# Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled
Painlevé V System

Shulin Lyu  School of Mathematics and Statistics, Qilu University of
Technology (Shandong Academy of Sciences), Jinan 250353, China  Yang Chen
Department of Mathematics, Faculty of Science and Technology, University of
Macau, Macau, China  Shuai-Xia Xu  Institut Franco-Chinois de l’Energie
Nucléaire, Sun Yat-sen University, Guangzhou 510275, China

###### Abstract

We study the Hankel determinant generated by the Laguerre weight with jump
discontinuities at $t_{k},k=1,\cdots,m$. By employing the ladder operator
approach to establish Riccati equations, we show that
$\sigma_{n}(t_{1},\cdots,t_{m})$, the logarithmic derivative of the
$n$-dimensional Hankel determinant, satisfies a generalization of the
$\sigma$-from of Painlevé V equation. Through investigating the Riemann-
Hilbert problem for the associated orthogonal polynomials and via Lax pair, we
express $\sigma_{n}$ in terms of solutions of a coupled Painlevé V system. We
also build relations between the auxiliary quantities introduced in the above
two methods, which provides connections between the Riccati equations and Lax
pair. In addition, when each $t_{k}$ tends to the hard edge of the spectrum
and $n$ goes to $\infty$, the scaled $\sigma_{n}$ is shown to satisfy a
generalized Painlevé III equation.

$\mathbf{Keywords}$: Laguerre unitary ensembles; Hankel determinant;
Orthogonal polynomials;

Painlevé equations; Riemann-Hilbert problems

$\mathbf{Mathematics\>\>Subject\>\>Classification\>\>2020}$: 33E17; 34M55;
41A60; 42C05

###### Contents

  1. 1 Introduction and Statement of Main Results
    1. 1.1 Statement of main results
      1. 1.1.1 Results for finite dimension $n$: PDEs and the coupled $P_{V}$ system
      2. 1.1.2 Double scaling analysis at the hard edge: generalized $P_{III}$
    2. 1.2 Outline of the paper
  2. 2 Finite Dimension Analysis – Ladder Operator Approach
    1. 2.1 Difference equations for auxiliary quantities
    2. 2.2 Toda equations for recurrence coefficients
    3. 2.3 Proof of Theorem 1.1
  3. 3 Lax Pair and Coupled $P_{V}$ System
    1. 3.1 RH problem for orthogonal polynomials
    2. 3.2 A model RH problem and Lax pair
    3. 3.3 Proof of Theorem 1.2
    4. 3.4 Proof of Theorem 1.3 by Lax pair
  4. 4 Double Scaling Analysis at the Hard Edge
    1. 4.1 Proof of Proposition 1.4
    2. 4.2 Proof of Theorem 1.5 and Corollary 1.6

##  1 Introduction and Statement of Main Results

The $n$-dimensional Laguerre unitary ensemble (LUE for short) is a group of
$n\times n$ Hermitian random matrices with eigenvalues having the following
joint probability density function

$\displaystyle
p(x_{1},x_{2},\cdots,x_{n})=\frac{1}{C_{n}}\cdot\frac{1}{n!}\prod_{1\leq
i<j\leq n}(x_{i}-x_{j})^{2}\prod_{k=1}^{n}x_{k}^{\alpha}{\rm e}^{-x_{k}},$

where $x_{k}\in[0,\infty)$ for $k=1,2,\cdots,n$ and $\alpha>-1$. See [23,
sections 2.5, 2.6 and 3.3] for more details. The normalization constant
$n!C_{n}$, which is also known as the partition function, has the following
explicit expression [23, equation (17.6.5)]

$\displaystyle n!C_{n}:=$ $\displaystyle\int_{[0,\infty)^{n}}\prod_{1\leq
i<j\leq n}(x_{i}-x_{j})^{2}\prod_{k=1}^{n}x_{k}^{\alpha}{\rm
e}^{-x_{k}}dx_{k}$ $\displaystyle=$
$\displaystyle\prod_{k=1}^{n}k!\cdot\Gamma(\alpha+k).$

For any interval $I\subset[0,\infty)$, the probability that all the
eigenvalues of LUE lie in $I$ is given by

$\displaystyle P\left(\\{x_{k}\in I:k=1,\cdots,n\\}\right)=$
$\displaystyle\frac{1}{C_{n}}\cdot\frac{1}{n!}\int_{I^{n}}\prod_{1\leq i<j\leq
n}(x_{i}-x_{j})^{2}\prod_{k=1}^{n}x_{k}^{\alpha}{\rm e}^{-x_{k}}dx_{k}$
$\displaystyle=$
$\displaystyle\frac{1}{C_{n}}\det\left(\int_{I}x^{i+j}x^{\alpha}{\rm
e}^{-x}dx\right)_{i,j=0}^{n-1},$ (1.1)

where the second equality is due to Heine’s formula (see e.g. [27, sections
2.1 and 2.2]) and the determinant is called the Hankel determinant.

In this paper, we are concerned with the Hankel determinant generated by the
moments of the Laguerre weight multiplied by a factor with $m$ jump
discontinuities, namely

$\displaystyle
D_{n}\left(\,\vec{t}\,\right):=\det\left(\int_{0}^{\infty}x^{i+j}w(x;\vec{t}\,)dx\right)_{i,j=0}^{n-1},$
(1.2)

where $\vec{t}=\left(t_{1},\cdots,t_{m}\right)$ with $0<t_{1}<\cdots<t_{m}$,
and the weight function reads

$w(x;\vec{t}\,):=x^{\alpha}{\rm
e}^{-x}\left(\omega_{0}+\sum_{k=1}^{m}\omega_{k}\theta(x-t_{k})\right),\qquad\qquad\alpha>-1.$
(1.3)

Here $\sum\limits_{k=0}^{\ell}\omega_{k}\geq 0$ for $\ell=0,1,\cdots,m$, and
$\theta(x)$ is the step function which is 1 for $x>0$ and 0 otherwise.

When $m=2$, according to (1.1) and (1.2), we find that the probability that
the interval $(t_{1},t_{2})$ has no or all eigenvalues of LUE is given by
$D_{n}(t_{1},t_{2})/C_{n}$ with $\omega_{0}=1,\omega_{1}=-1,\omega_{2}=1$ and
$\omega_{0}=0,\omega_{1}=1,\omega_{2}=-1$, respectively. The latter
probability was studied in [4] by using the ladder operator approach [17], a
formalism adapted to monic polynomials orthogonal with respect to
$w(x;t_{1},t_{2})$, and the logarithmic derivative of $D_{n}(t_{1},t_{2})$ was
shown to satisfy a second order partial differential equation (PDE for short)
which can be viewed as a two-variable generalization of the Painlevé V
equation ($P_{V}$ for short).

For $w(x;\vec{t}\,)$, we can define the associated monic orthogonal
polynomials by

$\displaystyle\int_{0}^{\infty}P_{n}(x;\vec{t}\,)P_{m}(x;\vec{t}\,)w(x;\vec{t}\,)dx=h_{n}\delta_{mn},\qquad
m,n\geq 0,$ (1.4)

where $\delta_{mn}=1$ for $m=n$ and 0 otherwise, and

$\displaystyle
P_{n}\left(x;\vec{t}\,\right):=x^{n}+p(n,\vec{t}\,)x^{n-1}+\cdots+P_{n}(0;\vec{t}\,).$
(1.5)

From this definition, there follows the three-term recurrence relation

$\displaystyle
xP_{n}(x;\vec{t}\,)=P_{n+1}(x;\vec{t}\,)+\alpha_{n}(\vec{t}\,)P_{n}(x;\vec{t}\,)+\beta_{n}(\vec{t}\,)P_{n-1}(x;\vec{t}\,),\qquad
n\geq 0,$ (1.6)

where $P_{0}(x;\vec{t}\,):=1$ and $\beta_{0}P_{-1}(x;\vec{t}\,):=0$, and the
recurrence coefficients are given by

$\displaystyle\alpha_{n}(\vec{t}\,)=$ $\displaystyle
p(n,\vec{t}\,)-p(n+1,\vec{t}\,),\qquad n\geq 0,$ (1.7)
$\displaystyle\beta_{n}(\vec{t}\,)=$
$\displaystyle\frac{h_{n}(\vec{t}\,)}{h_{n-1}(\vec{t}\,)},\qquad n\geq 1,$

with $p(0,\vec{t}\,):=0$. The recurrence relation indicates the following
Christoffel-Darboux formula

$\sum_{k=0}^{n-1}\frac{P_{k}(x;\vec{t}\,)P_{k}(y;\vec{t}\,)}{h_{k}(\vec{t}\,)}=\frac{P_{n}(x;\vec{t}\,)P_{n-1}(y;\vec{t}\,)-P_{n-1}(x;\vec{t}\,)P_{n}(y;\vec{t}\,)}{h_{n-1}(\vec{t}\,)(x-y)}.$

With all these properties, one can derive a pair of ladder operators satisfied
by $P_{n}(z;\vec{t})$ and three compatibility conditions, based on which the
ladder operator approach is formulated. We should point out that, given an
arbitrary positive function whose moments of all orders exist, the associated
monic orthogonal polynomials have the properties mentioned above. For more
details, the reader may consult [27, section 3.2] and [17].

The ladder operator approach was widely used to study problems related to
unitary ensembles which involve one or two perturbation variables or jump
discontinuites. For example, the Hankel determinant generated by the
singularly perturbed Laguerre weight $x^{\alpha}{\rm e}^{-x-t/x}$, which
arises from a finite-temperature integrable quantum field theory, was
investigated in [10] for $x\in(0,\infty)$ and in [22] for $x\in(s,\infty)$;
The Hankel determinant of a two-time deformation of the Laguerre weight
characterizes the multiple-input-multiple-output wireless communication system
[8], and its log-derivative was shown to satisfy a two-variable generalization
of the $\sigma$-form of $P_{V}$. Under double scaling, the log-derivative of
the gap probability on $(-a,a)$ for Gaussian and symmetric Jacobi unitary
ensembles were found to satisfy the $\sigma$-form of $P_{V}$[21, 24]; etc..
For problems involving more than two variables, the ladder operator approach
is rarely chosen, which may be caused due to the complexity and difficulty of
the derivation of the coupled PDEs.

It is well-known that orthogonal polynomials satisfy a Riemann-Hilbert (RH for
short) problem. This fact provides an alternative method to study unitary
ensembles. By conducting Deift-Zhou steepest descent analysis [14] to the RH
problem, one can derive asymptotics for orthogonal polynomials and quantities
relating unitary ensembles. For problems involving more than one variables, a
coupled Painlevé system is usually established. See, for instance [2, 11, 12,
13].

The Hankel determinant generated by the Gaussian weight with $m$ jumps was
studied in [29]. When the dimension of the Hankel matrix is finite, by
considering the RH problem for the associated monic orthogonal polynomials and
with the help of the Lax pair, a coupled $P_{IV}$ system was derived. As the
jump discontinuities tend to the edge of the spectrum and the dimension
becomes large, by carrying out Deift-Zhou steepest descent analysis to the RH
problem mentioned above, an asymptotic expression for the orthogonal
polynomials in terms of quantities satisfying a coupled $P_{II}$ system was
established. The two-jump case ($m=2$) was investigated in [25] via the ladder
operator approach and a second order PDE was deduced for the logarithmic
derivative of the Hankel determinant. By making use of the finite dimensional
results therein, the first and the second author of the present paper
reproduce the aforementioned coupled $P_{IV}$ and $P_{II}$ system of [29] with
$m=2$ [20]. See also [6, 11, 30] for the applications of coupled $P_{II}$
system in the studies Airy kernel determinants with discontinuities and the
Painlevé-type kernel determinants.

In this paper, we adopt both approaches mentioned above to study
$D_{n}\left(\vec{t}\right)$. We establish a coupled $P_{V}$ system and a
second order PDE for the logarithmic derivative of $D_{n}(\vec{t})$. By making
comparison and connection between the results obtained by these two different
methods, we find that the ladder operators are closely related to the Lax pair
of the RH problem for orthogonal polynomials. We also consider the behavior of
$D_{n}(\vec{t})$ under the double scaling that $\vec{t}\rightarrow\vec{0}$ and
$n\rightarrow\infty$ such that $\vec{t}=\frac{1}{4n}\vec{s}$, with
$\vec{s}=(s_{1},s_{2},\cdots,s_{m})$ fixed. We deduce the limiting PDE
satisfied by the logarithmic derivative of the double scaled Hankel
determinant, which can be viewed as an $m$-variable generalization of the
$\sigma$-form of $P_{III}$. If further $s_{k}=\nu_{k}\tau$ with $\nu_{k}$
arbitrary given constant, we obtain the expression for the scaled Hankel
determinant in terms of quantities that satisfy a coupled ODEs in variable
$\tau$ and whose square roots were shown in [7] to be approximated by the
Bessel function.

The main results of this paper are presented as below.

###  1.1 Statement of main results

####  1.1.1 Results for finite dimension $n$: PDEs and the coupled $P_{V}$
system

Define

$\sigma_{n}(\vec{t}):=\delta\,{\ln}D_{n}(\vec{t}),\qquad\delta:=\sum_{k=1}^{m}t_{k}\partial_{t_{k}},$
(1.8)

where $\partial_{t_{k}}$ denotes $\frac{\partial}{\partial t_{k}}$. For ease
of notations, we also represent $\frac{\partial^{2}}{\partial t_{k}^{2}}$ and
$\frac{\partial^{2}}{\partial t_{j}\partial t_{k}}(j\neq k)$ by
$\partial_{t_{k}t_{k}}^{2}$ and $\partial_{t_{j}t_{k}}^{2}$ respectively.

To take the ladder operator approach, we introduce $2m$ auxiliary quantities
allied to $P_{n}(x;\vec{t})$:

$\displaystyle R_{n,k}(\vec{t}):=$
$\displaystyle\omega_{k}\frac{w_{0}(t_{k})}{h_{n}(\vec{t})}P_{n}^{2}(t_{k};\vec{t}),$
(1.9) $\displaystyle r_{n,k}(\vec{t}):=$
$\displaystyle\omega_{k}\frac{w_{0}(t_{k})}{h_{n-1}(\vec{t})}P_{n}(t_{k};\vec{t})P_{n-1}(t_{k};\vec{t}),$
(1.10)

with $k=1,2,\cdots,m$ and $w_{0}(x)=x^{\alpha}{\rm e}^{-x}$. We first derive
the Riccati equations for $\\{R_{n,k}\\}$ and $\\{r_{n,k}\\}$, which give us a
system of PDEs satisfied by $\\{R_{n,k}\\}$. Then we establish the expression
for $\sigma_{n}$ in terms of $\\{R_{n,k}\\}$ and $\\{r_{n,k}\\}$, from which a
second order PDE satisfied by $\sigma_{n}$ is deduced.

######  Theorem 1.1.

$R_{n,k}(\vec{t})$ and $r_{n,k}(\vec{t})$, defined in (1.9) and (1.10),
respectively, satisfy the following analogs of Riccati equations

$\displaystyle\delta R_{n,k}=$ $\displaystyle
2r_{n,k}+\Bigg{(}\sum_{j=1}^{m}t_{j}R_{n,j}-t_{k}+2n+\alpha\Bigg{)}R_{n,k},$
(1.11a) $\displaystyle\delta r_{n,k}=$
$\displaystyle\frac{r_{n,k}^{2}}{R_{n,k}}-R_{n,k}\left(\dfrac{\bigg{(}n+\alpha+\sum\limits_{j=1}^{m}r_{n,j}\bigg{)}\bigg{(}n+\sum\limits_{j=1}^{m}r_{n,j}\bigg{)}}{1-\sum\limits_{j=1}^{m}R_{n,j}}+\sum\limits_{j=1}^{m}\frac{r_{n,j}^{2}}{R_{n,j}}\right),$
(1.11b)

for $k=1,\cdots,m$, from which we establish a coupled second order PDEs for
$\\{R_{n,k}(\vec{t})\\}:$

$\displaystyle\delta^{2}R_{n,k}+\bigg{(}t_{k}-\sum_{j\neq
k}t_{j}R_{n,j}-2t_{k}R_{n,k}-2n-\alpha\bigg{)}\delta
R_{n,k}-R_{n,k}\sum_{j\neq k}t_{j}\cdot\delta R_{n,j}$ (1.12) $\displaystyle=$
$\displaystyle\left(\frac{\bigg{(}2(n+\alpha)+\sum\limits_{j=1}^{m}2r_{n,j}\bigg{)}\bigg{(}2n+\sum\limits_{j=1}^{m}2r_{n,j}\bigg{)}}{2\bigg{(}\sum\limits_{j=1}^{m}R_{n,j}-1\bigg{)}}-\sum\limits_{j=1}^{m}\frac{(2r_{n,j})^{2}}{2R_{n,j}}+\sum_{j=1}^{m}t_{j}R_{n,j}-t_{k}\right)R_{n,k}+\frac{(2r_{n,k})^{2}}{2R_{n,k}},$

with $k=1,\cdots,m$, where $2r_{n,k}$ is given by (1.11a) and

$\displaystyle\delta^{2}=$
$\displaystyle\sum_{k=1}^{m}t_{k}^{2}\partial_{t_{k}t_{k}}^{2}+2\sum_{1\leq
j<k\leq m}t_{j}t_{k}\partial_{t_{j}t_{k}}^{2}+\delta.$

Moreover, $\sigma_{n}(\vec{t})$, defined in (1.8), is expressed in terms of
$\\{R_{n,k}(\vec{t})\\}$ and $\\{r_{n,k}(\vec{t})\\}$ by

$\displaystyle\sigma_{n}=\frac{\left(n+\alpha+\sum\limits_{k=1}^{m}r_{n,k}\right)\left(n+\sum\limits_{k=1}^{m}r_{n,k}\right)}{\sum\limits_{k=1}^{m}R_{n,k}-1}-\sum\limits_{k=1}^{m}\frac{r_{n,k}^{2}}{R_{n,k}}+\sum_{k=1}^{m}t_{k}r_{n,k}+n(n+\alpha),$
(1.13)

and $\sigma_{n}(\vec{t})$ satisfies the following PDE

$\displaystyle\bigg{(}\sum\limits_{k=1}^{m}{\rm
sgn}(\omega_{k})\sqrt{\Delta_{k}(\vec{t})}-2\left(\delta\sigma_{n}-\sigma_{n}+n(n+\alpha)\right)\bigg{)}^{2}-\bigg{(}\sum_{k=1}^{m}\sum_{j=1}^{m}t_{j}\cdot\partial_{t_{k}t_{j}}^{2}\sigma_{n}\bigg{)}^{2}$
(1.14)
$\displaystyle=4\left(\delta\sigma_{n}-\sigma_{n}+n(n+\alpha)\right)\bigg{(}n+\alpha+\sum\limits_{k=1}^{m}\partial_{t_{k}}\sigma_{n}\bigg{)}\bigg{(}n+\sum\limits_{k=1}^{m}\partial_{t_{k}}\sigma_{n}\bigg{)},$

where ${\rm sgn}(\omega_{k})$ is the sign function of $\omega_{k}$, which is
$1$ for $\omega_{k}>0$, $-1$ for $\omega_{k}<0$ and $0$ for $\omega_{k}=0$.
Here

$\displaystyle\Delta_{k}(\vec{t}):=$
$\displaystyle\bigg{(}\sum_{j=1}^{m}t_{j}\cdot\partial_{t_{k}t_{j}}^{2}\sigma_{n}\bigg{)}^{2}+4\left(\delta\sigma_{n}-\sigma_{n}+n(n+\alpha)\right)\left(\partial_{t_{k}}\sigma_{n}\right)^{2}.$

######  Remark 1.

Supposing $\vec{t}=\eta(b_{1},\cdots,b_{m})=:\eta\vec{b}$,
$\eta\in\mathbb{R}$, we have $\delta=\eta\frac{d}{d\eta}$ and (1.12) is
reduced to a second order ODE satisfied by $R_{n,k}$, reading

$\displaystyle\eta^{2}R_{n,k}^{\prime\prime}+\eta\bigg{(}\eta\Big{(}b_{k}-\sum_{j\neq
k}b_{j}R_{n,j}-2b_{k}R_{n,k}\Big{)}-2n+1-\alpha\bigg{)}R_{n,k}^{\prime}-\eta^{2}R_{n,k}\sum_{j\neq
k}b_{j}R_{n,j}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\frac{\bigg{(}2(n+\alpha)+\sum\limits_{j=1}^{m}2r_{n,j}\bigg{)}\bigg{(}2n+\sum\limits_{j=1}^{m}2r_{n,j}\bigg{)}}{2\bigg{(}\sum\limits_{j=1}^{m}R_{n,j}-1\bigg{)}}-\sum\limits_{j=1}^{m}\frac{(2r_{n,j})^{2}}{2R_{n,j}}+\eta\sum_{j=1}^{m}b_{j}R_{n,j}-\eta
b_{k}\right)R_{n,k}+\frac{(2r_{n,k})^{2}}{2R_{n,k}},$

for $k=1,\cdots,m$, where

$2r_{n,k}=\eta R_{n,k}^{\prime}-\Bigg{(}\eta\sum_{j=1}^{m}b_{j}R_{n,j}-\eta
b_{k}+2n+\alpha\Bigg{)}R_{n,k}.$

######  Remark 2.

When $m=1$, with $t_{1}$ replaced by $t$ and $R_{n,1}$ by $R_{n}$, equation
(1.12) is reduced to

$\displaystyle R_{n}^{\prime\prime}=$
$\displaystyle\frac{2R_{n}-1}{2R_{n}(R_{n}-1)}\left(R_{n}^{\prime}\right)^{2}-\frac{R_{n}^{\prime}}{t}+R_{n}^{3}+\left(\frac{2n+1+\alpha}{t}-\frac{3}{2}\right)R_{n}^{2}$
(1.15)
$\displaystyle+\left(-\frac{2n+1+\alpha}{t}+\frac{1}{2}\right)R_{n}-\frac{\alpha^{2}}{2t^{2}}\cdot\frac{R_{n}}{R_{n}-1},$

which can be transformed into
$P_{V}\left(0,-\alpha^{2}/2,2n+1+\alpha,-1/2\right)$ satisfied by
$1-1/R_{n}(t)$. Putting $m=1$ in (1.14), we get

$\displaystyle(t\sigma_{n}^{\prime\prime})^{2}=\left((2n+\alpha-t)\sigma_{n}^{\prime}+\sigma_{n}\right)^{2}-4\left(t\sigma_{n}^{\prime}-\sigma_{n}+n(n+\alpha)\right)(\sigma_{n}^{\prime})^{2},$
(1.16)

which is the Jimbo-Miwa-Okamoto (J-M-O for short) $\sigma$-form of the above-
mentioned $P_{V}$. Equations (1.15) and (1.16) agree with (4.8) and (4.11) of
[3], respectively.

With these discussions, we may treat (1.12) with $R_{n,k}$ replaced by
$1/(1-R_{n,k})$ and (1.14) as an $m$-variable generalization of $P_{V}$ and
the J-M-O $\sigma$-form of $P_{V}$, respectively.

Denote by $H_{V}$ the Hamiltonian for
$P_{V}\left(\frac{n^{2}}{2},-\frac{(n+\alpha)^{2}}{2},\alpha+1,-\frac{1}{2}\right)$,
i.e.

$tH_{V}(t,u,v)=u^{2}v(v-1)^{2}+tuv+(n+\alpha)u(v-1)^{2}-\alpha uv(v-1).$
(1.17)

See [26, equations (0.4) and (0.5) ] and [16, equations (2.1)–(2.5)]. By
establishing the RH problem for $P_{n}(x;\vec{t})$ and with the aid of Lax
pair, we show that $\sigma_{n}(\vec{t})$ can be expressed in terms of a
coupled $P_{V}$ system.

######  Theorem 1.2.

$\sigma_{n}(\vec{t})$ is connected with the Hamiltonian (1.17) by

$\sigma_{n}(\vec{t})=\sum_{k=1}^{m}t_{k}H_{V}(t_{k},u_{k},v_{k})+\frac{1}{2}\sum_{j\neq
k}u_{j}u_{k}(v_{j}+v_{k})(v_{j}-1)(v_{k}-1),$ (1.18)

where $u_{k}(\vec{t})$ and $v_{k}(\vec{t}),k=1,\cdots,m$, satisfy the coupled
$P_{V}$ system

$\left\\{\begin{array}[]{l}\delta
u_{k}=-t_{k}u_{k}-u_{k}\sum\limits_{j=1}^{m}u_{j}(v_{j}-1)(2v_{k}+v_{j}-1)-2nu_{k}v_{k}+(2n+\alpha)u_{k},\\\
\delta
v_{k}=t_{k}v_{k}+(v_{k}-1)\sum\limits_{j=1}^{m}u_{j}(v_{j}-1)(v_{k}+v_{j})+(n+\alpha)(v_{k}-1)^{2}-\alpha
v_{k}(v_{k}-1),\end{array}\right.$ (1.19)

which is equivalent to the following Hamiltonian formulation

$\delta v_{k}=\frac{\partial\sigma_{n}}{\partial u_{k}},\qquad\qquad\delta
u_{k}=-\frac{\partial\sigma_{n}}{\partial v_{k}},\qquad\qquad k=1,\cdots,m.$
(1.20)

######  Remark 3.

Setting $m=1$ in (1.19) and replacing $t_{1}$ by $t$, we have

$\Bigg{\\{}\begin{array}[]{l}tu_{1}^{\prime}=-tu_{1}-u_{1}^{2}(v_{1}-1)(3v_{1}-1)-2nu_{1}v_{1}+(2n+\alpha)u_{1},\\\
tv_{1}^{\prime}=tv_{1}+2u_{1}v_{1}(v_{1}-1)^{2}+\frac{\alpha}{2}(1-v_{1}^{2})+(n+\frac{\alpha}{2})(v_{1}-1)^{2}.\end{array}\Bigg{.}$

Solving $u_{1}$ from the second equation and substituting it into the first
one, we obtain

$\displaystyle v_{1}^{\prime\prime}=$
$\displaystyle\left(\frac{1}{2v_{1}}+\frac{1}{v_{1}-1}\right)(v_{1}^{\prime})^{2}-\frac{v_{1}^{\prime}}{t}+\frac{(v_{1}-1)^{2}}{t^{2}}\left(\frac{n^{2}}{2}v_{1}-\frac{(n+\alpha)^{2}}{2v_{1}}\right)$
$\displaystyle-(\alpha+1)\frac{v_{1}}{t}-\frac{1}{2}\frac{v_{1}(v_{1}+1)}{v_{1}-1},$

which is
$P_{V}\left(\frac{n^{2}}{2},-\frac{(n+\alpha)^{2}}{2},\alpha+1,-\frac{1}{2}\right)$.

Combining the results obtained by the ladder operator approach and by solving
the RH problem for $P_{n}(x;\vec{t})$, we build the relationship between
$\\{R_{n,k},r_{n,k}\\}$ which satisfy the $m$-variable generalization of
$P_{V}$ and $\\{u_{k},v_{k}\\}$ which are solutions of the coupled $P_{V}$
system. In addition, we obtain expressions for a few quantities allied with
$P_{n}(x;\vec{t})$ in terms of both $\\{R_{n,k},r_{n,k}\\}$ and
$\\{u_{k},v_{k}\\}$, including the recurrence coefficients
$\alpha_{n}(\vec{t})$ and $\beta_{n}(\vec{t})$, the $L^{2}$ norm of the monic
orthogonal polynomial and its derivative with respect to $t_{k}$. To continue,
we define $b_{1}(\vec{t})$ and $b_{2}(\vec{t})$ by

$\displaystyle b_{1}(\vec{t}):=$
$\displaystyle-\sum\limits_{k=1}^{m}u_{k}(v_{k}-1)-n,$ (1.21a) $\displaystyle
b_{2}(\vec{t}):=$
$\displaystyle-\sum\limits_{k=1}^{m}u_{k}v_{k}(v_{k}-1)+n+\alpha.$ (1.21b)

######  Theorem 1.3.

For $k=1,\cdots,m$, $R_{n,k}(\vec{t})$ and $r_{n,k}(\vec{t})$ defined in (1.9)
and (1.10) are connected with $u_{k}(\vec{t})$ and $v_{k}(\vec{t})$ by

$\displaystyle R_{n,k}=$ $\displaystyle\frac{u_{k}}{b_{1}},$ (1.22a)
$\displaystyle r_{n,k}=$ $\displaystyle u_{k}v_{k},$ (1.22b)

or equivalently,

$\displaystyle u_{k}=$ $\displaystyle
R_{n,k}\cdot\dfrac{\sum\limits_{j=1}^{m}r_{n,j}+n}{\sum\limits_{j=1}^{m}R_{n,j}-1},$
(1.23a) $\displaystyle v_{k}=$
$\displaystyle\dfrac{r_{n,k}}{R_{n,k}}\cdot\frac{\sum\limits_{j=1}^{m}R_{n,j}-1}{\sum\limits_{j=1}^{m}r_{n,j}+n}.$
(1.23b)

And we have the following expressions

$\displaystyle\alpha_{n}(\vec{t})=$
$\displaystyle\sum\limits_{k=1}^{m}t_{k}R_{n,k}+2n+1+\alpha$ (1.24a)
$\displaystyle=$
$\displaystyle\frac{1}{b_{1}}\sum\limits_{k=1}^{m}t_{k}u_{k}+2n+1+\alpha,$
(1.24b) $\displaystyle\beta_{n}(\vec{t})=$
$\displaystyle\dfrac{\left(n+\alpha+\sum\limits_{k=1}^{m}r_{n,k}\right)\cdot\left(n+\sum\limits_{k=1}^{m}r_{n,k}\right)}{1-\sum\limits_{k=1}^{m}R_{n,k}}+\sum\limits_{k=1}^{m}\frac{r_{n,k}^{2}}{R_{n,k}}$
(1.25a) $\displaystyle=$ $\displaystyle-b_{1}b_{2},$ (1.25b) $\displaystyle
h_{n}(\vec{t})=$ $\displaystyle-2\pi it_{1}^{2n+\alpha}b_{1}y,\qquad\qquad$
(1.26) $\displaystyle\partial_{t_{k}}\ln h_{n}(\vec{t})=$ $\displaystyle-
R_{n,k}=-\frac{u_{k}}{b_{1}},\qquad\qquad$ (1.27)

where $y(\vec{t})$ is connected with $u_{k}(\vec{t})$ and $v_{k}(\vec{t})$ by
$\delta\ln y=\sum\limits_{k=1}^{m}u_{k}(v_{k}-1)^{2}-2n-\alpha.$

######  Remark 4.

From the coupled $P_{V}$ system (1.19), we can derive the Riccati equations
(1.11) which give rise to the second order PDEs satisfied by $R_{n,k}$. In the
following derivation, we always bear (1.22) in mind.

Indeed, differentiating both sides of (1.22a), we have

$\displaystyle\delta R_{n,k}=$ $\displaystyle\frac{\delta
u_{k}}{b_{1}}-\frac{u_{k}\cdot\delta b_{1}}{b_{1}^{2}}.$ (1.28)

Taking the derivative of both sides of (1.21a), and using (1.19) to replace
$\delta u_{k}$ and $\delta v_{k}$ in the resulting expression, we find

$\displaystyle\delta
b_{1}=-\sum_{k=1}^{m}t_{k}u_{k}-b_{1}\sum_{k=1}^{m}u_{k}(v_{k}-1)^{2}.$

Inserting it and $\delta u_{k}$ given by the first equation of (1.19) into
(1.28), we come to the Riccati equation (1.11a).

To derive (1.11b), we apply $\delta$ to (1.22b) and get in view of (1.19)

$\displaystyle\delta r_{n,k}=$ $\displaystyle\delta u_{k}\cdot
v_{k}+u_{k}\cdot\delta v_{k}=\frac{r_{n,k}^{2}}{R_{n,k}}+b_{1}b_{2}R_{n,k}.$
(1.29)

From the definitions of $b_{1}$ and $b_{2}$, it follows that

$\displaystyle b_{1}b_{2}=$
$\displaystyle\dfrac{\left(n+\alpha+\sum\limits_{k=1}^{m}r_{n,k}\right)\cdot\left(n+\sum\limits_{k=1}^{m}r_{n,k}\right)}{\sum\limits_{k=1}^{m}R_{n,k}-1}-\sum_{k=1}^{m}\frac{r_{n,k}^{2}}{R_{n,k}}.$
(1.30)

Combining it with (1.29) gives us (1.11b).

####  1.1.2 Double scaling analysis at the hard edge: generalized $P_{III}$

We consider the hard edge scaling limit of the Hankel determinant
$D_{n}(\vec{t})$ under the assumption that $n\rightarrow\infty$ and
$t_{k}\rightarrow 0^{+}$ such that

$\displaystyle s_{k}=4nt_{k},\qquad k=1,2,\cdots,m,$

are fixed. This scaling is motivated by the following fact: if $t_{j}=0$ for
$j=1,\cdots,m-1$ while $t_{m}\neq 0$, then our weight function has only one
jump, i.e. $x^{\alpha}{\rm
e}^{-x}\left(\sum\limits_{k=0}^{m-1}\omega_{k}+\omega_{m}\theta(x-t_{m})\right)$,
and its associated Hankel determinant with the special cases
$\sum\limits_{k=0}^{m-1}\omega_{k}=0,\omega_{m}=1$ and
$\sum\limits_{k=0}^{m-1}\omega_{k}=1,\omega_{m}=-1$ correspond to the smallest
and largest eigenvalue distribution of LUE respectively, both of which under
the scaling that $s_{m}=4nt_{m}$ tend to the Fredholm determinant of the
Bessel kernel.

Assuming

$\vec{t}=\frac{1}{4n}\vec{s},\quad\vec{s}=(s_{1},s_{2},\cdots,s_{m}),$ (1.31)

with $0<s_{1}<\cdots<s_{m}$, we have

$\delta=\sum_{k=1}^{m}t_{k}\partial_{t_{k}}=\sum_{k=1}^{m}s_{k}\partial_{s_{k}}.$

Define

$\sigma(\vec{s}):=\lim\limits_{n\rightarrow\infty}\sigma_{n}\left(\frac{\vec{s}}{4n}\right)=\delta\ln\lim\limits_{n\rightarrow\infty}D_{n}\left(\frac{\vec{s}}{4n}\right).$
(1.32)

By using the finite-$n$ results obtained via the ladder operator approach, we
build direct relationships between the scaled $R_{n,k},r_{n,k}$ and
$\sigma(\vec{s})$. With them and the results presented in Theorem 1.1, we
establish the second order PDE satisfied by $\sigma(\vec{s})$.

######  Proposition 1.4.

Under the assumption (1.31), the uxiliary quantities $R_{n,k}(\vec{t})$ and
$r_{n,k}(\vec{t})$ are scaled as below:

$\displaystyle-\lim\limits_{n\rightarrow\infty}\frac{r_{n,k}\left(\frac{\vec{s}}{4n}\right)}{n}=$
$\displaystyle-4\partial_{s_{k}}\sigma(\vec{s})$ (1.33) $\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}R_{n,k}\left(\frac{\vec{s}}{4n}\right)=:R_{k}(\vec{s}).$
(1.34)

######  Theorem 1.5.

Under the assumption (1.31), the function $\sigma(\vec{s})$ specified by
(1.32) can be expressed in terms of $R_{k}(\vec{s})$ and $\delta
R_{k}(\vec{s})$ by

$\displaystyle 4\sigma=\frac{\left(\sum\limits_{k=1}^{m}\delta
R_{k}\right)^{2}-\alpha^{2}}{\sum\limits_{k=1}^{m}R_{k}-1}-\sum_{k=1}^{m}\frac{\left(\delta
R_{k}\right)^{2}}{R_{k}}-\sum_{k=1}^{m}s_{k}R_{k}-\alpha^{2},$ (1.35)

and $\\{R_{k}(\vec{s})\\}$ satisfy the following system of PDEs

$\displaystyle
2\left(\delta^{2}-\alpha\delta\right)R_{k}+\left(1-\frac{1}{R_{k}}\right)\left(\delta
R_{k}-\alpha R_{k}\right)^{2}+R_{k}\cdot\sum_{j\neq k}\frac{\left(\delta
R_{j}-\alpha R_{j}\right)^{2}}{R_{j}}$ (1.36)
$\displaystyle+\frac{R_{k}}{1-\sum\limits_{j=1}^{m}{R_{j}}}\cdot\sum_{j=1}^{m}\left(\delta
R_{j}-\alpha R_{j}\right)\cdot\bigg{(}\sum_{j=1}^{m}\left(\delta R_{j}-\alpha
R_{j}\right)+2\alpha\bigg{)}$
$\displaystyle+R_{k}\bigg{(}s_{k}-\sum_{j=1}^{m}s_{j}R_{j}\bigg{)}=0,\qquad\qquad
k=1,\cdots,m,$

where

$\displaystyle\delta^{2}=$
$\displaystyle\sum_{k=1}^{m}s_{k}^{2}\partial_{s_{k}s_{k}}^{2}+2\sum_{1\leq
j<k\leq m}s_{j}s_{k}\partial_{s_{j}s_{k}}^{2}+\delta.$

Moreover, $\sigma(\vec{s})$ satisfies the following PDE

$\displaystyle\frac{-4\left[\sum\limits_{k=1}^{m}s_{k}\partial_{s_{k}s_{k}}^{2}\sigma+\sum\limits_{1\leq
j<k\leq
m}(s_{j}+s_{k})\partial_{s_{j}s_{k}}^{2}\sigma\right]^{2}+\frac{\alpha^{2}}{4}}{4\sum\limits_{k=1}^{m}\partial_{s_{k}}\sigma+1}$
(1.37)
$\displaystyle+\sum_{k=1}^{m}\bigg{[}\frac{1}{\partial_{s_{k}}\sigma}\bigg{(}\sum\limits_{j=1}^{m}s_{j}\partial_{s_{j}s_{k}}^{2}\sigma\bigg{)}^{2}\bigg{]}$
$\displaystyle+\delta\sigma-\sigma-\frac{\alpha^{2}}{4}=0.$

######  Remark 5.

When $m=1$ and with $s_{1}$ replaced by $s$, expression (1.35) becomes

$\displaystyle
4\sigma=\frac{\left(sR_{1}^{\prime}\right)^{2}}{R_{1}(R_{1}-1)}-\frac{\alpha^{2}R_{1}}{R_{1}-1}-sR_{1}.$
(1.38)

Equations (1.36) are reduced to

$\displaystyle
R_{1}^{\prime\prime}=\frac{(2R_{1}-1)(R_{1}^{\prime})^{2}}{2(R_{1}-1)R_{1}}-\frac{R_{1}^{\prime}}{s}-\frac{\alpha^{2}R_{1}}{2s^{2}(R_{1}-1)}+\frac{R_{1}}{2s}(R_{1}-1),$
(1.39)

which can be transformed into $P_{V}\left(0,-\alpha^{2}/2,1,0\right)$
satisfied by

$Y(z):=1-1/R_{1}(s),\qquad\qquad\text{~{}with~{}}z=\frac{s}{2}.$

According to Theorem 3.9 of [1], this $P_{V}$ can be converted into
$P_{III}(-2(\sqrt{\alpha^{2}}+1),2\sqrt{\alpha^{2}},1,-1)$ satisfied by

$\nu(x)=\frac{\sqrt{2z}Y}{z\frac{d}{dz}Y+\sqrt{\alpha^{2}}Y-\sqrt{\alpha^{2}}}=\frac{\sqrt{s}R_{1}(R_{1}-1)}{s\frac{d}{ds}R_{1}-\sqrt{\alpha^{2}}R_{1}},\qquad\qquad
x^{2}=2z=s.$

In this sense, (1.36) can be viewed as an $m$-variable generalization of
$P_{III}$. To continue, (1.37) now reads

$\displaystyle(s\sigma^{\prime\prime})^{2}=$
$\displaystyle\sigma^{\prime}(4\sigma^{\prime}+1)(\sigma-s\sigma^{\prime})+\alpha^{2}(\sigma^{\prime})^{2},$
(1.40)

which is identified to be the J-M-O $\sigma$ form of $P_{III}$ satisfied by
$\sigma(-4\zeta)$ with $\zeta=-s/4$ (see [18, formula (3.13)] with the
parameters $\theta_{0}=\alpha,~{}\theta_{\infty}=-\alpha$ therein). This
result coincides with the one in [3].

In particular, for $m=1$ with $\omega_{0}=0$ and $\omega_{1}=1$, we know that
$D_{n}(t_{1})$ represents the smallest eigenvalue distribution of LUE up to a
constant term. The double scaling analysis of this case was carried out by
Tracy and Widom [28] where an auxiliary quantity $q(s)$ was introduced. By
combining equation (2.26) and the one below it in [28] with the definition of
$\sigma(s)$ and (1.38) in our paper, we find that $\sigma(s),R(s)$ and
$q^{2}(s)$ in [28] correspond to our $-\sigma(s),-\sigma(s)/s$ and $R_{1}(s)$
respectively. We now check the consistency between their results and our
equations (1.38)–(1.40). In fact, combining (1.20) of [28] with
$q(s)=cos\psi(s)$ therein, we obtain an expression that agrees with (1.38).
Substituting $R_{1}(s)=q^{2}(s)$ into (1.39), we are led to (1.16) of [28].
(1.21) of [28] is readily seen to be in accord with (1.40). The last thing to
remark is that, as was pointed out in [28], the quantity

$y(x):=\frac{q(s)-1}{q(s)+1},\qquad\qquad\text{~{}with~{}}x=\sqrt{s},$

satisfies $P_{V}\left(\alpha^{2}/8,-\alpha^{2}/8,0,-2\right)$.

To continue, we let $s_{i}=\nu_{i}\tau,\tau>0$. Then

$\vec{t}=(t_{1},t_{2},\cdots,t_{m})=\frac{\vec{s}}{4n}=\frac{\tau}{4n}\vec{\nu},\qquad\vec{\nu}=(\nu_{1},\cdots,\nu_{m}),$

and we now have only one scaled variable $\tau$. Supposing
$c_{k}=\sum\limits_{j=0}^{k-1}\omega_{j}/\omega,k=1,\cdots,m$, with
$\omega_{k}$ positive and $\omega=\sum\limits_{k=0}^{m}\omega_{k}$ strictly
positive, we find

$\displaystyle\frac{D_{n}(\vec{t})}{D_{n}(\vec{0})}$
$\displaystyle=\frac{\det\left(\displaystyle{\int_{0}^{+\infty}\bigg{(}1-\sum\limits_{k=1}^{m}(1-c_{k})\chi_{(t_{k-1},t_{k})}(x)\bigg{)}x^{i+j}x^{\alpha}{{\rm
e}}^{-x}dx}\right)_{i,j=0}^{n-1}}{\det\left(\displaystyle{\int}_{0}^{+\infty}x^{i+j}x^{\alpha}{{\rm
e}}^{-x}dx\right)_{i,j=0}^{n-1}}$
$\displaystyle=\det\big{(}I-\chi_{(0,t_{m})}\sum_{k=1}^{m}(1-c_{k})K_{n}\chi_{(t_{k-1},t_{k})}\big{)},$

where $t_{0}=0$ and

$\displaystyle
K_{n}(x,y)=(xy)^{\alpha/2}e^{-(x+y)}\sum_{j=0}^{n-1}L_{j}(x)L_{j}(y),$

with $L_{j}(x)$ being the normalized Laguerre polynomial of degree $j$. It is
known that for bounded $x,y\in(0,+\infty)$

$\lim_{n\to\infty}\frac{1}{4n}K_{n}\left(\frac{x}{4n},\frac{y}{4n}\right)=K_{Bes}(x,y),$

where $K_{\text{Bes}}(x,y)$ is the Bessel kernel

$K_{\text{Bes}}(x,y)=\frac{J_{\alpha}(\sqrt{x})\sqrt{y}J_{\alpha}^{\prime}(\sqrt{y})-\sqrt{x}J_{\alpha}^{\prime}(\sqrt{x})J_{\alpha}(\sqrt{y})}{2(x-y)},$

with $J_{\alpha}$ denoting the Bessel function of the first kind of order
$\alpha$. This limit implies

$\lim_{n\to\infty}\frac{D_{n}(\vec{t})}{D_{n}(\vec{0})}=\det\bigg{(}I-\chi_{(0,\nu_{m}\tau)}\sum_{j=1}^{m}(1-c_{j})K_{\text{Bes}}\chi_{(\nu_{j-1}\tau,\nu_{j}\tau)}\bigg{)}.$

According to the results presented in Theorem 1.5, we obtain the integral
representation of
$\lim\limits_{n\to\infty}\frac{D_{n}(\vec{t})}{D_{n}(\vec{0})}$, or
equivalently, the determinant of the Bessel kernel, in terms of
$R_{k}(\vec{s})$.

######  Corollary 1.6.

For $\vec{t}=\frac{\vec{s}}{4n}=\frac{\tau}{4n}\vec{\nu}$,
$\vec{\nu}=(\nu_{1},\cdots,\nu_{m})$, define

$\displaystyle\hat{R}_{k}(\tau):=R_{k}(\vec{s})=\lim\limits_{n\rightarrow\infty}R_{n,k}\left(\vec{t}\right).$
(1.41)

We have the following asymptotics

$\lim_{n\to\infty}\frac{D_{n}(\vec{t})}{D_{n}(\vec{0})}=\exp\bigg{(}-\frac{1}{4}\int_{0}^{\tau}\log\left(\frac{\tau}{\xi}\right)\sum\limits_{k=1}^{m}\nu_{k}\hat{R}_{k}(\xi)d\xi\bigg{)},$
(1.42)

where $\\{\hat{R}_{k}(\tau)\\}$ satisfy the following coupled ODEs:

$\displaystyle
2\tau^{2}\hat{R}_{k}^{\prime\prime}+2(1-\alpha)\tau\hat{R}_{k}^{\prime}+\left(1-\frac{1}{\hat{R}_{k}}\right)(\tau\hat{R}_{k}^{\prime}-\alpha\hat{R}_{k})^{2}+\hat{R}_{k}\sum_{j\neq
k}\frac{(\tau\hat{R}_{j}^{\prime}-\alpha\hat{R}_{j})^{2}}{\hat{R}_{j}}$
$\displaystyle+\frac{\hat{R}_{k}}{1-\sum\limits_{j=1}^{m}\hat{R}_{j}}\cdot\sum_{j=1}^{m}(\tau\hat{R}_{j}^{\prime}-\alpha\hat{R}_{j})\cdot\Bigg{(}\sum_{j=1}^{m}(\tau\hat{R}_{j}^{\prime}-\hat{R}_{j})+2\alpha\Bigg{)}+\tau\hat{R}_{k}(\nu_{k}-\sum_{j=1}^{m}\nu_{j}\hat{R}_{j})=0,$

for $k=1,\cdots,m$. Setting

$\hat{R}_{k}(\tau)=q_{k}^{2}(\tau),\qquad\qquad k=1,\cdots,m,$

we find that $q_{1}(\tau),\cdots,q_{m}(\tau)$ satisfy the differential
equations

$\tau q_{k}(1-\sum_{j=1}^{m}q_{j}^{2})\sum_{j=1}^{m}(\tau
q_{j}q_{j}^{\prime})^{\prime}+\tau\Bigg{(}1-\sum_{j=1}^{m}q_{j}^{2}\Bigg{)}^{2}((\tau
q^{\prime}_{k})^{\prime}+\frac{a_{k}q_{k}}{4})+\tau^{2}q_{k}\Bigg{(}\sum_{j=1}^{m}q_{j}q^{\prime}_{j}\Bigg{)}^{2}-\frac{\alpha^{2}}{4}q_{k}=0,$
(1.43)

for $k=1,\cdots,m$.

######  Remark 6.

Equation (1.43) is identical with (1.12) of [7] where the asymptotics of
$q_{1}(\tau),\cdots,q_{m}(\tau)$ as $\tau\to 0$ were provided:

$q_{k}(\tau)=\sqrt{c_{k+1}-c_{k}}J_{\alpha}(\sqrt{\nu_{k}\tau})(1+O(\tau)),$
(1.44)

where $c_{k}=\sum\limits_{j=0}^{k-1}\omega_{k}/\omega,k=1,\cdots,m$,
$c_{m+1}=1$ and $J_{\alpha}$ is the Bessel function of order $\alpha$. As
pointed out in [7], when $m=1$ and $a_{1}=1$, the equation (1.43) is reduced
to the $P_{V}$ equation. Corollary 1.6, together with (1.44), generalizes a
result of Tracy and Widom [28] for the determinant of the classical Bessel
kernel, which corresponds to the case $m=1$. See also [5, 7] for other
properties of the Bessel kernel determinants with discontinuities and further
applications.

###  1.2 Outline of the paper

The rest of the paper is organized as follows. In Section 2, we employ the
ladder operator approach to express the recurrence coefficients in terms of
$2m$ auxiliary quantities which satisfy a system of difference equations that
can be iterated. We also show that the recurrence coefficients satisfy Toda
equations and finally give the proof of Theorem 1.1. In Section 3, by
introducing the RH problem for the orthogonal polynomials and with the aid of
Lax pair, we establish the coupled $P_{V}$ system presented in Theorem 1.2,
and obtain the relations and expressions given in Theorem 1.3. Section 4 is
devoted to the double scaling analysis of the Hankel determinant, and the
proof of Proposition 1.4, Theorem 1.5 and Corollary 1.6.

##  2 Finite Dimension Analysis – Ladder Operator Approach

Write

$\displaystyle w_{0}(x):=$ $\displaystyle x^{\alpha}{\rm e}^{-x}=:{\rm
e}^{-{\rm v}_{0}(x)},$ $\displaystyle w_{J}(x;\vec{t}):=$
$\displaystyle\omega_{0}+\sum_{k=1}^{m}\omega_{k}\theta(x-t_{k}).$

Then the weight function (1.3) reads

$w(x;\vec{t}\,)=x^{\alpha}{\rm
e}^{-x}\left(\omega_{0}+\sum_{k=1}^{m}\omega_{k}\theta(x-t_{k})\right)=w_{0}(x)w_{J}(x;\vec{t}),\quad
x\in[0,\infty).$

Using the properties of orthogonal polynomials, as was pointed out in Lemma 1
and Remark 1 of [3], one can show that $\\{P_{n}(x;\vec{t}\,)\\}$ satisfy a
pair of ladder operators. For simplicity of presentation, in what follows, we
shall not display the dependence of $\vec{t}$ when not necessary.

######  Lemma 2.1.

The monic orthogonal polynomials $P_{n}(z):=P_{n}(z;\vec{t}\,)$ satisfy the
following lowering and raising operator

$\displaystyle P_{n}^{\prime}(z)=$
$\displaystyle\beta_{n}A_{n}(z)P_{n-1}(z)-B_{n}(z)P_{n}(z),$ $\displaystyle
P_{n-1}^{\prime}(z)=$ $\displaystyle\left(B_{n}(z)+{\rm
v}_{0}^{\prime}(z)\right)P_{n-1}(z)-A_{n-1}(z)P_{n}(z),$

where ${\rm v}_{0}(z)=z-\alpha\ln z$, and $A_{n}(z)$ and $B_{n}(z)$ are given
by

$\displaystyle A_{n}(z):=$
$\displaystyle\frac{1}{h_{n}}\int_{0}^{\infty}\frac{{\rm
v}_{0}^{\prime}(z)-{\rm
v}_{0}^{\prime}(y)}{z-y}P_{n}^{2}(y)w(y;\vec{t}\,)dy+\sum_{k=1}^{m}\frac{R_{n,k}(t_{k};\vec{t}\,)}{z-t_{k}},$
(2.1a) $\displaystyle B_{n}(z):=$
$\displaystyle\frac{1}{h_{n-1}}\int_{0}^{\infty}\frac{{\rm
v}_{0}^{\prime}(z)-{\rm
v}_{0}^{\prime}(y)}{z-y}P_{n}(y)P_{n-1}(y)w(y;\vec{t}\,)dy+\sum_{k=1}^{m}\frac{r_{n,k}(t_{k};\vec{t})}{z-t_{k}}.$
(2.1b)

Here the auxiliary quantities $R_{n,k}(\vec{t})$ and
$r_{n,k}(\vec{t}),k=1,\cdots,m$, are defined by

$\displaystyle R_{n,k}(\vec{t}):=$
$\displaystyle\omega_{k}\frac{w_{0}(t_{k})}{h_{n}(\vec{t})}P_{n}^{2}(t_{k};\vec{t}),$
$\displaystyle r_{n,k}(\vec{t}):=$
$\displaystyle\omega_{k}\frac{w_{0}(t_{k})}{h_{n-1}(\vec{t})}P_{n}(t_{k};\vec{t})P_{n-1}(t_{k};\vec{t}).$

######  Remark 7.

When $-1<\alpha\leq 0$, the coefficients $A_{n}(z)$ and $B_{n}(z)$ in the
ladder operators should be modified to

$\displaystyle A_{n}(z)=$
$\displaystyle\frac{1}{z}\left(\frac{1}{h_{n}}\int_{0}^{\infty}\frac{z{\rm
v}_{0}^{\prime}(z)-y{\rm
v}_{0}^{\prime}(y)}{z-y}P_{n}^{2}(y)w(y;\vec{t}\,)dy-\sum\limits_{k=1}^{m}R_{n,k}\right)+\sum_{k=1}^{m}\frac{R_{n,k}}{z-t_{k}},$
(2.2a) $\displaystyle B_{n}(z)=$
$\displaystyle\frac{1}{z}\left(\frac{1}{h_{n-1}}\int_{0}^{\infty}\frac{z{\rm
v}_{0}^{\prime}(z)-y{\rm
v}_{0}^{\prime}(y)}{z-y}P_{n}(y)P_{n-1}(y)w(y;\vec{t}\,)dy-\sum_{k=1}^{m}r_{n,k}-n\right)+\sum_{k=1}^{m}\frac{r_{n,k}}{z-t_{k}}.$
(2.2b)

They can be derived by using the framework presented in Section 4.3 of [1] and
[9]. It should be pointed out that (2.2) are also well-defined for $\alpha>0$
and in this case they are equivalent to (2.1).

With the three-term recurrence relation (1.6) and the above ladder operators,
we can derive the following two compatibility conditions for $A_{n}(z)$ and
$B_{n}(z)$.

######  Lemma 2.2.

$A_{n}(z)$ and $B_{n}(z)$ defined in (2.1a)-(2.1b) satisfy the following two
identities

$\displaystyle B_{n+1}(z)+B_{n}(z)=$
$\displaystyle\left(z-\alpha_{n}\right)A_{n}(z)-{\rm v}_{0}^{\prime}(z),$
($S_{1}$) $\displaystyle
1+\left(z-\alpha_{n}\right)\left(B_{n+1}(z)-B_{n}(z)\right)=$
$\displaystyle\beta_{n+1}A_{n+1}(z)-\beta_{n}A_{n-1}(z).$ ($S_{2}$)

Multiplying both sides of $(S_{2})$ by $A_{n}(z)$ and eliminating
$(z-\alpha_{n})A_{n}(z)$ in the resulting expression by using $(S_{1})$, we
find

$A_{n}(z)+B_{n+1}^{2}(z)-B_{n}^{2}(z)+{\rm
v}_{0}^{\prime}(z)\left(B_{n+1}(z)-B_{n}(z)\right)=\beta_{n+1}A_{n+1}(z)A_{n}(z)-\beta_{n}A_{n}(z)A_{n-1}(z).$

Replacing $n$ by $j$ in this equality and summing over $j$ from $0$ to $n-1$,
with the initial conditions $B_{0}(z)=A_{-1}(z)=0$, we obtain another identity
for $A_{n}(z)$ and $B_{n}(z)$.

######  Lemma 2.3.

We have

$\displaystyle B_{n}^{2}(z)+{\rm
v}_{0}^{\prime}(z)B_{n}(z)+\sum_{j=0}^{n-1}A_{j}(z)=$
$\displaystyle\beta_{n}A_{n}(z)A_{n-1}(z).$ ($S_{2}^{\prime}$)

Substituting ${\rm v}_{0}(z)=z-\alpha\ln z$ into (2.1), with the aid of (1.4)
and (1.5), we obtain the expressions for $A_{n}(z)$ and $B_{n}(z)$ in terms of
$R_{n,k}(\vec{t})$ and $r_{n,k}(\vec{t})$.

######  Lemma 2.4.

$A_{n}(z)$ and $B_{n}(z)$ defined in (2.1a)-(2.2b) are rational functions in
$z$ with simple poles at $0,t_{1},\cdots,t_{m}$, reading

$\displaystyle A_{n}(z)=$
$\displaystyle\dfrac{1-\sum\limits_{k=1}^{m}R_{n,k}}{z}+\sum\limits_{k=1}^{m}\frac{R_{n,k}}{z-t_{k}},$
(2.3a) $\displaystyle B_{n}(z)=$
$\displaystyle-\frac{n+\sum\limits_{k=1}^{m}r_{n,k}}{z}+\sum_{k=1}^{m}\frac{r_{n,k}}{z-t_{k}}.$
(2.3b)

###### Proof.

We first compute $A_{n}(z)$ and $B_{n}(z)$ for the case $\alpha>0$. Since

$\frac{{\rm v}_{0}^{\prime}(z)-{\rm
v}_{0}^{\prime}(y)}{z-y}=\frac{\alpha}{yz},$

the integral in (2.1a) is given by

$\displaystyle\frac{1}{h_{n}}\int_{0}^{\infty}\frac{{\rm
v}_{0}^{\prime}(z)-{\rm
v}_{0}^{\prime}(y)}{z-y}P_{n}^{2}(y)w(y;\vec{t})dy=\frac{1}{z}\cdot\frac{1}{h_{n}}\int_{0}^{\infty}P_{n}^{2}(y){\rm
e}^{-y}w_{J}(x;\vec{t})dy^{\alpha}.$

Through integration by parts and noting that
$\left(d/dx\right)\theta(x-t)=\delta(x-t)$ with $\delta(\cdot)$ denoting the
Dirac delta function, we find

$\displaystyle\frac{1}{h_{n}}\int_{0}^{\infty}\frac{{\rm
v}_{0}^{\prime}(z)-{\rm v}_{0}^{\prime}(y)}{z-y}P_{n}^{2}(y)w(y;\vec{t})dy$
$\displaystyle=$
$\displaystyle\frac{1}{z}\cdot\frac{1}{h_{n}}\left[P_{n}^{2}(y)w(y;\vec{t})\Big{|}_{y=0}^{y=\infty}-\int_{0}^{\infty}2P_{n}(y)P_{n}^{\prime}(y)w(y;\vec{t})dy\right.$
$\displaystyle\left.-\int_{0}^{\infty}P_{n}^{2}(y)y^{\alpha}(-{\rm
e}^{-y})w_{J}(x;\vec{t})dy-\sum_{k=1}^{m}w_{k}\int_{0}^{\infty}P_{n}^{2}(y)y^{\alpha}{\rm
e}^{-y}\delta(y-t_{k})dy\right].$ (2.4)

Now we look at the four terms in the square bracket. Noting that
$w(0;\vec{t})=w(\infty;\vec{t})=0$, the first term is equal to zero. Since

$\displaystyle
P_{n}^{\prime}(y)=ny^{n-1}+\cdots=nP_{n-1}(y)+\sum_{k=0}^{n-2}d_{k}P_{k}(y),$

according to the orthogonality (1.4), we find that the second term is also
zero and the third one is $h_{n}$. Due to the following property of the Dirac
delta function

$\int_{0}^{\infty}f(x)\delta(x-t)dx=f(t),$

the last term is seen to be $h_{n}\sum\limits_{k=1}^{m}R_{n,k}$. Hence we
conclude that $\eqref{m-An1}$ equals $(1-\sum\limits_{k=1}^{m}R_{n,k})/z$.
Plugging it back into (2.1a), we get (2.3a). The integral in (2.1b) can be
computed similarly and thus we omit the derivation of (2.3b).

For $-1<\alpha\leq 0$, we make use of (2.2) to calculate $A_{n}(z)$ and
$B_{n}(z)$, which gives us (2.3) immediately. ∎

Inserting (2.3) into the three identities satisfied by $A_{n}(z)$ and
$B_{n}(z)$, namely, $(S_{1}),(S_{2}),(S_{2}^{\prime})$, by equating the
residues on their both sides, we arrive at a series of difference equations.

###  2.1 Difference equations for auxiliary quantities

From $(S_{1})$, we get

$\displaystyle z^{-1}:\qquad\qquad
2n+1+\sum_{k=1}^{m}\left(r_{n+1,k}+r_{n,k}\right)=$
$\displaystyle\alpha_{n}\left(1-\sum_{k=1}^{m}R_{n,k}\right)-\alpha,$ (2.5)
$\displaystyle(z-t_{k})^{-1}:\qquad\qquad\qquad\qquad\qquad
r_{n+1,k}+r_{n,k}=$ $\displaystyle(t_{k}-\alpha_{n})R_{n,k},\qquad
k=1,\cdots,m.\qquad$ (2.6)

From $(S_{2})$, we find

$\displaystyle\quad
z^{-1}:\qquad\alpha_{n}\left(1+\sum_{k=1}^{m}\left(r_{n+1,k}-r_{n,k}\right)\right)=$
$\displaystyle\beta_{n+1}\left(1-\sum_{k=1}^{m}R_{n+1,k}\right)-\beta_{n}\left(1-\sum_{k=1}^{m}R_{n-1,k}\right),$
(2.7)
$\displaystyle(z-t_{k})^{-1}:\qquad\left(t_{k}-\alpha_{n}\right)\left(r_{n+1,k}-r_{n,k}\right)=$
$\displaystyle\beta_{n+1}R_{n+1,k}-\beta_{n}R_{n-1,k},\qquad k=1,\cdots,m.$
(2.8)

From $(S_{2}^{\prime})$, we obtain

$\displaystyle\quad
z^{-2}:\quad\qquad\Bigg{(}n+\alpha+\sum_{k=1}^{m}r_{n,k}\Bigg{)}\left(n+\sum_{k=1}^{m}r_{n,k}\right)=\beta_{n}\left(1-\sum_{k=1}^{m}R_{n,k}\right)\left(1-\sum_{k=1}^{m}R_{n-1,k}\right),$
(2.9) $\displaystyle(z-t_{k})^{-2}:\qquad\qquad\qquad\qquad
r_{n,k}^{2}=\beta_{n}R_{n,k}R_{n-1,k},\qquad
k=1,\cdots,m.\qquad\qquad\qquad\quad$ (2.10)

Using (2.5), (2.6), (2.9) and (2.10), we can express $\alpha_{n}(\vec{t})$ and
$\beta_{n}(\vec{t})$ in terms of $\\{R_{n,k}(\vec{t})\\}$ and
$\\{r_{n,k}(\vec{t})\\}$ which are shown to satisfy a system of difference
equations.

######  Lemma 2.5.

The recurrence coefficients are expressed in terms of the auxiliary quantities
by

$\displaystyle\alpha_{n}(\vec{t})=$
$\displaystyle\sum_{k=1}^{m}t_{k}R_{n,k}+2n+1+\alpha,$ (2.11)
$\displaystyle\beta_{n}(\vec{t})=$
$\displaystyle\frac{\left(n+\alpha+\sum\limits_{k=1}^{m}r_{n,k}\right)\left(n+\sum\limits_{k=1}^{m}r_{n,k}\right)}{1-\sum\limits_{k=1}^{m}R_{n,k}}+\sum\limits_{k=1}^{m}\frac{r_{n,k}^{2}}{R_{n,k}}.$
(2.12)

###### Proof.

Summing (2.6) over $k=1,\cdots,m$, and combining the resulting equality with
(2.5), we are led to (2.11). Eliminating $R_{n-1,k}$ in (2.9) by using (2.10),
after simplification, we obtain (2.12). ∎

######  Lemma 2.6.

The quantities $\\{R_{n,k}(\vec{t})\\}$ and $\\{r_{n,k}(\vec{t})\\}$ satisfy
the following system of difference equations

$\displaystyle r_{n+1,k}=$
$\displaystyle\Bigg{(}t_{k}-\sum_{j=1}^{m}t_{j}R_{n,j}-2n-1-\alpha\Bigg{)}R_{n,k}-r_{n,k},\qquad
k=1,\cdots,m,$ (2.13) $\displaystyle\frac{1}{R_{n,1}}=$ $\displaystyle
1+\frac{R_{n-1,1}}{r_{n,1}^{2}}\cdot\left[\sum_{k=2}^{m}\frac{r_{n,k}^{2}}{R_{n-1,k}}+\frac{\left(n+\alpha+\sum\limits_{k=1}^{m}r_{n,k}\right)\left(n+\sum\limits_{k=1}^{m}r_{n,k}\right)}{1-\sum\limits_{k=1}^{m}R_{n-1,k}}\right],$
(2.14) $\displaystyle R_{n,k}=$
$\displaystyle\frac{R_{n-1,1}}{R_{n-1,k}}\cdot\frac{r_{n,k}^{2}}{r_{n,1}^{2}}\cdot
R_{n,1},\qquad k=2,\cdots,m,$ (2.15)

which can be iterated in $n$ with initial conditions

$\displaystyle R_{0,k}=\frac{\omega_{k}t_{k}^{\alpha}{\rm
e}^{-t_{k}}}{\int_{0}^{\infty}w(x;\vec{t})dx},\qquad\qquad r_{0,k}=0.$

###### Proof.

Plugging (2.11) into (2.6) leads to (2.13) immediately. With $k=1$ in (2.10),
we have

$\displaystyle\beta_{n}=\frac{r_{n,1}^{2}}{R_{n,1}R_{n-1,1}}.$ (2.16)

Substituting it into (2.10) with $k=2,\cdots,m$, we come to (2.15). Combining
(2.16) with (2.12), we get (2.14). ∎

To conclude this subsection, we present expressions for $p(n,\vec{t})$, the
second leading coefficient of degree-$n$ monic orthogonal polynomial
$P_{n}(x;\vec{t})$, involving $\beta_{n}$ and the auxiliary quantities. We
will see that these relations play an important role in the study of the log
derivative of the Hankel determinant.

######  Lemma 2.7.

We have

$\displaystyle p(n,\vec{t})=$
$\displaystyle\sum_{k=1}^{m}t_{k}r_{n,k}-\beta_{n}$ (2.17) $\displaystyle=$
$\displaystyle-\sum_{k=1}^{m}t_{k}\sum_{j=0}^{n-1}R_{j,k}-n(n+\alpha).$ (2.18)

###### Proof.

From (1.7), it follows that

$\displaystyle\sum_{j=0}^{n-1}\alpha_{j}(\vec{t})=-p(n,\vec{t}).$ (2.19)

This motivates us to derive (2.17) and (2.18) using the expressions involving
$\alpha_{n}$.

Summing (2.8) over $k=1,\cdots,m$, and adding the resulting equality to (2.7),
we get

$\displaystyle\alpha_{n}+\sum_{k=1}^{m}t_{k}(r_{n+1,k}-r_{n,k})=\beta_{n+1}-\beta_{n}.$
(2.20)

Replacing $n$ by $j$ and summing it from $j=0$ to $n-1$, noting that
$r_{0,k}=0=\beta_{0}$ and in view of (2.19), we obtain (2.17). Replacing $n$
by $j$ in (2.11) and summing it over $j$ from $0$ to $n-1$, we are led to
(2.18). ∎

###  2.2 Toda equations for recurrence coefficients

We proceed to develop differential relations by differentiating the
orthogonality relation (1.4) with $m=n$ and $m=n-1$. It turns out that the
derivatives of $\ln h_{n}(\vec{t})$ and $p(n,\vec{t})$ are directly related to
the auxiliary quantities.

######  Lemma 2.8.

We have

$\displaystyle\partial_{t_{k}}\ln h_{n}(\vec{t})=$ $\displaystyle-
R_{n,k}(\vec{t}),$ (2.21) $\displaystyle\partial_{t_{k}}p(n,\vec{t})=$
$\displaystyle r_{n,k}(\vec{t}),$ (2.22)

which, according to $\beta_{n}=h_{n}/h_{n-1}$ and
$\alpha_{n}=p(n,\vec{t})-p(n+1,\vec{t})$, gives us

$\displaystyle\partial_{t_{k}}\ln\beta_{n}(\vec{t})=$ $\displaystyle
R_{n-1,k}(\vec{t})-R_{n,k}(\vec{t}),$ (2.23)
$\displaystyle\partial_{t_{k}}\alpha_{n}(\vec{t})=$ $\displaystyle
r_{n,k}(\vec{t})-r_{n+1,k}(\vec{t}).$ (2.24)

###### Proof.

Differentiating both sides of

$h_{n}(\vec{t})=\int_{0}^{\infty}P_{n}^{2}(x;\vec{t})w(x;\vec{t})dx$

over $t_{k}$, we get

$\displaystyle\partial_{t_{k}}h_{n}(\vec{t})=\int_{0}^{\infty}2P_{n}(x;\vec{t})\cdot\left(\partial_{t_{k}}P_{n}(x;\vec{t})\right)w(x;\vec{t}))dx+\int_{0}^{\infty}P_{n}^{2}(x;\vec{t})\cdot\left(\partial_{t_{k}}w(x;\vec{t})\right)dx.$
(2.25)

Applying $\partial_{t_{k}}$ to

$P_{n}\left(x;\vec{t}\,\right)=x^{n}+p(n,\vec{t}\,)x^{n-1}+\cdots+P_{n}(0;\vec{t}\,),$

we have

$\displaystyle\partial_{t_{k}}P_{n}(x;\vec{t})=$
$\displaystyle\partial_{t_{k}}p(n,\vec{t}\,)\cdot
x^{n-1}+\\{\text{lower~{}degree~{}of~{}}x\\}$ $\displaystyle=$
$\displaystyle\partial_{t_{k}}p(n,\vec{t}\,)\cdot
P_{n-1}+\\{\text{linear~{}combination~{}of~{}}P_{i},i=n-2,n-3,\cdots,1,0\\}.$

This together with (1.4) indicates that the first integral in (2.25) is zero
and hence

$\displaystyle\partial_{t_{k}}h_{n}(\vec{t})=$
$\displaystyle\int_{0}^{\infty}P_{n}^{2}(x;\vec{t})\cdot\left(\partial_{t_{k}}w(x;\vec{t})\right)dx$
$\displaystyle=$ $\displaystyle-
w_{k}\int_{0}^{\infty}P_{n}^{2}(x;\vec{t})x^{\alpha}{\rm
e}^{-x}\delta(x-t_{k})dx$ $\displaystyle=$ $\displaystyle-
R_{n,k}(\vec{t})\cdot h_{n}(\vec{t}),$

which gives us (2.21).

Taking the derivative of

$0=\int_{0}^{\infty}P_{n}(x;\vec{t})P_{n-1}(x;\vec{t})dx$

with respect to $t_{k}$, via an argument similar to the above, we can derive
(2.22) . ∎

By using the differential relations (2.23) and (2.24), and with the aid of
identities and expressions involving $\alpha_{n}$ and $\beta_{n}$ which are
presented in the previous subsection, we establish Toda equations for
$\alpha_{n}$ and $\beta_{n}$.

######  Proposition 2.9.

The recurrence coefficients $\alpha_{n}$ and $\beta_{n}$ satisfy the following
Toda equations

$\delta\ln\beta_{n}=\alpha_{n-1}-\alpha_{n}+2,$ (2.26a)
$\left(\delta-1\right)\alpha_{n}=\beta_{n}-\beta_{n+1},$ (2.26b) where
$\delta=\sum\limits_{k=1}^{m}t_{k}\partial_{t_{k}}$.

###### Proof.

Multiplying both sides of (2.23) by $t_{k}$ yields

$t_{k}\,\partial_{t_{k}}\ln\beta_{n}=t_{k}\left(R_{n-1,k}-R_{n,k}\right).$

Summing this equation over $k$ from $1$ to $m$, we have

$\displaystyle\delta\ln\beta_{n}=$
$\displaystyle\sum_{k=1}^{m}t_{k}R_{n-1,k}-\sum_{k=1}^{m}t_{k}R_{n,k}.$

Using (2.11) to remove the two summation terms, we get (2.26a). Due to (2.24),
we replace $r_{n+1,k}-r_{n,k}$ by $-\partial_{t_{k}}\alpha_{n}$ in (2.20) and
obtain (2.26b). ∎

###  2.3 Proof of Theorem 1.1

(a) Derivation of the Riccati equations (1.11) and the coupled PDEs (1.12)
We observe the following facts:

$\displaystyle\partial_{t_{k}}R_{n,j}=$
$\displaystyle\partial_{t_{j}}R_{n,k},$ (2.27)
$\displaystyle\partial_{t_{k}}r_{n,j}=$
$\displaystyle\partial_{t_{j}}r_{n,k},$ (2.28)

for $j,k=1,\cdots,m$. The first equality can be seen by combining the fact
that $\partial_{t_{j}t_{k}}^{2}\ln h_{n}=\partial_{t_{k}t_{j}}^{2}\ln h_{n}$
with (2.21), while the second one is a consequence of
$\partial_{t_{j}t_{k}}^{2}p(n,\vec{t})=\partial_{t_{k}t_{j}}^{2}p(n,\vec{t})$
and (2.22).

Now we go ahead to derive the Riccati equations satisfied by $R_{n,k}$ and
$r_{n,k}$. Eliminating $r_{n+1,k}$ from (2.6) and (2.24), we get

$\displaystyle\partial_{t_{k}}\alpha_{n}=2r_{n,k}+(\alpha_{n}-t_{k})R_{n,k}.$

Inserting (2.11) into this equation, we find

$\displaystyle\sum_{j\neq
k}t_{j}\cdot\partial_{t_{k}}R_{n,j}+t_{k}\cdot\partial_{t_{k}}R_{n,k}=2r_{n,k}+\Bigg{(}\sum_{j=1}^{m}t_{j}R_{n,j}-t_{k}+2n+\alpha\Bigg{)}R_{n,k}.$

According to (2.27), we replace $\partial_{t_{k}}R_{n,j}$ by
$\partial_{t_{j}}R_{n,k}$ in the above equation, which leads to the first
Riccati equation (1.11a). To derive (1.11b), we make use of (2.10) and (2.23),
both of which are related to $R_{n-1,k}$ and $R_{n,k}$:

$\displaystyle r_{n,k}^{2}=$ $\displaystyle\beta_{n}R_{n,k}R_{n-1,k},$
$\displaystyle\partial_{t_{k}}\beta_{n}=$
$\displaystyle\beta_{n}R_{n-1,k}-\beta_{n}R_{n,k}.$

Getting rid of $R_{n-1,k}$ from the second equality by using the first one, we
get

$\displaystyle\partial_{t_{k}}\beta_{n}=\frac{r_{n,k}^{2}}{R_{n,k}}-\beta_{n}R_{n,k}.$
(2.29)

Applying $\partial_{t_{k}}$ to (2.17), in view of (2.22) and (2.28), we find

$\displaystyle\partial_{t_{k}}\beta_{n}=\delta r_{n,k}.$

Combining it with (2.29), and replacing $\beta_{n}$ in the resulting equation
by using (2.12), we come to (1.11b), which completes the derivation of Riccati
equations.

Solving $r_{n,k}$ from (1.11a) and substituting it into (1.11b), noting that
$\delta(t_{k})=t_{k}$ and $\delta(t_{j}R_{n,j})=t_{j}R_{n,j}+t_{j}\cdot\delta
R_{n,j}$, after simplification, we arrive at the coupled PDEs satisfied by
$R_{n,k},k=1,2,\cdots,m$.
(b) Derivation of PDE (1.14) satisfied by $\sigma_{n}(\vec{t})$
Recall the Hankel determinant of our interest, i.e.

$\displaystyle
D_{n}\left(\,\vec{t}\,\right):=\det\left(\int_{0}^{\infty}x^{i+j}w(x;\vec{t}\,)dx\right)_{i,j=0}^{n-1}.$

It is well known that it can be represented as the product of $h_{j}$, the
square of the $L^{2}$ norm of the monic orthogonal polynomial
$P_{j}(x;\vec{t})$, that is,

$\displaystyle D_{n}(\vec{t})=\prod_{j=0}^{n-1}h_{j}(\vec{t}).$

Refer to [17, pp. 16–19]. Thus, it follows from (2.21) that

$\displaystyle\sigma_{n}(\vec{t})=\delta\,lnD_{n}(\vec{t})=$
$\displaystyle-\sum_{j=0}^{n-1}\sum_{k=1}^{m}t_{k}R_{j,k}(\vec{t}),$

where $\delta=\sum\limits_{k=1}^{m}t_{k}\partial_{t_{k}}$. According to (2.17)
and (2.18), we find

$\displaystyle\sigma_{n}=$ $\displaystyle p(n,\vec{t})+n(n+\alpha)$ (2.30)
$\displaystyle=$
$\displaystyle-\beta_{n}+\sum_{k=1}^{m}t_{k}r_{n,k}+n(n+\alpha).$ (2.31)

Inserting (2.12) into (2.31), we obtain the expression for
$\sigma_{n}(\vec{t})$ in terms of $R_{n,k}$ and $r_{n,k}$, namely (1.13). If,
in turn, we can express $R_{n,k}$ and $r_{n,k}$ by $\sigma_{n}$ or its
derivatives, then we will readily establish the PDE satisfied by $\sigma_{n}$.

######  Lemma 2.10.

$r_{n,k}$ and $R_{n,k}$ are connected with $\sigma_{n}$ and its derivatives by

$\displaystyle r_{n,k}=\partial_{t_{k}}\sigma_{n},$ (2.32)

and

$\displaystyle
R_{n,k}=\frac{1}{2\beta_{n}}\left(-\partial_{t_{k}}\beta_{n}+{\rm
sgn}(\omega_{k})\sqrt{(\partial_{t_{k}}\beta_{n})^{2}+4\beta_{n}r_{n,k}^{2}}\right),$
(2.33a) for $k=1,\cdots,m$, where ${\rm sgn}(\omega_{k})$ is the sign function
of $\omega_{k}$, which is $1$ for $\omega_{k}>0$, $-1$ for $\omega_{k}<0$ and
$0$ for $\omega_{k}=0$. Here $\displaystyle\beta_{n}=$
$\displaystyle\delta\sigma_{n}-\sigma_{n}+n(n+\alpha),$ (2.33b)
$\displaystyle\partial_{t_{k}}\beta_{n}=$
$\displaystyle\sum_{j=1}^{m}t_{j}\cdot\partial_{t_{k}t_{j}}^{2}\sigma_{n}.$
(2.33c)

###### Proof.

Differentiating both sides of (2.30) with respect to $t_{k}$, in light of
(2.22), we get (2.32). Plugging it into (2.31) gives us (2.33b) which
immediately leads to (2.33c).

Note that (2.29) is equivalent to the following second order linear equation
in $R_{n,k}$:

$\displaystyle\beta_{n}R_{n,k}^{2}+\left(\partial_{t_{k}}\beta_{n}\right)R_{n,k}-r_{n,k}^{2}=0.$

Solving for $R_{n,k}$ from it, we get two possible solutions

$\displaystyle
R_{n,k}=\frac{-\partial_{t_{k}}\beta_{n}\pm\sqrt{(\partial_{t_{k}}\beta_{n})^{2}+4\beta_{n}r_{n,k}^{2}}}{2\beta_{n}},$

or equivalently,

$\displaystyle
R_{n,k}+\frac{1}{2}\partial_{t_{k}}\ln\beta_{n}=\frac{\pm\sqrt{(\partial_{t_{k}}\beta_{n})^{2}+4\beta_{n}r_{n,k}^{2}}}{2\beta_{n}}.$
(2.34)

To determine which solution to choose, it suffices to discuss the sign
function of $R_{n,k}+\frac{1}{2}\partial_{t_{k}}\ln\beta_{n}$. According to
(2.23), we have

$\displaystyle
R_{n,k}+\frac{1}{2}\partial_{t_{k}}\ln\beta_{n}=\frac{1}{2}\left(R_{n-1,k}+R_{n,k}\right).$
(2.35)

From the definition of $R_{j,k}$, i.e.

$\displaystyle
R_{j,k}=\frac{\omega_{k}}{h_{j}}P_{j}^{2}(t_{k};\vec{t})t_{k}^{\alpha}{\rm
e}^{-t_{k}},\qquad\qquad j\geq 0,$

and the fact that $h_{j}>0$, we observe that for $t_{k}>0$,

$\displaystyle{\rm sgn}(R_{n-1,k})={\rm sgn}(R_{n,k})={\rm sgn}(\omega_{k}).$

Hence, it follows from (2.35) that

${\rm sgn}\left(R_{n,k}+\frac{1}{2}\partial_{t_{k}}\ln\beta_{n}\right)={\rm
sgn}(\omega_{k}).$

This combined with (2.34) gives us (2.33a). ∎

Substituting (2.32) and (2.33) into (1.13), after simplification, we arrive at
(1.14).

##  3 Lax Pair and Coupled $P_{V}$ System

We start with the RH problem for the monic polynomials $P_{n}(z;\vec{t})$
orthogonal with respect to the weight function (1.3). With the aid of Lax
pair, we can express $\sigma_{n}(\vec{t})$ in terms of quantities satisfying a
coupled $P_{V}$ system.

###  3.1 RH problem for orthogonal polynomials

Let

$Y(z;\vec{t})=\left(\begin{array}[]{cc}P_{n}(z)&\frac{1}{2\pi
i}\int_{0}^{+\infty}\frac{P_{n}(x)w(x;\vec{t})}{x-z}dx\\\ -2\pi
i\gamma_{n-1}^{2}\;P_{n-1}(z)&-\gamma_{n-1}^{2}\;\int_{0}^{+\infty}\frac{P_{n-1}(x)w(x;\vec{t})}{x-z}dx\end{array}\right),$
(3.1)

where $\gamma_{n-1}=h_{n-1}^{-1/2}$ with $h_{n-1}$ denoting the square of the
$L^{2}$-norm of $P_{n-1}(z)$ with respect to $w(x;\vec{t})$ defined in (1.3).

From (3.1), it is seen that the function $Y(z)=Y(z;\vec{t})$ is the unique
solution of the following RH problem (see Fokas, Its and Kitaev [15], and also
Deift [14]):

(a) $Y(z)$ is analytic in $\mathbb{C}\backslash[0,+\infty)$;

(b) $Y(z)$ satisfies the jump condition

$Y_{+}(x)=Y_{-}(x)\left(\begin{array}[]{cc}1&w(x;\vec{t})\\\ 0&1\\\
\end{array}\right),\qquad x\in(0,+\infty),$

where $w(x;\vec{t})$ is defined in (1.3);

(c) As $z\to\infty$, we have

$Y(z)=\left(I+\frac{Y_{1}}{z}+\frac{Y_{2}}{z^{2}}+O\left(\frac{1}{z^{3}}\right)\right)\left(\begin{array}[]{cc}z^{n}&0\\\
0&z^{-n}\\\ \end{array}\right),\quad\quad z\rightarrow\infty;$

(d) As $z\to 0$, we have

$Y(z)=\left\\{\begin{array}[]{ccc}\left(\begin{array}[]{cc}O(1)&O(1)\\\
O(1)&O(1)\\\ \end{array}\right),&\quad\alpha>0,\\\
\left(\begin{array}[]{cc}O(1)&O(\log z)\\\ O(1)&O(\log z)\\\
\end{array}\right),&\quad\alpha=0,\\\
\left(\begin{array}[]{cc}O(1)&O(z^{\alpha})\\\ O(1)&O(z^{\alpha})\\\
\end{array}\right),&\quad-1<\alpha<0;\end{array}\right.$

(e) As $z\to t_{k}$, we have

$Y(z)=\left(\begin{array}[]{cc}O(1)&O(\log(z-t_{k}))\\\
O(1)&O(\log(z-t_{k}))\\\ \end{array}\right),$

for $k=1,\cdots,m$.

###  3.2 A model RH problem and Lax pair

We define

$\Psi(z;\vec{t})=t_{1}^{-(n+\frac{\alpha}{2})\sigma_{3}}Y(t_{1}z;\vec{t})(t_{1}z)^{\frac{\alpha}{2}\sigma_{3}}e^{-\frac{t_{1}}{2}z\sigma_{3}},$
(3.2)

where $\arg z\in(-\pi,\pi)$ and $\sigma_{3}$ is the Pauli matrix

$\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\ \end{array}\right).$

Then, it follows from the RH problem for $Y$ that $\Psi(z)=\Psi(z;\vec{t})$
satisfies the following RH problem.

(a) $\Psi(z)$ is analytic in $\mathbb{C}\backslash[0,+\infty)$;

(b) $\Psi(z)$ satisfies the jump condition

$\left\\{\begin{array}[]{ll}\Psi_{+}(x)=\Psi_{-}(x)\begin{pmatrix}1&\omega_{0}+\sum_{k=1}^{m}\omega_{k}\theta(x-a_{k})\\\
0&1\\\ \end{pmatrix},&\quad x\in(0,+\infty),\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \Psi_{+}(x)=\Psi_{-}(x)e^{\pi i\alpha\sigma_{3}},&\quad
x\in(-\infty,0),\end{array}\right.$ (3.3)

where

$a_{k}:=\frac{t_{k}}{t_{1}},\qquad k=1,\cdots,m;$

(c) As $z\to\infty$, we have

$\Psi(z)=\left(I+\frac{\Psi_{1}}{z}+\frac{\Psi_{2}}{z^{2}}+O\left(\frac{1}{z^{3}}\right)\right)z^{(n+\frac{\alpha}{2})\sigma_{3}}{{\rm
e}}^{-\frac{t_{1}}{2}z\sigma_{3}};$ (3.4)

(d) As $z\to 0$, we have

$\Psi(z)=\Psi^{(0)}(z)z^{\frac{\alpha}{2}\sigma_{3}}\left(\begin{array}[]{cc}1&\frac{\omega_{0}}{1-{{\rm
e}}^{2\pi i\alpha}}\\\ 0&1\\\
\end{array}\right)E_{0},\quad\alpha\notin\mathbb{N},$ (3.5)

and

$\Psi(z)=\Psi^{(0)}(z)z^{\frac{\alpha}{2}\sigma_{3}}\left(\begin{array}[]{cc}1&-\frac{\omega_{0}}{2\pi
i}\log z\\\ 0&1\\\ \end{array}\right)E_{0},\quad\alpha\in\mathbb{N},$ (3.6)

where $\Psi^{(0)}(z)$ is analytic near the origin, the branch cut for
$z^{\alpha}$ and $\log z$ is taken along $(0,+\infty)$ such that $\arg
z\in(-\pi,\pi)$ and $E_{0}$ is a piecewise constant matrix such that $E_{0}=I$
for $\arg z\in(0,\pi)$ and $E_{0}=\left(\begin{array}[]{cc}1&-\omega_{0}\\\
0&1\\\ \end{array}\right)$ for $\arg z\in(-\pi,0)$;

(e) As $z\to a_{k}$, we have

$\Psi(z)=\Psi^{(k)}(z)\left(\begin{array}[]{cc}1&-\frac{\omega_{k}}{2\pi
i}\log(z-a_{k})\\\ 0&1\\\ \end{array}\right)E_{k},$ (3.7)

where $\Psi^{(k)}(z)$ is analytic near $a_{k}$, the branch cut for
$\log(z-a_{k})$ is taken along $(a_{k},+\infty)$ such that
$\arg(z-a_{k})\in(-\pi,\pi)$, for $k=1,\cdots,m$. The piecewise constant
matrix $E_{k}$ is defined by $E_{k}=I$ for $\arg z\in(0,\pi)$ and
$E_{k}=\left(\begin{array}[]{cc}1&-\omega_{k}\\\ 0&1\\\ \end{array}\right)$
for $\arg z\in(-\pi,0)$, for $k=1,\cdots,m$.

######  Proposition 3.1.

We have the following Lax pair

$\left\\{\begin{array}[]{l}\frac{d}{dz}\Psi(z;\vec{t})=L(z;\vec{t})\Psi(z;\vec{t}),\\\
\delta\Psi(z;\vec{t})=U(z;\vec{t})\Psi(z;\vec{t}),\\\
\partial_{t_{k}}\Psi(z;\vec{t})=\frac{\hat{A}_{k}(t)}{t_{k}-t_{1}z}\Psi(z;\vec{t}),\quad
k=2,\cdots,m,\end{array}\right.$ (3.8)

where $\delta=\sum\limits_{k=1}^{m}t_{k}\partial_{t_{k}}$ and

$L(z;\vec{t})=-\frac{t_{1}}{2}\sigma_{3}+\frac{\hat{A}_{0}(\vec{t})}{z}+\sum_{k=1}^{m}\frac{\hat{A}_{k}(\vec{t})}{z-a_{k}},$
(3.9) $U(z;\vec{t})=-\frac{t_{1}}{2}z\sigma_{3}+\hat{B}(\vec{t}),$ (3.10)

with $a_{k}=t_{k}/t_{1},k=1,\cdots,m$. Here the coefficients are given below

$\hat{A}_{0}=\begin{pmatrix}a&-\left(a-\frac{\alpha}{2}\right)y\\\
\left(a+\frac{\alpha}{2}\right)/y&-a\end{pmatrix},$ (3.11)
$\qquad\qquad\hat{A}_{k}=\begin{pmatrix}-u_{k}v_{k}&u_{k}y\\\
-u_{k}v_{k}^{2}/y&u_{k}v_{k}\end{pmatrix},\quad k=1,\cdots,m,$ (3.12)

and

$\hat{B}=\begin{pmatrix}0&b_{1}y\\\ b_{2}/y&0\end{pmatrix},$ (3.13)

with $b_{1}(\vec{t})$ and $b_{2}(\vec{t})$ defined by (1.21), and

$a(\vec{t}):=\sum\limits_{k=1}^{m}u_{k}(\vec{t})v_{k}(\vec{t})+n+\frac{\alpha}{2}.$

It is seen from (3.3) that $\Psi$, $\frac{d}{dz}\Psi$, $\partial_{t_{k}}\Psi$
and $\delta\Psi$ share the same jump (3.3) on the real axis. Thus,
$L=(\frac{d}{dz}\Psi)\Psi^{-1}$, $U=(\delta\Psi)\Psi^{-1}$ and
$(\partial_{t_{k}}\Psi)\Psi^{-1}$ are meromorphic for $z$ in the complex plane
with only possible isolate singularities at $z=0,a_{k}$. Furthermore, using
the asymptotic behaviors of $\Psi(z)$ given in (3.4)–(3.7), we see that $U$
and $L$ are of the form appearing in (3.9) and (3.10).

Next, we determine the coefficients in (3.9) and (3.10). It follows from the
fact $\det\Psi=1$ that ${\rm tr}L={\rm tr}U=0$. Hence all the coefficients
$\hat{A}_{k}$, $k=0,\cdots,m$, and $\hat{B}$ are trace-zero. Substituting
(3.4) into the first equation of the system (3.8), we find after comparing the
coefficient of $\frac{1}{z}$ on both sides of the equation that

$\sum_{k=0}^{m}\hat{A}_{k}=\begin{pmatrix}n+\frac{\alpha}{2}&t_{1}(\Psi_{1})_{12}\\\
-t_{1}(\Psi_{1})_{21}&-(n+\frac{\alpha}{2})\end{pmatrix},$ (3.14)

where $\Psi_{1}$ is the coefficient in the large $z$ expansion of $\Psi(z)$ in
(3.4). Similarly, inserting the behavior of $\Psi$ at infinity, i.e. (3.4),
into the second equation of the system (3.8), we find

$\hat{B}=\begin{pmatrix}0&t_{1}(\Psi_{1})_{12}\\\
-t_{1}(\Psi_{1})_{21}&0\end{pmatrix}.$ (3.15)

Using the behaviors of $\Psi(z)$ as $z\to 0$ and $z\to a_{k}$, given in
(3.5)–(3.7), we have

$\det\hat{A}_{0}=-\alpha^{2}/4\quad\mbox{and}\quad\det\hat{A}_{k}=0,\quad
k=1,\cdots,m.$ (3.16)

Hence, the equations (3.14) and (3.16) imply that the coefficients
$\hat{A}_{k}$, $k=1,\cdots,m$, can be parameterized in the form appearing in
(3.11) and (3.12). The equation (3.13) then follows from (3.11), (3.12),
(3.14) and (3.15).

######  Remark 8.

When $m=2$, the Lax pair (3.8) is equivalent to the Garnier system [19, (3.6)]
appeared in the studies of the degeneration scheme of 4-dimensional Painlevé-
type equations.

###  3.3 Proof of Theorem 1.2

The compatibility condition $\delta\frac{d}{dz}\Psi=\frac{d}{dz}\delta\Psi$
gives the zero-curve equation

$\delta L(z)-\frac{d}{dz}U(z)+[L(z),U(z)]=0,$

which is equivalent to

$\left\\{\begin{array}[]{ll}\delta\hat{A}_{0}=[\hat{B},\hat{A}_{0}],\\\
\delta\hat{A}_{k}=[\hat{B}-\frac{t_{k}}{2}\sigma_{3},\hat{A}_{k}],\quad
k=1,\cdots,m.\end{array}\right.$ (3.17)

Substituting (3.11)–(3.13) into (3.17), we arrive at

$\left\\{\begin{array}[]{ll}\delta((a-\frac{\alpha}{2})y)=2ab_{1}y,\\\
\delta((a+\frac{\alpha}{2})/y)=2ab_{2}/y,\\\
\delta(u_{k}v_{k})=b_{1}u_{k}v_{k}^{2}+b_{2}u_{k},\quad k=1,\cdots,m,\\\
\delta(u_{k}y)=-t_{k}u_{k}y+2b_{1}u_{k}v_{k}y,\quad
k=1,\cdots,m.\end{array}\right.$ (3.18)

We obtain from the first two equations that

$\delta\ln y=b_{1}-b_{2}=\sum_{k=1}^{m}u_{k}(v_{k}-1)^{2}-2n-\alpha.$ (3.19)

The system of nonlinear differential equations (1.19) then follows from (3.19)
and the last $2m$ equations of (3.18).

Substituting into the first equation of the Lax pair the expansion (3.4), we
obtain after comparing the coefficients of $1/z^{2}$ in the expansion

$-\frac{t_{1}}{2}[\sigma_{3}\Psi_{1},\Psi_{1}]-\frac{t_{1}}{2}[\Psi_{2},\sigma_{3}]+\left(n+\frac{\alpha}{2}\right)[\Psi_{1},\sigma_{3}]-\Psi_{1}=\sum_{k=1}^{m}a_{k}\hat{A}_{k}.$
(3.20)

This, together with (3.15) and (3.12), implies

$t_{1}(\Psi_{1})_{11}=-t_{1}^{2}(\Psi_{1})_{12}(\Psi_{1})_{21}-\sum_{k=1}^{m}t_{k}(\hat{A}_{k})_{11}=b_{1}b_{2}+\sum_{k=1}^{m}t_{k}u_{k}v_{k}.$
(3.21)

From (3.1) and (3.2), we find that

$p(n,\vec{t})=t_{1}\left(\Psi_{1}\right)_{11},$

which combined with (2.30) and (3.21) gives us

$\displaystyle\sigma_{n}=$ $\displaystyle
t_{1}\left(\Psi_{1}\right)_{11}+n(n+\alpha)$ (3.22) $\displaystyle=$
$\displaystyle b_{1}b_{2}+\sum_{k=1}^{m}t_{k}u_{k}v_{k}+n(n+\alpha).$

Replacing $b_{1}$ and $b_{2}$ by (1.21), in view of (1.17) , we come to
(1.18). It follows directly from (1.18) that (1.20) are equivalent to (1.19).

###  3.4 Proof of Theorem 1.3 by Lax pair

We aim to prove (1.22)–(1.27). Substituting in the Lax pair (3.8) the
expansion (3.7), we obtain

$\displaystyle\hat{A}_{k}=$ $\displaystyle-\frac{\omega_{k}}{2\pi
i}\Psi^{(k)}(a_{k})\left(\begin{array}[]{cc}0&1\\\ 0&0\\\
\end{array}\right)\Psi^{(k)}(a_{k})^{-1}$ (3.23)
$\displaystyle=-\frac{\omega_{k}}{2\pi
i}\left(\begin{array}[]{cc}-\Psi^{(k)}_{11}(a_{k})\Psi^{(k)}_{21}(a_{k})&\Psi^{(k)}_{11}(a_{k})^{2}\\\
-\Psi^{(k)}_{21}(a_{k})^{2}&\Psi^{(k)}_{11}(a_{k})\Psi^{(k)}_{21}(a_{k})\\\
\end{array}\right),$

where $\Psi^{(k)}(z)$ appears in (3.7) for $k=1,\cdots,m$. Using (3.1), (3.2),
(3.15) and (3.23), we obtain (1.22), (1.25b) and (1.26). (1.23) are direct
consequence of (1.22). Combining (1.25b) with (1.30) which results from the
definitions of $b_{1}$ and $b_{2}$, we get (1.25a).

Now we go ahead with the derivation of (1.27). Substituting the expansion
(3.4) into the last equation of (3.8) yields

$\partial_{t_{k}}(t_{1}\Psi_{1})=-\hat{A}_{k},\qquad k=2,\cdots,m,$

where $\Psi_{1}$ is the coefficient in the expansion (3.4). Recalling (3.12),
(3.15) and (3.22), we arrive at a system of differential equations with
respect to $t_{k}$

$\left\\{\begin{array}[]{ll}\partial_{t_{k}}\ln(b_{1}y)=-u_{k}/b_{1},\\\
\partial_{t_{k}}\ln(b_{1}b_{2})=u_{k}v_{k}^{2}/b_{1}-u_{k}/b_{1},\\\
\partial_{t_{k}}\sigma_{n}=u_{k}v_{k},\end{array}\right.$ (3.24)

for $k=2,\cdots,m$. To study the case $k=1$, we substitute the expansion (3.4)
into the identity

$z\frac{d}{dz}\Psi(z)-\delta\Psi(z)=(zL-U)\Psi(z)$

which comes from the first two equations of (3.8), and obtain after comparing
the coefficients of $1/z$ on both sides of the resulting equality

$\delta(t_{1}\Psi_{1})=\left(n+\frac{\alpha}{2}\right)[t_{1}\Psi_{1},\sigma_{3}]-\sum_{k=1}^{m}t_{k}\hat{A}_{k}.$

Then, it is seen from the above equation

$\delta\sigma_{n}=\delta(t_{1}(\Psi_{1})_{11})=\sum_{k=1}^{m}t_{k}u_{k}v_{k},$
(3.25)
$\delta(b_{1}y)=\delta(t_{1}(\Psi_{1})_{12})=-(2n+\alpha)b_{1}y-\sum_{k=1}^{m}t_{k}u_{k}y,$
(3.26)

and

$\delta(b_{2}/y)=-\delta(t_{1}(\Psi_{1})_{21})=(2n+\alpha)b_{2}/y+\sum_{k=1}^{m}t_{k}u_{k}^{2}v_{k}/y.$
(3.27)

Combining (3.24) with (3.25)–(3.27), we obtain

$\left\\{\begin{array}[]{ll}\partial_{t_{1}}\ln(t_{1}^{2n+\alpha}b_{1}y)=-u_{1}/b_{1},\\\
\partial_{t_{1}}\ln(b_{1}b_{2})=u_{1}v_{1}^{2}/b_{1}-u_{1}/b_{1},\\\
\partial_{t_{1}}\sigma_{n}=u_{1}v_{1}.\end{array}\right.$ (3.28)

A combination of (3.24) and (3.28) yields

$\left\\{\begin{array}[]{ll}\partial_{t_{k}}\ln(t_{1}^{2n+\alpha}b_{1}y)=-u_{k}/b_{1},\\\
\partial_{t_{k}}\ln(b_{1}b_{2})=u_{k}v_{k}^{2}/b_{1}-u_{k}/b_{1},\\\
\partial_{t_{k}}\sigma_{n}=u_{k}v_{k},\end{array}\right.$ (3.29)

for $k=1,\cdots,m$, and the first equation of (3.29) together with (1.26)
gives us (1.27).

It is seen from (3.1) that

$\alpha_{n}=(Y_{1})_{11}+\frac{(Y_{2})_{12}}{(Y_{1})_{12}},$ (3.30)

where $Y_{1}$ and $Y_{2}$ are the coefficients in the expansion of $Y(z)$ as
$z\to\infty$; see Deift [14]. Recalling the relation (3.2), we have

$\alpha_{n}=t_{1}(\Psi_{1})_{11}+t_{1}\frac{(\Psi_{2})_{12}}{(\Psi_{1})_{12}},$
(3.31)

where $\Psi_{1}$ and $\Psi_{2}$ are the coefficients in the expansion (3.4).
Now, the relation (1.24) follows directly from (3.31) and the $(12)$-entry of
(3.20).

######  Remark 9.

With the relations (1.22), (1.25b) and the definition (1.21), we can show that
the ladder operators given by Lemma 2.1 are indeed the first column of the Lax
pair relating $z$, i.e. the first equation of (3.8). Note that, to derive the
raising operator, we also make use of (2.10) which can be deduced by using the
definitions of $R_{n,k}$ and $r_{n,k}$ together with the fact that
$\beta_{n}=h_{n}/h_{n-1}$.

In addition to the Lax pair, by using the asymptotic behavior of $\Psi(z)$ at
$\infty$ given by (3.4), we can establish the following difference equation

$\displaystyle\Psi(z;t,n+1)=$
$\displaystyle\left(\begin{array}[]{cc}z+(\Psi_{1})_{11}(n+1)-(\Psi_{1})_{11}(n)&\qquad-(\Psi_{1})_{12}(n)\\\
\qquad(\Psi_{1})_{21}(n+1)&0\\\ \end{array}\right)\Psi(z;t,n),$

which is equivalent to the three-term recurrence relation (1.6) satisfied by
monic orthogonal polynomials. Moreover, the compatibility condition of the
above difference equation and the first equation of (3.8) lead to the
identities $(S_{1})$ and $(S_{2})$ given in Lemma 2.2.

######  Remark 10.

According to (1.22b), (1.25b) and (1.26), the differential equations (3.29)
are equivalent to

$\left\\{\begin{array}[]{ll}\partial_{t_{k}}\ln h_{n}=-R_{n,k},\\\
\partial_{t_{k}}\beta_{n}=\frac{r_{n,k}^{2}}{R_{n,k}}-\beta_{n}R_{n,k},\\\
\partial_{t_{k}}\sigma_{n}=r_{n,k},\\\ \end{array}\right.$ (3.32)

for $k=1,\cdots,m$. It is noted that these equations play an important role in
deriving the differential equation satisfied by $\sigma_{n}$, i.e. (1.14).

Furthermore, with the identities obtained by the Lax pair, we are able to
derive the Riccati equations deduced by using the ladder operator approach.
Indeed, in view of (1.22), we find from (3.18) that

$\delta r_{n,k}=\frac{r_{n,k}^{2}}{R_{n,k}}+(b_{1}b_{2})R_{n,k}.$

Using (1.30) to replace $b_{1}b_{2}$, we get the Riccati equation (1.11b).
Since $R_{n,k}=u_{k}/b_{1}$, we have

$\delta(R_{n,k})=\frac{1}{b_{1}y}\delta(u_{k}y)-\frac{u_{k}y}{(b_{1}y)^{2}}\delta(b_{1}y).$

Replacing the derivatives of $u_{k}y$ and $b_{1}y$ by the last equation of
(3.18) and (3.26) respectively, we come to the Riccati equation (1.11a).

##  4 Double Scaling Analysis at the Hard Edge

###  4.1 Proof of Proposition 1.4

Since $r_{n,k}$ and $R_{n,k}$ are connected with $\sigma_{n}$ by the relations
given in Lemma 2.10, we make use of them to derive what we want.

Changing variables $t_{k}=s_{k}/(4n)$ and $\vec{t}=\vec{s}/(4n)$ in (2.32)
yields

$\displaystyle
r_{n,k}\left(\frac{\vec{s}}{4n}\right)=4n\cdot\partial_{s_{k}}\sigma_{n}.$
(4.1)

Dividing its both sides by $4n$ and taking the limit $n\rightarrow\infty$, in
view of the definition of $\sigma(\vec{s})$, we get (1.33).

Replacing $r_{n,k}(\vec{t})$ by using (4.1) and $t_{k}$ by $s_{k}/(4n)$ in
(2.33), we have

$\displaystyle
R_{n,k}\left(\frac{\vec{s}}{4n}\right)=\frac{1}{2\beta_{n}}\left(-\partial_{t_{k}}\beta_{n}+{\rm
sgn}(\omega_{k})\sqrt{(\partial_{t_{k}}\beta_{n})^{2}+64n^{2}\beta_{n}\left(\partial_{s_{k}}\sigma_{n}\right)^{2}}\right),$
(4.2)

where

$\displaystyle\beta_{n}=$
$\displaystyle\delta\sigma_{n}-\sigma_{n}+n(n+\alpha),$
$\displaystyle\partial_{t_{k}}\beta_{n}=$ $\displaystyle
4n\sum_{j=1}^{m}s_{j}\cdot\partial_{s_{k}s_{j}}^{2}\sigma_{n},$

with $\delta=\sum\limits_{k=1}^{m}s_{k}\partial_{s_{k}}$. Taking the series
expansion in large $n$ of the right hand side of (4.2), we find that the
leading order term is given by $4\,{\rm
sgn}(\omega_{k})\sqrt{\left(\partial_{s_{k}}\sigma_{n}\right)^{2}}$, which
indicates that

$R_{n,k}\left(\frac{\vec{s}}{4n}\right)=O(1),$

as $n\to\infty$. With this fact, on replacing $r_{n,k}(\vec{t})$ by using
(4.1) and $t_{k}$ by $s_{k}/(4n)$ in (1.11a), we are led to (1.34).

###  4.2 Proof of Theorem 1.5 and Corollary 1.6

To derive the expression for $\sigma(\vec{s})$ in terms of $R_{k}(s)$ and
$\delta R_{k}(\vec{s})$, the PDE(s) satisfied by $R_{k}(\vec{s})$ and
$\sigma(\vec{s})$, we make use of the corresponding finite-$n$ results and the
relations given by Proposition 1.4.

Using (1.11a) to get rid of $r_{n,k}$ in (1.13), we come to an expression for
$\sigma_{n}$ in terms of $R_{n,k}$ and $\delta R_{n,k}$. Replacing
$R_{n,k},\delta R_{n,k}$ and $t_{k}$ by $R_{k},\delta R_{k}$ and $s_{k}/(4n)$
respectively in the expression, and sending $n$ to $\infty$ on both sides, we
obtain (1.35). Substituting $R_{k}=-4\partial_{s_{k}}\sigma$ which comes from
Proposition (1.4) into (1.35), we get (1.37), the PDE satisfied by $\sigma$.
Replacing $R_{n,k}$ by $R_{k}$ and $t_{k}$ by $s_{k}/(4n)$ in (1.12), and
taking the series expansion of both sides for large $n$, by comparing the
leading coefficients in $n$, we come to (1.36), which completes the proof of
Theorem 1.5.

For $\vec{t}=(t_{1},\cdots,t_{m})=\frac{\tau}{4n}(\nu_{1},\cdots,\nu_{m})$, we
have

$\tau\frac{d}{d\tau}\ln D_{n}(\vec{t})=\delta\ln
D_{n}(\vec{t})=\sigma_{n}(\vec{t}).$

Taking derivative once again yields

$\displaystyle\left(\tau\frac{d}{d\tau}\right)^{2}\ln D_{n}(\vec{t})=$
$\displaystyle\delta\sigma_{n}(\vec{t})=\sum_{k=1}^{m}t_{k}r_{n,k}(\vec{t}),$

where the second equality results from (3.25) and (1.22b). Taking the limit of
both sides as $n\rightarrow\infty$, we find

$\displaystyle\left(\tau\frac{d}{d\tau}\right)^{2}\ln\lim\limits_{n\rightarrow\infty}D_{n}(\vec{t})=$
$\displaystyle\frac{\tau}{4}\cdot\sum\limits_{k=1}^{m}\left(\nu_{k}\lim_{n\to\infty}\frac{r_{n,k}\left(\vec{t}\right)}{n}\right)$
$\displaystyle=$
$\displaystyle-\frac{\tau}{4}\sum_{k=1}^{m}\nu_{k}\hat{R}_{k}(\tau),$ (4.3)

where the second identity comes from (1.34) and (1.41). We readily see that
(1.42) is a solution of (4.3). This completes the proof of Corollary 1.6.

## Acknowledgments

Shulin Lyu was supported by National Natural Science Foundation of China under
grant number 12101343 and by Shandong Provincial Natural Science Foundation
with project number ZR2021QA061. Yang Chen was supported by the Macau Science
and Technology Development Fund under grant numbers FDCT 023/2017/A1 and FDCT
0079/2020/A2, and by the University of Macau under grant number MYRG
2018-00125-FST. Shuai-Xia Xu was partially supported by National Natural
Science Foundation of China under grant numbers 11971492 and 11571376.

## References

  * [1] W. Van Assche, Orthogonal Polynomials and Painlevé Equations, Australian Mathematical Society Lecture Series 27, Cambridge University Press, 2018.
  * [2] M. Atkin, T. Claeys and F. Mezzadri, Random matrix ensembles with singularities and a hierarchy of Painlevé III equations, Int. Math. Res. Notices 2016 (2016), 2320–2375.
  * [3] E. Basor and Y. Chen, Painlevé V and the distribuition function of a discontinuous linear statistic in the Laguerre unitary ensembels, J. Phys. A: Math. Theor. 42 (2009), 035203 (18pp).
  * [4] E. Basor, Y. Chen and L. Zhang, PDEs satisfied by extreme eigenvalues distributions of GUE and LUE, Random Matrices–Theory Appl., 1 (2012), 1150003 (21pp).
  * [5] C. Charlier, Exponential moments and piecewise thinning for the Bessel point process, Int. Math. Res. Notices 2021 (2021), 16009–16073.
  * [6] C. Charlier and T. Claeys, Large gap asymptotics for Airy kernel determinants with discontinuities, Commun. Math. Phys. 375(2019), no. 2, 1299–1339.
  * [7] C. Charlier and A. Doeraene, The generating function for the Bessel point process and a system of coupled Painlevé V equations, Random Matrices–Theory Appl. 8 (2019), 1950008 (31pp).
  * [8] Y. Chen, N. Haq and M. McKay, Random matrix models, double-time Painlevé equations, and wireless relaying, J. Math. Phys. 54 (2013), 063506 (55pp).
  * [9] Y. Chen and M. Ismail, Ladder operators and differential equations for orthogonal polynomials, J. Phys. A: Math. Gen. 30 (1997), 7817–7829.
  * [10] Y. Chen and A. Its, Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I, J. Approx. Theory 162 (2010), 270–297.
  * [11] T. Claeys and A. Doeraene, The generating function for the Airy point process and a system of coupled Painlevé II equations, Stud. Appl. Math. 140, 403–437.
  * [12] D. Dai, S. Xu and L. Zhang, Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential, SIAM J. Math. Anal. 50 (2018), 2233–2279.
  * [13] D. Dai, S. Xu and L. Zhang, Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations, Ann. Henri Poincaré 20 (2019), 3313–3364.
  * [14] P. Deift, _Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach_ , Courant Lecture Notes 3, New York University, Amer. Math. Soc., Providence, RI, 1999.
  * [15] A. Fokas, A. Its and A. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), 395–430.
  * [16] P. Forrester and N. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: $P_{V},P_{III},$ the LUE, JUE, and CUE, Commun. Pure Appl. Math. 55 (2002), 0679–0727.
  * [17] M. Ismail, _Classical and Quantum Orthogonal Polynomials in One Variable_ , Encyclopedia of Mathematics and its Applications 98, Cambridge University Press, Cambridge, 2005.
  * [18] M. Jimbo, Monodromy problem and the boundary condition for some Painleve equations, Publ. RIMS, Kyoto Univ. 18 (1982), 1137-1161.
  * [19] H. Kawakami, A. Nakamura and H. Sakai, Degeneration scheme of 4-dimensional Painlevé-type equations, arXiv:1209.3836.
  * [20] S. Lyu and Y. Chen, Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systems, Stud. Appl. Math. 146 (2021), 118–138.
  * [21] S. Lyu, Y. Chen and E. Fan, Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles, Nucl. Phys. B 926 (2018), 639–670.
  * [22] S. Lyu, J. Griffin and Y. Chen, The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble, J. Nonlinear Math. Phys. 26 (2019), 24–53.
  * [23] M. Mehta, _Random Matrices_ , 3rd edition, Elsevier, New York, 2004.
  * [24] C. Min and Y. Chen, Gap probability distribution of the Jacobi unitary ensembles: an elementary treatment, from finite $n$ to double scaling, Stud. Appl. Math. 140 (2018), 202–220.
  * [25] C. Min and Y. Chen, Painlevé transcendents and the Hankel determinants generated by a discontinuous Gaussian weight, Math. Meth. Appl. Sci. 42 (2019), 301–321.
  * [26] K. Okamoto, Studies on the Painlevé equations. II. Fifth Painlevé equation PV, Japan. J. Math. 13 (1987), no. 1, 47–76.
  * [27] G. Szegö, _Orthogonal Polynomials_ , American Mathematical Society Colloquium Publications, vol. 23, New York, 1939.
  * [28] C. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161 (1994), 289–309.
  * [29] X. Wu and S. Xu, Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems, Nonlinearity 34 (2021), 2070–2115.
  * [30] S.-X. Xu and D. Dai, Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system, Commun. Math. Phys. 365 (2019), no. 2, 515–567.


* [20] S. Gaspers, S. Ordyniak, and S. Szeider.  Backdoor sets for CSP.  In The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 137–157. 2017\.
  * [21] S. Gaspers and S. Szeider.  Backdoors to satisfaction.  In The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, pages 287–317. Springer, 2012.
  * [22] A. Goerdt.  On random ordering constraints.  In Proc. 4th International Computer Science Symposium in Russia (CSR-2009), volume 5675, pages 105–116, 2009.
  * [23] V. Guruswami, J. Håstad, R. Manokaran, P. Raghavendra, and M. Charikar.  Beating the random ordering is hard: Every ordering CSP is approximation resistant.  SIAM Journal on Computing, 40(3):878–914, 2011.
  * [24] W. Guttmann and M. Maucher.  Variations on an ordering theme with constraints.  In Proc. 4th IFIP International Conference on Theoretical Computer Science (TCS-2006), volume 209, pages 77–90, 2006.
  * [25] R. Hirsch.  Relation algebras of intervals.  Artificial intelligence, 83(2):267–295, 1996.
  * [26] W. Hodges.  Model theory.  Cambridge University Press, 1993.
  * [27] I. Holyer.  The NP-completeness of some edge-partition problems.  SIAM Journal on Computing, 10(4):713–717, 1981.
  * [28] J. Huang.  Compactness and its implications for qualitative spatial and temporal reasoning.  In Proc. 13th International Conference on Principles of Knowledge Representation and Reasoning (KR-2012), 2012.
  * [29] P. Jonsson.  Constants and finite unary relations in qualitative constraint reasoning.  Artificial Intelligence, 257:1–23, 2018.
  * [30] P. Jonsson and V. Lagerkvist.  An initial study of time complexity in infinite-domain constraint satisfaction.  Artificial Intelligence, 245:115–133, 2017.
  * [31] P. Jonsson and V. Lagerkvist.  Why are CSPs based on partition schemes computationally hard?  In Proc. 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS-2018), pages 43:1–43:15, 2018.
  * [32] P. Jonsson, V. Lagerkvist, and S. Ordyniak.  Reasoning short cuts in infinite domain constraint satisfaction: Algorithms and lower bounds for backdoors.  In Proc. 27th International Conference on Principles and Practice of Constraint Programming (CP-2021), pages 32:1–32:20, 2021.
  * [33] P. Jonsson, V. Lagerkvist, and S. Ordyniak.  Computational short cuts in infinite domain constraint satisfaction.  Journal of Artificial Intelligence Research, 75, 2022.
  * [34] P. Jonsson, V. Lagerkvist, and G. Osipov.  Acyclic orders, partition schemes and CSPs: Unified hardness proofs and improved algorithms.  Artificial Intelligence, 296:103505, 2021.
  * [35] D. Kavvadias and M. Sideri.  The inverse satisfiability problem.  SIAM Journal on Computing, 28:152–163, 1998.
  * [36] P. Kilby, J. Slaney, S. Thiébaux, and T. Walsh.  Backbones and backdoors in satisfiability.  In Proc. 20th National Conference on Artificial Intelligence (AAAI-2005), page 1368–1373, 2005.
  * [37] M. Kronegger, S. Ordyniak, and A. Pfandler.  Backdoors to planning.  Artificial Intelligence, 269:49–75, 2019.
  * [38] V. Lagerkvist and B. Roy.  Complexity of inverse constraint problems and a dichotomy for the inverse satisfiability problem.  Journal of Computer and System Sciences, 117:23–39, 2021.
  * [39] H. W. Lenstra, Jr.  Integer programming with a fixed number of variables.  Mathematics of Operations Research, 8(4):538–548, 1983.
  * [40] G. Ligozat and J. Renz.  What is a qualitative calculus? A general framework.  In Proc. 8th Pacific Rim International Conference on Artificial Intelligence (PRICAI-2004), pages 53–64, 2004.
  * [41] A. Meier, S. Ordyniak, M. S. Ramanujan, and I. Schindler.  Backdoors for linear temporal logic.  Algorithmica, 81(2):476–496, 2019.
  * [42] R. H. Möhring, M. Skutella, and F. Stork.  Scheduling with AND/OR precedence constraints.  SIAM Journal on Computing, 33(2):393–415, 2004.
  * [43] D. Morrison, S. Jacobson, J. Sauppe, and E. Sewell.  Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning.  Discrete Optimization, 19:79–102, 2016.
  * [44] B. Nebel and H. Bürckert.  Reasoning about temporal relations: A maximal tractable subclass of allen’s interval algebra.  Journal of the ACM, 42(1):43–66, 1995.
  * [45] S. Ordyniak, A. Schidler, and S. Szeider.  Backdoor DNFs.  In Proc. 30th International Joint Conference on Artificial Intelligence (IJCAI-2021), pages 1403–1409, 2021.
  * [46] A. Pfandler, S. Rümmele, and S. Szeider.  Backdoors to abduction.  In Proc. 23rd International Joint Conference on Artificial Intelligence (IJCAI-2013), pages 1046–1052, 2013.
  * [47] J. Renz and B. Nebel.  On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus.  Artificial Intelligence, 108(1-2):69–123, 1999.
  * [48] M. Samer and S. Szeider.  Backdoor trees.  In Proc. 23rd AAAI Conference on Artificial Intelligence (AAAI-2008), pages 363–368, 2008.
  * [49] M. Samer and S. Szeider.  Backdoor sets of quantified boolean formulas.  Journal of Automated Reasoning, 42(1):77–97, 2009.
  * [50] M. Samer and S. Szeider.  Fixed-parameter tractability.  In Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications, pages 425–454. IOS Press, 2009.
  * [51] M. Sioutis and T. Janhunen.  Towards leveraging backdoors in qualitative constraint networks.  In Proc. 42nd German Conference on AI (KI-2019), pages 308–315, 2019.
  * [52] M. Vilain and H. Kautz.  Constraint propagation algorithms for temporal reasoning.  In Proc. 5th National Conference on Artificial Intelligence (AAAI-1986), pages 377–382, 1986.
  * [53] R. Williams, C. Gomes, and B. Selman.  Backdoors to typical case complexity.  In Proc. 18th International Joint Conference on Artificial Intelligence (IJCAI-2003), pages 1173–1178, 2003.
  * [54] G. Woeginger.  Exact algorithms for NP-hard problems: a survey.  In M. Juenger, G. Reinelt, and G. Rinaldi, editors, Combinatorial Optimization – Eureka! You Shrink!, pages 185–207, 2000.
  * [55] D. Zhuk.  A proof of the CSP dichotomy conjecture.  Journal of the ACM, 67(5):30:1–30:78, 2020.