#  Isomonodromy sets of accessory parameters for Heun class equations

Jun Xia111 Department of Mathematics, Sun Yat-sen University, GuangZhou
510275, China., Shuai-Xia Xu222Institut Franco-Chinois de l’Energie Nucléaire,
Sun Yat-sen University, GuangZhou 510275, China. and Yu-Qiu
Zhao11footnotemark: 1

###### Abstract

In this paper, we consider the monodromy and, in particularly, the
isomonodromy sets of accessory parameters for the Heun class equations. We
show that the Heun class equations can be obtained as limits of the linear
systems associated with the Painlevé equations when the Painlevé transcendents
go to one of the actual singular points of the linear systems. While the
isomonodromy sets of accessory parameters for the Heun class equations are
described by the Taylor or Laurent coefficients of the corresponding Painlevé
functions, or the associated tau functions, at the positions of the critical
values. As an application of these results, we derive some asymptotic
approximations for the isomonodromy sets of accessory parameters in the Heun
class equations, including the confluent Heun equation, the doubly-confluent
Heun equation and the reduced biconfluent Heun equation.

2010 mathematics subject classification: 33E17; 34A30; 34E05; 34M55; 41A60
Keywords and phrases: Heun class equations, Painlevé equations, accessory
parameter, monodromy, isomonodromy deformation, asymptotic analysis.

###### Contents

  1. 1 Introduction and statement of results
  2. 2  Accessory parameters of RTHE
    1. 2.1 Monodromy of RTHE
    2. 2.2 Isomonodromy deformation and PI equation
    3. 2.3 Reduction of the linear system for PI to RTHE
    4. 2.4 Isomonodromy set of accessory parameters of RTHE
  3. 3 Accessory parameters of THE
    1. 3.1 Monodromy of THE
    2. 3.2 Isomonodromy deformation and PII equation
    3. 3.3 Reduction of the linear system for PII to THE
    4. 3.4 Isomonodromy set of accessory parameters of THE
  4. 4  Accessory parameters of RBHE
    1. 4.1 Monodromy of RBHE
    2. 4.2 Isomonodromy deformation and PXXXIV equation
    3. 4.3 Reduction of the linear system for PXXXIV to RBHE
    4. 4.4 Isomonodromy set of accessory parameters of RBHE
  5. 5  Accessory parameters of DHE
    1. 5.1 Monodromy of DHE
    2. 5.2 Isomonodromy deformation and PIII equation
    3. 5.3 Reduction of the linear system for PIII to DHE
    4. 5.4 Isomonodromy set of accessory parameters of DHE
  6. 6 Accessory parameters of BHE
    1. 6.1 Monodromy of BHE
    2. 6.2 Isomonodromy deformation and PIV equation
    3. 6.3 Reduction of the linear system for PIV to BHE
    4. 6.4 Isomonodromy set of accessory parameters of BHE
  7. 7  Accessory parameters of CHE
    1. 7.1 Monodromy of CHE
    2. 7.2 Isomonodromy deformation and PV equation
    3. 7.3 Reduction of the linear systems for PV to CHE
    4. 7.4 Isomonodromy set of accessory parameters of CHE
  8. 8  Accessory parameters of HE
    1. 8.1 Reduction of the linear system for PVI to HE
    2. 8.2 Isomonodromy set of accessory parameters of HE
  9. 9 Asymptotics of the accessory parameters
    1. 9.1 Asymptotics of the accessory parameter of RBHE
    2. 9.2 Asymptotics of the accessory parameter of CHE
    3. 9.3 Asymptotics of the accessory parameter of DHE

##  1 Introduction and statement of results

The Heun equation (HE) is the general second-order linear ODE having four
regular singular points, with the canonical form [25, Eq. (31.2.1)]

$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha\beta
z-q}{z(z-1)(z-a)}w=0,$ (1.1)

where $\alpha+\beta+1=\gamma+\delta+\epsilon$. The parameters
$\alpha,\beta,\gamma,\delta,\epsilon$ determine the characteristic exponents
of the regular singularities at $z=0,1,a,\infty$: the exponents are
$\\{0,1-\gamma\\}$, $\\{0,1-\delta\\}$, $\\{0,1-\epsilon\\}$ and
$\\{\alpha,\beta\\}$, respectively. While the remaining parameters $a$ and
$q$, known as the accessory parameters, involve global monodromy properties of
(1.1).

The same as the classical Gauss hypergeometric equation, the Heun equation has
several confluent forms. Indeed, there are four standard confluent forms when
two or more singularities merge into one or more irregular singularities (cf.
[25, Eqs. (31.12.1)-(31.12.4)]).
$(i)$ Confluent (or, singly-confluent) Heun equation (CHE):

$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(1+\frac{\gamma}{z}+\frac{\delta}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z(z-a)}w=0.$
(1.2)

This equation has two regular singularities, and an irregular singularity of
rank $1$ at infinity arising from the coalescing of two regular singularities.
$(ii)$ Doubly-confluent Heun equation (DHE):

$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(\frac{1}{2}+\frac{\gamma}{z}-\frac{a^{2}}{2z^{2}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z^{2}}w=0.$
(1.3)

This equation has two irregular singularities of rank $1$ at zero and
infinity, each originating from the confluence of two regular singularities.
$(iii)$ Biconfluent Heun equation (BHE):

$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(2z+2a+\frac{\gamma}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z}w=0.$
(1.4)

This equation possesses a regular singularity, and an irregular singularity of
rank $2$ at infinity arising from the coalescing of three regular
singularities.
$(iv)$ Triconfluent Heun equation (THE):

$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+(2z^{2}+a)\frac{\mathrm{d}w}{\mathrm{d}z}+(pz-q)w=0.$
(1.5)

This equation has one irregular singularity of rank $3$ at infinity, resulting
from the coalescing of all of the four singularities.

Modified or reduced forms of the confluent Heun equations are also available.
Five reduced confluent equations appear as a result of weak confluence
processes. For instance, we have the reduced triconfluent Heun equation (RTHE)

$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}-\left(4z^{3}+2az+q\right)w=0,$ (1.6)

which has one irregular singularity of rank $5/2$ at infinity. There is the
reduced biconfluent Heun equation (RBHE)

$\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\frac{2\alpha}{z}\frac{\mathrm{d}w}{\mathrm{d}z}-\left(z+a+\frac{q}{z}\right)w=0$
(1.7)

with one irregular singularity of rank $3/2$ at infinity and one regular
singularity at the origin. Also, ten more equations are to be added to the
list, if elementary singularities are regarded as special types of
singularities; see [31].

The HE, together with all of its confluent forms, is called the Heun class of
equations; see [29]. This class of equations has numerous applications in
theory of black holes, general relativity, polymer and chemical physics,
astrophysics, molecular physics, crystalline materials and cosmology, etc.;
see [17, 25, 26, 31] and the references therein.

Painlevé equations and Painlevé transcendents will be the main tools in the
present investigation. By Painlevé equations we mean six equations denoted by
PI-PVI. The six Painlevé equations were first introduced by Paul Painlevé and
his coworkers at the turn of the twentieth century in the classification of
second-order ODEs of the form

$y_{xx}=F(x,y,y_{x})$

with $F$ meromorphic in $x$ and rational in $y$ and $y_{x}$, whose solutions
have the Painlevé property, that is, the only movable singularities of their
solutions are poles or isolated essential singularities; see [16, 25]. The
Painlevé equations find important applications in various fields, such as
number theory, statistical mechanics, random matrix theory, orthogonal
polynomials, quantum gravity and quantum field theory, nonlinear optics and
fibre optics, etc.; see [5, 12, 13, 25] and the references therein.

In [14, 15], Fuchs discovered a remarkable connection between the HE and the
PVI. He added an extra apparent singularity at $z=y$ to the HE (1.1). The
apparent singularity is presented in the equation but is absent in the
solution. The position of the apparent singularity $y$ is deformed with the
regular singularity $a$ in (1.1). Then, the PVI equation arose as a
compatibility condition, of the deformed HE coupled with another linear
equation with differentiation in the variable $a$. Inspired by the works of
Fuchs, Slavyanov et al. studied the deformation of the Heun class of equations
by adding an apparent singularity. It was shown that each Painlevé equation
can be considered as the isomonodromy deformation condition for a deformed
equation of the Heun class; see [30]-[32]. In [29], Slavyanov also discovered
that the Heun class equations can be regarded as the quantization of the
classical Hamiltonian of the Painlevé equations. In this sense, there exists
the following correspondence between Heun class equations and Painlevé
equations:

$\mathrm{HE}\rightarrow\mathrm{PVI},\ \mathrm{CHE}\rightarrow\mathrm{PV},\
\mathrm{BHE}\rightarrow\mathrm{PIV},\ \mathrm{DHE}\rightarrow\mathrm{PIII},\
\mathrm{THE}\rightarrow\mathrm{PII},\ \mathrm{RTHE}\rightarrow\mathrm{PI}.$
(1.8)

Here ‘$\rightarrow$’ means that given an equation of Heun class there is a
Painlevé equation corresponding to it.

It is well-known that every Painlevé equation can be obtained as the
compatibility condition of a $2\times 2$ matrix linear system (also called Lax
pair) [20]:

$\frac{\partial\Phi(z,x)}{\partial z}=A(z,x)\Phi(z,x),$ (1.9)

and

$\frac{\partial\Phi(z,x)}{\partial x}=B(z,x)\Phi(z,x).$ (1.10)

The first-order matrix equation (1.9) is equivalent to two second-order linear
ODEs satisfied respectively by the elements of the first row and second row of
$\Phi(z,x)$. More precisely, if we denote the coefficient matrix $A(z,x)$ by

$A(z,x):=A=\begin{pmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{pmatrix},$

then each element of the first row of $\Phi(z,x)$ satisfies the ODE:

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}-\left(\mathrm{Tr}A+\frac{A_{12}^{\prime}}{A_{12}}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}+\left(\det
A-A_{11}^{\prime}+A_{11}\frac{A_{12}^{\prime}}{A_{12}}\right)\Phi_{1}=0,$
(1.11)

while the elements of the second row solve a similar ODE:

$\frac{\mathrm{d}^{2}\Phi_{2}}{\mathrm{d}z^{2}}-\left(\mathrm{Tr}A+\frac{A_{21}^{\prime}}{A_{21}}\right)\frac{\mathrm{d}\Phi_{2}}{\mathrm{d}z}+\left(\det
A-A_{22}^{\prime}+A_{22}\frac{A_{21}^{\prime}}{A_{21}}\right)\Phi_{2}=0.$
(1.12)

Here, the prime indicates the differentiation with respect to $z$.

It was observed in [12, p.86] that the linear equation (1.11) for PVI is
equivalent to HE when the solutions to PVI approach the critical values
$0,1,x,\infty$, which are the actual singular points of (1.11). Recently,
Dubrovin and Kapaev [10] studied in detail the equivalence of HE and the Lax
pair for PVI at the movable poles of the solutions of PVI. Moreover, they
established a connection between the accessory parameter of HE and the free
parameter of the Laurent expansion of PVI. Similar results were also derived
earlier in [22]. In [6, 7], the equivalence of CHE and the Lax pair for PV at
certain critical value of PV was shown and the accessory parameter was
expressed in terms of the $\tau$-function of PV. An application to black holes
was also addressed therein.

In this paper, we consider the monodromy and isomonodromy deformation of the
Heun class equations. Firstly, we describe the monodromy of Heun class
equations and consider the isomonodromy deformation by using the linear
systems for the corresponding Painlevé equations. We show that the Heun class
equations can be obtained as limits of the linear system (1.11) or (1.12)
associated with the Painlevé equations I-VI and XXXIV as the corresponding
Painlevé transcendents $y(x)$ approach one of the actual singular points of
the linear system:

$\mathrm{PI}\rightarrow\mathrm{RTHE},~{}~{}~{}~{}\mathrm{PII}\rightarrow\mathrm{THE},~{}~{}~{}~{}\mathrm{PXXXIV}\rightarrow\mathrm{RBHE},~{}~{}~{}~{}\mathrm{PIII}\rightarrow\mathrm{DHE},$
(1.13)

and

$\mathrm{PIV}\rightarrow\mathrm{BHE},~{}~{}~{}~{}\mathrm{PV}\rightarrow\mathrm{CHE},~{}~{}~{}~{}\mathrm{PVI}\rightarrow\mathrm{HE}.$
(1.14)

Moreover, the accessory parameters in these equations of Heun class are
determined by the corresponding Painlevé functions and $\tau$-functions.
Secondly, using the the limiting procedure in (1.13), (1.14) and the monodromy
of a specific Heun class equation, we show that there is a discrete set of
pairs of accessory parameters $(a_{n},q_{n})$, such that the equation of Heun
class corresponding to these parameters has the same monodromy data. Under a
bijection, the discrete set coincides with the set of parameters
$(a_{n},b_{n})$ in the Taylor or Laurent expansions, respectively at the zeros
or poles $a_{n}$, of the unique solution to the corresponding Painlevé
equation with the same monodromy data as the Heun class equation. Finally,
using known asymptotic expansions for the Painlevé transcendents and the
associated $\tau$-functions in the literature, we derive some asymptotic
approximations for the isomonodromy sets of accessory parameters in the Heun
class equations.

The rest of this paper is organized in the following way. We consider the
monodromy data and isomonodromy deformation of the Heun class equations RTHE-
HE in Sections 2-8, respectively. We describe the isomonodromy sets of
accessary parameters by the sets of parameters in the Taylor or Laurent
expansion near the zeros or poles of the solution of the corresponding
Painlevé equation with the same monodromy data as the Heun class equation. The
main results are stated in Theorems 1-7 at the end of each section. The proofs
of Theorem 1 and Theorem 7 are given in Section 2 and Section 8, respectively,
with details included. These proofs are concerned with two representative
examples of Heun class equations, namely the HE with four regular
singularities and the RTHE with an irregular singularity. While Theorems 2-6
can be proved in the same manner and we skip the detail. In the last section,
we derive asymptotic approximations for isomonodromy sets of accessory
parameters of some Heun class equations, expressed in terms of the monodromy
data. The equations of Heun class we considered in this section include the
RBHE, CHE, and DHE.

##  2  Accessory parameters of RTHE

###  2.1 Monodromy of RTHE

Consider the RTHE equation (1.6) with parameters $a$ and $q$. There exist
unique solutions, of the form $Y_{k}(z)=(y_{k1}(z),y_{k2}(z))$, which satisfy
the normalized asymptotic behavior as $z\to\infty$

$Y_{k}(z)\sim
z^{-\frac{3}{4}}(1,1)e^{(\frac{4}{5}z^{\frac{5}{2}}+az^{\frac{1}{2}})\sigma_{3}},\quad
z\in\Omega_{k},$ (2.1)

where $\Omega_{k}$, $k=-1,\cdots,4$, are the Stokes sectors defined by

$\Omega_{k}=\left\\{z\in\mathbb{C}:~{}~{}\frac{\pi}{5}(2k-5)<\mathop{\rm
arg}\nolimits z<\frac{\pi}{5}(2k-1)\right\\},$ (2.2)

and $\sigma_{3}$ is one of the Pauli matrices,

$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\quad\mbox{and}\quad\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$

The solutions are related to each other by the Stokes matrices

$Y_{k+1}(z)=Y_{k}(z)S_{k},\quad k=-1,\cdots,3,$ (2.3)

where

$S_{2k-1}=\begin{pmatrix}1&0\\\ s_{2k-1}&1\end{pmatrix},\quad
S_{2k}=\begin{pmatrix}1&s_{2k}\\\ 0&1\end{pmatrix}.$ (2.4)

Moreover, it follows from the asymptotic behavior in (2.1) that

$Y_{4}(e^{2\pi i}z)=iY_{-1}(z)\sigma_{1}.$ (2.5)

Using (2.3)-(2.5), we find the cyclic condition

$S_{-1}\cdots S_{3}=i\sigma_{1}.$ (2.6)

The restriction (2.6) contributes four scalar algebraic equations, three among
them are independent. Accordingly, the Stokes multipliers $s_{k}$ fulfill

$s_{k}=i(1+s_{k+2}s_{k+3})$ (2.7)

for all integer $k$, with $s_{k+5}=s_{k}$. The monodromy data is constituted
by the set of Stokes multipliers, which is a 2-dimensional surface in
$\mathbb{C}^{5}$ described by (2.7)

$\\{(s_{1},s_{2},s_{3},s_{4},s_{5})\in\mathbb{C}^{5}:s_{1},\dots
s_{5}~{}\mbox{satisfy}~{}\eqref{eq:RTHE-StokesEq}\\}.$ (2.8)

###  2.2 Isomonodromy deformation and PI equation

To study the isomonodromy deformation of the RTHE equation, it is convenient
to consider a $2\times 2$ matrix system, which has one irregular singular
point of rank 5/2 at infinity. In [12, 20], the isomonodromy deformation of
such a matrix system has been considered. We review it briefly below in this
section.

Consider the following Lax pair for PI parameterized in the way as [20, (C.2)]

$\left\\{\begin{aligned} &\frac{\partial\Phi(z,x)}{\partial
z}=A(z,x)\Phi(z,x),\\\ &\frac{\partial\Phi(z,x)}{\partial
x}=B(z,x)\Phi(z,x),\end{aligned}\right.$ (2.9)

where

$A(z,x)=\begin{pmatrix}-v&z^{2}+yz+y^{2}+\frac{x}{2}\\\
4(z-y)&v\end{pmatrix},\ \ \ \ \ B(z,x)=\begin{pmatrix}0&\frac{z}{2}+y\\\
2&0\end{pmatrix}.$

The compatibility condition of the Lax pair reads

$\left\\{\begin{aligned} &\frac{\mathrm{d}y}{\mathrm{d}x}=v,\\\
&\frac{\mathrm{d}v}{\mathrm{d}x}=6y^{2}+x,\end{aligned}\right.$ (2.10)

which leads to the PI equation

$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=6y^{2}+x.$

The $\tau$-function associated with PI is defined by (see [20, (C.7)])

$\sigma(x)=\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=\frac{1}{2}v^{2}-2y^{3}-xy.$
(2.11)

Then, we have (see [20, (C.9)])

$\left(\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}x^{2}}\right)^{2}+4\left(\frac{\mathrm{d}\sigma}{\mathrm{d}x}\right)^{2}+2x\frac{\mathrm{d}\sigma}{\mathrm{d}x}-2\sigma=0.$
(2.12)

There exist unique solutions $\Phi_{k}(z,x)$ of the equation (2.9), which
satisfy the normalized asymptotic behavior as $z\to\infty$

$\Phi_{k}(z,x)=\left(\frac{z}{4}\right)^{\frac{1}{4}\sigma_{3}}\frac{\sigma_{3}+\sigma_{1}}{i\sqrt{2}}\left(I+O\left(z^{-\frac{1}{2}}\right)\right)e^{\left(\frac{4}{5}z^{\frac{5}{2}}+xz^{\frac{1}{2}}\right)\sigma_{3}},\quad
z\in\Omega_{k},$ (2.13)

where $\Omega_{k}$ for $k=-1,\cdots,4$ are the Stokes sectors defined by
(2.2). From (2.13), we see that the solutions are related to each other by the
Stokes matrices

$\Phi_{k+1}(z)=\Phi_{k}(z)\hat{S}_{k},\quad k=-1,\cdots,3,$ (2.14)

where

$\hat{S}_{2k-1}=\begin{pmatrix}1&0\\\
\hat{s}_{2k-1}&1\end{pmatrix},\quad\hat{S}_{2k}=\begin{pmatrix}1&\hat{s}_{2k}\\\
0&1\end{pmatrix}.$ (2.15)

The behavior (2.13) also gives

$\Phi_{4}(e^{2\pi i}z)=i\Phi_{-1}(z)\sigma_{1}.$ (2.16)

From (2.14)-(2.16), we find the cyclic condition

$\hat{S}_{-1}\hat{S}_{0}\cdots\hat{S}_{3}=i\sigma_{1}.$ (2.17)

This condition implies the following algebraic equations

$\hat{s}_{k}=i(1+\hat{s}_{k+2}\hat{s}_{k+3}),$ (2.18)

with $k\in\mathbb{Z}$ and $\hat{s}_{k+5}=\hat{s}_{k}$. Thus, the Stokes
multipliers are described by the 2-dimensional surface (2.8) in
$\mathbb{C}^{5}$ with the coordinates $(\hat{s}_{k})_{1\leq k\leq 5}$.

###  2.3 Reduction of the linear system for PI to RTHE

In this subsection, we show that the RBHE can be obtained as a limit of the
linear system (1.12) associated with the Lax pair for the Painlevé I equation
when $x$ tends to one of the poles of the Painlevé I transcendents.

Substituting the enties of $A(z,x)$ into (1.12), we obtain

$\frac{\mathrm{d}^{2}\Phi_{2}}{\mathrm{d}z^{2}}-\frac{1}{z-y}\frac{\mathrm{d}\Phi_{2}}{\mathrm{d}z}-Q(z,x)\Phi_{2}=0,$
(2.19)

where

$\displaystyle Q(z,x)$
$\displaystyle=4z^{3}+2xz+v^{2}-4y^{3}-2xy-\frac{v}{z-y}$
$\displaystyle=4z^{3}+2xz+2\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{v}{z-y}.$

To obtain RTHE, we need to eliminate the singularity $z=y$ in (2.19). This can
be realized by considering the poles of $y$. It is known that $y$ admits the
following Laurent expansion near a pole (see [16, Eq. (1.1)] ):

$y(x)=\frac{1}{(x-a)^{2}}-\frac{a}{10}(x-a)^{2}-\frac{1}{6}(x-a)^{3}+b(x-a)^{4}+O((x-a)^{5}),$
(2.20)

where $b$ is an arbitrary parameter.

Using the first equation of (2.10) to replace $v$ by $y^{\prime}$ and
substituting (2.20) into the expression of $Q$ in (2.19), we obtain the RTHE
as $x\rightarrow a$

$\frac{\mathrm{d}^{2}\Phi_{2}}{\mathrm{d}z^{2}}-\left(4z^{3}+2az+q\right)\Phi_{2}=0,$
(2.21)

with the accessory parameter $q$ given by

$q=\lim_{x\rightarrow
a}2\left(\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{1}{x-a}\right)=-28b.$
(2.22)

###  2.4 Isomonodromy set of accessory parameters of RTHE

As shown in Section 2.3, the isomonodromy deformation of (2.9) is described by
the PI equation. While the RTHE can be obtained as a limit of the second row
of the isomonodromy family of $\Phi(z,x)$, when $x$ tends to one of the pole
of the Painlevé I transcendent. The accessory parameters of RTHE are then
expressed in terms of the parameters in the Laurent expansion near the poles
of the PI transcendents. As a consequence, we obtain a family of accessory
parameters such that the corresponding RTHE shares the same monodromy data, as
stated in the following theorem.

######  Theorem 1.

There is a discrete set of pairs of accessory parameters $(a_{n},q_{n})$ such
that the RTHE (1.6) corresponding to these parameters has the same monodromy
data (2.8) as the original one with the parameters $a$ and $q$. This set
coincides, via the transformation $q_{n}=-28b_{n}$, with the set of parameters
$(a_{n},b_{n})$ in the Laurent expansion (2.20) near the poles $a_{n}$, of the
unique solution of PI corresponding to the same monodromy data (2.8) as the
RTHE.

###### Proof.

Consider the RTHE (1.6) with given accessory parameters $a$ and $q$. There
exist unique solutions of (1.6) which satisfy the normalized asymptotic
behavior (2.1) as $z\to\infty$. The solutions are related by the Stokes
matrices $S_{k}$ as given in (2.3) and (2.4), while the independent Stokes
multipliers constitute the monodromy data (2.8) for the RTHE. Moreover, there
exist unique solutions $\Phi_{k}(z,x)$ of the system (2.9) with the normalized
behavior (2.13) at infinity, where the $x$-dependence of $\Phi_{k}(z,x)$ is
described by the unique solution of the PI equation having the Laurent
expansion (2.20). Here we let $a$ be a pole of the PI solution and choose the
parameter $b$ according to (2.22), namely $b=-\frac{q}{28}$. The solutions
$\Phi_{k}(z,x)$ are related by the Stokes matrices $\hat{S}_{k}$ defined in
(2.14) and (2.15), which are independent of $x$ and have the same triangular-
matrix forms as the Stokes matrices $S_{k}$ for the RTHE.

From the limiting procedure demonstrated in Section 2.3, we obtain the RTHE
equation with accessory parameters $a$ and $q$, as a limit of the second row
of the isomonodromy family $\Phi_{k}(z;x)$ when $x\to a$. It follows from the
$x$-independence of the monodromy data that the Stokes matrices
$S_{k}=\hat{S}_{k}$. Thus, we have shown that any given accessory parameters
$(a,q)$ is related, via $q=-28b$, to the pole parameters $(a,b)$ of the unique
solution of PI corresponding to the same monodromy data (2.8). It is known
that each Painlevé tanscendents has infinity many poles which are discrete in
the complex plane. Therefore the set of pairs of accessory parameters of RTHE
sharing the same monodromy data are also discrete. Thus, we complete the proof
of Theorem 1. ∎

##  3 Accessory parameters of THE

###  3.1 Monodromy of THE

Consider the THE (1.5) with the accessory parameters $a$ and $q$ and the fixed
parameter $p$. There exist unique solutions of (1.5) with normalized
asymptotic behavior as $z\to\infty$

$Y_{k}(z)\sim
e^{-(\frac{1}{3}z^{3}+\frac{a}{2}z)}z^{-1}(1,1)e^{(\frac{1}{3}z^{3}+\frac{a}{2}z)\sigma_{3}}z^{(\mu-\frac{1}{2})\sigma_{3}},\quad
z\in\Omega_{k},$ (3.1)

where the Stokes sectors are defined by

$\Omega_{k}=\left\\{z\in\mathbb{C}:~{}~{}\frac{\pi}{6}\left(2k-3\right)<\mathop{\rm
arg}\nolimits z<\frac{\pi}{6}\left(2k+1\right)\right\\},$ (3.2)

for $k=1,\cdots,7$. Here the parameter $\mu$ is given by $\mu=(3-p)/2$. The
solutions are related by the Stokes matrices

$Y_{k+1}(z)=Y_{k}(z)S_{k},\quad k=1,\dots,6,$ (3.3)

where

$S_{2k-1}=\begin{pmatrix}1&s_{2k-1}\\\ 0&1\end{pmatrix},\quad
S_{2k}=\begin{pmatrix}1&0\\\ s_{2k}&1\end{pmatrix}.$ (3.4)

The behavior (3.1) also gives

$Y_{7}(e^{2\pi i}z)=Y_{1}(z)e^{\pi i(2\mu-1)\sigma_{3}}.$ (3.5)

We define the monodromy data of THE by

$\left\\{S_{k}\right\\}_{k=1}^{6}.$ (3.6)

From (3.3)-(3.5), we see that the Stokes matrices fulfill the cyclic condition

$S_{1}S_{2}\cdots S_{6}=e^{\pi i(2\mu-1)\sigma_{3}}.$ (3.7)

Thus, the Stokes multipliers $s_{k}$, $k=1,\cdots,6$ are constrained by the
following algebraic equations

$\left\\{\begin{aligned} &1+s_{1}s_{2}+e^{2\pi i\mu}(1+s_{4}s_{5})=0,\\\
&1+s_{2}s_{3}+e^{-2\pi i\mu}(1+s_{5}s_{6})=0,\\\
&s_{1}+s_{3}+s_{1}s_{2}s_{3}-e^{2\pi i\mu}s_{5}=0.\end{aligned}\right.$ (3.8)

These equations describe a 3-dimensional surface in $\mathbb{C}^{6}$ with the
coordinates $(s_{k})_{1\leq k\leq 6}.$

###  3.2 Isomonodromy deformation and PII equation

To study the isomonodromy deformation of THE, it is convenient to consider a
$2\times 2$ matrix system, which has one irregular singular point of rank 3 at
infinity. In [20], the isomonodromy deformation of the matrix system with such
an irregular singular point has been considered.

Recall the following Lax pair for PII with the parameterization given in [20,
(C.10)-(C.11)]:

$\left\\{\begin{aligned} &\frac{\partial\Phi(z,x)}{\partial
z}=A(z,x)\Phi(z,x),\\\ &\frac{\partial\Phi(z,x)}{\partial
x}=B(z,x)\Phi(z,x),\end{aligned}\right.$ (3.9)

where

$A(z,x)=\begin{pmatrix}z^{2}+v+\frac{x}{2}&u(z-y)\\\
-\frac{2}{u}\big{(}vz+vy+\frac{1}{2}-\mu\big{)}&-z^{2}-v-\frac{x}{2}\end{pmatrix},\
\ \ \ \ B(z,x)=\begin{pmatrix}\frac{1}{2}z&\frac{1}{2}u\\\
-\frac{v}{u}&-\frac{1}{2}z\end{pmatrix}.$

The compatibility condition of the above Lax pair gives

$\left\\{\begin{aligned}
&\frac{\mathrm{d}y}{\mathrm{d}x}=y^{2}+v+\frac{x}{2},\\\
&\frac{\mathrm{d}v}{\mathrm{d}x}=-2yv-\frac{1}{2}+\mu,\\\
&\frac{\mathrm{d}u}{\mathrm{d}x}=-uy,\end{aligned}\right.$ (3.10)

which yields the PII equation

$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=2y^{3}+xy+\mu.$ (3.11)

The PII $\tau$-function is defined by (see [20, (C.15)])

$\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=\frac{1}{2}v^{2}+\left(y^{2}+\frac{x}{2}\right)v+\left(\frac{1}{2}-\mu\right)y.$
(3.12)

There exist unique solutions $\Phi_{k}(z,x)$ of the equation (3.9) specified
by the normalized asymptotic behavior as $z\to\infty$

$\Phi_{k}(z,x)=(I+O(1/z))e^{(\frac{1}{3}z^{3}+\frac{1}{2}xz)\sigma_{3}}z^{(\mu-\frac{1}{2})\sigma_{3}},\quad
z\in\Omega_{k},$ (3.13)

where $\Omega_{k}$ for $k=1,\cdots,7$ are the Stokes sectors defined in (3.2).
From (3.13), we see that the solutions are related to each other by the Stokes
matrices

$\Phi_{k+1}(z)=\Phi_{k}(z)\hat{S}_{k},\quad k=1,\cdots,6,$ (3.14)

where

$\hat{S}_{2k-1}=\begin{pmatrix}1&\hat{s}_{2k-1}\\\
0&1\end{pmatrix},\quad\hat{S}_{2k}=\begin{pmatrix}1&0\\\
\hat{s}_{2k}&1\end{pmatrix}.$ (3.15)

We define the monodromy data of the system (3.9) by

$\\{\hat{S}_{k}\\}_{k=1,\dots,6}.$ (3.16)

The behavior (3.13) also implies

$\Phi_{7}(e^{2\pi i}z)=\Phi_{1}(z)e^{\pi i(2\mu-1)\sigma_{3}}.$ (3.17)

From (3.14)-(3.17), we find the cyclic condition

$\hat{S}_{1}\hat{S}_{2}\cdots\hat{S}_{6}=e^{\pi i(2\mu-1)\sigma_{3}}.$ (3.18)

The equation gives us three independent equations of the Stokes multipliers
$\hat{s}_{k}$, $k=1,\cdots,6$ as given in (3.8); see also [12, Proposition
5.3].

###  3.3 Reduction of the linear system for PII to THE

In this subsection, we derive THE from the linear system (1.11) for PII at
each pole of the solutions of PII.

Substituting $A(z,x)$ into (1.11) gives

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}-\frac{1}{z-y}\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}-Q(z,x)\Phi_{1}=0,$
(3.19)

where

$\displaystyle Q(z,x)$
$\displaystyle=z^{4}+xz^{2}+(1+2\mu)z+\frac{x^{2}}{4}+v^{2}+\left(2y^{2}+x\right)v+\left(1-2\mu\right)y-\frac{z^{2}+\frac{x}{2}+v}{z-y}$
$\displaystyle=z^{4}+xz^{2}+(1+2\mu)z+\frac{x^{2}}{4}+2\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{z^{2}+\frac{x}{2}+v}{z-y}.$

To eliminate the singularity $z=y$ in (3.19), we consider the poles of $y$.
Let $x=a$ be a pole of $y$, then $y$ possesses the following Laurent expansion
(see [16, (17.1)]):

$y(x)=\frac{\varepsilon}{x-a}-\frac{\varepsilon
a}{6}(x-a)-\frac{\mu+\varepsilon}{4}(x-a)^{2}+b(x-a)^{3}+O((x-a)^{4}),$ (3.20)

where $\varepsilon=\varepsilon_{\pm}=\pm 1$ and $b\in\mathbb{C}$ is arbitrary.
From the first equation of the compatibility condition in (3.10) and (3.20),
we obtain

$v(x)=\left\\{\begin{aligned}
&-\frac{2}{(x-a)^{2}}+O(1),&\varepsilon=\varepsilon_{+},\\\ &O(x-a),\
&\varepsilon=\varepsilon_{-}.\end{aligned}\right.$ (3.21)

Hence, taking $x\rightarrow a$ in (3.19) and then making the transformation

$\Phi_{1}=w\exp\left(\frac{1}{3}z^{3}+\frac{a}{2}z\right),$

we obtain the THEs

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+(2z^{2}+a)\frac{\mathrm{d}w}{\mathrm{d}z}+(pz-q)w=0,\\\
&p=3-2\mu,\\\ &q=\lim_{x\rightarrow
a}2\left(\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{1}{x-a}\right)=-\frac{a^{2}}{18}-10b,\end{aligned}\right.$
(3.22)

for $\varepsilon=\varepsilon_{+}=1$,

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+(2z^{2}+a)\frac{\mathrm{d}w}{\mathrm{d}z}+(pz-q)w=0,\\\
&p=1-2\mu,\\\ &q=\lim_{x\rightarrow
a}2\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=-\frac{a^{2}}{18}+10b,\end{aligned}\right.$
(3.23)

for $\varepsilon=\varepsilon_{-}=-1$.

###  3.4 Isomonodromy set of accessory parameters of THE

Thus, we see that THE can be obtained as a limit of the first row of the
isomonodromy family of $\Phi(z,x)$, when $x\to a$, with $a$ being the pole of
the Painlevé II transcendent corresponding to the same monodromy data of THE.
While, the accessory parameters $(a,q)$ are expressed in terms of the
parameters in the Laurent expansion of the PII transcendent. Consequently, we
have the following description of the isomonodromy set of accessory parameters
of THE.

######  Theorem 2.

There is a discrete set of pairs of accessory parameters $(a_{n},q_{n})$ such
that the THE (1.5), corresponding to these parameters has the same monodromy
data (3.6) and (3.8) as the original equation with the parameters $a$ and $q$.
Under the bijection given in the third equation of (3.22), this set coincides
with the set of parameters $(a_{n},b_{n})$ in the Laurent expansion (cf.
(3.20)) near the poles $a_{n}$ of the unique solution of PII equation (3.11)
with the parameter $\mu=(3-p)/2$ and the same monodromy data (3.6) and (3.8)
as the THE.

##  4  Accessory parameters of RBHE

###  4.1 Monodromy of RBHE

Consider the RBHE (1.7) with the fixed parameter $2\alpha$ and the accessory
parameters $a$ and $q$. There exist two linear independent solutions
$Y_{k}=(y_{k1},y_{k2})$ of (1.7) normalizing the asymptotic behavior as
$z\to\infty$

$Y_{k}(z)\sim
z^{-(\alpha+\frac{1}{4})}(1,i)e^{-(\frac{2}{3}z^{\frac{3}{2}}+az^{\frac{1}{2}})\sigma_{3}},\quad
z\in\Omega_{k},$ (4.1)

where $\Omega_{k}$ for $k=-1,\cdots,2$ are the Stokes sectors

$\Omega_{k}=\left\\{z\in\mathbb{C}:~{}~{}\frac{\pi}{3}(2k-3)<\mathop{\rm
arg}\nolimits z<\frac{\pi}{3}(2k+1)\right\\}.$ (4.2)

The solutions are related to each other by the Stokes matrices

$Y_{k+1}(z)=Y_{k}(z)S_{k},\quad k=-1,0,1,$ (4.3)

where

$S_{\pm 1}=\begin{pmatrix}1&0\\\ s_{\pm 1}&1\end{pmatrix},\quad
S_{0}=\begin{pmatrix}1&s_{0}\\\ 0&1\end{pmatrix}.$ (4.4)

The behavior (4.1) also gives

$Y_{2}(e^{2\pi i}z)=e^{-2\pi i\alpha}Y_{-1}(z)i\sigma_{2}.$ (4.5)

Moreover, from (1.7) we see that the characteristic exponents at $z=0$ are
$\\{0,1-2\alpha\\}$. Hence, as $z\to 0$, we have the asymptotic behavior

$Y_{2}(z)\sim z^{-\alpha}(1,z)z^{\alpha\sigma_{3}}E$ (4.6)

for $2\alpha\not\in\mathbb{Z}$, with $E$ being some invertible constant
matrix. Using (4.3)-(4.6), we obtain the same cyclic condition as that for the
Lax pair of the PXXXIV equation

$S_{-1}S_{0}S_{1}=i\sigma_{2}E^{-1}e^{-2\pi i\alpha\sigma_{3}}E.$ (4.7)

This condition implies the following algebraic equation

$s_{-1}s_{0}s_{1}+s_{-1}+s_{1}-s_{0}=-2\cos(2\pi\alpha).$ (4.8)

Thus, the monodromy data for RBHE (1.7) are described by the 2-dimensional
complex surface

$\left\\{(s_{-1},s_{0},s_{1})\in\mathbb{C}^{3}:~{}~{}s_{-1}s_{0}s_{1}+s_{-1}+s_{1}-s_{0}=-2\cos(2\pi\alpha)\right\\}.$
(4.9)

###  4.2 Isomonodromy deformation and PXXXIV equation

To study the isomonodromy deformation of the RBHE (1.7), we regard
$z^{\alpha}Y_{k}(z)$ as the first row of a $2\times 2$ matrix system
$\Phi_{k}(z,x)$ when $x\to a$. Then, we look for the system $\Phi_{k}(z,x)$
such that $\Phi_{k}(z,x)$ satisfy the same jump relations (4.3)-(4.5) and the
following asymptotic behaviors

$\Phi_{k}(z,x)=z^{-\frac{1}{4}\sigma_{3}}\frac{I+i\sigma_{1}}{\sqrt{2}}\left(I+O\left(z^{-\frac{1}{2}}\right)\right)e^{-\left(\frac{2}{3}z^{\frac{3}{2}}+xz^{\frac{1}{2}}\right)\sigma_{3}},\quad
z\in\Omega_{k},$ (4.10)

as $z\to\infty$, where $\Omega_{k}$, $k=-1,\cdots,2$, are the Stokes sectors
defined in (4.2), and

$\Phi_{2}(z,x)=\Phi^{(0)}(x)\left(I+O(z)\right)z^{\alpha\sigma_{3}}E,~{}~{}z\to
0,$ (4.11)

where the parameter $2\alpha\not\in\mathbb{Z}$ and $\Phi^{(0)}(x)$ is a matrix
independent of $z$. The behavior of $\Phi_{k}(z,x)$ as $z\to 0$ are then
determined by (4.10) and the jump relations.

It can be shown that $\Phi_{k}(z,x)$ satisfies the following Lax pair with
rational coefficients (see [12, Chapter 5], [18, Lemma 3.2] and [34])

$\left\\{\begin{aligned} &\frac{\partial\Phi(z,x)}{\partial
z}=A(z,x)\Phi(z,x),\\\ &\frac{\partial\Phi(z,x)}{\partial
x}=B(z,x)\Phi(z,x),\end{aligned}\right.$ (4.12)

where

$A(z,x)=\begin{pmatrix}\frac{y^{\prime}}{2z}&i-i\frac{y}{z}\\\
-iz-i(y+x)-i\frac{(y^{\prime})^{2}-(2\alpha)^{2}}{4yz}&-\frac{y^{\prime}}{2z}\end{pmatrix},\
\ \ B(z,x)=\begin{pmatrix}0&i\\\ -iz-2i(y+\frac{x}{2})&0\end{pmatrix}.$

The Lax pair is related by a gauge transformation to the Lax pair for PII
found by Flaschka and Newell; see for instance [12, Chapter 5] and [18,
Equation (3.19)].

The compatibility condition of the above Lax pair is described by the PXXXIV
equation

$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{1}{2y}\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}+4y^{2}+2xy-\frac{(2\alpha)^{2}}{2y}.$
(4.13)

Let

$v(x)=\frac{y^{\prime}(x)-2\alpha}{2y(x)},$

then $-2^{-1/3}v(-2^{-1/3}x)$ satisfies the PII equation (3.11) with
$\mu=-(2\alpha+\frac{1}{2})$. The associated $\tau$-function can be defined as

$\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=-y(x)^{2}+\left(v(x)^{2}-x\right)y(x)+2\alpha
v(x),$ (4.14)

which is related simply to the standard Hamiltonian for the PII equation; cf.
[25, Equation (32.6.9)].

Thus, we see that the solutions $\Phi_{k}(z,x)$ to the Lax pair (4.12) subject
to the boundary conditions (4.10)-(4.11) share the same monodromy data as RBHE
(independent of $x$) given in (4.9). While the isomonodromy deformation of
$\Phi_{k}(z,x)$ with respect to $x$ are described by the PXXXIV equation.

In the next subsection, we show that the RBHE can actually be obtained as a
limit of the linear system (1.11), associated with the Lax pair for the
Painlevé XXXIV equation, when the Painlevé XXXIV transcendent $y(x)$ tends to
zero or infinity.

###  4.3 Reduction of the linear system for PXXXIV to RBHE

Substituting the elements of $A(z,x)$ into (1.11) gives

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}-\left(\frac{1}{z-y}-\frac{1}{z}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}-P(z,x)\Phi_{1}=0,$
(4.15)

where

$\displaystyle P(z,x)$
$\displaystyle=z+x+\frac{((y^{\prime})^{2}-4\alpha^{2})/(4y)-y^{2}-xy}{z}+\frac{\alpha^{2}}{z^{2}}-\frac{y^{\prime}}{2z(z-y)}$
$\displaystyle=z+x+\frac{1}{z}\left(\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)\right)+\frac{\alpha^{2}}{z^{2}}-\frac{y^{\prime}}{2z(z-y)}.$

To eliminate the singularity $z=y$, we consider the poles and zeros of $y$. It
is readily seen that $y$ solving (4.13) possesses the following Laurent
expansion at a pole

$y(x)=\frac{1}{(x-a)^{2}}-\frac{a}{3}-\frac{1}{2}(x-a)+b(x-a)^{2}+O\left((x-a)^{3}\right),$
(4.16)

and the Taylor expansion at a zero

$y(x)=\varepsilon(x-a)+b(x-a)^{2}+\frac{2a\varepsilon}{3}(x-a)^{3}+O\left((x-a)^{4}\right),$
(4.17)

where $\varepsilon=\varepsilon_{\pm}=\pm 2\alpha$ and $b$ is arbitrary.

Taking $x\rightarrow a$ in (4.15) and making the transformation

$\Phi_{1}=z^{\alpha}w,$ (4.18)

we see that $w(z)$ satisfies the following RBHEs

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\frac{2\alpha+1}{z}\frac{\mathrm{d}w}{\mathrm{d}z}-\left(z+a+\frac{q}{z}\right)w=0,\\\
&q=\lim_{x\rightarrow
a}\left(\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{1}{x-a}\right)=\frac{a^{2}}{3}-5b,\end{aligned}\right.$
(4.19)

when $a$ is a pole of $y(x)$, and

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\frac{2\alpha}{z}\frac{\mathrm{d}w}{\mathrm{d}z}-\left(z+a+\frac{q}{z}\right)w=0,\\\
&q=\lim_{x\rightarrow
a}\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=b,\end{aligned}\right.$ (4.20)

when $a$ is a zero of $y(x)$ with coefficient
$\varepsilon=\varepsilon_{+}=2\alpha$ in (4.17). When $a$ is a zero with
coefficient $\varepsilon=\varepsilon_{-}=-2\alpha$, taking $x\rightarrow a$
and the transformation $\Phi_{1}=z^{\alpha+1}w$ in (4.15) we arrive at

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\frac{2(\alpha+1)}{z}\frac{\mathrm{d}w}{\mathrm{d}z}-\left(z+a+\frac{q}{z}\right)w=0,\\\
&q=\lim_{x\rightarrow
a}\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=b.\end{aligned}\right.$ (4.21)

###  4.4 Isomonodromy set of accessory parameters of RBHE

Thus, we see that RBHE can be obtained as a limit of the first row of the
isomonodromy family of $\Phi(z,x)$ when $x\to a$, $a$ being a zero or pole of
the Painlevé XXXIV transcendents corresponding to the same monodromy data as
RBHE. While the accessory parameters $(a,q)$ are expressed in terms of the
parameters in the Taylor or Laurent expansion of the PXXXIV transcendents. As
a consequence, we have the following description of the isomonodromy accessory
parameters set of RBHE.

######  Theorem 3.

There is a discrete set of pairs of accessory parameters $(a_{n},q_{n})$ such
that the RBHE (1.7) corresponding to these parameters has the same monodromy
data (4.9) as the original equation with the parameters $a$ and $q$. This set
coincides with the set of parameters $(a_{n},b_{n})$ in the Taylor expansion
(4.17) near the zeros $a_{n}$ of the unique solution of the PXXXIV equation
(4.13) with the same monodromy data (4.9) as the RBHE.

##  5  Accessory parameters of DHE

###  5.1 Monodromy of DHE

Consider the DHE (1.3) with the accessory parameters $a$ and $q$ and the fixed
parameters $\gamma$ and $p$. Let $\Omega_{k}^{(\infty)}$, $k=1,2$, be the
Stokes sectors

$\Omega_{k}^{(\infty)}=\left\\{z\in\mathbb{C}:~{}~{}\frac{\pi}{2}(-2k+1)<\mathop{\rm
arg}\nolimits z<\frac{\pi}{2}(-2k+5)\right\\}.$ (5.1)

There exist unique linear independent solutions of (1.3), namely
$Y_{k}^{(\infty)}=(y_{k1}^{(\infty)},y_{k2}^{(\infty)})$, which satisfy the
normalized asymptotic behavior as $z\to\infty$

$Y_{k}^{(\infty)}(z)\sim
e^{-\frac{z}{4}}z^{-\frac{1}{2}\theta_{0}}\left(1,~{}1/z\right)e^{\frac{z}{4}\sigma_{3}}z^{-\frac{1}{2}\theta_{\infty}\sigma_{3}},\quad
z\in\Omega_{k}^{(\infty)},$ (5.2)

for $k=1,2$. Here the parameters $\theta_{0}=\gamma-1$ and
$\theta_{\infty}=4p-\gamma+1$. The solutions are related to each other through
the Stokes matrices

$Y_{2}^{(\infty)}(z)=Y_{1}^{(\infty)}(z)S_{1}^{(\infty)},\quad
S_{1}^{(\infty)}=\begin{pmatrix}1&0\\\ s_{1}^{(\infty)}&1\end{pmatrix},$ (5.3)
$Y_{1}^{(\infty)}(ze^{2\pi i})e^{\pi i\theta_{0}}e^{\pi
i\theta_{\infty}\sigma_{3}}=Y_{2}^{(\infty)}(z)S_{2}^{(\infty)},\quad
S_{2}^{(\infty)}=\begin{pmatrix}1&s_{2}^{(\infty)}\\\ 0&1\end{pmatrix}.$ (5.4)

Parallelly, there exist unique linear independent solutions of (1.3)
determined by the asymptotic behavior as $z\to 0$

$Y_{k}^{(0)}(z)\sim
e^{-\frac{a^{2}}{4z}}z^{-\frac{1}{2}\theta_{0}}(1,~{}z)e^{\frac{a^{2}}{4z}\sigma_{3}}z^{\frac{1}{2}\theta_{0}\sigma_{3}},\quad
z\in\Omega_{k}^{(0)},$ (5.5)

where $\Omega_{k}^{(0)}$ for $k=1,2$ are the Stokes sectors

$\Omega_{k}^{(0)}=\left\\{z\in\mathbb{C}:~{}~{}\frac{\pi}{2}(-2k+1)<\mathop{\rm
arg}\nolimits\left(a^{2}/z\right)<\frac{\pi}{2}(-2k+5)\right\\}.$ (5.6)

The solutions are related to each other by the Stokes matrices

$Y_{2}^{(0)}(z)=Y_{1}^{(0)}(z)S_{1}^{(0)},\quad
S_{1}^{(0)}=\begin{pmatrix}1&0\\\ s_{1}^{(0)}&1\end{pmatrix},$ (5.7)
$Y_{1}^{(0)}(ze^{-2\pi i})e^{-\pi i\theta_{0}}e^{\pi
i\theta_{0}\sigma_{3}}=Y_{2}^{(0)}(z)S_{2}^{(0)},\quad
S_{2}^{(0)}=\begin{pmatrix}1&s_{2}^{(0)}\\\ 0&1\end{pmatrix}.$ (5.8)

Moreover, the solutions $Y_{1}^{(\infty)}(z)$ and $Y_{2}^{(0)}(z)$ are
connected by

$Y_{1}^{(\infty)}(z)=Y_{2}^{(0)}(z)E,$ (5.9)

where $E$ is some invertible constant matrix. We define the monodromy data of
(1.3) by the set

$\left\\{e^{\pi i\theta_{\infty}},e^{\pi
i\theta_{0}};S^{(\infty)}_{1},S^{(\infty)}_{2};S^{(0)}_{1},S^{(0)}_{2};E\right\\}.$
(5.10)

Combining (5.3), (5.4), and (5.7)-(5.9), we have the cyclic condition

$S_{1}^{(\infty)}S_{2}^{(\infty)}e^{-\pi
i\theta_{\infty}\sigma_{3}}=E^{-1}\left(S_{1}^{(0)}\right)^{-1}e^{\pi
i\theta_{0}\sigma_{3}}\left(S_{2}^{(0)}\right)^{-1}E.$ (5.11)

The condition implies that the Stokes multipliers satisfy the equation

$e^{\pi
i\theta_{\infty}}s_{1}^{(\infty)}s_{2}^{(\infty)}+2\cos(\pi\theta_{\infty})=e^{\pi
i\theta_{0}}s_{1}^{(0)}s_{2}^{(0)}+2\cos(\pi\theta_{0}).$ (5.12)

###  5.2 Isomonodromy deformation and PIII equation

To study the isomonodromy deformation of the RTHE equation, it is convenient
to consider a $2\times 2$ matrix system, which has two irregular singular
points, both of rank $1$, at infinity and at the origin. The isomonodromy
deformation of a matrix system with such a singular pattern has been
considered in [12, 20].

Recall the following Lax pair for PIII (see [20, (C.18)-(C.20)]):

$\left\\{\begin{aligned} &\frac{\partial\Phi(z,x)}{\partial
z}=A(z,x)\Phi(z,x),\\\ &\frac{\partial\Phi(z,x)}{\partial
x}=B(z,x)\Phi(z,x),\end{aligned}\right.$ (5.13)

where

$\displaystyle
A(z,x)=\begin{pmatrix}\frac{x}{4}-\frac{\theta_{\infty}}{2z}+\frac{4v-x}{4z^{2}}&\frac{r}{z}-\frac{uv}{z^{2}}\\\
\frac{s}{z}+\frac{2v-x}{2uz^{2}}&-\frac{x}{4}+\frac{\theta_{\infty}}{2z}-\frac{4v-x}{4z^{2}}\end{pmatrix},$
$\displaystyle
B(z,x)=\begin{pmatrix}\frac{z}{4}-\frac{4v-x}{4xz}&\frac{r}{x}+\frac{uv}{xz}\\\
\frac{s}{x}-\frac{2v-x}{2xuz}&-\frac{z}{4}+\frac{4v-x}{4xz}\end{pmatrix}.$

If we set $y=-r/uv$ and

$\frac{\theta_{0}}{2}=\frac{v}{x}\left[\frac{\theta_{\infty}}{2}\left(4-\frac{x}{v}\right)+\frac{x-2v}{uv}r+2us\right],$

then the compatibility condition of the above Lax pair reads

$\left\\{\begin{aligned}
&x\frac{\mathrm{d}y}{\mathrm{d}x}=(4v-x)y^{2}+(2\theta_{\infty}-1)y+x,\\\
&x\frac{\mathrm{d}v}{\mathrm{d}x}=-4yv^{2}+(2xy-2\theta_{\infty}+1)v+\frac{1}{2}(\theta_{0}+\theta_{\infty})x,\\\
&x\frac{\mathrm{d}u}{\mathrm{d}x}=u\left(-\frac{x}{2v}(\theta_{0}+\theta_{\infty})-xy+\theta_{\infty}\right),\end{aligned}\right.$
(5.14)

which implies the PIII equation

$\displaystyle\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{1}{y}\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}-\frac{1}{x}\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2\theta_{0}y^{2}+2-2\theta_{\infty}}{x}+y^{3}-\frac{1}{y}.$

The PIII $\tau$-function is defined by (see [20, (C.27)])

$x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=2y^{2}v^{2}+\left(-xy^{2}+2\theta_{\infty}y+x\right)v-\frac{(\theta_{0}+\theta_{\infty})xy}{2}-\frac{x^{2}}{4}-\frac{\theta_{0}^{2}-\theta_{\infty}^{2}}{4}.$
(5.15)

Using (5.14) and (5.15) to eliminate $v(x)$, we have

$x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=\frac{x^{2}y^{\prime
2}}{8y^{2}}+\frac{xy^{\prime}}{4y}-\frac{x^{2}}{8y^{2}}-\frac{x^{2}y^{2}}{8}-\frac{\theta_{\infty}x}{2y}-\frac{\theta_{0}xy}{2}+\frac{1}{8}-\frac{\theta_{0}^{2}+\theta_{\infty}^{2}}{4}.$
(5.16)

There exist unique solutions $\Phi_{k}^{(\infty)}(z,x)$ of the equation
(5.13), which satisfy the normalized asymptotic behavior as $z\to\infty$

$\Phi_{k}^{(\infty)}(z,x)=(I+O(1/z))e^{\frac{xz}{4}\sigma_{3}}z^{-\frac{1}{2}\theta_{\infty}\sigma_{3}},\quad
xz\in\Omega_{k}^{(\infty)},$ (5.17)

where $\Omega_{k}^{(\infty)}$ for $k=1,2$ are the Stokes sectors defined in
(5.1). The solutions are related to each other by the Stokes matrices

$\Phi_{2}^{(\infty)}(z)=\Phi_{1}^{(\infty)}(z)\hat{S}_{1}^{(\infty)},\quad\hat{S}_{1}^{(\infty)}=\begin{pmatrix}1&0\\\
\hat{s}_{1}^{(\infty)}&1\end{pmatrix},$ (5.18) $\Phi_{1}^{(\infty)}(ze^{2\pi
i})e^{\pi
i\theta_{\infty}\sigma_{3}}=\Phi_{2}^{(\infty)}(z)\hat{S}_{2}^{(\infty)},\quad\hat{S}_{2}^{(\infty)}=\begin{pmatrix}1&\hat{s}_{2}^{(\infty)}\\\
0&1\end{pmatrix}.$ (5.19)

Similarly, there exist unique solutions $\Phi_{k}^{(0)}(z,x)$ of the equation
(5.13) such that

$\Phi_{k}^{(0)}(z,x)=G_{k}(x)(I+O(z))e^{\frac{x}{4z}\sigma_{3}}z^{\frac{1}{2}\theta_{0}\sigma_{3}},\quad
z\in\Omega_{k}^{(0)},~{}~{}z\to 0,$ (5.20)

where $G_{k}(x)$ are matrices independent of $z$, and the sectors
$\Omega_{k}^{(0)}$, $k=1,2$, are defined in (5.6) with $\mathop{\rm
arg}\nolimits(a^{2}/z)$ replaced by $\mathop{\rm arg}\nolimits(x/z)$. These
solutions are related to each other by the Stokes matrices

$\Phi_{2}^{(0)}(z)=\Phi_{1}^{(0)}(z)\hat{S}_{1}^{(0)},\quad\hat{S}_{1}^{(0)}=\begin{pmatrix}1&0\\\
\hat{s}_{1}^{(0)}&1\end{pmatrix},$ (5.21) $\Phi_{1}^{(0)}(ze^{-2\pi i})e^{\pi
i\theta_{0}\sigma_{3}}=\Phi_{2}^{(0)}(z)\hat{S}_{2}^{(0)},\quad\hat{S}_{2}^{(0)}=\begin{pmatrix}1&\hat{s}_{2}^{(0)}\\\
0&1\end{pmatrix}.$ (5.22)

Moreover, the solutions $\Phi_{1}^{(\infty)}(z)$ and $\Phi_{2}^{(0)}(z)$ are
connected through

$\Phi_{1}^{(\infty)}(z)=\Phi_{2}^{(0)}(z)\hat{E},$ (5.23)

where $\hat{E}$ is some invertible constant matrix. The monodromy data of the
system (5.13) is defined as the set

$\left\\{e^{\pi i\theta_{\infty}},e^{\pi
i\theta_{0}};\hat{S}^{(\infty)}_{1},\hat{S}^{(\infty)}_{2};\hat{S}^{(0)}_{1},\hat{S}^{(0)}_{2};\hat{E}\right\\}.$
(5.24)

Applying (5.18), (5.19), and (5.21)-(5.23), we find the cyclic condition for
the monodromy data

$\hat{S}_{1}^{(\infty)}\hat{S}_{2}^{(\infty)}e^{-\pi
i\theta_{\infty}\sigma_{3}}=\hat{E}^{-1}\left(\hat{S}_{1}^{(0)}\right)^{-1}e^{\pi
i\theta_{0}\sigma_{3}}\left(\hat{S}_{2}^{(0)}\right)^{-1}\hat{E}.$ (5.25)

Thus, the Stokes multipliers satisfy the equation

$e^{\pi
i\theta_{\infty}}\hat{s}_{1}^{(\infty)}\hat{s}_{2}^{(\infty)}+2\cos(\pi\theta_{\infty})=e^{\pi
i\theta_{0}}\hat{s}_{1}^{(0)}\hat{s}_{2}^{(0)}+2\cos(\pi\theta_{0}).$ (5.26)

###  5.3 Reduction of the linear system for PIII to DHE

In this section, we will show that the DHE (1.3) can be obtained as a limit of
the linear system (1.11) associated with the Lax pair for the Painlevé III
equation as $x\to a$, where $a$ is one of the poles or zeros of the solution
to the Painlevé III equation.

Substituting the elements of $A(z,x)$ into (1.11) gives

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}-\left(\frac{1}{z+1/y}-\frac{2}{z}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}-Q(z,x)\Phi_{1}=0,$
(5.27)

where

$\displaystyle Q(z,x)$
$\displaystyle=\frac{x^{2}}{16}+\frac{(2-\theta_{\infty})x}{4z}-\frac{\theta_{0}x}{4z^{3}}+\frac{x^{2}}{16z^{4}}-\frac{1}{z+1/y}\left(\frac{x}{4}-\frac{\theta_{\infty}}{2z}+\frac{v-\frac{x}{4}}{z^{2}}\right)$
$\displaystyle\ \ \
+\frac{1}{z^{2}}\left[y^{2}v^{2}+\frac{1}{2}(-xy^{2}+2\theta_{\infty}y+x)v-\frac{1}{4}(\theta_{0}+\theta_{\infty})xy-\frac{x^{2}}{8}+\frac{\theta_{\infty}^{2}}{4}-\frac{\theta_{\infty}}{2}\right]$
$\displaystyle=\frac{x^{2}}{16}+\frac{(2-\theta_{\infty})x}{4z}-\frac{\theta_{0}x}{4z^{3}}+\frac{x^{2}}{16z^{4}}-\frac{1}{z+1/y}\left(\frac{x}{4}-\frac{\theta_{\infty}}{2z}+\frac{v-\frac{x}{4}}{z^{2}}\right)$
$\displaystyle\ \ \
+\frac{1}{z^{2}}\left[\frac{1}{2}x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{\theta_{0}^{2}+\theta_{\infty}^{2}}{8}-\frac{\theta_{\infty}}{2}\right].$

To obtain DHE, we eliminate the extra singularity $z=-1/y$ in the above
equation by considering the poles and zeros of $y$. It is known that $y$
admits the following Laurent expansion near a pole

$y(x)=\frac{\varepsilon}{x-a}-\frac{2\theta_{0}+\varepsilon}{2a}+b(x-a)+O((x-a)^{2}),$
(5.28)

and the Taylor expansion near a zero

$y(x)=\sigma(x-a)+\frac{\sigma-2+2\theta_{\infty}}{2a}(x-a)^{2}+b(x-a)^{3}+O((x-a)^{4}),$
(5.29)

where $\varepsilon=\varepsilon_{\pm}=\pm 1$, $\sigma=\sigma_{\pm}=\pm 1$,
$a\neq 0$ and $b$ is arbitrary.

In view of the first equation of compatibility condition (5.14), (5.28) and
(5.29), we obtain that

$v(x)=\left\\{\begin{aligned} &O(x-a),&\varepsilon=\varepsilon_{+},\\\
&\textstyle\frac{a}{2}+O(x-a),&\varepsilon=\varepsilon_{-},\\\
&O(1),&\sigma=\sigma_{+},\\\
&\textstyle-\frac{a}{2}(x-a)^{-2}+O(1),&\sigma=\sigma_{-}.\end{aligned}\right.$
(5.30)

Thus, taking $x\rightarrow a$ in (5.27) and setting

$\Phi_{1}=z^{\frac{\theta_{0}}{2}}e^{\frac{a}{4}(z+\frac{1}{z})}w,\ \ \ s=az,$

we obtain the DHEs

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}s^{2}}+\left(\frac{1}{2}+\frac{1+\theta_{0}}{s}-\frac{a^{2}}{2s^{2}}\right)\frac{\mathrm{d}w}{\mathrm{d}s}+\frac{ps-q}{s^{2}}w=0,\\\
&p=\frac{1}{4}(\theta_{\infty}+\theta_{0}),\\\ &q=\lim_{x\rightarrow
a}\frac{1}{2}x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{a^{2}}{8}-\frac{\theta_{0}^{2}-\theta_{\infty}^{2}}{8}\\\
&~{}~{}=-\frac{3}{8}a^{2}b+\frac{a^{2}}{8}+\frac{1}{32}(2\theta_{0}+1)(2\theta_{0}+5)\end{aligned}\right.$
(5.31)

for $\varepsilon=\varepsilon_{+}$,

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}s^{2}}+\left(\frac{1}{2}+\frac{2+\theta_{0}}{s}-\frac{a^{2}}{2s^{2}}\right)\frac{\mathrm{d}w}{\mathrm{d}s}+\frac{ps-q}{s^{2}}w=0,\\\
&p=\frac{1}{4}(\theta_{\infty}+\theta_{0}),\\\ &q=\lim_{x\rightarrow
a}\frac{1}{2}x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{a^{2}}{8}-\frac{\theta_{0}^{2}-\theta_{\infty}^{2}}{8}-\frac{\theta_{0}+\theta_{\infty}}{2}\end{aligned}\right.$
(5.32)

for $\sigma=\sigma_{+}$, and

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}s^{2}}+\left(\frac{1}{2}+\frac{2+\theta_{0}}{s}-\frac{a^{2}}{2s^{2}}\right)\frac{\mathrm{d}w}{\mathrm{d}s}+\frac{ps-q}{s^{2}}w=0,\\\
&p=\frac{1}{4}(\theta_{\infty}+\theta_{0}+2),\\\ &q=\lim_{x\rightarrow
a}\frac{1}{2}\left(x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a}{x-a}\right)+\frac{a^{2}}{8}-\frac{\theta_{0}^{2}-\theta_{\infty}^{2}}{8}-\frac{\theta_{0}}{2}-\frac{3}{4}\end{aligned}\right.$
(5.33)

for $\sigma=\sigma_{-}$. For $\varepsilon=\varepsilon_{-}$ in (5.27), taking
$x\rightarrow a$ and setting

$\Phi_{1}=z^{-\frac{\theta_{0}}{2}}e^{\frac{a}{4}(z-\frac{1}{z})}w,\ \ \
s=az,$

we obtain the DHE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}s^{2}}+\left(\frac{1}{2}+\frac{1-\theta_{0}}{s}+\frac{a^{2}}{2s^{2}}\right)\frac{\mathrm{d}w}{\mathrm{d}s}+\frac{ps-q}{s^{2}}w=0,\\\
&p=\frac{1}{4}(\theta_{\infty}-\theta_{0}),\\\ &q=\lim_{x\rightarrow
a}\frac{1}{2}x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a^{2}}{8}-\frac{\theta_{0}^{2}-\theta_{\infty}^{2}}{8}.\end{aligned}\right.$
(5.34)

###  5.4 Isomonodromy set of accessory parameters of DHE

Consider the DHE (1.3) with the accessory parameters $a$ and $q$ and the fixed
parameters $\gamma$ and $p$. We have shown that the DHE can be obtained as a
limit of the first row of the isomonodromy family of $\Phi(z,x)$, when $x$
tends to one of the poles or zeros of the Painlevé III transcendents
corresponding to the same monodromy data of the DHE. While the accessory
parameters $(a,q)$ are expressed in terms of the parameters in the Laurent or
Taylor expansion of the PIII transcendents with the parameters
$\theta_{0}=\gamma-1$ and $\theta_{\infty}=4p-\gamma+1$. As a result, we have
the following description of the isomonodromy set of accessory parameters of
DHE.

######  Theorem 4.

There is a discrete set of pairs of accessory parameters $(a_{n},q_{n})$ such
that the DHE (1.3) corresponding to these parameters has the same monodromy
data as the original equation with the parameters $a$ and $q$. Under the
bijection given in the last equation of (5.31), this set coincides with the
set of parameters $(a_{n},b_{n})$ in the Laurent expansion near the poles
$a_{n}$ of the unique solution of PIII with the same monodromy data (5.10) as
the DHE.

##  6 Accessory parameters of BHE

###  6.1 Monodromy of BHE

Consider the BHE (1.4) with the accessory parameters $a$ and $q$, the other
fixed parameters $\gamma$ and $p$. There exist unique solutions of (1.4) which
satisfy the normalized asymptotic behavior as $z\to\infty$

$Y_{k}(z)\sim
e^{-\frac{z^{2}}{2}-az}z^{-(\theta_{0}+1)}(1,1)e^{\frac{z^{2}+2az}{2}\sigma_{3}}z^{-\theta_{\infty}\sigma_{3}},\quad
z\in\Omega_{k},$ (6.1)

where the parameters $\theta_{0}=(\gamma-1)/2$,
$\theta_{\infty}=(p-\gamma-1)/2$ and the Stokes sectors $\Omega_{k}$,
$k=1,2,3,4$ are given by

$\Omega_{k}=\left\\{z\in\mathbb{C}:~{}~{}\frac{\pi}{4}(2k-5)<\mathop{\rm
arg}\nolimits z<\frac{\pi}{4}(2k-1)\right\\}.$ (6.2)

The solutions are related to each other by the Stokes matrices

$Y_{k+1}(z)=Y_{k}(z)S_{k},\quad k=1,2,3,$ (6.3) $Y_{1}(z)=Y_{4}(ze^{2\pi
i})S_{4}e^{2\pi i\theta_{0}}e^{2\pi i\theta_{\infty}\sigma_{3}},$ (6.4)

where

$S_{2k-1}=\begin{pmatrix}1&0\\\ s_{2k-1}&1\end{pmatrix},\quad
S_{2k}=\begin{pmatrix}1&s_{2k}\\\ 0&1\end{pmatrix},\quad k=1,2.$ (6.5)

The solution $Y_{1}(z)$ has the asymptotic behavior as $z\to 0$

$Y_{1}(z)\sim z^{-\theta_{0}}(1,1)z^{\theta_{0}\sigma_{3}}E,$ (6.6)

with some invertible constant matrix $E$. The monodromy data of BHE (1.4) is
defined as the set

$\left\\{e^{2\pi i\theta_{\infty}},e^{2\pi
i\theta_{0}};S_{1},S_{2},S_{3},S_{4};E\right\\}.$ (6.7)

By (6.3)-(6.6), they satisfy the cyclic condition

$S_{1}S_{2}S_{3}S_{4}e^{2\pi i\theta_{\infty}\sigma_{3}}=E^{-1}e^{-2\pi
i\theta_{0}\sigma_{3}}E.$ (6.8)

Thus, the Stokes multipliers $s_{k}$, $k=1,2,3,4$, satisfy the algebraic
equation

$e^{2\pi i\theta_{\infty}}(1+s_{2}s_{3})+e^{-2\pi
i\theta_{\infty}}\left[s_{1}s_{4}+(1+s_{1}s_{2})(1+s_{3}s_{4})\right]=2\cos(2\pi\theta_{0}).$
(6.9)

It is direct to see that the cyclic condition (6.8) also specifies the
connection matrix $E$ to within a left-multiplicative diagonal matrix.

###  6.2 Isomonodromy deformation and PIV equation

To study the isomonodromy deformation of the BHE, it is convenient to consider
a $2\times 2$ matrix system, which has one irregular singular point of rank 2
at infinity and one regular singularity at the origin. In [12, 20], the
isomonodromy deformation of such a matrix system has been considered.

Consider the following Lax pair for PIV (see [20, (C.30)-(C.31)]):

$\left\\{\begin{aligned} &\frac{\partial\Phi(z,x)}{\partial
z}=A(z,x)\Phi(z,x),\\\ &\frac{\partial\Phi(z,x)}{\partial
x}=B(z,x)\Phi(z,x),\end{aligned}\right.$ (6.10)

where

$\displaystyle A(z,x)$
$\displaystyle=\begin{pmatrix}z+x+\frac{\theta_{0}-v}{z}&u\left(1-\frac{y}{2z}\right)\\\
\frac{2}{u}\left(v-\theta_{0}-\theta_{\infty}+\frac{v^{2}-2\theta_{0}v}{yz}\right)&-z-x-\frac{\theta_{0}-v}{z}\end{pmatrix},$
$\displaystyle B(z,x)$ $\displaystyle=\begin{pmatrix}z&u\\\
\frac{2}{u}(v-\theta_{0}-\theta_{\infty})&-z\end{pmatrix}.$

The compatibility condition of the Lax pair gives

$\left\\{\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}x}&=-u(y+2x),\\\
\frac{\mathrm{d}y}{\mathrm{d}x}&=-4v+y^{2}+2xy+4\theta_{0},\\\
\frac{\mathrm{d}v}{\mathrm{d}x}&=-(v-\theta_{0}-\theta_{\infty})y-\frac{2v(v-2\theta_{0})}{y},\end{aligned}\right.$
(6.11)

which leads to the PIV equation

$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{1}{2y}\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}+\frac{3}{2}y^{3}+4xy^{2}+2(x^{2}+1-2\theta_{\infty})y-\frac{8\theta_{0}^{2}}{y}.$
(6.12)

The PIV $\tau$-function is defined by (see [20, (C.35)])

$\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=\frac{2}{y}v^{2}-\bigg{(}y+2x+\frac{4\theta_{0}}{y}\bigg{)}v+(\theta_{0}+\theta_{\infty})(y+2x).$
(6.13)

There exist unique solutions $\Phi_{k}(z,x)$ of the equation (6.10) subject to
the normalized asymptotic behavior as $z\to\infty$

$\Phi_{k}(z,x)=(I+O(1/z))e^{\frac{z^{2}+2xz}{2}\sigma_{3}}z^{-\theta_{\infty}\sigma_{3}},\quad
z\in\Omega_{k},$ (6.14)

where $\Omega_{k}$, $k=1,2,3,4$, are the Stokes sectors defined in (6.2).
Using (6.14), it is seen that the solutions are related to each other by the
Stokes matrices

$\Phi_{k+1}(z)=\Phi_{k}(z)\hat{S}_{k},\quad k=1,2,3,$ (6.15)
$\Phi_{1}(z)=\Phi_{4}(ze^{2\pi i})\hat{S}_{4}e^{2\pi
i\theta_{\infty}\sigma_{3}},$ (6.16)

where

$\hat{S}_{2k-1}=\begin{pmatrix}1&0\\\
\hat{s}_{2k-1}&1\end{pmatrix},\quad\hat{S}_{2k}=\begin{pmatrix}1&\hat{s}_{2k}\\\
0&1\end{pmatrix},~{}~{}k=1,2.$ (6.17)

The solution $\Phi_{1}(z,x)$ has the asymptotic behaviors as $z\to 0$

$\Phi_{1}(z,x)=\Phi_{1}^{(0)}(x)(I+O(z))z^{\theta_{0}\sigma_{3}}\hat{E},$

with some invertible constant matrix $\hat{E}$. The monodromy data of the
system (6.10) is defined as the set

$\left\\{e^{2\pi i\theta_{\infty}},e^{\pi
i\theta_{0}};\hat{S}_{1},\hat{S}_{2},\hat{S}_{3},\hat{S}_{4};\hat{E}\right\\}.$
(6.18)

They satisfy the cyclic condition

$\hat{S}_{1}\hat{S}_{2}\hat{S}_{3}\hat{S}_{4}e^{2\pi
i\theta_{\infty}\sigma_{3}}=\hat{E}^{-1}e^{-2\pi
i\theta_{0}\sigma_{3}}\hat{E}.$ (6.19)

Thus, the Stokes multipliers $\hat{s}_{k}$, $k=1,2,3,4$ satisfy

$e^{2\pi i\theta_{\infty}}(1+\hat{s}_{2}\hat{s}_{3})+e^{-2\pi
i\theta_{\infty}}(\hat{s}_{1}\hat{s}_{4}+(1+\hat{s}_{1}\hat{s}_{2})(1+\hat{s}_{3}\hat{s}_{4}))=2\cos(2\pi\theta_{0}).$
(6.20)

Moreover, the cyclic condition (6.18) specifies the connection matrices
$\hat{E}$ to within a left-multiplicative diagonal matrices; see [12].

###  6.3 Reduction of the linear system for PIV to BHE

In this subsection, we will derive BHE from (1.11) associated with the
isomonodromy system of PIV when the PIV transcendents $y(x)$ tends to zero or
infinity.

Substituting $A(z,x)$ into (1.11) gives

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}-\left(\frac{1}{z-\lambda}-\frac{1}{z}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}-Q(z,x)\Phi_{1}=0,$
(6.21)

where

$\displaystyle Q(z,x)$
$\displaystyle=z^{2}+2xz+x^{2}+2(1-\theta_{\infty})+\frac{\theta_{0}^{2}}{z^{2}}-\frac{1}{z-\lambda}\left(z+x+\frac{\theta_{0}-v}{z}\right)$
$\displaystyle\ \ \ \ \ \ \
+\frac{1}{z}\left[\frac{2}{y}v^{2}-\bigg{(}y+2x+\frac{4\theta_{0}}{y}\bigg{)}v+(\theta_{0}+\theta_{\infty})y+(1+2\theta_{0})x\right]$
$\displaystyle=z^{2}+2xz+x^{2}+2(1-\theta_{\infty})+\frac{\theta_{0}^{2}}{z^{2}}-\frac{1}{z-\lambda}\left(z+x+\frac{\theta_{0}-v}{z}\right)$
$\displaystyle\ \ \ \ \ \ \
+\frac{1}{z}\left[\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+(1-2\theta_{\infty})x\right].$

To derive BHE, we need to eliminate the extra singularity $\lambda=y/2$. Thus
we consider the zeros and poles of $y$. According to [23, (1.4)-(1.5)], the
solution $y$ of PIV equation admits the Taylor expansion near a zero

$y(x)=\varepsilon(x-a)+b(x-a)^{2}+O((x-a)^{3}),$ (6.22)

the Laurent expansion near a pole

$y(x)=\sigma(x-a)^{-1}-a+a_{1}(x-a)+b(x-a)^{2}+O((x-a)^{3}),$ (6.23)

where $b$ is arbitrary and the parameters

$\varepsilon=\varepsilon_{\pm}=\pm 4\theta_{0},\ \ \ \sigma=\sigma_{\pm}=\pm
1,\ \ \
a_{1}=\frac{1}{3}\sigma\left(a^{2}-2+4(\theta_{\infty}-\sigma)\right).$

Let us first consider the zeros of $y$. It follows from (6.11) and (6.22) that

$v(x)=\left\\{\begin{aligned} &O(x-a),&\varepsilon=\varepsilon_{+},\\\
&2\theta_{0}+O(x-a),&\varepsilon=\varepsilon_{-}.\end{aligned}\right.$

For $\varepsilon=\varepsilon_{+}$, taking $x\rightarrow a$ in (6.21) and then
setting

$\Phi_{1}=z^{\theta_{0}}e^{\frac{1}{2}z^{2}+az}w,$

we obtain the BHE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(2z+2a+\frac{2\theta_{0}}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z}w=0,\\\
&p=2(\theta_{0}+\theta_{\infty}),\\\ &q=\lim_{x\rightarrow
a}\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-2(\theta_{0}+\theta_{\infty})a.\end{aligned}\right.$
(6.24)

For $\varepsilon=\varepsilon_{-}$, by taking $x\rightarrow a$ in (6.21) and
setting

$\Phi_{1}=z^{\theta_{0}+1}e^{\frac{1}{2}z^{2}+az}w,$

we obtain the BHE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(2z+2a+\frac{2\theta_{0}+2}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z}w=0,\\\
&p=2(\theta_{0}+\theta_{\infty}+1),\\\ &q=\lim_{x\rightarrow
a}\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-2(\theta_{0}+\theta_{\infty}+1)a.\end{aligned}\right.$
(6.25)

Next we focus on the poles of $y$. It is seen from (6.11) and (6.23) that

$v(x)=\left\\{\begin{aligned}
&\textstyle\frac{1}{2}(x-a)^{-2}+O(1),&\sigma=\sigma_{+},\\\
&\theta_{0}+\theta_{\infty}+O(x-a),&\sigma=\sigma_{-}.\end{aligned}\right.$
(6.26)

By taking the limit $x\rightarrow a$ in (6.21) and then making the
transformation

$\Phi_{1}=z^{\theta_{0}}e^{\frac{1}{2}z^{2}+az}w,$

we obtain another two BHEs

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(2z+2a+\frac{2\theta_{0}+1}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z}w=0,\\\
&p=2(\theta_{0}+\theta_{\infty}+1),\\\ &q=\lim_{x\rightarrow
a}\left(\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{1}{x-a}\right)-(2\theta_{0}+2\theta_{\infty}+1)a\\\
&~{}~{}=-b-\left(2\theta_{0}+2\theta_{\infty}-\frac{1}{2}\right)a\end{aligned}\right.$
(6.27)

for $\sigma=\sigma_{+}=1$,

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(2z+2a+\frac{2\theta_{0}+1}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z}w=0,\\\
&p=2(\theta_{0}+\theta_{\infty}),\\\ &q=\lim_{x\rightarrow
a}\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-2(\theta_{0}+\theta_{\infty})a\\\
&~{}=-b-\left(2\theta_{0}+2\theta_{\infty}+\frac{1}{2}\right)a\end{aligned}\right.$
(6.28)

for $\sigma=\sigma_{-}=-1$.

###  6.4 Isomonodromy set of accessory parameters of BHE

We have shown that the BHE (1.4), with the accessory parameters $a$ and $q$,
the other fixed parameters $\gamma$ and $p$ can be obtained as a limit of the
first row of the isomonodromy family of $\Phi(z,x)$, when $x\to a$, $a$ being
one of the poles or zeros of the Painlevé IV transcendents. While the
accessory parameters $(a,q)$ are expressed in terms of the parameters in the
Laurent or Taylor expansion of the PIV transcendents as given in
(6.24)-(6.28). Therefore, we have the following description of the
isomonodromy set of accessory parameters of BHE.

######  Theorem 5.

There is a discrete set of pairs of accessory parameters $(a_{n},q_{n})$ such
that the BHE (1.4) corresponding to these parameters has the same monodromy
data described by (6.9) as the original equation with the parameters $a$ and
$q$. Under the bijection given in the last equation of (6.27), this set
coincides with the set of parameters $(a_{n},b_{n})$ in the Laurent expansion
near the poles $a_{n}$ of the unique solution of PIV (6.12) with the parameter
$\theta_{0}=(\gamma-1)/2$, $\theta_{\infty}=(p-\gamma-1)/2$ and the same
monodromy data (6.9) as the BHE.

##  7  Accessory parameters of CHE

###  7.1 Monodromy of CHE

Consider the CHE (1.2) with the accessory parameters $a$ and $q$, the fixed
parameters $\gamma$, $\delta$ and $p$. There exist uniquely two linear
independent solutions of (1.2), namely $Y_{k}=(y_{k1},y_{k2})$, normalizing
the asymptotic behavior as $z\to\infty$

$Y_{k}(z)\sim
e^{-\frac{z}{2}}z^{-\frac{1}{2}(\theta_{0}+\theta_{1})}(1,~{}1/z)e^{\frac{z}{2}\sigma_{3}}z^{-\frac{1}{2}\theta_{\infty}\sigma_{3}},\quad
z\in\Omega_{k},$ (7.1)

where the parameters $\theta_{0}=\gamma$, $\theta_{1}=\delta-1$,
$\theta_{\infty}=2p-\gamma-\delta+1$ and the Stokes sectors $\Omega_{k}$,
$k=1,2$ are defined by

$\Omega_{k}=\left\\{z\in\mathbb{C}:~{}~{}\frac{\pi}{2}(-2k+1)<\mathop{\rm
arg}\nolimits z<\frac{\pi}{2}(-2k+5)\right\\}.$ (7.2)

The solutions are related to each other by the Stokes matrices

$Y_{2}(z)=Y_{1}(z)S_{1},\quad S_{1}=\begin{pmatrix}1&0\\\
s_{1}&1\end{pmatrix},$ (7.3) $Y_{1}(ze^{2\pi i})e^{\pi
i(\theta_{0}+\theta_{1})}e^{\pi
i\theta_{\infty}\sigma_{3}}=Y_{2}(z)S_{2},\quad
S_{2}=\begin{pmatrix}1&s_{2}\\\ 0&1\end{pmatrix}.$ (7.4)

Near the regular singular points, $Y_{1}$ has the asymptotic behaviors

$Y_{1}(z)\sim\left(z^{0},z^{1-\theta_{0}}\right)E_{0},\quad
Y_{1}(z)\sim\left((z-a)^{0},(z-a)^{-\theta_{1}}\right)E_{1},$

with some invertible constant matrices $E_{0}$ and $E_{1}$. After an analytic
continuation along a closed loop around a singular point, we obtain the
relations

$Y_{1}\left(z_{k}+e^{2\pi i}(z-z_{k})\right)=Y_{1}(z)e^{-\pi
i\theta_{k}}M_{k},\quad M_{k}=E_{k}^{-1}e^{\pi i\theta_{k}\sigma_{3}}E_{k}$
(7.5)

for $k=0,1$, where $z_{0}=0$ and $z_{1}=a$. We define the monodromy data of
the CHE equation as

$\left\\{e^{\pi i\theta_{\infty}},e^{\pi i\theta_{0}},e^{\pi
i\theta_{1}};S_{1},S_{1};E_{0},E_{1}\right\\}.$ (7.6)

According to (7.3)-(7.5), the monodromy matrices satisfies the cyclic
condition

$M_{1}M_{0}=S_{1}S_{2}e^{-\pi i\theta_{\infty}\sigma_{3}}.$ (7.7)

Suppose

$2\cos(\pi\sigma)=\mathop{\rm Tr}\nolimits
M_{1}M_{2}=2\cos(\pi\theta_{\infty})+s_{1}s_{2}e^{\pi i\theta_{\infty}},$
(7.8)

with $0\leq\mathop{\rm Re}\nolimits\sigma\leq 1$. Then, the connection
matrices can be parameterized in terms of $\sigma$ and the other free
parameter $s$; see [19]. For instance, we have for $\sigma\neq 0$ [19,
Equation (3.7)]:

$\displaystyle D_{1}E_{1}D$
$\displaystyle=\begin{pmatrix}\sin(\frac{\pi}{2}(\theta_{1}+\theta_{0}+\sigma))\sin(\frac{\pi}{2}(\theta_{1}-\theta_{0}+\sigma))&-s\sin(\frac{\pi}{2}(\theta_{1}+\theta_{0}-\sigma))\sin(\frac{\pi}{2}(\theta_{1}-\theta_{0}-\sigma))\\\
-s^{-1}&1\end{pmatrix}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\begin{pmatrix}e^{-\frac{1}{2}\pi
i\sigma}&\sin(\frac{\pi}{2}(\theta_{\infty}+\sigma))\\\ e^{\frac{1}{2}\pi
i\sigma}&\sin(\frac{\pi}{2}(\theta_{\infty}-\sigma))\end{pmatrix},$ (7.9)

and

$\displaystyle D_{0}E_{0}D$
$\displaystyle=\begin{pmatrix}\sin(\frac{\pi}{2}(\theta_{1}+\theta_{0}+\sigma))&-se^{-\pi
i\sigma}\sin(\frac{\pi}{2}(\theta_{1}+\theta_{0}-\sigma))\\\ -s^{-1}e^{\pi
i\sigma}\sin(\frac{\pi}{2}(\theta_{1}-\theta_{0}+\sigma))&\sin(\frac{\pi}{2}(\theta_{1}-\theta_{0}-\sigma))\end{pmatrix}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\begin{pmatrix}e^{-\frac{1}{2}\pi
i\sigma}&\sin(\frac{\pi}{2}(\theta_{\infty}+\sigma))\\\ e^{\frac{1}{2}\pi
i\sigma}&\sin(\frac{\pi}{2}(\theta_{\infty}-\sigma))\end{pmatrix},$ (7.10)

with some invertible diagonal matrices $D_{1},D_{0}$ and $D$.

###  7.2 Isomonodromy deformation and PV equation

To study the isomonodromy deformation of the CHE, it is convenient to consider
a $2\times 2$ matrix system, which has one irregular singular point of rank 1
at infinity and two regular singularities. In [12, 20], the isomonodromy
deformation of such a matrix system has been considered.

Recall the following Lax pair for PV (see [20, (C.38)-(C.39)]):

$\left\\{\begin{aligned} &\frac{\partial\Phi(z,x)}{\partial
z}=A(z,x)\Phi(z,x),\ \
A(z,x)=\frac{\sigma_{3}}{2}+\frac{A_{0}(x)}{z}+\frac{A_{1}(x)}{z-x},\\\
&\frac{\partial\Phi(z,x)}{\partial x}=B(z,x)\Phi(z,x),\ \
B(z,x)=-\frac{A_{1}(x)}{z-x},\end{aligned}\right.$ (7.11)

the coefficients

$\displaystyle A_{0}(x)$
$\displaystyle=\begin{pmatrix}v+\frac{\theta_{0}}{2}&-u(v+\theta_{0})\\\
\frac{v}{u}&-v-\frac{\theta_{0}}{2}\end{pmatrix},$ $\displaystyle A_{1}(x)$
$\displaystyle=\begin{pmatrix}-v-\frac{\theta_{0}+\theta_{\infty}}{2}&yu\big{(}v+\frac{\theta_{0}-\theta_{1}+\theta_{\infty}}{2}\big{)}\\\
-\frac{1}{yu}\big{(}v+\frac{\theta_{0}+\theta_{1}+\theta_{\infty}}{2}\big{)}&v+\frac{\theta_{0}+\theta_{\infty}}{2}\end{pmatrix}.$

The compatibility condition of the above Lax pair reads

$\left\\{\begin{aligned}
x\frac{\mathrm{d}y}{\mathrm{d}x}&=xy-2v(y-1)^{2}-(y-1)\left(\frac{\theta_{0}-\theta_{1}+\theta_{\infty}}{2}y-\frac{3\theta_{0}+\theta_{1}+\theta_{\infty}}{2}\right),\\\
x\frac{\mathrm{d}v}{\mathrm{d}x}&=yv\left(v+\frac{\theta_{0}-\theta_{1}+\theta_{\infty}}{2}\right)-\frac{1}{y}(v+\theta_{0})\left(v+\frac{\theta_{0}+\theta_{1}+\theta_{\infty}}{2}\right),\\\
x\frac{\mathrm{d}u}{\mathrm{d}x}&=u\left[-2v-\theta_{0}-\theta_{\infty}+y\left(v+\frac{\theta_{0}-\theta_{1}+\theta_{\infty}}{2}\right)+\frac{1}{y}\left(v+\frac{\theta_{0}+\theta_{1}+\theta_{\infty}}{2}\right)\right],\end{aligned}\right.$
(7.12)

which implies the PV equation

$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\left(\frac{1}{2y}+\frac{1}{y-1}\right)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}-\frac{1}{x}\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{(y-1)^{2}}{x^{2}}\left(\alpha
y+\frac{\beta}{y}\right)+\frac{\gamma y}{x}-\frac{y(y+1)}{2(y-1)},$ (7.13)

where

$\alpha=\frac{1}{8}(\theta_{0}-\theta_{1}+\theta_{\infty})^{2},\ \ \ \
\beta=-\frac{1}{8}(\theta_{0}-\theta_{1}-\theta_{\infty})^{2},\ \ \ \
\gamma=1-\theta_{0}-\theta_{1}.$

There exist unique solutions $\Phi_{k}(z,x)$ of the equation (7.11), which
satisfies the normalized asymptotic behavior as $z\to\infty$

$\Phi_{k}(z,x)=(I+O(1/z))e^{\frac{z}{2}\sigma_{3}}z^{-\frac{1}{2}\theta_{\infty}\sigma_{3}},\quad
z\in\Omega_{k},$ (7.14)

where $\Omega_{k}$, $k=1,2$, are the Stokes sectors defined in (7.2). The
solutions are related to each other by the Stokes matrices

$\Phi_{2}(z)=\Phi_{1}(z)\hat{S}_{1},\quad\hat{S}_{1}=\begin{pmatrix}1&0\\\
\hat{s}_{1}&1\end{pmatrix},$ (7.15) $\Phi_{1}(ze^{2\pi i})e^{\pi
i\theta_{\infty}\sigma_{3}}=\Phi_{2}(z)\hat{S}_{2},\quad\hat{S}_{2}=\begin{pmatrix}1&\hat{s}_{2}\\\
0&1\end{pmatrix}.$ (7.16)

Near the regular singular points $z_{0}=0$ and $z_{1}=a$, $\Phi_{1}(z,x)$ has
the asymptotic behaviors

$\Phi_{1}(z,x)=\Phi_{1}^{(k)}(x)\left(I+O\left(z-z_{k}\right)\right)(z-z_{k})^{\frac{1}{2}\theta_{k}\sigma_{3}}\hat{E}_{k},~{}~{}z\to
z_{k},$

with some invertible constant matrices $\hat{E}_{k}$ and $z$-independent
matrices $\Phi_{1}^{(k)}(x)$, $k=0,1$. The monodromy data of the system (7.11)
is defined as the set

$\left\\{e^{\pi i\theta_{\infty}},e^{\pi
i\theta_{k}};\hat{S}_{1},\hat{S}_{2};\hat{E}_{0},\hat{E}_{1}\right\\}.$ (7.17)

They satisfy the cyclic condition

$\hat{M}_{1}\hat{M}_{0}=\hat{S}_{1}\hat{S}_{2}e^{-\pi
i\theta_{\infty}\sigma_{3}},\quad\hat{M}_{k}=\hat{E}_{k}^{-1}e^{\pi
i\theta_{k}\sigma_{3}}\hat{E}_{k},\quad k=0,1.$ (7.18)

###  7.3 Reduction of the linear systems for PV to CHE

In this subsection, we will derive CHE from the linear systems (1.11) and
(1.12) for PV at the poles, zeros and 1-points (that is, $y(a)=1$) of the
solutions to PV.

Substituting the elements of $A(z,x)$ into (1.11) gives

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}-\left(\frac{1}{z-\lambda_{1}}-\frac{1}{z}-\frac{1}{z-x}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}-P(z,x)\Phi_{1}=0,$
(7.19)

where

$\left\\{\begin{aligned}
&\lambda_{1}=\lambda_{1}(x)=\frac{x}{1-\frac{y(v+\frac{1}{2}(\theta_{0}-\theta_{1}+\theta_{\infty}))}{v+\theta_{0}}},\\\
&P(z,x)=\frac{1}{4}+\frac{\theta_{0}^{2}}{4z^{2}}+\frac{M(x)}{z(z-x)}+\frac{\theta_{1}^{2}}{4(z-x)^{2}}+\frac{2-\theta_{\infty}}{2(z-x)}\\\
&\ \ \ \ \ \ \ \ \ \ \ \
-\frac{1}{z-\lambda_{1}}\left(\frac{1}{2}+\frac{2v+\theta_{0}}{2z}-\frac{2v+\theta_{0}+\theta_{\infty}}{2(z-x)}\right),\\\
&M(x)=yv\left(v+\frac{\theta_{0}-\theta_{1}+\theta_{\infty}}{2}\right)+\frac{1}{y}(v+\theta_{0})\left(v+\frac{\theta_{0}+\theta_{1}+\theta_{\infty}}{2}\right)\\\
&\ \ \ \ \ \ \ \ \ \
-2\left(v+\frac{\theta_{0}}{2}\right)\left(v+\frac{\theta_{0}+\theta_{\infty}}{2}\right)-\left(v+\frac{1+\theta_{0}}{2}\right)x-\frac{\theta_{\infty}}{2}.\end{aligned}\right.$

In virtue of the definition of the PV $\tau$-function (see [20, (C.43)]):

$\displaystyle x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=$
$\displaystyle-\left[v-\frac{1}{y}\left(v+\frac{\theta_{0}+\theta_{1}+\theta_{\infty}}{2}\right)\right]\left[v+\theta_{0}-y\left(v+\frac{\theta_{0}-\theta_{1}+\theta_{\infty}}{2}\right)\right]$
$\displaystyle-\left(v+\frac{\theta_{0}+\theta_{\infty}}{2}\right)x,$ (7.20)

the expression of $M(x)$ is simplified to

$M(x)=x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{(\theta_{\infty}-1)x}{2}-\frac{\theta_{0}^{2}+\theta_{1}^{2}-\theta_{\infty}^{2}}{4}-\frac{\theta_{\infty}}{2}.$

Similarly, substituting the elements of $A(z,x)$ into (1.12) gives

$\frac{\mathrm{d}^{2}\Phi_{2}}{\mathrm{d}z^{2}}-\left(\frac{1}{z-\lambda_{2}}-\frac{1}{z}-\frac{1}{z-x}\right)\frac{\mathrm{d}\Phi_{2}}{\mathrm{d}z}-Q(z,x)\Phi_{2}=0,$
(7.21)

where

$\left\\{\begin{aligned}
&\lambda_{2}=\lambda_{2}(x)=\frac{x}{1-\frac{v+\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty})}{yv}},\\\
&Q(z,x)=\frac{1}{4}+\frac{\theta_{0}^{2}}{4z^{2}}+\frac{N(x)}{z(z-x)}+\frac{\theta_{1}^{2}}{4(z-x)^{2}}-\frac{2+\theta_{\infty}}{2(z-x)}\\\
&\ \ \ \ \ \ \ \ \ \
+\frac{1}{z-\lambda_{2}}\left(\frac{1}{2}+\frac{2v+\theta_{0}}{2z}-\frac{2v+\theta_{0}+\theta_{\infty}}{2(z-x)}\right),\\\
&N(x)=x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{(\theta_{\infty}+1)x}{2}-\frac{\theta_{0}^{2}+\theta_{1}^{2}-\theta_{\infty}^{2}}{4}+\frac{\theta_{\infty}}{2}.\end{aligned}\right.$

To eliminate the singularities $z=\lambda_{1}$ and $z=\lambda_{2}$ and obtain
CHE, we need

$\displaystyle\lambda_{1}(x)=0,\ \ \mathrm{or}\ \ \lambda_{1}(x)=x,\ \
\mathrm{or}\ \ \lambda_{1}(x)=\infty,$ (7.22) $\displaystyle\lambda_{2}(x)=0,\
\ \mathrm{or}\ \ \lambda_{2}(x)=x,\ \ \mathrm{or}\ \ \lambda_{2}(x)=\infty,$
(7.23)

in (7.19) and (7.21), respectively. For this purpose, we consider the poles,
zeros and 1-points of the solution $y$.

According to [16, (36.4)-(36.7)], the solution $y$ admits the Laurent
expansion near a pole

$y(x)=\left\\{\begin{aligned} &\varepsilon(x-a)^{-1}+b+O(x-a),\ \ &\alpha\neq
0,\\\ &ab(x-a)^{-2}+b(x-a)^{-1}+O(1),\ \ &\alpha=0,\end{aligned}\right.$

and the following Taylor expansions near a zero and near a 1-point

$y(x)=\left\\{\begin{aligned} &\delta(x-a)+b(x-a)^{2}+O((x-a)^{3}),\ \
&\beta\neq 0,\\\ &b(x-a)^{2}+O((x-a)^{3}),\ \ &\beta=0,\end{aligned}\right.$
$y(x)=1+\omega(x-a)+\bigg{(}\frac{1}{2}+\frac{\omega-1+\theta_{0}+\theta_{1}}{2a}\bigg{)}(x-a)^{2}+O((x-a)^{3}),$

where $b$ is arbitrary and the parameters

$\left\\{\begin{aligned} \varepsilon&=\varepsilon_{\pm}=\pm
2a/(\theta_{0}-\theta_{1}+\theta_{\infty}),\\\
\delta&=\delta_{\pm}=\pm(\theta_{0}-\theta_{1}-\theta_{\infty})/2a,\\\
\omega&=\omega_{\pm}=\pm 1.\end{aligned}\right.$ (7.24)

From the first equation in the compatibility condition (7.12) and above
expansions, we derive the behaviors of $v,\lambda_{1},\lambda_{2}$ at each
pole, zero or 1-point of $y$, as given in Table 1.

Case | $y$ | $v$ | $\lambda_{1}$ | $\lambda_{2}$
---|---|---|---|---
$\varepsilon=\varepsilon_{+}$ |  simple pole | $0$ | 0 | $O(a)$
$\varepsilon=\varepsilon_{-}$ | simple pole | $-\frac{1}{2}(\theta_{0}-\theta_{1}+\theta_{\infty})$ | $~{}~{}~{}~{}O(a)~{}~{}~{}~{}$ | $a$
$\alpha=0$ | double pole | $0$ | 0 | $a$
$\delta=\delta_{+}$ | simple zero | $-\theta_{0}$ | $O(a)$ | 0
$\delta=\delta_{-}$ | simple zero | $-\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty})$ | $a$ | $O(a)$
$\beta=0$ | double zero | $-\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty})$ | $a$ | 0
$\omega=\omega_{+}$ | 1-point | $O(1)$ | $O(a)$ | $~{}~{}~{}~{}O(a)~{}~{}~{}~{}$
$\omega=\omega_{-}$ | 1-point | $a(x-a)^{-2}+O(1)$ | $\infty$ | $\infty$
Table 1: Behaviors of $y,v,\lambda_{1},\lambda_{2}$ at the critical point
$x=a$

It is readily seen from Table 1 that the condition (7.22) or (7.23) is
fulfilled when $a$ is a pole, zero or 1-point of $y$ with
$y^{\prime}(a)=\omega_{-}$. Thus, the equation (7.19) or (7.21) is equivalent
to CHE in these cases. For example, we consider the case
$\varepsilon=\varepsilon_{+}$. According to Table 1, we have $v(a)=0$,
$\lambda_{1}(a)=0$. Thus, by taking $x\rightarrow a$ in (7.19) and setting

$\Phi_{1}=z^{\frac{\theta_{0}}{2}}(z-a)^{\frac{\theta_{1}}{2}}e^{\frac{1}{2}z}w,$

we obtain the CHE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+\left(1+\frac{\theta_{0}}{z}+\frac{1+\theta_{1}}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{pz-q}{z(z-a)}w=0,\\\
&p=\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty}),\\\
&q=\lim_{x\rightarrow
a}x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{(\theta_{0}+\theta_{\infty})a}{2}-\frac{(\theta_{0}+\theta_{1})^{2}-\theta_{\infty}^{2}}{4}\\\
&~{}~{}=\frac{a}{4}(\theta_{0}-\theta_{1}+\theta_{\infty})-\frac{b}{4}(\theta_{0}-\theta_{1}+\theta_{\infty})^{2}+\frac{1}{4}(\theta_{\infty}+1)(\theta_{0}-\theta_{1}+\theta_{\infty})-\theta_{0}\theta_{1}.\end{aligned}\right.$
(7.25)

Actually, we obtain six different CHEs with the characteristic exponents at
the singular points shown in Table 2 and the accessory parameters given below:

$q=\lim_{x\rightarrow
a}x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{(\theta_{0}+\theta_{\infty})a}{2}-\frac{(\theta_{0}+\theta_{1})^{2}-\theta_{\infty}^{2}}{4},$
(7.26)

for $\varepsilon=\varepsilon_{+}$, $\delta=\delta_{+}$ and
$\delta=\delta_{-}$,

$q=\lim_{x\rightarrow
a}x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{(\theta_{0}+\theta_{\infty}+2)a}{2}-\frac{(\theta_{0}+\theta_{1})^{2}-\theta_{\infty}^{2}}{4},$
(7.27)

for $\varepsilon=\varepsilon_{-}$,

$q=\lim_{x\rightarrow
a}\left(x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a}{x-a}\right)+\frac{(\theta_{0}+\theta_{\infty})a}{2}-\frac{(\theta_{0}+\theta_{1})^{2}-\theta_{\infty}^{2}}{4}+1-\theta_{0}-\theta_{1},$
(7.28)

for $\omega=\omega_{-}$ and employing $\Phi_{1}$,

$q=\lim_{x\rightarrow
a}\left(x\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a}{x-a}\right)+\frac{(\theta_{0}+\theta_{\infty}+2)a}{2}-\frac{(\theta_{0}+\theta_{1})^{2}-\theta_{\infty}^{2}}{4}+1-\theta_{0}-\theta_{1},$
(7.29)

for $\omega=\omega_{-}$ and using $\Phi_{2}$.

Case | Equation | $0$ | $a$ | $\infty$ | $p$
---|---|---|---|---|---
$\varepsilon=\varepsilon_{+}$ | (7.19) | $\theta_{0}$ | $~{}~{}1+\theta_{1}~{}~{}$ |  1 | $\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty})$
$\varepsilon=\varepsilon_{-}$ | (7.21) |  $1+\theta_{0}$ | $\theta_{1}$ | 1 | $\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty}+2)$
$\delta=\delta_{+}$ | (7.21) | $\theta_{0}$ | $1+\theta_{1}$ | 1 | $\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty}+2)$
$\delta=\delta_{-}$ | (7.19) | $1+\theta_{0}$ | $\theta_{1}$ | 1 | $\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty})$
$\omega=\omega_{-}$ | (7.19) | $1+\theta_{0}$ | $1+\theta_{1}$ | 1 | $\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty}+2)$
$\omega=\omega_{-}$ | (7.21) | $1+\theta_{0}$ | $1+\theta_{1}$ | 1 | $\frac{1}{2}(\theta_{0}+\theta_{1}+\theta_{\infty}+2)$
Table 2: The characteristic exponents of CHEs obtained from equations (7.19)
and (7.21)

We mention that the case $\lambda_{1}(x)=x$ has been considered in [6, 7].
According to Table 1, the condition $\lambda_{1}(x)=x$ is equivalent to the
case that $y$ has simple zeros with coefficient $\delta=\delta_{-}$ or $y$ has
double zeros.

###  7.4 Isomonodromy set of accessory parameters of CHE

We have shown that the CHE can be obtained as a limit of the first row or
second row of the isomonodromy family of $\Phi(z,x)$ when $x\to a$, $a$ being
one of the poles, zeros or $1$-points of the Painlevé V transcendents
corresponding to the same monodromy data of CHE. While the accessory
parameters $(a,q)$ are expressed in terms of the parameters in the Laurent or
Taylor expansion of the PV transcendents. As a consequence, we have the
following description of the isomonodromy set of accessory parameters of CHE.

######  Theorem 6.

There is a discrete set of pairs of accessory parameters $(a_{n},q_{n})$ such
that the CHE (1.2) corresponding to these parameters and the fixed parameters
$\gamma$, $\delta$ and $p$, has the same monodromy data (7.6) as the original
one with the parameters $a$ and $q$. Under the relation given in the last
equation of (7.25), this set coincides with the set of parameters
$(a_{n},b_{n})$ in the Laurent expansion near the poles $a_{n}$ of the unique
solution of PV (7.13) with the parameters $\theta_{0}=\gamma$,
$\theta_{1}=\delta-1$, $\theta_{\infty}=2p-\gamma-\delta+1$ and corresponding
to the same monodromy data (7.6) as the CHE.

##  8  Accessory parameters of HE

###  8.1 Reduction of the linear system for PVI to HE

In this subsection, we will derive HE from the linear system (1.11) for PVI at
the poles, zeros, 1-points ($y(x)=1$) and fixed points ($y(x)=x$) of the
solutions of PVI.

Consider the following Lax pair for PVI equation (see [20, (C.46)-(C.47)]):

$\left\\{\begin{aligned} &\frac{\partial\Phi(z,x)}{\partial
z}=A(z,x)\Phi(z,x),\ \
A(z,x)=\frac{A_{0}}{z}+\frac{A_{1}}{z-1}+\frac{A_{2}}{z-x},\\\
&\frac{\partial\Phi(z,x)}{\partial x}=B(z,x)\Phi(z,x),\ \
B(z,x)=-\frac{A_{2}}{z-x},\end{aligned}\right.$ (8.1)

where

$A_{i}=\begin{pmatrix}v_{i}+\theta_{i}&-u_{i}v_{i}\\\
u_{i}^{-1}\left(v_{i}+\theta_{i}\right)&-v_{i}\end{pmatrix},\ \ \ i=0,1,2.$

If we set

$\left\\{\begin{aligned}
&\kappa_{1}=-\frac{1}{2}\left(\theta_{0}+\theta_{1}+\theta_{2}-\theta_{\infty}\right),\\\
&\kappa_{2}=-\frac{1}{2}\left(\theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}\right),\\\
&A_{0}+A_{1}+A_{2}=-\begin{pmatrix}\kappa_{1}&0\\\
0&\kappa_{2}\end{pmatrix},\\\
&A(z,x)_{12}=\frac{-u_{0}v_{0}}{z}+\frac{-u_{1}v_{1}}{z-1}+\frac{-u_{2}v_{2}}{z-x}=\frac{k(z-y)}{z(z-1)(z-x)},\\\
&v=\frac{v_{0}+\theta_{0}}{y}+\frac{v_{1}+\theta_{1}}{y-1}+\frac{v_{2}+\theta_{2}}{y-x},\end{aligned}\right.$
(8.2)

then the compatibility condition of the above Lax pair gives

$\left\\{\begin{aligned}
\frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{y(y-1)(y-x)}{x(x-1)}\left(2v-\frac{\theta_{0}}{y}-\frac{\theta_{1}}{y-1}-\frac{\theta_{2}-1}{y-x}\right),\\\
\frac{\mathrm{d}v}{\mathrm{d}x}&=\frac{1}{x(x-1)}\bigg{\\{}\left[-3y^{2}+2(1+x)y-x\right]v^{2}+\big{[}(2y-1-x)\theta_{0}+(2y-x)\theta_{1}\\\
&\ \ \
+(2y-1)\left(\theta_{2}-1\right)\big{]}v-\kappa_{1}\left(\kappa_{2}+1\right)\bigg{\\}},\\\
\frac{\mathrm{d}k}{\mathrm{d}x}&=k\left(\theta_{\infty}-1\right)\frac{y-x}{x(x-1)},\end{aligned}\right.$
(8.3)

which leads to the PVI equation

$\displaystyle\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\ $
$\displaystyle\frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-x}\right)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}-\left(\frac{1}{x}+\frac{1}{x-1}+\frac{1}{y-x}\right)\frac{\mathrm{d}y}{\mathrm{d}x}$
$\displaystyle+\frac{y(y-1)(y-x)}{x^{2}(x-1)^{2}}\left(\alpha_{0}+\beta_{0}\frac{x}{y^{2}}+\gamma_{0}\frac{x-1}{(y-1)^{2}}+\delta_{0}\frac{x(x-1)}{(y-x)^{2}}\right),$
(8.4)

where

$\alpha_{0}=\frac{1}{2}\left(\theta_{\infty}-1\right)^{2},\ \
\quad\beta_{0}=-\frac{1}{2}\theta_{0}^{2},\ \
\quad\gamma_{0}=\frac{1}{2}\theta_{1}^{2},\ \
\quad\delta_{0}=\frac{1}{2}\left(1-\theta_{2}^{2}\right).$

The PVI $\tau$-function is defined by (see [20, (C.57)]):

$\displaystyle x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=\ $
$\displaystyle
y(y-1)(y-x)\bigg{\\{}v^{2}-\bigg{(}\frac{\theta_{0}}{y}+\frac{\theta_{1}}{y-1}+\frac{\theta_{2}}{y-x}\bigg{)}v+\frac{\kappa_{1}\kappa_{2}}{y(y-1)}\bigg{\\}}$
$\displaystyle+\theta_{0}\theta_{2}(x-1)+\theta_{1}\theta_{2}x.$ (8.5)

Substituting the entries of $A(z,x)$ into (1.11) gives

$\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}+P(z,x)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}+Q(z,x)\Phi_{1}=0,$
(8.6)

where

$\left\\{\begin{aligned}
&P(z,x)=\frac{1-\theta_{0}}{z}+\frac{1-\theta_{1}}{z-1}+\frac{1-\theta_{2}}{z-x}-\frac{1}{z-y},\\\
&Q(z,x)=\frac{\kappa_{1}(\kappa_{2}+1)}{z(z-1)}-\frac{M(x)}{z(z-1)(z-x)}+\frac{y(y-1)v}{z(z-1)(z-y)},\\\
&M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\kappa_{1}(y-x)+y(y-1)v-\theta_{0}\theta_{2}(x-1)-\theta_{1}\theta_{2}x.\end{aligned}\right.$

It is clear that the extra singularity $z=y$ can be removed by considering the
critical values

$y(a)=0,\ \ \mathrm{or}\ \ y(a)=1,\ \ \mathrm{or}\ \ y(a)=a,\ \ \mathrm{or}\ \
y(a)=\infty.$

First, we consider the critical value $y(a)=a$. By taking the limit
$x\rightarrow a$ in (8.6), we obtain the following HE directly

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}+\left(\frac{1-\theta_{0}}{z}+\frac{1-\theta_{1}}{z-1}+\frac{-\theta_{2}}{z-a}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}+\frac{pz-q}{z(z-1)(z-a)}\Phi_{1}=0,\\\
&p=\frac{1}{4}(\theta_{0}+\theta_{1}+\theta_{2}-\theta_{\infty})(\theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}-2),\\\
&q=\lim_{x\rightarrow
a}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\kappa_{1}(\kappa_{2}+1)a-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a.\end{aligned}\right.$
(8.7)

The equation (8.7) has also been obtained in [1, 3, 8]. It was also noted
therein that the condition $y(a)=a$ can be expressed in terms of the
associated $\tau$-function.

Next, we consider the poles of $y$. Let $x=a$ ($a\neq 0,1$) be a movable pole
of $y(x)$. Then $y(x)$ possesses the following Laurent expansions (see [16,
(46.7)]):

$y(x)=\left\\{\begin{aligned} &\frac{\varepsilon}{x-a}+b+O(x-a),\ \
&\alpha_{0}\neq 0,\\\
&\frac{b}{(x-a)^{2}}+\frac{(2a-1)b}{a(a-1)(x-a)}+c_{0}+O(x-a),\ \
&\alpha_{0}=0,\end{aligned}\right.$ (8.8)

where $\varepsilon=\varepsilon_{\pm}=\pm a(a-1)/(\theta_{\infty}-1)$, $b$ is
arbitrary and

$c_{0}=\frac{1}{3}(a+1)+\frac{b}{12a^{2}(a-1)^{2}}\Big{\\{}12a(a-1)+1-a\theta_{0}^{2}+(a-1)\theta_{1}^{2}-a(a-1)(\theta_{2}^{2}-1)\Big{\\}}.$

We derive HE by considering each case separately.

### Case 1: $a$ is a simple pole with residue $\varepsilon=\varepsilon_{+}$
and $\theta_{\infty}\neq 0$

In this case, we have from (8.1), (8.6) and (8.8) that

$\displaystyle
M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+(\theta_{\infty}-1)b-\kappa_{1}a-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a$
$\displaystyle\ \ \ \ \ \ \ \ \ \
+\frac{1}{2}\big{[}-\theta_{0}+(\theta_{2}-1)(a-1)-2a+1-(\theta_{\infty}-1)a\big{]}+O(x-a),$
(8.9)

and

$\displaystyle
x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=\theta_{\infty}(1-\theta_{\infty})b$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\frac{1}{2}\theta_{\infty}\big{[}-\theta_{0}+(\theta_{2}-1)(a-1)-2a+1-(\theta_{\infty}-1)a\big{]}-\kappa_{1}^{2}(a-1)$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\kappa_{1}\kappa_{2}a-\kappa_{1}\big{[}a\theta_{0}+(a-1)\theta_{1}\big{]}+\theta_{0}\theta_{2}(a-1)+\theta_{1}\theta_{2}a+O(x-a).$
(8.10)

This, together with $\theta_{\infty}\neq 0$, implies that

$\displaystyle M(x)$
$\displaystyle=(1-\theta_{\infty}^{-1})x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\kappa_{1}a-\theta_{\infty}^{-1}\Big{\\{}\kappa_{1}^{2}(a-1)+\kappa_{1}\kappa_{2}a+\kappa_{1}\big{[}a\theta_{0}$
$\displaystyle\ \ \ \
+(a-1)\theta_{1}\big{]}\Big{\\}}-(1-\theta_{\infty}^{-1})\big{[}\theta_{0}\theta_{2}(a-1)+\theta_{1}\theta_{2}a\big{]}+O(x-a).$

Taking $x\rightarrow a$ in (8.6), we obtain the HE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}+\left(\frac{1-\theta_{0}}{z}+\frac{1-\theta_{1}}{z-1}+\frac{1-\theta_{2}}{z-a}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}+\frac{pz-q}{z(z-1)(z-a)}\Phi_{1}=0,\\\
&p=\frac{1}{4}(\theta_{0}+\theta_{1}+\theta_{2}-\theta_{\infty})(\theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}-4),\\\
&q=\lim_{x\rightarrow
a}(1-\theta_{\infty}^{-1})x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\kappa_{1}(\kappa_{2}+2)a-\theta_{\infty}^{-1}\Big{\\{}\kappa_{1}^{2}(a-1)\\\
&\ \ \ \ \
+\kappa_{1}\kappa_{2}a+\kappa_{1}\big{[}a\theta_{0}+(a-1)\theta_{1}\big{]}\Big{\\}}-(1-\theta_{\infty}^{-1})\big{[}\theta_{0}\theta_{2}(a-1)+\theta_{1}\theta_{2}a\big{]},\end{aligned}\right.$
(8.11)

where $\kappa_{1}$ and $\kappa_{2}$ are given in (8.2).

### Case 2: $a$ is a simple pole with residue $\varepsilon=\varepsilon_{+}$
and $\theta_{\infty}=0$

When $\theta_{\infty}=0$, it is seen from (8) and (8) that

$\displaystyle
M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-b+\frac{1}{2}\big{[}-\theta_{0}+(\theta_{2}-1)(a-1)-a\big{]}-\kappa_{1}a$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a),$

and

$\displaystyle x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)=$
$\displaystyle-\kappa_{1}^{2}(a-1)-\kappa_{1}\kappa_{2}a-\kappa_{1}\big{[}a\theta_{0}+(a-1)\theta_{1}\big{]}$
$\displaystyle+\theta_{0}\theta_{2}(a-1)+\theta_{1}\theta_{2}a+O(x-a).$

It is straightforward that the free parameter $b$ is independent of $\tau(x)$.
To express the accessory parameter in terms of the $\tau$-function as (8.11),
we need the Schlesinger transformation used by Dubrovin and Kapaev to shift
the formal monodromy at infinity $\theta_{\infty}$ by $-2$, i.e.
$\theta_{\infty}\mapsto\theta_{\infty}-2$; see [10, Sec. 2.3.2]. As a
consequence, we obtain the HE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}+\left(\frac{1-\theta_{0}}{z}+\frac{1-\theta_{1}}{z-1}+\frac{1-\theta_{2}}{z-a}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}+\frac{pz-q}{z(z-1)(z-a)}\Phi_{1}=0,\\\
&p=\frac{1}{4}(\theta_{0}+\theta_{1}+\theta_{2}+2)(\theta_{0}+\theta_{1}+\theta_{2}-6),\\\
&q=\lim_{x\rightarrow
a}\frac{3}{2}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tilde{\tau}(x)+\tilde{\kappa}_{1}(\tilde{\kappa}_{2}+2)a+\frac{1}{2}\Big{\\{}\tilde{\kappa}_{1}^{2}(a-1)\\\
&\ \ \ \ \
+\tilde{\kappa}_{1}\tilde{\kappa}_{2}a+\tilde{\kappa}_{1}\big{[}a\theta_{0}+(a-1)\theta_{1}\big{]}\Big{\\}}-\frac{3}{2}\big{[}\theta_{0}\theta_{2}(a-1)+\theta_{1}\theta_{2}a\big{]},\end{aligned}\right.$

where $\tilde{\tau}(x)$ is a new $\tau$-function defined by

$\displaystyle x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tilde{\tau}(x)=\ $
$\displaystyle
y(y-1)(y-x)\bigg{\\{}v^{2}-\bigg{(}\frac{\theta_{0}}{y}+\frac{\theta_{1}}{y-1}+\frac{\theta_{2}}{y-x}\bigg{)}v+\frac{\tilde{\kappa}_{1}\tilde{\kappa}_{2}}{y(y-1)}\bigg{\\}}$
$\displaystyle+\theta_{0}\theta_{2}(x-1)+\theta_{1}\theta_{2}x$

with

$\left\\{\begin{aligned}
&\tilde{\kappa}_{1}=-\frac{1}{2}\left(\theta_{0}+\theta_{1}+\theta_{2}+2\right),\\\
&\tilde{\kappa}_{2}=-\frac{1}{2}\left(\theta_{0}+\theta_{1}+\theta_{2}-2\right).\\\
\end{aligned}\right.$

### Case 3: $a$ is a simple pole with residue $\varepsilon=\varepsilon_{-}$

From (8.6) and (8.8), it follows that

$\displaystyle M(x)$
$\displaystyle=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a(a-1)}{x-a}+\frac{1}{2}(\theta_{0}+\theta_{1}+2\theta_{2}-4)a$
$\displaystyle\ \ \
+1-\frac{1}{2}(\theta_{0}+\theta_{2})-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a).$

By taking $x\rightarrow a$ in (8.6), we arrive at the HE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}+\left(\frac{1-\theta_{0}}{z}+\frac{1-\theta_{1}}{z-1}+\frac{1-\theta_{2}}{z-a}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}+\frac{pz-q}{z(z-1)(z-a)}\Phi_{1}=0,\\\
&p=\frac{1}{4}(\theta_{0}+\theta_{1}+\theta_{2}-\theta_{\infty}-2)(\theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}-2),\\\
&q=\lim_{x\rightarrow
a}\bigg{(}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a(a-1)}{x-a}\bigg{)}+(\kappa_{1}+1)(\kappa_{2}+1)a\\\
&\ \ \ \ \
+\frac{1}{2}(\theta_{0}+\theta_{1}+2\theta_{2}-4)a+1-\frac{1}{2}(\theta_{0}+\theta_{2})-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a\\\
&~{}~{}=-(1-\theta_{\infty})^{2}b-\frac{1}{4}\Big{(}2(\theta_{0}+\theta_{1})-(\theta_{0}+\theta_{1})^{2}+\theta_{2}^{2}-(\theta_{\infty}-2)^{2}+4(\theta_{\infty}-1)\Big{)}a\\\
&\qquad-\frac{1}{2}(\theta_{0}+\theta_{2})+\theta_{\infty}-1+\frac{1}{4}\Big{(}(\theta_{0}+\theta_{2})^{2}-\theta_{1}^{2}+(\theta_{\infty}-2)^{2}\Big{)},\end{aligned}\right.$
(8.12)

where $\kappa_{1}$ and $\kappa_{2}$ are given in (8.2).

### Case 4: $a$ is a double pole

We see from (8.6) and (8.8) that

$\displaystyle M(x)$
$\displaystyle=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a(a-1)}{x-a}-\frac{1}{2}(\theta_{0}+\theta_{1}-4)a-\frac{1}{2}(\theta_{0}+\theta_{2})$
$\displaystyle\ \ \ -\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a).$

By taking $x\rightarrow a$ in (8.6), we arrive at the HE

$\left\\{\begin{aligned}
&\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}z^{2}}+\left(\frac{1-\theta_{0}}{z}+\frac{1-\theta_{1}}{z-1}+\frac{1-\theta_{2}}{z-a}\right)\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}z}+\frac{pz-q}{z(z-1)(z-a)}\Phi_{1}=0,\\\
&p=\frac{1}{4}(\theta_{0}+\theta_{1}+\theta_{2}-1)(\theta_{0}+\theta_{1}+\theta_{2}-3),\\\
&q=\lim_{x\rightarrow
a}\bigg{(}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{a(a-1)}{x-a}\bigg{)}+\kappa_{1}(\kappa_{2}+2)a\\\
&\ \ \ \ \
-\frac{1}{2}(\theta_{0}+\theta_{1}-4)a-\frac{1}{2}(\theta_{0}+\theta_{2})-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a\\\
&~{}~{}=\frac{a^{2}(a-1)^{2}}{b}+\Big{(}\kappa_{1}\left(\kappa_{2}+\theta_{2}+3\right)+3\Big{)}a+\Big{(}\kappa_{1}\left(\kappa_{2}+\theta_{1}+1\right)-1\Big{)},\end{aligned}\right.$
(8.13)

where $\kappa_{1}$ and $\kappa_{2}$ are given in (8.2), with
$\theta_{\infty}=1$ in this case.

It is noted that the above HEs are also obtained from the linear system at the
poles of the solutions of PVI in [10], while the accessory parameters in these
equations are expressed in terms of the free parameter of the Laurent
expansion of $y$ at the poles.

Finally, we consider the critical values $y(x)=0$ and $y(x)=1$. It follows
from [16, (46.8)-(46.9)] that $y$ admits the following Taylor expansions:

$y(x)=\left\\{\begin{aligned} &\lambda(x-a)+b(x-a)^{2}+O((x-a)^{3}),\ \
&\beta\neq 0,\\\ &b(x-a)^{2}+O((x-a)^{3}),\ \ &\beta=0,\end{aligned}\right.$
(8.14) $y(x)=\left\\{\begin{aligned} &1+\omega(x-a)+b(x-a)^{2}+O((x-a)^{3}),\
\ &\gamma\neq 0,\\\ &1+b(x-a)^{2}+O((x-a)^{3}),\ \
&\gamma=0,\end{aligned}\right.$ (8.15)

where $b$ is arbitrary and

$\left\\{\begin{aligned} &\lambda=\lambda_{\pm}=\pm\frac{\theta_{0}}{a-1},\\\
&\omega=\omega_{\pm}=\pm\frac{\theta_{1}}{a}.\end{aligned}\right.$

It is seen from (8.6), (8.14) and (8.15) that

$M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\theta_{0}-\kappa_{1}a-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a),$

for $\lambda=\lambda_{+}$,

$M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\kappa_{1}a-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a),$

for $\lambda=\lambda_{-}$,

$M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)-\frac{\theta_{0}}{2}-\kappa_{1}a-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a),$

for $\beta=0$,

$M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\kappa_{1}(1-a)-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a),$

for $\omega=\omega_{+}$,

$M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\theta_{1}+\kappa_{1}(1-a)-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a),$

for $\omega=\omega_{-}$,

$M(x)=x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\frac{\theta_{1}}{2}+\kappa_{1}(1-a)-\theta_{0}\theta_{2}(a-1)-\theta_{1}\theta_{2}a+O(x-a),$

for $\gamma=0$.

Taking $x\rightarrow a$ in (8.6), we then obtain six HEs. We show the exponent
parameters at the singularities 0, 1, $a$ and $\infty$ in Table 3 and list the
accessory parameters below:

$\displaystyle q=\lim_{x\rightarrow
a}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\theta_{0}(\theta_{2}-1)+\left(\kappa_{1}\kappa_{2}+\theta_{0}-\theta_{0}\theta_{1}-\theta_{1}\theta_{2}-2\theta_{0}\theta_{2}\right)a,$

for $\lambda=\lambda_{+}$,

$q=\lim_{x\rightarrow
a}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\theta_{0}\theta_{2}+(\kappa_{1}\kappa_{2}-\theta_{0}\theta_{2}-\theta_{1}\theta_{2})a,$

for $\lambda=\lambda_{-}$,

$q=\lim_{x\rightarrow
a}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+(\kappa_{1}\kappa_{2}-\theta_{1}\theta_{2})a,$

for $\beta=0$,

$q=\lim_{x\rightarrow
a}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\kappa_{1}+\theta_{0}\theta_{2}+(\kappa_{1}\kappa_{2}-\theta_{0}\theta_{2}-\theta_{1}\theta_{2})a,$

for $\omega=\omega_{+}$,

$q=\lim_{x\rightarrow
a}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\theta_{1}+\theta_{0}\theta_{2}+\kappa_{1}+(\kappa_{1}\kappa_{2}+\theta_{1}-\theta_{0}\theta_{1}-\theta_{0}\theta_{2}-\theta_{1}\theta_{2})a,$

for $\omega=\omega_{-}$,

$q=\lim_{x\rightarrow
a}x(x-1)\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)+\kappa_{1}+\theta_{0}\theta_{2}+(\kappa_{1}\kappa_{2}-\theta_{0}\theta_{2})a,$

for $\gamma=0$.

Case | 0 | 1 | $a$ | $\infty$
---|---|---|---|---
$\lambda=\lambda_{+}$ | $\theta_{0}$ | $1-\theta_{1}$ | $1-\theta_{2}$ | $\frac{1}{4}(\theta_{0}-\theta_{1}-\theta_{2}+\theta_{\infty})(\theta_{0}-\theta_{1}-\theta_{2}-\theta_{\infty}+2)$
$\lambda=\lambda_{-}$ | $-\theta_{0}$ | $1-\theta_{1}$ | $1-\theta_{2}$ | $\frac{1}{4}(\theta_{0}+\theta_{1}+\theta_{2}-\theta_{\infty})(\theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}-2)$
$\beta=0$ | $0$ | $1-\theta_{1}$ | $1-\theta_{2}$ | $\frac{1}{4}(\theta_{1}+\theta_{2}-\theta_{\infty})(\theta_{1}+\theta_{2}+\theta_{\infty}-2)$
$\omega=\omega_{+}$ | $1-\theta_{0}$ | $-\theta_{1}$ | $1-\theta_{2}$ | $\frac{1}{4}(\theta_{0}+\theta_{1}+\theta_{2}-\theta_{\infty})(\theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}-2)$
$\omega=\omega_{-}$ | $1-\theta_{0}$ | $\theta_{1}$ | $1-\theta_{2}$ | $\frac{1}{4}(\theta_{1}-\theta_{0}-\theta_{2}+\theta_{\infty})(\theta_{1}-\theta_{0}-\theta_{2}-\theta_{\infty}+2)$
$\gamma=0$ | $1-\theta_{0}$ | $0$ | $1-\theta_{2}$ | $\frac{1}{4}(\theta_{0}+\theta_{2}-\theta_{\infty})(\theta_{0}+\theta_{2}+\theta_{\infty}-2)$
Table 3: The characteristic exponents at the singularities of HEs

We mentioned that, in the cases $\lambda=\lambda_{+}$ and $\omega=\omega_{-}$,
use has be made of the transformations $\Phi_{1}=z^{\theta_{0}}w$ and
$\Phi_{1}=(z-1)^{\theta_{1}}w$ to get the canonical form of HE (1.1).

###  8.2 Isomonodromy set of accessory parameters of HE

Consider the HE (1.1) with parameters $a$ and $q$. There exist uniquely two
linear independent solutions of (1.1), namely $(y_{1},y_{2})$, satisfied the
normalized asymptotic behavior as $z\to\infty$:

$\left(y_{1}(z),y_{2}(z)\right)\sim\left((1/z)^{\alpha},(1/z)^{\beta}\right),$
(8.16)

where $\alpha$ and $\beta$ are the characteristic exponents at $z=\infty$. We
have the asymptotic behaviors of $(y_{1},y_{2})$ near the singular points
$z_{0}=0,z_{1}=1,z_{2}=a$

$\left(y_{1}(z),y_{2}(z)\right)\sim\left((z-z_{k})^{\theta_{k}},(z-z_{k})^{0}\right)E_{k},~{}~{}k=0,1,2,$

with the characteristic exponents $\theta_{0}=1-\gamma$,
$\theta_{1}=1-\delta$, $\theta_{2}=1-\epsilon$ and some invertible constant
matrix $E_{k}$, $k=0,1,2$. Under an analytic continuation along a closed loop
around a singular point, we obtain another two linear independent solutions of
the same equation, which are therefore related to $(y_{1},y_{2})$ by

$\left(y_{1}\left(z_{k}+e^{2\pi i}(z-z_{k})\right),~{}y_{2}\left(z_{k}+e^{2\pi
i}(z-z_{k})\right)\right)=(y_{1}(z),y_{2}(z))e^{\pi i\theta_{k}}M_{k},$
$\left(y_{1}(e^{2\pi i}z),~{}y_{2}(e^{2\pi
i}z)\right)=\left(y_{1}(z),y_{2}(z)\right)e^{-\pi i(\alpha+\beta)}M_{\infty}.$

Here the constant matrices are known as the monodromy matrices and determined
by the connection matrices and the characteristic exponents

$M_{k}=E_{k}^{-1}e^{\pi i\theta_{k}\sigma_{3}}E_{k},\quad M_{\infty}=e^{\pi
i\theta_{\infty}\sigma_{3}},$

with $k=0,1,2$ and $\theta_{\infty}=-\alpha+\beta$. The monodromy data of HE
(1.1) is then constituted by

$\left\\{e^{\pi i\theta_{\infty}},e^{\pi i\theta_{0}},e^{\pi
i\theta_{1}},e^{\pi i\theta_{2}};E_{0},E_{1},E_{2}\right\\},$ (8.17)

where the characteristic exponents are related to the fixed parameters in
(1.1) by $\theta_{0}=1-\gamma$, $\theta_{1}=1-\delta$, $\theta_{2}=1-\epsilon$
and $\theta_{\infty}=-\alpha+\beta$. The monodromy matrices satisfies the
cyclic condition

$M_{\infty}M_{2}M_{1}M_{0}=I.$

According to [19], the monodromy matrices can be written explicitly in terms
of the characteristic exponents and the parameters

$2\cos(\pi\sigma_{jk})=\mathop{\rm Tr}\nolimits M_{j}M_{k}=\mathop{\rm
Tr}\nolimits M_{k}M_{j}$

with $j,k=0,1,2$ and $j<k$. Moreover, by the cyclic condition, only two of the
parameters $\\{\sigma_{01},\sigma_{02},\sigma_{12}\\}$ are independent.
Therefore, we get the same number of independent parameters of the monodromy
matrices as the parameters $a$ and $q$ in HE equation.

To study the isomonodromy deformation of the HE equation, it is convenient to
consider the matrix Fuchsian system with four regular singular points. As
shown in Section 8.1, the isomonodromy deformation of (LABEL:eq:P6) is
described by the PVI equation (8.3). Moreover, let $\Phi(z,x)$ be the
fundamental solution of (LABEL:eq:P6) normalized at infinity, then $\Phi(z,x)$
has the asymptotic behaviors near the singular points:

$\Phi(z,x)=\Phi_{0}^{(k)}(x)(I+O(z-z_{k}))(z-z_{k})^{\frac{1}{2}\theta_{k}}(z-z_{k})^{\frac{1}{2}\theta_{k}\sigma_{3}}\hat{E}_{k},$

and

$\Phi(z,x)=(I+O(1/z))z^{\frac{\alpha+\beta}{2}}z^{-\frac{1}{2}\theta_{\infty}\sigma_{3}},$

with $z_{0}=0$, $z_{1}=1$, $z_{2}=x$. Here the connection matrix $\hat{E}_{k}$
are certain invertible constant matrices. Similarly, the analytic continuation
of $\Phi(z,x)$ along a closed loop around the singular points are related to
$\Phi(z,x)$ by the monodromy matrices

$\hat{M}_{\infty}=e^{\pi
i\theta_{\infty}\sigma_{3}},\quad\hat{M}_{k}=\hat{E}_{k}^{-1}e^{\pi
i\theta_{k}\sigma_{3}}\hat{E}_{k},$

with $k=0,1,2$. While the HE can be obtained from a family of isomonodromy
deformation system by taking certain limit procedure at the poles of the
solution of PVI equation; see for instance (8.12). In this way, we obtain a
family of accessory parameters sharing the same monodromy data as stated in
the following theorem.

######  Theorem 7.

There is a discrete set of pairs of accessory parameters $(a_{n},q_{n})$ such
that the HE (1.1) corresponding to these parameters has the same monodromy
data (8.17) as the original one with the parameters $a$ and $q$. Under the
bijection given in the last equation of (8.12), and that of (8.13), this set
coincides with the set of parameters $(a_{n},b_{n})$ in the Laurent expansion
(8.8) near the poles $a_{n}$ of the unique solution of PVI equation (8.1) with
the same monodromy data (8.17) as the HE.

###### Proof.

Consider the HE (1.1) with given accessory parameters $a$ and $q$. There exist
unique fundamental solutions $(y_{1},y_{2})$ to this equation with the
normalized behavior at infinity (8.16). The solutions are corresponding to the
monodromy data (8.17) as mentioned at the beginning of this section. On the
other side, there is a unique solution $\Phi(z,x)$ normalized at infinity of
the system (LABEL:eq:P6) with the parameters related to the fixed parameters
of HE (1.1) by $\theta_{0}=1-\gamma$, $\theta_{1}=1-\delta$,
$\theta_{2}=1-\epsilon$ and $\theta_{\infty}=-\alpha+\beta$. Here the
$x$-dependence of $\Phi(z,x)$ is described by the unique solution of the PVI
equation with the Laurent expansion as given in (8.8) with the parameter $b$
given by the last equation of (8.12) and the parameter $\theta_{\infty}\neq 1$
therein. When $\theta_{\infty}=1$, we consider the Laurent expansion with a
double pole at $a$ in (8.8). The solution $\Phi(z,x)$ is corresponding to the
monodromy data of the form $\left\\{e^{\pi i\theta_{\infty}},e^{\pi
i\theta_{0}},e^{\pi i\theta_{1}},e^{\pi
i\theta_{2}};\hat{E}_{0},\hat{E}_{1},\hat{E}_{2}\right\\}$ which is
independent of $x$.

Using the limiting procedure shown in (8.12), we obtain the HE (1.1) with
accessory parameters $a$ and $q$ as limit of the first row of the isomonodromy
family $\Phi(z;x)$ as $x\to a$. It follows from the $x$-independent of the
monodromy data that $E_{k}=\hat{E}_{k}$, $k=0,1,2$. Thus, we have shown that
any given accessory parameters $(a,q)$ is related by the last equation of
(8.12) to the pole parameters $(a,b)$ of the unique solution of PVI
corresponding to the same monodromy data (8.17) when the parameter
$\theta_{\infty}\neq 1$. In the case $\theta_{\infty}=1$, we consider (8.13)
and similar analysis applies. By the meromorphic property of the PVI solution,
the poles of each PVI solution are discrete and hence the set of pairs of
accessory parameters of HE sharing the same monodromy data. We complete the
proof of Theorem 7. ∎

##  9 Asymptotics of the accessory parameters

In this section, we will derive some asymptotic approximations for the
isomonodromy sets of accessory parameters of some Heun class equations
expressed in terms of the monodromy data. The derivation are based on the
connection between the accessory parameters and the the Laurent or Taylor
coefficients for the corresponding Painlevé functions obtained in the previous
sections and the asymptotic expansions for the Painlevé transcendents known in
the literature. The equations of Heun class we considered in the section
include the RBHE, CHE, and DHE.

###  9.1 Asymptotics of the accessory parameter of RBHE

Consider the RBHE (1.7) with the accessory parameters $(a_{n},q_{n})$ such
that the equation corresponding to these parameters has the monodromy data
(4.9) specified by

$s_{-1}=-e^{-2\alpha\pi i},\ \ \ \ \ s_{0}=e^{-2\beta\pi i},\ \ \ \ \
s_{1}=-e^{2\alpha\pi i},$ (9.1)

with $i\beta\in\mathbb{R}$ and $\alpha>-1/2$. The corresponding Painlevé XXXIV
transcendents and their associated Lax pair play important roles in random
matrix theory when a Fisher-Hartwig singularity located at an interior point
where the density of the equilibrium density vanishes quadratically [4], or at
the right edge of the spectrum where typically the density vanishes like a
square root [18, 34]. The asymptotics of the Painlevé XXXIV transcendents and
their associated $\tau$-functions have been worked out in [9, 18, 34]. Using
these asympotics and the relation between the isomonodromy set of accessory
parameters and the corresponding Painlevé XXXIV transcendents given in Theorem
3, we derive the asymptotic behavior of the accessory parameters
$(a_{n},q_{n})$ expressed in terms of the given monodromy data (9.1).

######  Theorem 8.

Let $(a_{n},q_{n})$ be the sequence of accessory parameters such that the RBHE
(1.7) corresponding to these parameters has the same monodromy data (9.1),
then we have the asymptotic approximations as $n\to\infty$

$\displaystyle\frac{4}{3}|a_{n}|^{\frac{3}{2}}=2n\pi+2i\beta\ln
3n\pi+4i\beta\ln 2-2\mathop{\rm
arg}\nolimits\Gamma(\alpha-\beta)+\frac{1}{2}(2\alpha+1)\pi+O\left(\frac{\ln
n}{n}\right),$ (9.2)

and

$q(a_{n})=2\beta
i|a_{n}|^{\frac{1}{2}}-\frac{1}{2}\left(\alpha-\alpha^{2}+3\beta^{2}\right)a_{n}^{-1}+O\left(|a_{n}|^{-\frac{5}{2}}\right).$
(9.3)

###### Proof.

According to [34, Theorem 2] and [9, Proposition 3.4], we have the asymptotic
behaviors for the Painlevé XXXIV transcendents $y(x)$ and the associated
$\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)$, corresponding to the Stokes
multipliers (9.1), as $x\rightarrow-\infty$:

$\displaystyle y(x)$
$\displaystyle=\frac{2|\alpha-\beta|}{\sqrt{|x|}}\cos\left(\frac{\theta(x)}{2}+\mathop{\rm
arg}\nolimits\Gamma(1+\alpha-\beta)-\frac{\pi}{4}\right)$ (9.4)
$\displaystyle\quad\times\cos\left(\frac{\theta(x)}{2}+\mathop{\rm
arg}\nolimits\Gamma(\alpha-\beta)+\frac{\pi}{4}\right)+O(x^{-2}),$

and

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\log\tau(x)$ $\displaystyle=2\beta
i|x|^{\frac{1}{2}}-\frac{|\alpha-\beta|}{2x}\sin\left(\theta(x)+2\mathop{\rm
arg}\nolimits\Gamma(\alpha-\beta)+\mathop{\rm
arg}\nolimits(\alpha-\beta)\right)$
$\displaystyle\quad+\frac{\alpha^{2}-3\beta^{2}}{2x}+O(|x|^{-\frac{5}{2}}),$
(9.5)

where

$\theta(x)=\frac{4}{3}|x|^{\frac{3}{2}}-3i\beta\ln|x|-\alpha\pi-6i\beta\ln 2,$

and $\beta\in i\mathbb{R}$.

It follows from (9.4) that $y(x)$ admits a sequences of simple zeros lying on
the negative real axis and tending to infinity with the asymptotic
approximation given in (9.2). Moreover, the leading coefficient of the Taylor
expansion of $y(x)$ near the zero $a_{n}$ is $2\alpha$. The relation in the
second equation of (4.20), together with (9.2) and (9.1), then implies (9.3).
This completes the proof of Theorem 8.

∎

###  9.2 Asymptotics of the accessory parameter of CHE

In this subsection, we consider the CHE (7.25) with the accessory parameters
$(a_{n},q_{n})$ such that the equation corresponding to these parameters has
the monodromy data (7.6) parameterized in terms of $\sigma$ and $s$ as given
in (7.1)-(7.1). In the seminal work [19], Jimbo derived the asymptotic
expansion for the PV tanscendents and the associated $\tau$-function
corresponding to the monodromy data. Using these asympotics and the relation
between the isomonodromy set of accessory parameters and the corresponding
Painlevé V transcendents given in Theorem 6, we derive the asymptotic behavior
of the accessory parameters $(a_{n},q_{n})$ expressed in terms of the given
monodromy data.

######  Theorem 9.

Let $(a_{n},q_{n})$ be the sequence of accessory parameters such that the CHE
(7.25) corresponding to these parameters has the same monodromy data (7.6)
parameterized in terms of $\sigma$ and $s$ as given in (7.1)-(7.1), then we
have the asymptotic approximations as $n\to\infty$

$\ln|a_{n}|\sim\left(\mathop{\rm Re}\nolimits\sigma\ln|c_{0}|+\mathop{\rm
Im}\nolimits\sigma\mathop{\rm arg}\nolimits c_{0}-2\pi n|\mathop{\rm
Im}\nolimits\sigma|\right)/|\sigma|^{2},$ (9.6)

and

$q(a_{n})=\frac{\sigma^{2}-(\theta_{0}+\theta_{1})^{2}}{4}-\bigg{[}\frac{(\theta_{\infty}-1)(\theta_{1}^{2}-(\theta_{0}-1)^{2})}{4(\sigma^{2}-1)}+\frac{1-2\theta_{0}-\theta_{\infty}}{4}\bigg{]}a_{n}+O(a_{n}^{2}),$
(9.7)

where
$c_{0}=\frac{(\sigma+\theta_{\infty})(\sigma+\theta_{0}+\theta_{1})\Gamma(1+\sigma)^{2}\Gamma(\frac{1}{2}(\theta_{1}+\theta_{0}-\sigma)+1)\Gamma(\frac{1}{2}(\theta_{1}-\theta_{0}-\sigma)+1)\Gamma(\frac{1}{2}(\theta_{\infty}-\sigma)+1)}{(\sigma-\theta_{\infty})(\sigma-\theta_{0}-\theta_{1})\Gamma(1-\sigma)^{2}\Gamma(\frac{1}{2}(\theta_{1}+\theta_{0}+\sigma)+1)\Gamma(\frac{1}{2}(\theta_{1}-\theta_{0}+\sigma)+1)\Gamma(\frac{1}{2}(\theta_{\infty}+\sigma)+1)}\frac{1}{s}$
with $\mathop{\rm Im}\nolimits\sigma\neq 0$ and $s\neq 0$.

###### Proof.

According to [19], we have the following asymptotic expansion for the
$\tau$-function of PV as $x\rightarrow 0$

$\displaystyle\tau(x)\sim\ $ $\displaystyle\text{const.}\
x^{\frac{1}{4}(\sigma^{2}-\theta_{\infty}^{2})}$
$\displaystyle\times\Bigg{\\{}1-\frac{\theta_{\infty}\left(\sigma^{2}+\theta_{1}^{2}-\theta_{0}^{2}\right)}{4\sigma^{2}}x-\frac{\left(\sigma-\theta_{\infty}\right)\left[\left(\sigma-\theta_{1}\right)^{2}-\theta_{0}^{2}\right]}{8\sigma^{2}(1+\sigma)^{2}}x(\rho
x^{\sigma})$
$\displaystyle+\frac{\left(\sigma+\theta_{\infty}\right)\left[\left(\sigma+\theta_{1}\right)^{2}-\theta_{0}^{2}\right]}{8\sigma^{2}(1-\sigma)^{2}}x(\rho
x^{\sigma})^{-1}+\sum_{j=2}^{\infty}x^{j}\sum_{k=-j}^{j}c_{jk}x^{k\sigma}\Bigg{\\}},$
(9.8)
$\rho=\frac{\Gamma(1-\sigma)^{2}\Gamma(\frac{1}{2}(\theta_{1}+\theta_{0}+\sigma)+1)\Gamma(\frac{1}{2}(\theta_{1}-\theta_{0}+\sigma)+1)\Gamma(\frac{1}{2}(\theta_{\infty}+\sigma)+1)}{\Gamma(1+\sigma)^{2}\Gamma(\frac{1}{2}(\theta_{1}+\theta_{0}-\sigma)+1)\Gamma(\frac{1}{2}(\theta_{1}-\theta_{0}-\sigma)+1)\Gamma(\frac{1}{2}(\theta_{\infty}-\sigma)+1)}s.$
(9.9)

Here the parameters $\sigma\neq 0$ and $s$ are the parameterization of the
monodromy data of the corresponding CHE as given in (7.8)-(7.1).

As mentioned in [19, Remark 1], the asymptotics of the solution $y$ of PV can
also be obtained from the asymptotic analysis carried out therein. The
small-$x$ asymptotic expansion for $y$ was also derived in [27]. For
$\mathop{\rm Im}\nolimits\sigma\neq 0$, we see from the asymptotics of the PV
tanscendents $y$ that it admits a sequence of simple poles
$\\{a_{n}\\}_{n\in\mathbb{N}}$ such that

$a_{n}^{\sigma}\sim c_{0},\quad
c_{0}=\frac{(\sigma+\theta_{\infty})(\sigma+\theta_{0}+\theta_{1})}{(\sigma-\theta_{\infty})(\sigma-\theta_{0}-\theta_{1})}\rho^{-1},\quad\mbox{as}\quad
a_{n}\rightarrow 0.$ (9.10)

The sequence of poles are clustering at zero along the spiral described by

$|\sigma|^{2}\ln|a_{n}|-\mathop{\rm Re}\nolimits\sigma\ln|c_{0}|-\mathop{\rm
Im}\nolimits\sigma\mathop{\rm arg}\nolimits c_{0}\sim-2\pi n|\mathop{\rm
Im}\nolimits\sigma|,$ (9.11)

and

$|\sigma|^{2}\mathop{\rm arg}\nolimits a_{n}-\mathop{\rm
Re}\nolimits\sigma\mathop{\rm arg}\nolimits c_{0}+\mathop{\rm
Im}\nolimits\sigma\ln|c_{0}|\sim 2\pi n\mathop{\rm Re}\nolimits\sigma;$ (9.12)

see for instance [27, Theorem 2.8]. It can be checked that the residues of $y$
at $a_{n}$ equal $\varepsilon_{+}$ as given in (7.24).

Thus, from (9.9) and (9.11) we have (9.6). Substituting (9.2), (9.10) into
(7.26), we obtain (9.7). This completes the proof of Theorem 9.

∎

It should be mentioned that similar formulae for the asymptotic approximations
of the accessory parameters as $a\to 0$ have also been derived in [6] by
considering the zeros of a special $\tau$-function of PV, with applications in
black holes; see [6, (2.62a)-(2.62c)]. In general, the asymptotic expansions
of the PV solutions $y(x)$ and the associated $\tau$-functions near infinity
are rather complicated. There exists no general asymptotic expansions for
$y(x)$ or $\tau(x)$ as $x\rightarrow\infty$ except along some special rays:
$\mathrm{arg}x=0,\pi/2,\pi,3\pi/2$; see [2, 21, 28]. In [2, 28], the
asymptotics for $y(x)$ and the logarithmic derivative of the $\tau$-functions
as $x\rightarrow i\infty$ are established. From the asymptotic expansions, it
is shown in [28] that under certain conditions the PV solutions admit
sequences of poles and of zeros lying on the imaginary axis and tending to
$i\infty$. Combining these results with the expressions of the accessory
parameters of CHE in terms of the $\tau$-function given in (7.26)-(7.27), the
asymptotic approximations of the isomonodromic set of accessory parameters
$(a_{n},q_{n})$ corresponding to some special monodromy data may also be
obtained.

###  9.3 Asymptotics of the accessory parameter of DHE

In the pioneering work of McCoy, Tarcy and Wu [24], the asymptotics and the
connection formulae for one-parameter family of solutions to PIII were derived
rigorously. These solutions have important applications in the analysis of
two-dimensional Ising model [33]. More precisely, they showed in [24] that
there are one-parameter family of solutions of PIII with the asymptotics

$y(x;\nu,\lambda)\sim
1-\lambda\Gamma\left(\nu+\frac{1}{2}\right)2^{-2\nu}x^{-\nu-\frac{1}{2}}e^{-2x}\Big{(}1+\sum^{\infty}\limits_{j=1}\frac{c_{j}}{x^{j}}\Big{)},\
\ \ \ \ x\rightarrow+\infty,$ (9.13)

where parameters $\theta_{0}$, $\theta_{\infty}$ satisfy the relation

$\theta_{0}=\theta_{\infty}-1=\nu.$

For $|\lambda|<1/\pi$, the asymptotic behavior of $y(x;\nu,\lambda)$ as
$x\rightarrow 0^{+}$ is described by

$y(x;\nu,\lambda)\sim B(2x)^{\sigma}$

with the parameters $\sigma$ and $B$ are given as explicit functions of
$\lambda$ and $\nu$, which are now known as the connection formulae. When
$\lambda>1/\pi$ and the parameter $\nu=0$, the asymptotic behavior of $y(x)$
as $x\rightarrow 0^{+}$ was also derived in [24]

$y(x;0,\lambda)=\frac{x}{2\mu}\sin\left\\{2\mu\ln\frac{x}{4}-2\mathop{\rm
arg}\nolimits\Gamma(i\mu)\right\\}+O(x^{3}),\ \ \ \ x\rightarrow 0^{+},$
(9.14)

where

$\lambda=\frac{1}{\pi}\cosh(\pi\mu),\quad\mu>0.$

For $\lambda<-1/\pi$, the asymptotics of $y(x)$ as $x\rightarrow 0^{+}$
follows from (9.14) and the symmetry relation [24, (4.127)]

$y(x;\nu,\lambda)=\frac{1}{y(x;\nu,-\lambda)}.$ (9.15)

The asymptotic formula for general parameter $\nu\in\mathbb{R}$ was worked out
in [11]. When $\lambda>1/\pi$ and $\nu=0$, it is readily seen from (9.14) that
$y(x;0,\lambda)$ admits a sequence of zeros $\\{c_{n}\\}_{n\in\mathbb{N}}$
lying on the positive real axis with $x=0$ being a limiting point:

$c_{n}\sim 4\exp\left\\{-\frac{n\pi}{\mu}+\frac{\mathop{\rm
arg}\nolimits\Gamma(i\mu)}{\mu}\right\\}\to 0^{+},\ \ \ n\rightarrow\infty.$
(9.16)

Moreover, it is straightforward to check that the Taylor expansions of $y$ at
the zeros $\\{c_{2n}\\}_{n\in\mathbb{N}}$ and
$\\{c_{2n+1}\\}_{n\in\mathbb{N}}$ are corresponding to $\sigma=\sigma_{+}$ and
$\sigma=\sigma_{-}$ in (5.29), respectively. It is also seen from the relation
(9.15) that there are infinitely many poles of $y(x;0,\lambda)$ clustering at
$x=0$ on the positive real axis when $\lambda<-1/\pi$.

Let us consider the DHE equation (1.3) with the accessory parameters $a$ and
$q$ and the fixed parameters $\gamma=1+\theta_{0}$ and
$p=\frac{1}{4}(\theta_{\infty}+\theta_{0})$. Applying Theorem 4, there is an
isomonodromy set of pairs of accessory parameters $(a_{n},q_{n})$ such that
the DHE equation corresponding to these parameters have the same monodromy
data as the Painlevé III transcendent $y(x;0,\lambda)$ determined by the
asymptotic behavior (9.14). Moreover, the parameter $a_{n}=c_{2n}$ and the
accessory parameters $q_{n}$ are expressed in terms of the Laurent parameters
of the PIII transcendents as given in (5.31). Combining (5.16), (5.31), (9.14)
and (9.16), we obtain the asymptotic approximation of the accessory parameters
as stated in the following theorem.

######  Theorem 10.

Let $(a_{n},q_{n})$ be the sequence of accessory parameters such that the DHE
equation (1.3) corresponding to these parameters has the same monodromy data
as the Painlevé III transcendents $y(x;0,\lambda)$ determined by the
asymptotic behavior (9.14), then we have the asymptotic approximations as
$n\to\infty$

$a_{n}\sim 4\exp\left\\{-\frac{2n\pi}{\mu}+\frac{\mathop{\rm
arg}\nolimits\Gamma(i\mu)}{\mu}\right\\}\to 0^{+},$ (9.17)

and

$q(a_{n})=-\frac{\mu^{2}+1}{4}+O(a_{n}^{2}).$ (9.18)

Here the parameter $\lambda=\frac{1}{\pi}\cosh(\pi\mu)$ and $\mu>0$.

######  Remark 11.

In Theorem 10, we have derived the asymptotics of a isomonodromy sequence of
accessory parameters for DHE (1.3). The asymptotics are expressed in terms of
the parameters in the behavior (9.14) of the corresponding Painlevé III
transcendents. It would be desirable to describe the asymptotics via the
monodomy data as given in Theorem 8 and Theorem 9. However, to the best of our
knowledge the connection between the parameters in the asymptotic behavior
(9.14) for the Painlevé III transcendents and the monodomy data has not been
worked out in the literature. We expect that such a connection formula could
be derived, perhaps by using the Riemann-Hilbert method or Isomonodromy method
[12]. This, together with Theorem 10, would then give us a description of the
asymptotics of the accessory parameters for DHE via the monodomy data. We will
leave this problem to a future consideration.

## Acknowledgements

The authors are very grateful to the anonymous reviewers for their
constructive comments and suggestions. The work of Shuai-Xia Xu was supported
in part by the National Natural Science Foundation of China under grant
numbers 11571376 and 11971492. Yu-Qiu Zhao was supported in part by the
National Natural Science Foundation of China under grant numbers 11571375 and
11971489.

## References

  * [1] J.B. Amado, B.C. da Cunha and E. Pallante, On the Kerr-AdS/CFT correspondence, J. High Energy Phys., 2017 (2017), 08(2017)094.
  * [2] F.V. Andreev and A.V. Kitaev, Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis: I, Stud. Appl. Math., 145 (2020), 397–482.
  * [3] T. Anselmo, R. Nelson, B.C. da Cunha and D.G. Crowdy, Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI, Proc. Roy. Soc. Lond. A, 474 (2018), 20180080.
  * [4] T. Claeys, A.B.J. Kuijlaars and M. Vanlessen, Multi-critical unitary random matrix ensembles and the general Painlevé II equation, Ann. Math., 168 (2008), 601–641.
  * [5] P.A. Clarkson, Painlevé equations-nonlinear special functions, Orthogonal polynomials and special functions, 331–411, Lecture Notes in Math., 1883, Springer, Berlin, 2006.
  * [6] B.C. da Cunha and J.P. Cavalcante, Confluent conformal blocks and the Teukolsky master equation, to appear in Phys. Rev. D, arXiv:1906.10638.
  * [7] B.C. da Cunha and F. Novaes, Kerr scattering coefficients via isomonodromy, J. High Energy Phys., 2015 (2015), 11(2015)144.
  * [8] B.C. da Cunha and F. Novaes, Kerr-de Sitter greybody factors via isomonodromy, Phys. Rev. D, 93 (2016), 024045.
  * [9] D. Dai, S.-X. Xu and L. Zhang, On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation, J. Differential Equations, 269 (2020), 2430–2476.
  * [10] B. Dubrovin and A.A. Kapaev, A Riemann-Hilbert approach to the Heun equation, SIGMA, 14 (2018), 093.
  * [11] M. Fasondini, B. Fornberg and J.A.C. Weideman, A computational exploration of the McCoy-Tracy-Wu solutions of the third Painlevé equation, Phys. D, 363 (2018), 18–43.
  * [12] A.S. Fokas, A.R. Its, A.A. Kapaev and V.Y. Novokshenov, Painlevé transcendents: The Riemann-Hilbert approach, Math. Surv. Monog., Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
  * [13] P.J. Forrester, Log-gases and random matrices, London Mathematical Society Monographs Series, 34., Princeton University Press, Princeton, NJ, 2010.
  * [14] R. Fuchs, Sur quelques équations différentielles linéaires du second ordre, C. R. Acad. Sci. Paris, 141 (1906), 555–558.
  * [15] R. Fuchs, Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann., 63 (1907), 301–321
  * [16] V.I. Gromak, I. Laine and S. Shimomura, Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter GmbH $\&$ Co. KG, Berlin, 2002.
  * [17] M. Hortaçsu, Heun functions and their uses in physics, in Proceedings of the 13th Regional Conference on Mathematical Physics, Editors U. Camci, I. Semi, World Sci. Publ., Hackensack, NJ, 2013, 23–39.
  * [18] A.R. Its, A.B.J. Kuijlaars, J.Östensson, Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlevé transcendent, Int. Math. Res. Not., 2008 (2008), rnn017.
  * [19] M. Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci., 18 (1982), 1137–1161.
  * [20] M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D, 2 (1981), 407–448.
  * [21] O. Lisovyy, H. Nagoya and J. Roussillon, Irregular conformal blocks and connection formulae for Painlevé V functions, J. Math. Phys., 59 (2018), 091409.
  * [22] A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical conformal blocks and Painlevé VI, J. High Energy Phys., 2014 (2014), 07(2014)144.
  * [23] D. Masoero and P. Roffelsen, Poles of Painlevé IV rationals and their distribution, SIGMA, 14 (2018), 002.
  * [24] B.M. McCoy, C.A. Tracy and T.T. Wu, Painlevé functions of the third kind, J. Math. Phys., 18 (1977) 1058–1092.
  * [25] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (Eds.), NIST handbook of mathematical functions, Cambridge University Press, New York, 2010.
  * [26] A. Ronveaux (Ed.), Heun’s differential equations, Oxford University Press, Oxford-New York-Tokyo, 1995.
  * [27] S. Shimomura, Critical behaviours of the fifth Painlevé transcendents and the monodromy data, Kyushu J. Math., 71 (2017), 139–185.
  * [28] S. Shimomura, Three-parameter solutions of the PV Schlesinger-type equation near the point at infinity and the monodromy data, SIGMA, 14 (2018), 113.
  * [29] S. Yu. Slavyanov, Painlevé equations as classical analogues of Heun equations, J. Phys. A, 29 (1996), 7329–7335.
  * [30] S. Yu. Slavyanov, Isomonodromic deformations of Heun and Painlevé equations, Theor. Math. Phys., 123 (2000),744–753.
  * [31] S. Yu. Slavyanov and W. Lay, Special functions: a unified theory based on singularities, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
  * [32] S. Yu. Slavyanov, W. Lay and A. Seeger, Classification of Heun’s differential equations, Oxford University Press, Oxford, 1995.
  * [33] T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B, 13 (1976), 316–374.
  * [34] X.-B. Wu, S.-X. Xu and Y.-Q. Zhao, Gaussian unitary ensemble with boundary spectrum singularity and $\sigma$-form of Painlevé II, Stud. Appl. Math., 140 (2018), 221–251.