# On the Fredholm determinant of the confluent hypergeometric kernel with
discontinuities

Shuai-Xia Xu1, Shu-Quan Zhao2,∗, and Yu-Qiu Zhao2

1Institut Franco-Chinois de l’Energie Nucléaire, Sun Yat-sen University,
Guangzhou 510275, China
2Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
∗E-mail address<EMAIL_ADDRESS>
Abstract  We consider the determinantal point process with the confluent
hypergeometric kernel. This process is a universal point process in random
matrix theory and describes the distribution of eigenvalues of large random
Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-
Hilbert method, we study the generating function of this process on any given
number of intervals. It can be expressed as the Fredholm determinant of the
confluent hypergeometric kernel with $n$ discontinuities. In this paper, we
derive an integral representation for the determinant by using the Hamiltonian
of the coupled Painlevé V system. By evaluating the total integral of the
Hamiltonian, we obtain the asymptotics of the determinant as the $n$
discontinuities tend to infinity up to and including the constant term. Here
the constant term is expressed in terms of the Barnes $G$-function.

Keywords  Confluent hypergeometric kernel; Coupled Painlevé V system;
Asymptotic expansion; Fredholm determinant; Riemann-Hilbert approach.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Applications
  2. 2 Model RH problem and the coupled Painlevé V system
    1. 2.1 RH problem for the Fredholm determinant
    2. 2.2 Model RH problem
    3. 2.3 Lax pair and the coupled Painlevé V system
    4. 2.4 Differential identity
  3. 3 Asymptotic Analysis as $t$ $\rightarrow$ $0^{+}$
    1. 3.1 Local parametrix
    2. 3.2 Ratio RH problem
    3. 3.3 Asymptotic behavior as $t\rightarrow 0^{+}$
  4. 4 Asymptotic Analysis as $t$ $\rightarrow$ $+\infty$
    1. 4.1 Normalization
    2. 4.2 Opening lenses and deformation
    3. 4.3 Global parametrix
    4. 4.4 Local parametrices
    5. 4.5 Final transformation
    6. 4.6 Asymptotic behaviors as $t\rightarrow+\infty$
  5. 5 Proof of main theorems
    1. 5.1 Proof of Theorem 1
    2. 5.2 Proof of Theorem 2
  6. A The confluent hypergeometric parametrix

##  1 Introduction

In random matrix theory, one of the most important problems is to study the
local statistics of eigenvalues for large random matrices. Compared with the
global perspective in which the limiting average densities of eigenvalues are
diverse, the local statistics of eigenvalues for large random matrices display
remarkable universality. For instance, for general large unitary invariant
random matrices, the local eigenvalue distributions near a regular point in
the bulk, near the soft edge and the hard edge can be described by the
determinantal point processes associated with the sine kernel, Airy kernel and
Bessel kernel, respectively; see [1, 34, 47]. The gap probabilities in these
processes can then be expressed in terms of the Fredholm determinants with the
kernels associated. It is remarkable that the Fredholm determinants admit
integral representations in terms of the Painlevé transcendents; [1, 34, 35,
42, 49, 50].

A different type of local statistics will arise if we consider the unitary
ensemble with Fisher-Hartwig singularity as follows [28, 37, 43]:

$p_{n}\left(x_{1},\dots,x_{n}\right)=\frac{1}{Z_{n}}\prod_{i=1}^{n}e^{-V(x_{i})}\left|x_{i}\right|^{2\alpha}\chi_{\beta}\left(x_{i}\right)\prod_{i<j}\left(x_{i}-x_{j}\right)^{2},$
(1.1)

where $Z_{n}$ is the normalization constant, the potential $V$ is a general
real valued function with enough increase at infinity. The parameters satisfy
$\alpha>-\frac{1}{2}$, $\beta\in i\mathbb{R}$, and

$\chi_{\beta}(x)=\left\\{\begin{array}[]{ll}e^{\beta\pi i},&x<0,\\\
e^{-\beta\pi i},&x>0.\end{array}\right.$

The root-type singularity of the form $|x-a|^{2\alpha}$ and the jump-type
singularity such as $\chi_{\beta}(x-a)$ are known as Fisher-Hartwig (FH)
singularities in the literature. The asymptotics of the large Toeplitz
determinants with these types of singularities were first investigated and
some formulas on the asymptotics were conjectured by Fisher and Hartwig in
[33] and subsequently in [45] by Lenard, motivated by their applications in
Ising models, random walks and one-dimensional Bose gases.

When the limiting average density function of the ensemble (1.1) is positive
at the origin, the scaling limit of the correlation function near the origin
is now described by the following confluent hypergeometric kernel

$K^{(\alpha,\beta)}(x,y)=\frac{1}{2\pi
i}\frac{\Gamma(1+\alpha+\beta)\Gamma(1+\alpha-\beta)}{\Gamma(1+2\alpha)^{2}}\frac{\mathbb{A}(x)\mathbb{B}(y)-\mathbb{A}(y)\mathbb{B}(x)}{x-y},$
(1.2)

where

$\mathbb{A}(x)=\chi_{\beta}(x)^{1/2}|2x|^{\alpha}e^{-ix}\phi(1+\alpha+\beta,1+2\alpha,2ix),$
(1.3)

and $\mathbb{B}(x)$ is the complex conjugation of $\mathbb{A}(x)$. Here
$\phi(a,b,z)$ denotes the confluent hypergeometric function

$\phi(a,b,z)=1+\sum_{k=1}^{\infty}\frac{a(a+1)\dots(a+k-1)z^{k}}{b(b+1)\dots(b+k-1)k!}.$
(1.4)

For special parameters, the confluent hypergeometric kernel reduces to the
sine kernel and Bessel kernel:

$K^{(0,0)}(x,y)=K_{\sin}(x,y)=\frac{\sin(x-y)}{\pi(x-y)},$ (1.5)
$K^{(\alpha,0)}(x,y)=K_{\text{Bess,1
}}(x,y)=\frac{|x|^{\alpha}|y|^{\alpha}}{x^{\alpha}y^{\alpha}}\frac{\sqrt{xy}}{2}\frac{J_{\alpha+\frac{1}{2}}(x)J_{\alpha-\frac{1}{2}}(y)-J_{\alpha-\frac{1}{2}}(x)J_{\alpha+\frac{1}{2}}(y)}{x-y}.$
(1.6)

Analogously to the sine, Airy and Bessel point processes, the determinantal
point process defined by the confluent hypergeometric kernel represents
another universality class in random matrix theory. It describes the local
statistics of the eigenvalues of general random matrices near the Fisher-
Hartwig singularity as shown in (1.1). This process arises also in the studies
of the representation theory of the infinite-dimensional unitary group [4, 5].

In the present work, we consider the Fredholm determinant of the confluent
hypergeometric kernel (1.2) with discontinuities

$F(t\vec{r},\vec{\gamma};\alpha,\beta)=\det\left(I-K_{\sigma}^{(\alpha,\beta)}\right),$
(1.7)

where $t>0$, and $K_{\sigma}^{(\alpha,\beta)}$ denotes the integral operator
acting on $L^{2}(tr_{0},tr_{n})$ with the confluent hypergeometric kernel with
discontinuities $\sigma(y)K^{(\alpha,\beta)}(x,y)$. Here the function
$\sigma(x)$ is a step function defined by

$\sigma(x)=\sum_{k=0}^{n-1}\gamma_{k}\chi_{\left(r_{k}t,r_{k+1}t\right)}(x).$
(1.8)

The parameters
$\vec{\gamma}=\left(\gamma_{0},\dots,\gamma_{n-1}\right)\in[0,1]^{n}$,
$\vec{r}=\left(r_{0},r_{1},\dots,r_{n}\right)\in\mathbb{R}^{n+1}$ satisfy
$r_{0}<r_{1}<\dots<r_{n}$ and $r_{m}=0$ for some $0\leq m\leq n$. The
determinant represents the moment generating function of the confluent
hypergeometric process. To see this, we denote by $N(s)$ the random variable
counting the number of particles in this process falling into the interval
$[0,s]$ for $s>0$ and $[s,0]$ for $s<0$. The determinant (1.7) can be written
as the moment generating function

$F(t\vec{r},\vec{\gamma};\alpha,\beta)=\mathbb{E}\left(\prod_{j=0}^{n}e^{2\pi
ih_{j}N(tr_{j})}\right),$ (1.9)

where $h_{j}$ is related to the parameters $\vec{\gamma}$ by
$h_{j}=\frac{1}{2\pi i}\ln\frac{1-\gamma_{j-1}}{1-\gamma_{j}}$ for $m<j\leq n$
and $h_{j}=\frac{1}{2\pi i}\ln\frac{1-\gamma_{j}}{1-\gamma_{j-1}}$ for $0\leq
j\leq m$ with $\gamma_{-1}=\gamma_{n}=0$.

The Fredholm determinant (1.7), or equivalently the moment generating function
(1.9), contains information about the statistics of the confluent
hypergeometric process. For example, by taking $n=2$ in (1.9) we have

$F((r_{0}t,r_{1}t,r_{2}t),\gamma(1,1);\alpha,\beta)=\det\left(1-\gamma\left.K^{(\alpha,\beta)}\right|_{(r_{0}t,r_{2}t)}\right),$
(1.10)

where $\left.K^{(\alpha,\beta)}\right|_{(r_{0}t,r_{2}t)}$ denotes the integral
operator acting on $L^{2}(r_{0}t,r_{2}t)$ with the confluent hypergeometric
kernel $K^{(\alpha,\beta)}(x,y)$. Therefore, the quantity
$F((r_{0}t,r_{1}t,r_{2}t),(1,1);\alpha,\beta)$ gives the gap probability of
finding no particles of the confluent hypergeometric process on the interval
$(r_{0}t,r_{2}t)$. While $F((r_{0}t,r_{1}t,r_{2}t),\gamma(1,1);\alpha,\beta)$
is the gap probability of having no particles falling in the interval
$(r_{0}t,r_{2}t)$ in the thinned confluent hypergeometric process, where each
of the particles in the process is removed independently with the probability
$1-\gamma\in[0,1)$ [3]. Generally,
$F((r_{0}t,0,r_{2}t),(\gamma_{1},\gamma_{2});\alpha,\beta)$ with $r_{0}<0$ and
$r_{2}>0$, describes the probability to find no particles in the interval
$(r_{0}t,r_{2}t)$ in the thinned confluent hypergeometric process, where the
positive particles in the process are removed independently with the
probability $1-\gamma_{2}\in[0,1)$ and the negative particles in the process
are removed independently with the probability $1-\gamma_{1}\in[0,1)$.

In [30], the large gap asymptotics of the confluent hypergeometric process,
which is given by $F((-t,0,t),(1,1);\alpha,\beta)$, was derived by Deift,
Krasovsky, and Vasilevska by considering the asymptotics of the Toeplitz
determinant generated by certain symbols with Fisher-Hartwig singularity on
the unit circle. Later in [55], the first and third authors of the present
paper established an integral representation of the Fredholm determinant
$F((-t,0,t),(1,1);\alpha,\beta)$ in terms of the Hamiltonian of a coupled
Painlevé V system by considering the double scaling limit of the Toeplitz
determinant with merging singularitites. By evaluating the total integral of
the Hamiltonian, the large gap asymptotics was then derived therein. Recently,
the large gap probability asymptotics of the deformed Fredholm determinant
$F((-t,0,t),(\gamma,\gamma);\alpha,\beta)$ was derived by Dai and Zhai in [25]
by evaluating again the total integral of the Hamiltonian of the coupled
Painlevé V system. It is also worth mentioning that the integral
representations of the gap probabilities and their asymptotics of some other
determinantal processes arising in random matrix theory have been considered,
such as the Pearcey process and the hard edge Pearcey process in [17, 22, 23,
24], the higher order Airy processes [8, 46, 53] and the process defined by
the Painlevé II kernel [7, 54]. The integral representations and the
asymptotics of the moment generating functions for the sine, Airy and Bessel
point processes are considered in [10, 11, 12, 14, 18].

The studies of the asymptotics of the Fredholm determinants with jump-type FH
singularities similar to (1.7) are closely related to the problem of deriving
the asymptotics of the Toeplitz and Hankel determinants with FH singularities.
The problem has a long history tracing back to the pioneering work of Fisher
and Hartwig [33], and has been widely explored due partially to their various
important applications of which we have mentioned a few in the previous
context. For example, the conjectures proposed in [33] was proved later by
Widom in [51]. The asymptotics of the Toeplitz determinants with several
separate FH singularities have been obtained by Deift, Its and Krasovsky in
[28, 29]. Recently, the asymptotics of the Toeplitz determinants with merging
singularities have been derived in [20, 21, 32]. There are also important
progresses in the studies of the asymptotics of the Hankel determinants with
FH singularities [44, 39, 2, 19, 13, 52, 9, 16, 15].

In the present work, we formulate a Riemann-Hilbert (RH, for short)
representation for the determinant (1.7) of the confluent hypergeometric
kernel (1.2) with $n$ discontinuities by using the general theory of
integrable operator developed in [38]. By exploring the RH problem, we obtain
an integral representation for the determinant (1.7) in terms of the
Hamiltonian of the coupled Painlevé V system of $2n$ dimensions. The large gap
asymptotics of the determinant up to and including the constant term are then
derived by evaluating the integral of the Hamiltonian.

###  1.1 Main results

To state our main results, it is convenient to introduce the couple Painlevé V
system defined by the following Hamiltonian dynamic system

$\frac{dv_{k}}{dt}=\frac{\partial H}{\partial
u_{k}},\quad\frac{du_{k}}{dt}=-\frac{\partial H}{\partial v_{k}},\quad
k=0,1,\dots,n,\quad k\neq m$ (1.11)

for any specific $m$ and $0\leq m\leq n$, where the Hamiltonian $H$ is defined
by

$t{H(t)}=\sum_{k\neq
m}-2itr_{k}H_{V}(u_{k},v_{k},-2itr_{k};\alpha,\beta)+\frac{1}{2}\sum_{j\neq
k;\;j,k\neq m}u_{j}u_{k}(v_{j}+v_{k})(v_{j}-1)(v_{k}-1),$ (1.12)

with $r_{k}\neq 0$ for $k\neq m$. Here $H_{V}$ denotes the Hamiltonian for the
Painlevé V equation defined in [36, 41]

${sH_{V}(u,v,s;\alpha,\beta)}=-suv-\alpha u\left(v^{2}-1\right)-\beta
u(v-1)^{2}+u^{2}v(v-1)^{2}.$ (1.13)

From (1.11)-(1.13), the coupled Painlevé V system can be expressed explicitly
as follows

$\left\\{\begin{array}[]{l}\begin{aligned}
t\frac{du_{k}}{dt}=&-2itu_{k}r_{k}-u_{k}\sum_{j\neq
m}u_{j}\left(v_{j}-1\right)^{2}-2u_{k}v_{k}\left(\sum_{j\neq
m}u_{j}\left(v_{j}-1\right)\right)\\\ &+2(\alpha+\beta)u_{k}v_{k}-2\beta
u_{k},\end{aligned}\\\ \begin{aligned}
t\frac{dv_{k}}{dt}=&2itv_{k}r_{k}+v_{k}\left(\sum_{j\neq
m}u_{j}\left(v_{j}-1\right)^{2}\right)+v_{k}^{2}\left(\sum_{j\neq
m}u_{j}\left(v_{j}-1\right)\right)\\\ &-\sum_{j\neq
m}u_{j}v_{j}\left(v_{j}-1\right)-\alpha\left(v_{k}^{2}-1\right)-\beta\left(v_{k}-1\right)^{2}.\end{aligned}\end{array}\right.$
(1.14)

When $n=2$, the above system of equations reduce to the one used in [25, 55].

As the first result, we obtain an integral representation of the determinant
(1.7) by using the Hamiltonian associated with a family of solutions of the
coupled Painlevé V system (1.14).

######  THEOREM 1.

Let $\alpha>-\frac{1}{2}$, $\beta\in i\mathbb{R}$, $n\geq 1$,
$\vec{\gamma}=\left(\gamma_{0},\gamma_{1},\dots,\gamma_{n-1}\right)\in[0,1)^{n}$
and $\vec{r}=\left(r_{0},r_{1},\dots,r_{n}\right)\in\mathbb{R}^{n+1}$ be such
that $r_{0}<\dots<r_{m}=0<\dots<r_{n}$, we have the following integral
expression of the Fredholm determinant defined in (1.7)

$\ln F(t\vec{r},\vec{\gamma};\alpha,\beta)=\int_{0}^{t}H(s)ds,$ (1.15)

where the Hamiltonian $H$ is defined by (1.11) and (1.12) subject to the
following asymptotics: as $t\rightarrow 0^{+}$

$H(t)=i\sum_{k=0}^{n}\frac{c_{k}\cdot\Gamma(1+\alpha-\beta)\Gamma(1+\alpha+\beta)\left(2|r_{j}|\right)^{2\alpha+1}}{(2\alpha+1)\Gamma^{2}(1+2\alpha)}t^{2\alpha}+\mathcal{O}\left(t^{2\alpha+1}\right),$
(1.16)

with

$c_{k}=\begin{cases}\frac{\gamma_{k-1}-\gamma_{k}}{2\pi i}e^{\beta\pi
i},&0\leq k<m,\\\ \frac{\gamma_{k}-\gamma_{k-1}}{2\pi i}e^{-\beta\pi
i},&m<k\leq n,\end{cases}$ (1.17)

and as $t\rightarrow+\infty$

$H(t)=\sum_{k=0}^{n}2ib_{k}r_{k}-\left({\sum_{k=0}^{n}b_{k}^{2}+2\beta
b_{m}}\right){t^{-1}}+\mathcal{O}\left(t^{-2}\right),$ (1.18)

with

$b_{k}=\frac{1}{2\pi i}\ln\frac{1-\gamma_{k-1}}{1-\gamma_{k}},$ (1.19)

where $\gamma_{-1}=\gamma_{n}=0$.

######  REMARK 1.

For $n=2$, with the special parameter $r_{0}=-1,r_{1}=0,r_{2}=1$ and
$\gamma_{0}=\gamma_{1}=\gamma$, (1.8) is reduced into
$\sigma(x)=\gamma_{0}\chi_{(-t,0)}(x)+\gamma_{1}\chi_{(0,t)}(x)=\gamma\chi_{(-t,t)}(x)$.
The equations (1.15) and (1.18) recover the integral representation of the
determinant $F((-t,0,t),(\gamma,\gamma);\alpha,\beta)$, which was derived by
Dai and Zhai in [25]. While the determinant $F((-t,0,t),(1,1);\alpha,\beta)$
was expressed in terms of another solution of the coupled Pailevé V system
characterized by different asymptotic behavior as $t\to+\infty$, as shown by
the first and third authors of the paper in [55]. For general $n$, the
integral representation (1.15) is also true when some of the parameters
$\gamma_{j}=1$, but the asymptotic behavior would be quite different from
(1.18). We will consider this problem in a separate paper.

######  THEOREM 2.

Under the same assumptions as in Theorem 1, we have the asymptotic
approximation as $t\rightarrow+\infty$

$\displaystyle\ln F(t\vec{r},\vec{\gamma};\alpha,\beta)=$
$\displaystyle\sum_{k\neq m}2ib_{k}r_{k}t+\sum_{k\neq m}\left(2\beta
b_{k}-2b_{k}^{2}\right)\ln\left|2r_{k}t\right|-\sum_{0\leq j<k\leq n;\;j,k\neq
m}2b_{j}b_{k}\ln\left|\frac{2r_{j}r_{k}t}{r_{k}-r_{j}}\right|$ (1.20)
$\displaystyle-\frac{\alpha}{2}\ln\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]+\ln\frac{G\left(1+\alpha+\beta+b_{m}\right)G\left(1+\alpha-\beta-
b_{m}\right)}{G(1+\alpha+\beta)G(1+\alpha-\beta)}$ $\displaystyle+\sum_{k\neq
m}\ln\left[G\left(1+b_{k}\right)G\left(1-b_{k}\right)\right]+\mathcal{O}\left(\frac{1}{t}\right),$

where $b_{k}$ is defined by (1.19) and $G$ is the Barnes $G$-function. The
asymptotics (1.20) are uniform for $\gamma_{k}$ in compact subsets of $[0,1)$
and uniform for $r_{k}$ in compact subsets of $\mathbb{R}$, as long as there
exists $\epsilon>0$ such that $\min_{0\leq j<k\leq n}(r_{k}-r_{j})>\epsilon$.

Furthermore, (1.20) can be differentiated any $k$ times with respect to
$\gamma_{j}$ in the form $\prod_{j=0}^{n-1}\partial_{\gamma_{j}}^{k_{j}}$,
where $k_{0},\dots,k_{n-1}\in\mathbb{N}\cup\\{0\\}$ and
$k=\sum_{j=0}^{n-1}k_{j}$, with the error term replaced by
$\mathcal{O}\left(\frac{(\ln t)^{k}}{t}\right)$.

For $n=2$ with the special parameter $r_{0}=-1,r_{1}=0,r_{2}=1$ and
$\gamma_{0}=\gamma_{1}=\gamma\in[0,1)$, we recover the asymptotics of the
determinant $F((-t,0,t),(\gamma,\gamma);\alpha,\beta)$ in [25].

######  COROLLARY 1.

(Dai and Zhai [25]) For $\alpha>-\frac{1}{2}$, $\beta\in i\mathbb{R}$ and
$\gamma\in[0,1)$, let $\left.K^{(\alpha,\beta)}\right|_{(-t,t)}$ be the
integral operator acting on $L^{2}(-t,t)$ with the confluent hypergeometric
kernel defined in (1.2). Then, as $t\rightarrow+\infty$, we have

$\ln\det\left(1-\gamma\left.K^{(\alpha,\beta)}\right|_{(-t,t)}\right)=-4ct+2c^{2}\ln(4t)+2\alpha\pi
c+2\ln\left[G(1+ic)G(1-ic)\right]+\mathcal{O}\left(\frac{1}{t}\right),$ (1.21)

where $c=-\frac{1}{2\pi}\ln(1-\gamma)$.

While for the special parameter $\alpha=\beta=0$, we recover the asymptotics
of the determinant of the sine kernel with several discontinuities in [11].

######  COROLLARY 2.

(Charlier [11]) Let $K_{\sigma}^{(0,0)}$ denotes the integral operator acting
on $L^{2}(tr_{0},tr_{n})$ with the sine kernel multiplied by the function
$\sigma$ given in (1.8). Then, as $t\rightarrow+\infty$, we have

$\displaystyle\ln\det(I-K_{\sigma}^{(0,0)})=$ $\displaystyle\sum_{k\neq
m}2ib_{k}r_{k}t-\sum_{k\neq m}2b_{k}^{2}\ln\left|2r_{k}t\right|-\sum_{0\leq
j<k\leq n;\;j,k\neq
m}2b_{j}b_{k}\ln\left|\frac{2r_{j}r_{k}t}{r_{k}-r_{j}}\right|$ (1.22)
$\displaystyle+\sum_{k=0}^{n}\ln\left[G\left(1+b_{k}\right)G\left(1-b_{k}\right)\right]+\mathcal{O}\left(\frac{1}{t}\right).$

###  1.2 Applications

Consider the counting function $N(t)$ of the number of particles in the
confluent hypergeometric process falling into the interval $[0,t]$ for $t>0$
and $[t,0]$ for $t<0$. As shown in (1.9), the moment generating function for
the counting functions can be expressed in terms of the confluent
hypergeometric-kernel determinant. By using Theorem 2, we derive the
asymptotics for the mean, variance and covariance of the counting functions.

######  COROLLARY 3.

For $r_{2}>r_{1}>0$ fixed, as $t\rightarrow+\infty$, we have

$\mathbb{E}\left(N(tr_{1})\right)={\mu_{\alpha}(tr_{1})+\frac{\beta}{i\pi}\delta(tr_{1})}+\theta_{\alpha,\beta,1}+\mathcal{O}\left(\frac{\ln
t}{t}\right),$ (1.23)
$\mathbb{E}\left(N(-tr_{1})\right)=\mu_{\alpha}(tr_{1})-\frac{\beta}{i\pi}\delta(tr_{1})-\theta_{\alpha,\beta,1}+\mathcal{O}\left(\frac{\ln
t}{t}\right),$ (1.24) $\operatorname{Var}\left(N(\pm
tr_{1})\right)=\frac{\delta(tr_{1})-{(\ln
G)}^{{}^{\prime\prime}}(1)}{\pi^{2}}+\theta_{\alpha,\beta,2}+\mathcal{O}\left(\frac{{(\ln
t)^{2}}}{t}\right),$ (1.25)
$\operatorname{Cov}\left(N(tr_{1}),N(tr_{2})\right)=\Sigma(tr_{1},tr_{2})+\theta_{\alpha,\beta,2}+\mathcal{O}\left(\frac{(\ln
t)^{2}}{t}\right),$ (1.26)
$\operatorname{Cov}\left(N(tr_{1}),N(-tr_{2})\right)=-\Sigma(tr_{1},tr_{2})-\theta_{\alpha,\beta,2}+\mathcal{O}\left(\frac{(\ln
t)^{2}}{t}\right),$ (1.27)

where

$\mu_{\alpha}(x)=\frac{x}{\pi}-\frac{\alpha}{2},\quad\delta(x)=\ln
2x,\quad\Sigma(x,y)=\frac{1}{2\pi^{2}}\ln\frac{2xy}{|x-y|},{\quad\text{for
}x,y>0.}$ (1.28)

Here the parameters

$\theta_{\alpha,\beta,1}=\frac{1}{2\pi i}\left({(\ln
G)}^{{}^{\prime}}(1+\alpha-\beta)-(\ln
G)^{{}^{\prime}}(1+\alpha+\beta)\right),$ (1.29)
$\theta_{\alpha,\beta,2}=-\frac{1}{4\pi^{2}}\left((\ln
G)^{{}^{\prime\prime}}(1+\alpha+\beta)+(\ln
G)^{{}^{\prime\prime}}(1+\alpha-\beta)\right),$ (1.30)

and $-{(\ln G)}^{{}^{\prime\prime}}(1)-1=\gamma_{E}\approx 0.5772$ is Euler’s
constant.

###### Proof.

From (1.9) with $n=1$ and the parameters $0=r_{0}<r_{1}$, $b_{m}=b_{0}=-b$, we
obtain the following equation

$F(t(0,r_{1}),1-e^{2\pi ib};\alpha,\beta)=\mathbb{E}\left(e^{2\pi
ibN(tr_{1})}\right).$ (1.31)

Thus, we derive that

${\mathbb{E}\left(N(tr_{1})\right)=\frac{1}{2\pi i}\left.\partial_{b}\ln
F(t(0,r_{1}),1-e^{2\pi ib};\alpha,\beta)\right|_{b=0},}$ (1.32)

and

${\operatorname{Var}\left(N(tr_{1})\right)=-\frac{1}{4\pi^{2}}\left.\partial_{b}^{2}\ln
F(t(0,r_{1}),1-e^{2\pi ib};\alpha,\beta)\right|_{b=0}.}$ (1.33)

Similarly, we compute the covariance between $N(tr_{1})$ and $N(tr_{2})$ as
follows

${\operatorname{Cov}\left(N(tr_{1}),N(tr_{2})\right)=-\frac{1}{8\pi^{2}}\left.\partial_{b}^{2}\ln\left(\frac{F\left(t(0,r_{1},r_{2}),(1-e^{4\pi
ib},1-e^{2\pi ib});\alpha,\beta\right)}{F\left(t(0,r_{1}),1-e^{2\pi
ib};\alpha,\beta\right)F\left(t(0,r_{2}),1-e^{2\pi
ib};\alpha,\beta\right)}\right)\right|_{b=0}.}$ (1.34)

By using the definition of the Barnes $G$-function ([48, (5.17.3)]), we have

$G(1+z)=(2\pi)^{z/2}\exp\left(-\frac{1}{2}z(z+1)-\frac{1}{2}\gamma_{E}z^{2}\right)\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left(-z+\frac{z^{2}}{2k}\right)\right).$
(1.35)

From Theorem 2 and (1.32)-(1.35), we obtain after some direct calculations the
mean, variance of $N(r_{1}t)$ and the covariance between $N(tr_{1})$ and
$N(tr_{2})$ as given in (1.23), (1.25) and (1.26).

The mean and variance of $N(-r_{1}t)$ in (1.24) and (1.25) can be derived in
the same manner by considering (1.9) with $n=1$, specifying the parameters
$t(-r_{1},0)$ and $b_{m}=b_{1}=-b$. The formula (1.27) can be derived
similarly by considering $F((-tr_{2},0,tr_{1}),(1-e^{2\pi ib},1-e^{2\pi
ib});\alpha,\beta)$ like (1.34). We complete the proof of the corollary. ∎

From the definition of the counting function $N(t)$, the number of particles
in the confluent hypergeometric process falling into the interval $[-t,t]$ is
given by $N_{0}(t):=N(t)+N(-t)$. Combining (1.23), (1.25), (1.24) and (1.27),
we immediately derive the asymptotics of the mean and variance of the counting
function $N_{0}(t)$, which recovers the one established by Dai and Zhai in
[25].

######  COROLLARY 4.

(Dai and Zhai [25]) We have as $t\rightarrow+\infty$,

$\mathbb{E}(N_{0}(t))=\frac{2t}{\pi}-\alpha+\mathcal{O}\left(\frac{\ln
t}{t}\right),\quad\operatorname{Var}(N_{0}(t))=\frac{\ln
4t+1+\gamma_{E}}{\pi^{2}}+\mathcal{O}\left(\frac{(\ln t)^{2}}{t}\right),$
(1.36)

where $\gamma_{E}\approx 0.5772$ is Euler’s constant.

The rest of the paper is organized as follows. In Section 2, we formulate an
RH problem for the Fredholm determinant defined in (1.7) by using the general
theory of integrable operator developed in [38]. Making use of the confluent
hypergeometric functions, we transform the RH problem to a model RH problem
$Y(\eta,t)$ with constant jumps. We then derive a Lax pair for $Y(\eta,t)$ and
the compatibility conditions of the Lax pair lead us to the coupled Painlevé V
system. The Hamiltonian formulation of the system and several useful
differential identities are also established in this section. In Sections 3
and 4, we derive the asymptotic approximations of a family of solutions to the
coupled Painlevé V system and the associated Hamiltonian as $t\rightarrow
0^{+}$ and $t\to+\infty$ by performing Deift-Zhou nonlinear steepest descent
analysis [26, 27] of the RH problem for $Y(\eta,t)$. The results are stated in
Proposition 4 and 5 at the end of Sections 3 and 4, respectively. The last
section, Section 5, is devoted to the proofs of Theorems 1 and 2. For the
convenience of the reader, a brief description of the confluent hypergeometric
parametrix is put in the Appendix.

##  2 Model RH problem and the coupled Painlevé V system

In this section, we first relate the determinant (1.7) to the solution of an
RH problem by using the general theory of integrable operator developed in
[38]. Then, we derive from the RH problem the Lax pair and the Hamiltonian for
the coupled Painlevé V system (1.14). These results in the present section
will be used in Section 5 to prove Theorem 1.

###  2.1 RH problem for the Fredholm determinant

The kernel of the operator $K_{t\vec{r},\vec{\gamma},\alpha,\beta}$ can be
expressed in the following from

$\sigma(y)K^{(\alpha,\beta)}(x,y)=\frac{\vec{f}(x)^{T}\vec{g}(y)}{x-y},$ (2.1)

where

$\vec{f}(z)=\frac{1}{\sqrt{2\pi
i}}\left[\begin{array}[]{c}\frac{\Gamma(1+\alpha-\beta)}{\Gamma(1+2\alpha)}\mathbb{B}(z)\\\
\frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+2\alpha)}\mathbb{A}(z)\end{array}\right],\quad\vec{g}(z)=\frac{\sigma(z)}{\sqrt{2\pi
i}}\left[\begin{array}[]{c}-\frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+2\alpha)}\mathbb{A}(z)\\\
\frac{\Gamma(1+\alpha-\beta)}{\Gamma(1+2\alpha)}\mathbb{B}(z)\end{array}\right],$
(2.2)

with $\mathbb{A}(z)$ and $\mathbb{B}(z)$ defined in (1.3) and (1.4).
Therefore, the kernel is integrable in the sense of [38]. By standard
properties of trace class operators, we have

$\displaystyle\frac{d}{dt}\ln F$
$\displaystyle=\frac{d}{dt}\operatorname{tr}\left(\ln\left(1-K_{\sigma}^{(\alpha,\beta)}\right)\right)$
(2.3)
$\displaystyle=-\operatorname{tr}\left(\left(1-K_{\sigma}^{(\alpha,\beta)}\right)^{-1}\frac{d}{dt}K_{\sigma}^{(\alpha,\beta)}\right)$
$\displaystyle=\sum_{k=0}^{n}r_{k}\left(\lim_{z\rightarrow(r_{k}t)^{+}}R(z,z)-\lim_{z\rightarrow(r_{k}t)^{-}}R(z,z)\right),$

where $R(x,y)$ denotes the kernel of the resolvent operator
$(1-K_{\sigma}^{(\alpha,\beta)})^{-1}K_{\sigma}^{(\alpha,\beta)}$. According
to [38] and [31], the kernel is also of the integrable form

$R(x,y)=\frac{\left(\vec{F}(x),\vec{G}(y)\right)}{x-y},$ (2.4)

with

$\vec{F}(z)=\left(1-K_{\sigma}^{(\alpha,\beta)}\right)^{-1}\vec{f}(z),\quad\vec{G}(z)=\left(1-K_{\sigma}^{(\alpha,\beta)}\right)^{-1}\vec{g}(z).$
(2.5)

Furthermore, $\vec{F}$ and $\vec{G}$ can be expressed as

$\vec{F}(z)=m_{+}(z)\vec{f}(z),\quad\vec{G}(z)=m_{+}(z)^{-T}\vec{g}(z).$ (2.6)

Here $m(z)$ solves the following RH problem.

### RH problem for $m(z)$

  * (1)

$m(z)$ analytic in $\mathbb{C}\backslash[r_{0}t,r_{n}t]$,

  * (2)

$m_{+}(z)=m_{-}(z)\left(I-2\pi i\vec{f}(z){\vec{g}(z)}^{T}\right)$, on
$\displaystyle{\cup_{k=0}^{n-1}\left(r_{k}t,r_{k+1}t\right)}$,

  * (3)

$m(z)\rightarrow I$ , as $z\rightarrow\infty$.

Conversely, we have [38]

$m(z)=I-\int_{r_{0}t}^{r_{n}t}\frac{\vec{F}(s){\vec{g}(s)}^{T}}{s-z}ds,$ (2.7)

with $\vec{F}(s)$ defined by using the inverse of the operator
$I-K_{\sigma}^{(\alpha,\beta)}$ as given in (2.5). The determinant
$\det(1-K_{\sigma}^{(\alpha,\beta)})$ is strictly positive since it represents
the gap probability of the thinned confluent hypergeometric process.
Therefore, the operator $I-K_{\sigma}^{(\alpha,\beta)}$ is invertible and
hence the existence of solution to the RH for $Y(z)$ is justified.

###  2.2 Model RH problem

Next, we transform the RH problem for $m(z)$ to a new RH problem with constant
jumps. To begin with, we note that the function $\vec{f}(z)$ and $\vec{g}(z)$
defined in (2.2) can be expressed as

$\vec{f}(z)=e^{\frac{\beta}{2}\pi
i\sigma_{3}}\Phi(2z)e^{\pm\frac{\alpha}{2}\pi
i\sigma_{3}}e^{-\frac{\beta}{2}\pi i\sigma_{3}}\left(\frac{1}{\sqrt{2\pi
i}}\right)^{\sigma_{3}}\left[\begin{array}[]{l}1\\\ 0\end{array}\right],$
(2.8)

and

$\vec{g}(z)=\sigma(z)e^{-\frac{\beta}{2}\pi
i\sigma_{3}}\left(\Phi(2z)\right)^{-T}e^{\mp\frac{\alpha}{2}\pi
i\sigma_{3}}e^{\frac{\beta}{2}\pi i\sigma_{3}}\left(\frac{1}{\sqrt{2\pi
i}}\right)^{-\sigma_{3}}\left[\begin{array}[]{l}0\\\ 1\end{array}\right],$
(2.9)

for $z\in\Omega_{\Phi,1}$ and $\pm\mathtt{Re}z>0$. Here $\Phi(z)$ is the
solution of the RH problem for the confluent hypergeometric parametrix given
in the Appendix.

We introduce the transformation

$Y(z,t)=(2t)^{\beta\sigma_{3}}e^{-\frac{\beta}{2}\pi
i\sigma_{3}}m(tz)e^{\frac{\beta}{2}\pi i\sigma_{3}}\left\\{\begin{aligned}
&\Phi(2tz),\quad z\in\Omega_{Y,j},~{}~{}j=2,5,\\\
&\widehat{\Phi}_{j}(2tz),\quad
z\in\Omega_{Y,j},~{}~{}j=1,3,4,\end{aligned}\right.$ (2.10)

where $\Phi(z)$ is the solution of the RH problem for the confluent
hypergeometric parametrix given in the Appendix. The functions
$\widehat{\Phi}_{j}(z)$ denote the analytic extension of $\Phi(z)$ from the
regions $\Omega_{\Phi,j}$ to $\Omega_{Y,j}$, for $j=1,3,4$, respectively; see
Figures 1 and 5 for the regions.

Then $Y(z)=Y(z,t)$ satisfies the following RH problem.

### The model RH problem for $Y(z)$

  * (1)

$Y(z)$ is analytic for $z\in\mathbb{C}\backslash\cup{{}_{j=1}^{7}\Gamma_{j}}$,
where the oriented contour are given by

$\displaystyle\Gamma_{1}=r_{n}+e^{\frac{\pi
i}{4}}\mathbb{R}^{+},~{}~{}\Gamma_{2}=r_{0}+e^{\frac{3\pi
i}{4}}\mathbb{R}^{+},~{}~{}\Gamma_{3}=r_{0}+e^{-\frac{3\pi
i}{4}}\mathbb{R}^{+},~{}~{}\Gamma_{4}=e^{-\frac{\pi i}{2}}\mathbb{R}^{+},$
$\displaystyle\Gamma_{5}=r_{n}+e^{-\frac{\pi
i}{4}}\mathbb{R}^{+},~{}~{}\Gamma_{6}=\cup_{k=0}^{m-1}\left(r_{k},r_{k+1}\right),~{}~{}\Gamma_{7}=\cup_{k=m}^{n-1}\left(r_{k},r_{k+1}\right),$

as shown in Figure 1.

Figure 1: The jump contours and regions of the model RH problem for $Y$

  * (2)

$Y_{+}(z)=Y_{-}(z)J_{Y}(z)$ on $\cup{{}_{j=1}^{7}\Gamma_{j}}$ , where

$J_{Y}(z)=\left\\{\begin{array}[]{ll}{\left[\begin{array}[]{cc}1&0\\\
e^{-(\alpha-\beta)\pi i}&1\end{array}\right]},&{z\in\Gamma_{1}},\\\
{\left[\begin{array}[]{cc}1&0\\\ e^{(\alpha-\beta)\pi
i}&1\end{array}\right]},&{z\in\Gamma_{2}},\\\
{\left[\begin{array}[]{cc}1&-e^{-(\alpha-\beta)\pi i}\\\
0&1\end{array}\right]},&{z\in\Gamma_{3}},\\\ {e^{2\beta\pi
i\sigma_{3}}},&{z\in\Gamma_{4}},\\\
{\left[\begin{array}[]{cc}1&-e^{(\alpha-\beta)\pi i}\\\
0&1\end{array}\right]},&{z\in\Gamma_{5}},\\\
{\left[\begin{array}[]{cc}0&-e^{-(\alpha-\beta)\pi i}\\\ e^{(\alpha-\beta)\pi
i}&1-\sigma(zt)\end{array}\right]},&{z\in\Gamma_{6}},\\\
{\left[\begin{array}[]{cc}0&-e^{(\alpha-\beta)\pi i}\\\ e^{-(\alpha-\beta)\pi
i}&1-\sigma(zt)\end{array}\right]},&{z\in\Gamma_{7}}.\end{array}\right.$
(2.11)

  * (3)

As $z\rightarrow\infty$, we have for $\arg
z\in(-\frac{\pi}{2},\frac{3\pi}{2})$

$Y(z)=\left(I+\frac{Y_{1}(t)}{z}+\frac{Y_{2}(t)}{z^{2}}+\mathcal{O}\left(z^{-3}\right)\right)z^{-\beta\sigma_{3}}e^{-izt\sigma_{3}}.$
(2.12)

  * (4)

As $z\rightarrow r_{k}$ , $k\neq m$ , we have

$Y(z)=Y_{0}^{(k)}(t)\left(I+Y_{1}^{(k)}(t)\left(z-r_{k}\right)+Y_{2}^{(k)}\left(t\right)\left(z-r_{k}\right)^{2}+\ldots\right)\left[\begin{array}[]{cc}1&\tilde{c}_{k}\ln\left(z-r_{k}\right)\\\
0&1\end{array}\right],$ (2.13)

where
$\tilde{c}_{k}=\operatorname{sgn}(r_{k})c_{k}e^{\operatorname{sgn}(r_{k})\alpha\pi
i}$ and $c_{k}$ is defined in (1.17) and $z\in\Omega_{Y,1}$. The branch for
$\ln\left(z-r_{k}\right)$ is taken to be $\arg(z-r_{k})\in(0,2\pi)$ if
$r_{k}<0$, or $\arg(z-r_{k})\in(-\pi,\pi)$ if $r_{k}>0$. The behavior of
$Y(z)$ as $z\rightarrow r_{k}$ for $z$ in other regions is determined by using
the jump condition (2.11).

  * (5)

As $z\rightarrow 0$ , we have

$Y(z)=Y_{0}^{(m)}(t)\left(I+Y_{1}^{(m)}(t)z+Y_{2}^{(m)}(t)z^{2}+\ldots\right)z^{\alpha\sigma_{3}}\left\\{\begin{array}[]{l}{C_{m,1}},2\alpha\notin\mathbb{Z}^{+}\cup\\{0\\},\\\
{C_{m,2},2\alpha\in\mathbb{Z}^{+}\cup\\{0\\}},\end{array}\right.$ (2.14)

where $z\in\Omega_{Y,1}$,

$C_{m,1}=\left[\begin{array}[]{cc}1&\frac{c_{m}}{2i\sin 2\alpha\pi}\\\
0&1\end{array}\right],\quad
C_{m,2}=\left[\begin{array}[]{cc}1&\frac{e^{-2\alpha\pi i}}{2\pi i}c_{m}\ln
z\\\ 0&1\end{array}\right],$

and

$c_{m}=\left(1-\gamma_{m-1}\right)e^{(\alpha+\beta)\pi
i}-\left(1-\gamma_{m}\right)e^{-(\alpha+\beta)\pi i}.$

Both $z^{\alpha}$ and $\ln z$ take the principal branches. The behavior of
$Y(z)$ as $z\rightarrow 0$ for $z$ in other regions is determined by using the
jump condition (2.11).

###  2.3 Lax pair and the coupled Painlevé V system

In this section, we derive a Lax pair from the above model RH problem, of
which the compatibility condition gives us the coupled Painlevé V system. The
Hamiltonian for the coupled Painlevé V system is also derived.

######  PROPOSITION 1.

We have the Lax pair

$\frac{\partial}{\partial
z}Y(z,t)=\mathcal{A}(z,t)Y(z,t),\quad\frac{\partial}{\partial
t}Y(z,t)=\mathcal{B}(z,t)Y(z,t),$ (2.15)

where

$\mathcal{A}(z,t)=\sum_{k=0}^{n}\frac{A_{k}(t)}{z-r_{k}}-it\sigma_{3},$ (2.16)
$\mathcal{B}(z,t)=-iz\sigma_{3}+B(t),$ (2.17)

with the coefficients being given as

$A_{m}(t)=\left[\begin{array}[]{cc}-\beta-\sum_{k\neq
m}u_{k}(t)v_{k}(t)&\left(\alpha+\beta+\sum_{k\neq
m}u_{k}(t)v_{k}(t)\right)y(t)\\\ \frac{\alpha-\beta-\sum_{k\neq
m}u_{k}(t)v_{k}(t)}{y(t)}&\beta+\sum_{k\neq
m}u_{k}(t)v_{k}(t)\end{array}\right],$ (2.18)
$A_{k}(t)=\left[\begin{array}[]{cc}-u_{k}(t)v_{k}(t)&u_{k}(t)y(t)\\\
-\frac{u_{k}(t)v_{k}^{2}(t)}{y(t)}&u_{k}(t)v_{k}(t)\end{array}\right],\quad
k\neq m,$ (2.19)

and

$B(t)=\frac{1}{t}\left[\begin{array}[]{cc}0&d_{1}(t)y(t)\\\
\frac{d_{2}(t)}{y(t)}&0\end{array}\right],$ (2.20)

where

$\displaystyle d_{1}(t)=\alpha+\beta-\sum_{k\neq
m}u_{k}(t)\left(v_{k}(t)-1\right),$ (2.21) $\displaystyle
d_{2}(t)=\alpha-\beta-\sum_{k\neq m}u_{k}(t)v_{k}(t)\left(v_{k}(t)-1\right).$

The compatibility condition of the Lax pair is expressed as the coupled
Painlevé V system (1.14) and the following differential equation for $y(t)$

$t\frac{d}{dt}y=y\left(d_{1}-d_{2}\right).$ (2.22)

Defining the Hamiltonian by

$tH=2it{(Y_{1})}_{11}-(\alpha^{2}-\beta^{2}),$ (2.23)

where ${(Y_{1})}_{11}$ is the $(1,1)$-entry of the coefficient matrix $Y_{1}$
in the asymptotic approximation (2.12) as $z\rightarrow\infty$, then we have

$tH=\sum_{k\neq
m}-2itr_{k}H_{V}\left(u_{k},v_{k},-2itr_{k};\alpha,\beta\right)+\frac{1}{2}\sum_{j\neq
k;\;j,k\neq
m}u_{j}u_{k}\left(v_{j}+v_{k}\right)\left(v_{j}-1\right)\left(v_{k}-1\right),$
(2.24)

where $H_{V}$ is the classical Hamiltonian for the Painlevé V equation defined
in (1.13). The Hamiltonian system (1.11) is equivalent to the coupled Painlevé
V system (1.14).

###### Proof.

Observing that the jumping matrices in (2.11) are independent of both the
variables $z$ and $t$ , $Y(z,t)$ , $\frac{d}{dz}Y(z,t)$ and
$\frac{d}{dt}Y(z,t)$ have the same jump conditions. Thus, $\mathcal{A}$ and
$\mathcal{B}$ are meromorphic functions in the $z$-plane with the only
possible isolated singularities at $z=r_{k}$, $k=0,\dots,n$. Taking
(2.12),(2.13) and (2.14) into account, we then obtain that $\mathcal{A}$ and
$\mathcal{B}$ are of the form given in (2.16) and (2.17).

Next, we derive the coefficients in the $\mathcal{A}$ and $\mathcal{B}$. It is
seen from (2.12), (2.13)) and (2.14) that

$\det A_{m}=-\alpha^{2},\qquad\det A_{k}=0,~{}k\neq m.$ (2.25)

Moreover, in view of (2.12), we obtain by comparing the coefficients of $1/z$
on both sides of the first equation in (2.15)

$\sum_{k=0}^{n}A_{k}=\left[\begin{array}[]{cc}-\beta&2it(Y_{1})_{12}\\\
-2it(Y_{1})_{21}&\beta\end{array}\right].$ (2.26)

Therefore, $A_{k}$, $k=0,\dots n$ can be parametrized as (2.18) and (2.19).
Similarly, we obtain (2.20) by substituting (2.12) into the second equation of
(2.15) and using (2.26).

The compatibility condition of the Lax pair gives

$\displaystyle\frac{d}{dt}A_{k}(t)=\left[-ir_{k}\sigma_{3}+B(t),A_{k}(t)\right],\quad
k\neq m,$ (2.27)

and

$\frac{d}{dt}A_{m}(t)=\left[B(t),A_{m}(t)\right].$ (2.28)

We then have from (2.27)

$\left\\{\begin{array}[]{l}t\frac{d}{dt}\left(u_{k}v_{k}\right)=d_{1}u_{k}v_{k}^{2}+d_{2}u_{k},\quad
k\neq m,\\\\[8.5359pt]
t\frac{d}{dt}\left(u_{k}y\right)=-2itr_{k}u_{k}y+2d_{1}u_{k}v_{k}y,\quad k\neq
m.\end{array}\right.$ (2.29)

While the equation (2.28) gives us

$\left\\{\begin{array}[]{l}t\frac{d}{dt}{(A_{m})}_{11}={(A_{m})}_{11}\left(d_{1}+d_{2}\right)+\alpha\left(d_{1}-d_{2}\right),\\\\[8.5359pt]
t\frac{d}{dt}y=y\left(d_{1}-d_{2}\right),\\\ \end{array}\right.$ (2.30)

with $d_{1}$ and $d_{2}$ given in (2.21). Combining (2.29) and (2.30), we
arrive at the coupled Painlevé V system (1.14) and (2.22).

To derive the expression of the Hamiltonian, we compare the coefficient of
$\frac{1}{z^{2}}$ in the large $z$ expansion of the first equation in (2.14)
and obtain

$\left[-it\sigma_{3}Y_{1},Y_{1}\right]+\left[Y_{2},-it\sigma_{3}\right]-\beta\left[Y_{1},\sigma_{3}\right]-Y_{1}=\sum_{k=0}^{n}A_{k}r_{k}.$
(2.31)

A combination of (2.23) and (2.31) then gives (2.24). From (2.24), it is
direct to see that the Hamiltonian system (1.11) is equivalent to the coupled
Painlevé V system (1.14). This completes the proof of the proposition. ∎

For later use, we also collect several relations as what follows.

######  PROPOSITION 2.

The function $y(t)$ appears in the coefficient (2.18) is related to the
solution to the RH problem for $Y(z)$ via

$y(t)=\frac{\left(Y_{0}^{(m)}\right)_{11}}{\left(Y_{0}^{(m)}\right)_{21}},$
(2.32)

where $Y_{0}^{(m)}$ is given in the asymptotic expansion of $Y(z)$ in (2.14).
Moreover, let

$d(t)=2\alpha\left(Y_{0}^{(m)}\right)_{11}\left(Y_{0}^{(m)}\right)_{21},$
(2.33)

then $d(t)$ satisfies the differential equation

$t\frac{d}{dt}d=d\left(d_{1}+d_{2}\right),$ (2.34)

with $d_{1}$ and $d_{2}$ given in (2.21).

###### Proof.

From (2.18), we have

$y=\frac{\left(A_{m}\right)_{11}+\alpha}{\left(A_{m}\right)}.$ (2.35)

Then, by (2.14), (2.25) and (2.35), we obtain (2.32).

To derive (2.34), we obtain from (2.14) and the second equation in the Lax
pair (2.15) that

$\frac{d}{dt}Y^{(m)}_{0}(t)=B(t)Y^{(m)}_{0}(t).$ (2.36)

This, together with (2.33) and (2.22), implies (2.34).

∎

###  2.4 Differential identity

Recently, the authors in [6, 40] have developed a new method to evaluate the
total integral of the Hamiltonian. In this method, the differential identities
for Hamiltonian play an important role. In the present subsection, we derive
the differential identities for the Hamiltonian which will be used in Section
5, there we applied the method developed in [6, 40] to calculate the constant
term in the large gap asymptotics.

######  PROPOSITION 3.

We have the relation

$H=\left(\sum_{k\neq
m}u_{k}\frac{dv_{k}}{dt}-H\right)+\frac{d}{dt}\left(tH+\alpha\ln d-\beta\ln
y-2\left(\alpha^{2}-\beta^{2}\right)\ln t\right),$ (2.37)

and the following differential identities

$\frac{d}{d\gamma_{j}}\left(\sum_{k\neq
m}u_{k}\frac{dv_{k}}{dt}-H\right)=\frac{d}{dt}\left(\sum_{k\neq
m}u_{k}\frac{dv_{k}}{d\gamma_{j}}\right),$ (2.38)

with $j=0,\dots,n-1$.

###### Proof.

By using the Hamiltonian dynamic system (1.11), (2.22) and (2.34), we have

$\displaystyle\sum_{k\neq m}u_{k}\frac{dv_{k}}{dt}-H$
$\displaystyle=\sum_{k\neq m}u_{k}\frac{\partial H}{\partial u_{k}}-H$ (2.39)
$\displaystyle=\frac{1}{t}\sum_{k\neq
m}u_{k}^{2}v_{k}\left(v_{k}-1\right)^{2}+\frac{1}{t}\sum_{j<k}u_{j}u_{k}\left(v_{j}+v_{k}\right)\left(v_{j}-1\right)\left(v_{k}-1\right),$

and

$\displaystyle\frac{d}{dt}\left(tH+\alpha\ln d(t)\right.$
$\displaystyle\left.-\beta\ln y(t)-2\left(\alpha^{2}-\beta^{2}\right)\ln
t\right)$ (2.40) $\displaystyle=2i\sum_{k\neq
m}u_{k}v_{k}r_{k}-\frac{\alpha}{t}\sum_{k\neq
m}u_{k}\left(v_{k}^{2}-1\right)-\frac{\beta}{t}\sum_{k\neq
m}u_{k}\left(v_{k}-1\right)^{2}.$

Combining (2.24) with (2.39) and (2.40), we obtain (2.37).

Applying again the Hamiltonian dynamic system (1.11), we derive the
differential identity with respect to the parameters

$\displaystyle\frac{d}{d\gamma_{j}}\left(\sum_{k\neq
m}u_{k}\frac{dv_{k}}{dt}-H\right)$ $\displaystyle=\sum_{k\neq
m}\left(\frac{du_{k}}{d\gamma_{j}}\frac{dv_{k}}{dt}+u_{k}\frac{d^{2}v_{k}}{d\gamma_{j}dt}-\frac{\partial
H}{\partial u_{k}}\frac{du_{k}}{d\gamma_{j}}-\frac{\partial H}{\partial
v_{k}}\frac{dv_{k}}{d\gamma_{j}}\right)$ (2.41)
$\displaystyle=\frac{d}{dt}\left(\sum_{k\neq
m}u_{k}\frac{dv_{k}}{d\gamma_{j}}\right).$

This completes the proof of the proposition. ∎

##  3 Asymptotic Analysis as $t$ $\rightarrow$ $0^{+}$

In this section, we derive the asymptotic approximations of a family of
solutions to the coupled Painlevé V system (1.14) and the associated
Hamiltonian as $t\rightarrow 0^{+}$ by performing the Deift-Zhou nonlinear
steepest descent analysis [26] of the RH problem for $Y(z)$.

For the purpose, we consider

$X(z)=(2t)^{-\beta\sigma_{3}}Y\left(\frac{z}{2t},t\right).$ (3.1)

It is readily seen from the RH problem for $Y(z)$ that $X(z)$ shares the same
jump matrices with $\Phi(z)$ given in the Appendix, for
$|z|>\max\left\\{2\left|r_{0}t\right|,2\left|r_{n}t\right|\right\\}$.
Therefore, the solution to the RH problem for $X(z)$ can be approximated by
$\Phi(z)$ for
$|z|>\max\left\\{2\left|r_{0}t\right|,2\left|r_{n}t\right|\right\\}$ bounded
away from the origin as $t\to 0$.

In the next subsection, we construct a local parametrix $L(z)$ which has the
same jump condition as $X(z)$ in a neighborhood of the origin containing the
interval $[2r_{0}t,2r_{n}t]$.

###  3.1 Local parametrix

### RH problem for $L(z)$

  * (1)

$L(z)$ is analytic for $z\in U(0,\varepsilon)\setminus\Sigma_{X}$ with some
constant $\varepsilon>0$.

  * (2)

We have the jump relation

$L_{+}(z)=L_{-}(z)J_{Y}(z/2t)$ (3.2)

for $z\in\Sigma_{X}\cap U(0,\varepsilon)$, where $J_{Y}(z)$ is given in
(2.11).

  * (2)

As $t\rightarrow 0^{+}$, we have

$L(z)\Phi(z)^{-1}=I+O\left(t^{2\alpha+1}\right),$ (3.3)

where the error term is uniform for $z\in\partial U(0,\varepsilon)$.

Inspired by the asymptotic behavior of $\Phi(z)$ near the origin given in
(A.6) and (A.7), we construct $L(z)$ of the following form

$L(z)=\Phi^{(0)}(z)\left[\begin{array}[]{cc}1&t^{2\alpha}k\left(\frac{z}{t}\right)\\\
0&1\end{array}\right]z^{\alpha\sigma_{3}}\left\\{\begin{array}[]{ll}{\left[\begin{array}[]{cc}1&\frac{\sin(\alpha+\beta)\pi}{\sin
2\alpha\pi}\\\
0&1\end{array}\right]C_{j}},&2\alpha\notin\mathbb{N},\\\\[11.38092pt]
{\left[\begin{array}[]{cc}1&(-1)^{2\alpha}\frac{\sin(\alpha+\beta)\pi}{\pi}\ln
z\\\ 0&1\end{array}\right]C_{j}},&2\alpha\in\mathbb{N},\end{array}\right.$
(3.4)

where $\alpha>-\frac{1}{2}$ and the branches for $z^{\alpha}$ and $\ln z$ are
chosen such that $\arg z\in(-\frac{\pi}{2},\frac{3}{2}\pi)$. Here the function
$k(z)$ is analytic for $z\in\mathbb{C}\setminus[2r_{0},2r_{n}]$ with the
following jumps on $(2r_{0},2r_{n})$

$k_{+}(x)-k_{-}(x)=-|x|^{2\alpha}\sigma\left(\frac{xt}{2}\right)e^{(\alpha+\beta)\pi
i},\quad 2r_{0}<x<0,$ (3.5)

and

$k_{+}(x)-k_{-}(x)=-x^{2\alpha}\sigma\left(\frac{xt}{2}\right)e^{(\alpha-\beta)\pi
i},\quad 0<x<2r_{n},$ (3.6)

in which $\sigma(x)$ is the step function defined in (1.8). Applying the
Plemelj formula, we have

$k(z)=-\frac{e^{(\alpha+\beta)\pi i}}{2\pi
i}\int_{2r_{0}}^{0}\frac{|x|^{2\alpha}\sigma\left(\frac{xt}{2}\right)}{x-z}dx-\frac{e^{(\alpha-\beta)\pi
i}}{2\pi
i}\int_{0}^{2r_{n}}\frac{x^{2\alpha}\sigma\left(\frac{xt}{2}\right)}{x-z}dx,\quad
z\in\mathbb{C}\setminus[2r_{0},2r_{n}].$ (3.7)

It is seen from the construction (3.4) and the jump relations (3.5) and (3.6)
that the jump condition (3.2) is fulfilled.

To show (3.3), we expand $k(z)$ as $z\rightarrow\infty$:

$k(z)=-e^{\alpha\pi
i}\sum_{k=0}^{n}\frac{c_{k}}{2\alpha+1}\left(2|r_{k}|\right)^{2\alpha+1}\frac{1}{z}+\mathcal{O}\left(\frac{1}{z^{2}}\right).$
(3.8)

Substituting (3.8) into (3.4), we have as $t\to 0$

$L(z)\Phi{(z)}^{-1}=I-e^{\alpha\pi
i}\sum_{k=0}^{n}\frac{c_{k}}{2\alpha+1}\left(2|r_{k}|\right)^{2\alpha+1}\frac{t^{2\alpha+1}}{z}\Phi^{(0)}(z)\left[\begin{array}[]{ll}0&1\\\
0&0\end{array}\right]{\Phi^{(0)}(z)}^{-1}+\mathcal{O}\left(t^{2\alpha+2}\right),$
(3.9)

where the error term is uniform for $z\in\partial U(0,\varepsilon)$.
Therefore, we have the matching condition (3.3).

###  3.2 Ratio RH problem

We define the following ratio RH problem

$E(z)=\begin{cases}X(z)L(z)^{-1},&|z|<\varepsilon,\\\
X(z)\Phi{(z)}^{-1},&|z|>\varepsilon.\end{cases}$ (3.10)

Then, $E(z)$ satisfies the following RH problem.

### RH problem for $E(z)$

  * (1)

$E(z)$ is analytic for $z\in\mathbb{C}\setminus U(0,\varepsilon)$ with some
constant $\varepsilon>0$.

  * (2)

$E(z)$ satisfies the jump condition

$E_{+}(z)=E_{-}(z)J_{E}(z),\quad J_{E}(z)=\Phi{(z)}L(z)^{-1},$ (3.11)

for $z\in\partial U(0,\varepsilon)$ with the orientation clockwise.

  * (3)

$E(z)=I+\mathcal{O}\left(1/z\right)$ as $z\to\infty$.

Using (3.3) and a standard argument for the small norm RH problem as given in
[26] , we have the expansion

$E(z)=I-e^{\alpha\pi
i}\sum_{k=0}^{n}\frac{c_{k}}{2\alpha+1}\frac{(2|r_{k}|t)^{2\alpha+1}}{z}\Phi^{(0)}(0)\left[\begin{array}[]{ll}0&1\\\
0&0\end{array}\right]{\Phi^{(0)}(0)}^{-1}+\mathcal{O}\left(t^{2\alpha+2}\right),\quad|z|>\varepsilon,$
(3.12)

and

$E(z)=I+\mathcal{O}\left(t^{2\alpha+1}\right)$ (3.13)

for $|z|<\varepsilon$.

###  3.3 Asymptotic behavior as $t\rightarrow 0^{+}$

As a consequence of nonlinear steepest descent analysis of the RH problem for
$Y(z)$, we obtain the asymptotics of a family of special solutions to the
coupled Painlevé V system and the Hamiltonian associated.

######  PROPOSITION 4.

For $\alpha>-\frac{1}{2}$, $\beta\in i\mathbb{R}$ and $\gamma_{k}\in[0,1)$,
there exist solutions to the coupled Painlevé V system (1.14) with the
following asymptotic behaviors as $t\rightarrow 0^{+}$

$\left\\{\begin{array}[]{l}u_{k}(t)=\frac{\operatorname{sgn}(r_{k})c_{k}\Gamma(1+\alpha-\beta)\Gamma(1+\alpha+\beta)(2|r_{k}|t)^{2\alpha}}{\Gamma^{2}(1+2\alpha)}\left(1+\mathcal{O}\left(t^{2\alpha+1}\right)\right),\\\
v_{k}(t)=1+\mathcal{O}\left(t^{2\alpha+1}\right),\end{array}\right.$ (3.14)

with $c_{k}$ given in (1.17) for $k\neq m$. Furthermore, the Hamiltonian
$H(t)$ associated with the solutions has the asymptotic behavior (1.16) as
$t\to 0^{+}$.

###### Proof.

Tracing back the invertible transformations $Y\to X\to E$ given in (3.1) and
(3.10), we obtain

$Y(z/2t)=\frac{1}{2t}(2t)^{\beta\sigma_{3}}E(z)\begin{cases}L(z),&|z|<\varepsilon,\\\
\Phi(z),&|z|>\varepsilon,\end{cases}$ (3.15)

where $L(z)$, $\Phi(z)$ and $E(z)$ are given in (3.4), (3.12), (3.13) and
(A.8).

Comparing the large-$z$ expansion on both sides of the second equation in
(3.15), we have

$Y_{1}=\frac{1}{2t}(2t)^{\beta\sigma_{3}}(E_{1}+\Phi_{1})(2t)^{-\beta\sigma_{3}},$
(3.16)

where $Y_{1}$, $E_{1}$ and $\Phi_{1}$ are the coefficients of $1/z$ in the
large-$z$ expansion of $Y(z)$, $E(z)$ and $\Phi{(z)}$, respectively.
Substituting (3.12) and (A.5) into (3.16) yields

${(Y_{1})}_{11}=\sum_{k=0}^{n}\frac{c_{k}\Gamma(1+\alpha-\beta)\Gamma(1+\alpha+\beta)\left(2|r_{k}|\right)^{2\alpha+1}}{2(2\alpha+1)\Gamma^{2}(1+2\alpha)}t^{2\alpha}+\frac{\alpha^{2}-\beta^{2}}{2it}+\mathcal{O}\left(t^{2\alpha+1}\right),$
(3.17)

as $t\to 0^{+}$. This, together with (2.23), implies the asymptotics of the
Hamiltonian as given in (1.16).

To derive the asymptotics of $y(t)$ and $d(t)$, we consider the asymptotic
expansion of $Y(z)$ as $t\to 0^{+}$ for $z$ near the origin. Plugging (2.14),
(3.4) and (3.13) into the first equation in (3.15), we have

$Y_{0}^{(m)}\left[\begin{array}[]{l}1\\\
0\end{array}\right]=(2t)^{\alpha}(2t)^{\beta\sigma_{3}}\left(I+\mathcal{O}\left(t^{2\alpha+1}\right)\right)\Phi^{(0)}(0)\left[\begin{array}[]{l}1\\\
0\end{array}\right],$ (3.18)

where $\Phi^{(0)}(0)$ is given in (A.6). Using (A.8), we have from (3.18) that

$Y_{0}^{(m)}=\left[\begin{array}[]{ll}(2t)^{\alpha+\beta}e^{-\frac{\alpha+\beta}{2}\pi
i}\frac{\Gamma(1+\alpha-\beta)}{\Gamma(1+2\alpha)}&*\\\
(2t)^{\alpha-\beta}e^{-\frac{\alpha-\beta}{2}\pi
i}\frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+2\alpha)}&*\end{array}\right]\left(I+\mathcal{O}(t^{2\alpha+1})\right),$
(3.19)

for $\alpha>-1/2$, $2\alpha\not\in\mathbb{N}$ and $\beta\in i\mathbb{R}$. The
case $2\alpha\in\mathbb{N}$ can be derived in a similar way and we skip the
details. Recalling (2.32) and (2.33), we derive the asymptotics of $y(t)$ and
$d(t)$ as $t\rightarrow 0^{+}$:

$y(t)=\frac{\Gamma(1+\alpha-\beta)}{\Gamma(1+\alpha+\beta)}e^{-\beta\pi
i}(2t)^{2\beta}\left(1+\mathcal{O}\left(t^{2\alpha+1}\right)\right),$ (3.20)
$d(t)=2\alpha\frac{\Gamma(1+\alpha-\beta)\Gamma(1+\alpha+\beta)}{\Gamma^{2}(1+2\alpha)}e^{-\alpha\pi
i}(2t)^{2\alpha}\left(1+\mathcal{O}\left(t^{2\alpha+1}\right)\right).$ (3.21)

Finally, we derive the asymptotics of $u_{k}$ and $v_{k}$ for $k\neq m$ by
using (3.20) and the following relations

$v_{k}(t)=\frac{\left(Y_{0}^{(k)}(t)\right)_{21}}{\left(Y_{0}^{(k)}(t)\right)_{11}}y(t),$
(3.22)

and

$u_{k}(t)=\frac{\tilde{c_{k}}\left(Y_{0}^{(k)}\right)_{11}\left(Y_{0}^{(k)}\right)_{21}}{v(t)},$
(3.23)

which can be seen from Proposition 1 and (2.13). Similarly, we obtain after
substituting (3.4) and (3.13) into the first equation in (3.15) that

$Y(z)=(2t)^{\beta\sigma_{3}}\left(I+\mathcal{O}\left(t^{2\alpha+1}\right)\right)\Phi^{(0)}(z)\left[\begin{array}[]{cc}1&t^{2\alpha}k\left(\frac{z}{t}\right)\\\
0&1\end{array}\right]z^{\alpha\sigma_{3}}\left[\begin{array}[]{cc}1&\frac{\sin(\alpha+\beta)\pi}{\sin
2\alpha\pi}\\\ 0&1\end{array}\right].$ (3.24)

From the integral expression of $k(z)$ in (3.7), we have

$k(z)\sim\operatorname{sgn}(r_{k})c_{k}e^{\alpha\pi
i}\left(2|r_{k}|\right)^{2\alpha}\ln\left(z-2r_{k}\right),$ (3.25)

as $z\rightarrow 2r_{k}$ for $k\neq m$. Comparing (2.13) with (3.24) and
(3.25), we obtain

$Y_{0}^{(k)}=\left[\begin{array}[]{ll}(2t)^{\alpha+\beta}e^{-\frac{\alpha+\beta}{2}\pi
i}\frac{\Gamma(1+\alpha-\beta)}{\Gamma(1+2\alpha)}r_{k}^{\alpha}&*\\\
(2t)^{\alpha-\beta}e^{-\frac{\alpha-\beta}{2}\pi
i}\frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+2\alpha)}r_{k}^{\alpha}&*\end{array}\right](I+\mathcal{O}(t^{2\alpha+1}))$
(3.26)

for $r_{k}>0$, and

$Y_{0}^{(k)}=\left[\begin{array}[]{ll}(2t)^{\alpha+\beta}e^{-\frac{\alpha+\beta}{2}\pi
i}\frac{\Gamma(1+\alpha-\beta)}{\Gamma(1+2\alpha)}\left(|r_{k}|e^{\pi
i}\right)^{\alpha}&*\\\ (2t)^{\alpha-\beta}e^{-\frac{\alpha-\beta}{2}\pi
i}\frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+2\alpha)}\left(|r_{k}|e^{\pi
i}\right)^{\alpha}&*\end{array}\right](I+\mathcal{O}(t^{2\alpha+1}))$ (3.27)

for $r_{k}<0$. Using (3.20), (3.22), (3.23), (3.26) and (3.27), we have
(3.14). This completes the proof of the proposition. ∎

##  4 Asymptotic Analysis as $t$ $\rightarrow$ $+\infty$

In this section, we carry out the Deift-Zhou nonlinear steepest descent
analysis of the RH problem for $Y(z)$ as $t\rightarrow+\infty$. Based on this,
we obtain the asymptotic behavior of a family of solutions to the coupled
Painlevé V system and the associated Hamiltonian as $t\rightarrow+\infty$.

###  4.1 Normalization

In the first transformation, we normalize the behavior of $Y(z)$ at infinity.

$T(z)=Y(z,t)e^{izt\sigma_{3}},$ (4.1)

Then, $T(z)$ satisfies the following RH problem.

### RH problem for $T(z)$

  * (1)

$T(z)$ is analytic in $\mathbb{C}\backslash\cup_{j=1}^{7}\Gamma_{j}$. The jump
contours are shown in Figure 1.

  * (2)

$T(z)$ satisfies the jump condition

$T_{+}(z)=T_{-}(z)J_{T}(z),$ (4.2)

where

$J_{T}(z)=\begin{cases}{\left[\begin{array}[]{cc}1&0\\\ e^{-(\alpha-\beta)\pi
i}e^{2izt}&1\end{array}\right]},&{z\in\Gamma_{1},}\\\\[11.38092pt]
{\left[\begin{array}[]{cc}1&0\\\ e^{(\alpha-\beta)\pi
i}e^{2izt}&1\end{array}\right],}&{z\in\Gamma_{2},}\\\\[11.38092pt]
{\left[\begin{array}[]{cc}1&-e^{-(\alpha-\beta)\pi i}e^{-2izt}\\\
0&1\end{array}\right],}&{z\in\Gamma_{3},}\\\\[11.38092pt] {e^{2\beta\pi
i\sigma_{3}},}&{z\in\Gamma_{4},}\\\\[11.38092pt]
{\left[\begin{array}[]{cc}1&-e^{(\alpha-\beta)\pi i}e^{-2izt}\\\
0&1\end{array}\right],}&{z\in\Gamma_{5},}\\\\[11.38092pt]
{\left[\begin{array}[]{cc}0&-e^{-(\alpha-\beta)\pi i}e^{-2izt}\\\
e^{(\alpha-\beta)\pi
i}e^{2izt}&1-\sigma(zt)\end{array}\right],}&{z\in\Gamma_{6},}\\\\[11.38092pt]
{\left[\begin{array}[]{cc}0&-e^{(\alpha-\beta)\pi i}e^{-2izt}\\\
e^{-(\alpha-\beta)\pi
i}e^{2izt}&1-\sigma(zt)\end{array}\right],}&{z\in\Gamma_{7}.}\end{cases}$
(4.3)

  * (3)

As $z\rightarrow\infty$,

$T(z)=\left(I+\frac{Y_{1}(t)}{z}+\mathcal{O}\left(\frac{1}{z^{2}}\right)\right)z^{-\beta\sigma_{3}}.$
(4.4)

  * (4)

The behavior of $T(z)$ near $r_{k}$, $k=0,\dots,n$, are the same as that of
$Y(z)$, as given in (2.13) and (2.14).

###  4.2 Opening lenses and deformation

It is readily observed that the off-diagonal entries in the jump matrices on
$\Gamma_{6}$ and $\Gamma_{7}$ are highly oscillatory as $t\rightarrow+\infty$.
To eliminate the oscillation, we factorize the jump matrices on $\Gamma_{6}$
and $\Gamma_{7}$ as follow:

$\displaystyle\left[\begin{array}[]{cc}0&-e^{-(\alpha-\beta)\pi i}e^{-2izt}\\\
e^{(\alpha-\beta)\pi i}e^{2izt}&1-\sigma\end{array}\right]=$ (4.5)
$\displaystyle\left[\begin{array}[]{cc}1&-\frac{1}{1-\sigma}e^{-(\alpha-\beta)\pi
i}e^{-2izt}\\\
0&1\end{array}\right]\left[\begin{array}[]{cc}\frac{1}{1-\sigma}&0\\\
0&1-\sigma\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
\frac{1}{1-\sigma}e^{(\alpha-\beta)\pi i}e^{2izt}&1\end{array}\right],\quad
z\in\Gamma_{6},$

$\displaystyle\left[\begin{array}[]{cc}0&-e^{(\alpha-\beta)\pi i}e^{-2izt}\\\
e^{-(\alpha-\beta)\pi i}e^{2izt}&1-\sigma\end{array}\right]=$ (4.6)
$\displaystyle\left[\begin{array}[]{cc}1&-\frac{1}{1-\sigma}e^{(\alpha-\beta)\pi
i}e^{-2izt}\\\
0&1\end{array}\right]\left[\begin{array}[]{cc}\frac{1}{1-\sigma}&0\\\
0&1-\sigma\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
\frac{1}{1-\sigma}e^{-(\alpha-\beta)\pi i}e^{2izt}&1\end{array}\right],\quad
z\in\Gamma_{7}.$

We also deform the contour $\Gamma_{6}$ and $\Gamma_{7}$ into lens-shape
regions as shown in Figure 2.

Figure 2: Jump contours and regions for $S$

Based on the factorization of jumps and deformation of contours, we introduce
the second transformation

$S(z)=T(z)\left\\{\begin{array}[]{ll}{\left[\begin{array}[]{cc}1&0\\\
-\frac{1}{1-\sigma}e^{(\alpha-\beta)\pi i}e^{2izt}&1\end{array}\right]},&{z\in
H_{k},0\leq k<m,}\\\ {\left[\begin{array}[]{cc}1&0\\\
-\frac{1}{1-\sigma}e^{-(\alpha-\beta)\pi
i}e^{2izt}&1\end{array}\right]},&{z\in H_{k},m\leq k<n,}\\\
{\left[\begin{array}[]{cc}1&-\frac{1}{1-\sigma}e^{-(\alpha-\beta)\pi
i}e^{-2izt}\\\ 0&1\end{array}\right]},&{z\in L_{k},0\leq k<m,}\\\
{\left[\begin{array}[]{cc}1&-\frac{1}{1-\sigma}e^{(\alpha-\beta)\pi
i}e^{-2izt}\\\ 0&1\end{array}\right]},&{z\in L_{k},m\leq k<n,}\\\ I&\text{
elsewhere. }\end{array}\right.$ (4.7)

Then, $S(z)$ satisfies the following RH problem.

### RH problem for $S(z)$

  * (1)

$S(z)$ is analytic in $\mathbb{C}\backslash\Gamma_{S}$ as shown in Figure 2.

  * (2)

$S(z)$ satisfies the jump condition

$S_{+}(z)=S_{-}(z)J_{S}(z),$ (4.8)

where

$J_{S}(z)=\left\\{\begin{array}[]{ll}J_{T}(z),&z\in\bigcup_{j=1}^{5}\Gamma_{S,j},\\\
{\left[\begin{array}[]{cc}1&0\\\ \frac{1}{1-\gamma_{k}}e^{(\alpha-\beta)\pi
i}e^{2izt}&1\end{array}\right]},&{\text{on }\Gamma_{S,8,k},k<m},\\\
{\left[\begin{array}[]{cc}1&0\\\ \frac{1}{1-\gamma_{k}}e^{-(\alpha-\beta)\pi
i}e^{2izt}&1\end{array}\right]},&{\text{on }\Gamma_{S,8,k},k\geqslant m},\\\
{\left[\begin{array}[]{cc}1&-\frac{1}{1-\gamma_{k}}e^{-(\alpha-\beta)\pi
i}e^{-2izt}\\\ 0&1\end{array}\right]},&{\text{on }\Gamma_{S,9,k},k<m},\\\
{\left[\begin{array}[]{cc}1&-\frac{1}{1-\gamma_{k}}e^{(\alpha-\beta)\pi
i}e^{-2izt}\\\ 0&1\end{array}\right]},&{\text{on }\Gamma_{S,9,k},k\geqslant
m},\\\ {\left[\begin{array}[]{cc}\frac{1}{1-\sigma(zt)}&0\\\
0&1-\sigma(zt)\end{array}\right]},&{\text{on
}\Gamma_{S,6}\cup\Gamma_{S,7}.}\end{array}\right.$ (4.9)

  * (3)

As $z\to\infty$, we have

$S(z)=\left(I+\frac{Y_{1}(t)}{z}+\mathcal{O}\left(\frac{1}{z^{2}}\right)\right)z^{-\beta\sigma_{3}}.$
(4.10)

  * (4)

The asymptotics of $S(z)$ are the same as $T(z)$ near $r_{k}$, $k=0,\dots,n$.

###  4.3 Global parametrix

The jumps for $S(z)$ tend to the identity matrix exponentially fast except the
ones on the real axis as $t\to\infty$. Therefore, we arrive at the following
approximate RH problem as $t\rightarrow+\infty$.

### RH problem for $P^{(\infty)}(z)$

  * (1)

$P^{(\infty)}(z)$ is analytic for
$z\in\mathbb{C}\backslash\\{[r_{0},r_{n}]\cup\Gamma_{S,4}\\}$.

  * (2)

$P^{(\infty)}(z)$ satisfies the following jump condition

$P_{+}^{(\infty)}(z)=P_{-}^{(\infty)}(z)\left\\{\begin{array}[]{ll}{\left[\begin{array}[]{cc}\frac{1}{1-\sigma}&0\\\
0&1-\sigma\end{array}\right],}&z\in\left(r_{0},r_{n}\right),\\\\[11.38092pt]
e^{2\beta\pi i\sigma_{3}},&z\in\Gamma_{S,4},\end{array}\right.$ (4.11)

  * (3)

As $z\rightarrow\infty$,

$P^{(\infty)}(z)=\left(I+\mathcal{O}\left(\frac{1}{z}\right)\right)z^{-\beta\sigma_{3}}.$
(4.12)

The solution $P^{(\infty)}(z)$ can be constructed explicitly as below

$P^{(\infty)}(z)=\prod_{k=0}^{n-1}\left(\frac{z-r_{k+1}}{z-r_{k}}\right)^{-\frac{\ln\left(1-\gamma_{k}\right)}{2\pi
i}\sigma_{3}}z^{-\beta\sigma_{3}},$ (4.13)

where
$\left(\frac{z-r_{k+1}}{z-r_{k}}\right)^{-\frac{\ln\left(1-\gamma_{k}\right)}{2\pi
i}}$ takes the principal branch for $0\leq k\leq n-1$ and the branch of
$z^{-\beta}$ is taken such that arg$z\in(-\frac{\pi}{2},\frac{3\pi}{2})$.

###  4.4 Local parametrices

Near the points $r_{k}$, $k=0,\dots,n$, the solution of the RH problem $S(z)$
can not be approximated by the global parametrix $P^{(\infty)}(z)$. In this
section, we construct the local parametrices in the neighborhoods of $r_{k}$,
$k=0,\dots,n$.

### RH problem for $P^{(r_{k})}(z)$

  * (1)

$P^{(r_{k})}(z)$ is analytic for $z\in
U(r_{k},\varepsilon)\setminus\Gamma_{S}$ with some constant $\varepsilon>0$.

  * (2)

$P^{(r_{k})}(z)$ has the same jumps as $S(z)$ for $z\in\Gamma_{S}\cap
U(r_{k},\varepsilon)$.

  * (3)

As $t\rightarrow+\infty$, we have

$S(z)=\left(I+O\left(\frac{1}{t}\right)\right)P^{(r_{k})}(z),$ (4.14)

where the error term is uniform for $z\in\partial U(r_{k},\varepsilon)$.

For $k\neq m$, the solution can be constructed by using the confluent
hypergeometric parametrix $\Phi^{(\mathrm{CHF})}$

$\displaystyle P^{\left(r_{k}\right)}(z)=$ $\displaystyle
E^{\left(r_{k}\right)}(z)\Phi^{(\mathrm{CHF})}\left(2it(z-r_{k});0,b_{k}\right)\left[\left(1-\gamma_{k}\right)\left(1-\gamma_{k-1}\right)\right]^{-\frac{\sigma_{3}}{4}}$
(4.15)
$\displaystyle\times\begin{cases}e^{-\operatorname{sgn}(r_{k})\frac{\alpha-\beta}{2}\pi
i\sigma_{3}}e^{izt\sigma_{3}},&\operatorname{Im}z>0,\\\
{\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right]e^{-\operatorname{sgn}(r_{k})\frac{\alpha-\beta}{2}\pi
i\sigma_{3}}e^{izt\sigma_{3}}},&\operatorname{Im}z<0,\end{cases}$

where

$\displaystyle E^{\left(r_{k}\right)}(z)=$ $\displaystyle\prod_{j\neq
k}\left(z-r_{j}\right)^{-b_{j}\sigma_{3}}z^{-\beta\sigma_{3}}(2it)^{b_{k}\sigma_{3}}e^{-ir_{k}t\sigma_{3}}e^{\operatorname{sgn}(r_{k})\frac{\alpha-\beta}{2}\pi
i\sigma_{3}}$ (4.16)
$\displaystyle\times\begin{cases}\left(1-\gamma_{k-1}\right)^{-\frac{\sigma_{3}}{4}}\left(1-\gamma_{k}\right)^{\frac{3}{4}\sigma_{3}},&\operatorname{Im}z>0,\\\
{\left[\left(1-\gamma_{k}\right)\left(1-\gamma_{k-1}\right)\right]^{-\frac{\sigma_{3}}{4}},}&\operatorname{Im}z<0,\end{cases}$

with $\gamma_{-1}=\gamma_{n}=0$. While for $r_{m}=0$, we construct
$P^{(r_{m})}$ as follows

$\displaystyle P^{\left(r_{m}\right)}(z)=$ $\displaystyle
E^{\left(r_{m}\right)}(z)\Phi^{(\mathrm{CHF})}\left(2itz;\alpha,\beta+b_{m}\right)\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]^{-\frac{\sigma_{3}}{4}}$
(4.17) $\displaystyle\times\begin{cases}e^{\frac{\alpha}{2}\pi
i\sigma_{3}}e^{izt\sigma_{3}},&\arg z\in\left(0,\frac{\pi}{2}\right),\\\
e^{-\frac{\alpha}{2}\pi i\sigma_{3}}e^{izt\sigma_{3}},&\arg
z\in\left(\frac{\pi}{2},\pi\right),\\\ e^{\frac{\alpha+2\beta}{2}\pi
i\sigma_{3}}\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right]e^{izt\sigma_{3}},&\arg
z\in\left(-\pi,\frac{\pi}{2}\right),\\\ e^{-\frac{\alpha+2\beta}{2}\pi
i\sigma_{3}}\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right]e^{izt\sigma_{3}},&\arg
z\in\left(-\frac{\pi}{2},0\right),\end{cases}$

where

$\displaystyle E^{\left(r_{m}\right)}(z)=$ $\displaystyle\prod_{j\neq
m}\left(z-r_{j}\right)^{-b_{j}\sigma_{3}}(2it)^{\left(\beta+b_{m}\right)\sigma_{3}}$
(4.18) $\displaystyle\times\begin{cases}e^{-\beta\pi
i\sigma_{3}}\left(1-\gamma_{m-1}\right)^{-\frac{\sigma_{3}}{4}}\left(1-\gamma_{m}\right)^{\frac{3}{4}\sigma_{3}},&\arg
z\in(0,\pi),\\\ e^{\beta\pi
i\sigma_{3}}\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]^{-\frac{\sigma_{3}}{4}},&\arg
z\in\left(-\pi,-\frac{\pi}{2}\right),\\\ e^{-\beta\pi
i\sigma_{3}}\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]^{-\frac{\sigma_{3}}{4}},&\arg
z\in\left(-\frac{\pi}{2},0\right).\end{cases}$

The branch for $z^{\beta}$ is taken such that $\arg
z\in(-\frac{\pi}{2},\frac{3\pi}{2})$. While the branches of the other power
functions in (4.15), (4.16), (4.17) and (4.18) take the principal branches.

###  4.5 Final transformation

In the final transformation, we define

$R(z)=\begin{cases}{S(z)P^{\left(r_{k}\right)}(z)}^{-1},&z\in
U\left(r_{k},\varepsilon\right),~{}~{}0\leq k\leq n,\\\
S(z){P^{(\infty)}(z)}^{-1}&\text{ elsewhere.}\end{cases}$ (4.19)

### RH problem for $R(z)$

  * (1)

The jump contours for $R$ are shown in Figure 3

Figure 3: Jump contours $\Gamma_{R}$ and regions for $R$

  * (2)

$R(z)$ satisfies the jump condition

$R_{+}(z)=R_{-}(z)J_{R}(z),$ (4.20)

where

$J_{R}(z)=\begin{cases}{P^{(r_{k})}(z){P^{(\infty)}(z)}^{-1}},&z\in\partial
U(r_{k},\varepsilon),~{}~{}0\leq k\leq n,\\\
P^{(\infty)}(z)J_{S}(z){P^{(\infty)}(z)}^{-1}&\text{elsewhere}.\end{cases}$
(4.21)

  * (3)

As $z\rightarrow\infty$,

$R(z)=I+\mathcal{O}(\frac{1}{z}).$ (4.22)

From (4.9) and (4.14), we have

$J_{R}(z)=I+\mathcal{O}\left(\frac{1}{t}\right),$ (4.23)

where the error term is uniform for $z$ on the jump contours for $R(z)$,
$\gamma_{j}$ in any compact subset of $[0,1)$, and
$r_{0}<\dots<r_{m}=0<\dots<r_{n}$ in any compact subset of $\mathbb{R}$ and
bounded away from each other. Therefore, we have as $t\rightarrow+\infty$

$R(z)=I+\mathcal{O}\left(\frac{1}{t}\right),$ (4.24)

where the error term is uniform for $z$ in the complex plane. For later use,
we derive from (4.13), (4.15), (4.16) and (A.5) the following expansion

$J_{R}(z)=I+\frac{J_{R,1}(z)}{t}+\mathcal{O}\left(\frac{1}{t^{2}}\right),$
(4.25)

where

$J_{R,1}(z)=-\frac{b_{k}^{2}}{2i\left(z-r_{k}\right)}E^{\left(r_{k}\right)}(z)\left[\begin{array}[]{cc}1&\frac{\Gamma\left(-b_{k}\right)}{\Gamma\left(1+b_{k}\right)}\\\
\frac{-\Gamma\left(b_{k}\right)}{\Gamma\left(1-b_{k}\right)}&-1\end{array}\right]{E^{\left(r_{k}\right)}(z)}^{-1},$
(4.26)

for $z\in\partial U(r_{k},\varepsilon)$, $k\neq m$ and

$J_{R,1}(z)=\frac{\alpha^{2}-\left(\beta+b_{m}\right)^{2}}{2iz}E^{\left(r_{m}\right)}(z)\left[\begin{array}[]{cc}1&\frac{\Gamma\left(\alpha-\beta-
b_{m}\right)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)}\\\
\frac{-\Gamma\left(\alpha+\beta+b_{m}\right)}{\Gamma\left(1+\alpha-\beta-
b_{m}\right)}&-1\end{array}\right]{E^{(r_{m})}(z)}^{-1},$ (4.27)

for $z\in\partial U(0,\varepsilon)$. Therefore, we have as
$t\rightarrow+\infty$

$R(z)=I+\frac{R_{1}(z)}{t}+\mathcal{O}\left(\frac{1}{t^{2}}\right),$ (4.28)

where

$R_{1}(z)=\frac{1}{2\pi i}\sum_{k=0}^{n}\int_{\partial
U\left(r_{k},\xi\right)}\frac{J_{R,1}(x)}{x-z}dx.$ (4.29)

After some residue calculation, we have

$\left(R_{1}(z)\right)_{11}=\begin{cases}-\sum_{j=0}^{n}\frac{b_{j}^{2}}{2i\left(z-r_{j}\right)}+\frac{\alpha^{2}-\beta^{2}-2\beta
b_{m}}{2iz},&z\in\mathbb{C}\backslash\bigcup^{n}_{j=0}{U(r_{j},\varepsilon),}\\\
-\sum_{j\neq
k}\frac{b_{j}^{2}}{2i\left(z-r_{j}\right)}+\frac{\alpha^{2}-\beta^{2}-2\beta
b_{m}}{2iz},&z\in U\left(r_{k},\varepsilon\right),~{}k\neq m,\\\ -\sum_{j\neq
m}\frac{b_{j}^{2}}{2i\left(z-r_{j}\right)},&z\in
U\left(r_{m},\varepsilon\right).\end{cases}$ (4.30)

For later use, we derive the following estimate of derivatives of $J_{R}(z)$
with respect to $\gamma_{j}$ as $t\rightarrow+\infty$,

${\left\|\left(\prod_{j=0}^{n-1}\partial_{\gamma_{j}}^{k_{j}}\right)(J_{R}(z)-I)\right\|_{L^{2}(\Gamma_{R})\cap
L^{\infty}(\Gamma_{R})}\leq c\frac{(\ln t)^{k}}{t},}$ (4.31)

where $k=\sum_{j=0}^{n-1}k_{j}$ and the constant $c>0$. The factor $(\ln
t)^{k}$ is due to the factors $t^{\pm b_{j}}$ in the entries of $J_{R}(z)$,
cf. (4.15)-(4.18) and (4.21). It is worth pointing out that the estimate holds
uniformly for $\gamma_{j}$ in any compact subset of $[0,1)$, and uniformly for
$r_{0}<\dots<r_{m}=0<\dots<r_{n}$ in any compact subset of $\mathbb{R}$ and
bounded away from each other. Then, by (4.29), we obtain

$\left(\prod_{j=0}^{n-1}\partial_{\gamma_{j}}^{k_{j}}\right)R_{1}(z)=\mathcal{O}((\ln
t)^{k}),\quad\text{as }t\rightarrow+\infty.$ (4.32)

###  4.6 Asymptotic behaviors as $t\rightarrow+\infty$

Using the results of the nonlinear steepest descent analysis of the RH problem
for $Y(z)$, we obtain the asymptotics of a family of special solutions to the
coupled Painlevé V system and the Hamiltonian associated as
$t\rightarrow+\infty$.

######  PROPOSITION 5.

Under the same conditions as in Proposition 4, the family of solutions of the
coupled Painlevé V system (1.14) with the asymptotic behaviors as $t\to 0^{+}$
given in Proposition 4 have the asymptotics behaviors as $t\rightarrow+\infty$

$\left\\{\begin{array}[]{l}\begin{aligned}
u_{k}(t)=&\operatorname{sgn}(r_{k})~{}c_{k}\cdot\frac{\Gamma^{2}\left(1-b_{k}\right)\Gamma(1+\alpha+\beta+b_{m})}{\Gamma\left(1+\alpha-\beta-
b_{m}\right)}\prod_{j\neq
m,k}\left(\frac{r_{k}-r_{j}}{r_{m}-r_{j}}\right)_{+}^{-2b_{j}}|r_{k}|^{2\left(b_{k}-b_{m}-\beta\right)}\\\
&\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]^{-\frac{1}{2}}e^{\operatorname{sgn}(r_{k})\pi
i\left(b_{k}+b_{m}+\alpha+\beta\right)}(2t)^{2\left(b_{k}-b_{m}-\beta\right)}e^{-2itr_{k}}(1+\mathcal{O}(t^{-1})),\end{aligned}\\\
\begin{aligned}
v_{k}(t)=&\operatorname{sgn}(r_{k})~{}\frac{(\gamma_{k}-\gamma_{k-1})}{2\pi
ic_{k}}\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)\Gamma\left(1+b_{k}\right)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)\Gamma\left(1-b_{k}\right)}\prod_{j\neq
m,k}\left(\frac{r_{k}-r_{j}}{r_{m}-r_{j}}\right)_{+}^{2b_{j}}|r_{k}|^{2\left(\beta+b_{m}-b_{k}\right)}\\\
&\left[\frac{\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)}{\left(1-\gamma_{k-1}\right)\left(1-\gamma_{k}\right)}\right]^{\frac{1}{2}}e^{-\operatorname{sgn}(r_{k})\pi
i\left(b_{k}+b_{m}+\alpha+\beta\right)}(2t)^{2\left(\beta+b_{m}-b_{k}\right)}e^{2itr_{k}}\left(1+\mathcal{O}\left(t^{-1}\right)\right),\end{aligned}\end{array}\right.$
(4.33)

for $k\neq m$ and $c_{k}$ is defined in (1.17). Here the power functions take
the principal branches. Moreover, the Hamiltonian associated with this family
of solutions has the asymptotics behavior as $t\rightarrow+\infty$

$H(t)=\sum_{k=0}^{n}2ib_{k}r_{k}-\frac{\sum_{k=0}^{n}b_{k}^{2}+2\beta
b_{m}}{t}+\mathcal{O}\left(t^{-2}\right).$ (4.34)

###### Proof.

Tracing back the transformations $Y\longmapsto N\longmapsto S\longmapsto R$,
we can derive the asymptotics of $Y(z)$ uniformly for $z$ in several different
regions covering the whole complex plane as $t\rightarrow+\infty$.

As $t\rightarrow+\infty$, we obtain from (4.1), (4.7), (4.13), (4.19) and
(4.28) that

$\displaystyle Y(z)$ $\displaystyle=R(z)P^{(\infty)}(z)e^{-izt\sigma_{3}}$
(4.35)
$\displaystyle=\left(I+\frac{R_{1}(z)}{t}+\mathcal{O}\left(\frac{1}{t^{2}}\right)\right)\prod_{k=0}^{n-1}\left(\frac{z-r_{k+1}}{z-r_{k}}\right)^{-\frac{\ln\left(1-\gamma_{k}\right)}{2\pi
i}\sigma_{3}}z^{-\beta\sigma_{3}}e^{-izt\sigma_{3}},$

where the error term is uniform for $|z-r_{k}|>\varepsilon$, $k=1,\dots,n$.
Here, the branch for $z^{\beta}$ is taken such that $\arg
z\in(-\frac{\pi}{2},\frac{3\pi}{2})$ and the other power functions take the
principal branches. Using(2.12), (4.30) and (4.35), we have

$\left(Y_{1}\right)_{11}=-\frac{1}{2\pi
i}\sum_{k=0}^{n-1}\left(r_{k}-r_{k+1}\right)\ln\left(1-\gamma_{k}\right)+\frac{\alpha^{2}-\beta^{2}-2\beta
b_{m}-\sum_{k=0}^{n}b_{k}^{2}}{2it}+\mathcal{O}\left(\frac{1}{t^{2}}\right).$
(4.36)

Substitute (4.36) into (2.23), we obtain (4.34).

From (4.1), (4.7), (4.17), (4.19) and (4.28), we derive as
$t\rightarrow+\infty$

$\displaystyle Y(z)=$ $\displaystyle
R(z)P^{\left(r_{m}\right)}(z)e^{-izt\sigma_{3}}$ (4.37) $\displaystyle=$
$\displaystyle\left(I+\frac{R_{1}(z)}{t}+\mathcal{O}\left(\frac{1}{t^{2}})\right)\right)E^{\left(r_{m}\right)}(z)\Phi\left(2itz;\alpha,\beta+b_{m}\right)$
$\displaystyle{\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]^{-\frac{\sigma_{3}}{4}}e^{\frac{\alpha}{2}\pi
i\sigma_{3}},}$

where $\arg z\in\left(0,\frac{\pi}{2}\right)$ and the error term is uniform
for $|z|<\varepsilon$. Using (2.14), (4.18) and (4.30), we obtain from (4.37)
that

$\displaystyle\left(Y_{0}^{(m)}\right)_{11}=$
$\displaystyle\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)}{\Gamma(1+2\alpha)}\prod_{j\neq
m}\left(-r_{j}\right)_{+}^{-b_{j}}e^{-i\frac{\pi}{2}(\alpha+\beta+b_{m})}(2t)^{\alpha+\beta+b_{m}}\left(1+\mathcal{O}\left(\frac{1}{t}\right)\right),$
(4.38) $\displaystyle\left(Y_{0}^{(m)}\right)_{21}=$
$\displaystyle\frac{\Gamma\left(1+\alpha+\beta+b_{m}\right)}{\Gamma(1+2\alpha)}\prod_{j\neq
m}\left(-r_{j}\right)_{+}^{b_{j}}e^{-i\frac{\pi}{2}(\alpha-\beta-
b_{m})}(2t)^{\alpha-\beta-b_{m}}$
$\displaystyle\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]^{-\frac{1}{2}}\left(1+\mathcal{O}\left(\frac{1}{t}\right)\right).$

Therefore, we have from (2.32), (2.33) and (4.38) that as
$t\rightarrow+\infty$

$y(t)=\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)}\prod_{j\neq
m}\left(-r_{j}\right)_{+}^{-2b_{j}}e^{-(\beta+b_{m})\pi
i}(2t)^{2\left(\beta+b_{m}\right)}{\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]^{\frac{1}{2}}\left(1+\mathcal{O}\left(\frac{1}{t}\right)\right),}$
(4.39)

and

$d(t)=2\alpha\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)\Gamma\left(1+\alpha+\beta+b_{m}\right)}{\Gamma^{2}(1+2\alpha)}e^{-\alpha\pi
i}(2t)^{2\alpha}\left[\left(1-\gamma_{m}\right)\left(1-\gamma_{m-1}\right)\right]^{-\frac{1}{2}}\left(1+\mathcal{O}\left(\frac{1}{t}\right)\right).$
(4.40)

Similarly, we have from (4.1), (4.7) and (4.19) that as $t\rightarrow+\infty$

$Y(z)=R(z)P^{\left(r_{k}\right)}(z)e^{-izt\sigma_{3}},$ (4.41)

where the error term is uniform for $|z-r_{k}|<\varepsilon$. From (4.1),
(4.7), (4.17), (4.18) and (4.28)), we obtain

$\displaystyle\left(Y_{0}^{(k)}(t)\right)_{11}=$
$\displaystyle\Gamma\left(1-b_{k}\right)\prod_{j\neq
k}\left(r_{k}-r_{j}\right)_{+}^{-b_{j}}r_{k}^{-\beta}e^{-\frac{b_{k}}{2}\pi
i}(2t)^{b_{k}}e^{-itr_{k}}(1+\mathcal{O}(\frac{1}{t})),$ (4.42)
$\displaystyle\left(Y_{0}^{(k)}(t)\right)_{21}=$
$\displaystyle\Gamma\left(1+b_{k}\right)\prod_{j\neq
k}\left(r_{k}-r_{j}\right)_{+}^{b_{j}}r_{k}^{\beta}e^{\frac{b_{k}}{2}\pi
i}(2t)^{-b_{k}}e^{-\operatorname{sgn}\left(r_{k}\right)(\alpha-\beta)\pi i}$
$\displaystyle\left[\left(1-\gamma_{k-1}\right)\left(1-\gamma_{k}\right)\right]^{-\frac{1}{2}}(1+\mathcal{O}(\frac{1}{t})).$

From (3.22), (3.23) and (4.42), we have (4.33). This completes the proof of
the proposition. ∎

##  5 Proof of main theorems

###  5.1 Proof of Theorem 1

From (2.3), (2.4), (2.6), (2.10) and (2.13), we have

$\frac{d}{dt}\ln
F=\frac{1}{t}\sum_{k=0}^{n}\tilde{c}_{k}r_{k}\left(Y_{1}^{(k)}\right)_{21},$
(5.1)

where $r_{m}=0$, $Y_{1}^{(k)}$ is the coefficient appears in the expansion
(2.13) for $k=0,\dots,n$. Here the coefficients
$\tilde{c}_{k}=c_{k}e^{\operatorname{sgn}(r_{k})(\alpha+1)\pi i}$ and $c_{k}$
is defined in (1.17). By (2.13) and (2.15), we have

$\left(\frac{d}{dt}Y_{1}^{(k)}\right)_{21}=2i\left(Y_{0}^{(k)}\right)_{11}\left(Y_{0}^{(k)}\right)_{21}=\frac{2iu_{k}v_{k}}{\tilde{c}_{k}}.$
(5.2)

Therefore, we have

$\frac{d}{dt}\left(t\frac{d}{dt}\ln F\right)=2i\sum_{k=1}^{n}r_{k}u_{k}v_{k}.$
(5.3)

On the other side, we get from (1.13), (1.12) (1.11) that

$\frac{d}{dt}\left(tH(t)\right)=2i\sum_{k=1}^{n}r_{k}u_{k}v_{k},$ (5.4)

with $r_{m}=0$. From (1.16), (5.3) and (5.4), we have

$\frac{d}{dt}\ln F=H(t).$ (5.5)

By (1.16), the Hamiltonian function $H$ is integrable near zero. By using the
fact that $F(0)=1$, we obtain the integral expression (1.15) by taking
integration on both sides of the above equation. This completes the proof of
the theorem. $\square$

###  5.2 Proof of Theorem 2

From the asymptotic behavior of the Hamiltonian function (1.18), we obtain the
asymptotics of the integral except the constant term as $t\to+\infty$

$\int_{0}^{t}H(s)ds=\left(\sum_{k=0}^{n}2ib_{k}r_{k}\right)t-\left(\sum_{k=0}^{n}b_{k}^{2}+2\beta
b_{m}\right)\ln t+C_{1}+\mathcal{O}\left(\frac{1}{t}\right),$ (5.6)

where $C_{1}=C_{1}(\alpha,\beta,\vec{\gamma})$ is some constant depending on
the parameters.

To determine the constant $C_{1}$, we integrate on both sides of (2.37)

$\int_{0}^{t}H(s)ds=\int_{0}^{t}\sum_{k\neq
m}u_{k}\frac{dv_{k}}{ds}-H(s)ds+\left.(sH(s)+\alpha\ln d(s)-\beta\ln
y(s)-2(\alpha^{2}-\beta^{2})\ln s)\right|_{0}^{t}.$ (5.7)

Substituting the asymptotics (1.16), (3.20), (3.21), (4.34), (4.39) and (4.40)
into the second term on the right-hand side of the above equation, we have

$\displaystyle\left.\left(sH+\alpha\ln d-\beta\ln
y-2\left(\alpha^{2}-\beta^{2}\right)\ln s\right)\right|_{0}^{t}$ (5.8)
$\displaystyle=\sum_{k=0}^{n}2ib_{k}r_{k}t-\sum_{k=0}^{n}b_{k}^{2}-2\beta
b_{m}+\alpha\ln\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)\Gamma\left(1+\alpha+\beta+b_{m}\right)}{\Gamma(1+\alpha-\beta)\Gamma(1+\alpha+\beta)}$
$\displaystyle~{}~{}-\beta\ln\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)\Gamma(1+\alpha+\beta)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)\Gamma(1+\alpha+\beta)}+\beta\sum_{j\neq
m}2b_{j}\ln|r_{j}|-2\beta b_{m}\ln(2t)$
$\displaystyle~{}~{}-\frac{\alpha}{2}\ln\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]+\mathcal{O}\left(\frac{1}{t}\right).$

Comparing (5.6) with (5.7) and (5.8), we have

$\int_{0}^{t}\sum_{k\neq
m}u_{k}\frac{dv_{k}}{ds}-Hds=-\sum_{k=0}^{n}b_{k}^{2}\ln
t+C_{2}+\mathcal{O}\left(\frac{1}{t}\right),$ (5.9)

with

$\displaystyle C_{1}=$ $\displaystyle
C_{2}+\alpha\ln\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)\Gamma\left(1+\alpha+\beta+b_{m}\right)}{\Gamma(1+\alpha-\beta)\Gamma(1+\alpha+\beta)}-\beta\ln\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)\Gamma(1+\alpha+\beta)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)\Gamma(1+\alpha-\beta)}$
(5.10) $\displaystyle+\beta\sum_{k\neq m}2b_{k}\ln|r_{k}|-2\beta
b_{m}\ln(2)-\frac{\alpha}{2}\ln\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right]-\Sigma
b^{2}_{k}-2\beta b_{m}.$

Therefore, the remaining task is to derive the constant $C_{2}$. It is more
convenient to evaluate the integral constant $C_{2}$ since the derivatives of
the integrand in (5.9) with respect to the parameters $\vec{\gamma}$ is
totally differential in $t$ as shown in (2.38).

Actually, we obtain from (2.38) that

$\frac{d}{db_{j}}C_{2}=\lim_{t\to+\infty}\left(\sum_{k\neq
m}u_{k}(t)\frac{dv_{k}(t)}{db_{j}}+2b_{j}\ln t-2b_{m}\ln t\right)-\lim_{t\to
0^{+}}\left(\sum_{k\neq m}u_{k}(t)\frac{dv_{k}(t)}{db_{j}}+2b_{j}\ln
t-2b_{m}\ln t\right).$ (5.11)

Substituting into the above formula the asymptotics (3.14), (3.21), (4.33) and
(4.40), we have

$\displaystyle\frac{d}{db_{j}}C_{2}=$ $\displaystyle
b_{j}\left(\partial_{b_{j}}\ln\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)}+\partial_{b_{j}}\ln\frac{\Gamma\left(1+b_{j}\right)}{\Gamma\left(1-b_{j}\right)}-4\ln\left|2r_{j}\right|\right)$
(5.12) $\displaystyle+\sum_{k\neq
m,j}b_{k}\left(\partial_{b_{j}}\ln\frac{\Gamma\left(1+\alpha-\beta-
b_{m}\right)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)}-2\ln\left|\frac{2r_{j}r_{k}}{r_{j}-r_{k}}\right|\right),$

with $j\neq m$ and $b_{m}=-\sum_{j\neq m}b_{j}$.

When the parameters $\vec{b}=\vec{0}$, we have $\ln F(s)=0$ and the constant
$C_{1}(0,\dots,0;\alpha,\beta)=C_{2}(0,\dots,0;\alpha,\beta)=0$. Using this
initial value and the differential identity (5.12), we evaluate the constant
$C_{2}$ by taking integration with respect to the variables $b_{j}$, $j\neq m$
successively. For the purpose, we recall the integral representation for the
Barnes G-function (see [48])

$\ln
G(z+1)=\frac{1}{2}z\ln(2\pi)-\frac{1}{2}z(z+1)+z\ln\Gamma(z+1)-\int_{0}^{z}\ln\Gamma(t+1)\mathrm{d}t.$
(5.13)

Using (5.13), we obtain by integrating by parts that

$\displaystyle\int_{0}^{b_{j}}x\partial_{x}\ln\frac{\Gamma(1+x)}{\Gamma(1-x)}dx=b_{j}^{2}+\ln\left[G\left(1+b_{j}\right)G\left(1-b_{j}\right)\right].$
(5.14)

In the $j$-th step, taking $b_{j+1}=\dots=b_{n}=0$, we have $b_{m}=-s_{j}$
with $s_{j}=\sum_{l=0,l\neq m}^{j}b_{l}$. Then, we have from integration by
parts and (5.13) that

$\displaystyle\int_{0}^{b_{j}}(s_{j}+x)\partial_{x}\ln\frac{\Gamma\left(1+\alpha-\beta+s_{j-1}+x\right)}{\Gamma\left(1+\alpha+\beta-
s_{j-1}-x\right)}dx$ (5.15)
$\displaystyle=\left[x\ln\frac{\Gamma(1+\alpha-\beta+x)}{\Gamma(1+\alpha+\beta-x)}\right]_{s_{j-1}}^{s_{j}}-\int_{s_{j-1}}^{s_{j}}\partial_{x}\ln\frac{\Gamma(1+\alpha-\beta+x)}{\Gamma(1+\alpha+\beta-x)}dx$
$\displaystyle=b_{j}^{2}+2b_{j}s_{j-1}-(\alpha+\beta)\ln\frac{\Gamma(1+\alpha+\beta-
s_{j})}{\Gamma\left(1+\alpha+\beta-
s_{j-1}\right)}-(\alpha-\beta)\ln\frac{\Gamma(1+\alpha-\beta+s_{j})}{\Gamma\left(1+\alpha-\beta+s_{j-1}\right)}$
$\displaystyle~{}~{}+\ln\frac{G\left(1+\alpha-\beta+s_{j}\right)G\left(1+\alpha+\beta-
s_{j}\right)}{G(1+\alpha-\beta+s_{j-1})G(1+\alpha+\beta-s_{j-1})}.$

By (5.12), (5.15), (5.14) and the initial value
$C_{2}(0,\dots,0;\alpha,\beta)=0$, we derive

$\displaystyle C_{2}(\vec{b};\alpha,\beta)=\sum_{k\neq m}b_{k}^{2}+2\beta
b_{m}+\alpha\ln\frac{\Gamma(1+\alpha+\beta)\Gamma(1+\alpha-\beta)}{\Gamma\left(1+\alpha+\beta+b_{m}\right)\Gamma\left(1+\alpha-\beta-
b_{m}\right)}$ (5.16)
$\displaystyle+\beta\ln\frac{\Gamma(1+\alpha+\beta)\Gamma\left(1+\alpha-\beta-
b_{m}\right)}{\Gamma(1+\alpha-\beta)\Gamma\left(1+\alpha+\beta+b_{m}\right)}+\sum_{j\neq
m}\ln\left[G\left(1+b_{j}\right)G\left(1-b_{j}\right)\right]$
$\displaystyle+\ln\frac{G\left(1+\alpha+\beta+b_{m}\right)G\left(1+\alpha-\beta-
b_{m}\right)}{G(1+\alpha+\beta)G(1+\alpha-\beta)}-\sum_{\begin{subarray}{c}0\leq
j\neq k\leq n\\\ j,k\neq
m\end{subarray}}2b_{j}b_{k}\ln\left|\frac{2r_{j}r_{k}}{r_{j}-r_{k}}\right|$
$\displaystyle-\sum_{j\neq m}2b_{j}^{2}\ln\left|2r_{j}\right|.$

This, together with (5.10), implies

$\displaystyle C_{1}=$ $\displaystyle-\sum_{k\neq
m}2b_{k}^{2}\ln\left|2r_{k}\right|+\sum_{k\neq
m}2b_{k}\beta\ln\left|2r_{k}\right|-\sum_{\begin{subarray}{c}0\leq
j<k\leqslant n\\\ j,k\neq
m\end{subarray}}2b_{j}b_{k}\ln\left|\frac{2r_{j}r_{k}}{r_{j}-r_{k}}\right|$
(5.17) $\displaystyle+\sum_{k\neq
m}\ln\left[G\left(1+b_{j}\right)G\left(1-b_{j}\right)\right]+\ln\frac{G\left(1+\alpha+\beta+b_{m}\right)G\left(1+\alpha-\beta-
b_{m}\right)}{G(1+\alpha+\beta)G(1+\alpha-\beta)}$
$\displaystyle-\frac{\alpha}{2}\ln\left[\left(1-\gamma_{m-1}\right)\left(1-\gamma_{m}\right)\right].$

Substitute (5.17) into (5.6), we obtain (1.20).

Finally, from (1.15), (2.23), (2.38), (4.32) and (4.35), we see that (1.20)
can be be differentiated with respect to $\gamma_{0},\dots,\gamma_{n-1}$ at
the cost of increasing the error term by a factor $\ln t$ for each
differentiation. This completes the proof of the theorem.

##  Appendix A The confluent hypergeometric parametrix

### RH problem for $\Phi^{(\mathrm{CHF})}$

(1)

$\Phi^{(\mathrm{CHF})}(z;\alpha,\beta)$ ($\Phi^{(\mathrm{CHF})}(z)$, for
short) is analytic in $\mathbb{C}\setminus\\{\cup^{8}_{j=1}\Sigma_{j}\\}$,
where the contours $\Sigma_{j}$, $j=1,\ldots,8,$ are shown in Fig. 4.

Figure 4: The jump contours and regions of the RH problem for
$\Phi^{(\mathrm{CHF})}(z)$.

(2)

$\Phi^{(\mathrm{CHF})}(z)$ satisfies the jump condition

$\left(\Phi^{(\mathrm{CHF})}\right)_{+}(z)=\left(\Phi^{(\mathrm{CHF})}\right)_{-}(z)J_{j}(z),\quad
z\in\Sigma_{i},\quad j=1,\ldots,8,$ (A.1)

where

$J_{1}(z)=\begin{bmatrix}0&e^{-\pi i\beta}\\\ -e^{\pi
i\beta}&0\end{bmatrix},\quad J_{2}(z)=\begin{bmatrix}1&0\\\ e^{\pi
i(\beta-2\alpha)}&1\end{bmatrix},\quad J_{3}(z)=J_{7}(z)=e^{\pi
i\alpha\sigma_{3}},$ $J_{4}(z)=\begin{bmatrix}1&0\\\ e^{-\pi
i(\beta-2\alpha)}&1\end{bmatrix},\quad J_{5}(z)=\begin{bmatrix}0&e^{\pi
i\beta}\\\ -e^{-\pi i\beta}&0\end{bmatrix},\quad
J_{6}(z)=\begin{bmatrix}1&0\\\ e^{-\pi i(\beta+2\alpha)}&1\end{bmatrix},$

and

$J_{8}(z)=\begin{bmatrix}1&0\\\ e^{\pi i(\beta+2\alpha)}&1\end{bmatrix}.$

(3)

As $z\to\infty$, $\Phi^{(\mathrm{CHF})}(z)$ satisfies the asymptotic behavior

$\displaystyle\Phi^{(\mathrm{CHF})}(z)=$
$\displaystyle\left(I+\frac{\alpha^{2}-\beta^{2}}{z}\left[\begin{array}[]{cc}1&\frac{\Gamma(\alpha-\beta)}{\Gamma(1+\alpha+\beta)}\\\
-\frac{\Gamma(\alpha+\beta)}{\Gamma(1+\alpha-\beta)}&-1\end{array}\right]+\ldots\right)z^{-\beta\sigma_{3}}e^{-\frac{1}{2}z\sigma_{3}}$
(A.2) $\displaystyle\begin{cases}e^{-\frac{1}{2}\alpha\pi
i\sigma_{3}}e^{\beta\pi i\sigma_{3}},&\frac{\pi}{2}<\arg z<\pi,\\\
e^{\frac{1}{2}\alpha\pi i\sigma_{3}}e^{\beta\pi i\sigma_{3}},&\pi<\arg
z<\frac{3\pi}{2},\\\ {\left[\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right]e^{-\frac{1}{2}\alpha\pi
i\sigma_{3}}},&-\frac{\pi}{2}<\arg z<0,\\\\[11.38092pt]
{\left[\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right]e^{\frac{1}{2}\alpha\pi
i\sigma_{3}}},&0<\arg z<\frac{\pi}{2}.\end{cases}$

(4)

As $z\to 0$, $\Phi^{(\mathrm{CHF})}(z)$ has the following asymptotic behaviors

$\Phi^{(\mathrm{CHF})}(z)=\begin{bmatrix}O(|z|^{\alpha})&O(|z|^{-|\alpha|})\\\
O(|z|^{\alpha})&O(|z|^{-|\alpha|})\end{bmatrix},\quad\alpha>0,$
$\Phi^{(\mathrm{CHF})}(z)=\begin{bmatrix}O(1)&O(\ln|z|)\\\
O(1)&O(\ln|z|)\end{bmatrix},\quad\alpha=0,$

and

$\Phi^{(\mathrm{CHF})}(z)=\begin{bmatrix}O(|z|^{\alpha})&O(|z|^{|\alpha|})\\\
O(|z|^{\alpha})&O(|z|^{|\alpha|})\end{bmatrix},\quad\alpha<0.$

For the application in the present paper, we introduce a matrix-valued
function $\Phi(z)$ defined via $\Phi^{(\mathrm{CHF})}(z)$ by the
transformation:

$\Phi(z)=e^{-\frac{1}{2}\beta\pi
i\sigma_{3}}\Phi^{(\mathrm{CHF})}(e^{\frac{1}{2}\pi
i}z;\alpha,\beta)\left\\{\begin{array}[]{ll}e^{\frac{\alpha}{2}\pi
i\sigma_{3}},&\arg z\in\left(0,\frac{\pi}{2}\right),\\\
e^{-\frac{\alpha}{2}\pi i\sigma_{3}},&\arg
z\in\left(\frac{\pi}{2},\pi\right),\\\ e^{\frac{\alpha+2\beta}{2}\pi
i\sigma_{3}}{\left[\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right]},&\arg
z\in\left(-\pi,-\frac{\pi}{2}\right),\\\ e^{-\frac{\alpha+2\beta}{2}\pi
i\sigma_{3}}{\left[\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right]},&\arg
z\in\left(-\frac{\pi}{2},0\right).\end{array}\right.$ (A.3)

Then $\Phi(z)$ satisfies the following RH problem.

  * (1)

$\Phi(z)$ is analytic in $\mathbb{C}\backslash\cup_{j=1}^{5}\Sigma_{\Phi,j}$,
where the oriented contours

$\displaystyle\Sigma_{\Phi,1}=e^{\frac{\pi
i}{4}}\mathbb{R}^{+},\quad\Sigma_{\Phi,2}=e^{\frac{3\pi
i}{4}}\mathbb{R}^{+},\quad\Sigma_{\Phi,3}=e^{-\frac{3\pi
i}{4}}\mathbb{R}^{+},$ $\displaystyle\Sigma_{\Phi,4}=e^{-\frac{\pi
i}{2}}\mathbb{R}^{+},\quad\Sigma_{\Phi,5}=e^{-\frac{\pi i}{4}}\mathbb{R}^{+},$

are shown in Figure 5.

Figure 5: The jump contours and regions of the RH problem for $\Phi(z)$.

  * (2)

$\Phi(z)$ satisfies the jump condition $\Phi_{+}(z)=\Phi_{-}(z)J_{\Phi}(z)$ on
$\cup_{j=1}^{5}\Sigma_{\Phi,j}$, where

$J_{\Phi}(z)=\begin{cases}\left[\begin{array}[]{cc}1&0\\\ e^{-\pi
i(\alpha-\beta)}&1\end{array}\right],&z\in\Sigma_{\Phi,1},\\\\[11.38092pt]
\left[\begin{array}[]{cc}1&0\\\ e^{\pi
i(\alpha-\beta)}&1\end{array}\right],&z\in\Sigma_{\Phi,2},\\\\[11.38092pt]
\left[\begin{array}[]{cc}1&-e^{-\pi i(\alpha-\beta)}\\\
0&1\end{array}\right],&z\in\Sigma_{\Phi,3},\\\\[11.38092pt] e^{2\pi
i\beta\sigma_{3}},&z\in\Sigma_{\Phi,4},\\\ \left[\begin{array}[]{cc}1&-e^{\pi
i(\alpha-\beta)}\\\ 0&1\end{array}\right],&z\in\Sigma_{\Phi,5}.\end{cases}$
(A.4)

  * (3)

As $z\rightarrow\infty$, we have

$\Phi(z)=\left(I+\frac{\Phi_{1}}{z}+\mathcal{O}\left(\frac{1}{z^{2}}\right)\right)z^{-\beta\sigma_{3}}e^{-\frac{i}{2}z\sigma_{3}},$
(A.5)

where
$\Phi_{1}=\left[\begin{array}[]{cc}-\left(\alpha^{2}-\beta^{2}\right)i&-e^{-\pi
i\beta}\frac{\Gamma(1+\alpha-\beta)}{i\Gamma(\alpha+\beta)}\\\ e^{\pi
i\beta}\frac{\Gamma(1+\alpha+\beta)}{i\Gamma(\alpha-\beta)}&\left(\alpha^{2}-\beta^{2}\right)i\end{array}\right]$,
and arg $z\in(-\frac{\pi}{2},\frac{3\pi}{2})$.

  * (4)

If $\alpha>-\frac{1}{2}$ and $2\alpha\not\in\mathbb{N}$, we have as $z\to 0$

$\Phi(z)=\Phi^{(0)}(z)z^{\alpha\sigma_{3}}C_{j},~{}~{}~{}z\to
0,~{}z\in\Omega_{\Phi,j},~{}j=1,2,3,4,5,$ (A.6)

where $\Phi^{(0)}(z)$ is analytic near the origin, the regions
$\Omega_{\Phi,j}$ are shown in Figure 5, and the branch for
$z^{\alpha\sigma_{3}}$ is chosen such that $\arg z\in(-\pi/2,3\pi/2)$. The
constant matrix

$C_{1}=\begin{bmatrix}1&\frac{\sin(\pi(\alpha+\beta))}{\sin(2\pi\alpha)}\\\
0&1\end{bmatrix},$

and the constant matrices $C_{j}$, $j=2,3,4,5$ are determined by $C_{1}$ and
the jump condition. If $2\alpha\in\mathbb{N}$, we have as $z\to 0$

$\Phi(z)=\widehat{\Phi}^{(0)}(z)z^{\alpha\sigma_{3}}\begin{bmatrix}1&(-1)^{2\alpha}\frac{\sin(\pi(\alpha+\beta))}{\pi}\ln
z\\\ 0&1\end{bmatrix}\widehat{C}_{j},~{}z\in\Omega_{\Phi,j},~{}j=1,2,3,4,5,$
(A.7)

where $\widehat{\Phi}^{(0)}(z)$ is analytic near the origin, the regions
$\Omega_{\Phi,j}$ are shown in Figure 5, and the branches for
$z^{\alpha\sigma_{3}}$ and $\ln z$ are chosen such that $\arg
z\in(-\pi/2,3\pi/2)$, the constant matrix $\hat{C}_{1}$ is the identity matrix
and the other constant matrices are determined by $\hat{C}_{1}$ and the jump
condition.

The solution to the RH problem can be constructed explicitly by using the
confluent hypergeometric function

$\displaystyle\Phi(z)=$ $\displaystyle
e^{-\frac{iz}{2}}\left[\begin{array}[]{cc}e^{-\frac{\pi
i(\alpha+\beta)}{2}}\frac{\Gamma(1+\alpha-\beta)}{\Gamma(1+2\alpha)}\phi(\alpha+\beta,1+2\alpha,iz)&-e^{\frac{\pi
i(\alpha-\beta)}{2}}\frac{\Gamma(2\alpha)}{\Gamma(\alpha+\beta)}\phi(-\alpha+\beta,1-2\alpha,iz)\\\
e^{-\frac{\pi
i(\alpha-\beta)}{2}}\frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+2\alpha)}\phi(1+\alpha+\beta,1+2\alpha,iz)&e^{\frac{\pi
i(\alpha+\beta)}{2}}\frac{\Gamma(2\alpha)}{\Gamma(\alpha-\beta)}\phi(1-\alpha+\beta,1-2\alpha,iz)\end{array}\right]$
(A.8) $\displaystyle
z^{\alpha\sigma_{3}}\left[\begin{array}[]{cc}1&\frac{\sin(\pi(\alpha+\beta))}{\sin(2\pi\alpha)}\\\
0&1\end{array}\right]$

for $z\in\Omega_{\Phi,1}$, $\alpha>-1/2$ and $2\alpha\notin\mathbb{N}$; see
[20, 39]. Here, $\phi(a,b,z)$ is the confluent hypergeometric function defined
in (1.4). The expression of $\Phi(z)$ in the other regions is then determined
by using (A.8) and the jump condition (A.4). The case $2\alpha\in\mathbb{N}$
can be constructed in a similar way and particularly the first column of
$\Phi(z)$ is given by the same formula in (A.8).

## Acknowledgements

The work of Shuai-Xia Xu was supported in part by the National Natural Science
Foundation of China under grant numbers 11971492 and 12371257, and by
Guangdong Basic and Applied Basic Research Foundation (Grant No.
2022B1515020063). Yu-Qiu Zhao was supported in part by the National Natural
Science Foundation of China under grant numbers 11971489 and 12371077.

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