\left[{\
}_{i_{1}+1}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{1}}{\bm{b}_{1}};(-1)^{l_{1}}x\right)}\boxplus_{n}{\
}_{i_{2}+1}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{2}}{\bm{b}_{2}};(-1)^{l_{2}}x\right)}\right].$

###### Proof.

From the hypothesis viewed as a differential function we get

$p(\mathfrak{d})={\
}_{i_{1}}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-\bm{b}_{1}-n+1}{-\bm{a}_{1}-n+1};(-1)^{i_{1}+j_{1}+l_{1}+1}\mathfrak{d}\right)}{\
}_{i_{2}}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-\bm{b}_{2}-n+1}{-\bm{a}_{2}-n+1};(-1)^{i_{2}+j_{2}+l_{2}+1}\mathfrak{d}\right)}.$

By Theorem B,

$p\left(\mathfrak{d}\right)[x^{n}]\simeq{\
}_{i_{1}+1}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{1}}{\bm{b}_{1}};(-1)^{l_{1}}x\right)}\boxplus_{n}{\
}_{i_{2}+1}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{2}}{\bm{b}_{2}};(-1)^{l_{2}}x\right)}.$ (23)

On the other hand, for a polynomial

$q(x)=\sum_{j=0}^{n}x^{n-j}(-1)^{j}e_{j}(q)$

it holds that

$q\left(\mathfrak{d}\right)[x^{n}]=\sum_{k=0}^{n}(-1)^{k}\left(n\right)^{\underline{n-k}}e_{k}(q)x^{k}=(-1)^{n}n!\sum_{k=0}^{n}(-1)^{k}\frac{e_{k}(q^{*})}{(n-k)!}x^{n-k}.$
(24)

Applying the multiplicative convolution by

${\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\,1}{\cdot};x\right)}=\sum_{k=0}^{n}(-1)^{k}\,\binom{n}{k}(n-k)!\,x^{n-k}$

on both sides of (24) and using the definition (15), we get the identity

${\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\,1}{\cdot};x\right)}\boxtimes_{n}q\left(\mathfrak{d}\right)[x^{n}]\simeq
q^{*}.$

Applying it to (23) it yields

$p^{*}\simeq{\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n,1}{\cdot};x\right)}\boxtimes_{n}\left[{\
}_{i_{1}+1}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{1}}{\bm{b}_{1}};(-1)^{l_{1}}x\right)}\boxplus_{n}{\
}_{i_{2}+1}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\
\bm{a}_{2}}{\bm{b}_{2}};(-1)^{l_{2}}x\right)}\right],$

which concludes the proof. ∎

###  2.4. Real roots, interlacing, and free finite convolution

A very important fact is that in many circumstances the finite free
convolution of two polynomials with real roots also has all its roots real.
Here, we use the notation introduced at the beginning of Section 2.

######  Proposition 2.4 (Szegő [48], Walsh [54], see also [38]).

Let $p,q\in\mathbb{P}_{n}$. Then

  1. (i)

$p,q\in\mathbb{P}_{n}(\mathbb{R})\ \Rightarrow\
p\boxplus_{n}q\in\mathbb{P}_{n}(\mathbb{R})$.

  2. (ii)

$p\in\mathbb{P}_{n}(\mathbb{R}),\ q\in\mathbb{P}_{n}(\mathbb{R}_{\geq 0})\
\Rightarrow\ p\boxtimes_{n}q\in\mathbb{P}(\mathbb{R})$.

  3. (iii)

$p,q\in\mathbb{P}_{n}(\mathbb{R}_{\geq 0})\ \Rightarrow\
p\boxtimes_{n}q\in\mathbb{P}(\mathbb{R}_{\geq 0})$.

  4. (iv)

$p,q\in\mathbb{P}_{n}(\mathbb{R}_{\leq 0})\ \Rightarrow\
p\boxtimes_{n}q\in\mathbb{P}(\mathbb{R}_{\geq 0})$

  5. (v)

$p\in\mathbb{P}_{n}(\mathbb{R}_{\leq 0}),\ q\in\mathbb{P}_{n}(\mathbb{R}_{\geq
0})\ \Rightarrow\ p\boxtimes_{n}q\in\mathbb{P}(\mathbb{R}_{\leq 0})$.

If we replace above the sets $\mathbb{R}_{\geq 0}$ and $\mathbb{R}_{\leq 0}$
by strict inclusions, $\mathbb{R}_{>0}$ and $\mathbb{R}_{<0}$, respectively,
the statements of Proposition 2.4 remain valid.

######  Definition 2.5 (Interlacing).

Let

$p(x)=e_{0}(p)\prod_{j=1}^{n}\left(x-\lambda_{j}(p)\right)\in\mathbb{P}_{n}(\mathbb{R}),\quad\lambda_{1}(p)\leq\dots\leq\lambda_{n}(p),$

and

$q(x)=e_{0}(q)\prod_{j=1}^{m}\left(x-\lambda_{j}(q)\right)\in\mathbb{P}_{m}(\mathbb{R}),\quad\lambda_{1}(q)\leq\dots\leq\lambda_{m}(q).$

We say that $q$ interlaces $p$ (or, equivalently, that zeros of $q$ interlace
zeros of $p$, see, e.g., [10]), and denote it $p\preccurlyeq q$, if

$m=n\quad\text{and}\quad\lambda_{1}(p)\leq\lambda_{1}(q)\leq\lambda_{2}(p)\leq\lambda_{2}(q)\leq\cdots\leq\lambda_{n}(p)\leq\lambda_{n}(q),$
(25)

or if

$m=n-1\quad\text{and}\quad\lambda_{1}(p)\leq\lambda_{1}(q)\leq\lambda_{2}(p)\leq\lambda_{2}(q)\leq\cdots\leq\lambda_{n-1}(p)\leq\lambda_{n-1}(q)\leq\lambda_{n}(p).$
(26)

Furthermore, we use the notation $p\prec q$ when all inequalities in (25) or
(26) are strict.

From the real-root preservation and the linearity of the free finite
convolution one easily obtains the following interlacing-preservation
property:

######  Proposition 2.6 (Preservation of interlacing).

If $p,\widetilde{p}\in\mathbb{P}_{n}(\mathbb{R})$ be of degree exactly $n$
such that $p\preccurlyeq\widetilde{p}$, then for a polynomial $q$ of degree
$n$,

$q\in\mathbb{P}_{n}(\mathbb{R})\quad\Rightarrow\quad
p\boxplus_{n}q\preccurlyeq\widetilde{p}\boxplus_{n}q$

and

$q\in\mathbb{P}_{n}(\mathbb{R}_{\geq 0})\quad\Rightarrow\quad
p\boxtimes_{n}q\preccurlyeq\widetilde{p}\boxtimes_{n}q.$

The same statements hold if we replace all $\preccurlyeq$ by $\prec$.

For a proof of this result, see, for instance, [27, Theorems 1.7 and 1.8].

######  Remark 2.7.

Noticing that $p\preccurlyeq q$ if and only if $q(-x)\preccurlyeq p(-x)$ and
that $p(x)\boxtimes_{n}q(-x)=[p\boxtimes_{n}q](-x)$, we can easily extend
Proposition 2.6 to include polynomials with negative real roots. Namely, if
$p,\widetilde{p}\in\mathbb{P}_{n}^{*}(\mathbb{R})$ and
$p\preccurlyeq\widetilde{p}$, then for a polynomial $q$ of degree $n$,

$q\in\mathbb{P}_{n}(\mathbb{R}_{\leq
0})\quad\Rightarrow\quad\widetilde{p}\boxtimes_{n}q\preccurlyeq
p\boxtimes_{n}q.$

###  2.5. Integral transforms and free convolution of measures

We follow the standard notation from the literature on free probability (see,
e.g. [4, 36, 41, 43]) and define several transforms for a Borel measure $\mu$.

The Cauchy transform of $\mu$,

$\mathcal{G}_{\mu}(z)\overset{\mathrm{def}}{=\joinrel=}\int\frac{d\mu(y)}{z-y},$
(27)

is well defined and analytic on $\mathbb{C}\setminus\operatorname{supp}(\mu)$,
even in the case of a signed measure. If $\mu$ is compactly supported, then
$\mathcal{G}_{\mu}$ has a Laurent expansion

$\mathcal{G}_{\mu}(z)=\sum_{k=0}^{\infty}\frac{m_{k}}{z^{k+1}},$ (28)

convergent in a neighborhood of infinity, where

$m_{j}=\int z^{j}\,d\mu(z),\quad j=0,1,2,\dots$

are the moments of the measure $\mu$. Unless specified otherwise, we assume in
what follows that $\mu$ is a compactly supported positive probability measure,
so that

$m_{0}=1.$ (29)

The $K$-transform of the compactly supported probability measure $\mu$ is the
functional inverse of its Cauchy transform, that is,

$\mathcal{K}_{\mu}(w)\overset{\mathrm{def}}{=\joinrel=}\mathcal{G}^{-1}_{\mu}(w),$

analytic in a punctured neighborhood of the origin, while its $R$-transform

$\mathcal{R}_{\mu}(w)\overset{\mathrm{def}}{=\joinrel=}\mathcal{K}_{\mu}(w)-\frac{1}{w}=\sum_{n=0}^{\infty}\kappa_{n+1}w^{n},$

is analytic in a neighborhood of the origin222 Clearly, the analyticity of
$\mathcal{R}_{\mu}$ at the origin for a compactly supported measure is
equivalent to condition (29).. The coefficients $\kappa_{n}$ are called the
free cumulants of $\mu$. Notice that $\mathcal{R}_{\mu}\equiv 0$ if and only
if

$\mathcal{G}_{\mu}(z)=1/z$

in a neighborhood of infinity; this is the case of $\mu=\delta_{0}$, the Dirac
delta or unit mass point at the origin, but not only: any unit Lebesgue
measure on a circle or a disk centered at the origin has this property.

Closely related to the Cauchy transform is the generating function of the
moments (for short, the $M$-transform) of a compactly-supported probability
measure $\mu$,

$\mathcal{M}_{\mu}(z)\overset{\mathrm{def}}{=\joinrel=}\frac{1}{z}\mathcal{G}_{\mu}\left(\frac{1}{z}\right)-1=\int\frac{zy\,d\mu(y)}{1-zy},$
(30)

for which the expansion

$\mathcal{M}_{\mu}(z)=\sum_{k=1}^{\infty}m_{k}\,z^{k}$ (31)

converges in a neighborhood of the origin. We finally define the $S$-transform
of $\mu$ as

$\mathcal{S}_{\mu}(w)\overset{\mathrm{def}}{=\joinrel=}\frac{w+1}{w}\mathcal{M}_{\mu}^{-1}(w),$
(32)

where $\mathcal{M}_{\mu}^{-1}$ is the functional inverse of
$\mathcal{M}_{\mu}$ (when it exists). Notice that once again
$\mathcal{M}_{\mu}\equiv 0$ if and only if $\mathcal{G}_{\mu}(z)=1/z$;
furthermore, if $\mu$ is compactly supported and $m_{1}\neq 0$ then
$\mathcal{S}_{\mu}$ is analytic in a neighborhood of origin and non-vanishing
at $w=0$, with $\mathcal{S}_{\mu}(0)=1/m_{1}.$ We can guarantee that
$m_{1}\neq 0$ if $\mu$ is a positive probability measure supported on the
positive (or negative) semiaxis.

All these transforms determine each other via the identities

$\left(z\mathcal{M}_{\mu}(z)+z\right)\mathcal{R}_{\mu}\left(z\mathcal{M}_{\mu}(z)+z\right)=\mathcal{M}_{\mu}(z),\qquad\mathcal{G}_{\mu}\left(\frac{z+1}{z\mathcal{S}_{\mu}(z)}\right)=z\mathcal{S}_{\mu}(z),$
(33)

which follow directly from the definitions above. Combining (32) and (33) we
obtain also a direct relation between the $R$-transform and the $S$-transforms
(see for instance [43, Definition 18.15 and Remark 18.16]):

$\mathcal{S}_{\mu}(w)=\frac{1}{w}\mathcal{R}_{\mu}^{-1}(w),$ (34)

where $\mathcal{R}_{\mu}^{-1}$ is the functional inverse of
$\mathcal{R}_{\mu}$ (when it exists).

We will later use the well-known fact, see for instance [17, Proposition
3.13], that for a measure $\mu$ such that $\mu(\\{0\\})=0$ it holds that

$S_{\mu}(z)S_{\mu^{*}}(-z-1)=1,$ (35)

where $\mu^{*}$ is the reversed measure of $\mu$ (equivalently, the push-
forward measure under the mapping $t\mapsto 1/t$), so that its density is
$d\mu^{*}(t)=t^{-2}d\mu(t^{-1})$). The identity (35) is valid in a punctured
neighborhood of origin, in the domain of analyticity of both functions on the
left-hand side.

######  Remark 2.8.

When $\mu$ is not compactly supported, expressions (27) and (30) make sense
and define analytic functions in the open set
$\mathbb{C}\setminus\operatorname{supp}(\mu)$ and in its image by $z\mapsto
1/z$, respectively. Formula (32) is also well-posed, at least where
$\mathcal{M}_{\mu}$ is locally invertible. In this case, we cannot expect the
series in (28) and (31) to converge.

On the other hand, the formulas in (28) and (31) can be considered as formal
power series associated with a “moment” sequence $m_{0},m_{1},\dots,$ of
complex numbers. In that case, if we denote the sequence by $\mu$, we can use
these formulas as a definition of $\mathcal{G}_{\mu}$ and $\mathcal{M}_{\mu}$,
as well as define $\mathcal{R}_{\mu}$ and $\mathcal{S}_{\mu}$ through (33). In
other words, all the transforms can be defined for any complex sequence but
only as formal power series (also called moment series), with no analyticity
or convergence assumed a priori.

As we will see in Section 3.2, the advantage of this approach is that many
asymptotic results can be established even for formal moment sequences,
without assuming any underlying measure.

Using these definitions, we can introduce two important operations on
compactly supported probability measures $\mu$ and $\nu$ (or, as we have just
discussed, on their moment sequences):

  1. (i)

the free additive convolution $\mu\boxplus\nu$ via the identity

$\mathcal{R}_{\mu\boxplus\nu}(w)=\mathcal{R}_{\mu}(w)+\mathcal{R}_{\nu}(w);$
(36)

  2. (ii)

the free multiplicative convolution $\mu\boxtimes\nu$ via the identity

$\mathcal{S}_{\mu\boxtimes\nu}(w)=\mathcal{S}_{\mu}(w)\mathcal{S}_{\nu}(w).$
(37)

Clearly, identities (36)–(37) do not define the resulting measures
$\mu\boxplus\nu$ and $\mu\boxtimes\nu$ uniquely, unless they are determined by
the analytic expression of their $R-$ and $S-$transforms, respectively. This
is the case of probability measures compactly supported on $\mathbb{R}$.

##  3\. General results

###  3.1. Kampé de Fériet polynomials and finite free convolution

Recall the definition of the Kampé de Fériet polynomials in (13). The
following factorization holds:

######  Theorem 3.1.

For $i_{k},j_{k}\in\mathbb{N}$ and multi-indices
$\bm{a}_{k}\in\mathbb{R}^{i_{k}}$, $\bm{b}_{k}\in\mathbb{R}^{j_{k}}$,
$k=1,2,3$, and a non-zero constant $c$,

$\displaystyle
F_{j_{1}\mathrel{\mathop{\ordinarycolon}}j_{2};j_{3}}^{i_{1}+1\mathrel{\mathop{\ordinarycolon}}i_{2};i_{3}}\left[\left.\begin{array}[]{c}-n,\bm{a}_{1}\mathrel{\mathop{\ordinarycolon}}\bm{a}_{2};\bm{a}_{3}\\\
\bm{b}_{1}\mathrel{\mathop{\ordinarycolon}}\bm{b}_{2};\bm{b}_{3}\end{array}\right\rvert\,c,x\right]\simeq\left(p(x)\boxplus_{n}q(x)\right)\boxtimes_{n}r(x),$

where

$\displaystyle p(x)$ $\displaystyle\overset{\mathrm{def}}{=\joinrel=}{\
}_{i_{1}+1}F_{j_{1}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{1}}{\bm{b}_{1}};-x\right)},\qquad
q(x)\overset{\mathrm{def}}{=\joinrel=}{\
}_{i_{2}+1}F_{j_{2}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,-\bm{b}_{2}-n+1}{-\bm{a}_{2}-n+1};\frac{x}{c}\right)},$
$\displaystyle r(x)$ $\displaystyle\overset{\mathrm{def}}{=\joinrel=}{\
}_{i_{3}+1}F_{j_{3}}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{3}}{\bm{b}_{3}};-x\right)}.$

The proportionality constant can be computed explicitly in terms of all the
parameters.

###### Proof.

Using the notation in (14), we can write the coefficients of $p$, $q$, and $r$
(up to a multiplicative constant) as

$\displaystyle e_{k}(p)$
$\displaystyle=\binom{n}{k}(-1)^{k}\frac{\left(\bm{a}_{1}\right)^{\overline{n-k}}}{\left(\bm{b}_{1}\right)^{\overline{n-k}}},\qquad
e_{k}(q)=\binom{n}{k}\frac{\left(\bm{a}_{2}\right)^{\overline{k}}}{\left(\bm{b}_{2}\right)^{\overline{k}}}\,c^{k},$
$\displaystyle e_{k}(r)$
$\displaystyle=\binom{n}{k}(-1)^{k}\frac{\left(\bm{a}_{3}\right)^{\overline{n-k}}}{\left(\bm{b}_{3}\right)^{\overline{n-k}}}.$

By (20),

$e_{k}(p\boxplus_{n}q)=\sum_{i+j=k}\left(n\right)^{\underline{k}}\frac{(-1)^{i}\left(\bm{a}_{1}\right)^{\overline{n-i}}}{i!\left(\bm{b}_{1}\right)^{\overline{n-i}}}\frac{\left(\bm{a}_{2}\right)^{\overline{j}}}{j!\left(\bm{b}_{2}\right)^{\overline{j}}}\,c^{j}$

and by (15),

$e_{k}((p\boxplus_{n}q)\boxtimes_{n}r)=(-1)^{k}\frac{\left(\bm{a}_{3}\right)^{\overline{n-k}}}{\left(\bm{b}_{3}\right)^{\overline{n-k}}}\sum_{i+j=k}\left(n\right)^{\underline{k}}\frac{(-1)^{i}\left(\bm{a}_{1}\right)^{\overline{n-i}}}{i!\left(\bm{b}_{1}\right)^{\overline{n-i}}}\frac{\left(\bm{a}_{2}\right)^{\overline{j}}}{j!\left(\bm{b}_{2}\right)^{\overline{j}}}\,c^{j}.$

Thus,

$\displaystyle(p\boxplus_{n}q)\boxtimes_{n}r$
$\displaystyle\simeq\sum_{k^{\prime}=0}^{n}\sum_{i^{\prime}+j=k^{\prime}}x^{n-k^{\prime}}\frac{\left(\bm{a}_{3}\right)^{\overline{n-k^{\prime}}}}{\left(\bm{b}_{3}\right)^{\overline{n-k^{\prime}}}}\frac{n!}{(n-k^{\prime})!}\frac{(-1)^{i^{\prime}}\left(\bm{a}_{1}\right)^{\overline{n-i^{\prime}}}}{i^{\prime}!\left(\bm{b}_{1}\right)^{\overline{n-i^{\prime}}}}\frac{\left(\bm{a}_{2}\right)^{\overline{j}}}{j!\left(\bm{b}_{2}\right)^{\overline{j}}}\,c^{j}$
$\displaystyle=\sum_{i=0}^{n}\sum_{j+k=i}(-1)^{i}\frac{n!}{k!}\frac{\left(\bm{a}_{1}\right)^{\overline{i}}}{(n-i)!\left(\bm{b}_{1}\right)^{\overline{i}}}\frac{\left(\bm{a}_{2}\right)^{\overline{j}}}{\left(\bm{b}_{2}\right)^{\overline{j}}}\frac{\left(\bm{a}_{3}\right)^{\overline{k}}}{\left(\bm{b}_{3}\right)^{\overline{k}}}\frac{c^{j}x^{k}}{j!}$
$\displaystyle=\sum_{i=0}^{n}\sum_{j+k=i}\frac{\left(-n\right)^{\overline{i}}\left(\bm{a}_{1}\right)^{\overline{i}}}{\left(\bm{b}_{1}\right)^{\overline{i}}}\frac{\left(\bm{a}_{2}\right)^{\overline{j}}}{\left(\bm{b}_{2}\right)^{\overline{j}}}\frac{\left(\bm{a}_{3}\right)^{\overline{k}}}{\left(\bm{b}_{3}\right)^{\overline{k}}}\frac{c^{j}x^{k}}{j!k!}=F_{j_{1}\mathrel{\mathop{\ordinarycolon}}j_{2};j_{3}}^{i_{1}+1\mathrel{\mathop{\ordinarycolon}}i_{2};i_{3}}\left[\left.\begin{array}[]{c}-n,\bm{a}_{1}\mathrel{\mathop{\ordinarycolon}}\bm{a}_{2};\bm{a}_{3}\\\
\bm{b}_{1}\mathrel{\mathop{\ordinarycolon}}\bm{b}_{2};\bm{b}_{3}\end{array}\right\rvert\,c,x\right],$

where in the second line, we did the change of variables $i=n-i^{\prime}$ and
$k=n-k^{\prime}$ and changed the order of summation. ∎

###  3.2. Finite free convolutions in the asymptotic regime

We have already seen that free finite convolution is useful when studying the
zeros of polynomials. An additional crucial advantage is that finite free
convolutions tend to free convolution in the asymptotic regime when the degree
$n\to\infty$.

The connection between the convolutions of polynomials and free probability
(reason for the name of “finite free” convolutions) was first noticed by
Marcus, Spielman, and Srivastava in [36] when they used Voiculescu’s $R$\- and
$S$-transform to improve the bounds on the largest root of a convolution of
two real-rooted polynomials. This connection was explored further in [35],
where Marcus defined a finite analog of the $R$\- and $S$-transform that are
related to Voiculescu’s transforms in the limit. Using finite free cumulants,
Arizmendi and Perales [5] showed that in the asymptotic regime, a finite free
additive convolution becomes a free additive convolution. This was later
extended to the multiplicative convolution by Arizmendi, Garza-Vargas and
Perales [4].

Since the proofs of these facts in [5, 4] are essentially algebraic, it turns
out that they are applicable in a broader context of moment sequences instead
of measures; see Remark 2.8. More precisely, we need to consider the
normalized zero-counting measures for a sequence of polynomials, real-rooted
or not, under the only assumptions that their moments converge.

Given a polynomial $p$ of degree $n$ and roots $\lambda_{j}(p)$, $j=1,\dots,n$
(not necessarily all distinct or real), its (normalized) zero counting measure
(or empirical root distribution of $p$) is

$\chi(p)\overset{\mathrm{def}}{=\joinrel=}\frac{1}{n}\sum_{j=1}^{n}\delta_{\lambda_{j}(p)},$
(38)

where $\delta_{z}$ is the Dirac delta (unit mass) placed at the point $z$. The
corresponding moments of $\chi(p)$ (which we also call the moments of $p$,
stretching the terminology a bit) are

$m_{k}(p)\overset{\mathrm{def}}{=\joinrel=}\frac{1}{n}\sum_{j=1}^{n}\lambda_{j}^{k}(p)=\int
x^{k}\,d\chi(p),\quad k=0,1,2,\dots$

As mentioned above, the connection between finite and standard free
probability is revealed in the asymptotic regime, when we let the degree
$n\to\infty$. We say that the sequence of polynomials
$\mathfrak{p}=\left(p_{n}\right)_{n=1}^{\infty}$ such that each $p_{n}$ is
real-rooted and of degree exactly $n$ converges in moments if all finite
limits

$m_{k}=\lim_{n\to\infty}m_{k}(p_{n}),\qquad k=0,1,2,\dots$ (39)

exist. If the supports of $\chi(p_{n})$ are contained in a compact set of the
complex plane, then the sequence $\chi(p_{n})$ is weakly compact and
$m_{0},m_{1},\dots$ are the moment of any of its accumulation points. In this
case, (39) is equivalent to existence of a probability measure
$\nu(\mathfrak{p})$, compactly supported on $\mathbb{C}$, with all its moments
finite such that

$\lim_{n\to\infty}m_{k}(p_{n})=m_{k}(\nu(\mathfrak{p})),\qquad k=0,1,2,\dots$

If we know additionally that $\chi(p_{n})$ are on $\mathbb{R}$, with their
supports contained in the same compact, and if the moment problem for
$\nu(\mathfrak{p})$ is determined, then (39) is equivalent to the weak-*
convergence of the sequence $\chi(p_{n})$ to $\nu(\mathfrak{p})$.

For real-rooted polynomials, the following proposition is a direct consequence
of [5, Corollary 5.5] and [4, Theorem 1.4]:

######  Proposition 3.2.

Let
$\mathfrak{p}\mathrel{\mathop{\ordinarycolon}}=\left(p_{n}\right)_{n=1}^{\infty}$
and
$\mathfrak{q}\mathrel{\mathop{\ordinarycolon}}=\left(q_{n}\right)_{n=1}^{\infty}$
be two sequences of real-rooted polynomials as above, and let
$\nu(\mathfrak{p})$ and $\nu(\mathfrak{q})$ be two compactly supported
probability Borel measures on $\mathbb{R}$ such that $\mathfrak{p}$
(respectively, $\mathfrak{q}$) converges in moments to $\nu(\mathfrak{p})$
(respectively, $\nu(\mathfrak{q})$). Then

  1. (i)

the sequence $\left(p_{n}\boxplus_{n}q_{n}\right)_{n=1}^{\infty}$ converges in
moments to $\nu(\mathfrak{p})\boxplus\nu(\mathfrak{q})$;

  2. (ii)

if, additionally, for all sufficiently large $n$,
$p_{n},q_{n}\subset\mathbb{P}_{n}(\mathbb{R}_{>0})$ (or if
$p_{n},q_{n}\subset\mathbb{P}_{n}(\mathbb{R}_{<0})$) then the sequence
$\left(p_{n}\boxtimes_{n}q_{n}\right)_{n=1}^{\infty}$ converges in moments to
$\nu(\mathfrak{p})\boxtimes\nu(\mathfrak{q})$.

In other words, in the case of real-rooted polynomials, as $n\to\infty$, we
can replace the finite free convolution of polynomials by the standard free
convolution of measures supported on $\mathbb{R}$. The motivation to consider
only real-rooted polynomials is given by their application in free
probability. However, the proof of Proposition 3.2 is basically algebraic,
based on the notion of convergence in moments, and can be easily extended to
more general cases, such as polynomials with uniformly bounded, but not
necessarily real, roots.

Namely, we have the following result:

######  Theorem 3.3.

Let
$\mathfrak{p}\mathrel{\mathop{\ordinarycolon}}=\left(p_{n}\right)_{n=1}^{\infty}$
and
$\mathfrak{q}\mathrel{\mathop{\ordinarycolon}}=\left(q_{n}\right)_{n=1}^{\infty}$
be two sequences of polynomials, such that $\deg p_{n}=\deg q_{n}=n$,
$n\in\mathbb{Z}_{\geq 0}$, and all zeros of both $\mathfrak{p}$ and
$\mathfrak{q}$ are uniformly bounded.

If, additionally, all finite limits

$\lim_{n\to\infty}m_{k}(p_{n})=\alpha_{k}\in\mathbb{C}\qquad\text{and}\qquad\lim_{n\to\infty}m_{k}(q_{n})=\beta_{k}\in\mathbb{C},\qquad
k=0,1,2,\dots,$

exist, then

  1. (i)

all moments of the sequence of polynomials
$\left(p_{n}\boxplus_{n}q_{n}\right)_{n=1}^{\infty}$ have a finite limit,

$\gamma_{k}\overset{\mathrm{def}}{=\joinrel=}\lim_{n\to\infty}m_{k}(p_{n}\boxplus_{n}q_{n})\in\mathbb{C}\qquad
k=0,1,2,\dots,$

and the $R$-transform associated to the sequence
$\left(\gamma_{k}\right)_{k=1}^{\infty}$ satisfies

$\mathcal{R}_{\gamma}(z)=\mathcal{R}_{\alpha}(z)+\mathcal{R}_{\beta}(z).$

  2. (ii)

All moments of the sequence
$\left(p_{n}\boxtimes_{n}q_{n}\right)_{n=1}^{\infty}$ have a limit

$\theta_{k}\mathrel{\mathop{\ordinarycolon}}=\lim_{n\to\infty}m_{k}(p_{n}\boxtimes_{n}q_{n})\in\mathbb{C}\qquad
k=0,1,2,\dots,$

and the $S$-transform associated to the sequence
$\left(\theta_{k}\right)_{k=1}^{\infty}$ satisfies

$\SS_{\theta}(z)=\SS_{\alpha}(z)\SS_{\beta}(z).$

######  Remark 3.4.

For the notion of the $S$\- and $R$-transforms associated to a sequence, see
Remark 2.8.

This theorem can be established following the arguments leading to Proposition
3.2. As noted, the proofs of the results in [5, 4] rely on the fact that the
combinatorial structure behind the finite convolutions of polynomials tends
(as $n\to\infty$) to the combinatorial formulas that relate the coefficients
of the series in the $M$-, $R$-, and $S$-transforms of the limit sequence.
This fact is established regardless of whether this sequence is in fact a
moment sequence of a probability measure. When this is the case, the notions
of the $R$\- and other transforms for a sequence and for the underlying
measure match, see, for instance, [43, Lectures 16 and 18].

The reader interested in further details can check Appendix A.

###  3.3. S-transform for a hypergeometric polynomial

In this section, we show that although the Cauchy transform of any limit of
zero-counting measures of hypergeometric polynomials are, generally speaking,
algebraic functions whose explicit expressions are not available, their
$S$-transforms are straightforward rational functions.

Let us first discuss the case of ${}_{2}F_{1}$ polynomials.

######  Proposition 3.5.

Let

$p_{n}(x)={\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\alpha_{n}}{\beta_{n}};x\right)},$

be a sequence of hypergeometric polynomials such that the finite limits

$\lim_{n}\frac{\alpha_{n}}{n}=A,\quad\lim_{n}\frac{\beta_{n}}{n}=B$

exist. If additionally

$A\notin[-1,0),\quad B\neq-1,\quad\text{and}\quad A\neq B,$ (40)

then any weak-* limit $\mu$ of the normalized zero-counting measures
$\chi(p_{n})$ is a positive probability measure compactly supported on
$\mathbb{C}$, for which in a neighborhood of the origin,

$\mathcal{S}_{\mu}(z)=\frac{z+A+1}{z+B+1}.$ (41)

###### Proof.

Denote by $\mu_{n}=\chi(p_{n})$ the probability zero-counting measure of
$p_{n}$.

We use well-known identities expressing ${}_{2}F_{1}$ polynomials in standard
normalization in terms of Jacobi polynomials,

$\displaystyle\left(\beta\right)^{\overline{n}}{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\alpha}{\beta};x\right)}$
$\displaystyle=n!\,P^{(\beta-1,-n+\alpha-\beta)}_{n}(1-2x)=(-1)^{n}n!\,P^{(-n+\alpha-\beta,\beta-1)}_{n}(2x-1).$
(42)

Jacobi polynomials also exhibit several transformation formulas in the cases
when they can have multiple zeros (at $\pm 1$) or have a degree reduction,
such as

$P_{n}^{(-k-1,\beta)}(z)=\frac{\Gamma(n+\beta+1)}{\Gamma(n+\beta-k)}\frac{(n-k-1)!}{n!}\left(\frac{z-1}{2}\right)^{k+1}P_{n-k-1}^{(k+1,\beta)}(z),\qquad
k\in\mathbb{Z}_{n},$

and several more. For a more detailed discussion, see [49, §4.22] or [25].
From these identities it follows that the assumption $A\notin[-1,0)$ is
sufficient to guarantee that the zeros of $p_{n}$ are uniformly bounded, while
with $B\neq-1$, its zero-counting measures $\mu_{n}$ do not collapse to
$\delta_{0}$.

Thus, standard arguments on weak compactness of the sequence $\mu_{n}$ show
that under these assumptions, there exist $\Lambda\subset\mathbb{N}$ and a
unit measure $\mu\neq\delta_{0}$ such that

$\mu_{n}\xrightarrow{*}\mu,\quad n\in\Lambda.$

As a consequence, $\mathcal{G}_{\chi(p_{n})}\to\mathcal{G}_{\mu}$,
$n\in\Lambda$, in a neighborhood of infinity.

Polynomials $p_{n}$ satisfy the ODE

$z(1-z)p_{n}^{\prime\prime}(z)+\left(\beta_{n}-(\alpha_{n}-n+1)z\right)p_{n}^{\prime}(z)+n\alpha_{n}p_{n}(z)=0.$

Rewriting it in terms of $h_{n}=p_{n}^{\prime}/p_{n}$ and noticing that
$h_{n}/n\to G=\mathcal{G}_{\mu}$, we get an equation for $G$,

$z(1-z)G^{2}(z)+\left(B-(A-1)z\right)G(z)+A=0.$ (43)

Observe again that for $B\neq-1$, $G(z)\neq 1/z$.

With the notation $w=\mathcal{S}_{\mu}(z)$, the second identity in (33) takes
the form

$G\left(\frac{z+1}{zw}\right)=zw.$ (44)

Thus, making the change of variable $z\mapsto(z+1)/(wz)$ in (43) and using the
identity above, we arrive at

$z(w(z+B+1)-(z+A+1))=0,$

or

$w=\frac{z+A+1}{z+B+1},$

which proves (41). ∎

######  Remark 3.6.

It is interesting to observe that expression (41) is meaningful even if we
drop the assumptions (40). It is worth discussing then the possible
consequences of relaxing these restrictions.

As we have seen in the proof, if $B=-1$, then $\mathcal{G}_{\mu}(z)=1/z$,
which happens, for instance, if $\mu=\delta_{0}$. If the zeros of $p_{n}$’s
are real, it means that they asymptotically collapse to the origin.

The consequence of dropping the assumption $A\neq B$ is also clear: if $A=B$,
we have $\mathcal{G}_{\mu}(z)=1/(z-1)$, which happens, for example, if
$\mu=\delta_{1}$.

If $-1<A<0$, the explicit formula (9) shows that selecting the sequence
$\alpha_{n}$ appropriately, we can make the degree of $p_{n}$ be of
$\mathcal{O}(-An)$. This would mean that non-trivial part of the measures
$\mu_{n}$ “escapes” to infinity. Any possible weak-* limit could have a
bounded support, but it will be of a total mass $|A|$. Notice that it would
mean that for this $\mu$, $m_{0}=-A<1$, which is compatible with the algebraic
equation (43).

Finally, if $A=-1$, the $S$-transform in (41) is still rational, but now it
vanishes at $z=0$. In this case, one of the branch points of the Cauchy
transform $\mathcal{G}_{\mu}$, solution of (43), is at infinity. This shows
that if the limit measure $\mu$ exists, it is a probability measure with
unbounded support.

Now we formulate the general result of this section:

######  Theorem 3.7.

For $s,t\in\mathbb{Z}_{\geq 0}$, let $\bm{a}_{n}\in\mathbb{R}^{s}$,
$\bm{b}_{n}\in\mathbb{R}^{t}$ be such that finite limits

$\lim_{n}\frac{\bm{a}_{n}}{n}=\bm{A}=(A_{1},\dots
A_{s})\in\mathbb{R}^{s},\quad\lim_{n}\frac{\bm{b}_{n}}{n}=\bm{B}=(B_{1},\dots,B_{t})\in\mathbb{R}^{t}$
(45)

exist. Assume additionally that

$A_{j}\notin[-1,0),\quad B_{k}\neq-1,\quad A_{j}\neq B_{k},\qquad
j\in\\{1,\dots,s\\},\quad k\in\\{1,\dots,t\\}.$ (46)

Then any weak-* limit $\mu$ of the normalized zero-counting measures
$\chi(p_{n})$ of the sequence

$p_{n}(x)={\
}_{s+1}F_{t}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{n}}{\bm{b}_{n}};n^{t-s}x\right)}$
(47)

is a positive probability measure compactly supported on $\mathbb{C}$, for
which in a neighborhood of the origin,

$\mathcal{S}_{\mu}(z)=\frac{\prod_{i=1}^{s}(z+A_{i}+1)}{\prod_{j=1}^{t}(z+B_{j}+1)}.$
(48)

###### Proof.

Consider the sequence

$q_{n}(x)={\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{\beta_{n}};x\right)},$

such that

$\lim_{n}\frac{\beta_{n}}{n}=B.$

Polynomials $q_{n}$ are, in fact, Laguerre polynomials,

$q_{n}(x)=\frac{n!}{\left(\beta_{n}\right)^{\overline{n}}}\,L_{n}^{(\beta_{n}-1)}(x).$

We are interested in the sequence $\mu_{n}=\chi(\widehat{q}_{n})$ of
probability zero-counting measures for rescaled polynomials

$\widehat{q}_{n}(x)={\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{\beta_{n}};nx\right)}.$

It is well known that for each fixed $\beta$, the sequence of monic Laguerre
polynomials
$\left\\{\widetilde{L}_{n}^{\left(\beta\right)}\right\\},n\in\mathbb{N}$,
satisfies the three-term recurrence relation

$\widetilde{L}_{n+1}^{\left(\beta\right)}(x)=\left(x-(2n+\beta+1)\right)\widetilde{L}_{n}^{\left(\beta\right)}(x)+n(n+\beta)\widetilde{L}_{n-1}^{\left(\beta\right)}(x).$

In particular, the monic polynomials

$n^{-n}\widetilde{L}_{n}^{(\beta-1)}(nx)$

satisfy

$n^{-n-1}L_{n+1}^{\left(\beta-1\right)}(nx)=\left(x-\frac{2n+\beta}{n}\right)n^{-n}L_{n}^{\left(\beta-1\right)}(nx)+\frac{n+\beta-1}{n}\left(n^{-n+1}L_{n-1}^{\left(\beta-1\right)}(nx)\right).$

Writing these relations as an eigenvalue problem for a Jacobi matrix and
applying Gershgorin’s theorem, it is easy to prove that under the assumption
that the finite limit $B$ of $\beta_{n}/n$ exists, the zeros of
$\widehat{q}_{n}$ are uniformly bounded. Moreover, the well-known identity

$L_{n}^{(-k)}(x)=\frac{(n-k)!}{n!}(-x)^{k}L_{n-k}^{(k)}(x),\quad
k\in\mathbb{Z}_{n},$

see e.g. [46, Section 18.5] or [38, Section 2.3], indicates that $\mu_{n}$’s
do not converge to $\delta_{0}$, unless $B\neq-1$. This is indeed the case;
see the discussion in [37] or [26].

Polynomials $q_{n}$ are solution of Kummer’s differential

$zq_{n}^{\prime\prime}(z)+(\beta_{n}-z)q_{n}^{\prime}(z)+nq_{n}(z)=0,$

see, e.g. [24, Chapter 7]. Hence, $\widehat{q}_{n}$ satisfy

$z\widehat{q}_{n}^{\prime\prime}(z)+(\beta_{n}-nz)\widehat{q}_{n}^{\prime}(z)+n^{2}\widehat{q}_{n}(z)=0.$
(49)

As before, by weak compactness there exist $\Lambda\subset\mathbb{N}$ and a
unit measure $\mu$ such that

$\mu_{n}\xrightarrow{*}\mu,\quad n\in\Lambda.$

As a consequence, $\mathcal{G}_{\chi(q_{n})}\to\mathcal{G}_{\mu}$,
$n\in\Lambda$, in a neighborhood of infinity. Rewriting (49) in terms of
$h_{n}=\widehat{q}_{n}^{\prime}/\widehat{q}_{n}$ and using that $h_{n}/n\to
G=\mathcal{G}_{\mu}$, we get an equation for $G$,

$zG^{2}(z)+\left(B-z\right)G(z)+1=0$

(notice again that for $B\neq-1$, $G(x)\neq 1/x$). Now we proceed as in the
proof of Proposition 3.5: making the change of variable $z\mapsto(z+1)/(wz)$
in this quadratic equation on $G$ and using (44), we get for
$w=\mathcal{S}_{\mu}(z)$ the equation

$z\left(w(z+B+1)-1\right)=0,$

and we conclude that

$\mathcal{S}_{\mu}(z)=\frac{1}{z+B+1}.$ (50)

Same arguments, or using identity (35) and the fact that

${\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\alpha_{n}}{\cdot};x\right)}=(-x)^{n}\left(\alpha_{n}\right)^{\overline{n}}{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{-n-\alpha_{n}+1};-1/x\right)}$

(see, e.g., [38, Lemma 2.1]) allows us to derive that for the rescaled
polynomials

$\widehat{q}_{n}(x)={\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\alpha_{n}}{\cdot};x/n\right)}$

under assumption

$\lim_{n}\frac{\alpha_{n}}{n}=A\notin[-1,0),$

for any accumulation point $\mu$ of the normalized zero-counting measures
$\chi(\widehat{q}_{n})$ it holds that

$\mathcal{S}_{\mu}(z)=z+A+1.$ (51)

We “assemble” the general case (47), using Proposition 3.5 and the building
blocks above, appealing to the identity (16) and Theorem A.

Indeed, the polynomial in (47) can be written as

$p_{n}(x)={\
}_{s+1}F_{t}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{n}}{\bm{b}_{n}};n^{t-s}x\right)}\simeq\text{Dil}_{n^{t-s}}\left[{\
}_{s+1}F_{t}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{n}}{\bm{b}_{n}};x\right)}\right].$

Denote $\bm{a}_{n}=\left(a_{1}^{(n)},\dots,a_{s}^{(n)}\right)$,
$\bm{b}_{n}=\left(b_{1}^{(n)},\dots,b_{t}^{(n)}\right)$. If $s=t$, by Theorem
A,

$p_{n}(x)={\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n,a_{1}^{(n)}}{b_{1}^{(n)}};x\right)}\boxtimes_{n}{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n,a_{1}^{(n)}}{b_{1}^{(n)}};x\right)}\boxtimes_{n}\dots\boxtimes_{n}{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n,a_{s}^{(n)}}{b_{s}^{(n)}};x\right)},$
(52)

and in this case, (48) follows from (41) and Theorem 3.3.

Assume that $s<t$. Then

$\displaystyle p_{n}(x)$ $\displaystyle\simeq
q_{n,1}(x)\boxtimes_{n}q_{n,2}(x),$

where $q_{n,1}(x)$ is the polynomial in the right hand side of (52), while

$\displaystyle q_{n,2}(x)$ $\displaystyle=\text{Dil}_{n^{s-t}}\left[{\
}_{1}F_{t-s}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{b_{s+1}^{(n)},\dots,b_{t}^{(n)}};x\right)}\right]$
$\displaystyle=\text{Dil}_{n^{-1}}\left[{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{b_{s+1}^{(n)}};x\right)}\right]\boxtimes_{n}\dots\boxtimes_{n}\text{Dil}_{n^{-1}}\left[{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{b_{t}^{(n)}};x\right)}\right]$
$\displaystyle={\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{b_{s+1}^{(n)}};nx\right)}\boxtimes_{n}\dots\boxtimes_{n}{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n}{b_{t}^{(n)}};nx\right)}.$

It remains to use (41), (50), and Theorems A and 3.3 to obtain (48).

The case $s>t$ is analyzed in the same way, using (51). ∎

######  Remark 3.8.

From the proof above it is clear that (48) follows directly from (50)–(51) and
Theorem A. However, we found the analysis of the case of ${}_{2}F_{1}$
polynomials illuminating and having an independent interest.

We can analyze the “pathological” situations of Theorem 3.7 that arise if we
relax the conditions (46) using the arguments from Remark 3.6.

######  Theorem 3.9.

Under the assumptions of Theorem 3.7, the Cauchy transform
$\mathcal{G}_{\mu}(u)$ of the limit measure $\mu$ in a neighborhood of
infinity is an algebraic function $y/u$, where $y=y(u)$ satisfies the equation

$y\prod_{j=1}^{t}(y+B_{j})=u(y-1)\prod_{i=1}^{s}(y+A_{i}),$ (53)

whose Riemann surface is a ramified covering of $\mathbb{C}$ of genus $0$.

###### Proof.

By (32) and (48) we can find $z=\mathcal{M}_{\mu}(u)$ by solving

$u=\frac{z}{z+1}\frac{\prod_{i=1}^{s}(z+A_{i}+1)}{\prod_{j=1}^{t}(z+B_{j}+1)}.$

Recalling the definition (30) of the $M$-transform in terms of the Cauchy
transform $\mathcal{G}_{\mu}$ of $\mu$, we have that

$z=\frac{1}{u}\mathcal{G}_{\mu}\left(\frac{1}{u}\right)-1,$

and we can rewrite the equation above as

$u=\frac{u^{-1}\mathcal{G}_{\mu}\left(u^{-1}\right)-1}{u^{-1}\mathcal{G}_{\mu}\left(u^{-1}\right)}\frac{\prod_{i=1}^{s}(u^{-1}\mathcal{G}_{\mu}\left(u^{-1}\right)+A_{i})}{\prod_{j=1}^{t}(u^{-1}\mathcal{G}_{\mu}\left(u^{-1}\right)+B_{j})}.$

Replacing $u\mapsto 1/u$ and denoting $y=u\mathcal{G}_{\mu}(u)$ we arrive at
equation (53). Since it defines $u$ as a rational function of $y$, the Riemann
surface of $y=y(u)$ (and hence, of $w=w(u)$) has genus $0$. ∎

For future reference, it will be convenient to formulate the following
immediate consequence:

######  Corollary 3.10.

Let $c,d\in\mathbb{R}$, $c\neq 0$. Then, under the assumptions of Theorem 3.7,
any weak-* limit $\mu$ of the normalized zero-counting measures $\chi(p_{n})$
of the sequence

$p_{n}(x)={\
}_{s+1}F_{t}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{n}}{\bm{b}_{n}};n^{t-s}(cx+d)\right)}$

is a positive probability measure $\mu$ compactly supported on $\mathbb{C}$,
whose $S$-transform $w=\mathcal{S}_{\mu}(z)$ satisfies the equation

$w\prod_{j=1}^{t}\left[duw+c(u+1+B_{j})\right]=c^{t-s}\left[dw+c\right]\prod_{i=1}^{s}\left(duw+c(u+1+A_{i})\right).$
(54)

###### Proof.

Let

$p_{n}(x)={\
}_{s+1}F_{t}{\left(\genfrac{.}{.}{0.0pt}{}{-n,\bm{a}_{n}}{\bm{b}_{n}};n^{t-s}x\right)}$

and let $\widetilde{\mu}$ be a weak-* limit of the normalized zero-counting
measures $\chi(q_{n})$. It is easy to see that the Cauchy transforms of
$\widetilde{\mu}$ and $\mu$ are related by

$\mathcal{G}_{\mu}(z)=c\,\mathcal{G}_{\widetilde{\mu}}(cz+d).$

Hence, using (53), we find that $G=\mathcal{G}_{\mu}$ satisfies the equation

$G\prod_{j=1}^{t}\left(\left(cu+d\right)G+cB_{j}\right)=c^{t-s}\left(\left(cu+d\right)G-c\right)\prod_{i=1}^{s}\left(\left(cu+d\right)G+cA_{i}\right).$

The change of variable $u\mapsto(u+1)/(wu)$, along with the identity (44),
yield (54) with $w=\mathcal{S}_{\mu}(z)$. ∎

##  4\. Type I Multiple orthogonal polynomials

Recall that Type I multiple orthogonal polynomials are defined by the
orthogonality and normalization conditions (1)–(3). The potential-theoretic
techniques allow to describe the asymptotics of the zeros of function
$Q_{\bm{n}}$ in (1). The recently obtained expressions for the actual type I
polynomials allow us to perform this analysis for polynomial coefficients of
$Q_{\bm{n}}$.

###  4.1. Type I Jacobi-Piñeiro polynomials

The AT-system of Jacobi-Piñeiro weights was introduced in (5): for $\beta>-1$,
$\bm{n}=(n_{1},\dots,n_{r})\in\mathbb{N}^{r}$,
$\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ such that, without loss of
generality,

$\alpha_{1}>\alpha_{2}>\cdots>\alpha_{r}>-1\quad\text{and}\quad\alpha_{i}-\alpha_{j}\not\in\mathbb{Z}\text{
for }i\not=j,$ (55)

the Type I Jacobi-Piñeiro polynomials of Type I,
$P^{(\bm{\alpha};\beta)}_{\bm{n},i}$, $\deg
P^{(\bm{\alpha};\beta)}_{\bm{n},i}\leq n_{j}-1$, $i=1,\dots,r$, are given by
the orthogonality conditions on the function

$Q_{\bm{n}}^{(\bm{\alpha};\beta)}(x)\overset{\mathrm{def}}{=\joinrel=}\sum_{j=1}^{r}P^{(\bm{\alpha};\beta)}_{\bm{n},i}(x)w_{j}(x),\quad
w_{j}(x)\overset{\mathrm{def}}{=\joinrel=}x^{\alpha_{j}}(1-x)^{\beta},$ (56)

namely,

$\int_{0}^{1}Q_{\bm{n}}^{(\bm{\alpha};\beta)}(x)x^{k}\,dx=0,\qquad 0\leq
k\leq|\bm{n}|-2,$ (57)

together with the normalization

$\int_{0}^{1}Q_{\bm{n}}^{(\bm{\alpha};\beta)}(x)x^{|\bm{n}|-1}\,dx=1.$

Recently, it was shown that these are hypergeometric polynomials. In order to
write the explicit expression, we borrow notation from [8]: for
$\bm{a}\in\mathbb{R}^{r}$, let $\bm{a}^{(i)}\in\mathbb{R}^{r-1}$ stand for the
vector obtained from $\bm{a}$ by deleting its $i$-th entry; also, in a slight
abuse of notation, we also understand that adding a scalar to a multi-index
means adding this scalar to each one of its components. Then, see [8],

$P^{(\bm{\alpha};\beta)}_{\bm{n},i}(x)=c^{(\bm{\alpha};\beta)}_{\bm{n},i}{\
}_{r+1}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|,\alpha_{i}+1-\bm{\alpha}^{(i)}-\bm{n}^{(i)}}{\alpha_{i}+1,\alpha_{i}+1-\bm{\alpha}^{(i)}};x\right)},$
(58)

where

$c^{(\bm{\alpha};\beta)}_{\bm{n},i}\overset{\mathrm{def}}{=\joinrel=}(-1)^{\left|\bm{n}\right|-1}\frac{\Gamma(\alpha_{i}+\beta+|\bm{n}|)\prod_{k=1}^{r}\left(\alpha_{k}+\beta+|\bm{n}|\right)^{\overline{n_{k}}}}{(n_{i}-1)!\,\Gamma(\beta+|\bm{n}|)\Gamma(\alpha_{i}+1)\prod_{\begin{subarray}{c}1\leq
k\leq r\\\ k\neq
i\end{subarray}}\left(\alpha_{k}-\alpha_{i}\right)^{\overline{n_{k}}}}$

is just a normalizing constant that does not affect the zeros of the
polynomial. For $r=2$, this formula was obtained first in [7].

Using this expression and Theorem A, we get the following representation for
polynomials (57) for $i=1,\dots,r$:

$P^{(\bm{\alpha};\beta)}_{\bm{n},i}=c^{(\bm{\alpha};\beta)}_{\bm{n},i}p_{i,1}\boxtimes_{n_{i}-1}p_{i,2}\boxtimes_{n_{i}-1}\cdots\boxtimes_{n_{i}-1}p_{i,r},$
(59)

where

$p_{i,j}(x)=\begin{cases}\displaystyle{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}-\alpha_{j}-n_{j}+1}{\alpha_{i}-\alpha_{j}+1};x\right)},&j\in\\{1,\cdots,r\\}\setminus\\{i\\},\\\\[11.38109pt]
\displaystyle{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|}{\alpha_{i}+1};x\right)},&j=i.\end{cases}$
(60)

####  4.1.1. Real zeros: monotonicity and interlacing

Since the Jacobi-Piñeiro weights with $\bm{\alpha}$ satisfying (55) form an
AT-system, it is known that the corresponding function
$Q_{\bm{n}}^{(\bm{\alpha};\beta)}$ in (56) has $|\bm{n}|-1$ zeros on $(0,1)$
for every multi-index $\bm{n}$, see, e.g. [50, Theorem 2.3]. Although the
potential-theoretic description of the asymptotics of
$Q_{\bm{n}}^{(\bm{\alpha};\beta)}$ is relatively well studied, the behavior of
the zeros of the Type I Jacobi-Piñeiro polynomials
$P^{(\bm{\alpha};\beta)}_{\bm{n},i}$, $i=1,\dots,r$, is much less known.

Let us denote the open intervals $\Delta_{1},\dots,\Delta_{r}$, with

$\Delta_{1}=(0,1),\quad\Delta_{k}=\begin{cases}\mathbb{R}_{<0},&\text{if
$2\leq k\leq r$ is even,}\\\ \mathbb{R}_{>0},&\text{if $2\leq k\leq r$ is
odd.}\end{cases}$ (61)

A simple calculation shows that for $-1<\alpha<0$,

$\int_{\Delta_{k}}\frac{|t|^{\alpha}}{t-x}dt=(-1)^{k}\frac{\pi}{\sin(\pi\alpha)}\,|x|^{\alpha},\quad
x\in\Delta_{k-1},\quad k\geq 2.$ (62)

Let

$s_{k}(x)\overset{\mathrm{def}}{=\joinrel=}|x|^{\alpha_{k}-\alpha_{k-1}},\quad
x\in\Delta_{k-1},\quad 2\leq k\leq r.$

If together with (55) we assume that

$\alpha_{1}-1<\alpha_{r}=\min_{1\leq k\leq r}\alpha_{k},$ (63)

then by (62), each $s_{k}$ is equal, up to a constant factor, to its Cauchy
transform on $\Delta_{k}$:

$s_{k}(x)=\text{const}_{k}\int_{\Delta_{k}}\frac{|t|^{\alpha_{k}-\alpha_{k-1}}}{t-x}\,dt,\quad
x\in\Delta_{k-1},\quad 2\leq k\leq r,$

which means that the weights $w_{j}$ in (56) form a Nikishin system. By [34],
under the additional constraint that if the sequence of multi-indices
$\bm{n}=(n_{1},\dots,n_{r})$ is such that

$\left(\max_{1\leq k\leq r}n_{k}\right)-\left(\min_{1\leq k\leq
r}n_{k}\right)$

is uniformly bounded, then the number of zeros of each
$P^{(\bm{\alpha};\beta)}_{\bm{n},i}$, $i=1,\dots,r$, outside
$\overline{\Delta}_{r}$ is also uniformly bounded, and the zeros accumulate on
$\overline{\Delta}_{r}\cup\\{\infty\\}$ as $|\bm{n}|\to\infty$.

If we drop the restriction (63), the weights $w_{i}$ still form a generalized
Nikishin system, with

$w_{j+1}(x)=x^{m_{j}}s_{j}(x)w_{j}(x),\quad j=1,\dots,r-1,$

for some $m_{j}\in\mathbb{Z}$. Such a situation is considered in [33]; the
results therein are not immediately applicable due to the assumptions the
authors impose on the polynomial factors. However, we conjecture that the
zeros of the Jacobi-Piñeiro polynomials $P^{(\bm{\alpha};\beta)}_{\bm{n},i}$
accumulate on $\overline{\Delta}_{r}\cup\\{\infty\\}$ as $|\bm{n}|\to\infty$
under very mild assumptions on the sequence $\bm{n}$.

Using the approach that we have already described in [38], we can give some
additional sufficient conditions for real-rootedness of Jacobi-Piñeiro
polynomials, as well as deduce the interlacing and monotonicity of zeros of
polynomials with respect to the parameters.

######  Theorem 4.1.

Let $\beta>-1$ and $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ satisfy (55).
Assume that for a multi-index $\bm{n}\in\mathbb{N}^{r}$ and for a specific
index $i\in\\{1,2,\dots,r\\}$,

$\alpha_{1}-1<\alpha_{i}<\min_{1\leq j\leq r}\\{\alpha_{j}+n_{j}\\}-n_{i}+1.$
(64)

Then $P^{(\bm{\alpha};\beta)}_{\bm{n},i}\in\mathbb{P}_{n_{i}-1}(\Delta_{r})$.

Additionally, for $0<t\leq 2$,

  1. (i)

$P^{(\bm{\alpha}+t;\beta)}_{\bm{n},i}\prec P^{(\bm{\alpha};\beta)}_{\bm{n},i}$
and $P^{(\bm{\alpha};\beta)}_{\bm{n},i}\prec
P^{(\bm{\alpha};\beta+t)}_{\bm{n},i}$, if $r$ is even,

  2. (ii)

$P^{(\bm{\alpha};\beta)}_{\bm{n},i}\prec P^{(\bm{\alpha}+t;\beta)}_{\bm{n},i}$
and $P^{(\bm{\alpha};\beta+t)}_{\bm{n},i}\prec
P^{(\bm{\alpha};\beta)}_{\bm{n},i}$, if $r$ is odd.

Observe that if condition (64) is satisfied for certain multi-indices
$\bm{\alpha}$ and $\bm{n}\in\mathbb{N}^{r}$, it is also satisfied for
$\bm{\alpha}+t$, with $t>0$.

###### Proof.

Let us show first that

$p_{i,i}\in\mathbb{P}(\mathbb{R}_{>0})\quad\text{and}\quad
p_{i,j}(x)\in\mathbb{P}(\mathbb{R}_{<0}),\quad
j\in\\{1,\dots,r\\}\backslash\\{i\\}.$

We can establish it using some results from [11], many of them re-proved in
[38]. Indeed, since by assumptions, $\alpha_{i},\beta>-1$,
$\bm{n}\in\mathbb{N}^{r}$, and thus,

$\alpha_{i}+\beta+|\bm{n}|>\alpha_{i}+n_{i}-1,$

we have by [11, Theorem 1, (ii)] that $p_{i,i}\in\mathbb{P}_{n_{i}-1}((0,1))$.

Similarly, by (55) and (64), for every $j\in\\{1,\dots,r\\}\backslash\\{i\\}$,

$\alpha_{j}-1<\alpha_{i}<\alpha_{j}+n_{j}-n_{i}+1,$

or equivalently,

$-n_{i}+2>\alpha_{i}-\alpha_{j}-n_{j}+1,\quad\alpha_{i}-\alpha_{j}+1>0.$

It remains to use [11, Theorem 1, (i)] to conclude that
$p_{i,j}\in\mathbb{P}_{n_{i}-1}(\mathbb{R}_{<0})$. By Proposition 2.4, for

$q_{i}\mathrel{\mathop{\ordinarycolon}}=p_{i,1}\boxtimes_{n_{i}-1}\cdots\boxtimes_{n_{i}-1}p_{i,i-1}\boxtimes_{n_{i}-1}p_{i,i+1}\boxtimes_{n_{i}-1}\cdots\boxtimes_{n_{i}-1}p_{{i},r},$
(65)

we have that

$q_{i}\in\begin{dcases}\mathbb{P}(\mathbb{R}_{>0})&\text{for $r$ odd,}\\\
\mathbb{P}(\mathbb{R}_{<0})&\text{for $r$ even.}\end{dcases}$ (66)

Since the decomposition (59) of $P^{(\bm{\alpha};\beta)}_{\bm{n},i}$ can be
written as

$P^{(\bm{\alpha};\beta)}_{\bm{n},i}=c^{(\bm{\alpha};\beta)}_{\bm{n},i}p_{i,i}\boxtimes_{n_{i}-1}q_{i},$
(67)

it remain to Proposition 2.4 again to establish the first assertion of the
theorem (about the location of the zeros).

A key observation to prove interlacing is that polynomials $p_{i,j}$ in (60),
with $j\neq i$, are independent of the parameter $\beta>-1$, and invariant
with respect to the scalar shift $\bm{\alpha}\mapsto\bm{\alpha}+t$, $t>0$.
Thus, the same property is shared by the polynomial $q_{i}$ in (65).

On the other hand, by [38, Eq. (61)], for $0\leq t_{1},t_{2}\leq 2$ and
$t_{1}+t_{2}\not=0$.

${\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|+t_{1}}{\alpha_{i}+1};x\right)}\prec{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|+t_{2}}{\alpha_{i}+1+t_{2}};x\right)}.$
(68)

For $r$ odd, using (66) and Proposition 2.6, we get that

$\begin{split}{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|+t_{1}}{\alpha_{i}+1};x\right)}&\boxtimes_{n_{i}-1}q_{i}\\\
&\prec{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|+t_{2}}{\alpha_{i}+1+t_{2}};x\right)}\boxtimes_{n_{i}-1}q_{i},\end{split}$
(69)

while for $r$ even, using now Remark 2.7, we obtain

$\begin{split}{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|+t_{2}}{\alpha_{i}+1+t_{2}};x\right)}&\boxtimes_{n_{i}-1}q_{i}\\\
&\prec{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+\beta+|\bm{n}|+t_{1}}{\alpha_{i}+1};x\right)}\boxtimes_{n_{i}-1}q_{i}.\end{split}$
(70)

In particular, setting in (69)-(70), $t_{1}=0$ and $t_{2}=t$, and noticing
that a multiplicative constant does not affect interlacing, we conclude that

$P^{(\bm{\alpha};\beta)}_{\bm{n},i}\prec
P^{(\bm{\alpha}+t;\beta)}_{\bm{n},i}\quad\text{for }r\text{
even}\quad\text{and}\quad P^{(\bm{\alpha}+t;\beta)}_{\bm{n},i}\prec
P^{(\bm{\alpha};\beta)}_{\bm{n},i}\quad\text{for }r\text{ odd},$

while taking $t_{1}=t$ and $t_{2}=0$ in (69)-(70) yields

$P^{(\bm{\alpha};\beta+t)}_{\bm{n},i}\prec
P^{(\bm{\alpha};\beta)}_{\bm{n},i}\quad\text{for }r\text{
even}\quad\text{and}\quad P^{(\bm{\alpha};\beta)}_{\bm{n},i}\prec
P^{(\bm{\alpha};\beta+t)}_{\bm{n},i}\quad\text{for }r\text{ odd}.$

∎

######  Remark 4.2.

The statement about zero interlacing contained in Theorem 4.1 implies that for
$r$ even, the zeros of the polynomial $P^{(\bm{\alpha}+t;\beta)}_{\bm{n},i}$
are decreasing with respect to $t$ and increasing with respect to $\beta$;
this monotonicity is exactly the opposite if $r$ is odd.

As it follows from the proof of Theorem 4.1, conditions (64) can be weakened.
For instance, if instead of (64) we assume the existence of an index $j\neq i$
such that

$\alpha_{j}-2<\alpha_{i}<\alpha_{j}+n_{j}-n_{i}+2,\quad\max_{k\not=i,j}\\{\alpha_{k}\\}-1<\alpha_{i}<\min_{k\not=i,j}\\{\alpha_{k}+n_{k}\\}-n_{i}+1,$
(71)

then we arrive at a weaker conclusion that
$P^{(\bm{\alpha};\beta)}_{\bm{n},i}\in\mathbb{P}_{n_{i}-1}(\mathbb{R})$.

Notice that additionally to (i)–(ii) we could state that
$P^{(\bm{\alpha};\beta)}_{\bm{n},i}\preccurlyeq
P^{(\bm{\alpha}+\bm{e}_{i};\beta+1)}_{\bm{n}-\bm{e}_{i},i}$, where
$\bm{e}_{i}\in\mathbb{N}^{r}$ is the multi-index that has all entries equal to
0 except the entry of index $i$ which is equal to 1. This observation is based
on the simple fact that, by (12),

$\frac{d}{dx}\left(P^{(\bm{\alpha};\beta)}_{\bm{n},i}(x)\right)\simeq
P^{(\bm{\alpha}+\bm{e}_{i};\beta+1)}_{\bm{n}-\bm{e}_{i},i}(x).$

If we try to enforce the assumptions (64) to be valid for every index $i$, we
obtain the following simple consequence of Theorem 4.1:

######  Corollary 4.3.

Let $\beta>-1$, $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ satisfy (55) and
(63). Then for any multi-index $\bm{n}\in\mathbb{N}^{r}$ of the form

$\bm{n}=(n,\dots,n,n+1,\dots,n+1)$ (72)

(that is, such that for $i<j$, $0\leq n_{j}-n_{i}\leq 1$), conclusions of
Theorem 4.1 hold for all $i\in\\{1,\dots,r\\}$.

From our discussion above it is clear that the assumptions of Corollary 4.3
reduce the situation to a step-line (as the one studied already in [44]) for a
Nikishin system of weights. In this sense, the reality of zeros is not
surprising, although the interlacing properties seem to be new even in this
case.

###### Proof.

Fix $i\in\\{1,\dots,r\\}$. Taking into account (55), condition (63) implies
that $\alpha_{1}-1<\alpha_{i}$.

On the other hand, for any index $j<i$, by (72), $n_{i}\leq n_{j}+1$, and

$-1<\alpha_{i}-\alpha_{j}<0\leq n_{j}-n_{i}+1.$

Analogously, for any index $j>i$, by (72), $n_{i}\leq n_{j}$, and

$0<\alpha_{i}-\alpha_{j}<1\leq n_{j}-n_{i}+1.$

These two sets of inequalities are equivalent to (64). On the other hand, they
hold for every $i\in\\{1,\dots,r\\}$ if and only if $0\leq n_{j}-n_{i}\leq 1$
for $i<j$, and the assertion follows. ∎

####  4.1.2. Zero asymptotics of Jacobi-Piñeiro polynomials of Type I

We can describe the weak zero asymptotics of a sequence of Type I Jacobi-
Piñeiro polynomials,

$p_{\bm{n},i}=P^{(\bm{\alpha}_{\bm{n}};\beta_{\bm{n}})}_{\bm{n},i},\quad
i=1,\dots,r,$

where
$\bm{\alpha}_{\bm{n}}=\left(\alpha_{1}^{(\bm{n})},\dots,\alpha_{r}^{(\bm{n})}\right)$,
$\bm{n}=(n_{1},\dots,n_{r})\in\mathbb{N}^{r}$, under assumption

$\lim_{|\bm{n}|\to\infty}\frac{\alpha_{i}^{(\bm{n})}}{|\bm{n}|}=A_{i}\geq
0,\quad\lim_{|\bm{n}|\to\infty}\frac{\beta_{(\bm{n})}}{|\bm{n}|}=B\geq
0,\quad\text{and}\quad\lim_{|\bm{n}|\to\infty}\frac{n_{i}}{|\bm{n}|}=\theta_{i}>0,\qquad
i=1,\dots,r.$ (73)

Clearly, $\theta_{1}+\dots+\theta_{r}=1$. Formula (58) indicates that we need
to consider the parameters

$\mathfrak{a}_{j}^{(i)}\overset{\mathrm{def}}{=\joinrel=}\begin{cases}(A_{i}+B+1)/\theta_{i},&\text{if
}j=i,\\\ (A_{i}-A_{j}-\theta_{j})/\theta_{i}&\text{if }j\neq
i,\end{cases}\quad\text{and}\quad\mathfrak{b}_{k}^{(i)}\overset{\mathrm{def}}{=\joinrel=}\begin{cases}A_{i}/\theta_{i},&\text{if
}k=i,\\\ (A_{i}-A_{k})/\theta_{i}&\text{if }k\neq i.\end{cases}$ (74)

Restrictions $\mathfrak{a}_{j}^{(i)}\notin[-1,0)$ and
$\mathfrak{b}_{k}^{(i)}\neq-1$ imply that additionally to (73) we need to
assume that

$A_{j}\neq A_{i}+\theta_{i},\quad
A_{j}+\theta_{j}\notin(A_{i},A_{i}+\theta_{i}],\qquad
j\in\\{1,\dots,r\\}\setminus\\{i\\}.$ (75)

Notice that the assumption that
$\mathfrak{a}_{j}^{(i)}\neq\mathfrak{b}_{k}^{(i)}$ holds automatically.

By Theorem 3.7, for any weak-* accumulation point $\mu$ of the normalized
zero-counting measures $\chi(p_{\bm{n},i})$,

$\mathcal{S}_{\mu}(z)=\prod_{j=1}^{r}\frac{z+\mathfrak{a}_{j}^{(i)}+1}{z+\mathfrak{b}_{j}^{(i)}+1},$
(76)

and by Theorem 3.9, its Cauchy transform $\mathcal{G}_{\mu}(u)$ in a
neighborhood of infinity is an algebraic function $y/u$, where $y=y(u)$
satisfies the equation

$y\prod_{j=1}^{r}(y+\mathfrak{b}_{j}^{(i)})=u(y-1)\prod_{i=1}^{r}(y+\mathfrak{a}_{j}^{(i)}).$

Let us discuss the case when $\bm{\alpha}$ and $\beta$ do not depend on
$\bm{n}$, so that

$\mathfrak{a}_{j}^{(i)}\overset{\mathrm{def}}{=\joinrel=}\begin{cases}1/\theta_{i},&\text{if
}j=i,\\\ -\theta_{j}/\theta_{i}&\text{if }j\neq
i,\end{cases}\quad\text{and}\quad\mathfrak{b}_{j}^{(i)}=0,$ (77)

and constrains (75) boil down to

$\theta_{j}\notin(0,\theta_{i}],\qquad j\in\\{1,\dots,r\\}\setminus\\{i\\}.$
(78)

If we assume that there exists an index $i\in\\{1,\dots,r\\}$ such that

$\theta_{i}<\min\left\\{\theta_{j}\mathrel{\mathop{\ordinarycolon}}\,j\in\\{1,\dots,r\\}\setminus\\{i\\}\right\\},$

then for all sufficiently large values of $|\bm{n}|$, the inequality in the
right-hand side of (64) holds. Thus, imposing additionally that
$|\alpha_{i}-\alpha_{j}|<1$ for all $j$’s, we conclude that any weak-*
accumulation point $\mu$ of the normalized zero-counting measures
$\chi(p_{\bm{n},i})$ is a compactly supported probability measure on
$\mathbb{R}_{\geq 0}$, if $r$ is odd, and on $\mathbb{R}_{\leq 0}$, otherwise,
and such that

$\mathcal{S}_{\mu}(z)=\frac{z+1+1/\theta_{i}}{(z+1)^{r}}\prod_{j\neq
i}(z+1-\theta_{j}/\theta_{i}),$

while $y=u\mathcal{G}_{\mu}(u)$ is a solution of

$y^{r+1}=u\left(y+\frac{1}{\theta_{i}}\right)\prod_{j=1}^{r}\left(y-\frac{\theta_{j}}{\theta_{i}}\right).$

In the asymptotically diagonal situation, when all $\theta_{j}=1/r$, we have
$\mathfrak{a}_{j}^{(i)}=-1$ if $j\neq i$, which is one of the degenerate cases
discussed in Remark 3.6. However, the expression for the $S$-transform of the
limit measure is still valid:

$\mathcal{S}_{\mu}(z)=\frac{z^{r-1}}{(z+1)^{r}}(z+1+1/r),$

as well as for the Cauchy transform $\mathcal{G}_{\mu}(u)=y/u$, where $y=y(u)$
solves

$y^{r+1}=u(y-1)^{r}\left(y+r\right).$ (79)

We conjecture that, in this case, the support of any limit measure $\mu$ is
unbounded. If the parameters $\bm{\alpha}$ satisfy (64), it follows from
Theorem 4.1 that all zeros of the corresponding polynomial
$P^{(\bm{\alpha};\beta)}_{\bm{n},i}$ are on $\Delta_{r}$. Furthermore, as
follows from the method of proof of [33, Theorem 3.2], even when (64) is not
satisfied, the zeros still accumulate on $\Delta_{r}\cup\\{\infty\\}$.

######  Example 4.4.

If $r=2$ and $\theta_{1}=\theta<1/2$, the Cauchy transform
$\mathcal{G}_{\mu}(u)$ of the limit measure $\mu$ is given by $y/u$, where
$y=y(u)$ solves the equation

$y^{3}=u(y-1)\left(y+\frac{1}{\theta}\right)\left(y-\frac{1}{\theta}+1\right)=u\left(y^{3}-\left(1+\nu\right)y+\nu\right),\quad\nu=\frac{1}{\theta}\left(\frac{1}{\theta}-1\right)>0.$
(80)

We can take advantage that this is a depressed cubic and use the Cardano
formula for its solution, selecting the correct branch noticing that, by the
definition of the Cauchy transform, $y(u)>0$ for $u>0$. Namely, consider the
functions

$f(u)=\left(\frac{u}{1-u}\right)^{1/3},$

analytic in $\mathbb{C}\setminus\left[(-\infty,0]\cup[1,+\infty)\right]$,
whose single-valued branch is fixed by requiring $f(x)>0$ for $x\in(0,1)$.
Then we can write

$y(u)=\sqrt[3]{\frac{\nu}{2}}\,f(u)\left(\left(\sqrt{1+\frac{\kappa\,u}{1-u}}+1\right)^{1/3}-\left(\sqrt{1+\frac{\kappa\,u}{1-u}}-1\right)^{1/3}\right),$

where

$\kappa=\kappa(\nu)=\frac{4}{27}\,\frac{(1+\nu)^{3}}{\nu^{2}}$

and with the branches of the roots taken always positive for
$u\in(1,1+\varepsilon)$ for a sufficiently small $\varepsilon>0$. Notice that
$\kappa(\nu)$ is a strictly increasing function on $[2,+\infty)$, with
$\kappa(2)=1$.

In particular, for $u=-x<0$, denote by $y_{\pm}$ the boundary values of $y$ on
$\mathbb{R}_{<0}$. By the symmetry principle, $y_{-}=\overline{y_{+}}$ on
$\mathbb{R}_{<0}$.

By the Sokhotski-Plemelj Formula (or Stieltjes’ inversion theorem), the
density of the limit measure $\mu$ can be recovered as

$\displaystyle\mu^{\prime}(-x)$ $\displaystyle=\frac{1}{2\pi
i(-x)}\left(y_{-}(-x)-y_{+}(-x)\right)=\frac{1}{\pi x}\mathop{\rm
Im}\left(y_{+}(-x)\right).$

We have that

$\left(\sqrt{1+\frac{\kappa\,u}{1-u}}+1\right)^{1/3}_{+}\bigg{|}_{u=-x}=\left(\sqrt{1-\frac{\kappa\,x}{1+x}}+1\right)^{1/3}>0\text{
for }x\in(-c^{*},0),$

where $c^{*}=c^{*}(\nu)>0$ solves $\kappa\,c^{*}/(1+c^{*})=1$, that is,

$c^{*}=\frac{1}{\kappa-1}=27\left(\frac{\theta(1-\theta)}{(1-2\theta)(2-\theta)(1+\theta)}\right)^{2}.$
(81)

Analogously,

$\left(\sqrt{1+\frac{\kappa\,u}{1-u}}-1\right)^{1/3}_{+}\bigg{|}_{u=-x}=\left(1-\sqrt{1-\frac{\kappa\,x}{1+x}}\right)^{1/3}e^{2\pi
i/3}\text{ for }x\in(-c^{*},0).$

Finally,

$f_{+}(-x)=\sqrt[3]{\frac{x}{1+x}}\,e^{2\pi i/3},\quad x<0.$

Putting all together, we get that $\mu$ is supported on $[-c^{*},0]$, $\beta$
given in (81), with

$\begin{split}\mu^{\prime}(-x)=&\frac{\sqrt{3}}{2\pi}\,\sqrt[3]{\frac{\nu}{2}}\,\frac{\left(\sqrt{1+x}+\sqrt{(c^{*}-x)/c^{*}}\right)^{1/3}-\left(\sqrt{1+x}-\sqrt{(c^{*}-x)/c^{*}}\right)^{1/3}}{x^{2/3}\sqrt{1+x}}.\end{split}$
(82)

Symbolic integration allows us to check that $\mu^{\prime}(-x)$ is the density
of a positive probability measure on $[-\beta,0]$, and that
$\mu^{\prime}(-x)=\frac{\sqrt{3}\nu^{1/3}}{2\pi x^{2/3}}(1+o(1))$ as $x\to
0+$, and with a square root decay as $x\to c^{*}-$.

Figure 1. Left: histogram of the zeros of the polynomial
$P^{(\bm{\alpha};\beta)}_{\bm{n},1}$, with $\bm{\alpha}=(1/2,3/7)$, $\beta=1$,
and $\bm{n}=(300,600)$, along with the corresponding density (82). Notice that
for $\theta=1/3$, measure $\mu$ is supported on $[-2.43,0]$. The smallest zero
of $P^{(\bm{\alpha};\beta)}_{\bm{n},1}$ in this case is approximately $-2.2$.
Right: histogram of the cube roots of the same zeros.

As an illustration, we plot the histogram of the zeros, all real and negative,
of the polynomial $P^{(\bm{\alpha};\beta)}_{\bm{n},1}$, with
$\bm{\alpha}=(1/2,3/7)$, $\beta=1$, and $\bm{n}=(300,900)$, along with the
corresponding density (82), see Figure 1. For $\theta=1/3$, the asymptotic
distribution of zeros is supported on $[-2.43,0]$.

The diagonal case can be considered taking $\theta\to(1/2)_{-}$, when (82)
becomes

$\mu^{\prime}(-x)=\frac{\sqrt{3}}{2\pi}\,\frac{\left(\sqrt{1+x}+1\right)^{1/3}-\left(\sqrt{1+x}-1\right)^{1/3}}{x^{2/3}\sqrt{1+x}},\quad
x>0.$

This is the density of a positive probability measure on $\mathbb{R}_{>0}$,
such that $\mu^{\prime}(-x)=\frac{\sqrt{3}}{\pi(2x)^{2/3}}(1+o(1))$ as $x\to
0+$.

###  4.2. Type I Multiple Laguerre polynomials of the first kind

Let $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ be a multi-index satisfying
conditions (55). Then for $\bm{n}=(n_{1},\dots,n_{r})\in\mathbb{N}^{r}$, the
type I multiple Laguerre polynomials of the first kind,
$L_{\bm{n},i}^{(\bm{\alpha})}$, $\deg L_{\bm{n},i}^{(\bm{\alpha})}\leq
n_{j}-1$, $i=1,\dots,r$, are given by the orthogonality conditions on the
function

$Q_{\bm{n}}^{(\bm{\alpha})}(x)\overset{\mathrm{def}}{=\joinrel=}\sum_{j=1}^{r}L_{\bm{n},i}^{(\bm{\alpha})}(x)\,x^{\alpha_{j}}e^{-x},$
(83)

namely,

$\int_{0}^{+\infty}Q_{\bm{n}}^{(\bm{\alpha})}(x)x^{k}\,dx=0,\qquad 0\leq
k\leq|\bm{n}|-2,$ (84)

together with the normalization

$\int_{0}^{+\infty}Q_{\bm{n}}^{(\bm{\alpha})}(x)x^{|\bm{n}|-1}\,dx=1.$

In [8], it was shown that these are hypergeometric polynomials: using the
notation introduced before (58),

$L_{\bm{n},i}^{(\bm{\alpha})}(x)=c_{\bm{n},i}^{(\bm{\alpha})}{\
}_{r}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}+1-\bm{\alpha}^{(i)}-\bm{n}^{(i)}}{\alpha_{i}+1,\alpha_{i}+1-\bm{\alpha}^{(i)}};x\right)}$
(85)

where

$c_{\bm{n},i}^{(\bm{\alpha})}=(-1)^{|\bm{n}|-1}\left((n_{i}-1)!\Gamma(\alpha_{i}+1)\prod_{\begin{subarray}{c}1\leq
i\leq r\\\ j\neq
i\end{subarray}}\left(\alpha_{j}-\alpha_{i}\right)^{\overline{n_{j}}}\right)^{-1}.$

For $r=2$, this formula was previously derived in [7]).

Using this expression and Theorem A, we get the following representation for
polynomials (85) for $i=1,\dots,r$:

$L^{(\bm{\alpha})}_{\bm{n},i}=c^{(\bm{\alpha})}_{\bm{n},i}p_{i,1}\boxtimes_{n_{i}-1}p_{i,2}\boxtimes_{n_{i}-1}\cdots\boxtimes_{n_{i}-1}p_{i,r},$
(86)

where

$p_{i,j}(x)=\begin{cases}\displaystyle{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha_{i}-\alpha_{j}-n_{j}+1}{\alpha_{i}-\alpha_{j}+1};x\right)},&j\in\\{1,\cdots,r\\}\setminus\\{i\\},\\\\[11.38109pt]
\displaystyle{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1}{\alpha_{i}+1};x\right)},&j=i.\end{cases}$
(87)

Notice that for $i\neq j$ the $p_{i,j}$ are exactly the same polynomials
appearing in (60), thus we have the relation

$L^{(\bm{\alpha})}_{\bm{n},i}\simeq\left(v_{i}(x)\boxtimes_{n_{i}-1}P^{(\bm{\alpha};\beta)}_{\bm{n},i}\right)$
(88)

with

$v_{i}(x)={\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1}{\alpha_{i}+\beta+|\bm{n}|};x\right)}=\frac{(n_{i}-1)!}{\left(\alpha_{i}+\beta+|\bm{n}|\right)^{\overline{n_{i}-1}}}\,L_{n_{i}-1}^{(\alpha_{i}+\beta+|\bm{n}|-1)}(x),$
(89)

where $L_{n}^{(\alpha)}$ is the Laguerre polynomial. In other words, the Type
I multiple Laguerre polynomial of the first kind can be obtained as a finite
multiplicative convolution of the Type I Jacobi-Piñeiro polynomials with the
same parameters and a standard Laguerre polynomial.

####  4.2.1. Real zeros: monotonicity and interlacing

The discussion about real zeros carried out in Section 4.1.1 applies to Type I
multiple Laguerre polynomials, with the only modification that now in (61),
$\Delta_{1}=(0,+\infty)$. In particular, under assumption (63), these
polynomials form a Nikishin system, and thus, for close-to-diagonal multi-
indices $\bm{n}$ their zeros will be on $\Delta_{r}$.

In addition, we have the following

######  Theorem 4.5.

Let $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ satisfy (55). Assume that for
a multi-index $\bm{n}\in\mathbb{N}^{r}$ and for a specific index
$i\in\\{1,2,\dots,r\\}$, condition (64) is satisfied. Then
$L^{(\bm{\alpha})}_{\bm{n},i}\in\mathbb{P}_{n_{i}-1}(\Delta_{r})$.

Additionally, for $0<t\leq 2$,

  1. (i)

$L^{(\bm{\alpha}+t)}_{\bm{n},i}\prec L^{(\bm{\alpha})}_{\bm{n},i}$, if $r$ is
even, and

  2. (ii)

$L^{(\bm{\alpha})}_{\bm{n},i}\prec L^{(\bm{\alpha}+t)}_{\bm{n},i}$, if $r$ is
odd.

###### Proof.

The statement follows immediately from representation (88) and Theorem 4.1,
and the fact that all zeros of the Laguerre polynomial in (89) are positive. ∎

Additionally, as for the Jacobi-Piñeiro polynomials, from (12) it follows that
$L^{(\bm{\alpha})}_{\bm{n},i}\preccurlyeq
L^{(\bm{\alpha}+\bm{e}_{i})}_{\bm{n}-\bm{e}_{i},i}$.

######  Corollary 4.6.

Let $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ satisfy (55) and (63). Then
for any multi-index $\bm{n}\in\mathbb{N}^{r}$ on the step-line, i.e., of the
form (72), conclusions of Theorem 4.5 hold for all $i\in\\{1,\dots,r\\}$.

####  4.2.2. Zero asymptotics of Type I Multiple Laguerre polynomials of first
kind

As we did for the Type I Jacobi-Piñeiro polynomials, we can describe the weak
zero asymptotics of a sequence of rescaled Type I Multiple Laguerre
polynomials of first kind, namely,

$p_{\bm{n},i}(x)=L^{(\bm{\alpha}_{\bm{n}})}_{\bm{n},i}(n_{i}\,x),\quad
i=1,\dots,r,$

where
$\bm{\alpha}_{\bm{n}}=\left(\alpha_{1}^{(\bm{n})},\dots,\alpha_{r}^{(\bm{n})}\right)$,
$\bm{n}=(n_{1},\dots,n_{r})\in\mathbb{N}^{r}$, under assumption

$\lim_{|\bm{n}|\to\infty}\frac{\alpha_{i}^{(\bm{n})}}{|\bm{n}|}=A_{i}\geq
0,\quad\text{and}\quad\lim_{|\bm{n}|\to\infty}\frac{n_{i}}{|\bm{n}|}=\theta_{i}>0,\qquad
i=1,\dots,r.$ (90)

Clearly, $\theta_{1}+\dots+\theta_{r}=1$. Formula (85) indicates that with the
parameters $\mathfrak{a}_{j}^{(i)}$ and $\mathfrak{b}_{j}^{(i)}$ defined in
(74), satisfying the same restrictions (75), namely,

$A_{j}\neq A_{i}+\theta_{i},\quad
A_{j}+\theta_{j}\notin(A_{i},A_{i}+\theta_{i}],\qquad
j\in\\{1,\dots,r\\}\setminus\\{i\\},$

Theorem 3.7 implies that for any weak-* accumulation point $\mu$ of the
normalized zero-counting measures $\chi(p_{\bm{n},i})$,

$\mathcal{S}_{\mu}(z)=\frac{\prod_{j=1,\,j\neq
i}^{r}\left(z+\mathfrak{a}_{j}^{(i)}+1\right)}{\prod_{j=1}^{r}\left(z+\mathfrak{b}_{j}^{(i)}+1\right)},$
(91)

and by Theorem 3.9, the Cauchy transform $\mathcal{G}_{\mu}(u)$ of the limit
measure $\mu$ in a neighborhood of infinity is an algebraic function $y/u$,
where $y=y(u)$ satisfies the equation

$y\prod_{j=1}^{r}(y+\mathfrak{b}_{j}^{(i)})=u(y-1)\prod_{i=1,\,j\neq
i}^{r}(y+\mathfrak{a}_{j}^{(i)}).$

Let us discuss the case when $\bm{\alpha}$ does not depend on $\bm{n}$, so
that identities (77) hold, and constrains (75) once again boil down to (78).
If we assume that there exists an index $i\in\\{1,\dots,r\\}$ such that

$\theta_{i}<\min\left\\{\theta_{j}\mathrel{\mathop{\ordinarycolon}}\,j\in\\{1,\dots,r\\}\setminus\\{i\\}\right\\},$

imposing additionally that $|\alpha_{i}-\alpha_{j}|<1$ for all $j$’s, then as
in Section 4.1.2 we conclude that any weak-* accumulation point $\mu$ of the
normalized zero-counting measures $\chi(p_{\bm{n},i})$ is a compactly
supported probability measure on $\mathbb{R}_{\geq 0}$, if $r$ is odd, and on
$\mathbb{R}_{\leq 0}$, otherwise, and such that

$\mathcal{S}_{\mu}(z)=\frac{1}{(z+1)^{r}}\prod_{j\neq
i}(z+1-\theta_{j}/\theta_{i}).$

######  Example 4.7.

Like in Example 4.4, if $r=2$, and $\theta_{1}=\theta<1/2$, the Cauchy
transform $\mathcal{G}_{\mu}(u)$ of the limit measure $\mu$ is given by $y/u$,
where $y=y(u)$ solves the equation

$y^{3}=u(y-1)\left(y-\frac{1}{\theta}+1\right).$

With the change $y=3w-u/3$ we get

$27w^{3}+\left(\frac{3}{\theta}-u\right)uw+\left(\frac{u}{3\theta}+1-\frac{1}{\theta}-\frac{2u^{2}}{27}\right)u=0.$

Proceeding as in Example 4.4, we can use Cardano’s formula and select the
right branch of the solution observing that $w>0$ for $u>0$. The
Sokhotski–Plemelj formula shows then that the limit measure $\mu$ lives on the
interval $[-c^{*},0]$, with

$c^{*}=\frac{9(1-\theta)\theta-2+2\sqrt{(1-3(1-\theta)\theta)^{3}}}{\theta(1-2\theta)^{2}}>0.$

See Figure 2 for a comparison of some actual values of the smallest zeros with
the predicted value of $-c^{*}$.

Figure 2. Values of the smallest zeros (marked with $*$) and of the
corresponding values of $-c^{*}$ (left end-point of the limit zero
distribution, marked with ’o’) for $L^{(\bm{\alpha})}_{(n_{1},n_{2}),1}$, with
$\bm{\alpha}=(1/2,3/7)$, $n_{1}=250$, and different values of $n_{2}/n_{1}$.

###  4.3. Type I Multiple Laguerre of the second kind

Let $\alpha>-1$ and $\bm{c}=(c_{1},\dots,c_{r})\in\mathbb{R}_{>0}^{r}$ be a
multi-index such that $c_{i}\neq c_{j}$ for $i\neq j$. Then, for
$\bm{n}=(n_{1},\dots,n_{r})\in\mathbb{N}^{r}$, the type I multiple Laguerre
polynomials of the second kind, $\mathcal{L}_{\bm{n},i}^{(\alpha,\bm{c})}$,
$\deg\mathcal{L}_{\bm{n},i}^{(\alpha,\bm{c})}\leq n_{j}-1$, $i=1,\dots,r$, are
given by the orthogonality conditions on the function

$Q_{\bm{n}}^{(\alpha,\bm{c})}(x)\overset{\mathrm{def}}{=\joinrel=}\sum_{j=1}^{r}\mathcal{L}_{\bm{n},i}^{(\alpha,\bm{c})}(x)\,x^{\alpha}e^{-c_{j}x},$

namely,

$\int_{0}^{+\infty}Q_{\bm{n}}^{(\alpha,\bm{c})}(x)x^{k}\,dx=0,\qquad 0\leq
k\leq|\bm{n}|-2,$ (92)

together with the normalization

$\int_{0}^{+\infty}Q_{\bm{n}}^{(\alpha,\bm{c})}(x)x^{|\bm{n}|-1}\,dx=1.$

Notice that in this case, the weights $w_{j}$ in (7) no longer form a Nikishin
system, although they are an AT-system, see, e.g. [21, §23.4].

We are aware of an explicit formula for these polynomials only in the case of
$r=2$: it was shown in [7, Proposition 8.1] that the Type I Laguerre
polynomials of the second kind are also Kampé de Fériet polynomials333 For
convenience, we changed the order of second and third vectors of parameters
with respect to the notation used therein.: for $\bm{c}=(c_{1},c_{2})$ and
$i\in\\{0,1\\}$,

$\mathcal{L}_{\bm{n},i}^{(\alpha,\bm{c})}(x)\simeq
F_{1\mathrel{\mathop{\ordinarycolon}}0;1}^{2\mathrel{\mathop{\ordinarycolon}}0;0}\left[\left.\begin{array}[]{c}(-n_{i}+1,\alpha+1)\mathrel{\mathop{\ordinarycolon}}\cdot\
;\cdot\\\ -n_{1}-n_{2}+2\mathrel{\mathop{\ordinarycolon}}\cdot\
;\alpha+1\end{array}\right\rvert\,d_{i},-c_{i}d_{i}x\right],\qquad
d_{i}\overset{\mathrm{def}}{=\joinrel=}1-\frac{c_{i}}{c_{2-i}}\neq 0.$ (93)

Hence, as an application of Theorem 3.1 we conclude that

######  Proposition 4.8.

The Type I Laguerre polynomials of the second kind can be expressed as a
multiplicative convolution of a Jacobi polynomial with a Laguerre polynomial

$\mathcal{L}_{\bm{n},i}^{(\alpha,\bm{c})}(x)\simeq{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha+1}{-n_{1}-n_{2}+2};-x+d_{i}\right)}\boxtimes_{n_{i}-1}{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1}{\alpha+1};c_{i}d_{i}x\right)}.$
(94)

###### Proof.

By Theorem 3.1,

$\displaystyle
F_{1\mathrel{\mathop{\ordinarycolon}}0;1}^{2\mathrel{\mathop{\ordinarycolon}}0;0}\left[\left.\begin{array}[]{c}(-n_{i}+1,\alpha+1)\mathrel{\mathop{\ordinarycolon}}\cdot\
;\cdot\\\ -n_{1}-n_{2}+2\mathrel{\mathop{\ordinarycolon}}\cdot\
;\alpha+1\end{array}\right\rvert\,d_{i},y\right]\simeq\left(p\boxplus_{n_{i}-1}q\right)\boxtimes_{n_{i}-1}r,$

where

$\displaystyle p(y)$ $\displaystyle={\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha+1}{-n_{1}-n_{2}+2};-y\right)},\qquad
q(y)={\
}_{1}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1}{\cdot};y/d_{i}\right)},$
$\displaystyle r(y)$ $\displaystyle={\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1}{\alpha+1};-y\right)}.$

Thus, by (17),

$\displaystyle
F_{1\mathrel{\mathop{\ordinarycolon}}0;1}^{2\mathrel{\mathop{\ordinarycolon}}0;0}\left[\left.\begin{array}[]{c}(-n_{i}+1,\alpha+1)\mathrel{\mathop{\ordinarycolon}}\cdot\
;\cdot\\\ -n_{1}-n_{2}+2\mathrel{\mathop{\ordinarycolon}}\cdot\
;\alpha+1\end{array}\right\rvert\,d_{i},-c_{i}d_{i}x\right]\simeq\text{Dil}_{\frac{1}{-c_{i}d_{i}}}\left[\left(p\boxplus_{n_{i}-1}q\right)\boxtimes_{n_{i}-1}r\right]$
$\displaystyle=\left(p\boxplus_{n_{i}-1}q\right)\boxtimes_{n_{i}-1}\text{Dil}_{\frac{1}{-c_{i}d_{i}}}(r)\simeq\left[p\boxplus_{n_{i}-1}q\right](x)\boxtimes_{n_{i}-1}r(-c_{i}d_{i}x).$

Notice that $q(x)={\
}_{1}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1}{\cdot};x/d_{i}\right)}\simeq\left(x-d_{i}\right)^{n_{i}-1}$,
so that by (21),

$\left[p\boxplus_{n_{i}-1}q\right](x)=p\left(x-d_{i}\right),$

and formula (94) follows. ∎

####  4.3.1. Zero asymptotics of Type I Multiple Laguerre polynomials of the
second kind

Unfortunately, the decomposition in Proposition 4.8 does not allow us to
conclude that the zeros of $\mathcal{L}_{\bm{n},i}^{(\alpha,\bm{c})}$ are
real. However, using the arguments of Section 3.2, we can find at least the
$S$-transform (and, eventually, an equation on the Cauchy transform) of any
accumulation point of the normalized zero-counting measures of these
polynomials.

Let $\bm{n}=(n_{1},n_{2})\in\mathbb{N}^{2}$ and $\alpha_{\bm{n}}>-1$ be such
that

$\lim_{|\bm{n}|\to\infty}\frac{\alpha_{\bm{n}}}{|\bm{n}|}=A\geq
0,\quad\text{and}\quad\lim_{|\bm{n}|\to\infty}\frac{n_{i}}{|\bm{n}|}=\theta_{i}>0,\qquad
i=1,2.$ (95)

Moreover, assume that
$\bm{c}_{\bm{n}}=\left(c_{1}^{(\bm{n})},c_{2}^{(\bm{n})}\right)\in\mathbb{R}_{>0}^{2}$,
with $c_{1}^{(\bm{n})}\neq c_{2}^{(\bm{n})}$, depends on the multi-index
$\bm{n}$ in such a way that the limit

$\lim_{\bm{n}}\bm{c}_{\bm{n}}=\bm{c}=(c_{1},c_{2})\in\mathbb{R}_{>0}^{2},\quad
c_{1}\neq c_{2},$ (96)

exists. Let us also define

$d_{1}^{(\bm{n})}\overset{\mathrm{def}}{=\joinrel=}1-\frac{c_{1}^{(\bm{n})}}{c_{2}^{(\bm{n})}},\quad
d_{2}^{(\bm{n})}\overset{\mathrm{def}}{=\joinrel=}1-\frac{c_{2}^{(\bm{n})}}{c_{1}^{(\bm{n})}},$

so that $d_{i}^{(\bm{n})}\to d_{i}$, $i=1,2$, with $d_{i}$ defined in (93).

######  Remark 4.9.

It is easy to see that a simple re-scaling $x\mapsto x/N$, $N=N(\bm{n})$, in
(92) allows us to tackle the zero asymptotics also in the case of parameters
$\bm{c}$ satisfying

$\lim_{\bm{n}}\frac{\bm{c}_{\bm{n}}}{N}=\bm{c}=(c_{1},c_{2})\in\mathbb{R}_{>0}^{2},\quad
c_{1}\neq c_{2}.$

Notice that under this assumption, the limits $d_{i}^{(\bm{n})}\to d_{i}$,
$i=1,2$, still hold.

######  Theorem 4.10.

Under the assumptions (95)–(96), the Cauchy transform $y=\mathcal{G}_{\mu}(u)$
of any limiting zero distribution $\mu$ of polynomials
$\mathcal{L}_{\bm{n},i}^{(\alpha_{\bm{n}},\bm{c}_{\bm{n}})}(n_{i}x/(c_{i}d_{i}))$
satisfies the cubic equation

$d_{i}(d_{i}-1)\theta_{i}u\,y^{3}+\left(A_{i}d_{i}(d_{i}-1)+\theta_{i}(1-2d_{i})u\right)y^{2}+\left(-A_{i}d_{i}+\theta_{i}(d_{i}+u)-1\right)\,y-\theta_{i}=0.$
(97)

Notice that for the case $A_{i}=0$, (97) simplifies to

$d_{i}(d_{i}-1)u\,y^{3}+(1-2d_{i})u\,y^{2}+\left(u+d_{i}-1/\theta_{i}\right)\,y-1=0.$

###### Proof.

By (94),

$\begin{split}\mathcal{L}_{\bm{n},i}^{(\alpha_{\bm{n}},\bm{c}_{\bm{n}})}&(n_{i}x/(c_{i}d_{i}))\simeq{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1,\alpha+1}{-n_{1}-n_{2}+2};-x+d_{i}\right)}\boxtimes_{n_{i}-1}{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i}+1}{\alpha+1};n_{i}x\right)}\end{split}$
(98)

(for the sake of brevity, we have omitted writing the super-index $\bm{n}$ in
$c_{i}$ and $d_{i}$; recall that under assumption (95) they all have finite
and non-zero limits).

By Theorem 3.7, the $S$-transform for a limiting zero-counting measure of the
${}_{1}F_{1}$ polynomials on the right-hand side of (98) is

$\frac{1}{z+A_{i}/\theta_{i}+1}.$

It means that if we denote by $\widetilde{\mu}$ the limiting zero distribution
of the ${}_{2}F_{1}$ polynomials on the right-hand side of (98), then the
$S$-transforms of $\widetilde{\mu}$ and the limiting zero distribution $\mu$
of the polynomials
$\mathcal{L}_{\bm{n},i}^{(\alpha_{\bm{n}},\bm{c}_{\bm{n}})}(n_{i}x/(c_{i}d_{i}))$
are related by

$\mathcal{S}_{\mu}(z)=\frac{\mathcal{S}_{\widetilde{\mu}}(z)}{z+A/\theta_{i}+1}.$

By Corollary 3.10, $\widetilde{w}=\mathcal{S}_{\widetilde{\mu}}(z)$ satisfies
(with $c=-1$, $A=A_{i}/\theta_{i}$ and $B=-1/\theta_{i}$)

$\widetilde{w}\left[d_{i}z\widetilde{w}-(z+1-1/\theta_{i})\right]=\left(d_{i}\widetilde{w}-1\right)\left(d_{i}z\widetilde{w}-(z+1+A_{i}/\theta_{i})\right).$

Taking $\widetilde{w}=(z+A/\theta_{i}+1)w$ above and simplifying, we get that
$w=\mathcal{S}_{\mu}(z)$ is a solution of

$\begin{split}d_{i}(d_{i}-1)(A_{i}+\theta_{i}(z+1))zw^{2}+\left(-A_{i}d_{i}+\theta_{i}(-d_{i}(2z+1)+z+1)-1\right)w+\theta_{i}=0.\end{split}$
(99)

We can get an equation on $z=\mathcal{M}_{\mu}(u)$ by replacing
$zw\mapsto(z+1)u$ in (99):

$\begin{split}d_{i}(d_{i}-1)(A_{i}+\theta_{i}(z+1))(z+1)^{2}u^{2}+\left(-A_{i}d_{i}+\theta_{i}(-d_{i}(2z+1)+z+1)-1\right)(z+1)u+\theta_{i}z=0.\end{split}$

Recalling the definition (30) of the $M$-transform in terms of the Cauchy
transform $\mathcal{G}_{\mu}$ of $\mu$, we have that

$z=\frac{1}{u}\mathcal{G}_{\mu}\left(\frac{1}{u}\right)-1,$

and replacing $u\mapsto 1/u$, we can rewrite the equation above to obtain a
cubic equation on $y=\mathcal{G}_{\mu}(u)$:

$\displaystyle
d_{i}(d_{i}-1)(A_{i}+\theta_{i}uy)\,y^{2}+\left(-A_{i}d_{i}+\theta_{i}(-d_{i}(2uy-1)+uy)-1\right)y+\theta_{i}(uy-1)=0,$

which is equivalent to (97). ∎

##  5\. Type II Multiple orthogonal polynomials

Recall that Type II multiple orthogonal polynomial $P_{\bm{n}}$ are monic
polynomial of degree $|\bm{n}|$ that satisfy the orthogonality conditions (4),
namely,

$\int P_{\bm{n}}(x)x^{k}w_{j}(x)\,dx=0,\qquad 0\leq k\leq n_{j}-1,\quad 1\leq
j\leq r.$

These polynomials are better studied than their Type I counterparts. In
particular, it is known that their asymptotic zero distribution can be
described via a solution of a vector equilibrium problem. Now we can perform
this analysis using our free probability tools.

###  5.1. Type II Jacobi-Piñeiro polynomials

We consider again the AT-system of Jacobi-Piñeiro weights on $[0,1]$ that
satisfy the conditions

$\alpha_{1},\dots,\alpha_{r}>-1\quad\text{and}\quad\alpha_{i}-\alpha_{j}\not\in\mathbb{Z}\text{
for }i\not=j.$ (100)

The Type II (monic) Jacobi-Piñeiro polynomials,
$P^{(\bm{\alpha};\beta)}_{\bm{n}}$, $\deg
P^{(\bm{\alpha};\beta)}_{\bm{n}}=|\bm{n}|$, are given by the orthogonality
conditions

$\int_{0}^{1}P_{\bm{n}}^{(\bm{\alpha};\beta)}(x)\,x^{\alpha_{j}+k}(1-x)^{\beta}\,dx=0,\qquad
0\leq k\leq n_{j}-1,\quad j=1,\dots,r.$ (101)

Once again, these are hypergeometric polynomials (see [21, §23.3.2]):

$\begin{split}(-1)^{|\bm{n}|}\prod_{j=1}^{r}&\frac{\left(|\bm{n}|+\alpha_{j}+\beta+1\right)^{\overline{n_{j}}}}{\left(\alpha_{j}+1\right)^{\overline{n_{j}}}}\,(1-x)^{\beta}P_{\bm{n}}^{(\bm{a};\beta)}(x)={\
}_{r+1}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|-\beta,\bm{\alpha}+\bm{n}+1}{\bm{\alpha}+1};x\right)}.\end{split}$
(102)

Taking $\beta\in\mathbb{Z}_{\geq 0}$ on (102) and using Theorem A from Section
2.3, we obtain the following representation:

$(1-x)^{\beta}P_{\bm{n}}^{(\bm{a};\beta)}(x)\simeq
p_{1}(x)\boxtimes_{|\bm{n}|+\beta}\dots\boxtimes_{|\bm{n}|+\beta}p_{r}(x),$
(103)

where

$\begin{split}p_{j}(x)&=\frac{\left(\alpha_{j}+1\right)^{\overline{n_{j}}}}{(n_{j})!}\,{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|-\beta,\alpha_{j}+n_{j}+1}{\alpha_{j}+1};x\right)}\\\
&=(1-x)^{|\bm{n}|+\beta-n_{j}}\,{\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{j},|\bm{n}|+\beta+\alpha_{j}+1}{\alpha_{j}+1};x\right)},\quad
j=1,\dots,r.\end{split}$ (104)

These identities allow us to follow the methodology described above to find
the asymptotic zero distribution of these polynomials. To deal with the case
of noninteger $\beta$, we appeal to an alternative expression for the
reciprocals of the Type II Jacobi-Piñeiro polynomials, consequence of Lemma
2.3:

######  Proposition 5.1.

For the Type II Jacobi-Piñeiro polynomials,

$\begin{split}P_{\bm{n}}^{(\bm{a};\beta)}(x)\simeq\,p^{*}(x)\boxtimes_{|\bm{n}|}\left[{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{-\beta-|\bm{n}|+1};|\bm{n}|x\right)}\boxplus_{|\bm{n}|}\left({\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{\beta+1};|\bm{n}|x\right)}\boxtimes_{|\bm{n}|}q\right)\right]^{*},\end{split}$
(105)

where

$p(x)={\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,1}{\cdot};-\frac{x}{|\bm{n}|}\right)},\quad
q(x)={\ }_{r+1}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,\
-|\bm{n}|-\bm{\alpha}}{-|\bm{n}|-\bm{n}-\bm{\alpha}};-x\right)}.$ (106)

###### Proof.

By (102), and using the hypergeometric representation of $(1-x)^{\beta}$, we
have that

$P_{\bm{n}}^{(\bm{a};\beta)}(x)\simeq{\
}_{1}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{\beta}{\cdot};x\right)}{\
}_{r+1}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|-\beta,\alpha_{1}+n_{1}+1,\ldots,\alpha_{r}+n_{r}+1}{\alpha_{1}+1,\ldots,\alpha_{r}+1};x\right)}.$

Notice that $P_{\bm{n}}^{(\bm{\alpha};\beta)}(0)\neq 0$, so its reciprocals
has the same degree. Applying Lemma 2.3, we get that

$\displaystyle\left(P_{\bm{n}}^{(\bm{a};\beta)}\right)^{*}(x)$
$\displaystyle\simeq{\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,1}{\cdot};x\right)}$
$\displaystyle\boxtimes_{|\bm{n}|}\left[{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{-\beta-|\bm{n}|+1};-x\right)}\boxplus_{|\bm{n}|}{\
}_{r+1}F_{r+1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,\
-|\bm{n}|-\bm{\alpha}}{\beta+1,-|\bm{n}|-\bm{n}-\bm{\alpha}};-x\right)}\right].$

Using that
$(x-1)^{|\bm{n}|}=(x+|\bm{n}|)^{|\bm{n}|}\boxtimes_{|\bm{n}|}(x+|\bm{n}|^{-1})^{|\bm{n}|}$
is an identity under the finite free multiplicative convolution (see (16)), we
can rewrite $\left(P_{\bm{n}}^{(\bm{a};\beta)}\right)^{*}$ as the
multiplicative convolution of two polynomials,

$p(x)={\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,1}{\cdot};x\right)}\boxtimes_{|\bm{n}|}(x+|\bm{n}|)^{|\bm{n}|}\simeq{\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,1}{\cdot};-|\bm{n}|^{-1}x\right)}$

and

$\displaystyle\widetilde{p}(x)$
$\displaystyle=(x+|\bm{n}|^{-1})^{|\bm{n}|}\boxtimes_{|\bm{n}|}\left[{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{-\beta-|\bm{n}|+1};-x\right)}\boxplus_{|\bm{n}|}{\
}_{r+1}F_{r+1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,\
-|\bm{n}|-\bm{\alpha}}{\beta+1,-|\bm{n}|-\bm{n}-\bm{\alpha}};-x\right)}\right]$
$\displaystyle={\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{-\beta-|\bm{n}|+1};|\bm{n}|x\right)}\boxplus_{|\bm{n}|}{\
}_{r+1}F_{r+1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,\
-|\bm{n}|-\bm{\alpha}}{\beta+1,-|\bm{n}|-\bm{n}-\bm{\alpha}};|\bm{n}|x\right)}$
$\displaystyle={\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{-\beta-|\bm{n}|+1};|\bm{n}|x\right)}\boxplus_{|\bm{n}|}\left[{\
}_{1}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{\beta+1};|\bm{n}|x\right)}\boxtimes_{|\bm{n}|}q(x)\right],$

where $q$ was defined in (106).

It remains to observe that by the definition (15), the operation ∗ acts
distributively on the multiplicative convolution $\boxtimes_{|\bm{n}|}$. ∎

####  5.1.1. Real zeros: monotonicity and interlacing

######  Theorem 5.2.

Let a multi-index $\bm{n}\in\mathbb{N}^{r}$, $\beta\in\mathbb{Z}_{\geq 0}$ and
$\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ satisfy (100). For each
$i\in\\{1,\dots,r\\}$ and $0\leq t\leq 2$ such that

$\alpha_{i}-\alpha_{j}+t\not\in\mathbb{Z},\quad j\not=i,$ (107)

the following interlacing holds:

$P_{\bm{n}+\bm{e}_{i}}^{(\bm{\alpha},\beta)}(x)\preccurlyeq
P_{\bm{n}}^{(\bm{\alpha},\beta)}\preccurlyeq
P_{\bm{n}}^{(\bm{\alpha}+t\bm{e}_{i},\beta)}.$ (108)

Recall that $\bm{e}_{i}\in\mathbb{N}^{r}$ is the multi-index whose only non-
zero entry (equal to $1$) is in the position $i$.

###### Proof.

Take $i\in\\{1,\dots,r\\}$ and define the following polynomial,

$Q_{n_{i}}^{(\alpha_{i},\beta)}(x)={\
}_{2}F_{1}{\left(\genfrac{.}{.}{0.0pt}{}{-n_{i},|\bm{n}|+\beta+\alpha_{i}+1}{\alpha_{i}+1};x\right)}$
(109)

and using the decomposition (104) of $P_{\bm{n}}^{(\bm{\alpha},\beta)}$ define
the following polynomial,

$q_{i}(x)\mathrel{\mathop{\ordinarycolon}}=p_{1}\boxtimes_{|\bm{n}|+\beta}\cdots\boxtimes_{|\bm{n}|+\beta}p_{i-1}\boxtimes_{|\bm{n}|+\beta}p_{i+1}\boxtimes_{|\bm{n}|+\beta}\cdots\boxtimes_{|\bm{n}|+\beta}p_{r}$
(110)

By hypothesis, we have, $\alpha_{j},\beta>-1$ and $\bm{n}\in\mathbb{N}^{r}$,
thus

$\alpha_{j}+\beta+|\bm{n}|>\alpha_{j}+n_{j}-1,\qquad j\in\\{1,\dots,r\\}$

then we have by [11, Theorem 1, (ii)] that
$Q_{n_{i}}^{\alpha_{i},\beta},p_{j}\in\mathbb{P}_{|\bm{n}|+\beta}((0,1))$.
Applying Proposition 2.4 to polynomials $q_{i}$ defined in (110), we get that
$q_{i}\in\mathbb{P}_{|\bm{n}|+\beta}((0,1))$.

Using the the interpretation of Jacobi Polynomials as ${}_{2}F_{1}$, we can
extract from [12, Theorem 2.2] that for $0<t\leq 2$,

$Q_{n_{i}+1}^{\alpha_{i},\beta}\prec Q_{n_{i}}^{\alpha_{i},\beta}\prec
Q_{n_{i}}^{\alpha_{i}+t,\beta}$

Using Proposition 2.6, we get that

$\begin{split}(1-x)^{|\bm{n}|+\beta-
n_{i}-1}Q_{n_{i}+1}^{\alpha_{i},\beta}\boxtimes_{|\bm{n}|+\beta}q_{i}&\preccurlyeq(1-x)^{|\bm{n}|+\beta-
n_{i}}Q_{n_{i}}^{\alpha_{i},\beta}\boxtimes_{|\bm{n}|+\beta}q_{i}\\\
&\preccurlyeq(1-x)^{|\bm{n}|+\beta-
n_{i}}Q_{n_{i}}^{\alpha_{i}+t,\beta}\boxtimes_{|\bm{n}|+\beta}q_{i}.\end{split}$

For $0<t\leq 2$ satisfying (107), we conclude that

$P_{\bm{n}+\bm{e}_{i}}^{(\bm{\alpha},\beta)}(x)\preccurlyeq
P_{\bm{n}}^{(\bm{\alpha},\beta)}\preccurlyeq
P_{\bm{n}}^{(\bm{\alpha}+t\bm{e}_{i},\beta)}.$

∎

######  Remark 5.3.

From [12, Theorem 2.2], the polynomial defined in (109) satisfies

$Q_{n_{i}+1}^{\alpha_{i},\beta}\prec Q_{n_{i}}^{\alpha_{i},\beta+s}\prec
Q_{n_{i}}^{\alpha_{i},\beta}\quad\text{ for }0\leq s\leq 2.$

Using similar ideas from the proof of Theorem 5.2, we can obtain

$P_{\bm{n}+\bm{e}_{i}}^{(\bm{\alpha},\beta)}(x)\preccurlyeq
P_{\bm{n}}^{(\bm{\alpha},\beta+s)}\preccurlyeq
P_{\bm{n}}^{(\bm{\alpha},\beta)}\quad\text{ and }\quad s=0,1,2.$ (111)

The restriction of $s$ to entire numbers is necessary in order to use (103).
The interlacing stated in Theorem 5.2 and in (111) was partially proved in
[18, Theorem 2.2], where it was shown that

$P_{\bm{n}+\bm{e}_{i}}^{(\bm{\alpha},\beta)}(x)\preccurlyeq
P_{\bm{n}}^{(\bm{\alpha},\beta)}.$

####  5.1.2. Zero asymptotics of Type II Jacobi-Piñeiro polynomials

We can describe the weak zero asymptotics of a sequence of Type II Jacobi-
Piñeiro polynomials,

$p_{\bm{n}}=P_{\bm{n}}^{(\bm{a};\beta)}(x),$

where
$\bm{\alpha}_{\bm{n}}=\left(\alpha_{1}^{(\bm{n})},\dots,\alpha_{r}^{(\bm{n})}\right)$,
$\bm{n}=(n_{1},\dots,n_{r})\in\mathbb{N}^{r}$, under assumption (100), that
is,

$\lim_{|\bm{n}|\to\infty}\frac{\alpha_{i}^{(\bm{n})}}{|\bm{n}|}=A_{i}\geq
0,\quad\lim_{|\bm{n}|\to\infty}\frac{\beta^{(\bm{n})}}{|\bm{n}|}=B\geq
0,\quad\text{and}\quad\lim_{|\bm{n}|\to\infty}\frac{n_{i}}{|\bm{n}|}=\theta_{i}>0,\qquad
i=1,\dots,r.$ (112)

Let us suppose initially that for all $\bm{n}$,

$\beta^{(\bm{n})}\in\mathbb{Z}_{\geq 0}.$ (113)

Representation (102) indicates that we need to consider the parameters

$\mathfrak{a}_{j}\overset{\mathrm{def}}{=\joinrel=}(A_{j}+\theta_{j})/(1+B),\quad\text{and}\quad\mathfrak{b}_{j}\overset{\mathrm{def}}{=\joinrel=}A_{j}/(1+B),\quad
j=1,\dots,r.$ (114)

Restrictions $\mathfrak{a}_{j}\notin[-1,0)$ and $\mathfrak{b}_{j}\neq-1$ and
the assumption that $\mathfrak{a}_{i}\neq\mathfrak{b}_{j}$ for $i\neq j$ hold
automatically.

Let $\mu$ be a weak-* accumulation point $\mu$ of the normalized zero-counting
measures $\chi(p_{\bm{n},i})$, and denote

$\widetilde{\mu}=\frac{1}{1+B}\left(B\delta_{0}+\mu\right).$

By Theorem 3.7, the $S$-transform of $\widetilde{\mu}$ is

$\mathcal{S}_{\widetilde{\mu}}(z)=\prod_{j=1}^{r}\frac{z+\mathfrak{a}_{j}^{(i)}+1}{z+\mathfrak{b}_{j}^{(i)}+1}=\prod_{j=1}^{r}\frac{(1+B)z+\theta_{j}+A_{j}+B+1}{(1+B)z+A_{j}+B+1},$
(115)

while, by Theorem 3.9,

$y=u\,\mathcal{G}_{\widetilde{\mu}}(u)=\frac{u}{1+B}\left(\frac{B}{u}+\mathcal{G}_{\mu}(u)\right)=\frac{1}{1+B}\left(B+u\,\mathcal{G}_{\mu}(u)\right)$

is a solution of

$y\prod_{j=1}^{r}\left(y+\frac{A_{j}}{1+B}\right)=u(y-1)\prod_{j=1}^{r}\left(y+\frac{A_{j}+\theta_{j}}{1+B}\right).$

Let us discuss the case when $\bm{\alpha}$ and $\beta$ do not depend on
$\bm{n}$, so that $A=B=0$ and $\widetilde{\mu}=\mu$. In this case,

$\mathcal{S}_{\mu}(z)=\frac{1}{(z+1)^{r}}\,\prod_{j=1}^{r}\left(z+\theta_{j}+1\right),$

and $y=u\,\mathcal{G}_{\nu}(u)$ is a solution of

$y^{r+1}=u(y-1)\prod_{j=1}^{r}\left(y+\theta_{j}\right).$

In the asymptotically diagonal situation, when all $\theta_{j}=1/r$, this
equation boils down to

$y^{r+1}=u(y-1)\left(y+1/r\right)^{r}$

(compare it to (79); these equations can be reduced to each other with the
change $y\mapsto-y/r$; an equivalent equation appeared in [42]). By the
Sokhotski-Plemelj Formula, the density $\mu^{\prime}$ of $\mu$ on $(0,1)$ can
be recovered as

$\mu^{\prime}(x)=\frac{1}{2\pi ix}\left(y_{-}(x)-y_{+}(x)\right)=-\frac{1}{\pi
x}\mathop{\rm Im}\left(y_{+}(x)\right),\quad x\in(0,1).$

######  Example 5.4.

If $r=2$, and $\theta_{1}=\theta\leq 1/2$, the Cauchy transform
$\mathcal{G}_{\mu}(u)$ of the limit measure $\mu$ is given by $y/u$, where
$y=y(u)$ solves the equation

$y^{3}=u(y-1)\left(y+\theta\right)\left(y+1-\theta\right)=u\left(y^{3}-\left(1+\nu\right)y+\nu\right),\quad\nu=\theta(\theta-1)\in(-1/4,0).$
(116)

This is the same cubic equation (80) we obtained in Example 4.4, up to the
modification $\theta\mapsto 1/\theta$, and the corresponding change of the
value of $\nu$. Thus, we can use our previous calculations, selecting the
branch such that $y(u)>0$ for $u<0$. Namely, if now

$f(u)=\left(\frac{\theta(\theta-1)u}{2(1-u)}\right)^{1/3}$

is taken analytic in $\mathbb{C}\setminus\left[0,+\infty\right)$, whose
single-valued branch is fixed by requiring $f(x)>0$ for $x<0$, then the
desired solution of (116) is

$y(u)=f(u)\left(\left(1+\sqrt{1+\frac{\kappa\,u}{1-u}}\right)^{1/3}+\left(1-\sqrt{1+\frac{\kappa\,u}{1-u}}\right)^{1/3}\right),$

where

$\kappa=\kappa(\nu)=\frac{4}{27}\,\frac{(1+\nu)^{3}}{\nu^{2}}=\frac{4(1+(\theta-1)\theta)^{3}}{27\theta^{2}(1-\theta)^{2}}\geq
1$

and with the branches of the roots taken always positive for
$u\in(-\varepsilon,0)$ for a sufficiently small $\varepsilon>0$. Notice that
$\kappa(\nu)$ is a strictly increasing function on $(-1/4,0))$, with
$\kappa(-1/4)=1$.

For $u=x>0$, denote by $y_{\pm}$ the boundary values of $y$ on $(0,1)$. As
before, the Sokhotski-Plemelj Formula gives us the density of the limit
measure $\mu$ as

$\displaystyle\mu^{\prime}(x)$ $\displaystyle=\frac{1}{2\pi
ix}\left(y_{-}(x)-y_{+}(x)\right)=-\frac{1}{\pi x}\mathop{\rm
Im}\left(y_{+}(x)\right).$

We have that

$\left(\sqrt{1+\frac{\kappa\,u}{1-u}}+1\right)^{1/3}_{+}\bigg{|}_{u=x}=\left(\sqrt{1+\frac{\kappa\,x}{1-x}}+1\right)^{1/3}>0\text{
for }x\in(0,1),$

and, taking into account that
$1-\sqrt{1+u}=-\frac{u}{2}\left(1+\mathcal{O}(u)\right)$ as $u\to 0$,

$\left(1-\sqrt{1+\frac{\kappa\,u}{1-u}}\right)^{1/3}_{+}\bigg{|}_{u=x}=\left(\sqrt{1+\frac{\kappa\,x}{1-x}}-1\right)^{1/3}e^{-\pi
i/3}\text{ for }x\in(0,1).$

Since

$f_{+}(x)=\sqrt[3]{\frac{-\nu x}{2(1-x)}}\,e^{-\pi i/3},\quad x\in(0,1),$

we get that on $(0,1)$,

$y_{+}(x)=\sqrt[3]{\frac{-\nu x}{2(1-x)}}\,e^{-\pi
i/3}\left(\left(\sqrt{1+\frac{\kappa\,x}{1-x}}+1\right)^{1/3}+\left(\sqrt{1+\frac{\kappa\,x}{1-x}}-1\right)^{1/3}e^{-\pi
i/3}\right).$

Putting all together, we conclude that $\mu$ is supported on $[0,1]$, with

$\begin{split}\mu^{\prime}(x)&=\frac{\sqrt{3}}{2\pi}\,\sqrt[3]{\frac{\theta(1-\theta)}{2}}\,\,\frac{\left(\sqrt{1+(\kappa-1)x}+\sqrt{1-x}\right)^{1/3}+\left(\sqrt{1+(\kappa-1)x}-\sqrt{1-x}\right)^{1/3}}{x^{2/3}\sqrt{1-x}}.\end{split}$
(117)

Symbolic integration allows us to check that $\mu^{\prime}(x)$ is the density
of a positive probability measure on $[0,1]$, and that

$\mu^{\prime}(x)=\frac{\sqrt{3}\sqrt[3]{\theta(1-\theta))}}{2\pi\,x^{2/3}}(1+o(1))\text{
as }x\to
0+,\quad\text{and}\quad\mu^{\prime}(x)=\frac{\sqrt{1-\theta(1-\theta)}}{\pi\sqrt{1-x}}(1+o(1))\text{
as }x\to 1-.$

See Figure 3 for a histogram of the zeros for $\theta=1/3$.

In the diagonal case, when $\theta=1/2$ (and $\kappa=1$), this formula
coincides with the expression

$\mu^{\prime}(x)=\frac{\sqrt{3}}{4\pi}\,\,\frac{\left(1+\sqrt{1-x}\right)^{1/3}+\left(1-\sqrt{1-x}\right)^{1/3}}{x^{2/3}\sqrt{1-x}},$
(118)

found in [42], see also [28, §8.4]. On the other hand, as $\theta\to 0+$ or
$\theta\to 1-$, $\mu$ converges to the equilibrium measure (arcsine
distribution) on $[0,1]$.

Figure 3. Histogram of the zeros of the polynomial
$P^{(\bm{\alpha};\beta)}_{\bm{n}}$, with $\bm{\alpha}=(1/2,3/7)$, $\beta=1$,
and $\bm{n}=(300,600)$, along with the corresponding density.

######  Example 5.5.

Still in the case of $r=2$ and asymptotically diagonal multi-indices
($\theta_{1}=\theta_{2}=1/2$), similar ideas allow us to tackle the case of
varying parameters $\bm{\alpha}_{\bm{n}}$ and $\beta^{(\bm{n})}$, satisfying
(112). We avoid performing cumbersome calculations here. Instead, we point out
that if the case of $B=0$ and $A_{1}=A_{2}=A\geq 0$, the support of the limit
measure $\mu$ is $[a^{*},1]$, with

$a^{*}=\frac{A^{3}(A+1)}{\left(A+\frac{3}{2}\right)^{3}\left(A+\frac{1}{2}\right)}\in[0,1).$
(119)

On the other hand, if $A_{1}=A_{2}=0$ and $B\geq 0$, the measure is supported
on $[0,b^{*}]$, with

$b^{*}=\frac{27(B+1)^{2}}{\left(2B+3\right)^{3}}\in(0,1].$ (120)

See Figure 4 for the result of some numerical experiments.

Figure 4. Left: plot of the smallest zeros of the polynomial
$P^{(\bm{\alpha};\beta)}_{\bm{n}}$, with $\bm{n}=(300,300)$, $\beta=0$, and
$\bm{\alpha}=(600A,600A)$, for $A=1,\dots,15$, along with the predicted
leftmost end-point of the support, given by (119). Right: plot of the largest
zeros of the polynomial $P^{(\bm{\alpha};\beta)}_{\bm{n}}$, with
$\bm{\alpha}=(0,3/2)$, $\bm{n}=(300,300)$, and $\beta=600B$, for
$B=1,\dots,15$, along with the predicted rightmost end-point of the support,
given by (120).

All considerations above have been carried out under the assumption (113) that
all $\beta$’s are integers. Consider the representation (105) from Proposition
5.1. We observe that for both polynomials in its right-hand side depending on
$\beta$, their limit zero distribution depends only on the value of the limit
$\beta/|\bm{n}|$, and not on the concrete values of $\beta$. Hence, the same
is true for the expression of the $S$-transforms of the limit measures for $p$
and for the polynomial within the brackets on the right-hand side of (105).
Finally, using (35), we can conclude that the asymptotic zero distribution of
$P_{\bm{n}}^{(\bm{a};\beta)}$ depends only on the limits (112) but not on the
actual values of $\beta$. In other words, expression (117) is valid under the
general assumptions (112).

###  5.2. Type II Multiple Laguerre polynomials of the first kind

For $\bm{n}\in\mathbb{N}^{r}$, the Type II Multiple Laguerre polynomials of
the first kind, $L_{\bm{n}}^{\bm{\alpha}}$, corresponding to the weights (6),
are polynomial of degree $|\bm{n}|-1$, satisfying

$\int_{0}^{\infty}L_{\bm{n}}^{\bm{\alpha}}(x)x^{\alpha_{j}}e^{-x}x^{k}dx=0,\quad
k=0,1,\ldots,n_{j}-1,\quad j=1,\dots,r,$

where parameters $\alpha_{1},\ldots,\alpha_{r}>-1$ are such that
$\alpha_{i}-\alpha_{j}\notin\mathbb{Z}$ whenever $i\neq j$. These are
hypergeometric polynomials (see [21, §23.4.2]):

$\begin{split}(-1)^{|\bm{n}|}e^{-x}L_{\bm{n}}^{\bm{\alpha}}(x)=\prod_{j=1}^{r}\left(\alpha_{j}+1\right)^{\overline{n_{j}}}\,{\
}_{r}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{\bm{\alpha}+\bm{n}+1}{\bm{\alpha}+1};-x\right)}.\end{split}$
(121)

Notice that in this identity, neither the left-hand nor the right-hand side is
a polynomial. However, the polynomials $L_{\bm{n}}^{\bm{\alpha}}$ do not
vanish at the origin, and we can write their reciprocal as a finite
convolution:

######  Proposition 5.6.

For the reciprocal $p^{*}$ of the Type II Multiple Laguerre polynomials of the
first kind $p(x)=L_{\bm{n}}^{\bm{\alpha}}$,

$p^{*}\simeq{\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,1}{\cdot};x\right)}\boxtimes_{|\bm{n}|}{\
}_{r+1}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,\
-|\bm{n}|-\bm{\alpha}}{-|\bm{n}|-\bm{n}-\bm{\alpha}};x+1\right)}.$ (122)

###### Proof.

Since

$e^{x}={\ }_{0}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{\cdot}{\cdot};x\right)},$

we can rewrite (121) as

$p(x)\simeq{\
}_{0}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{\cdot}{\cdot};x\right)}{\
}_{r}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{\bm{\alpha}+\bm{n}+1}{\bm{\alpha}+1};-x\right)}.$

Applying the Lemma 2.3 we get that

$p^{*}\simeq{\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,1}{\cdot};x\right)}\boxtimes_{|\bm{n}|}\left({\
}_{1}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{\cdot};-x\right)}\boxplus_{|\bm{n}|}{\
}_{r+1}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,\
-|\bm{n}|-\bm{\alpha}}{-|\bm{n}|-\bm{n}-\bm{\alpha}};x\right)}\right).$

On the other hand,

${\
}_{1}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|}{\cdot};-x\right)}\simeq(x+1)^{|\bm{n}|},$

and from (21) we get (122). ∎

####  5.2.1. Real zeros: monotonicity and interlacing

The Type II Multiple Laguerre polynomials of the first kind
$L_{\bm{n}}^{\bm{\alpha}}$ can be obtained from the Jacobi-Piñeiro polynomials
$P_{\bm{n}}^{(\bm{\alpha};\beta)}$ by the limit process,

$L_{\bm{n}}^{\bm{\alpha}}(x)\simeq\lim_{\beta\to+\infty}\beta^{|\bm{n}|}P_{\bm{n}}^{(\bm{\alpha};\beta)}(x/\beta),$

which can be easily established by taking the limit directly in the
orthogonality relations (101). In consequence, interlacing properties of the
zeros of $L_{\bm{n}}^{\bm{\alpha}}$ follow from Theorem 5.2:

######  Theorem 5.7.

Let a multi-index $\bm{n}\in\mathbb{N}^{r}$, and
$\bm{\alpha}=(\alpha_{1},\dots,\alpha_{r})$ satisfy (100). For each
$i\in\\{1,\dots,r\\}$ and $0\leq t\leq 2$ satisfying (107), the following
interlacing holds,

$L_{\bm{n}+\bm{e}_{i}}^{(\bm{\alpha})}(x)\preccurlyeq
L_{\bm{n}}^{(\bm{\alpha})}\preccurlyeq
L_{\bm{n}}^{(\bm{\alpha}+t\bm{e}_{i})},$

where $\bm{e}_{i}\in\mathbb{N}^{r}$ is the multi-index whose only non-zero
entry (equal to $1$) is in the position $i$.

####  5.2.2. Zero asymptotics of Type II Multiple Laguerre polynomials of the
first kind

We can use Proposition 5.6 to describe the asymptotic zero distribution of the
rescaled polynomials $L^{(\bm{\alpha}_{\bm{n}})}_{\bm{n}}(|\bm{n}|\,x)$, under
assumptions (90), that is,

$\lim_{|\bm{n}|\to\infty}\frac{\alpha_{i}^{(\bm{n})}}{|\bm{n}|}=A_{i}\geq
0,\quad\text{and}\quad\lim_{|\bm{n}|\to\infty}\frac{n_{i}}{|\bm{n}|}=\theta_{i}>0,\qquad
i=1,\dots,r,$ (123)

where
$\bm{\alpha}_{\bm{n}}=\left(\alpha_{1}^{(\bm{n})},\dots,\alpha_{r}^{(\bm{n})}\right)$,
$\bm{n}=(n_{1},\dots,n_{r})\in\mathbb{N}^{r}$.

######  Theorem 5.8.

Under the assumptions (123), function $y=u\mathcal{G}_{\mu}(u)$, where
$\mathcal{G}_{\mu}$ is the Cauchy transform of any limiting zero distribution
$\mu$ of polynomials $L^{(\bm{\alpha}_{\bm{n}})}_{\bm{n}}(|\bm{n}|\,x)$,
satisfies the algebraic equation

$u\prod_{i=1}^{r}\left(y+A_{i}+\theta_{i}-u\right)=\left(u-y\right)\prod_{i=1}^{r}\left(y+A_{i}-u\right).$
(124)

In the particular case of $\bm{\alpha}$ independent on $\bm{n}$, so that all
$A_{j}=0$, it simplifies to

$u\prod_{i=1}^{r}\left(y+\theta_{i}-u\right)=-\left(y-u\right)^{r+1}.$ (125)

###### Proof.

If

$p_{\bm{n}}(x)=L^{(\bm{\alpha}_{\bm{n}})}_{\bm{n}}(x),$

then

$\left(L^{(\bm{\alpha}_{\bm{n}})}_{\bm{n}}(|\bm{n}|\,x)\right)^{*}\simeq
p^{*}_{\bm{n}}(x/|\bm{n}|).$

By (122),

$p^{*}(x/|\bm{n}|)\simeq{\
}_{2}F_{0}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,1}{\cdot};x/|\bm{n}|\right)}\boxtimes_{|\bm{n}|}{\
}_{r+1}F_{r}{\left(\genfrac{.}{.}{0.0pt}{}{-|\bm{n}|,\
-|\bm{n}|-\bm{\alpha}_{\bm{n}}}{-|\bm{n}|-\bm{n}-\bm{\alpha}_{\bm{n}}};x+1\right)}.$
(126)

By Corollary 3.10, the $S$-transform
$\widetilde{w}=\mathcal{S}_{\widetilde{\mu}}(z)$ of a weak-* limit
$\widetilde{\mu}$ of the normalized zero-counting measures of the
${}_{r+1}F_{r}$ polynomial on the right-hand side satisfies the equation

$\widetilde{w}\prod_{i=1}^{r}\left(z\widetilde{w}+z-A_{i}-\theta_{i}\right)=\left(\widetilde{w}+1\right)\prod_{i=1}^{r}\left(z\widetilde{w}+z-A_{i}\right).$
(127)

Taking into account (51) and the representation (126), we see that the
$S$-transform $w=\mathcal{S}_{\mu^{*}}(z)$ of the normalized zero-counting
measure of reversed scaled polynomials
$\left(L^{(\bm{\alpha}_{\bm{n}})}_{\bm{n}}(|\bm{n}|\,x)\right)^{*}$ is

$\mathcal{S}_{\mu^{*}}(z)=(z+1)\mathcal{S}_{\widetilde{\mu}}(z).$

Thus, making the change of variables $\widetilde{w}=w/(z+1)$ in (127), we get
an equation for $w=\mathcal{S}_{\mu^{*}}(z)$:

$w\prod_{i=1}^{r}\left(zw+(z+1)\left(z-A_{i}-\theta_{i}\right)\right)=\left(w+z+1\right)\prod_{i=1}^{r}\left(zw+(z+1)\left(z-A_{i}\right)\right).$
(128)

In particular, the equation for the shifted $S$-transform
$w=\mathcal{S}_{\mu^{*}}(-z-1)$ is

$w\prod_{i=1}^{r}\left(-(z+1)w+z\left(z+1+A_{i}+\theta_{i}\right)\right)=\left(w-z\right)\prod_{i=1}^{r}\left(-(z+1)w+z\left(z+1+A_{i}\right)\right).$

By (35), the $S$-transform $w=\mathcal{S}_{\mu}(z)$ of the normalized zero-
counting measure of the scaled polynomials
$L^{(\bm{\alpha}_{\bm{n}})}_{\bm{n}}(|\bm{n}|\,x)$ satisfies the equation

$\prod_{i=1}^{r}\left(\left(z+1+A_{i}+\theta_{i}\right)zw-(z+1)\right)=\left(1-zw\right)\prod_{i=1}^{r}\left(\left(z+1+A_{i}\right)zw-(z+1)\right).$

By (32), the equation for $u=\mathcal{M}_{\mu}^{-1}(z)$ is

$\prod_{i=1}^{r}\left(\left(z+1+A_{i}+\theta_{i}\right)u-1\right)=\left(1-(z+1)u\right)\prod_{i=1}^{r}\left(\left(z+1+A_{i}\right)u-1\right).$

Recalling the definition (30) of the $M$-transform in terms of the Cauchy
transform $\mathcal{G}_{\mu}$ of $\mu$, we have that

$z=\frac{1}{u}\mathcal{G}_{\mu}\left(\frac{1}{u}\right)-1;$

with $u\mapsto 1/u$, we arrive at the algebraic equation (124) for
$y=u\mathcal{G}_{\mu}(u)$. ∎

######  Example 5.9.

For $r=1$ (so that $\theta_{1}=1$) and $\bm{\alpha}$ independent on $\bm{n}$,
we get the equation

$y^{2}-uy+u=0,$

which yields that

$\mathcal{G}_{\mu}(u)=\frac{1}{2}\left(1-\sqrt{\frac{u-4}{u}}\right),$

which corresponds to the Marchenko-Pastur distribution, supported on $[0,4]$.

For $r=2$ and $\bm{\alpha}$ independent of $\bm{n}$, denoting
$\theta_{1}=\theta$, the equation (125) reduces to

$y^{3}-2uy^{2}+\left(u+1\right)uy+u\left(\theta(1-\theta)-u\right)=0.$

The support of $\mu$ in this case is the interval $[0,c^{*}]$, where $c=c^{*}$
is the positive solution of the equation

$(1-2\theta)^{2}c^{2}-2(2-9(1-\theta)\theta)c-27(\theta-1)^{2}\theta^{2}=0,$

Figure 5. Graph of the predicted end-point $c^{*}$, given in (129), as
function of $\theta$, along with the largest zero of the polynomial
$L^{(\bm{\alpha})}_{\bm{n}}(500\,x)$, with $\bm{\alpha}=(0,1/3)$ and
$\bm{n}=(500\,\theta,500\,(1-\theta))$, for $\theta=k/10$, $k=0,1,\dots,10$.

that is

$c^{*}=c^{*}(\theta)=\frac{27\theta^{2}(1-\theta)^{2}}{9\theta(1-\theta)-2+2(1-3\theta(1-\theta))^{3/2}}\in[27/8,4],$
(129)

see Figure 5.

###  5.3. Type II Multiple Laguerre of the second kind

Let $\alpha>-1$ and $\bm{c}=(c_{1},\dots,c_{r})\in\mathbb{R}_{>0}^{r}$ be such
that $c_{i}\not=c_{j}$ for $i\not=j$. For $\bm{n}\in\mathbb{N}^{r}$, the Type
II Multiple Laguerre polynomials of the second kind,
$\mathcal{L}_{\bm{n}}^{\alpha,\bm{c}}$, corresponding to the weights (7), are
polynomial of degree $|\bm{n}|-1$, satisfying

$\int_{0}^{\infty}\mathcal{L}_{\bm{n}}^{\alpha,\bm{c}}(x)\,x^{k+\alpha}e^{-c_{j}x}\,dx=0,\quad
k=0,1,\ldots,n_{j}-1,\quad j=1,\dots,r.$

Explicitly, see [21, §23.4.2],

$\mathcal{L}_{\bm{n}}^{(\alpha,\bm{c})}(x)=\sum_{k_{1}=0}^{n_{1}}\cdots\sum_{k_{r}=0}^{n_{r}}(-1)^{|\bm{k}|}\binom{\mathinner{\\!\left\lvert\bm{n}\right\rvert}+\alpha}{|\bm{k}|}\prod_{j=1}^{r}\binom{n_{j}}{k_{j}}\frac{|\bm{k}|!}{c_{1}^{k_{1}}\cdots
c_{r}^{k_{r}}}\,x^{\mathinner{\\!\left\lvert\bm{n}\right\rvert}-|\bm{k}|}.$
(130)

Alternatively, we can express them in terms of the finite free convlution of
simpler polynomials:

######  Theorem 5.10.

For $\bm{n}\in\mathbb{N}^{r}$, $\alpha>-1$ and $c_{1},\dots,c_{r}>0$ such that
$c_{i}\not=c_{j}$ for $i\not=j$. The Laguerre polynomials of second kind of