# Christoffel deformations of discrete ensembles related to random partitions

Pierre Lazag  I2M, CNRS, Aix-Marseille Université
Technopôle de Château-Gombert
13453 Marseille cedex 13
France<EMAIL_ADDRESS>
###### Abstract.

Christoffel deformation of a measure on the real line consists of multipying
this measure by a positive polynomial. We introduce Christoffel deformations
of discrete orthogonal polynomial ensembles by considering the Christoffel
deformations of the underlying measure, and prove that this construction
extends to more general point processes describing distribution on partitions
: the poissonized Plancherel measure and the $z$-measures. The extension to
the Plancherel measure is obtained via a limit transition from the Charlier
ensemble, while the extension to the $z$-measures follows from an analytic
continuation argument. A limit procedure starting from the non-degenerate
$z$-measures leads to a deformation of the process with the Gamma kernel
introduced by Borodin and Olshanski.

##  1\. Introduction

###  1.1. Random partitions as point processes

The aim of this article is to study deformations of discrete determinantal
point processes describing distributions or limit distributions on partitions,
namely the poissonized Plancherel measure, the $z$-measures and its limiting
process withe the Gamma kernel (see e.g. [5], [18], [23], [6], [7] and [9]).
Recall that a partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq...)$ is a
finite non-increasing sequence of non-negative integers, and that it can be
identified with a Young diagram ([22]). The set of all partitions will be
denoted by $\mathbb{Y}$. As it is usual, for a partition
$\lambda\in\mathbb{Y}$, we define its length $l(\lambda)$ as being its last
non-zero entry :

$\displaystyle l(\lambda):=\max\\{\lambda_{i},\hskip 2.84544pt\lambda_{i}\neq
0,\hskip 2.84544pti=1,2,...\\},$

and denote its size by $|\lambda|$ :

$\displaystyle|\lambda|:=\sum_{i\geq 1}\lambda_{i}.$

A Young diagram $\lambda\in\mathbb{Y}$ can be seen as a subset of $\mathbb{Z}$
via the map $\lambda\mapsto\\{\lambda_{i}-i,i=1,2,...\\}$ or some shifted
version of it. More precisely, when we consider partitions
$\lambda\in\mathbb{Y}$ with fixed length $N$, it is convenient to consider the
finite set of positive integers :

$\displaystyle\\{\lambda_{i}-i+N,\hskip 2.84544pti=1,...,N\\},$

while when the lengths may vary, it is convenient to consider the infinite
sets

$\displaystyle\\{\lambda_{i}-i,\hskip 2.84544pti=1,2,...\\}\text{ or
}\\{\lambda_{i}-i+1/2,\hskip 2.84544pti=1,2,...\\},$

the latter providing a more symmetric picture. In any case, the pushforward of
a probability measure on $\mathbb{Y}$ by one of these maps is thus a point
process on a discrete subset of $\mathbb{R}$. In the cases we deal with,
namely the poissonized Plancherel measure, the $z$-measures and its limiting
process with the Gamma kernel, these point processes are determinantal point
processes (see definition 2.3.2 below).

###  1.2. Determinantal point processes and Christoffel deformations

A specific and important class of determinantal point processes are the so
called polynomial orthogonal ensembles, see e.g. the survey [20] and
references therein. They are $N$-points random configurations (for some fixed
deterministic $N$) governed by the kernel of the orthogonal projection onto
the first $N$ polynomials in the Hilbert space $L^{2}(\mathbb{R}^{d},\mu)$ for
some weight $\mu$ with finite moments of all orders. This kernel is namely the
$N$-th Christoffel-Darboux kernel of the polynomials orthogonal with respect
to $\mu$. For exemple, the classical G.U.E. ensemble from random matrix theory
is the orthogonal polynomial ensemble with the gaussian weight on
$\mathbb{R}$, and is thus described by the Hermite polynomials, see e.g. [20]
and references therein.

Multiplying the weight $\mu$ by a positive polynomial leads to the so-called
Christoffel deformation of the system and an explicit description of the new
family of orthogonal polynomials as well as the corresponding deformed point
process.

###  1.3. Background and motivations

Such deformations have been considered e.g. in [14] in order to describe the
ergodic decomposition of inifinite Hua-Pickrell measures, and also in [2] in
connection with averages of characteristic polynomials of random matrices, the
study of the latter having connections with number theory, see for exemple
[19]. There are lots of studies of averages (or more generally moments) of
characteristic polynomials in the contiuous setting. For the discrete one, in
[10], the authors develop the notion of Giambelli compatible point processes
which provides a suitable formalism in the study of characteristic polynomials
of general point processes. Among others, the authors show that the
$z$-measures are Giambelli compatible, which means that characteristic
polynomials of these processes are well defined and involve determinatal
formulas. However, we were not able to perform the limit transition to the
process with the Gamma kernel by means of this formalism, which is one reason
why we consider Christoffel deformations of the $z$-measures : they lead to a
deformation of the process with the Gamma kernel and thus give a meaning of
characteristic polynomial of this ensemble. Thus, rather than directly
studying characteristic polynomials, from which formulas for the correlations
can be obtained (see [17], [25], [11], and [10]), we go the reverse way and
study Christoffel deformations of these processes. Direct information on the
characteristic polynomials of our processes of interest can be obtained from
formulas from [2], [17] or [25].

Our results also connect with those of [12] and [15] concerning Palm
deformations of determinantal point processes (see also [13]). For a given
point process $\mathbb{P}$ on a set $E$, the Palm deformation $\mathbb{P}^{u}$
of $\mathbb{P}$ at $u\in E$ is the same point process conditionned to have a
particle at $u$. In these papers, the author shows that conditionning at
different points leads to equivalent point processes and that the Radon-
Nikodym derivatives are explicitly given by regularized multiplicative
functionals. The connection with our setting is that Christoffel deformations
of a point process is precisely the multiplication of this point process by a
multiplicative functional analogous of the one from [12], [15] and [13].

###  1.4. Main results

As mentioned above, it is easy to define Christoffel deformations of
orthotogonal polynomial ensembles, but neither the Plancherel nor the
$z$-measures are orthogonal polynomial ensembles. As for the characteristic
polynomials, an obstacle is that, unlike for orthogonal polynomial ensembles,
the number of points might be random or infinite. We still manage to give a
meaning to Christoffel deformations of these processes, and also for the
process with the Gamma kernel, and we now briefly explain our arguments and
state our main results in an informal way.

On the one hand, we know from Johansson’s paper [18] that the poissonized
Plancherel measure is the limit of the Charlier ensemble, i.e. the orthogonal
polynomial ensemble with the Poisson weight. The poissonized Plancherel
measure is a determinantal point process described by Bessel functions, the
discrete Bessel kernel ([4]). Refining asymptotic results of [18], we proove
the convergence of the Charlier polynomials towards the Bessel functions, and
obtain the following theorem, (see theorem 3.2.1 for a precise statement) :

######  Theorem.

The Christoffel deformation of the Charlier ensemble converges to a
determinantal point process described by discrete Wronskians of Bessel
functions, the latter being a deformation of the poissonized Plancherel
measure.

On the other hand, the $z$-measures are determinantal point processes
described by specific Gauss hypergeometric functions. The analytic
continuation argument presented in [9] also allows us to describe the
deformation of all $z$-measures, a particular case of them being the Meixner
ensemble, and we obtain the following theorem (see theorem 4.1.3) :

######  Theorem.

The Christoffel deformations of the $z$-measures are determinantal point
proccesses governed by a kernel involving discrete Wronskians of Gauss
hypergeometric functions.

Moreover, a somehow tricky but elementary asymptotic analysis, similar to the
one performed in [7], leads to a deformation of the process with the Gamma
kernel (see theorem 4.2.1) :

######  Theorem.

The one-point deformations of the non-degenerate $z$-measures converge to a
determinantal point process described by the Gamma function and its first
derivative, wich might be considered as a deformation of the process with the
Gamma kernel.

Although we focus on just three exemples, our method is based on the algebraic
structure of both the deformed orthogonal polynomials and the integrable form
of the correlation kernels and we hope that it could be applied to more
general determinantal processes having an integrable kernel.

We have been recently informed by M. Bertola and G. Ruzza that a similar
formula as the one found for the modified Plancherel measure, involving
discrete Wronskians of Bessel functions, appears in the description of Gromov-
Witten theory of $\mathbb{P}^{1}$ [3].

###  1.5. Organization of the paper

The paper is organized as follows. In section 2, we collect definitions and
basic properties of Christoffel deformations, orthogonal polnomial ensembles
and determinantal point processes. In section 3, we introduce the Charlier
ensemble and the Plancherel measure. We describe the Christoffel deformation
of the Charlier ensemble and pass to the limit, in the same regime as in [18].
This leads to a deformation of the poissonized Plancherel measure, i.e. the
process with the discrete Bessel kernel, see theorem 3.2.1. In section 4, we
introduce the $z$-measures and their Christoffel deformation. We describe the
Christoffel deformation in the case where the parameters belong to the
degenerate series (wich is nothing but the Meixner ensemble) and prove that a
similar determinantal formula also holds for the other series, see theorem
4.1.3. This allows us to pass to the limit towards the deformed Gamma kernel
in the case of the one point deformation, see theorem 4.2.1.

#### Acknowledgements.

I am deeply grateful to Alexander Bufetov who posed the problem to me and for
helpful discussions. I also would like to thank Alexander Boritchev, Pascal
Hubert and Nizar Demni for helpful discussions and comments. This project has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement
N°647133).

##  2\. Christoffel deformations of polynomial ensembles

###  2.1. Notation

The set of non-negative integers is denoted by $\mathbb{N}$. For
$n\in\mathbb{N}$ and $a\in\mathbb{C}$, $(a)_{n}$ is the Pochhammer symbol :

$\displaystyle(a)_{0}=1\quad;\quad(a)_{n}:=a(a+1)...(a+n-1)=\frac{\Gamma(a+n)}{\Gamma(a)},\quad
n=1,2,...$

where $\Gamma$ is the Euler Gamma function.

####

For $a_{1},a_{2},...,a_{p}\in\mathbb{C}$,
$b_{1},...,b_{q}\in\mathbb{C}\setminus(-\mathbb{N})$ the hypergeometric
function $\left.{}_{p}F_{q}\right.(a_{1},...,a_{p};b_{1},...,b_{q};.)$ is
defined by :

(1)
$\displaystyle\left.{}_{p}F_{q}\right.(a_{1},...,a_{p};b_{1},...,b_{q};z)=\sum_{n=0}^{+\infty}\frac{(a_{1})_{n}...(a_{p})_{n}}{(b_{1})_{n}...(b_{q})_{n}}\frac{z^{n}}{n!},\quad
z\in\mathbb{C}$

provided the series converges. The function $\left.{}_{2}F_{1}\right.$ is the
Gauss hypergeometric function.

If $\lambda=(\lambda_{1}\geq\lambda_{2}\geq...)\in\mathbb{Y}$ is a partition
and $a\in\mathbb{C}$ is a complex number, we define the generelized Pochhammer
symbol $(a)_{\lambda}$ by :

$\displaystyle(a)_{\lambda}:=\prod_{i=1}^{l(\lambda)}(a-i+1)_{\lambda_{i}}=\prod_{(i,j)\in\lambda}(a-i+j)$

where we write :

$\displaystyle(i,j)\in\lambda\Leftrightarrow 1\leq j\leq\lambda_{i}.$

###  2.2. Christoffel deformations of discrete orthogonal polynomials

Let $\omega$ be a discrete measure on $\mathbb{R}$ with infinite support and
finite moments of all orders :

$\displaystyle\int_{\mathbb{R}}x^{n}\omega(dx)<+\infty\quad\text{for all
$n\in\mathbb{N}$}.$

We denote by $\\{P_{n}\\}_{n\in\mathbb{N}}$ a family of orthogonal polynomials
with respect to $\omega$. The number $c_{n}$ stands for the leading
coefficient of $P_{n}$, and its squared norm will be denoted by $h_{n}$ :

$\displaystyle P_{n}(x)$ $\displaystyle=c_{n}x^{n}+\text{terms of degree less
than $n$}$ $\displaystyle\int_{\mathbb{R}}P_{n}(x)P_{m}(x)\omega(dx)$
$\displaystyle=0\quad\text{if $n\neq m$},$ $\displaystyle=h_{n}\quad\text{if
$n=m$}.$

Let $k$ be a non-negative integer and let $u_{1}$,…,$u_{k}$ be real numbers
which lie outside the support of $\omega$.

######  Definition 2.2.1.

The Christoffel deformation $\omega^{k}$ of $\omega$ of order $k$ at points
$u_{1}$,…,$u_{k}$ is the discrete measure on $\mathbb{R}$ defined by :

$\displaystyle\omega^{k}(x)=\prod_{j=1}^{k}(x-u_{j})^{2}\omega(x)$

Let $\\{p_{n}^{k}\\}_{n\in\mathbb{N}}$ be the family of monic orthogonal
polynomials with respect to $\omega^{k}$. We have the following :

######  Proposition 2.2.2.

The following explicit formula holds :

$\displaystyle
p_{n}^{k}(x)=\left(\prod_{i=1}^{k}(x-u_{i})^{2}.\delta_{n}^{k}.c_{n+2k}\right)^{-1}D_{n}^{k}(x)$

where :

$\displaystyle D_{n}^{k}(x)$
$\displaystyle=\begin{vmatrix}P_{n}(u_{1})&P_{n+1}(u_{1})&...&P_{n+2k}(u_{1})\\\
...&&&...\\\ P_{n}(u_{k})&...&...&P_{n+2k}(u_{k})\\\
P^{\prime}_{n}(u_{1})&P^{\prime}_{n+1}(u_{1})&...&P^{\prime}_{n+2k}(u_{1})\\\
...&&&...\\\ P^{\prime}_{n}(u_{k})&...&...&P^{\prime}_{n+2k}(u_{k})\\\
P_{n}(x)&P_{n+1}(x)&...&P_{n+2k}(x)\end{vmatrix};$
$\displaystyle\delta_{n}^{k}$
$\displaystyle=\begin{vmatrix}P_{n}(u_{1})&P_{n+1}(u_{1})&...&P_{n+2k-1}(u_{1})\\\
...&&&...\\\ P_{n}(u_{k})&...&...&P_{n+2k-1}(u_{k})\\\
P^{\prime}_{n}(u_{1})&P^{\prime}_{n+1}(u_{1})&...&P^{\prime}_{n+2k-1}(u_{1})\\\
...&&&...\\\
P^{\prime}_{n}(u_{k})&P^{\prime}_{n+1}(u_{k})&...&P^{\prime}_{n+2k-1}(u_{k})\end{vmatrix}.$

Its squared $l_{2}(\omega^{k})$-norm is given by :

$\displaystyle
h_{n}^{k}=\frac{h_{n}\delta_{n+1}^{k}}{\delta_{n}^{k}.c_{n+2k}.c_{n}}$

#### Proof :

See e.g. [26], [16] or [2] for the explicit expression. The squared norm
$h_{n}^{k}$ is computed in [14], proposition 4.2. $\square$

###  2.3. Discrete orthogonal polynomial ensembles and determinantal point
processes

Let $E$ be an infinite countable discrete subset of $\mathbb{R}$. We first
recall the definition of an orthogonal polynomial ensemble in our setting. Let
$\omega$ be a discrete weight as in the previous subsection. We assume that
its support lies inside $E$. Let $N$ be a positive integer.

######  Definition 2.3.1.

An $N$-orthogonal polynomial ensemble is a probability measure
$\mathbb{P}_{N}$ on $E^{N}$ given by the following formula :

(2) $\mathbb{P}_{N}(x_{1},...x_{N})=C_{N}\prod_{1\leq i<j\leq
N}(x_{i}-x_{j})^{2}\prod_{i=1}^{N}\omega(x_{i})$

where $C_{N}$ is the normalization constant :

$\displaystyle C_{N}^{-1}=\sum_{x_{1},...,x_{N}\in E}\prod_{1\leq i<j\leq
N}(x_{i}-x_{j})^{2}\prod_{i=1}^{N}\omega(x_{i})$

We now recall the definition of a determinantal point process on $E$. We
denote by $\text{Conf}(E)$ the space of configuration in $E$ which is in this
context simply the set of all subsets of $E$ :

$\displaystyle\text{Conf}(E):=2^{E}.$

######  Definition 2.3.2.

A point process is a probability measure $\mathbb{P}$ on $\text{Conf}(E)$. It
is said to be determinantal if there exists a function :

$\displaystyle K:E\times E\rightarrow\mathbb{R}$

such that for any $n\in\mathbb{N}$ and any $\\{a_{1},...,a_{n}\\}\subset E$,
one has :

$\displaystyle\mathbb{P}(X\in\text{Conf}(E),\hskip
2.84544pt\\{a_{1},...,a_{n}\\}\subset
X)=\det\left(K(a_{i},a_{j})\right)_{i,j=1}^{n}.$

The function $K$ is called the correlation kernel of the determinantal point
process.

######  Remark 2.3.3.

The correlation kernel of a given determinantal point process is not unique.
Indeed, if one has a determinantal point process with kernel $K$, for any non-
vanishing function $f:E\rightarrow\mathbb{R}$ the function :

$\displaystyle\tilde{K}(x,y)=\frac{f(x)}{f(y)}K(x,y)$

can also serve as a kernel for the same determinantal point process.

As a matter of fact, any orthogonal polynomial ensemble gives rise to a
determinantal point process, what we resume in the following proposition (see
e.g. [20] and references therein). Note that since the measure (2) does not
charge $N$-tuples which contain two equal coordinates, it can and it will be
identified with a measure on the space of configurations.

######  Proposition 2.3.4.

An orthogonal polynomial ensemble given by (2) is a determinantal point
process on $E$. Its correlation kernel is given by the formula :

$\displaystyle K_{N}(x,y)$
$\displaystyle=\sqrt{\omega(x)\omega(y)}\sum_{n=0}^{N-1}\frac{p_{n}(x)p_{n}(y)}{h_{n}}$
$\displaystyle=\sqrt{\omega(x)\omega(y)}\frac{c_{N-1}}{h_{N-1}c_{N}}\frac{p_{N}(x)p_{N-1}(y)-p_{N}(y)p_{N-1}(x)}{x-y},\quad\text{for
$x\neq y$},$
$\displaystyle=\omega(x)\frac{c_{N-1}}{h_{N-1}c_{N}}\left(p_{N}^{\prime}(x)p_{N-1}(x)-p_{N}(x)p_{N-1}^{\prime}(x)\right),\quad\text{for
$x=y$}.$

The second equality is just the Christoffel-Darboux formula, while the last
one follows from L’Hospital rule.

######  Remarks.

Observe that $K_{N}$ is the kernel of the orthogonal projection from
$l^{2}(E)$ (where $E$ is equipped with the counting measure) onto

$\displaystyle\sqrt{\omega}\mathbb{R}_{N-1}[x]=\\{x\mapsto\sqrt{\omega(x)}p(x),\hskip
2.84544ptp\text{ is a polynomial of degree less than $N$}\\}.$

The main object of our study is given by the following definition.

######  Definition 2.3.5 (Christoffel deformations of orthogonal polynomial
ensembles).

Let $k\in\mathbb{N}$ be a non-negative integer. Given an orthogonal polynomial
ensemble $\mathbb{P}_{N}$ with weight $\omega$, its Christoffel deformation of
order $k$, denoted by $\mathbb{P}_{N}^{k}$, is the orthogonal polynomial
ensemble with weight $\omega^{k}$.

The next proposition gives a compact formula for the kernel of the Christoffel
deformation which will be suitable for asymptotic analysis.

######  Proposition 2.3.6.

The Christoffel deformation of order $k$, $\mathbb{P}_{N}^{k}$, is a
determinantal point process with kernel $K_{N}^{k}$ given by :

$\displaystyle
K^{k}_{N}(x,y)=\frac{\sqrt{\omega(x)\omega(y)}}{\prod_{i=1}^{k}|(x-u_{i})(y-u_{i})|}\frac{c_{N-1}}{h_{N-1}c_{N+2k}}\frac{D_{N}^{k}(x)D_{N-1}^{k}(y)-D_{N}^{k}(y)D_{N-1}^{k}(x)}{(\delta_{N}^{k})^{2}(x-y)}.$

Proof : This is a straightforward computation, using propositions 2.3.4 and
2.2.2. Indeed, proposition 2.3.4 states that $\mathbb{P}_{N}^{k}$ is a
determinantal point process with kernel :

$\displaystyle
K_{N}^{k}(x,y)=\sqrt{\omega^{k}(x)\omega^{k}(y)}\frac{p_{N}^{k}(x)p_{N-1}^{k}(y)-p_{N}^{k}(y)p_{N-1}^{k}(x)}{h_{N-1}^{k}(x-y)}.$

By the definition of the measure $\omega^{k}$ and using proposition 2.2.2 and
the explicit expression of the polynomials $p_{n}^{k}$, this kernel can be
written as :

$\displaystyle
K_{N}^{k}(x,y)=\frac{\sqrt{\omega(x)\omega(y)}}{\prod_{i=1}^{k}|(x-u_{i})(y-u_{i})|}\frac{1}{\delta_{N}^{k}\delta_{N-1}^{k}c_{N+2k}c_{N+2k-1}h_{N-1}^{k}}\frac{D_{N}^{k}(x)D_{N-1}^{k}(y)-D_{N}^{k}(y)D_{N-1}^{k}(x)}{x-y}.$

The expression for the squared norm $h_{N-1}^{k}$ given in proposition 2.2.2
leads to the desired result. $\square$

##  3\. The Charlier ensemble and a deformation of the Plancherel measure

###  3.1. On the Plancherel measure on partitions

The Plancherel measure is a probability measure $\text{Pl}_{n}$ on the set of
Young diagrams of fixed size $n$,
$\mathbb{Y}_{n}=\\{\lambda\in\mathbb{Y},\hskip 2.84544pt|\lambda|=n\\}$. It is
given by :

$\displaystyle\text{Pl}_{n}(\lambda)=\frac{\dim(\lambda)^{2}}{n!},\quad\lambda\in\mathbb{Y}_{n}$

where $\dim(\lambda)$ is the dimension of the irreducible representation of
the $n$-th symmetric group parametrized by $\lambda$, or equivalentely, the
number of stantard Young tableaux of shape $\lambda$ (see [22] or [24]). It is
well known that the poissonization of the Plancherel measure is a
determinantal point process. More precisely, let $\alpha>0$ be a positive
parameter and form the poissonized Plancherel measure $\mathbb{P}_{\alpha}$ on
$\mathbb{Y}$ by imposing $n$ to have the Poisson distribution with parameter
$\alpha$:

$\displaystyle\mathbb{P}_{\alpha}(\lambda)=e^{-\alpha}\alpha^{|\lambda|}\left(\frac{\dim(\lambda)}{|\lambda|!}\right)^{2},\quad\lambda\in\mathbb{Y}.$

For $\lambda\in\mathbb{Y}$, we set :

$\displaystyle\mathfrak{S}(\lambda):=\\{\lambda_{i}-i,\hskip
2.84544pti=1,2,...\\}\in\text{Conf}(\mathbb{Z})$

Then the pushforward of $\mathbb{P}_{\alpha}$ under the map $\mathfrak{S}$ is
a determinantal point process with the discrete Bessel kernel :

$\displaystyle
K_{\alpha}(x,y)=\sqrt{\alpha}\frac{J_{x}(2\sqrt{\alpha})J_{y+1}(2\sqrt{\alpha})-J_{y}(2\sqrt{\alpha})J_{x+1}(2\sqrt{\alpha})}{x-y}$

where the Bessel functions $J_{x}(2\sqrt{\alpha}),\hskip
2.84544ptx\in\mathbb{Z}$ are defined via their generating Laurent series ([1])
:

(3)
$e^{\sqrt{\alpha}(z-z^{-1})}=\sum_{x\in\mathbb{Z}}J_{x}(2\sqrt{\alpha})z^{x}$

This definition leads to an expression for the Bessel functions that can be
analytically continuated for all values of the indices $x$. We can thus
introduce the derivatives of the Bessel functions with respect to their index
:

$\displaystyle\frac{d}{dx}J_{x}(2\sqrt{\alpha})=:L_{x}(2\sqrt{\alpha})$

###  3.2. Christoffel deformations of the Charlier ensemble and main result

We now define the Christoffel deformation of the Charlier ensemble. We know
from [18] that this ensemble approximates the poissonized Plancherel measure,
and the aim of this section is to generalize this fact by proving that
Christoffel deformations of this ensemble admit a limit that is expressed
through Bessel functions. Our discrete measure is :

$\displaystyle\omega(x)=\omega_{a}(x)=e^{-a}\frac{a^{x}}{x!},\quad
x\in\mathbb{N},$

where $a>0$ is a parameter, and the $N$-th orthogonal polynomial ensemble that
this measure defines is denoted by $\mathbb{P}_{N,a}$. The corresponding
orthonormal polynomials are the Charlier polynomials ([16], [1]) :

$\displaystyle
C_{n}(x)=C_{n}(x;a)=\frac{a^{\frac{n}{2}}}{\sqrt{n!}}\left.{}_{2}F_{0}\right.(-n,-x;;-1/a)\hskip
2.84544pt,\quad n\in\mathbb{N}.$

As in the previous section, we define the Christoffel deformations of the
Charlier ensemble. Let $k\in\mathbb{N}$ be a non-negative integer, and
$\tilde{u}_{i}\in\mathbb{R}\backslash\mathbb{N}$ be non-integer numbers. Let
$\alpha>0$ be a positive real number, and for each $N\in\mathbb{N}$, consider
the Christoffel deformation of the $N$-th Charlier ensemble at points
$u_{1}=\tilde{u}_{1}+N,...,u_{k}=\tilde{u}_{k}+N$, with parameter
$a=\alpha/N$, according to definition 2.3.5. This determinantal point process
is denoted by $\mathbb{P}_{N,\alpha/N}^{k}$, and its correlation kernel is
denoted by $K_{N}^{\alpha/N,k}$. The main result of this section is the
following theorem :

######  Theorem 3.2.1.

In the regime described above, we have that, for all $x,y\in\mathbb{R}$,
$x\neq y$ :

$\displaystyle\lim_{N\rightarrow\infty}K_{N}^{\alpha/N,k}(x+N,y+N)$
$\displaystyle=\frac{\alpha^{\frac{2k+1}{2}}}{C_{k}^{2}\prod_{i=1}^{k}|(x-\tilde{u}_{i})(y-\tilde{u}_{i})|}\frac{A_{k,0}(x)A_{k,1}(y)-A_{k,0}(y)A_{k,1}(x)}{x-y}$
where $\displaystyle A_{k,p}(x)$
$\displaystyle=\begin{vmatrix}J_{\tilde{u}_{1}+p}(2\sqrt{\alpha})&...&J_{\tilde{u}_{1}-2k+p}(2\sqrt{\alpha})\\\
...&...&...\\\
J_{\tilde{u}_{k}+p}(2\sqrt{\alpha})&...&J_{\tilde{u}_{k}-2k+p}(2\sqrt{\alpha})\\\
L_{\tilde{u}_{1}+p}(2\sqrt{\alpha})&...&L_{\tilde{u}_{1}-2k+p}(2\sqrt{\alpha})\\\
...&...&...\\\
L_{\tilde{u}_{k}+p}(2\sqrt{\alpha})&...&L_{\tilde{u}_{k}-2k+p}(2\sqrt{\alpha})\\\
J_{x+p}(2\sqrt{\alpha})&...&J_{x-2k+p}(2\sqrt{\alpha})\end{vmatrix}\hskip
2.84544pt,\quad p=0,1\hskip 2.84544pt,$ $\displaystyle C_{k}$
$\displaystyle=\begin{vmatrix}J_{\tilde{u}_{1}}(2\sqrt{\alpha})&...&J_{\tilde{u}_{1}-2k+1}(2\sqrt{\alpha})\\\
...&...&...\\\
J_{\tilde{u}_{k}}(2\sqrt{\alpha})&...&J_{\tilde{u}_{k}-2k+1}(2\sqrt{\alpha})\\\
L_{\tilde{u}_{1}}(2\sqrt{\alpha})&...&L_{\tilde{u}_{1}-2k+1}(2\sqrt{\alpha})\\\
...&...&...\\\
L_{\tilde{u}_{k}}(2\sqrt{\alpha})&...&L_{\tilde{u}_{k}-2k+1}(2\sqrt{\alpha})\end{vmatrix}.$

When $x=y$, L’Hospital rule entails :

$\displaystyle\lim_{N\rightarrow\infty}K_{N}^{\alpha/N,k}(x+N,x+N)$
$\displaystyle=\frac{\alpha^{\frac{2k+1}{2}}}{C_{k}^{2}\prod_{i=1}^{k}(x-\tilde{u}_{i})^{2}}\left(A_{k,0}^{\prime}(x)A_{k,1}(x)-A_{k,0}(x)A_{k,1}^{\prime}(x)\right).$

This implies that the Christoffel deformation of the Charlier ensemble
$\mathbb{P}_{N,\alpha/N}^{k}$, shifted by $-N$, weakly converges, as a
probability measure on $\text{Conf}(\mathbb{Z})$, to a determinantal point
process governed by the limit correlation kernel of the theorem. In
particular, if we take $k=0$, one recovers Johansson’s result ([18], theorem
1.2).

###  3.3. Proof of Theorem 3.2.1

We will use the following lemma, which we prove at the end of this section.
This lemma gives simple asymptotic results on the Charlier polynomials
involving Bessel functions and their derivatives.

######  Lemma 3.3.1.

For any $l\in\mathbb{Z}$ and any $u\in\mathbb{R}$, we have

$\displaystyle\lim_{N\rightarrow\infty}\sqrt{\omega_{\alpha/n}(u+N)}C_{N+l}(u+N;\alpha/N)=J_{u-l}(2\sqrt{\alpha})$

and

$\displaystyle\lim_{N\rightarrow\infty}\sqrt{\omega_{\alpha/n}(u+N)}\left(C^{\prime}_{N+l}(u+n;\alpha/N)-\log\left(N/\sqrt{\alpha}\right)C_{N+l}(u+N;\alpha/N)\right)=L_{u-l}(2\sqrt{\alpha}).$

The proof of theorem 3.2.1 goes as follows. We use the formula from
proposition 2.3.6 for the kernel $K_{N}^{\alpha/N,k}$ and proceed to the
analysis of each factor. First, a straightforward computation shows that :

(4) $\frac{c_{N-1}}{c_{N+2k}}\rightarrow\alpha^{\frac{2k+1}{2}}$

when $N\rightarrow+\infty$. We now analyse the ratio :

$\displaystyle\sqrt{\omega_{\alpha/N}(x+N)\omega_{\alpha/N}(y+N)}\frac{D_{N}^{k}(x+N)D_{N-1}^{k}(y+N)-D_{N}^{k}(y+N)D_{N-1}^{k}(x+N)}{\left(\delta_{N}^{k}\right)^{2}(x-y)}$

To this end, we multiply both sides of this ratio by :

$\displaystyle m:=\prod_{i=1}^{k}\omega_{\alpha/N}(\tilde{u}_{i}+N)^{2},$

and in each determinant $D_{N}^{k}(.)$, $D_{N-1}^{k}(.)$ and $\delta_{N}^{k}$,
we perform the following operations on rows :

$\displaystyle R_{k+i}\leftarrow R_{k+i}-\log\left(N/\alpha\right)R_{i}$

for $i=1,...,k$. Let $M$ be the diagonal $2k\times 2k$ matrix :

$\displaystyle M=$
$\displaystyle\text{diag}\left(\sqrt{\omega_{\alpha/N}(u_{1})},...,\sqrt{\omega_{\alpha/N}(u_{k})},\sqrt{\omega_{\alpha/N}(u_{1})},...,\sqrt{\omega_{\alpha/N}(u_{k})}\right),$

then :

$\displaystyle m=\det(M^{2})=\det(M)^{2}.$

Let us denote by $\hat{\delta_{N}^{k}}$ the matrix such that
$\det\left(\hat{\delta_{N}^{k}}\right)=\delta_{N}^{k}$ with changed rows as
described above and use the same notation for the other determinants. We have
:

$\displaystyle
m\left(\delta_{N}^{k}\right)^{2}=\det\left(M\hat{\delta_{N}^{k}}\right)^{2}$

and each term of this matrix is precisely of the type we can analyse by lemma
3.3.1. In the same fashion, if we denote by $M_{z}$ the $(2k+1)\times(2k+1)$
matrix :

$\displaystyle M_{z}=\text{diag}\left(M,\sqrt{\omega_{\alpha/N}(z+N)}\right),$

we have that :

$\displaystyle\sqrt{\omega_{\alpha/N}(x+N)\omega_{\alpha/N}(y+N)}$
$\displaystyle.m.\left(D_{N}^{k}(x+N)D_{N-1}^{k}(y+N)-D_{N}^{k}(x+N)D_{N-1}^{k}(y+N)\right)$
$\displaystyle=\det(M_{x})\det(M_{y}).\left(D_{N}^{k}(x+N)D_{N-1}^{k}(y+N)-D_{N}^{k}(x+N)D_{N-1}^{k}(y+N)\right)$
$\displaystyle=\det\left(M_{x}\hat{D}_{N}^{k}(x+N)\right)\det\left(M_{y}\hat{D}_{N-1}^{k}(y+N)\right)$
$\displaystyle-\det\left(M_{y}\hat{D}_{N}^{k}(y+N)\right)\det\left(M_{x}\hat{D}_{N-1}^{k}(x+N)\right),$

and again, we know the asymptotics for the components of each matrix thanks to
lemma 3.3.1. Using the continuity of the determinant, we have established
that, for all $x\neq y$ :

$\displaystyle\lim_{N\rightarrow+\infty}\sqrt{\omega_{\alpha/N}(x+N)\omega_{\alpha/N}(y+N)}$
$\displaystyle\frac{D_{N}^{k}(x+N)D_{N-1}^{k}(y+N)-D_{N}^{k}(y+N)D_{N-1}^{k}(x+N)}{\left(\delta_{N}^{k}\right)^{k}(x-y)}$
$\displaystyle=\frac{A_{k,0}(x)A_{k,1}(y)-A_{k,0}(y)A_{k,1}(x)}{C_{k}(x-y)}.$

The case $x=y$ goes as follows. For a $(2k+1)\times(2k+1)$ determinant $A$,
and for $i=0,...,2k$ we denote by $[A]^{i}$ its $\left((i+1),(2k+1)\right)$
cofactor, i.e. the same determinant with the $(i+1)$th column and the last
line removed, multiplied by $(-1)^{i+1+2k+1}=(-1)^{i}$. We have :

(5)
$\begin{split}D_{N}^{k}&(x+N)^{\prime}D_{N-1}^{k}(x+N)-D_{N}^{k}(x+N)D_{N-1}^{k}(x+N)^{\prime}\\\
&=\left(\sum_{i=0}^{2k}\left[D_{N}^{k}(x+N)\right]^{i}C_{N+i}^{\prime}(x+N)\right)\left(\sum_{i=0}^{2k}\left[D_{N-1}^{k}(x+N)\right]^{i}C_{N-1+i}(x+N)\right)\\\
&-\left(\sum_{i=0}^{2k}\left[D_{N}^{k}(x+N)\right]^{i}C_{N+i}(x+N)\right)\left(\sum_{i=0}^{2k}\left[D_{N-1}^{k}(x+N)\right]^{i}C_{N-1+i}^{\prime}(x+N)\right)\\\
&=\sum_{i,j=0}^{2k}\left[D_{N}^{k}(x+N)\right]^{i}\left[D_{N-1}^{k}(x+N)\right]^{j}\left(C_{N+i}^{\prime}(x+N)C_{N-1+j}(x+N)-C_{N+i}(x+N)C_{N-1+j}^{\prime}(x+N)\right).\end{split}$

Remark that the analysis of the cofactors has been done, and observe now that
we have nice symmetries in this expression. Indeed, for any $i,j=0,...,2k$,
second part of lemma 3.3.1 entails :

$\displaystyle C_{N+i}$
${}^{\prime}(x+N)C_{N-1+j}(x+N)-C_{N+i}(x+N)C_{N-1+j}^{\prime}(x+N)$
$\displaystyle=\omega_{\alpha/N}(x+N)^{-1/2}$
$\displaystyle\times\left\\{\left(L_{x-i}(2\sqrt{\alpha})+\log\left(N/\sqrt{\alpha}\right)C_{N+i}(x+N)+o(1)\right)C_{N-1+j}(x+N)\right.$
$\displaystyle\left.-C_{N+i}(x+N)\left(L_{x-1+j}(2\sqrt{\alpha})+\log\left(N/\sqrt{\alpha}\right)C_{N-1+j}(x+N)+o(1)\right)\right\\}.$

We see that the term involving the factor $\log(N/\sqrt{\alpha})$ vanishes.
Multiplying (5) by $\omega_{\alpha/N}(x+N)$ and applying the first part of
lemma 3.3.1, we are left with :

$\displaystyle\omega_{\alpha/N}(x+N)$
$\displaystyle\left(D_{N}^{k}(x+N)^{\prime}D_{N-1}^{k}(x+N)-D_{N}^{k}(x+N)D_{N-1}^{k}(x+N)^{\prime}\right)$
$\displaystyle=\sum_{i,j=0}^{2k}\left[D_{N}^{k}(x+N)\right]^{i}\left[D_{N-1}^{k}(x+N)\right]^{j}$
$\displaystyle\times\left(L_{-x-i}(2\sqrt{\alpha})J_{-x+1-j}(2\sqrt{\alpha})-J_{-x-i}(2\sqrt{\alpha})L_{-x+1-j}(2\sqrt{\alpha})+o(1)\right).$

This sum can be factorized, and we obtain that :

$\displaystyle\omega_{\alpha/N}(x+N)$
$\displaystyle.m.\left(D_{N}^{k}(x+N)^{\prime}D_{N-1}^{k}(x+N)-D_{N}^{k}(x+N)D_{N-1}^{k}(x+N)^{\prime}\right)$

tends to

$\displaystyle A_{k,0}^{\prime}(x)A_{k,1}(x)-A_{k,0}(x)A_{k,1}^{\prime}(x)$

as $N$ tends to $+\infty$, which is the desired result. Recalling (4), the
theorem is proved. $\square$

Proof of the lemma : Following [18], we will use the integral representation
for Charlier polynomials. Since the generating series for the normalized
Charlier polynomials is :

$\displaystyle\sum_{m=0}^{\infty}\left(\frac{\alpha}{N}\right)^{m/2}\frac{1}{\sqrt{m!}}C_{m}(x;\alpha/N)w^{m}=e^{-\alpha
w/N}\left(1+w\right)^{x},$

we have by Cauchy’s formula, for $r>1$ :

$\displaystyle C_{m}(x;\alpha/N)$
$\displaystyle=\sqrt{m!}\left(\frac{N}{\alpha}\right)^{m/2}\left(\frac{1}{2i\pi}\int_{|w|=r}e^{-\alpha
w/N}(1+w)^{x}w^{-m}\frac{dw}{w}\right.$
$\displaystyle\left.-(-1)^{m}\frac{\sin\pi x}{\pi}\int_{1}^{r}e^{\alpha
u/N}(u-1)^{x}u^{-m}\frac{du}{u}\right).$

Performing the change of variables $w=Nz/\sqrt{\alpha}$, $u=Ns/\sqrt{\alpha}$,
one obtains, for $r>\sqrt{\alpha}/N$ :

$\displaystyle C_{m}(x;\alpha/N)$
$\displaystyle=\sqrt{\frac{m!}{N^{m}}}\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-\sqrt{\alpha}re^{i\theta}}\left(1+\frac{Nre^{i\theta}}{\sqrt{\alpha}}\right)^{x}\frac{d\theta}{(re^{i\theta})^{m}}\right.$
$\displaystyle\left.-(-1)^{m}\frac{\sin\pi
x}{\pi}\int_{\sqrt{\alpha}/N}^{r}e^{\sqrt{\alpha}s}\left(\frac{Ns}{\sqrt{\alpha}}-1\right)^{x}s^{-m}\frac{ds}{s}\right).$

This expression can be suitably factorized :

(6)
$\begin{split}C_{m}(x;\alpha/N)&=e^{-\sqrt{\alpha}}\sqrt{\frac{m!}{N^{m}}}\left(1+\frac{N}{\sqrt{\alpha}}\right)^{x}\\\
&\times\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{\sqrt{\alpha}(1-re^{i\theta})}\left(\frac{\sqrt{\alpha}+Nre^{i\theta}}{\sqrt{\alpha}+N}\right)^{x}\frac{d\theta}{(re^{i\theta})^{m}}\right.\\\
&\left.-(-1)^{m}\frac{\sin\pi
x}{\pi}\int_{\sqrt{\alpha}/N}^{r}e^{\sqrt{\alpha}(s+1)}\left(\frac{Ns-\sqrt{\alpha}}{\sqrt{\alpha}+N}\right)^{x}s^{-m}\frac{ds}{s}\right).\end{split}$

Take $m=N+l$ and denote by $I_{N}^{l}(x)$ the sum of the integrals :

(7)
$\begin{split}I_{N}^{l}(x)&:=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{\sqrt{\alpha}(1-re^{i\theta})}\left(\frac{\sqrt{\alpha}+Nre^{i\theta}}{\sqrt{\alpha}+N}\right)^{x}\frac{d\theta}{(re^{i\theta})^{N+l}}\\\
&-(-1)^{N+l}\frac{\sin\pi
x}{\pi}\int_{\sqrt{\alpha}/N}^{r}e^{\sqrt{\alpha}(s+1)}\left(\frac{Ns-\sqrt{\alpha}}{\sqrt{\alpha}+N}\right)^{x}s^{-N-l}\frac{ds}{s}\end{split}$

for any $l\in\mathbb{Z}$, and define :

$\displaystyle
A_{\alpha}^{N}(x)=e^{-2\sqrt{\alpha}}\frac{N!}{N^{N}}\left(1+\frac{N}{\sqrt{\alpha}}\right)^{2x}\omega_{\alpha/N}(x).$

We have :

(8)
$\sqrt{\omega_{\alpha/N}(x)}C_{N+l}(x;\alpha/N)=\sqrt{\frac{(N+1)...(N+l)}{N^{l}}A_{\alpha}^{N}(x)}I_{N}^{l}(x).$

Recall the integral representation for the Bessel function, which follows from
(3) :

(9)
$J_{x}(2\sqrt{\alpha})=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{\sqrt{\alpha}(\frac{1}{r}e^{-i\theta}-re^{i\theta})}(re^{i\theta})^{x}d\theta-\frac{\sin\pi
x}{\pi}\int_{0}^{r}e^{\sqrt{\alpha}(-1/s+s)}s^{x}\frac{ds}{s}.$

We first show that the integrals (7) converges to the integrals (9), with a
speed of order at least $1/N$, showing that the integrands converge. For the
first integrand, we have by definition :

$\displaystyle e^{\sqrt{\alpha}(1-z)}$
$\displaystyle\left(\frac{\sqrt{\alpha}+Nz}{\sqrt{\alpha}+N}\right)^{x+N}z^{-N-l}$
$\displaystyle=\exp\left(\sqrt{\alpha}(1-z)-(N+l)\log
z+(x+N)(\log(\sqrt{\alpha}+Nz)-\log(\sqrt{\alpha}+N))\right).$ Factorizing
inside the logarithms leads to : $\displaystyle e^{\sqrt{\alpha}(1-z)}$
$\displaystyle\left(\frac{\sqrt{\alpha}+Nz}{\sqrt{\alpha}+N}\right)^{x+N}z^{-N-l}$
$\displaystyle=\exp\left(\sqrt{\alpha}(1-z)-(l-x)\log
z+(x+N)\left(\log\left(1+\frac{\sqrt{\alpha}}{Nz}\right)-\log\left(1+\frac{\sqrt{\alpha}}{N}\right)\right)\right)$
$\displaystyle=\exp\left(\sqrt{\alpha}(1/z-z)+O(1/N)\right)z^{x-l},$

where the last line follows from straightforward simplifications and first
order expansions of the logarithms near $1$. In the same way, we have for the
second integrand :

$\displaystyle
e^{\sqrt{\alpha}(s+1)}\left(\frac{Ns-\sqrt{\alpha}}{\sqrt{\alpha}+N}\right)^{x+N}s^{-N-l}=\exp\left(\sqrt{\alpha}(-1/s+s)+O(1/N)\right)s^{x-l}.$

We have shown that there exists a constant $K=K(\alpha,x,r,l)$ such that :

$\displaystyle\left|e^{\sqrt{\alpha}(1-z)}\left(\frac{\sqrt{\alpha}+Nz}{\sqrt{\alpha}+N}\right)^{x+N}z^{-N-l}-\exp\left(\sqrt{\alpha}(1/z-z)\right)z^{x-l}\right|\leq\frac{K}{N}$
and
$\displaystyle\left|e^{\sqrt{\alpha}(s+1)}\left(\frac{Ns-\sqrt{\alpha}}{\sqrt{\alpha}+N}\right)^{x+N}s^{-N-l}-e^{\sqrt{\alpha}(-1/s+s)}s^{x-l}\right|\leq\frac{K}{N}$

for all involved $z$ and $s$. This implies :

$\displaystyle|I_{N}^{l}(x+N)-J_{x-l}(2\sqrt{\alpha})|\leq\frac{K}{N},$

as announced. Since as $N\rightarrow+\infty$, we have that :

$\displaystyle A_{\alpha}^{N}(x+N)\rightarrow 1,$

and recalling (8), we have established the first part of the lemma.

We now move to the proof of the second statement of the lemma. Since the
techniques are very similar, we will give less details in our computations.
Differentating expression (6), we obtain :

$\displaystyle C_{m}^{\prime}(x;\alpha/N)$
$\displaystyle=\log\left(1+\frac{N}{\sqrt{\alpha}}\right)C_{m}(x;\alpha/N)$
$\displaystyle+e^{-\sqrt{\alpha}}\sqrt{\frac{m!}{N^{m}}}\left(1+\frac{N}{\sqrt{\alpha}}\right)^{x}\frac{d}{dx}\left(I^{m-N}_{N}(x)\right).$

We thus only need to proof that the derivative of the integrals converges to
the derivative of the Bessel function. Derivations under the sign of the
integrals give :

$\displaystyle\frac{d}{dx}\left(I^{l}_{N}(x)\right)$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\log\left(\frac{\sqrt{\alpha}+Nre^{i\theta}}{\sqrt{\alpha}+N}\right)e^{\sqrt{\alpha}(1-re^{i\theta})}\left(\frac{\sqrt{\alpha}+Nre^{i\theta}}{\sqrt{\alpha}+N}\right)^{x}\frac{d\theta}{(re^{i\theta})^{N+l}}$
$\displaystyle-(-1)^{N+l}\frac{\sin\pi
x}{\pi}\int_{\sqrt{\alpha}/N}^{r}\log\left(\frac{Ns-\sqrt{\alpha}}{\sqrt{\alpha}+N}\right)e^{\sqrt{\alpha}(s+1)}\left(\frac{Ns-\sqrt{\alpha}}{\sqrt{\alpha}+N}\right)^{x}s^{-N-l}\frac{ds}{s}$
$\displaystyle-(-1)^{N+l}\frac{\cos\pi
x}{\pi}\int_{\sqrt{\alpha}/N}^{r}e^{\sqrt{\alpha}(s+1)}\left(\frac{Ns-\sqrt{\alpha}}{\sqrt{\alpha}+N}\right)^{x}s^{-N-l}\frac{ds}{s}$

and :

$\displaystyle L_{x}(2\sqrt{\alpha})$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\log(re^{i\theta})e^{\sqrt{\alpha}(\frac{1}{r}e^{-i\theta}-re^{i\theta})}(re^{i\theta})^{x}d\theta$
$\displaystyle-\frac{\sin\pi
x}{\pi}\int_{0}^{r}\log(s)e^{\sqrt{\alpha}(-1/s+s)}s^{x}\frac{ds}{s}$
$\displaystyle-\frac{\cos\pi
x}{\pi}\int_{0}^{r}e^{\sqrt{\alpha}(-1/s+s)}s^{x}\frac{ds}{s}.$

Analogous computations lead to :

$\displaystyle\lim_{N\rightarrow+\infty}\frac{d}{dx}\left(I^{l}_{N}(x+N)\right)=L_{x-l}(2\sqrt{\alpha}).$

This completes the proof of the lemma. $\square$

##  4\. The $z$-measures and a deformation of the gamma process

###  4.1. Christoffel deformations of the $z$-measures

The $z$-measures on partitions were introduced and extensively studied by
Borodin and Olshanski in a series of papers, see e.g. [6], [7], [8] and [9]
and references therein. We first briefly explain their construcion. The
$z$-measures are defined on the set $\mathbb{Y}$ of all partitions by :

$\displaystyle
M_{z,z^{\prime},\xi}(\lambda)=(1-\xi)^{zz^{\prime}}\xi^{|\lambda|}(z)_{\lambda}(z^{\prime})_{\lambda}\left(\frac{\dim(\lambda)}{|\lambda|!}\right)^{2}.$

They depend on three complex parameters, $z$, $z^{\prime}$ and $\xi$ and they
are in general complex measures. The following choices ensure that it is a
probability measure (see e.g. [8] proposition 1.2 for a proof). First of all,
the parameter $\xi$ is real and lies in the open unit interval $(0,1)$. Next,
we distinguish three cases :

  1. $\bullet$

[Principal series] The parameters $z$ and $z^{\prime}$ are conjugate to each
other.

  2. $\bullet$

[Complementary series] The parameters $z$ and $z^{\prime}$ are both real and
belong to the same interval $(m,m+1)$ where $m\in\mathbb{Z}$.

  3. $\bullet$

[Degenerate series] One of the parameter, say $z$, is a non-zero integer,
while the other has the same sign and satisfy $|z^{\prime}|>|z|-1$.

If $z$ and $z^{\prime}$ satisfy one of these three conditions, we say that
they are admissible parameters. The Christoffel defomations of the $z$-measure
are then defined as follows :

######  Definition 4.1.1.

Let $z$, $z^{\prime}$ be admissible parameters and $M_{z,z^{\prime},\xi}$ the
corresponding $z$-measure. Let $u_{1}$,…,$u_{k}$,
$v_{1}$,…,$v_{k}\in\mathbb{C}$ be $2k$ complex numbers which satisfy :

$\displaystyle\prod_{i=1}^{k}(x-u_{i}+z)(x-v_{i}+z^{\prime})>0$

for any $x\in\mathbb{Z}$. We define the Christoffel deformation of
$M_{z,z^{\prime},\xi}$ by :

(10) $\displaystyle
M_{z,z^{\prime},\xi}^{k}(\lambda)=C\prod_{j=1}^{l(\lambda)}\prod_{i=1}^{k}(\lambda_{j}-j-u_{i}+z)(\lambda_{j}-j-v_{i}+z^{\prime})M_{z,z^{\prime}\xi}(\lambda),$

where $C$ is the normalisation constant chosen such that
$M_{z,z^{\prime},\xi}^{k}$ is a probability measure.

We first have to focus on the degenerate case. For simplicity, we assume that
$z=N$ is a positive integer and $z^{\prime}>z-1$. The $z$-measure
$M_{z,z^{\prime}\xi}$ is then supported on the subset of Young diagrams of
length less than or equal to $N$, and we map $\mathbb{Y}$ on
$\text{Conf}(\mathbb{N})\subset\text{Conf}(\mathbb{Z})$ via :

$\displaystyle\lambda\mapsto\mathfrak{S}_{N}(\lambda):=\\{\lambda_{i}-i+N,\hskip
2.84544pti=1,...,N\\}.$

The pushforward of the $z$-measure $M_{z,z^{\prime}\xi}$ under
$\mathfrak{S}_{N}$ is then the $N$-th Meixner ensemble (see e.g. [8],
proposition 1.4), i.e. the $N$-th orthogonal ensemble with weight :

$\displaystyle\omega_{\beta,\xi}(x)=\frac{(\beta)_{x}\xi^{x}}{x!},$

where $\beta=z^{\prime}-z+1>0$. The corresponding orthogonal polynomials are
the Meixner polynomials defined by :

$\displaystyle M_{n}(x;\beta,\xi)=F(-n,-x;\beta;1-1/\xi),\hskip
2.84544ptn\in\mathbb{N}.$

Their leading coefficient $c_{n}$ and their squared norm $h_{n}$ are given by
:

$\displaystyle
c_{n}=\frac{\left(1-\frac{1}{\xi}\right)^{n}}{n!(\beta)_{n}}\hskip
2.84544pt,\quad h_{n}=\frac{n!}{\xi^{n}(1-\xi)^{\beta}(\beta)_{n}}.$

We now choose $k$ real numbers
$u_{1},...,u_{k}\in\mathbb{R}\setminus\mathbb{N}$ which are not integers, and
set $v_{i}=u_{i}+\beta-1$. We thus have :

$\displaystyle\prod_{j=1}^{l(\lambda)}\prod_{i=1}^{k}(\lambda_{j}-j-u_{i}+z)(\lambda_{j}-j-v_{i}+z^{\prime})=\prod_{x\in\mathfrak{S}_{N}(\lambda)}\prod_{j=1}^{k}(x-u_{i})^{2}$

from which we deduce that the pushforward of $M_{z,z^{\prime},\xi}^{k}$ under
$\mathfrak{S}_{N}$ is the Christoffel deformation of the $N$-th Meixner
ensemble at points $u_{1},...,u_{k}$ as defined in definition 2.3.5. As a
consequence, it defines a determinantal point process with an explicit kernel
given by proposition 2.3.6. We now prepare a similar statement for other
choices of admissible parameters $z$ and $z^{\prime}$. For this aim, we will
reproduce the analytic continuation argument developed in [8] and [9],
although the simplifications we need are less obvious. We will note :
$\mathbb{Z}^{\prime}=\mathbb{Z}+1/2$ the set of proper half integers. For
$a\in\mathbb{Z}^{\prime}$, $x\in\mathbb{Z}^{\prime}$ and $z$ and $z^{\prime}$
admissible parameters that are not in the degenerate case, we define :

(11)
$\begin{split}\psi_{a}(x;z,&z^{\prime},\xi):=\left(\frac{\Gamma(x+z+1/2)\Gamma(x+z^{\prime}+1/2)}{\Gamma(z-a+1/2)\Gamma(z^{\prime}-a+1/2)}\right)^{1/2}\xi^{1/2(x+a)}(1-\xi)^{1/2(z+z^{\prime})-a}\\\
&\times\frac{\left.{}_{2}F_{1}\right.\left(-z+a+1/2,-z^{\prime}+a+1/2;x+a+1;\frac{\xi}{\xi-1}\right)}{\Gamma(x+a+1)}.\end{split}$

Observe that these functions are symmetric in $z,z^{\prime}$. We now recall
the following integral representation, which is due to Borodin-Olshanski (see
[8], proposition 2.3):

(12)
$\begin{split}\psi_{a}(x;z,z^{\prime},\xi)&=\left(\frac{\Gamma(x+z+1/2)\Gamma(x+z^{\prime}+1/2)}{\Gamma(z-a+1/2)\Gamma(z^{\prime}-a+1/2)}\right)^{1/2}\frac{\Gamma(z^{\prime}-a+1/2)}{\Gamma(z^{\prime}+x+1/2)}(1-\xi)^{\frac{z^{\prime}-z+1}{2}}\\\
&\times\frac{1}{2i\pi}\oint(1-\sqrt{\xi}\omega)^{-z^{\prime}+a-1/2}\left(1-\frac{\sqrt{\xi}}{\omega}\right)^{z-a-1/2}\omega^{-x-a}\frac{d\omega}{\omega}.\end{split}$

Observe that it is now not obvious that such an expression is symmetric in
$z,z^{\prime}$, i.e. none of its factor is symmetric in $z,z^{\prime}$, but
since expression (11) is, it has to be symmetric in $z,z^{\prime}$. We will
also need the following proposition, relating the functions $\psi_{a}$ to
Meixner polynomials, which comes from Borodin and Olshanski ([8], proposition
2.8).

######  Proposition 4.1.2.

Suppose $z=N\in\mathbb{N}$ and $z^{\prime}=N+\beta-1$. Suppose that :

$\displaystyle\tilde{x}:=x+N-1/2\in\mathbb{N},\quad\quad
n:=N-a-1/2\in\mathbb{N}$

are non-negative integers. Then, $\psi_{a}(x;z,z^{\prime},\xi)$ is well
defined and we have :

$\displaystyle\psi_{a}(x;z,z^{\prime},\xi)=\sqrt{\frac{\omega_{\beta,\xi}(\tilde{x})}{h_{n}}}M_{n}(\tilde{x};\beta,\xi).$

In what follows, we will sometimes omit the dependance on the parameters $z$,
$z^{\prime}$ and $\xi$ when the context is clear. The first theorem of this
section states that, for any pair of admissible parameters $(z,z^{\prime})$,
the modified $z$-measure given by formula (10) gives rise to a determinantal
point process with an explicit kernel expressed through the functions
$\psi_{a}$ and their derivatives.

######  Theorem 4.1.3.

For any admissible parameters $z$ and $z^{\prime}$ which are not in the
degenerate case, and any $u_{i}\in\mathbb{R}\setminus\mathbb{Z}^{\prime}$,
$v_{i}=u_{i}+z^{\prime}-z-1/2$, the pushforward of the measure
$M_{z,z^{\prime},\xi}^{k}$ under

$\displaystyle\mathfrak{S}:\lambda\mapsto\\{\lambda_{i}-i+1/2\\}\in\text{Conf}(\mathbb{Z}^{\prime}),$

is a determinantal point process on $\mathbb{Z}^{\prime}$. Its correlation
kernel is given by :

(13)
$K_{z,z^{\prime},\xi}^{k}(x,y):=\frac{C_{z,z^{\prime},k}}{\prod_{i=1}^{k}|(x-u_{i})(y-u_{i})|}\frac{B_{k,0}(x;z,z^{\prime},\xi)B_{k,1}(y;z,z^{\prime},\xi)-B_{k,0}(y;z,z^{\prime},\xi)B_{k,1}(x;z,z^{\prime},\xi)}{D_{k}(z,z^{\prime},\xi)^{2}(x-y)}$

with :

$\displaystyle B_{k,p}(x)$
$\displaystyle=\begin{vmatrix}\psi_{-1/2+p}(u_{1})&\psi_{-3/2+p}(u_{1})&...&\psi_{-1/2-2k+p}(u_{1})\\\
...&...&...&...\\\ \psi_{-1/2+p}(u_{k})&...&...&\psi_{-1/2-2k+p}(u_{k})\\\
\psi^{\prime}_{-1/2+p}(u_{1})&...&...&\psi^{\prime}_{-1/2-2k+p}(u_{1})\\\
...&...&...&...\\\
\psi^{\prime}_{-1/2+p}(u_{k})&...&...&\psi^{\prime}_{-1/2-2k+p}(u_{k})\\\
\psi_{-1/2+p}(x)&...&...&\psi_{-1/2-2k+p}(x)\end{vmatrix},\quad p=0,1\hskip
2.84544pt,$ $\displaystyle D_{k}$
$\displaystyle=\begin{vmatrix}\psi_{-1/2}(u_{1})&\psi_{-3/2}(u_{1})&...&\psi_{1/2-2k}(u_{1})\\\
...&...&...&...\\\ \psi_{-1/2}(u_{k})&...&...&\psi_{1/2-2k}(u_{k})\\\
\psi^{\prime}_{-1/2}(u_{1})&...&...&\psi^{\prime}_{1/2-2k}(u_{1})\\\
...&...&...&...\\\
\psi^{\prime}_{-1/2}(u_{k})&...&...&\psi^{\prime}_{1/2-2k}(u_{k})\\\
\end{vmatrix}$ and $\displaystyle C_{z,z^{\prime},k}$
$\displaystyle=\left(\frac{\xi}{\xi-1}\right)^{1+2k}\frac{\Gamma(z+2k+1)}{\Gamma(z)}\frac{\Gamma(z^{\prime}+2k)}{\Gamma(z^{\prime}-1)}.$

#### Proof :

The proof requires some steps. We first establish in step 1 that the statement
of the Theorem holds for the degenerate series case. This fact is precisely
formulated in lemma 4.1.4 below. In step 2, we prove that quantities of
interest involving the modified $z$-measures have an analytic expension in
$\xi$ (and thus in $\sqrt{\xi}$), with polynomial coefficients in
$(z,z^{\prime})$, for all values of the admissible parameters. In step 3, we
prove that a modification of the kernel also have an analytic expension in
$\sqrt{\xi}$ with polynomials coefficients in $(z,z^{\prime})$. We here use
the integral reprensentations of the functions $\psi_{a}$. We conclude the
proof in the final step.

#### Step 1.

Assume that $z=N\in\mathbb{N}$ and $z^{\prime}=N+\beta-1$, and :

$\displaystyle\tilde{x}:=x+N-1/2,\quad\tilde{y}:=y+N-1/2\quad\quad\tilde{u}_{i}\in\mathbb{R}\setminus\mathbb{N},\quad\tilde{v}_{i}=\tilde{u}_{i}+\beta-1,\quad
i=1,...,k.$

Then, by means of propositions 2.3.6 and 4.1.2, and computations similar to
those performed at the beginning of the proof of theorem 3.2.1, formula (13)
makes sense and holds with :

$\displaystyle\tilde{u}_{i}=u_{i}+N-1/2.$

Indeed, the factor $C_{z,z^{\prime},k}$ comes from the ratio
$\frac{c_{N-1}}{c_{N+2k}}$ appearing in proposition 2.3.6, while the
expressions for $A_{k}$, $B_{k}$ and $D_{k}$ are obtained from proposition
2.3.6 after performing operations similar to those in the proof of theorem
3.2.1 : the polynomials $M_{n}$ are replaced by the functions $\psi_{a}$
thanks to homogeneity, and their derivatives $M_{n}^{\prime}$ are replaced by
$\psi_{a}^{\prime}$ by operations on rows. We resume this fact in the
following lemma :

######  Lemma 4.1.4.

Suppose that $z=N\in\mathbb{N}$ is a positive integer, $\beta>0$ is a positive
number and that $z^{\prime}=N+\beta-1$. Let
$\tilde{u}_{1},...,\tilde{u}_{k}\in\mathbb{R}\setminus\mathbb{N}$ be $k$ real
and non-integer numbers, and set :

$\displaystyle\tilde{v}_{i}=\tilde{u}_{i}+\beta-1,\quad i=1,...,k.$

Let $M_{z,z^{\prime},\xi}^{k}$ be the Christoffel deformation of the
$z$-measure $M_{z,z^{\prime},\xi}$ defined by (10), with $u_{i}$ (resp.
$v_{i}$) replaced by $\tilde{u}_{i}$ (resp. $\tilde{v}_{i}$). We then have,
for every $n\in\mathbb{N}$ and every
$\\{\tilde{x}_{1},...,\tilde{x}_{n}\\}\subset\mathbb{N}$ :

$\displaystyle
M_{z,z^{\prime},\xi}^{k}\left(\\{x_{1},...,x_{n}\\}\subset\mathfrak{S}(\lambda)\right)=M_{z,z^{\prime},\xi}^{k}\left(\\{\tilde{x}_{1},...,\tilde{x}_{n}\\}\subset\mathfrak{S}_{N}(\lambda)\right)=\det\left(K_{z,z^{\prime},\xi}^{k}(x_{i},x_{j})\right)_{i,j=1}^{n}$

where $K_{z,z^{\prime},\xi}^{k}$ is defined by (13) and :

$\displaystyle x_{j}$ $\displaystyle=\tilde{x}_{j}-N+1/2,\quad j=1,...,n$ and
$\displaystyle u_{i}$ $\displaystyle=\tilde{u}_{i}-N+1/2,\quad i=1,...,k.$

#### Step 2.

For any values of the parameters $z$ and $z^{\prime}$, and for any
$n\in\mathbb{N}$, and any $\\{x_{1},..,x_{n}\\}\subset\mathbb{Z}^{\prime}$,
Borodin and Olshanski ([8], lemma 3.9) show that the quantity :

$\displaystyle
M_{z,z^{\prime},\xi}\left(\\{x_{1},...,x_{n}\\}\subset\mathfrak{S}(\lambda)\right),$

as a function of $\xi$, can be expanded in series around zero, and its
coefficients are polynomials in $(z,z^{\prime})$. As an immediate consequence,
we also have that, viewed as a function of $\xi$, the quantity :

$\displaystyle
M_{z,z^{\prime},\xi}^{k}\left(\\{x_{1},...,x_{n}\\}\subset\mathfrak{S}(\lambda)\right)$

has an analytic expansion around zero with coefficients that are polynomials
in $(z,z^{\prime})$.

#### Step 3

We will prove that the same statement holds for a slight modification of the
kernel $K_{z,z^{\prime},\xi}(x,y)$. We first assume that $z$ and $z^{\prime}$
are admissible parameters not in the degenerate series case. Following [8] (or
[9]), we define :

$\displaystyle
f_{z,z^{\prime}}(x)=\frac{\Gamma(x+z^{\prime}+1/2)}{\sqrt{\Gamma(x+z+1/2)\Gamma(x+z^{\prime}+1/2)}}$

where $x\in\mathbb{Z}^{\prime}$. By the assumptions on $z$ and $z^{\prime}$,
this function is well defined. Indeed, the Gamma functions are well defined
and the fact that :

$\displaystyle\Gamma(x+z+1/2)\Gamma(x+z^{\prime}+1/2)>0$

allows to define the square root. We also assume that $u_{1},...u_{k}$ are
such that $f_{z,z^{\prime}}(x)$ is well defined for
$x\in\\{u_{1},...,u_{k}\\}$. Observe that interchanging $z$ and $z^{\prime}$
corresponds to turning $f$ into $1/f$. We will also use the notation :

$\displaystyle
I_{a}(x;z,z^{\prime},\xi)=\frac{1}{2i\pi}\oint(1-\sqrt{\xi}\omega)^{-z^{\prime}+a-1/2}\left(1-\frac{\sqrt{\xi}}{\omega}\right)^{z-a-1/2}\omega^{-x-a}\frac{d\omega}{\omega}$

for any $a\in\mathbb{Z}^{\prime}$ and
$x\in\mathbb{Z}^{\prime}\cup\\{u_{1},...u_{k}\\}$. By means of the integral
representation (12), we can decompose :

(14)
$\displaystyle\psi_{a}(x;z,z^{\prime},\xi)=(1-\xi)^{\frac{z-z^{\prime}+1}{2}}\frac{f_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(-a)}I_{a}(x;z,z^{\prime},\xi)$

Since the initial definition (11) is symmetric in $(z,z^{\prime})$, we can
also write :

(15)
$\displaystyle\psi_{a}(x;z,z^{\prime},\xi)=(1-\xi)^{\frac{z^{\prime}-z+1}{2}}\frac{f_{z,z^{\prime}}(-a)}{f_{z,z^{\prime}}(x)}I_{a}(x;z^{\prime},z,\xi),$

Similarly, we have for the derivatives :

(16)
$\displaystyle\psi_{a}^{\prime}(x;z,z^{\prime},\xi)-\frac{f^{\prime}_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(x)}\psi_{a}(x;z,z^{\prime},\xi)=(1-\xi)^{\frac{z-z^{\prime}+1}{2}}\frac{f_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(-a)}I_{a}^{\prime}(x;z,z^{\prime},\xi)$

and :

(17)
$\displaystyle\psi_{a}^{\prime}(x;z,z^{\prime},\xi)+\frac{f^{\prime}_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(x)}\psi_{a}(x;z,z^{\prime},\xi)=(1-\xi)^{\frac{z^{\prime}-z+1}{2}}\frac{f_{z,z^{\prime}}(-a)}{f_{z,z^{\prime}}(x)}I_{a}^{\prime}(x;z^{\prime},z,\xi).$

Observe that the integrals $I_{a}$ and their derivatives $I_{a}^{\prime}$ have
an analytic expansion in $\sqrt{\xi}$ with coefficients polynomial in
$(z,z^{\prime})$. We thus want to get rid of the factors involving
$f_{z,z^{\prime}}$ and $f_{z^{\prime},z}$ appearing in formula (13) through
the functions $\psi_{a}$. We begin with the constant factor $D_{k}^{2}$. By
(14) and (16) and their multiplicative structure, we can write :

$\displaystyle
D_{k}=(1-\xi)^{k(z^{\prime}-z+1)}\left(\prod_{i=1}^{k}f_{z,z^{\prime}}(u_{i})\right)^{2}\left(\prod_{i=1}^{2k}f_{z,z^{\prime}}(-1/2-i+1)\right)^{-1}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\end{pmatrix}.$

But using (15) and (17), we see that $D_{k}$ can also be expressed in the
following way :

$\displaystyle
D_{k}=(1-\xi)^{k(z-z^{\prime}+1)}\left(\prod_{i=1}^{k}f_{z,z^{\prime}}(u_{i})\right)^{-2}\left(\prod_{i=1}^{2k}f_{z,z^{\prime}}(-1/2-i+1)\right)\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\end{pmatrix}.$

Thus, writing the determinants $D_{k}$ in $D_{k}^{2}$ in these two different
ways leads to :

$\displaystyle
D_{k}^{2}=(1-\xi)^{2k}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\end{pmatrix}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k\end{subarray}}\end{pmatrix}.$

For the products $B_{k,0}(x)B_{k,1}(y)$ and $B_{k,1}(x)B_{k,0}(y)$, we also
write the functions $\psi_{a}$ and $\psi_{a}^{\prime}$ in two different ways,
but now, we have to take care of the ”boundary factors”, i.e. the factors that
are not shared by both determinants. For instance, we have :

$\displaystyle B_{k,0}(x)B_{k,1}(y)$
$\displaystyle=(1-\xi)^{2k+1}\frac{f_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(y)}\frac{f_{z,z^{\prime}}(1/2)}{f_{z,z^{\prime}}(-1/2-2k)}$
$\displaystyle\times\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}(x;z,z^{\prime},\xi)\right)_{j=1,..2k+1}\end{pmatrix}\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}(y;z^{\prime},z,\xi)\right)_{j=1,...,2k+1}\end{pmatrix},$

and :

$\displaystyle B_{k,0}(y)B_{k,1}(x)$
$\displaystyle=(1-\xi)^{2k+1}\frac{f_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(y)}\frac{f_{z,z^{\prime}}(-1/2-2k)}{f_{z,z^{\prime}}(1/2)}$
$\displaystyle\times\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}(x;z,z^{\prime},\xi)\right)_{j=1,..2k+1}\end{pmatrix}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}(y;z^{\prime},z,\xi)\right)_{j=1,...,2k+1}\end{pmatrix}.$

This gives :

$\displaystyle B$
${}_{k,0}(x)B_{k,1}(y)-B_{k,0}(y)B_{k,1}(x)=(1-\xi)^{2k+1}\frac{f_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(y)}$
$\displaystyle\times\left(\frac{f_{z,z^{\prime}}(1/2)}{f_{z,z^{\prime}}(-1/2-2k)}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}(x;z,z^{\prime},\xi)\right)_{j=1,..2k+1}\end{pmatrix}\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}(y;z^{\prime},z,\xi)\right)_{j=1,...,2k+1}\end{pmatrix}\right.$
$\displaystyle\left.-\frac{f_{z,z^{\prime}}(-1/2-2k)}{f_{z,z^{\prime}}(1/2)}\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{1/2-j+1}(x;z,z^{\prime},\xi)\right)_{j=1,..2k+1}\end{pmatrix}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)_{\begin{subarray}{c}i=1,...,k\\\
j=1,...,2k+1\end{subarray}}\\\
\left(I_{-1/2-j+1}(y;z^{\prime},z,\xi)\right)_{j=1,...,2k+1}\end{pmatrix}\right)$
$\displaystyle=\frac{(1-\xi)^{2k+1}}{\sqrt{(z^{\prime}+1)z^{\prime}...(z^{\prime}-2k-1)(z+1)z...(z-2k-1)}}\frac{f_{z,z^{\prime}}(x)}{f_{z,z^{\prime}}(y)}$
$\displaystyle\times\left((z^{\prime}+1)z^{\prime}...(z^{\prime}-2k-1)\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{-1/2-j+1}(x;z,z^{\prime},\xi)\right)\end{pmatrix}\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{1/2-j+1}(y;z^{\prime},z,\xi)\right)\end{pmatrix}\right.$
$\displaystyle\left.-(z+1)z...(z-2k-1)\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{1/2-j+1}(x;z,z^{\prime},\xi)\right)\end{pmatrix}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{-1/2-j+1}(y;z^{\prime},z,\xi)\right)\end{pmatrix}\right),$

where we used :

$\displaystyle\frac{a}{b}C+\frac{b}{a}D=\frac{a^{2}C+b^{2}D}{ab}$

and direct simplifications due to the standard property of the Gamma function
:

$\displaystyle\Gamma(A+1)=A\Gamma(A).$

Let us introduce the auxilliary kernel $\tilde{K}_{z,z^{\prime},\xi}^{k}(x,y)$
:

$\displaystyle\tilde{K}_{z,z^{\prime},\xi}^{k}(x,y)$
$\displaystyle=\frac{f_{z,z^{\prime}}(y)}{f_{z,z^{\prime}}(x)}D_{k}^{2}\sqrt{(z^{\prime}+1)z^{\prime}...(z^{\prime}-2k-1)(z+1)z...(z-2k-1)}K_{z,z^{\prime},\xi}^{k}(x,y),$
i.e. $\displaystyle\tilde{K}_{z,z^{\prime},\xi}^{k}(x,y)$
$\displaystyle=\left((z^{\prime}+1)z^{\prime}...(z^{\prime}-2k-1)\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{-1/2-j+1}(x;z,z^{\prime},\xi)\right)\end{pmatrix}\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{1/2-j+1}(y;z^{\prime},z,\xi)\right)\end{pmatrix}\right.$
$\displaystyle\left.-(z+1)z...(z-2k-1)\det\begin{pmatrix}\left(I_{1/2-j+1}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{1/2-j+1}^{\prime}(u_{i};z,z^{\prime},\xi)\right)\\\
\left(I_{1/2-j+1}(x;z,z^{\prime},\xi)\right)\end{pmatrix}\det\begin{pmatrix}\left(I_{-1/2-j+1}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{-1/2-j+1}^{\prime}(u_{i};z^{\prime},z,\xi)\right)\\\
\left(I_{-1/2-j+1}(y;z^{\prime},z,\xi)\right)\end{pmatrix}\right),$

and observe that it has an expansion in $\sqrt{\xi}$ around zero with
coefficients that are polynomials in $(z,z^{\prime})$, since it is the case
for the integrals $I_{a}$ and their derivatives. Observe also that, for any
$\\{x_{1},...,x_{n}\\}\subset\mathbb{Z}^{\prime}$, we have :

$\displaystyle\det\left(\tilde{K}_{z,z^{\prime},\xi}^{k}(x_{i},x_{j})\right)_{i,j=1}^{n}=D_{k}^{2n}\sqrt{(z^{\prime}+1)z^{\prime}...(z^{\prime}-2k-1)(z+1)z...(z-2k-1)}^{n}\det\left(K_{z,z^{\prime},\xi}^{k}(x_{i},x_{j})\right)_{i,j=1}^{n}.$

#### Final step.

Our computations above remain valid for $z$ and $z^{\prime}$ belonging to the
degenerate series case, provided $z>2k+2$. Assume now that we are in this
case, and let $\\{x_{1},...,x_{n}\\}\subset\mathbb{Z}^{\prime}$ be a finite
subset of $\mathbb{Z}^{\prime}$. We have established the following equality
between series in $\sqrt{\xi}$ with coefficients that are polynomials in
$(z,z^{\prime})$:

$\displaystyle\left(\det\left(\tilde{K}_{z,z^{\prime},\xi}^{k}(x_{i},x_{j})\right)_{i,j=1,...,n}\right)^{2}$
$\displaystyle=D_{k}^{4n}\left((z^{\prime}+1)z^{\prime}...(z^{\prime}-2k-1)(z+1)z...(z-2k-1)\right)^{n}$
$\displaystyle\times
M_{z,z^{\prime},\xi}^{k}\left(\\{x_{1},...,x_{n}\\}\subset\mathfrak{S}(\lambda)\right)^{2}.$

Indeed, the equality follows from the first step and lemma 4.1.4. The right
hand side has an analytic expansion in $\xi$ with polynomial coefficients in
$(z,z^{\prime})$, by step 2 and 3, while it also holds for the left hand side
by step 3. Since the set :

$\displaystyle\\{(z,z^{\prime})\in\mathbb{C}^{2},z\in\mathbb{N},\hskip
2.84544ptz>2k+2\text{ and }z^{\prime}\in\mathbb{R},\hskip
2.84544ptz^{\prime}>z-1\\}$

is a set of uniqueness for polynomials in two variables, the latter equality
holds for all values of $(z,z^{\prime})$, in particular for all admissible
values. This proves the theorem. $\square$.

###  4.2. A deformation of the Gamma kernel

Let $(z,z^{\prime})$ be admissible parameters, either in the principal or in
the complementary series cases. In [7] and [9], the authors show that with
such a choice of parameters, the $z$-measure, viewed as a point process on
$\mathbb{Z}^{\prime}$, converges as $\xi$ tends to $1$, without any scaling,
to a determinantal point process with a correlation kernel expressed only
through the Gamma function. They call this kernel the ”Gamma kernel”. The goal
of this section is to show that such a statement still holds for the
Christoffel deformation of the $z$-measure of order one, i.e. with $k=1$. Our
argument is a slight modification of the one presented in [7], theorem 2.3. As
mentioned in the introduction, our result must be compared with the one from
[15], where a hierarchy of Palm deformations of the process with the Gamma
kernel is established.

Before stating the theorem, we give some definitions : for
$a\in\mathbb{Z}^{\prime}$, we define :

$\displaystyle
g_{z,z^{\prime}}(a):=\sqrt{\Gamma(z-a+1/2)\Gamma(z^{\prime}-a+1/2)}\Gamma(-z+a+1/2),$

and for $u\in\mathbb{R}$ and $a\in\mathbb{Z}^{\prime}$, we set :

$\displaystyle
h_{z,z^{\prime}}(u,a)=\frac{\Gamma(z^{\prime}-z)}{f_{z,z^{\prime}}(u)g_{z,z^{\prime}}(a)}.$

We will denote by $h_{z,z^{\prime}}^{\prime}(u,a)$ the derivative of
$h_{z,z^{\prime}}(u,a)$ with respect to $u$. We will also use the notation :

$\displaystyle\phi_{a}(u;z,z^{\prime},\xi)=\psi^{\prime}_{a}(u;z,z^{\prime},\xi)-\frac{\log(\xi)}{2}\psi_{a}(u;z,z^{\prime},\xi).$

######  Theorem 4.2.1.

Let $(z,z^{\prime})$ be admissible parameters either in the principal or in
the complementary series cases, and let $K_{z,z^{\prime},\xi}^{1}$ be the
correlation kernel of the Christoffel deformation of order $1$ at
$u\in\mathbb{R}\setminus\mathbb{Z}^{\prime}$ defined by (13). We have, for all
$x,y\in\mathbb{Z}^{\prime}$, $x\neq y$ :

$\displaystyle\lim_{\xi\rightarrow
1}(1-\xi)^{2}K_{z,z^{\prime},\xi}^{1}(x,y)=\frac{\mathcal{C}_{z,z^{\prime}}}{|(x-u)(y-u)|}\frac{\mathcal{A}_{z,z^{\prime}}(x,y)}{\Phi_{z,z^{\prime},0,-1}(u)^{2}(x-y)}$

where :

$\displaystyle\mathcal{A}_{z,z^{\prime}}(x,y)$
$\displaystyle=\sum_{i,j=0}^{2}\Phi_{z,z^{\prime},a(i)b(i)}(u)\Phi_{z,z^{\prime},a(j)+1,b(j)+1}(u)\Psi_{z,z^{\prime},i,j}(x,y),$
$\displaystyle\Psi_{z,z^{\prime},i,j}(x,y)$
$\displaystyle=h_{z,z^{\prime}}(x,-1/2+i)h_{z^{\prime},z}(y,1/2+j)+h_{z^{\prime},z}(x,-1/2+i)h_{z,z^{\prime}}(y,1/2+j)$
$\displaystyle-
h_{z,z^{\prime}}(x,1/2+j)h_{z^{\prime},z}(y,-1/2+i)-h_{z^{\prime},z}(x,1/2+j)h_{z,z^{\prime}}(y,-1/2+i),$
$\displaystyle\Phi_{z,z^{\prime},a(i),b(i)}(u)$
$\displaystyle=h_{z,z^{\prime}}(u,-1/2+a(i))h_{z^{\prime},z}^{\prime}(u,-1/2+b(i))+h_{z^{\prime},z}(u,-1/2+a(i))h_{z,z^{\prime}}^{\prime}(u,1/2+b(i))$
$\displaystyle-
h_{z,z^{\prime}}(u,-1/2+b(i))h_{z^{\prime},z}^{\prime}(u,-1/2+a(i))-h_{z^{\prime},z}(u,1/2+b(i))h_{z,z^{\prime}}^{\prime}(u,-1/2+a(i)),$
$\displaystyle a(0)$ $\displaystyle=-1,\hskip 2.84544pta(1)=a(2)=0\quad;\quad
b(0)=b(1)=-2,\hskip 2.84544ptb(2)=-1,$
$\displaystyle\mathcal{C}_{z,z^{\prime}}$
$\displaystyle=(z+3)(z+2)(z+1)(z^{\prime}+2)(z^{\prime}+1)z^{\prime}.$

The diagonal entries are obtained by l’Hospital rule.

######  Remark 4.2.2.

An explicit expression for the function $\Psi_{z,z^{\prime},i,j}(x,y)$ may be
found from :

$\displaystyle h_{z,z^{\prime}}(x,a)h_{z^{\prime},z}(y,b)$
$\displaystyle+h_{z^{\prime},z}(x,a)h_{z,z^{\prime}}(y,b)-h_{z,z^{\prime}}(x,b)h_{z^{\prime},z}(y,a)-h_{z^{\prime},z}(x,b)h_{z,z^{\prime}}(y,a)$
$\displaystyle=\Gamma(z-z^{\prime})\Gamma(z^{\prime}-z)$
$\displaystyle\times\left(\frac{\Gamma(x+z+1/2)\Gamma(x+z^{\prime}+1/2)\Gamma(y+z+1/2)\Gamma(y+z^{\prime}+1/2)}{\Gamma(z-a+1/2)\Gamma(z^{\prime}-a+1/2)\Gamma(z-b+1/2)\Gamma(z^{\prime}-b+1/2)}\right)^{1/2}$
$\displaystyle\times\left\\{\frac{1}{\Gamma(-z+a+1/2)\Gamma(-z^{\prime}+b+1/2)\Gamma(x+z^{\prime}+1/2)\Gamma(y+z+1/2)}\right.$
$\displaystyle+\frac{1}{\Gamma(-z^{\prime}+a+1/2)\Gamma(-z+b+1/2)\Gamma(x+z+1/2)\Gamma(y+z^{\prime}+1/2)}$
$\displaystyle-\frac{1}{\Gamma(-z^{\prime}+a+1/2)\Gamma(-z+b+1/2)\Gamma(x+z^{\prime}+1/2)\Gamma(y+z+1/2)}$
$\displaystyle\left.-\frac{1}{\Gamma(-z+a+1/2)\Gamma(-z^{\prime}+b+1/2)\Gamma(x+z+1/2)\Gamma(y+z^{\prime}+1/2)}\right\\}.$

It only involves the Gamma function. A similar expression for the function
$\Phi_{z,z^{\prime},j}$ can be obtained in the same way, and would involve the
Gamma function and its first derivative.

######  Remark 4.2.3.

The factor $(1-\xi)^{2}$ appearing in our statement should be interpreted as
$(1-\xi)^{2k}$ in order to recover Borodin-Olshanski theorem from [7] and [9]
in the case $k=0$.

###  4.3. Proof of theorem 4.2.1

We begin with the following proposition which gives the first asymptotic
results we need.

######  Proposition 4.3.1.

For any $u\in\mathbb{R}$, $a\in\mathbb{Z}^{\prime}$, we have :

$\displaystyle(1-\xi)^{-\frac{1}{2}}\psi_{a}(u;z,z^{\prime},\xi)$
$\displaystyle=h_{z,z^{\prime}}(u,a)(1-\xi)^{\frac{z^{\prime}-z}{2}}\left(1+O(1-\xi)\right)$
$\displaystyle+h_{z^{\prime},z}(u,a)(1-\xi)^{\frac{z-z^{\prime}}{2}}\left(1+O(1-\xi)\right)$
and $\displaystyle(1-\xi)^{-\frac{1}{2}}\phi_{a}(u;z,z^{\prime},\xi)$
$\displaystyle=h_{z,z^{\prime}}^{\prime}(u,a)(1-\xi)^{\frac{z^{\prime}-z}{2}}\left(1+O(1-\xi)\right)$
$\displaystyle+h_{z^{\prime},z}^{\prime}(u,a)(1-\xi)^{\frac{z-z^{\prime}}{2}}\left(1+O(1-\xi)\right)$

as $\xi$ tends to $1$.

#### Proof :

This is similar to that in [7], theorem 2.3 . We apply the formula :

$\displaystyle\frac{1}{\Gamma(C)}\left.{}_{2}F_{1}\right.(A,B;C;w)$
$\displaystyle=\frac{\Gamma(B-A)(-w)^{-A}}{\Gamma(B)\Gamma(C-A)}\left.{}_{2}F_{1}\right.(A,1-C+A;1-B+A;w^{-1})$
$\displaystyle+\frac{\Gamma(A-B)(-w)^{-B}}{\Gamma(A)\Gamma(C-B)}\left.{}_{2}F_{1}\right.(B,1-C+B;1-A+B;w^{-1})$

We specialize this results with :

$\displaystyle A$ $\displaystyle=-z+a+1/2\quad;\quad
B=-z^{\prime}+a+1/2\quad;\quad C=u+a+1\quad;$ $\displaystyle w$
$\displaystyle=\frac{\xi}{\xi-1},$

and plug it into (11) to obtain :

$\displaystyle(1-\xi)^{-\frac{1}{2}}\psi_{a}(u;z,z^{\prime},\xi)$
$\displaystyle=\xi^{\frac{u-a+1}{2}}\left\\{h_{z,z^{\prime}}(u,a)\xi^{z}(1-\xi)^{\frac{z^{\prime}-z}{2}}\left.{}_{2}F_{1}\right.\left(-z+a+1/2,-z-u+1/2;1+z^{\prime}-z;\frac{\xi-1}{\xi}\right)\right.$
$\displaystyle\left.+h_{z^{\prime},z}(u,a)\xi^{z^{\prime}}(1-\xi)^{\frac{z-z^{\prime}}{2}}\left.{}_{2}F_{1}\right.\left(-z^{\prime}+a+1/2,-z^{\prime}-u+1/2;1+z-z^{\prime};\frac{\xi-1}{\xi}\right)\right\\}.$

By means of the asymptotics :

$\displaystyle\left.{}_{2}F_{1}\right.(A,1-C+A;1-B+A;w^{-1})$
$\displaystyle=1+O(w^{-1}),$
$\displaystyle\left.{}_{2}F_{1}\right.(B,1-C+B;1-A+B;w^{-1})$
$\displaystyle=1+O(w^{-1}),$
$\displaystyle\frac{d}{dC}\left.{}_{2}F_{1}\right.(A,1-C+A;1-B+A;w^{-1})$
$\displaystyle=O(w^{-1}),$
$\displaystyle\frac{d}{dC}\left.{}_{2}F_{1}\right.(B,1-C+B;1-A+B;w^{-1})$
$\displaystyle=O(w^{-1}),$

which stand for large negative values of $w$ and directly follow from
definition (1) for the Gauss hypergeometric function, we obtain the desired
asymptotics. $\square$

These estimates imply the following lemma :

######  Lemma 4.3.2.

For any $a,b\in\mathbb{Z}+1/2$, and any $x,y\in\mathbb{R}$, we have :

$\displaystyle\lim_{\xi\rightarrow
1}(1-\xi)^{-1}\left(\psi_{a}(x)\psi_{b}(y)-\psi_{a}(y)\psi_{b}(x)\right)$
$\displaystyle=h_{z,z^{\prime}}(x,a)h_{z^{\prime},z}(y,b)+h_{z^{\prime},z}(x,a)h_{z,z^{\prime}}(y,b)$
$\displaystyle-
h_{z,z^{\prime}}(x,b)h_{z^{\prime},z}(y,a)-h_{z^{\prime},z}(x,b)h_{z,z^{\prime}}(y,a)$
and $\displaystyle\lim_{\xi\rightarrow
1}(1-\xi)^{-1}(\psi_{a}(x)\phi_{b}(x)-\psi_{b}(x)\phi_{a}(x))$
$\displaystyle=h_{z,z^{\prime}}(x,a)h_{z^{\prime},z}^{\prime}(x,b)+h_{z^{\prime},z}(x,a)h_{z,z^{\prime}}^{\prime}(x,b)$
$\displaystyle-
h_{z,z^{\prime}}(x,b)h_{z^{\prime},z}^{\prime}(x,a)-h_{z^{\prime},z}(x,b)h_{z,z^{\prime}}^{\prime}(x,a).$

#### Proof :

It is similar to that in [7], theorem 2.3 and we use the formulas from
proposition 4.3.1. Since

$\displaystyle
h_{z,z^{\prime}}(x,a)h_{z,z^{\prime}}(y,b)=h_{z,z^{\prime}}(x,b)h_{z,z^{\prime}}(y,a)$
and $\displaystyle
h_{z^{\prime},z}(x,a)h_{z^{\prime},z}(y,b)=h_{z^{\prime},z}(x,b)h_{z^{\prime},z}(y,a),$

the terms involving the factors $(1-\xi)^{\pm(z-z^{\prime})}$ will vanish
within a term of the type $(1-\xi)^{\pm(z-z^{\prime})}O(1-\xi)$ in the
difference $\psi_{a}(x)\psi_{b}(y)-\psi_{a}(y)\psi_{b}(x)$. This term is
negligeable because $\mathfrak{R}(z-z^{\prime})<1$ by our assumption on $z$
and $z^{\prime}$. Writing down the other terms, we obtain the first part of
the lemma. The proof of the second part is analogous, once we remarked that :

$\displaystyle
h_{z,z^{\prime}}(x,a)h_{z,z^{\prime}}^{\prime}(x,b)=h_{z,z^{\prime}}(x,b)h_{z,z^{\prime}}^{\prime}(x,a)$
and that $\displaystyle
h_{z^{\prime},z}(x,a)h_{z^{\prime},z}^{\prime}(x,b)=h_{z^{\prime},z}(x,b)h_{z^{\prime},z}^{\prime}(x,a).$

$\square$

We can now move to the conclusion of the proof of theorem 4.2.1. We consider
expression (13), and first write, as in the proof of theorem 3.2.1 :

$\displaystyle
B_{1,0}(x)B_{1,1}(y)-B_{1,0}(y)B_{1,1}(x)=\sum_{i,j=0}^{2}\left[B_{1,0}(x)\right]^{i}\left[B_{1,1}(y)\right]^{j}\left(\psi_{-1/2-i}(x)\psi_{1/2-j}(y)-\psi_{-1/2-i}(y)\psi_{1/2-j}(x)\right).$

By the first part of lemma 4.3.2, we have :

(18) $\displaystyle\lim_{\xi\rightarrow
1}(1-\xi)^{-1}\left(\psi_{-1/2-i}(x)\psi_{1/2-j}(y)-\psi_{-1/2-i}(y)\psi_{1/2-j}(x)\right)=\Psi_{z,z^{\prime},i,j}(x,y),$

while its second part entails :

(19) $\displaystyle\lim_{\xi\rightarrow
1}(1-\xi)^{-1}\left[B_{1,p}(x)\right]^{i}=\lim_{\xi\rightarrow
1}(1-\xi)^{-1}\left[B_{1,p}(y)\right]^{i}=\Phi_{z,z^{\prime},a(i)+p,b(i)+p}(u).$

Putting (18) and (19) together, we obtain :

$\displaystyle\lim_{\xi\rightarrow
1}(1-\xi)^{-3}\left(B_{1,0}(x)B_{1,1}(y)-B_{1,0}(y)B_{1,1}(x)\right)=\mathcal{A}_{z,z^{\prime}}(x,y).$

Second part of lemma 4.3.2 also implies that :

$\displaystyle\lim_{\xi\rightarrow
1}(1-\xi)^{-1}D_{k}=\Phi_{z,z^{\prime},0,-1}(u),$

which concludes the proof of the theorem for the off-diagonal part of the
kernel. The diagonal entries are treated in a manner similar to that of the
proof of theorem 3.2.1, by means of the second part of lemma 4.3.2. $\square$

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