# Non-Hermitian Orthogonal Polynomials on a Trefoil

Ahmad B. Barhoumi  Department of Mathematics, University of Michigan, 530
Church Street, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>and  Maxim L.
Yattselev  Department of Mathematical Sciences, Indiana University-Purdue
University Indianapolis, 402 North Blackford Street, Indianapolis, IN 46202,
USA<EMAIL_ADDRESS>
###### Abstract.

We investigate asymptotic behavior of polynomials $Q_{n}(z)$ satisfying non-
Hermitian orthogonality relations

$\int_{\Delta}s^{k}Q_{n}(s)\rho(s)\mathrm{d}s=0,\quad k\in\\{0,\ldots,n-1\\},$

where $\Delta$ is a Chebotarëv (minimal capacity) contour connecting three
non-collinear points and $\rho(s)$ is a Jacobi-type weight including a
possible power-type singularity at the Chebotarëv center of $\Delta$.

###### Key words and phrases:

Non-Hermitian orthogonality, strong asymptotics, Padé approximation, Riemann-
Hilbert analysis

###### 1991 Mathematics Subject Classification:

42C05, 41A20, 41A21

The first author was partially supported by the National Science Foundation
under DMS-1812625. The research of the second author was supported in part by
a grant from the Simons Foundation, CGM-706591.

###### Contents

  1. 1 Introduction
  2. 2 Nuttall-Szegő Functions
    1. 2.1 Riemann surface
    2. 2.2 Geometric Term
    3. 2.3 Szegő Function
    4. 2.4 Adjustment Terms
    5. 2.5 Nuttall-Szegő Functions
    6. 2.6 Local Obstruction
  3. 3 Main Results
  4. 4 Proof of Propositions 1–5 and Theorem 1
    1. 4.1 Proof of Proposition 1
    2. 4.2 Proof of Proposition 2
    3. 4.3 Proof of Theorem 1
    4. 4.4 Proof of Proposition 3
    5. 4.5 Proof of Proposition 4
    6. 4.6 Proof of Proposition 5
  5. 5 Proof of Theorem 2
    1. 5.1 Initial Riemann-Hilbert Problem
    2. 5.2 Opening of Lenses
    3. 5.3 Parametrices
    4. 5.4 Approximate Local Parametrix around $a_{0}$
    5. 5.5 Small-Norm Riemann-Hilbert Problem
    6. 5.6 Proof of Theorem 2
  6. 6 Proof of Theorem 3
    1. 6.1 On Matrices $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$
    2. 6.2 Global and Local Parametrices
    3. 6.3 A Discussion
    4. 6.4 Proof of Theorem 3
  7. A Riemann-Hilbert Problem for Painlevé II
    1. A.1 Lax Pair and the Corresponding Riemann-Hilbert Problem
    2. A.2 Asymptotic Expansion
  8. B Riemann-Hilbert Problem for Painlevé XXXIV
    1. B.1 Riemann-Hilbert Problem
    2. B.2 Lax Pair
    3. B.3 Modified Riemann-Hilbert Problem
  9. C An Example of Weights in $W^{\mathsf{sing}}$
    1. C.1 Orthogonal Polynomials on a Segment
    2. C.2 Rotationally Symmetric Weights
    3. C.3 Sequence $\mathbb{N}^{\mathsf{reg}}$
    4. C.4 Sequence $\mathbb{N}^{\mathsf{sing}}$
  10. D Examples of Approximated Functions

##  1\. Introduction

Let $f(z)=\sum_{k=1}^{\infty}f_{k}z^{-k}$ be a convergent power series. The
$n$-th diagonal Padé approximant of $f(z)$ is a rational function
$[n/n]_{f}(z)=P_{n}(z)/Q_{n}(z)$, where $\deg(P_{n}),\deg(Q_{n})\leq n$ and
$Q_{n}(z)$ is not identically zero, such that the linearized error function
$R_{n}(z)$ satisfies

(1.1) $R_{n}(z):=(Q_{n}f-P_{n})(z)=\mathcal{O}\left(z^{-(n+1)}\right)$

as $z\to\infty$. One can readily check that the above condition is nothing but
a system of linear equations on the coefficients of the polynomials of
$P_{n}(z),Q_{n}(z)$ that always has a non-trivial solution. This system is not
necessarily uniquely solvable, but it is known that the rational function
$[n/n]_{f}(z)$ is indeed unique. Hereafter, we shall understand that
$Q_{n}(z)$ in (1.1) is monic and of minimal possible degree, which does make
it unique.

Our goal is to understand convergence properties of $\smash{[n/n]_{f}(z)}$. It
was shown by Herbert Stahl [23, 24, 25] that if $f(z)$ can be meromorphically
continued along any path $\overline{\mathbb{C}}\setminus A$ for a polar set
$A$ and there exists a point in $\mathbb{C}\setminus A$ with at least two
distinct continuations, then there exists a compact set $\Delta_{f}$ such that
$f(z)$ has a single-valued meromorphic continuation into the complement of
$\Delta_{f}$ (which is a branch cut for $f(z)$) and the diagonal Padé
approximants converge to this continuation in logarithmic capacity. The set
$\Delta_{f}$ is uniquely characterized as the branch cut of smallest
logarithmic capacity that is set-theoretically minimal (if $K$ is a branch cut
with the same logarithmic capacity, then $\Delta_{f}\subseteq K$).

The minimal capacity contour $\Delta_{f}$ can also be characterized from the
point of view of quadratic differentials. Assume for simplicity that
$A=\\{a_{1},\ldots,a_{m}\\}$, $m\geq 2$, is a finite set. Suppose that every
element of $A$ is a branch point of $f(z)$. Then there exist auxiliary points
$b_{i}$, $i\in\\{1,\ldots,m-2\\}$, sometimes called Chebotarëv centers111In a
somewhat different language, Chebotarëv posed a problem of finding a connected
set of minimal logarithmic capacity containing a given finite set of points;
descriptions of this set were independently given by Grötzsch [14] and
Lavrentiev [19, 20]., which are not necessarily distinct nor disjoint from the
elements of $A$, such that the set $\Delta_{f}$ consists of the critical
trajectories of a rational quadratic differential

$\frac{\prod_{i=1}^{m-2}(z-b_{i})}{\prod_{i=1}^{m}(z-a_{i})}\mathrm{d}z^{2}.$

Suppose that all the points $b_{i}$ are disjoint from $A$ and that each
$b_{i}$ appears either once or an even number of times and in the latter case
does not belong to $\Delta_{f}$ (a generic situation). Assume further that
$f(z)$ has either logarithmic or power branching at each $a_{i}$ (i.e.,
behaves like $\log(z-a_{i})$ or $(z-a_{i})^{\alpha_{i}}$, $\alpha_{i}>-1$).
Then the strong asymptotics of the corresponding diagonal Padé approximants
was investigated by Aptekarev and the second author in [2]. Our overarching
goal is to remove all the assumptions on contours $\Delta_{f}$ corresponding
to finite sets $A$. As will become clear later, the main difficulty lies in
the local analysis of the polynomials $Q_{n}(z)$ around the auxiliary points
$b_{i}$. The first step in this direction was taken by the authors in [4]
where we considered the case $m=4$ and $b_{1}=b_{2}\in\Delta_{f}\setminus A$.
Here, we consider that case $m=4$ and $b_{1}=b_{2}\in A$.

$a_{0}$$a_{1}$$a_{3}$$a_{2}$$\pi(\boldsymbol{\alpha})$$\pi(\boldsymbol{\beta})$

Figure 1. Contour $\Delta$.

Let us now change the notation slightly. Given three distinct non-collinear
points $a_{1},a_{2},a_{3}$, denote by $a_{0}$ their Chebotarëv center. That
is, there are three disjoint, except for $a_{0}$, analytic arcs, say
$\Delta_{1},\Delta_{2},\Delta_{3}$ ($\Delta_{i}$ has endpoints $a_{i}$ and
$a_{0}$) such that

(1.2)
$\frac{z(t)-a_{0}}{(z(t)-a_{1})(z(t)-a_{2})(z(t)-a_{3})}\big{(}z^{\prime}(t)\big{)}^{2}<0$

for any smooth parametrization $z(t)$ of any of the arcs $\Delta_{i}$. Let
$\Delta:=\Delta_{1}\cup\Delta_{2}\cup\Delta_{3}$ (in our preceding notation
$A=\\{a_{0},a_{1},a_{2},a_{3}\\}$, $b_{1}=b_{2}=a_{0}$, and
$\Delta_{f}=\Delta$), see Figure 1. Examples of functions $f(z)$ that lead to
such minimal capacity contours include

(1.3) $\prod(z-a_{i})^{\alpha_{i}}\quad\text{and}\quad\sum
c_{i}\log(z-a_{i}),$

where $\alpha_{i}\not\in\mathbb{Z}$ while
$\sum_{i=0}^{4}\alpha_{i}\in\mathbb{Z}$ and $c_{i}\neq 0$ while
$\sum_{i=0}^{4}c_{i}=0$. Let us point out that if $f(z)$ was given by either
of the expressions above, but with $\alpha_{0}=c_{0}=0$, then the asymptotics
of the diagonal Padé approximants to $f(z)$ was obtained in [2], see also
[21]. Moreover, asymptotics of the approximants for the case
$\alpha_{0}=\alpha_{1}=\alpha_{2}=\alpha_{3}=-1/2$ is contained [27], see also
[3, 26]. The reader might want to consult the Appendix D to see how these
functions relate to the definition below.

In this work, we shall consider the following class of functions. Orient each
arc $\Delta_{i}$ towards $a_{0}$. Assume that the points $a_{1},a_{2},a_{3}$
are labeled counter-clockwise around $a_{0}$.

######  Definition.

We are interested in functions $f(z)$ of the form

(1.4)
$f(z)=\frac{1}{2\pi\mathrm{i}}\int_{\Delta}\frac{\rho(s)\mathrm{d}s}{s-z},$

where there exist exponents $\alpha_{i}>-1$, $i\in\\{0,1,2,3\\}$, and branches
of $(z-a_{i})^{\alpha_{i}}$ holomorphic across $\Delta\setminus\\{a_{i}\\}$
for which the restriction $\rho_{i}(s)$ of $\rho(s)$ to
$\Delta_{i}^{\circ}:=\Delta_{i}\setminus\\{a_{0},a_{i}\\}$ is such that
$\rho_{i}(s)(s-a_{0})^{-\alpha_{0}}(s-a_{i})^{-\alpha_{i}}$ extends to a
holomorphic and non-vanishing function in some neighborhood of $\Delta_{i}$.

Notice that $\rho(s)$ is not defined at $a_{0}$ even if $\alpha_{0}=0$.
However, in the latter case the values $\rho_{i}(a_{0})$ are well-defined. It
was assumed in [2] that $\rho_{1}(z)+\rho_{2}(z)+\rho_{3}(z)\equiv 0$ in some
neighborhood of $a_{0}$ (no branching assumption). No such supposition is made
here, neither in some neighborhood of $a_{0}$ nor at $a_{0}$ itself. The main
advantage of the class of functions introduced in (1.4) is that the
denominator polynomials $Q_{n}(z)$ can be equivalently characterized as non-
Hermitian orthogonal polynomials satisfying

(1.5) $\int_{\Delta}s^{k}Q_{n}(s)\rho(s)\mathrm{d}s=0,\quad
k\in\\{0,\ldots,n-1\\}.$

This paper is organized as follows. The next section is devoted to the
description of the main term of the asymptotics of the polynomials $Q_{n}(z)$.
The functions constructed in that section are known in integrable systems
literature as Baker-Akhiezer functions and in the literature on non-Hermitian
orthogonal polynomials are sometimes called Nuttall-Szegő functions. The
propositions stated in the next section are proven in Section 4. Our main
results on the asymptotics of the polynomials $Q_{n}(z)$ are stated in Section
3. Their proofs are given in Sections 5 and 6. As it happens, local behavior
of the polynomials $Q_{n}(z)$ around $a_{0}$ is described by certain special
functions that come from a $2\times 2$ matrix function solving the so-called
Painvlevé XXXIV Riemann-Hilbert problem with Stokes parameters that depend on
the weight $\rho(s)$ in a transcendental way. This connection is described in
Appendices A and B. Appendix C illustrates by an example that on the
rotationally symmetric contour $\Delta$ asymptotics of $Q_{n}(z)$ can be very
different for certain classes of weights from the rest of the cases.

##  2\. Nuttall-Szegő Functions

Similarly to orthogonal polynomials on an interval, the asymptotics of the
polynomials $Q_{n}(z)$ is described by the term that captures the geometric
rate of their growth, see (2.4), and a Szegő function of the weight $\rho(s)$,
see (2.11). Both of these functions naturally live on a genus one Riemann
surface associated with the contour $\Delta$, see (2.1). However, the nature
of the theory of functions on a Riemann surface leads to the introduction of
the third term, in fact, a normal family of functions that depends on $n$,
which are essentially ratios of the Riemann theta functions, see (2.17) (the
necessity of these terms was already demonstrated by Akhiezer [1]).

$a_{0}$$a_{0}$$a_{0}$$a_{0}$$a_{1}$$a_{2}$$a_{2}$$a_{3}$$a_{3}$$\boldsymbol{\beta}$$\boldsymbol{\beta}$$\boldsymbol{\alpha}$$\boldsymbol{\alpha}$$\mathfrak{S}^{(0)}$$\mathfrak{S}^{(1)}$$\boldsymbol{\Delta}_{1}$$\boldsymbol{\Delta}_{1}$$\boldsymbol{\Delta}_{2}$$\boldsymbol{\Delta}_{2}$$\boldsymbol{\Delta}_{3}$$\boldsymbol{\Delta}_{3}$

Figure 2. Surface $\mathfrak{S}$.

###  2.1. Riemann surface

Let $\mathfrak{S}$ be a genus one two-sheeted Riemann surface defined by

(2.1)
$\mathfrak{S}:=\left\\{{\boldsymbol{z}}=(z,w):~{}w^{2}=(z-a_{0})(z-a_{1})(z-a_{2})(z-a_{3})\right\\}.$

We realize $\mathfrak{S}$ as a ramified cover of $\overline{\mathbb{C}}$
constructed in the following manner. Two copies of the extended complex plane
are cut along each arc $\Delta_{i}^{\circ}$. These copies are glued together
along the cuts in such a manner that the right (resp. left) side of the arc
$\Delta_{i}^{\circ}$ belonging to one copy is joined with the left (resp.
right) side of the same arc $\Delta_{i}^{\circ}$ only belonging to the other
copy. Let

$\pi,w:\mathfrak{S}\to\overline{\mathbb{C}},\quad\pi({\boldsymbol{z}})\mapsto
z\quad\text{and}\quad w({\boldsymbol{z}})\mapsto w.$

We call $\pi$ the canonical projection and label the sheets
$\mathfrak{S}^{(0)}$ and $\mathfrak{S}^{(1)}$ (cut copies of the extended
complex plane) so that $w({\boldsymbol{z}})=w$ behaves like $z^{2}$ as
${\boldsymbol{z}}\to\infty^{(0)}$. In our definition, the sheets are
topologically closed and their intersection is equal to $\boldsymbol{\Delta}$,
where

$\boldsymbol{\Delta}:=\pi^{-1}(\Delta),\quad\boldsymbol{\Delta}_{i}:=\pi^{-1}(\Delta_{i}),\quad\text{and}\quad\boldsymbol{A}:=\\{\boldsymbol{a}_{0},\boldsymbol{a}_{1},\boldsymbol{a}_{2},\boldsymbol{a}_{3}\\},\quad\pi(\boldsymbol{a}_{l})=a_{l},$

$i\in\\{1,2,3\\}$ and $l\in\\{0,1,2,3\\}$. Thus, $\boldsymbol{A}$ is the set
of the ramification points of $\mathfrak{S}$ and $\boldsymbol{\Delta}$ is a
union of three simple cycles that all intersect each other at
$\boldsymbol{a}_{0}$. We orient each $\boldsymbol{\Delta}_{i}$ so that
$\mathfrak{S}^{(0)}$ remains on the left when $\boldsymbol{\Delta}_{i}$ is
traversed in the positive direction. We write

$z^{(k)}:=\pi^{-1}(z)\cap\mathfrak{S}^{(k)},\quad
z\in\overline{\mathbb{C}}\setminus\Delta,$

where $k\in\\{0,1\\}$, and use bold lower case letters such as
${\boldsymbol{z}},{\boldsymbol{t}},{\boldsymbol{s}}$ to indicate generic
points on $\mathfrak{S}$ with canonical projections $z,t,s$. We designate the
symbol $\cdot^{*}$ to stand for the conformal involution that sends $z^{(k)}$
into $z^{(1-k)}$, $k\in\\{0,1\\}$ (we then extend this notion to
$\boldsymbol{\Delta}$ by continuity). Since $\mathfrak{S}$ has genus $1$, we
need to select a homology basis. We choose cycles
$\boldsymbol{\alpha},\boldsymbol{\beta}$ to be involution-symmetric, i.e.,
${\boldsymbol{s}}\in\boldsymbol{\alpha}$ if and only if
${\boldsymbol{s}}^{*}\in\boldsymbol{\alpha}$, and such that
$\pi(\boldsymbol{\alpha}),\pi(\boldsymbol{\beta})$ are smooth Jordan arcs
joining $a_{1},a_{2}$ and $a_{1},a_{3}$ respectively, while
$\boldsymbol{\alpha}$ is oriented towards $\boldsymbol{a}_{1}$ and
$\boldsymbol{\beta}$ away from $\boldsymbol{a}_{1}$ in $\mathfrak{S}^{(0)}$,
see Figures 1 and 2. In what follows it will be sometimes convenient to set

$\mathfrak{S}_{\boldsymbol{\alpha}}:=\mathfrak{S}\setminus\boldsymbol{\alpha}\quad\text{and}\quad\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}:=\mathfrak{S}\setminus(\boldsymbol{\alpha}\cup\boldsymbol{\beta}).$

Finally, given a function $F({\boldsymbol{z}})$ defined on
$\mathfrak{S}\setminus\boldsymbol{\Delta}$, or on its subset, we shall set
$F(z):=F(z^{(0)})$ and $F^{*}(z):=F(z^{(1)})$. That is, with a slight abuse of
notation we use the same letter for a function on the top sheet as well as its
pull-back to the complex plane while we use the involution symbol ∗ to mark
the pull-back to the complex plane from the bottom sheet. In particular,

(2.2) $w(z)=\sqrt{(z-a_{0})(z-a_{1})(z-a_{2})(z-a_{3})},\quad
z\in\mathbb{C}\setminus\Delta,$

is the branch such that $w(z)=z^{2}+\mathcal{O}(z)$ as $z\to\infty$ and
$w^{*}(z)=-w(z)$.

As it happens, the asymptotics of the polynomials $Q_{n}(z)$ as well as of the
linearized error functions $R_{n}(z)$, see (1.1), is described by the solution
of the following boundary value problem on $\boldsymbol{\Delta}$.

######  Boundary Value Problem: BVP-$\Psi$.

Given $n\in\mathbb{N}$ and a function $\rho(s)$ on $\Delta$ as described after
(1.4), find a function $\Psi({\boldsymbol{z}})$ that is holomorphic in
$\mathfrak{S}\setminus\big{(}\boldsymbol{\Delta}\cup\big{\\{}\infty^{(0)}\big{\\}}\big{)}$,
has a pole of order $n$ at $\infty^{(0)}$ and a zero of order at least $n-1$
at $\infty^{(1)}$, and whose traces on $\boldsymbol{\Delta}$ are continuous
away from $\boldsymbol{A}$ and satisfy

(2.3)
$\left\\{\begin{array}[]{l}\Psi_{+}({\boldsymbol{s}})=\Psi_{-}({\boldsymbol{s}})/(\rho
w_{+})(s),\quad{\boldsymbol{s}}\in\boldsymbol{\Delta},\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\big{|}\Psi(z)\big{|}=\mathcal{O}\left(|z-a_{l}|^{-(2\alpha_{l}+1)/4}\right)\quad\text{as}\quad
z\to a_{l},\;\;l\in\\{0,1,2,3\\}.\end{array}\right.$

The solution of this boundary value problem is given in Theorem 1 further
below. It is rather explicit and constructed as a product of three terms that
are introduced in the following three subsections.

###  2.2. Geometric Term

Let

(2.4)
$\Phi({\boldsymbol{z}}):=\exp\left\\{\int_{\boldsymbol{a}_{0}}^{\boldsymbol{z}}G\right\\},\quad
G({\boldsymbol{z}}):=\frac{(z-a_{0})\mathrm{d}z}{w({\boldsymbol{z}})},$

for
${\boldsymbol{z}}\in\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$,
where the path of integration lies entirely in
$\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$ and
$G({\boldsymbol{z}})$ is a meromorphic differential on $\mathfrak{S}$ having
two simple poles at $\infty^{(1)}$ and $\infty^{(0)}$ with respective residues
$1$ and $-1$, whose period over any cycle on $\mathfrak{S}$ is purely
imaginary (the latter claim follows from (1.2)). The function
$\Phi({\boldsymbol{z}})$ is holomorphic and non-vanishing on
$\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$ except for a simple
pole at $\infty^{(0)}$ and a simple zero at $\infty^{(1)}$. Define (real)
constants

(2.5)
$\omega:=-\frac{1}{2\pi\mathrm{i}}\oint_{\boldsymbol{\beta}}G\quad\text{and}\quad\tau:=\frac{1}{2\pi\mathrm{i}}\oint_{\boldsymbol{\alpha}}G.$

One can readily check that $\Phi({\boldsymbol{z}})$ possesses continuous
traces on both sides of each cycle of the canonical basis (in fact, it extends
to a multiplicatively multi-valued function on the entire surface) that
satisfy

(2.6)
$\Phi_{+}({\boldsymbol{s}})=\Phi_{-}({\boldsymbol{s}})\left\\{\begin{array}[]{ll}e^{2\pi\mathrm{i}\omega},&{\boldsymbol{s}}\in\boldsymbol{\alpha}\setminus\\{\boldsymbol{a}_{1}\\},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
e^{2\pi\mathrm{i}\tau},&{\boldsymbol{s}}\in\boldsymbol{\beta}\setminus\\{\boldsymbol{a}_{1}\\}.\end{array}\right.$

Since $w(z)=-w^{*}(z)$, it follows from (2.4) that $\Phi(z)\Phi^{*}(z)\equiv
1$. Moreover, since the constants in (2.5) are real, it also holds that
$\log|\Phi({\boldsymbol{z}})|$ is harmonic on
$\mathfrak{S}\setminus\\{\infty^{(0)},\infty^{(1)}\\}$. Furthermore, since
$\Delta$ is the set of critical trajectories of the quadratic differential
$-(z-a_{0})^{2}w^{-2}(z)\mathrm{d}z^{2}$ and the domain
$\overline{\mathbb{C}}\setminus\Delta$ contains a single critical point of
this differential, namely, double pole at infinity, it follows from Basic
Structure Theorem [16, Theorem 3.5] that
$\overline{\mathbb{C}}\setminus\Delta$ is a circle domain for this
differential, see [16, Definition 3.9]. In particular, it must be true that

(2.7) $-\log|\Phi^{*}(z)|=\log|\Phi(z)|>0,\quad
z\in\overline{\mathbb{C}}\setminus\Delta.$

In particular, this means that $\log|\Phi(s)|\equiv 0$, $s\in\Delta$, and that
$\log|\Phi(z)|$ is the Green function for
$\overline{\mathbb{C}}\setminus\Delta$ with pole at infinity.

###  2.3. Szegő Function

Since $\mathfrak{S}$ has genus one, it has one (up to a constant factor)
holomorphic differential. Since $\mathfrak{S}$ is a two-sheeted surface, it is
a standard fact of the theory of Riemann surfaces that

(2.8)
$\Omega({\boldsymbol{z}}):=\left(\oint_{\boldsymbol{\alpha}}\frac{\mathrm{d}s}{w({\boldsymbol{s}})}\right)^{-1}\frac{\mathrm{d}z}{w({\boldsymbol{z}})},\quad\mathsf{B}:=\oint_{\boldsymbol{\beta}}\Omega,$

is the holomorphic differential on $\mathfrak{S}$ normalized to have unit
period on the $\boldsymbol{\alpha}$ cycle of the homology basis (it was shown
by Riemann that the constant $\mathsf{B}$ has positive imaginary part). In
particular, the function

(2.9)
$a({\boldsymbol{z}}):=\int_{\boldsymbol{a}_{0}}^{\boldsymbol{z}}\Omega,\quad{\boldsymbol{z}}\in\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}},$

where the path of integration lies entirely in
$\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$, extends to additively
multi-valued analytic function on the entire surface $\mathfrak{S}$. Denote
further by $\Omega_{{\boldsymbol{z}},{\boldsymbol{z}}^{*}}({\boldsymbol{s}})$
the meromorphic differential with two simple poles at ${\boldsymbol{z}}$ and
${\boldsymbol{z}}^{*}$ with respective residues $1$ and $-1$ normalized to
have a zero period on the $\boldsymbol{\alpha}$-cycle. When ${\boldsymbol{z}}$
does not lie on top of the point at infinity, it can be readily verified that

(2.10)
$\Omega_{{\boldsymbol{z}},{\boldsymbol{z}}^{*}}({\boldsymbol{s}})=\dfrac{w(\boldsymbol{z})}{s-z}\dfrac{\mathrm{d}s}{w({\boldsymbol{s}})}-\left(\oint_{\boldsymbol{\alpha}}\dfrac{w({\boldsymbol{z}})}{t-z}\dfrac{\mathrm{d}t}{w({\boldsymbol{t}})}\right)\Omega({\boldsymbol{s}}).$

The third kind differential
$\Omega_{{\boldsymbol{z}},{\boldsymbol{z}}^{*}}({\boldsymbol{s}})$ can be
thought of as a discontinuous Cauchy kernel on $\mathfrak{S}$, see [28]. The
following proposition elaborates this point.

######  Proposition 1.

For a function $\rho(s)$ as described after (1.4) fix a branch of
$\log(\rho_{i}w_{+})(s)$ that is continuous on $\Delta_{i}^{\circ}$,
$i\in\\{1,2,3\\}$. Put

(2.11)
$S_{\rho}({\boldsymbol{z}}):=\exp\left\\{-\dfrac{1}{4\pi\mathrm{i}}\oint_{\boldsymbol{\Delta}}\log(\rho
w_{+})(s)\Omega_{{\boldsymbol{z}},{\boldsymbol{z}}^{*}}({\boldsymbol{s}})\right\\}.$

Then $S_{\rho}({\boldsymbol{z}})$ is a holomorphic and non-vanishing function
in $\mathfrak{S}_{\boldsymbol{\alpha}}\setminus\boldsymbol{\Delta}$ with
continuous traces on
$(\boldsymbol{\Delta}\cup\boldsymbol{\alpha})\setminus\boldsymbol{A}$ that
satisfy

(2.12)
$S_{\rho+}({\boldsymbol{s}})=S_{\rho-}({\boldsymbol{s}})\left\\{\begin{array}[]{rl}\displaystyle\exp\big{\\{}2\pi\mathrm{i}c_{\rho}\big{\\}},&{\boldsymbol{s}}\in\boldsymbol{\alpha}\setminus\boldsymbol{A},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ 1/(\rho
w_{+})(s),&{\boldsymbol{s}}\in\boldsymbol{\Delta}\setminus\boldsymbol{A},\end{array}\right.$

where the constant $c_{\rho}$ is given by

(2.13) $c_{\rho}:=\frac{1}{2\pi\mathrm{i}}\oint_{\boldsymbol{\Delta}}\log(\rho
w_{+})\Omega.$

It also holds that
$S_{\rho}({\boldsymbol{z}})S_{\rho}({\boldsymbol{z}}^{*})\equiv 1$,
${\boldsymbol{z}}\in\mathfrak{S}_{\boldsymbol{\alpha}}\setminus\boldsymbol{\Delta}$,
and

(2.14)
$\big{|}S_{\rho}(z)\big{|}\sim|z-a_{l}|^{-(2\alpha_{l}+1)/4}\quad\text{as}\quad
z\to a_{l},\quad l\in\\{0,1,2,3\\}.$

We call $S_{\rho}({\boldsymbol{z}})$ the Szegő function of the weight
$\rho(s)$. The Cauchy kernel
$\Omega_{{\boldsymbol{z}},{\boldsymbol{z}}^{*}}({\boldsymbol{s}})$ is called
discontinuous due to the presence of the jump of $S_{\rho}({\boldsymbol{z}})$
across $\boldsymbol{\alpha}$. We prove Proposition 1 in Section 4.1.

###  2.4. Adjustment Terms

Due to the presence of the jumps of both $\Phi({\boldsymbol{z}})$ and
$S_{\rho}({\boldsymbol{z}})$ across the cycles of the homology basis, we need
to introduce an adjustment term that will cancel out these jumps. To this end,
let

$\theta(u):=\sum_{n\in\mathbb{Z}}\exp\big{\\{}\pi\mathrm{i}\mathsf{B}n^{2}+2\pi\mathrm{i}nu\big{\\}},\quad
u\in\mathbb{C},$

be the Riemann theta function associated with $\mathsf{B}$. Further, let
$\mathsf{Jac}(\mathfrak{S}):=\mathbb{C}/\\{\mathbb{Z}+\mathsf{B}\mathbb{Z}\\}$
be the Jacobi variety of $\mathfrak{S}$, where $\mathsf{B}$ is given by (2.8).
We shall represent elements of $\mathsf{Jac}(\mathfrak{S})$ as equivalence
classes $[s\,]=\\{s+l+\mathsf{B}m:l,m\in\mathbb{Z}\\}$, where
$s\in\mathbb{C}$. It is known that Abel’s map

(2.15)
${\boldsymbol{z}}\in\mathfrak{S}\mapsto\mathfrak{a}({\boldsymbol{z}})\in\mathsf{Jac}(\mathfrak{S}),\quad\mathfrak{a}({\boldsymbol{z}}):=\left[\int_{\boldsymbol{a}_{0}}^{\boldsymbol{z}}\Omega\right],$

is a bijection (clearly, it holds that
$[a({\boldsymbol{z}})]=\mathfrak{a}({\boldsymbol{z}})$ for
${\boldsymbol{z}}\in\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$,
see (2.9)). Inverting Abel’s map (2.15) is known as the Jacobi inversion
problem, see [7, Section III.6]. Since $\mathfrak{S}$ has genus one, every
Jacobi inversion problem has a unique solution.

######  Proposition 2.

Let $\omega,\tau$ and $c_{\rho}$ be as in (2.5) and (2.13), respectively. Set

(2.16) $x_{n}:=n\omega-\lfloor n\omega\rfloor,\quad y_{n}:=n\tau-\lfloor
n\tau\rfloor,\quad\text{and}\quad\mathsf{A}:=(1+\omega)/2+\mathsf{B}(1+\tau)/2,$

where $\lfloor\cdot\rfloor$ is equal to the largest integer smaller or equal
to its argument. Define

(2.17)
$T_{n}({\boldsymbol{z}}):=\exp\big{\\{}-2\pi\mathrm{i}y_{n}a({\boldsymbol{z}})\big{\\}}\dfrac{\theta\big{(}a({\boldsymbol{z}})-c_{\rho}-x_{n}-\mathsf{B}y_{n}+\mathsf{A}\big{)}}{\theta\big{(}a({\boldsymbol{z}})+\mathsf{A}\big{)}},\quad{\boldsymbol{z}}\in\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}.$

Each function $T_{n}({\boldsymbol{z}})$ is meromorphic in the domain of its
definition with continuous traces on
$(\boldsymbol{\alpha}\cup\boldsymbol{\beta})\setminus\\{\boldsymbol{a}_{1}\\}$
that satisfy

(2.18)
$T_{n+}({\boldsymbol{s}})=T_{n-}({\boldsymbol{s}})\left\\{\begin{array}[]{rl}\displaystyle\exp\big{\\{}-2\pi\mathrm{i}\big{(}c_{\rho}+n\omega\big{)}\big{\\}},&{\boldsymbol{s}}\in\boldsymbol{\alpha}\setminus\\{\boldsymbol{a}_{1}\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\exp\big{\\{}-2\pi\mathrm{i}n\tau\big{\\}},&{\boldsymbol{s}}\in\boldsymbol{\beta}\setminus\\{\boldsymbol{a}_{1}\\}.\end{array}\right.$

In fact, $T_{n}({\boldsymbol{z}})$ can be holomorphically continued across
each of these cycles. It has precisely one pole, namely $\infty^{(1)}$, and
exactly one zero, namely ${\boldsymbol{z}}_{n}$, both simple (the function
will become analytic and non-vanishing if
${\boldsymbol{z}}_{n}=\infty^{(1)}$), where
${\boldsymbol{z}}_{n}={\boldsymbol{z}}_{n}(\rho)\in\mathfrak{S}$ is the unique
solution of the Jacobi inversion problem

(2.19)
$\mathfrak{a}({\boldsymbol{z}}_{n})=\big{[}c_{\rho}+(n-1/2)(\omega+\mathsf{B}\tau)\big{]}.$

We prove Proposition 2 in Section 4.2.

###  2.5. Nuttall-Szegő Functions

Now we are ready to described the solutions of BVP-$\Psi$.

######  Theorem 1.

Fix $n\in\mathbb{N}$. Let $c_{\rho}$ be given by (2.13) and
${\boldsymbol{z}}_{n}$ be the solution of (2.19). If
${\boldsymbol{z}}_{n}=\infty^{(0)}$, then the boundary value problem
BVP-$\Psi$ does not have a solution. Otherwise, any solution of BVP-$\Psi$ is
of the form $c\Psi_{n}({\boldsymbol{z}})$ for a non-zero constant $c$, where

(2.20)
$\Psi_{n}({\boldsymbol{z}}):=\big{(}\Phi^{n}S_{\rho}T_{n}\big{)}({\boldsymbol{z}}),\quad{\boldsymbol{z}}\in\mathfrak{S}\setminus\boldsymbol{\Delta},$

and the functions $\Phi({\boldsymbol{z}})$, $S_{\rho}({\boldsymbol{z}})$, and
$T_{n}({\boldsymbol{z}})$ are given by (2.4), (2.11), and (2.17),
respectively.

We prove Theorem 1 in Section 4.3. Notice that in addition to the zero of
order $n-1$ at $\infty^{(1)}$, $\Psi_{n}({\boldsymbol{z}})$ also has a simple
zero at ${\boldsymbol{z}}_{n}$ (if ${\boldsymbol{z}}_{n}$ belongs to
$\boldsymbol{\Delta}\setminus\boldsymbol{A}$ the simplicity of the zero is
understood as the simplicity of the zero of an analytic continuation of
$\Psi_{n}({\boldsymbol{z}})$ to a neighborhood of ${\boldsymbol{z}}_{n}$; of
course, if ${\boldsymbol{z}}_{n}=\infty^{(1)}$, then the zero at
$\infty^{(1)}$ has order $n$) and the asymptotics of the behavior of
$\Psi_{n}({\boldsymbol{z}})$ around the points in $\boldsymbol{A}$ can be
improved to

(2.21)
$\big{|}\Psi_{n}(z)\big{|}\sim|z-a_{l}|^{m_{n}(\boldsymbol{a}_{l})/2-\alpha_{l}/2-1/4}\quad\text{as}\quad
z\to a_{l},$

$l\in\\{0,1,2,3\\}$, where $m_{n}(\boldsymbol{a})=1$ if
${\boldsymbol{z}}_{n}=\boldsymbol{a}$ and $m_{n}(\boldsymbol{a})=0$ otherwise.

Of course, the function $\Psi_{n}({\boldsymbol{z}})$ is defined even when
${\boldsymbol{z}}_{n}=\infty^{(0)}$. In this case it simply has a pole of
order $n-1$ at $\infty^{(0)}$ and therefore does not solve BVP-$\Psi$. It
readily follows from (2.19) that if
${\boldsymbol{z}}_{n}={\boldsymbol{z}}_{m}$ for some distinct indices $n$ and
$m$, then $[0]=[(n-m)(\omega+\mathsf{B}\tau)]$, which means that $\omega$ and
$\tau$ are rational numbers with the denominators that divide $n-m$. Thus,
either $\omega$ and $\tau$ are rational numbers, in which case
$\\{{\boldsymbol{z}}_{n}\\}$ is a periodic sequence, or
${\boldsymbol{z}}_{n}=\infty^{(0)}$ for at most one index $n$. Unfortunately,
for our asymptotic analysis we need to exclude not only indices for which
${\boldsymbol{z}}_{n}=\infty^{(0)}$, but also those for which
${\boldsymbol{z}}_{n}$ is close $\infty^{(0)}$. To this end, given
$\varepsilon>0$ such that $\max_{s\in\Delta}|s|<1/\varepsilon$, we let

(2.22)
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}:=\left\\{n\in\mathbb{N}:~{}{\boldsymbol{z}}_{n}\not\in\boldsymbol{U}_{\varepsilon}^{(0)}\right\\},$

where $\boldsymbol{U}_{\varepsilon}^{(k)}$ is a neighborhood of $\infty^{(k)}$
with natural projection equal to $\\{|z|>1/\varepsilon\\}$.

######  Proposition 3.

If $\omega$ and $\tau$ are rational numbers with the smallest common
denominator $d$, then there exists $\varepsilon_{0}>0$ such that either
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}=\mathbb{N}$ or
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}=\mathbb{N}\setminus(n_{0}+d\mathbb{N})$
for some $n_{0}\in\\{0,\ldots,d-1\\}$ and all $\varepsilon<\varepsilon_{0}$.
In the latter case it necessarily holds that $c_{\rho}=(p+\mathsf{B}q)/d$ for
some integers $p$ and $q$. Moreover, it holds that $d\geq 3$ and $d=3$ only if
$\Delta$ is the image under a linear map of

(2.23) $\Delta_{\mathsf{sym}}=\big{\\{}s:s^{3}\in[-1,0]\big{\\}}.$

On the other hand, if at least one of the numbers $\omega$ and $\tau$ is
irrational, then for any $N\in\mathbb{N}$ there exists $\varepsilon_{N}>0$
such that

(2.24) $n\notin\mathbb{N}_{\varepsilon}^{\mathsf{reg}}\quad\Rightarrow\quad
n+1,\ldots,n+N\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$

for any $0<\varepsilon\leq\varepsilon_{N}$. Moreover, there exists
$\delta=\delta(\varepsilon)$ such that

(2.25)
$n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}\quad\Rightarrow\quad{\boldsymbol{z}}_{n-1}\not\in\boldsymbol{U}_{\delta}^{(1)}.$

Furthermore, it is always true that ${\boldsymbol{z}}_{n}=\infty^{(0)}$ if and
only if ${\boldsymbol{z}}_{n-1}=\infty^{(1)}$.

We prove Proposition 3 in Section 4.4.

The reason we need to go from excluding indices for which
${\boldsymbol{z}}_{n}$ is equal to $\infty^{(0)}$ to sequences
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ is that we need to control
asymptotic behavior of $\Psi_{n}({\boldsymbol{z}})$. More precisely, for our
analysis it is important to know that functions $T_{n}({\boldsymbol{z}})$ and
their reciprocals form normal families in certain subdomains of
$\mathfrak{S}$.

######  Proposition 4.

Let $R>\max_{s\in\Delta}|s|$. There exists a constant $M_{R}>0$ such that

(2.26) $\left\\{\begin{array}[]{ll}|T_{n}({\boldsymbol{z}})|\leq
M_{R},&{\boldsymbol{z}}\not\in\boldsymbol{U}_{1/R}^{(1)},\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ |T_{n}({\boldsymbol{z}})|\leq
M_{R}\frac{|z|}{R},&{\boldsymbol{z}}\in\boldsymbol{U}_{1/R}^{(1)},\end{array}\right.$

for all $n\in\mathbb{N}$. Furthermore, let
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ and $\delta(\varepsilon)$ be as in
Proposition 3. Then there exists a constant $M_{\varepsilon}>0$ such that

(2.27) $\left\\{\begin{array}[]{ll}|T_{n}^{-1}({\boldsymbol{z}})|\leq
M_{\varepsilon},&{\boldsymbol{z}}\in\boldsymbol{U}_{\varepsilon/2}^{(0)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ |T_{n-1}^{-1}({\boldsymbol{z}})|\leq
M_{\varepsilon}|z|^{-1},&{\boldsymbol{z}}\in\boldsymbol{U}_{\delta(\varepsilon)/2}^{(1)},\end{array}\right.$

for $n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$.

We prove Proposition 4 in Section 4.5.

###  2.6. Local Obstruction

To prove our main results on asymptotic behavior of orthogonal polynomials we
utilize matrix Riemann-Hilbert analysis. As typical for such an approach, we
need to construct a local parametrix around $a_{0}$ that models the behavior
of the polynomials and their functions of the second kind there. In our case,
this is done with the help of the matrix that solves Riemann-Hilbert problems
that characterizes solutions of Painlevé XXXIV equation with the Painlevé
variable equal to zero. It is known that such a matrix does not exist when the
corresponding Painlevé function has a pole at zero. In this case we need to
modify our construction and this modification requires placing an additional
requirement on the sequence of allowable indices. More precisely, let us write

(2.28)
$T_{n}(z):=t_{0}^{(n)}+t_{1}^{(n)}(z-a_{0})^{1/2}+t_{2}^{(n)}(z-a_{0})+\cdots$

for $z$ close to $a_{0}$ lying in the sector between $\Delta_{1}$ and
$\Delta_{3}$ for some fixed determination of $(z-a_{0})^{1/2}$ that is
analytic in this sector. Let $h_{1}$ be the constant defined further below in
(6.2) and (6.4) (it is the second coefficient of the Puiseux series at $a_{0}$
of a certain function that depends on the geometry via $w(z)$ and the weight
$\rho(s)$ via $S_{\rho}(z)$ and the function $r(z)$ defined in (5.12)). Define

(2.29)
$\boldsymbol{T}_{n}:=\left(\begin{matrix}2t_{2}^{(n)}-h_{1}t_{1}^{(n)}&2t_{2}^{(n-1)}-h_{1}t_{1}^{(n-1)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ t_{0}^{(n)}&t_{0}^{(n-1)}\end{matrix}\right).$

We need to require that the determinants of these matrices are separated away
from zero.

######  Proposition 5.

For each $\varepsilon>0$ small enough there exists
$\boldsymbol{V}_{\varepsilon}$, a neighborhood of $\boldsymbol{a}_{0}$ that
can possibly be empty, such that if
${\boldsymbol{z}}_{n}\notin\boldsymbol{V}_{\varepsilon}$, then either $n$ or
$n+1$ belongs to $\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$, where

$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}:=\Big{\\{}n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}:~{}|\mathrm{det}(\boldsymbol{T}_{n})|\geq\varepsilon\Big{\\}}.$

We prove Proposition 5 in Section 4.6. Let us note that there are no
particular reasons to couple the constants in the definition of
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$ and
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ except esthetic ones. That is, we
could have defined
$\mathbb{N}_{\varepsilon_{1},\varepsilon_{2}}^{\mathsf{sing}}\subseteq\mathbb{N}_{\varepsilon_{1}}^{\mathsf{reg}}$.

Proposition 5 has the following implications. When at least one of the
constants $\omega$ and $\tau$ is irrational, given a natural number $N$,
$\varepsilon$ can be taken small enough so that
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ contains a consecutive block of $N$
integers after each missing index by Proposition 3. The same arguments as in
the proof of that proposition can be employed to argue that
$\boldsymbol{z}_{n}\in\boldsymbol{V}_{\varepsilon}$ for at most one index in
this block. Thus, according to Proposition 5, essentially at least half of the
indices in this block will belong to
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$. If the constants $\omega$ and
$\tau$ are rational with the smallest common denominator $d\geq 3$, then the
sequence $\\{{\boldsymbol{z}}_{n}\\}$ is $d$-periodic and at most one index in
this period can be such that the corresponding solution of the Jacobi
inversion problem (2.19) is equal to $\boldsymbol{a}_{0}$. Hence, it is always
true that

$\mathbb{N}\setminus\big{(}n_{0}+d\mathbb{N})\cup(n_{1}+d\mathbb{N})\cup(n_{1}+1+d\mathbb{N})\big{)}\subseteq\mathbb{N}_{\varepsilon}^{\mathsf{sing}},$

where $n_{0},n_{1}\in\\{0,\ldots,d-1\\}$ are such that
${\boldsymbol{z}}_{n_{0}}=\infty^{(0)}$ and
${\boldsymbol{z}}_{n_{1}}=\boldsymbol{a}_{0}$. Thus,
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$ is again necessarily infinite,
unless of course $d=3$. As we prove in Proposition 3, in the latter case
$\Delta$ is a linear image of $\Delta_{\mathsf{sym}}$. Further below in
Appendix C we construct a class of examples of weights of orthogonality on
$\Delta_{\mathsf{sym}}$ that do require the restriction to
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$ and for which it is indeed the case
that ${\boldsymbol{z}}_{0}=\infty^{(0)}$,
${\boldsymbol{z}}_{1}=\boldsymbol{a}_{0}$, and
${\boldsymbol{z}}_{2}=\infty^{(1)}$, while
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}=\varnothing$. Therefore neither of
the forthcoming asymptotic results applies. Of course, one might surmise that
this is simply due to the technical limitations of our methods. However, we do
analyze the orthogonal polynomials for this class of weights and show that the
main term of their asymptotics is actually _different_ from $\Psi_{n}(z)$
constructed in Theorem 1. That is, the polynomials considered in Appendix C do
have different asymptotic behavior as compared to the polynomials in the rest
of the cases, which does indicate that the obstruction that occurs during the
construction of local parametrices is not a mere technicality.

It is also curious to observe that the condition defining
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ can be equivalently restated, see
(5.5), (5.6), and (6.7) further below, as

$\varepsilon\leq\left|\mathrm{det}\left(\begin{matrix}t_{1}^{(n)}&t_{1}^{(n-1)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
t_{0}^{(n)}&t_{0}^{(n-1)}\end{matrix}\right)\right|\leq\varepsilon^{-1}.$

##  3\. Main Results

To state our main results on the asymptotics of the diagonal Padé approximants
to functions of the form (1.4), we need to separate the weights $\rho(s)$ in
(1.4) into two classes. To this end, let $\rho_{i}(s)$ be the restriction to
$\Delta_{i}^{\circ}$ of $\rho(s)$ and $(z-a_{0})^{\alpha_{0}}$ be a fixed
branch whose branch cut splits the sector between $\Delta_{2}$ and
$\Delta_{3}$. Define

(3.1) $\varrho_{i}(s):=\rho_{i}(s)/(s-a_{0})^{\alpha_{0}},\quad
s\in\Delta_{i}^{\circ},\quad i\in\\{1,2,3\\},$

which, according to our assumptions stated right after (1.4), extend to
analytic non-vanishing functions in some neighborhood of $a_{0}$. Define
constants

(3.2) $\begin{cases}\displaystyle
b_{1}(\rho):=-\frac{e^{\pi\mathrm{i}\alpha_{0}}\varrho_{2}(a_{0})+e^{-\pi\mathrm{i}\alpha_{0}}\varrho_{3}(a_{0})}{\varrho_{1}(a_{0})},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle
b_{i}(\rho):=-\frac{\varrho_{i-1}(a_{0})+\varrho_{i+1}(a_{0})}{e^{(-1)^{i}\pi\mathrm{i}\alpha_{0}}\varrho_{i}(a_{0})},\quad
i\in\\{2,3\\},\end{cases}$

where subindices are understood cyclically in $\\{1,2,3\\}$. The parameters
$\alpha_{0}$ and $b_{i}(\rho)$, $i\in\\{1,2,3\\}$, uniquely determine a
$2\times 2$ sectionally analytic matrix function
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;x)$ as a solution of the Riemann-Hilbert
problem RHP-${\boldsymbol{\Phi}}_{\alpha}$, see Appendix B, where
$x\in\mathbb{C}$ is a parameter that appears in the description of the
asymptotic behavior at infinity. As we explain in Appendix B,
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;x)$ is meromorphic in $x$.

######  Definition.

In what follows we shall say that a weight $\rho(s)$ on $\Delta$ belongs to
the class $W^{\mathsf{reg}}$ if the matrix function
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;0)$ solving Riemann-Hilbert problem
RHP-${\boldsymbol{\Phi}}_{\alpha}$ with $\alpha=\alpha_{0}$,
$b_{i}=b_{i}(\rho)$, $i\in\\{1,2,3\\}$, and $x=0$ exists. That is, if
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;x)$ does not have a pole at $x=0$.
Otherwise, we shall say that the weight $\rho(s)$ belongs to the class
$W^{\mathsf{sing}}$.

It can be readily verified that parameters (3.2) satisfy the relation

(3.3) $b_{1}+b_{2}+b_{3}-b_{1}b_{2}b_{3}=2\cos(\pi\alpha_{0}).$

We shall call any solution of (3.3) a set of Stokes parameters. Any set of
Stokes parameters satisfying (3.3) leads to its own matrix
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;x)$. Some of these matrices will have a
pole at $x=0$ and some will not. Essentially any such set can be obtained via
(3.2) (in particular, this means that the class $W^{\mathsf{sing}}$ is non-
empty). Indeed, given a set of Stokes parameters (3.3) for which none is equal
to $e^{\alpha_{0}\pi\mathrm{i}}$ (in this case it is also true that none is
equal to $e^{-\alpha_{0}\pi\mathrm{i}}$ and a product of no two parameters is
equal to $1$, hence, the third one is uniquely determined by the other two),
they are realized by a weight $\rho(s)$ such that

$\frac{\varrho_{1}(a_{0})}{\varrho_{2}(a_{0})}=-\frac{1-b_{2}b_{3}}{1-e^{-\alpha_{0}\pi\mathrm{i}}b_{3}}\quad\text{and}\quad\frac{\varrho_{3}(a_{0})}{\varrho_{2}(a_{0})}=\frac{1-e^{\alpha_{0}\pi\mathrm{i}}b_{2}}{1-e^{-\alpha_{0}\pi\mathrm{i}}b_{3}}.$

On the other hand, if one of the Stokes parameters in (3.3) is equal to
$e^{\alpha_{0}\pi\mathrm{i}}$ or $e^{-\alpha_{0}\pi\mathrm{i}}$, then one
(whichever) of the other two must be equal to $e^{-\alpha_{0}\pi\mathrm{i}}$
or $e^{\alpha_{0}\pi\mathrm{i}}$, respectively, and the third one could be
absolutely arbitrary. However, this is not the case for the parameters
$b_{i}(\rho)$. If one of them happens to be $e^{\alpha_{0}\pi\mathrm{i}}$ or
$e^{-\alpha_{0}\pi\mathrm{i}}$, then the other two are uniquely determined.

Unfortunately, currently it is unknown how to determine solely from the Stokes
parameters $b_{i}(\rho)$ whether $\rho\in W^{\mathsf{reg}}$ or $\rho\in
W^{\mathsf{sing}}$. Such a determination amounts to distinguishing Stokes
parameters that lead to finite initial conditions at $x=0$ for the
corresponding solution of Painlevé XXXIV equation, see (B.3) and (B.8),
further below, and those leading to solutions with a pole at $x=0$.

######  Theorem 2.

Let $f(z)$ be a function given by (1.4) with $\alpha_{0}\in(-1,1)$ and
$\rho\in W^{\mathsf{reg}}$. Let $Q_{n}(z)$ be the minimal degree monic
denominator of the diagonal Padé approximants $[n/n]_{f}(z)$ and $R_{n}(z)$ be
the corresponding linearized error function (1.1). Further, let
$\Psi_{n}({\boldsymbol{z}})$ be given by (2.20) and
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ be as in Proposition 3. Then it
holds for all $n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ large enough that

(3.4)
$\begin{cases}Q_{n}(z)&=\gamma_{n}\big{(}1+\upsilon_{n1}(z)\big{)}\Psi_{n}(z)+\gamma_{n}\upsilon_{n2}(z)\Psi_{n-1}(z),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
(wR_{n})(z)&=\gamma_{n}\big{(}1+\upsilon_{n1}(z)\big{)}\Psi_{n}^{*}(z)+\gamma_{n}\upsilon_{n2}(z)\Psi_{n-1}^{*}(z),\end{cases}$

for $z\in\overline{\mathbb{C}}\setminus\Delta$ (in particular, $\deg(Q_{n})=n$
for such indices $n$), where
$\gamma_{n}:=\lim_{z\to\infty}z^{n}\Psi_{n}^{-1}(z)$ is the normalizing
constant, $w(z)$ is given by (2.2), and the functions $\upsilon_{ni}(s)$
vanish at infinity and satisfy222The notation
$f(n)=\mathcal{O}_{\varepsilon}(g(n))$ means that there exists a constant
$C_{\varepsilon}$ depending on $\varepsilon$ such that $|f(n)|\leq
C_{\varepsilon}|g(n)|$.

(3.5) $\upsilon_{ni}(z)=\mathcal{O}_{\varepsilon}\big{(}n^{-1/3}\big{)}$

locally uniformly on $\overline{\mathbb{C}}\setminus\Delta$. Moreover, it
holds for all $n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ large enough that

(3.6)
$\begin{cases}Q_{n}(s)&=\gamma_{n}\big{(}1+\upsilon_{n1}(s)\big{)}\left(\Psi_{n+}(s)+\Psi_{n-}(s)\right)+\gamma_{n}\upsilon_{n2}(s)\left(\Psi_{n-1+}(s)+\Psi_{n-1-}(s)\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
(wR_{n})_{\pm}(s)&=\gamma_{n}\big{(}1+\upsilon_{n1}(s)\big{)}\Psi_{n\pm}^{*}(s)+\gamma_{n}\upsilon_{n2}(s)\Psi_{n-1\pm}^{*}(s),\end{cases}$

for $s\in\Delta^{\circ}$, where the functions $\upsilon_{ni}(s)$ can be
analytically continued into a neighborhood of $\Delta^{\circ}$ and satisfy
(3.5) locally uniformly on $\Delta^{\circ}$.

Theorem 2 is proved in Section 5. To prove it we use by now classical approach
of Fokas, Its, and Kitaev [10, 11] connecting orthogonal polynomials to matrix
Riemann-Hilbert problems. The RH problem is then analyzed via the nonlinear
steepest descent method of Deift and Zhou [6].

The error estimates in (3.5) can be improved in some cases.

  * •

It is known, see [2], that if $\alpha_{0}=0$ and
$\rho_{1}(z)+\rho_{2}(z)+\rho_{3}(z)\equiv 0$ in some neighborhood of $a_{0}$,
then the error estimates improve to $\mathcal{O}_{\varepsilon}(n^{-1})$ (in
this case $b_{i}(\rho)=1$ and $\boldsymbol{\Phi}_{0}(\zeta;0)$ admits an
explicit construction that uses Airy functions, so such weights do belong to
$W^{\mathsf{reg}}$).

  * •

If $\rho(s)=\rho^{*}(s)/w_{+}(s)$ for some function $\rho^{*}(s)$ that has an
analytic extension to some neighborhood of $\Delta$, then the error estimates
are geometric, see [27] (in this case $b_{i}(\rho)=0$ and
$\boldsymbol{\Phi}_{-1/2}(\zeta;x)$ is equal to the right-hand side of
RHP-${\boldsymbol{\Phi}}_{\alpha}$(d) with $U_{n}=V_{n}\equiv 0$).

In the case of singular weights, the following theorem takes place.

######  Theorem 3.

Let $f(z)$ be a function given by (1.4) with $\alpha_{0}\in(-1,1)$ and
$\rho\in W^{\mathsf{sing}}$. Assume further that

(3.7)
$\varrho_{1}^{\prime}(a_{0})/\varrho_{1}(a_{0})=\varrho_{2}^{\prime}(a_{0})/\varrho_{2}(a_{0})=\varrho_{3}^{\prime}(a_{0})/\varrho_{3}(a_{0}).$

Let $Q_{n}(z)$, $R_{n}(z)$, and $\Psi_{n}({\boldsymbol{z}})$ be as in Theorem
2. Let the matrices $\boldsymbol{T}_{n}$ be as in (2.29) and
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$ be as in Proposition 5. Then it
holds for all $n\in\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$ large enough that
asymptotic formulae (3.4) and (3.6) remain valid but with $\upsilon_{n1}(z)$
and $\upsilon_{n2}(z)$ replaced by

$\upsilon_{n1}(z)-\frac{2}{z-a_{0}}\frac{t_{0}^{(n-1)}t_{0}^{(n)}}{\mathrm{det}(\boldsymbol{T}_{n})}\quad\text{and}\quad\upsilon_{n2}(z)+\frac{2}{z-a_{0}}\frac{(t_{0}^{(n)})^{2}}{\mathrm{det}(\boldsymbol{T}_{n})},$

respectively, where the error functions $\upsilon_{ni}(z)$ possess the same
properties as in Theorem 2.

Let us point out that the new term added to the error functions
$\upsilon_{ni}(z)$ is not asymptotically small. The presence of the condition
(3.7) is connected to the local analysis around $a_{0}$ and seems to be
inescapable for our techniques. In particular, we needed this condition for
our analysis of non-Hermitian orthogonal polynomials on a cross in [4], see
the definitions of classes $\mathcal{W}_{\ell}$ on page 2. Its nature is not
entirely transparent to us at this stage. However, as we explain in Appendix
D, functions of the form (1.3) do satisfy (3.7). Theorem 3 is proved in
Section 6.

##  4\. Proof of Propositions 1–5 and Theorem 1

###  4.1. Proof of Proposition 1

Orient $\pi(\boldsymbol{\alpha})$ towards $a_{1}$. Let

$H(z):=\frac{w(z)}{2\pi\mathrm{i}}\int_{\pi(\boldsymbol{\alpha})}\frac{1}{t-z}\frac{\mathrm{d}t}{w(t)},\quad
z\in\mathbb{C}\setminus\big{(}\Delta\cup\pi(\boldsymbol{\alpha})\big{)}.$

Then $H(z)$ is an analytic function in its domain of definition (the integral
itself is analytic in
$\overline{\mathbb{C}}\setminus\pi(\boldsymbol{\alpha})$) that behaves like

(4.1)
$H(z)=-\frac{z}{2\pi\mathrm{i}}\int_{\pi(\boldsymbol{\alpha})}\frac{\mathrm{d}s}{w(s)}+\mathcal{O}(1)=-\frac{z}{4\pi\mathrm{i}}\oint_{\boldsymbol{\alpha}}\frac{\mathrm{d}s}{w({\boldsymbol{s}})}+\mathcal{O}(1)$

as $z\to\infty$. We get from Plemelj-Sokhotski formulae [12, Section I.4.2]
that

(4.2) $H_{+}(s)-H_{-}(s)=1,\quad
s\in\pi(\boldsymbol{\alpha})\setminus\\{a_{1},a_{3}\\}.$

It further follows from [12, Section 8.3] that $H(z)$ is bounded around
$a_{1}$ and $a_{3}$. Set

(4.3) $\Lambda(z):=\frac{w(z)}{2\pi\mathrm{i}}\int_{\Delta}\frac{\log(\rho
w_{+})(s)}{s-z}\frac{\mathrm{d}s}{w_{+}(s)},\quad
z\in\mathbb{C}\setminus\Delta.$

This is a holomorphic function in its domain of definition that satisfies

(4.4) $\Lambda(z)=-\frac{z}{2\pi\mathrm{i}}\int_{\Delta}\frac{\log(\rho
w_{+})(s)}{w_{+}(s)}\mathrm{d}s+\mathcal{O}(1)=-\frac{zc_{\rho}}{2}\oint_{\boldsymbol{\alpha}}\frac{\mathrm{d}s}{w({\boldsymbol{s}})}+\mathcal{O}(1)$

as $z\to\infty$, see (2.13) and (2.8). It follows from Plemelj-Sokhotski
formulae that

(4.5) $\Lambda_{+}(s)+\Lambda_{-}(s)=\log(\rho_{i}w_{+})(s),\quad
s\in\Delta_{i}^{\circ},~{}~{}i\in\\{1,2,3\\}.$

Write $\Lambda(z)=\Lambda_{1}(z)+\Lambda_{2}(z)+\Lambda_{3}(z)$, where
$\Lambda_{i}(z)$ is given by (4.3) with $\Delta$ replaced by $\Delta_{i}$. Fix
$i\in\\{1,2,3\\}$. Let $\sqrt{z-a_{i}}$ and $(z-a_{i})^{\alpha_{i}}$ be
branches whose branch cuts include $\Delta_{i}$ and such that
$w(z)=v_{i}(z)\sqrt{z-a_{i}}$ and
$\rho_{i}(s)=\tilde{\rho}_{i}(s)(s-a_{i})^{\alpha_{i}}_{+}$ for some functions
$v_{i}(z)$ and $\tilde{\rho}_{i}(z)$ analytic in a vicinity of $a_{i}$. Then
we get from [12, Sections I.8.3-6] that

(4.6) $\displaystyle\Lambda_{i}(z)$ $\displaystyle=$
$\displaystyle\frac{v_{i}(z)\sqrt{z-a_{i}}}{2\pi\mathrm{i}}\int_{\Delta_{i}}\frac{(\alpha_{i}+1/2)\log_{+}(s-a_{i})+\log(\tilde{\rho}_{i}v_{i})(s)}{s-z}\frac{\mathrm{d}s}{v_{i}(s)\sqrt{s-a_{i}}_{+}}$
$\displaystyle=$
$\displaystyle\frac{v_{i}(z)\sqrt{z-a_{i}}}{2\pi\mathrm{i}}\int_{\Delta_{i}}\frac{(\alpha_{i}+1/2)\log_{+}(s-a_{i})}{s-z}\frac{\mathrm{d}s}{v_{i}(s)\sqrt{s-a_{i}}_{+}}+\mathcal{O}(1)$
$\displaystyle=$ $\displaystyle(\alpha_{i}/2+1/4)\log(z-a_{i})+\mathcal{O}(1)$

as $z\to a_{i}$, where $\log(z-a_{i})$ has a branch cut that includes
$\Delta_{i}$.

Similarly, it follows from the conditions placed on $\rho_{i}(s)$ that
$\rho_{i}(s)/(s-a_{0})_{+}^{\alpha_{0}}$ extends to an analytic function in a
vicinity of $a_{0}$, where $(z-a_{0})^{\alpha_{0}}$ is a branch with the cut
along $\Delta_{i}$. Moreover, to get a function analytic in a vicinity of
$a_{0}$ out of $w(z)$, we need to divide it by $\sqrt{z-a_{0}}$ with the jump
along $\Delta_{i}$ and then multiply by $-1$ in the sector opposite to
$\Delta_{i}$. Hence, it holds that

(4.7) $\Lambda_{i}(z)=\nu_{i}(\alpha_{0}/2+1/4)L_{i}(z)+\mathcal{O}(1)$

as $z\to a_{0}$, where $L_{i}(z):=\log(z-a_{0})$ has a branch cut that
includes $\Delta_{i}$ and $\nu_{i}$ is equal to $1$ in the sectors adjacent to
$\Delta_{i}$ and is equal to $-1$ in the sector opposite to $\Delta_{i}$.
Since

$S_{\rho}\big{(}z^{(k)}\big{)}=\exp\left\\{(-1)^{k+1}\left(\Lambda(z)-2\pi\mathrm{i}c_{\rho}H(z)\right)\right\\},$

see (2.10) and (2.11), the claims of the proposition follow ((4.1) and (4.4)
give analyticity at infinity; (4.2) and (4.5) yield (2.12); (4.6) and (4.7)
give (2.14)).

###  4.2. Proof of Proposition 2

Before we start proving this proposition in earnest, let us make an
observation. Recall the differentials
$\Omega_{{\boldsymbol{z}},{\boldsymbol{z}}^{*}}({\boldsymbol{s}})$ introduced
in (2.11) and the differential $G({\boldsymbol{s}})$ defined in (2.4). One can
easily check that

$G({\boldsymbol{s}})=\Omega_{\infty^{(1)},\infty^{(0)}}({\boldsymbol{s}})+2\pi\mathrm{i}\tau\Omega({\boldsymbol{s}}),$

where $\Omega({\boldsymbol{s}})$ is given by (2.8). It further follows from
Riemann’s relations [7, Section III.3.6] that

$\oint_{\boldsymbol{\beta}}\Omega_{\infty^{(1)},\infty^{(0)}}=2\pi\mathrm{i}\int_{\infty^{(0)}}^{\infty^{(1)}}\Omega,$

where the path of integration lies entirely in
$\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$. Therefore, we can
immediately deduce from the last two identities, (2.8), and (2.5) that

(4.8)
$\int_{\infty^{(1)}}^{\infty^{(0)}}\Omega=\omega+\mathsf{B}\tau\quad\text{and}\quad\int_{\boldsymbol{a}_{0}}^{\infty^{(k)}}\Omega=(-1)^{k}(\omega+\mathsf{B}\tau)/2,\quad
k\in\\{0,1\\},$

where we used symmetry
$\Omega({\boldsymbol{z}}^{*})=-\Omega({\boldsymbol{z}})$ for the last
conclusion.

It is well known and can be checked directly that $\theta(u)$ enjoys the
following periodicity properties:

(4.9)
$\theta(u+l+\mathsf{B}m)=\exp\big{\\{}-\pi\mathrm{i}\mathsf{B}m^{2}-2\pi\mathrm{i}mu\big{\\}}\theta(u),\quad
l,m\in\mathbb{Z}.$

It is further known that $\theta(u)=0$ if and only if
$[u]=[(\mathsf{B}+1)/2]$. As already mentioned, the function
$a({\boldsymbol{z}})$ from (2.9) extends to a multi-valued analytic function
on $\mathfrak{S}$. Therefore, it has continuous traces on
$(\boldsymbol{\alpha}\cup\boldsymbol{\beta})\setminus\\{\boldsymbol{a}_{1}\\}$
and these traces satisfy

(4.10)
$a_{+}({\boldsymbol{s}})-a_{-}({\boldsymbol{s}})=\left\\{\begin{array}[]{rl}-\mathsf{B},&{\boldsymbol{s}}\in\boldsymbol{\alpha}\setminus\\{\boldsymbol{a}_{1}\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
1,&{\boldsymbol{s}}\in\boldsymbol{\beta}\setminus\\{\boldsymbol{a}_{1}\\},\end{array}\right.$

due to normalization of $\Omega({\boldsymbol{z}})$ and the definition of
$\mathsf{B}$, see (2.8). Relations (4.9) and (4.10) imply that
$T_{n}({\boldsymbol{z}})$ does indeed extend to a multiplicatively multi-
valued function on $\mathfrak{S}$. Furthermore, they also yield that

$\displaystyle T_{n+}({\boldsymbol{s}})$ $\displaystyle=$
$\displaystyle\exp\big{\\{}-2\pi\mathrm{i}y_{n}(a_{-}({\boldsymbol{s}})-\mathsf{B})\big{\\}}\frac{\theta(a_{-}({\boldsymbol{s}})-c_{\rho}-x_{n}-\mathsf{B}y_{n}+\mathsf{A}-\mathsf{B})}{\theta(a_{-}({\boldsymbol{s}})+\mathsf{A}-\mathsf{B})}$
$\displaystyle=$ $\displaystyle
T_{n-}({\boldsymbol{s}})\exp\\{2\pi\mathrm{i}(-c_{\rho}-x_{n})\\},\quad{\boldsymbol{s}}\in\boldsymbol{\alpha}\setminus\\{\boldsymbol{a}_{1}\\},$

from which the first relation in (2.18) follows (recall that $x_{n}$ and
$n\omega$ differ by an integer). The proof of the second one is analogous.
Moreover, since

$[(1+\mathsf{B})/2]=[a({\boldsymbol{z}})-c_{\rho}-x_{n}-\mathsf{B}y_{n}+\mathsf{A}]=[a({\boldsymbol{z}})-a({\boldsymbol{z}}_{n})+(1+\mathsf{B})/2]$

holds only when ${\boldsymbol{z}}={\boldsymbol{z}}_{n}$ by the unique
solvability of the Jacobi inversion problem (we used (2.19) and (2.16) for the
second equality), $T_{n}({\boldsymbol{z}})$ does indeed vanish solely at
${\boldsymbol{z}}_{n}$. The location of the pole of $T_{n}({\boldsymbol{z}})$
can be identified similarly using (4.8).

###  4.3. Proof of Theorem 1

Fix $n\in\mathbb{N}$ and let $\Psi_{n}({\boldsymbol{z}})$ be given by (2.20).
It follows from the properties of $\Phi({\boldsymbol{z}})$,
$S_{\rho}({\boldsymbol{z}})$, and $T_{n}({\boldsymbol{z}})$, including jump
relations (2.6), (2.12), and (2.18), and local behavior (2.14) that
$\Psi_{n}({\boldsymbol{z}})$ solves BVP-$\Psi$ when
${\boldsymbol{z}}_{n}\neq\infty^{(0)}$ (when
${\boldsymbol{z}}_{n}=\infty^{(0)}$, the pole at $\infty^{(0)}$ is of order
$n-1$). Let $\Psi({\boldsymbol{z}})$ be any solution of BVP-$\Psi$, if it
exists. Consider the function
$F({\boldsymbol{z}}):=\Psi({\boldsymbol{z}})/\Psi_{n}({\boldsymbol{z}})$. Then
$F({\boldsymbol{z}})$ is meromorphic in
$\mathfrak{S}\setminus\boldsymbol{\Delta}$ with at most one pole, namely
${\boldsymbol{z}}_{n}$, which is simple if present. It also holds that
$F_{+}({\boldsymbol{s}})=F_{-}({\boldsymbol{s}})$ for
${\boldsymbol{s}}\in\boldsymbol{\Delta}\setminus\boldsymbol{A}$. Since these
traces are continuous on $\boldsymbol{\Delta}\setminus\boldsymbol{A}$ (apart
from a possible simple pole at ${\boldsymbol{z}}_{n}$ when the latter lies on
$\boldsymbol{\Delta}\setminus\boldsymbol{A}$), the analytic continuation
principle yields that $F({\boldsymbol{z}})$ is analytic in
$\mathfrak{S}\setminus(\\{{\boldsymbol{z}}_{n}\\}\cup\boldsymbol{A})$ with at
most a simple pole at ${\boldsymbol{z}}_{n}$ when
${\boldsymbol{z}}_{n}\not\in\boldsymbol{A}$. Finally, it follows from (2.3)
and (2.21) that every $\boldsymbol{a}_{i}$ is a removable singularity of
$F({\boldsymbol{z}})$ unless it coincides with ${\boldsymbol{z}}_{n}$, in
which case it can be a simple pole. Altogether, $F({\boldsymbol{z}})$ is a
rational function on $\mathfrak{S}$ with at most one pole, which is simple.
Hence, $F({\boldsymbol{z}})$ is a constant. That is,
$\Psi({\boldsymbol{z}})=c\Psi_{n}({\boldsymbol{z}})$, which finishes the proof
of the theorem.

###  4.4. Proof of Proposition 3

Below we always assume that all $\varepsilon_{N}$ satisfy
$\displaystyle\max_{s\in\Delta}|s|<\varepsilon_{N}^{-1}$.

Because Jacobi inversion problems are uniquely solvable in the considered
case, it follows from (2.15) and (4.8) that if
${\boldsymbol{z}}_{n}=\infty^{(k)}$, then

$\big{[}c_{\rho}+(n-1/2)(\omega+\mathsf{B}\tau)\big{]}=[(-1)^{k}(\omega+\mathsf{B}\tau)/2]$

and therefore ${\boldsymbol{z}}_{n-(-1)^{k}}=\infty^{(1-k)}$, again by (2.15)
and (4.8), since

$\big{[}c_{\rho}+(n-1/2-(-1)^{k})(\omega+\mathsf{B}\tau)\big{]}=[-(-1)^{k}(\omega+\mathsf{B}\tau)/2].$

This proves the very last claim of the proposition.

If $\omega$ and $\tau$ are rational numbers with the smallest common
denominator $d$, then

$\big{[}c_{\rho}+(n+d-1/2)(\omega+\mathsf{B}\tau)\big{]}=\big{[}c_{\rho}+(n-1/2)(\omega+\mathsf{B}\tau)\big{]}$

and therefore the sequence $\\{{\boldsymbol{z}}_{n}\\}$ is $d$-periodic. Thus,
if none of ${\boldsymbol{z}}_{0},\ldots,{\boldsymbol{z}}_{d-1}$ coincides with
$\infty^{(0)}$, choose $\varepsilon_{0}$ small enough so that
${\boldsymbol{z}}_{0},\ldots,{\boldsymbol{z}}_{d-1}\not\in\boldsymbol{U}_{\varepsilon_{0}}^{(0)}$,
in which case $\mathbb{N}_{\varepsilon}^{\mathsf{reg}}=\mathbb{N}$ for all
$\varepsilon<\varepsilon_{0}$. If ${\boldsymbol{z}}_{n_{0}}=\infty^{(0)}$ for
some $n_{0}\in\\{0,\ldots,d-1\\}$, then of course
${\boldsymbol{z}}_{n_{0}+kd}=\infty^{(0)}$ for all $k\geq 0$ and
${\boldsymbol{z}}_{l+kd}\neq\infty^{(0)}$ for
$l\in\\{0,\ldots,d-1\\}\setminus\\{n_{0}\\}$, since otherwise $l-n_{0}$ would
be divisible by $d$, which is impossible. Hence, by periodicity,
$\mathbb{N}_{\varepsilon}^{\mathsf{reg}}=\mathbb{N}\setminus(n_{0}+d\mathbb{N})$
for all $\varepsilon<\varepsilon_{0}$, where $\varepsilon_{0}$ is such that
${\boldsymbol{z}}_{l}\not\in\boldsymbol{U}_{\varepsilon_{0}}^{(0)}$ for
$l\in\\{0,\ldots,d-1\\}\setminus\\{n_{0}\\}$. Moreover, in this case we get
that there exist integers $l$ and $m$ such that

$c_{\rho}=\big{(}(1-n_{0})\omega+l\big{)}+\mathsf{B}\big{(}(1-n_{0})\tau+m\big{)}=(p/d)+\mathsf{B}(q/d)$

by (2.15) and (4.8), where $p$ and $q$ are again integers.

Now, recall the definition of the constants $\omega$ and $\tau$ in (2.5). We
can equivalently write

(4.11) $\omega=I_{1}+I_{3}\quad\text{and}\quad\tau=-(I_{1}+I_{2}),\quad
I_{i}:=-\frac{1}{\pi\mathrm{i}}\int_{\Delta_{i}}\frac{(s-a_{0})\mathrm{d}s}{w_{+}(s)}$

(recall that each $\Delta_{i}$ is oriented towards $a_{0}$). It follows from
the Cauchy integral formula that

(4.12)
$1=\frac{1}{2\pi\mathrm{i}}\oint_{L}\frac{(s-a_{0})\mathrm{d}s}{w(s)}=-\frac{1}{\pi\mathrm{i}}\int_{\Delta}\frac{(s-a_{0})\mathrm{d}s}{w_{+}(s)}=I_{1}+I_{2}+I_{3},$

where $L$ is any smooth Jordan curve that encircles $\Delta$ in the counter-
clockwise direction. We also get from (1.2) that the integrand in (4.12),
i.e.,

(4.13) $\frac{(s-a_{0})}{w_{+}(s)}\frac{\mathrm{d}s}{\pi\mathrm{i}}$

is real and non-vanishing on each $\Delta_{i}^{\circ}$. In particular,
$I_{i}\neq 0$, $i\in\\{1,2,3\\}$. Moreover, it is known, see [22, Theorem
8.1], that there exists a conformal map $\phi(z)$ in a neighborhood of $a_{0}$
such that the differential (1.2) can be locally written as
$\phi\mathrm{d}\phi^{2}$. If we normalize $\phi(z)$ so that it maps
$\Delta_{1}$ into non-positive reals (locally around $a_{0}$), then
$\sqrt{\phi}\mathrm{d}\phi$ is equal to (4.13) in the sector delimited by
$\Delta_{2}$ and $\Delta_{3}$ that contains $\Delta_{1}$ and minus the
differential in (4.13) in the other sector. This yields that $I_{i}>0$,
$i\in\\{1,2,3\\}$. In particular, the latter claim implies that
$\omega\in(0,1)$, $\tau\in(-1,0)$, and for no $\Delta$ it can hold that
$\omega=-\tau=1/2$ as $\omega-\tau=1+I_{1}>1$, that is, $d\geq 3$.

Assume that $d=3$. Then $I_{i}=1/3$, $i\in\\{1,2,3\\}$, as these are positive
rational numbers smaller than $1$ with denominator $3$ by (4.11) and (4.12).
It readily follows from its symmetries that this is indeed the case for
$\Delta_{\mathsf{sym}}$. Let as before $\Phi(z)$ stand for the pull-back of
$\Phi({\boldsymbol{z}})$ from $\mathfrak{S}^{(0)}$ onto
$\mathbb{C}\setminus\Delta$. Denote by $\Phi_{\Delta}(z)$ the analytic
continuation of $\Phi(z)$ from infinity across $\pi(\boldsymbol{\alpha})$ and
$\pi(\boldsymbol{\beta})$ to a holomorphic function in
$\mathbb{C}\setminus\Delta$. In fact, $\Phi_{\Delta}(z)$ is again given by
(2.4), where
${\boldsymbol{z}}\in\mathfrak{S}^{(0)}\setminus\boldsymbol{\Delta}$ and the
path of integration also belongs to
$\mathfrak{S}^{(0)}\setminus\boldsymbol{\Delta}$, except for
$\boldsymbol{a}_{0}$, and proceeds from $\boldsymbol{a}_{0}$ into the sector
delimited by $\boldsymbol{\Delta}_{2}$ and $\boldsymbol{\Delta}_{3}$, recall
Figure 1. As was explained after (2.6), $\Phi_{\Delta}(z)$ is a conformal map
of $\mathbb{C}\setminus\Delta$ onto $\\{|z|>1\\}$ that maps infinity into
infinity. This map also sends $\partial(\mathbb{C}\setminus\Delta)$, the
boundary of $\mathbb{C}\setminus\Delta$ understood as the collection of limit
points of sequences in $\mathbb{C}\setminus\Delta$, bijectively onto the unit
circle. Clearly, $\partial(\mathbb{C}\setminus\Delta)$ consists of two copies
of each $s\in\Delta_{i}^{\circ}$, say $s_{+}$ and $s_{-}$, as accessed from
the $+$ and $-$ sides of $\Delta_{i}$, one copy of each $a_{1},a_{2},a_{3}$,
and three copies of $a_{0}$, say $a_{0,12}$, $a_{0,23}$, and $a_{0,31}$, where
$a_{0,ij}$ is accessed from the sector delimited by $\Delta_{i}$ and
$\Delta_{j}$. Then

$\begin{cases}\Phi_{\Delta}(a_{0,23})=1,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\Phi_{\Delta}(a_{0,12})=e^{\pi\mathrm{i}(-2I_{2})}=e^{-2\pi\mathrm{i}/3},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Phi_{\Delta}(a_{0,31})=e^{\pi\mathrm{i}(2I_{3})}=e^{2\pi\mathrm{i}/3},\end{cases}\quad\text{and}\quad\begin{cases}\Phi_{\Delta}(a_{1})=e^{\pi\mathrm{i}(2I_{3}+I_{1})}=-1,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Phi_{\Delta}(a_{2})=e^{\pi\mathrm{i}(-I_{2})}=e^{-\pi\mathrm{i}/3},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Phi_{\Delta}(a_{3})=e^{\pi\mathrm{i}I_{3}}=e^{\pi\mathrm{i}/3}.\end{cases}$

Moreover, as $w_{+}(s)=-w_{-}(s)$, it also holds that
$\Phi_{\Delta}(s_{+})/\Phi_{\Delta}(a_{i})=\overline{\Phi_{\Delta}(s_{-})/\Phi_{\Delta}(a_{i})}$
for any $s\in\Delta_{i}^{\circ}$. So, if $\Delta^{1}$ and $\Delta^{2}$ are two
such contours, then
$F(z):=\big{(}\Phi_{\Delta^{2}}^{-1}\circ\Phi_{\Delta^{1}}\big{)}(z)$ is a
conformal map of $\mathbb{C}\setminus\Delta^{1}$ onto
$\mathbb{C}\setminus\Delta^{2}$ that maps infinity into infinity and
$\partial(\mathbb{C}\setminus\Delta^{1})$ continuously and bijectively onto
$\partial(\mathbb{C}\setminus\Delta^{2})$. The just described properties also
yield that

$F(s_{+})=F(s_{-}),\quad s\in(\Delta_{i}^{1})^{\circ},\quad\text{and}\quad
F(a_{l}(\Delta^{1}))=a_{l}(\Delta^{2})$

for $i\in\\{1,2,3\\}$ and $l\in\\{0,1,2,3\\}$. Thus, $F(z)$ is continuous and
therefore holomorphic in the entire complex plane. Since $F(z)$ is conformal
around infinity, it has a simple pole there, which means that it is a linear
function that maps $\Delta^{1}$ onto $\Delta^{2}$, as claimed.

Assume now that at least one the numbers $\omega$ and $\tau$ is irrational. As
explained before the statement of the proposition, ${\boldsymbol{z}}_{n}$ can
be equal to $\infty^{(0)}$ for at most one index $n$. Fix $N\geq 1$. Let
${\boldsymbol{x}}_{-1},{\boldsymbol{x}}_{0},{\boldsymbol{x}}_{1},\ldots,{\boldsymbol{x}}_{N}$
be the unique points on $\mathfrak{S}$ such that

$\mathfrak{a}({\boldsymbol{x}}_{l})=\left[\frac{\omega+\mathsf{B}\tau}{2}+\big{(}\omega+\mathsf{B}\tau\big{)}l\right],\quad
l\in\\{-1,0,\ldots,N\\}.$

It follows from (4.8) that ${\boldsymbol{x}}_{-1}=\infty^{(1)}$ and
${\boldsymbol{x}}_{0}=\infty^{(0)}$. It also holds that none of
${\boldsymbol{x}}_{1},\ldots,{\boldsymbol{x}}_{N}$ is equal to $\infty^{(0)}$
and they are all distinct (if ${\boldsymbol{x}}_{i}={\boldsymbol{x}}_{j}$ for
$i\neq j$, then $\omega$ and $\tau$ are necessarily rational). Fix
$\epsilon_{*}>0$ small enough so that the sets

$\mathsf{O}_{l}:=\Big{\\{}[s]~{}\big{|}~{}\exists
x:~{}[x]=\mathfrak{a}({\boldsymbol{x}}_{l}),~{}~{}|s-x|<\epsilon_{*}\Big{\\}}\subset\mathsf{Jac}(\mathfrak{S})$

are disjoint. Recall that the Abel’s map $\mathfrak{a}({\boldsymbol{z}})$ is a
holomorphic bijection between $\mathfrak{S}$ and $\mathsf{Jac}(\mathfrak{S})$.
Set

$\boldsymbol{U}_{l}:=\mathfrak{a}^{-1}(\mathsf{O}_{l}),\quad
l\in\\{-1,0,\ldots,N\\}.$

Notice that if $[s]\in\mathsf{O}_{l}$, then
$[s+\omega+\mathsf{B}\tau]\in\mathsf{O}_{l+1}$. Hence, the domains
$\boldsymbol{U}_{l}$ are disjoint and

${\boldsymbol{z}}_{n+k}\in\boldsymbol{U}_{k},\quad
k\in\\{-1,\ldots,N-1\\}\quad\Rightarrow\quad{\boldsymbol{z}}_{n+l}\in\boldsymbol{U}_{l},\quad
l\in\\{k+1,\ldots,N\\}.$

Thus, to prove (2.24) it remains to choose $\varepsilon_{N}$ small enough so
that $\boldsymbol{U}_{\varepsilon_{N}}^{(0)}\subseteq\boldsymbol{U}_{0}$, and
to prove (2.25), given $0<\varepsilon<\varepsilon_{N}$, to choose
$\epsilon_{*}$ and $\delta(\varepsilon)$ so that
$\boldsymbol{U}_{0}\subseteq\boldsymbol{U}_{\varepsilon}^{(0)}$ and
$\boldsymbol{U}_{\delta(\varepsilon)}^{(1)}\subseteq\boldsymbol{U}_{-1}$.

###  4.5. Proof of Proposition 4

Since $a({\boldsymbol{z}})$ is a branch of an additively multi-valued function
on $\mathfrak{S}$, see (4.10), it holds that $K_{R}$, the closure of
$a(\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}\setminus\boldsymbol{U}_{1/R}^{(1)})$
is a compact set. Moreover, $K_{R}$ is necessarily disjoint from any point on
the lattice $[-(\omega+\mathsf{B}\tau)/2]$ by (4.8). Hence, there exists a
constant $M_{1,R}$ such that

$\Big{|}e^{-2\pi\mathrm{i}y_{n}u}\theta^{-1}(u+\mathsf{A})\Big{|}\leq
M_{1,R},\quad u\in K_{R}$

(recall also that $y_{n}\in[0,1)$). There also exists a constant $M_{2,R}$
such that

$|\theta(u+\mathsf{A})|\leq M_{2,R},\quad
u\in\Big{\\{}v-x-\mathsf{B}y-c_{\rho}:~{}v\in K_{R},~{}x,y\in[0,1]\Big{\\}},$

by compactness of the latter set. The first inequality in (2.26) now holds
with $M_{R}=M_{1,R}M_{2,R}$. The second inequality follows from the maximum
modulus principle applied in $\boldsymbol{U}_{1/R}^{(1)}$. The proof of (2.27)
is identical.

###  4.6. Proof of Proposition 5

Given
${\boldsymbol{x}}\in\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$,
let $x({\boldsymbol{x}})$, $y({\boldsymbol{x}})$ be real numbers such that

$\mathfrak{a}({\boldsymbol{x}})=\big{[}c_{\rho}+x({\boldsymbol{x}})+\mathsf{B}y({\boldsymbol{x}})-(\omega+\mathsf{B})/2\big{]},\quad
x({\boldsymbol{z}}_{0})=y({\boldsymbol{z}}_{0})=0,$

recall (2.19), and that continuously depend on ${\boldsymbol{x}}$. Observe
that these functions continuously extend to the boundary of
$\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}$, $x({\boldsymbol{x}})$
is in fact continuous across $\boldsymbol{\alpha}$ and
$x_{+}({\boldsymbol{x}})-x_{-}({\boldsymbol{x}})=1$ for
${\boldsymbol{x}}\in\boldsymbol{\beta}$ while $y({\boldsymbol{x}})$ is
continuous across $\boldsymbol{\beta}$ and satisfies
$y_{+}({\boldsymbol{x}})-y_{-}({\boldsymbol{x}})=-1$ for
${\boldsymbol{x}}\in\boldsymbol{\alpha}$, see (4.10). In particular, it
follows from the chosen normalization, (2.19), and (2.16) that either
$x({\boldsymbol{z}}_{n})=x_{n}$ or $x({\boldsymbol{z}}_{n})=x_{n}-1$ and the
same is true about $y({\boldsymbol{z}}_{n})$ and $y_{n}$. Define

$T({\boldsymbol{z}};{\boldsymbol{x}}):=\exp\big{\\{}-2\pi\mathrm{i}y({\boldsymbol{x}})a({\boldsymbol{z}})\big{\\}}\dfrac{\theta\big{(}a({\boldsymbol{z}})-c_{\rho}-x({\boldsymbol{x}})-\mathsf{B}y({\boldsymbol{x}})+\mathsf{A}\big{)}}{\theta\big{(}a({\boldsymbol{z}})+\mathsf{A}\big{)}},\quad{\boldsymbol{z}}\in\mathfrak{S}_{\boldsymbol{\alpha},\boldsymbol{\beta}}.$

In particular, it follows from (4.9) that

(4.14)
$\begin{cases}T({\boldsymbol{z}};{\boldsymbol{z}}_{n})=T_{n}({\boldsymbol{z}}),&y({\boldsymbol{z}}_{n})=y_{n},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
T({\boldsymbol{z}};{\boldsymbol{z}}_{n})=e^{2\pi\mathrm{i}(c_{\rho}+x_{n}+\mathsf{B}y_{n}-\mathsf{A}-\mathsf{B}/2)}T_{n}({\boldsymbol{z}}),&y({\boldsymbol{z}}_{n})=y_{n}-1.\end{cases}$

Furthermore, similarly to (2.18), it holds that

(4.15)
$T_{+}({\boldsymbol{s}};{\boldsymbol{x}})=T_{-}({\boldsymbol{s}};{\boldsymbol{x}})\left\\{\begin{array}[]{rl}\displaystyle\exp\big{\\{}-2\pi\mathrm{i}\big{(}c_{\rho}+x({\boldsymbol{x}})\big{)}\big{\\}},&{\boldsymbol{s}}\in\boldsymbol{\alpha}\setminus\\{\boldsymbol{a}_{1}\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\exp\big{\\{}-2\pi\mathrm{i}y({\boldsymbol{x}})\big{\\}},&{\boldsymbol{s}}\in\boldsymbol{\beta}\setminus\\{\boldsymbol{a}_{1}\\}.\end{array}\right.$

It is also true that $T({\boldsymbol{s}};{\boldsymbol{x}})$ is a holomorphic
non-vanishing function except for a simple zero at ${\boldsymbol{x}}$ and a
simple pole at $\infty^{(1)}$. Finally, with the same determination of
$(z-a_{0})^{1/2}$ as in (2.28) we can write

$T(z;{\boldsymbol{x}}):=t_{0}({\boldsymbol{x}})+t_{1}({\boldsymbol{x}})(z-a_{0})^{1/2}+t_{2}({\boldsymbol{x}})(z-a_{0})+\cdots$

for $z$ close to $a_{0}$ and lying in the sector between $\Delta_{1}$ and
$\Delta_{3}$. Notice that the coefficients $t_{l}({\boldsymbol{x}})$ are
continuous functions of
${\boldsymbol{x}}\in\mathfrak{S}_{\boldsymbol{\alpha}}$ (the change of
$x({\boldsymbol{x}})$ by an integer does not affect
$T({\boldsymbol{z}};{\boldsymbol{x}})$) that continuously extend to both sides
of $\boldsymbol{\alpha}$.

Recall (2.9). Given ${\boldsymbol{x}}\in\mathfrak{S}$, let
${\boldsymbol{x}}_{\pm 1}\in\mathfrak{S}$ be the unique points such that

(4.16) $\mathfrak{a}({\boldsymbol{x}}_{\pm
1})=\left[a({\boldsymbol{x}})\pm\big{(}\omega+\mathsf{B}\tau\big{)}\right].$

Clearly, ${\boldsymbol{x}}_{1}$ and ${\boldsymbol{x}}_{-1}$ continuously
depend on ${\boldsymbol{x}}$. Notice that
$({\boldsymbol{x}}_{1})_{-1}={\boldsymbol{x}}$ and that
$({\boldsymbol{z}}_{n})_{\pm 1}={\boldsymbol{z}}_{n\pm 1}$. Define

$\boldsymbol{T}({\boldsymbol{x}}):=\begin{pmatrix}2t_{2}({\boldsymbol{x}})-h_{1}t_{1}({\boldsymbol{x}})&2t_{2}({\boldsymbol{x}}_{-1})-h_{1}t_{1}({\boldsymbol{x}}_{-1})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
t_{0}({\boldsymbol{x}})&t_{0}({\boldsymbol{x}}_{-1})\end{pmatrix},$

where the constant $h_{1}$ is the same as in (2.29), whatever it might be.
Consider the function

$d({\boldsymbol{x}}):=\left|\mathrm{det}(\boldsymbol{T}({\boldsymbol{x}}))\right|+\left|\mathrm{det}(\boldsymbol{T}({\boldsymbol{x}}_{1}))\right|.$

As we have explained, this is a continuous function of
${\boldsymbol{x}}\in\mathfrak{S}_{\boldsymbol{\alpha}}$ that continuously
extends to both sides of $\boldsymbol{\alpha}$. Our next goal is to prove that
${\boldsymbol{x}}=\boldsymbol{a}_{0}$ is the only possible zero of
$d({\boldsymbol{x}})$ (it might happen that this function does not vanish at
all).

Assume that $d({\boldsymbol{x}})=0$, in which case
$\mathrm{det}(\boldsymbol{T}({\boldsymbol{x}}))=\mathrm{det}(\boldsymbol{T}({\boldsymbol{x}}_{1}))=0$.
This means that rows of $\boldsymbol{T}({\boldsymbol{x}})$ and
$\boldsymbol{T}({\boldsymbol{x}}_{1})$ are linearly dependent and since these
matrices share a column, there exists a constant $c$ such that

$2t_{2}(\boldsymbol{y})-h_{1}t_{1}(\boldsymbol{y})+ct_{0}(\boldsymbol{y})=0,\quad\boldsymbol{y}\in\\{{\boldsymbol{x}}_{-1},{\boldsymbol{x}},{\boldsymbol{x}}_{1}\\}.$

Hence, the matrix $[t_{l}(\boldsymbol{y})]$ has zero determinant, where $l$ is
the column index and
$\boldsymbol{y}\in\\{{\boldsymbol{x}}_{-1},{\boldsymbol{x}},{\boldsymbol{x}}_{1}\\}$
is the row one. By taking linear combinations of rows rather than columns, we
get that there exist constants $c_{-1},c_{0},c_{1}$, not all equal to zero,
such that

(4.17)
$c_{-1}t_{l}({\boldsymbol{x}}_{-1})+c_{0}t_{l}({\boldsymbol{x}})+c_{1}t_{l}({\boldsymbol{x}}_{1})=0,\quad
l\in\\{0,1,2\\}.$

In another connection, since the integrand in (2.4) behaves like
$\sqrt{z-a_{0}}$ around $\boldsymbol{a}_{0}$, it holds that

(4.18) $\Phi(z)=1+\mathcal{O}\left((z-a_{0})^{3/2}\right)$

as $z\to a_{0}$. Consider the function

$\displaystyle F({\boldsymbol{z}})$ $\displaystyle:=$ $\displaystyle
c_{-1}\frac{T({\boldsymbol{z}};{\boldsymbol{x}}_{-1})}{\Phi({\boldsymbol{z}})T({\boldsymbol{z}};{\boldsymbol{x}})}+c_{0}+c_{1}\frac{\Phi({\boldsymbol{z}})T({\boldsymbol{z}};{\boldsymbol{x}}_{1})}{T({\boldsymbol{z}};{\boldsymbol{x}})}$
$\displaystyle=$
$\displaystyle\frac{c_{-1}T({\boldsymbol{z}};{\boldsymbol{x}}_{-1})+c_{0}\Phi({\boldsymbol{z}})T({\boldsymbol{z}};{\boldsymbol{x}})+c_{1}\Phi^{2}({\boldsymbol{z}})T({\boldsymbol{z}};{\boldsymbol{x}}_{1})}{\Phi({\boldsymbol{z}})T({\boldsymbol{z}};{\boldsymbol{x}})}.$

It follows from (2.6), (4.15), and (4.16) that this is a rational function on
the entire surface $\mathfrak{S}$. Moreover, the first representation readily
yields that $F({\boldsymbol{z}})$ has exactly three poles, all simple, at
${\boldsymbol{x}}$, $\infty^{(0)}$, and $\infty^{(1)}$, while the second
formula together with (4.17) and (4.18) clearly shows that it has at least a
triple zero at $\boldsymbol{a}_{0}$ (unless
${\boldsymbol{x}}=\boldsymbol{a}_{0}$, in which case it is at least a double
zero). Since $F({\boldsymbol{z}})$ must have equal number of poles and zeros,
its zero/pole divisor is equal to

$3\boldsymbol{a}_{0}-{\boldsymbol{x}}-\infty^{(0)}-\infty^{(1)}.$

However, this means that the function $F({\boldsymbol{z}})/(z-a_{0})$ has a
zero/pole divisor $\boldsymbol{a}_{0}-{\boldsymbol{x}}$. Since there are no
rational functions with at most one pole besides constants,
$F({\boldsymbol{z}})=C(z-a_{0})$ for some constant $C$ and it must hold that
$\boldsymbol{a}_{0}={\boldsymbol{x}}$ as claimed.

Now, it follows from (2.29) and (4.14) that

$\big{|}\mathrm{det}(\boldsymbol{T}_{n})\big{|}=Y_{n-1}Y_{n}\big{|}\mathrm{det}(\boldsymbol{T}({\boldsymbol{z}}_{n}))\big{|},\quad
Y_{n}=\begin{cases}1,&y({\boldsymbol{z}}_{n})=y_{n},\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
e^{\pi\mathsf{Im}((2-2y_{n}+\tau)\mathsf{B}-2c_{\rho})},&y({\boldsymbol{z}}_{n})=y_{n}-1.\end{cases}$

Since $\mathsf{Im}(\mathsf{B})>0$, $\tau\in(-1,0)$, and $y_{n}\in[0,1)$, we
have that $Y_{n}\geq Y:=e^{-\pi\mathsf{Im}(2c_{\rho}+\mathsf{B})}$ for all
$n\in\mathbb{N}$. Let
$\boldsymbol{V}_{\varepsilon}:=d^{-1}([0,2Y^{-2}\varepsilon))$. Then for all
$\varepsilon$ small enough this set is either empty (when
$d({\boldsymbol{x}})$ does not vanish) or it is a neighborhood of
$\boldsymbol{a}_{0}$ by continuity of $d({\boldsymbol{x}})$. Then if
${\boldsymbol{z}}_{n}\notin\boldsymbol{V}_{\varepsilon}$, it holds that

$|\mathrm{det}(\boldsymbol{T}_{n})|+|\mathrm{det}(\boldsymbol{T}_{n+1})|\geq
Y^{2}d({\boldsymbol{z}}_{n})\geq 2\varepsilon,$

which yields that
$|\mathrm{det}(\boldsymbol{T}_{m})|\geq|\mathrm{det}(\boldsymbol{T}({\boldsymbol{z}}_{m}))|\geq\varepsilon$
for either $m=n$ or $m=n+1$.

##  5\. Proof of Theorem 2

###  5.1. Initial Riemann-Hilbert Problem

Recall the definition of $R_{n}(z)$ in (1.1) as well as the definition of
$f(z)$ in (1.4). It follows from the known behavior of Cauchy integrals around
the endpoints of arcs of integration [12, Sections 8.2-4] (we can consider
$f(z)$ as a sum of three Cauchy integrals over the arcs
$\Delta_{1},\Delta_{2},\Delta_{3}$) that

$|f(z)|,|R_{n}(z)|=\mathcal{O}(\psi_{\alpha_{l}}(z-a_{l})),\quad\psi_{\alpha}(z):=\left\\{\begin{array}[]{rl}1,&\alpha>0,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \log|z|,&\alpha=0,\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ |z|^{\alpha},&\alpha<0,\end{array}\right.$

for $l\in\\{0,1,2,3\\}$ (it was pointed out in [2] that these functions are
bounded around $a_{0}$ when $\alpha_{0}=0$ and
$\rho_{1}(0)+\rho_{2}(0)+\rho_{3}(0)=0$, where, as before, $\rho_{i}(s)$ is
the restriction of $\rho(s)$ to
$\Delta_{i}^{\circ}=\Delta_{i}\setminus\\{a_{0},a_{i}\\}$). Assume now that

(5.1) $\deg(Q_{n})=n\quad\text{and}\quad
R_{n-1}(z)=k_{n}^{-1}z^{-n}+\mathcal{O}\left(z^{-n-1}\right),$

where the second relation must hold as $z\to\infty$ for some finite non-zero
constant $k_{n}$. Let

(5.2) $\boldsymbol{Y}(z):=\begin{pmatrix}Q_{n}(z)&R_{n}(z)\\\
k_{n-1}Q_{n-1}(z)&k_{n-1}R_{n-1}(z)\end{pmatrix}.$

Then $\boldsymbol{Y}(z)$ solves the following Riemann-Hilbert problem
(RHP-${\boldsymbol{Y}}$):

  * (a)

$\boldsymbol{Y}(z)$ is analytic in $\mathbb{C}\setminus\Delta$ and
$\displaystyle\lim_{z\to\infty}\boldsymbol{Y}(z)z^{-n\sigma_{3}}=\boldsymbol{I}$,
where $\sigma_{3}:=\left(\begin{matrix}1&0\\\ 0&-1\end{matrix}\right)$ and
$\boldsymbol{I}=\left(\begin{matrix}1&0\\\ 0&1\end{matrix}\right)$;

  * (b)

$\boldsymbol{Y}(z)$ has continuous traces on each $\Delta_{i}^{\circ}$,
$i\in\\{1,2,3\\}$, that satisfy

$\boldsymbol{Y}_{+}(s)=\boldsymbol{Y}_{-}(s)\left(\begin{matrix}1&\rho_{i}(s)\\\
0&1\end{matrix}\right),\quad s\in\Delta_{i}^{\circ};$

  * (c)

$\displaystyle\boldsymbol{Y}(z)=\mathcal{O}\left(\begin{matrix}1&\psi_{\alpha_{l}}(z-a_{l})\\\
1&\psi_{\alpha_{l}}(z-a_{l})\end{matrix}\right)$ as $\Delta\not\ni z\to
a_{l}$, $l\in\\{0,1,2,3\\}$.

It is by now standard to show, see for example [2, 27], that if a solution of
RHP-${\boldsymbol{Y}}$ exists then it has the form (5.2) and (5.1) holds.
Notice that $\mathrm{det}(\boldsymbol{Y}(z))$ is holomorphic in
$\mathbb{C}\setminus\Delta$, equal to $1$ at infinity, is continuous across
$\Delta^{\circ}$ and can have at most removable singularity at each $a_{l}$.
Hence, $\mathrm{det}(\boldsymbol{Y}(z))\equiv 1$.

###  5.2. Opening of Lenses

Now, we construct a system of arcs, the lens, around $\Delta$ as on Figure 3.

$a_{1}$$\Delta_{1}^{\star}$$\Omega_{1}^{+}$$\Omega_{1}^{-}$$\Gamma_{1}^{+}$$\Gamma_{1}^{-}$$a_{2}$$\Delta_{2}^{\star}$$\Omega_{2}^{+}$$\Omega_{2}^{-}$$\Gamma_{2}^{+}$$\Gamma_{2}^{-}$$a_{3}$$\Delta_{3}^{\star}$$\Omega_{3}^{-}$$\Omega_{3}^{+}$$\Gamma_{3}^{-}$$\Gamma_{3}^{+}$

Figure 3. Contour $\Gamma$. The dotted circle represents $\partial U_{0}$.

Fix $\delta>0$ such that $\Delta\cap U_{0}$ is a smaller trefoil, where
$U_{0}:=\\{z:|z-a_{0}|<\delta\\}$, see the dotted circle on Figure 3. We are
assuming that $\delta$ is small enough so that
$\overline{U}_{0}\cap(\pi(\boldsymbol{\alpha})\cup\pi(\boldsymbol{\beta}))=\varnothing$.
Denote by $S_{i}$ the sector of $U_{0}\setminus\Delta$ for which $\partial
S_{i}\cap\Delta_{i}=\\{a_{0}\\}$ (i.e., the sector opposite to $\Delta_{i}$).
Further, let $U_{0}^{\pm}$ be the connected components of
$U_{0}\setminus(\Delta_{1}\cup\Delta_{1}^{\star})$ that border
$\Delta_{1}^{\pm}$ ($U_{0}^{+}$ intersects $\Delta_{3}$ and $U_{0}^{-}$
intersects $\Delta_{2}$), where $\Delta_{i}^{\star}$ is an open analytic arc
that splits the sector $S_{i}$ into two halves (its endpoints are $a_{0}$ and
some point on $\partial S_{i}\setminus\Delta$) and has the same tangent vector
at $a_{0}$ as $\Delta_{i}$ (we shall completely fix $\Delta_{i}^{\star}$ later
in Section 5.4, see (5.9)), see Figure 3. We orient each $\Delta_{i}^{\star}$
towards $a_{0}$. Further, let $\Gamma_{i}^{+}$ and $\Gamma_{i}^{-}$ be open
smooth arcs that connect $a_{i}$ to $\Delta_{i-1}^{\star}$ and
$\Delta_{i+1}^{\star}$, respectively, where the subindices are understood
cyclically within $\\{1,2,3\\}$, a convention we abide by throughout the rest
of the paper. We assume that all these arcs do not intersect except for the
common endpoints and let

$\Gamma:=\Delta\bigcup\big{(}\cup_{i}\Delta_{i}^{\star}\big{)}\bigcup\big{(}\cup_{i}(\Gamma_{i}^{+}\cup\Gamma_{i}^{-})\big{)}.$

We let $\Omega_{i}^{\pm}$ to denote bounded components of the complement of
$\Gamma$ that contain $\Gamma_{i}^{\pm}$ as a part of their boundary and
$\Gamma^{\circ}$ be the subset of those points in $\Gamma$ that possess a well
defined tangent.

Recall our assumption made after (1.4) that
$\rho_{i}(s)(s-a_{0})^{-\alpha_{0}}(s-a_{i})^{-\alpha_{i}}$ extends to an
analytic function in some neighborhood of $\Delta_{i}$. We shall suppose that
this neighborhood contains the closure of $\Omega_{i}^{+}\cup\Omega_{i}^{-}$
and that the branch of $(z-a_{i})^{\alpha_{i}}$ is analytic throughout this
closure, except, of course, at $a_{i}$. Recall also that we fixed a branch of
$(z-a_{0})^{\alpha_{0}}$ whose branch cut lies, in part, in $S_{1}$ and we
shall further assume that this cut contains $\Delta_{1}^{\star}$. Finally,
recall (3.1) and suppose that $U_{0}$ is small enough so that every
$\varrho_{i}(z)$ is analytic in its closure.

Define

(5.3)
$\boldsymbol{X}(z):=\boldsymbol{Y}(z)\left\\{\begin{array}[]{rl}\left(\begin{matrix}1&0\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \mp
1/\rho_{i}(z)&1\end{matrix}\right),&z\in\Omega_{i}^{\pm},\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \boldsymbol{I},&\text{otherwise}.\end{array}\right.$

Clearly, if $\boldsymbol{Y}(z)$ is a solution of RHP-${\boldsymbol{Y}}$, then
$\boldsymbol{X}(z)$ is a solution of the following Riemann-Hilbert problem
(RHP-${\boldsymbol{X}}$):

  * (a)

$\boldsymbol{X}(z)$ is analytic in $\mathbb{C}\setminus\Delta$ and
$\lim_{z\to\infty}\boldsymbol{X}(z)z^{-n\sigma_{3}}=\boldsymbol{I}$;

  * (b)

$\boldsymbol{X}(z)$ has continuous traces on $\Gamma^{\circ}$ that satisfy

$\boldsymbol{X}_{+}(s)=\boldsymbol{X}_{-}(s)\left\\{\begin{array}[]{rl}\left(\begin{matrix}0&\rho_{i}(s)\\\
-\rho_{i}^{-1}(s)&0\end{matrix}\right),&s\in\Delta_{i}^{\circ},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \left(\begin{matrix}1&0\\\
\rho_{i}^{-1}(s)&1\end{matrix}\right),&s\in\Gamma_{i}^{\pm},\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \left(\begin{matrix}1&0\\\
\rho_{i+1+}^{-1}(s)+\rho_{i-1-}^{-1}(s)&1\end{matrix}\right),&s\in\Delta_{i}^{\star},\end{array}\right.$

where taking traces is important only on $\Delta_{1}^{\star}$;

  * (c)

for each $l\in\\{0,1,2,3\\}$ it holds that

$\boldsymbol{X}(z)=\mathcal{O}\left(\begin{matrix}1&|z-a_{l}|^{\alpha_{l}}\\\
1&|z-a_{l}|^{\alpha_{l}}\end{matrix}\right)\quad\text{and}\quad\boldsymbol{X}(z)=\mathcal{O}\left(\begin{matrix}\log|z-a_{l}|&\log|z-a_{l}|\\\
\log|z-a_{l}|&\log|z-a_{l}|\end{matrix}\right)$

as $\Gamma\not\ni z\to a_{l}$ when $\alpha_{l}<0$ and $\alpha_{l}=0$,
respectively, and

$\boldsymbol{X}(z)=\mathcal{O}\left(\begin{matrix}1&1\\\
1&1\end{matrix}\right)\quad\text{and}\quad\boldsymbol{X}(z)=\mathcal{O}\left(\begin{matrix}|z-a_{l}|^{-\alpha_{l}}&1\\\
|z-a_{l}|^{-\alpha_{l}}&1\end{matrix}\right)$

as $\Gamma\not\ni z\to a_{l}$, from within the unbounded and bounded
components of the complement of $\Gamma$, respectively, when $\alpha_{l}>0$.

Conversely, assume that $\boldsymbol{X}(z)$ is a solution of
RHP-${\boldsymbol{X}}$ and $\boldsymbol{Y}(z)$ is defined by inverting (5.3).
It is quite easy to see that thus defined $\boldsymbol{Y}(z)$ satisfies
RHP-${\boldsymbol{Y}}$(a,b). Furthermore, since the second columns of
$\boldsymbol{X}(z)$ and $\boldsymbol{Y}(z)$ coincide, the second column of
$\boldsymbol{Y}(z)$ satisfies RHP-${\boldsymbol{Y}}$(c) as well. In fact, the
same holds for the first column of $\boldsymbol{Y}(z)$ if $\alpha_{l}<0$. In
other cases, we see from RHP-${\boldsymbol{Y}}$(b) that the first column of
$\boldsymbol{Y}(z)$ can have at most an isolated singularity at $a_{l}$.
However, if $\alpha_{l}=0$, then blowing up at $a_{l}$ is at most logarithmic
and therefore singularity must be removable. Similarly, when $\alpha_{l}>0$
and $l\in\\{1,2,3\\}$, the first column of $\boldsymbol{Y}(z)$ remains bounded
as $z\to a_{l}$ from outside the lens and therefore the singularity at $a_{l}$
cannot be polar. Furthermore, in this case
$|z-a_{l}|^{\alpha_{l}}\boldsymbol{Y}(z)$ is bounded around $a_{l}$ and
therefore the singularity cannot be essential. Thus, it is again removable.
Finally, since $a_{0}$ cannot be approached from outside the lens, we rule out
polar (and essential) singularity at $a_{0}$ due to the requirement
$\alpha_{0}<1$.

###  5.3. Parametrices

Below we assume that the index $n$ is such that
${\boldsymbol{z}}_{n}\neq\infty^{(0)}$, see (2.19).

The global parametrix is obtained from RHP-${\boldsymbol{X}}$ by ignoring
local behavior around $a_{i}$ and discarding the jumps on $\Delta_{i}^{\star}$
and $\Gamma_{i}^{\pm}$. Namely, let $\Psi_{n}({\boldsymbol{z}})$ be given by
(2.20). Recall our convention that $\Psi_{n}(z)$ and $\Psi_{n}^{*}(z)$ stand
for the pull-backs of $\Psi_{n}({\boldsymbol{z}})$ from $\mathfrak{S}^{(0)}$
and $\mathfrak{S}^{(1)}$, respectively. Define

$\boldsymbol{N}(z):=\begin{pmatrix}\gamma_{n}\Psi_{n}(z)&\gamma_{n}\Psi_{n}^{*}(z)/w(z)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\gamma_{n-1}^{*}\Psi_{n-1}(z)&\gamma_{n-1}^{*}\Psi_{n-1}^{*}(z)/w(z)\end{pmatrix}$

for $z\not\in\Delta$, where the constants $\gamma_{n},\gamma_{n-1}^{*}$ are
chosen so that
$\lim_{z\to\infty}\boldsymbol{N}(z)z^{-n\sigma_{3}}=\boldsymbol{I}$. By the
restriction placed on the indices $n$, these constants are finite and non-
zero, see the last claim of Proposition 3. Then $\boldsymbol{N}(z)$ is a
solution of the following Riemann-Hilbert problem (RHP-${\boldsymbol{N}}$):

  * (a)

$\boldsymbol{N}(z)$ is analytic in $\mathbb{C}\setminus\Delta$ and
$\lim_{z\to\infty}\boldsymbol{N}(z)z^{-n\sigma_{3}}=\boldsymbol{I}$;

  * (b)

$\boldsymbol{N}(z)$ has continuous traces on each $\Delta_{i}^{\circ}$,
$i\in\\{1,2,3\\}$, that satisfy

$\boldsymbol{N}_{+}(s)=\boldsymbol{N}_{-}(s)\left(\begin{matrix}0&\rho_{i}(s)\\\
-\rho_{i}^{-1}(s)&0\end{matrix}\right),\quad s\in\Delta_{i}^{\circ};$

  * (c)

$\displaystyle\boldsymbol{N}(z)=\mathcal{O}\left(\begin{matrix}|z-a_{l}|^{-(2\alpha_{l}+1)/4}&|z-a_{l}|^{(2\alpha_{l}-1)/4}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
|z-a_{l}|^{-(2\alpha_{l}+1)/4}&|z-a_{l}|^{(2\alpha_{l}-1)/4}\end{matrix}\right)$
as $z\to a_{l}$, $l\in\\{0,1,2,3\\}$.

The properties of $\boldsymbol{N}(z)$ described above follow easily from the
requirements of BVP-$\Psi$.

In fact, it will be more convenient for us later to work with separate factors
of $\boldsymbol{N}(z)$ rather than $\boldsymbol{N}(z)$ itself. To this end we
write
$\boldsymbol{N}(z):=\boldsymbol{C}\boldsymbol{M}^{\mathsf{reg}}(z)\Phi^{n\sigma_{3}}(z)$,
where

(5.4) $\boldsymbol{C}:=\begin{pmatrix}\gamma_{n}&0\\\
0&\gamma_{n-1}^{*}\end{pmatrix}\quad\text{and}\quad\displaystyle\boldsymbol{M}^{\mathsf{reg}}(z):=\begin{pmatrix}T_{n}(z)&T_{n}^{*}(z)/w(z)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
(T_{n-1}\Phi^{*})(z)&(T_{n-1}^{*}\Phi)(z)/w(z)\end{pmatrix}S_{\rho}^{\sigma_{3}}(z),$

where we used the facts that $\Phi(z)\Phi^{*}(z)\equiv 1$ and
$S_{\rho}(z)S_{\rho}^{*}(z)\equiv 1$, see Proposition 1. As in the case of
$\boldsymbol{Y}(z)$, it is straightforward to argue that
$\mathrm{det}(\boldsymbol{N}(z))\equiv 1$ in $\mathbb{C}$. Hence, it holds
that

$\mathrm{det}(\boldsymbol{M}^{\mathsf{reg}}(z))\equiv
1/\mathrm{det}(\boldsymbol{C})=1/(\gamma_{n}\gamma_{n-1}^{*}).$

Let $\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ be as in (2.22). Then it follows
from the above identity, (5.4), and (2.20) that

(5.5)
$(\gamma_{n}\gamma_{n-1}^{*})^{-1}=T_{n}(\infty)\lim_{z\to\infty}\frac{T_{n-1}^{*}(z)\Phi(z)}{w(z)}.$

It is well known (and is not very important for us here) that
$\lim_{z\to\infty}|\Phi(z)/z|=1/\mathrm{cap}(\Delta)$ is the reciprocal of the
logarithmic capacity of $\Delta$ (in any case, it is a non-zero and finite
constant). Hence, it follows from (2.26)–(2.27) that

(5.6)
$\mathrm{cap}(\Delta)RM_{R}^{-2}\leq|\gamma_{n}\gamma_{n-1}^{*}|\leq\mathrm{cap}(\Delta)M_{\varepsilon}^{2},\quad
n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}},$

for any given $R>\max_{s\in\Delta}|s|$. We further get from (2.26)–(2.27) and
the above inequalities that

(5.7)
$\begin{cases}\boldsymbol{M}^{\mathsf{reg}}(z)&=\boldsymbol{\mathcal{O}}_{K,\varepsilon}(1)\mathrm{diag}(1,1/w(z))S_{\rho}^{\sigma_{3}}(z)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\boldsymbol{M}^{\mathsf{reg}}(z)^{-1}&=S_{\rho}^{-\sigma_{3}}(z)\mathrm{diag}(1/w(z),1)\boldsymbol{\mathcal{O}}_{K,\varepsilon}(1)\end{cases}$

on any compact set $K$ as $\mathbb{N}_{\varepsilon}^{\mathsf{reg}}\ni
n\to\infty$.

Assume now that $\delta>0$ is small enough so that
$U_{i}:=\\{z:|z-a_{i}|<\delta\\}$, $i\in\\{1,2,3\\}$, intersects only
$\Delta_{i}$, i.e., $U_{i}\cap\Delta_{j}=\varnothing$ for $j\neq i$, and that
$\rho_{i}(s)(s-a_{i})^{-\alpha_{i}}$ extends analytically to its closure,
which is disjoint from the closure of $U_{0}$. Local parametrix around $a_{i}$
is obtained by solving RHP-${\boldsymbol{X}}$ in $U_{i}$. That is, we need to
solve the following Riemann-Hilbert problem (RHP-${\boldsymbol{P}_{i}}$):

  * (a)

$\boldsymbol{P}_{i}(z)$ is analytic in $U_{i}\setminus\Gamma$;

  * (b,c)

$\boldsymbol{P}_{i}(z)$ satisfies RHP-${\boldsymbol{X}}$(b,c) within $U_{i}$;

  * (d)

it holds that
$\boldsymbol{P}_{i}(s)=(\boldsymbol{I}+\boldsymbol{\mathcal{O}}(1/n))\boldsymbol{M}^{\mathsf{reg}}(s)\Phi^{n\sigma_{3}}(s)$
uniformly on $\partial U_{i}$.

The solution of RHP-${\boldsymbol{P}_{i}}$ is well known and was first derived
in [18] in the context of orthogonal polynomials on a segment with the help of
the modified Bessel and Hankel functions. Since then it appeared in countless
papers and was adopted to the context of non-Hermitian orthogonal polynomials
on symmetric contours in [2]. Therefore, we shall skip the explicit
construction.

Around $a_{0}$, we look only for an approximate parametrix. More precisely, we
would like to solve the following Riemann-Hilbert problem
(RHP-${\boldsymbol{P}_{0}}$):

  * (a)

$\boldsymbol{P}_{0}(z)$ is analytic in $U_{0}\setminus\Gamma$;

  * (b)

$\boldsymbol{P}_{0}(z)$ satisfies RHP-${\boldsymbol{X}}$(b) within $U_{0}$
except the $21$-entries of the jump matrices on $\Delta_{i}^{\star}$,
$i\in\\{1,2,3\\}$, are replaced by

$\kappa_{i,i-1}(s)\rho_{i+1+}^{-1}(s)+\kappa_{i,i+1}(s)\rho_{i-1-}^{-1}(s),$

where
$\kappa_{i,j}(z):=(\varrho_{i}(z)\varrho_{j}(a_{0}))/(\varrho_{i}(a_{0})\varrho_{j}(z))$
(recall (3.1));

  * (c)

$\boldsymbol{P}_{0}(z)$ satisfies RHP-${\boldsymbol{X}}$(c) within $U_{0}$;

  * (d)

it holds that
$\boldsymbol{P}_{0}(s)=\big{(}\boldsymbol{I}+\boldsymbol{\mathcal{O}}\big{(}n^{-1/3}\big{)}\big{)}\boldsymbol{M}^{\mathsf{reg}}(s)\Phi^{n\sigma_{3}}(s)$
uniformly on $\partial U_{0}$.

We solve RHP-${\boldsymbol{P}_{0}}$ in the next subsection. The presence of
the functions $\kappa_{i,j}(z)$ is precisely what makes this parametrix
approximate. Of course, it would be preferable to solve RHP-${\boldsymbol{X}}$
locally around $a_{0}$ exactly, that is, with the jumps appearing in
RHP-${\boldsymbol{X}}$(b). However, currently, we do not know how to achieve
this. Notice also that if $\varrho_{i}(s)=c_{i}\varrho(s)$ for some function
$\varrho(z)$ analytic around $a_{0}$ and possibly different constants $c_{i}$,
which is exactly the case when the approximated function has the form (1.3),
see Appendix D, then $\kappa_{i,j}(z)\equiv 1$ and the parametrix is exact.

###  5.4. Approximate Local Parametrix around $a_{0}$

Solution of RHP-${\boldsymbol{P}_{0}}$ is based on the solution of
RHP-${\boldsymbol{\Phi}}_{\alpha}$ from Appendix B. The knowledge of the
statement of that Riemann-Hilbert problem is sufficient for understanding the
construction further below.

Recall the definition of $\Sigma_{2},\Sigma_{3}$ in Appendix B. Define

(5.8)
$\zeta^{3}(z):=\left(\frac{3}{2}\int_{a_{0}}^{z}\frac{(s-a_{0})}{w(s)}\mathrm{d}s\right)^{2},\quad
z\in\big{(}S_{1}\cup S_{2}\cup S_{3}\big{)}\cap U_{0}.$

Since $w_{+}(s)=-w_{-}(s)$ for $s\in\Delta$, $\zeta^{3}(z)$ extends to a
holomorphic function in $U_{0}$, which, upon easy verification, has a triple
zero at $a_{0}$ (we already used this observation in (4.18)). Furthermore, it
readily follows from (1.2) that $\zeta^{3}(s)$ is negative for $s\in\Delta\cap
U_{0}$ (except for $a_{0}$, of course). Hence, we can select a branch
$\zeta(z)$ which is holomorphic in $U_{0}$ and such that $\zeta(s)<0$ when
$s\in\Delta_{1}\cap U_{0}$. Since $\zeta(z)$ has a simple zero at $a_{0}$, we
can decrease the radius of $U_{0}$ if necessary so that $\zeta(z)$ is
univalent in $U_{0}$. Moreover, as we had some freedom in choosing the arcs
$\Delta_{i}^{\star}$, we now fix them so that

(5.9)
$\zeta(\Delta_{1}^{\star})\subset(0,\infty)\quad\text{and}\quad\zeta(\Delta_{i}^{\star})\subset\Sigma_{i},\quad
i\in\\{2,3\\}.$

In what follows, all the roots of $\zeta(z)$ are principal, that is, they
remain positive on $\Delta_{1}^{\star}$ and have a branch cut along
$\Delta_{1}$. In particular,

(5.10) $\zeta_{+}^{1/4}(s)=\mathrm{i}\zeta_{-}^{1/4}(s),\quad
s\in\Delta_{1}\cap U_{0}.$

Recall the definition of $\Phi({\boldsymbol{z}})$ in (2.4) and that
$|\Phi(z)|>1$ for $z\in U_{0}\setminus\Delta$, see (2.7). Thus, it also holds
that

(5.11) $e^{-(2/3)\zeta^{3/2}(z)}=\left\\{\begin{array}[]{rl}\Phi^{*}(z),&z\in
S_{1}\cap U_{0},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \Phi(z),&z\in(S_{2}\cup
S_{3})\cap U_{0}.\end{array}\right.$

Let $\varrho_{i}(s)$ be given by (3.1), where for definiteness we fix a branch
of $(z-a_{0})^{\alpha_{0}}$ that is analytic in
$U_{0}\setminus\overline{\Delta_{1}^{\star}}$. Define

$E_{\alpha_{0}}(z):=\left\\{\begin{array}[]{ll}e^{\pm\pi\mathrm{i}\alpha_{0}/2},&z\in
S_{1}\cap U_{0}^{\pm},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
e^{\mp\pi\mathrm{i}\alpha_{0}/2},&z\in(S_{2}\cup S_{3})\cap
U_{0}^{\pm}.\end{array}\right.$

Further, let $r_{i}(z)$, $i\in\\{1,2,3\\}$, be a function holomorphic in
$S_{i}$ given by

(5.12)
$r_{i}(z):=\varsigma_{i}\mathrm{i}(z-a_{0})^{\alpha_{0}/2}E_{\alpha_{0}}(z)\left(\frac{\varrho_{i-1}(z)\varrho_{i+1}(z)}{\varrho_{i}(z)}\right)^{1/2},$

where $\varsigma_{1}=\varsigma_{2}=-\varsigma_{3}=1$ and we use the same
square root branches of $\varrho_{j}(z)$ in all $r_{i}(z)$. Since
$E_{\alpha_{0}+}(s)E_{\alpha_{0}-}(s)\equiv 1$ for $s\in\Delta_{i}^{\circ}\cap
U_{0}$, it holds that

(5.13) $r_{i-1-}(s)r_{i+1+}(s)=\varsigma_{i}\rho_{i}(s),\quad
s\in\Delta_{i}^{\circ}\cap U_{0}.$

We further set $r(z):=r_{i}(z)$ for $z\in S_{i}$.

Recall that $\rho\in W^{\mathsf{reg}}$. Let
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;0)$ be the solution of
RHP-${\boldsymbol{\Phi}}_{\alpha}$ with $\alpha=\alpha_{0}$ and the Stokes
parameters given by (3.2), see Appendix B (it exists by the very definition of
the class $W^{\mathsf{reg}}$). Then a solution of RHP-${\boldsymbol{P}_{0}}$
is given by

(5.14)
$\boldsymbol{P}_{0}(z):=\boldsymbol{E}_{0}^{\mathsf{reg}}(z)\boldsymbol{\Phi}_{\alpha_{0}}\big{(}n^{2/3}\zeta(z);0\big{)}\boldsymbol{J}(z)r^{-\sigma_{3}}(z),$

where $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$ is a holomorphic matrix function
in $U_{0}$ that we shall specify further below in (5.16) and

(5.15)
$\boldsymbol{J}(z):=\left\\{\begin{array}[]{rl}\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right),&z\in S_{1}\cap U_{0},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ \boldsymbol{I},&z\in(S_{2}\cup S_{3})\cap U_{0}.\end{array}\right.$

Indeed, it follows from (5.9) that $\boldsymbol{P}_{0}(z)$ is a holomorphic
matrix function in $U_{0}\setminus\Gamma$, i.e., RHP-${\boldsymbol{P}_{0}}$(a)
is satisfied. We further get from RHP-${\boldsymbol{\Phi}}_{\alpha}$(b) and
(5.13) that

$\boldsymbol{P}_{0-}^{-1}(s)\boldsymbol{P}_{0+}(s)=r_{3-}^{\sigma_{3}}(s)\left(\begin{matrix}0&1\\\
-1&0\end{matrix}\right)r_{2+}^{-\sigma_{3}}(s)=\left(\begin{matrix}0&\rho_{1}(s)\\\
-1/\rho_{1}(s)&0\end{matrix}\right)$

for $s\in\Delta_{1}^{\circ}\cap U_{0}$,

$\boldsymbol{P}_{0-}^{-1}(s)\boldsymbol{P}_{0+}(s)=r_{1-}^{\sigma_{3}}(s)\boldsymbol{J}_{-}^{-1}(s)r_{3+}^{-\sigma_{3}}(s)=\left(\begin{matrix}0&\rho_{2}(s)\\\
-1/\rho_{2}(s)&0\end{matrix}\right)$

for $s\in\Delta_{2}^{\circ}\cap U_{0}$, and

$\boldsymbol{P}_{0-}^{-1}(s)\boldsymbol{P}_{0+}(s)=r_{2-}^{\sigma_{3}}(s)\boldsymbol{J}_{+}(s)r_{1+}^{-\sigma_{3}}(s)=\left(\begin{matrix}0&\rho_{3}(s)\\\
-1/\rho_{3}(s)&0\end{matrix}\right)$

for $s\in\Delta_{3}^{\circ}\cap U_{0}$. Moreover, using (3.2) we get that

$\boldsymbol{P}_{0-}^{-1}(s)\boldsymbol{P}_{0+}(s)=r_{1}^{\sigma_{3}}(s)\boldsymbol{J}^{-1}(s)\left(\begin{matrix}1&-b_{1}(\rho)\\\
0&1\end{matrix}\right)\boldsymbol{J}(s)r_{1}^{-\sigma_{3}}(s)=\left(\begin{matrix}1&0\\\
\frac{\kappa_{1,2}(s)}{\rho_{2+}(s)}+\frac{\kappa_{1,3}(s)}{\rho_{3-}(s)}&1\end{matrix}\right)$

for $s\in\Delta_{1}^{\star}\cap U_{0}$ (notice that the orientations of
$\zeta(\Delta_{1}^{\star})$ and $\Sigma_{1}$ are opposite to each other), and

$\boldsymbol{P}_{0-}^{-1}(s)\boldsymbol{P}_{0+}(s)=r_{l}^{\sigma_{3}}(s)\left(\begin{matrix}1&0\\\
b_{l}(\rho)&1\end{matrix}\right)r_{l}^{-\sigma_{3}}(s)=\left(\begin{matrix}1&0\\\
\frac{\kappa_{l,l-1}(s)}{\rho_{l+1}(s)}+\frac{\kappa_{l,l+1}(s)}{\rho_{l-1}(s)}&1\end{matrix}\right)$

for $s\in\Delta_{l}^{\star}\cap U_{0}$, $l\in\\{2,3\\}$. Hence,
$\boldsymbol{P}_{0}(z)$, defined in (5.14), fulfills
RHP-${\boldsymbol{P}_{0}}$(b). Next, it follows from
RHP-${\boldsymbol{\Phi}}_{\alpha}$(c), (5.14), and (5.12) that

$\boldsymbol{P}_{0}(z)=\boldsymbol{E}_{0}^{\mathsf{reg}}(z)\left\\{\begin{array}[]{rl}\boldsymbol{\mathcal{O}}\left(|\zeta(z)|^{-|\alpha_{0}|/2}\right),&\alpha_{0}\neq
0\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\boldsymbol{\mathcal{O}}\left(\log|\zeta(z)|\right),&\alpha_{0}=0\end{array}\right\\}\mathcal{O}\left(|z-a_{0}|^{-\alpha_{0}\sigma_{3}/2}\right).$

Since multiplication by $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$ on the left
does not mix the entries from different columns of the product of the other
two factors above, $\boldsymbol{P}_{0}(z)$ does indeed satisfy
RHP-${\boldsymbol{P}_{0}}$(c).

Finally, let

(5.16)
$\boldsymbol{E}_{0}^{\mathsf{reg}}(z):=\boldsymbol{M}^{\mathsf{reg}}(z)r^{\sigma_{3}}(z)\boldsymbol{J}^{-1}(z)\left(\begin{matrix}1&-\mathrm{i}\\\
-\mathrm{i}&1\end{matrix}\right)\frac{\big{(}n^{2/3}\zeta(z)\big{)}^{\sigma_{3}/4}}{\sqrt{2}}.$

We get from RHP-${\boldsymbol{\Phi}}_{\alpha}$(d), (5.7), (5.11), and (5.15)
that

(5.17) $\displaystyle\boldsymbol{P}_{0}(s)$ $\displaystyle=$
$\displaystyle\boldsymbol{M}^{\mathsf{reg}}(s)r^{\sigma_{3}}(s)\left(\boldsymbol{I}+\boldsymbol{\mathcal{O}}\big{(}n^{-1/3}\big{)}\right)r^{-\sigma_{3}}(s)\Phi^{n\sigma_{3}}(s)$
$\displaystyle=$
$\displaystyle\left(\boldsymbol{I}+\boldsymbol{\mathcal{O}}\big{(}n^{-1/3}\big{)}\right)\boldsymbol{M}^{\mathsf{reg}}(s)\Phi^{n\sigma_{3}}(s),$

which is exactly what was required in RHP-${\boldsymbol{P}_{0}}$(d). Thus, it
only remains to prove that $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$ is analytic
in $U_{0}$. To this end, this matrix is clearly analytic in
$U_{0}\setminus\Delta$. It also holds that

$\displaystyle(\boldsymbol{E}_{0-}^{\mathsf{reg}})^{-1}(s)\boldsymbol{E}_{0+}^{\mathsf{reg}}(s)$
$\displaystyle=$ $\displaystyle
n^{-\sigma_{3}/6}\frac{\zeta_{-}^{-\sigma_{3}/4}(s)}{\sqrt{2}}\left(\begin{matrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{matrix}\right)\left(\begin{matrix}0&1\\\
-1&0\end{matrix}\right)\left(\begin{matrix}1&-\mathrm{i}\\\
-\mathrm{i}&1\end{matrix}\right)\frac{\zeta_{+}^{\sigma_{3}/4}(s)}{\sqrt{2}}n^{\sigma_{3}/6}$
$\displaystyle=$ $\displaystyle
n^{-\sigma_{3}/6}\zeta_{-}^{-\sigma_{3}/4}(s)\left(\begin{matrix}-\mathrm{i}&0\\\
0&\mathrm{i}\end{matrix}\right)\zeta_{+}^{\sigma_{3}/4}(s)n^{\sigma_{3}/6}=\boldsymbol{I}$

for $s\in\Delta_{1}^{\circ}\cap U_{0}$, where we used
RHP-${\boldsymbol{N}}$(b), (5.13), (5.15), and (5.10). Similarly, we have for
$s\in\Delta_{l}^{\circ}\cap U_{0}$, $l\in\\{2,3\\}$, that
$(\boldsymbol{E}_{0-}^{\mathsf{reg}})^{-1}(s)\boldsymbol{E}_{0+}^{\mathsf{reg}}(s)=\boldsymbol{I}$,
because it holds there that

$\boldsymbol{J}_{-}(s)\left(\begin{matrix}0&(-1)^{l}\\\
(-1)^{l-1}&0\end{matrix}\right)\boldsymbol{J}_{+}^{-1}(s)=\boldsymbol{I}.$

Hence, $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$ is analytic in
$U_{0}\setminus\\{a_{0}\\}$. Furthermore, we get from
RHP-${\boldsymbol{N}}$(c) and (5.12) that

$\boldsymbol{E}_{0}^{\mathsf{reg}}(z)=\boldsymbol{\mathcal{O}}\left(|z-a_{0}|^{-1/4}\right)\boldsymbol{J}^{-1}(s)\left(\begin{matrix}1&-\mathrm{i}\\\
-\mathrm{i}&1\end{matrix}\right)\frac{\zeta^{\sigma_{3}/4}(z)}{\sqrt{2}}n^{\sigma_{3}/6}=\boldsymbol{\mathcal{O}}\left(|z-a_{0}|^{-1/2}\right).$

Therefore, $a_{0}$ cannot be a polar singularity of
$\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$ and must be a point of analyticity.

###  5.5. Small-Norm Riemann-Hilbert Problem

Let $\boldsymbol{C}$ and $\boldsymbol{M}^{\mathsf{reg}}(z)$ be matrices
defined in (5.4) and $\boldsymbol{P}_{l}(z)$, $l\in\\{0,1,2,3\\}$, be the
matrix functions solving RHP-${\boldsymbol{P}_{0}}$ and
RHP-${\boldsymbol{P}_{i}}$, $i\in\\{1,2,3\\}$. Further, let $U:=U_{0}\cup
U_{1}\cup U_{2}\cup U_{3}$, whose boundary we orient clockwise.

$\partial U_{1}$$\partial U_{2}$$\partial U_{3}$$\partial U_{0}$

Figure 4. Contour $\Sigma$ and the circles $\partial U_{l}$.

We are looking for the solution of RHP-${\boldsymbol{X}}$ in the form

(5.18)
$\boldsymbol{X}(z)=\boldsymbol{C}\boldsymbol{Z}(z)\left\\{\begin{array}[]{rl}\boldsymbol{M}^{\mathsf{reg}}(z)\Phi^{n\sigma_{3}},&z\not\in\left(\Gamma\cup\overline{U}\right),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \boldsymbol{P}_{l}(z),&z\in
U_{l}\setminus\Gamma,\quad l\in\\{0,1,2,3\\}.\end{array}\right.$

Let $\Sigma:=\partial U\cup\big{(}\Gamma\setminus(\Delta^{\circ}\cup U_{1}\cup
U_{2}\cup U_{3})\big{)}$, see Figure 4. Combining the above equation with
RHP-${\boldsymbol{X}}$, we see that $\boldsymbol{Z}(z)$ should solve the
following Riemann-Hilbert problem (RHP-${\boldsymbol{Z}}$):

  * (a)

$\boldsymbol{Z}(z)$ is analytic in $\mathbb{C}\setminus\Sigma$ and
$\lim_{z\to\infty}\boldsymbol{Z}(z)=\boldsymbol{I}$;

  * (b)

$\boldsymbol{Z}(z)$ has continuous traces on $\Sigma^{\circ}$ that satisfy

$\boldsymbol{Z}_{+}(s)=\boldsymbol{Z}_{-}(s)\left\\{\begin{array}[]{rl}\boldsymbol{P}_{l}(s)\boldsymbol{N}_{*}^{-1}(s),&s\in\partial
U,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\boldsymbol{N}_{*}(s)\left(\begin{matrix}1&0\\\
\rho_{i}^{-1}(s)&1\end{matrix}\right)\boldsymbol{N}_{*}^{-1}(s),&s\in\Gamma_{i}^{\pm}\setminus\overline{U},\end{array}\right.$

where
$\boldsymbol{N}_{*}(z):=\boldsymbol{C}^{-1}\boldsymbol{N}^{\mathsf{reg}}(z)=\boldsymbol{M}^{\mathsf{reg}}(z)\Phi^{n\sigma_{3}}(z)$
and we understand that $\boldsymbol{P}_{l}(s)$ is used when $s\in\partial
U_{l}$, $l\in\\{0,1,2,3\\}$, in the first jump relation, and

$\boldsymbol{Z}_{+}(s)=\boldsymbol{Z}_{-}(s)\boldsymbol{P}_{0-}(s)\left(\begin{matrix}1&0\\\
\rho_{i+1+}^{-1}(s)+\rho_{i-1-}^{-1}(s)&1\end{matrix}\right)\boldsymbol{P}_{0+}^{-1}(s),\quad
s\in\Delta_{i}^{\star}.$

Note that if the jump relations RHP-${\boldsymbol{P}_{0}}$(b) on
$\Delta_{i}^{\star}$ are exactly the same as the ones in
RHP-${\boldsymbol{X}}$(b), then the jump relations on $\Delta_{i}^{\star}\cap
U_{0}$ in RHP-${\boldsymbol{Z}}$(b) above are absent.

As usual in this line of arguments, we shall show that all the jump matrices
in RHP-${\boldsymbol{Z}}$(b) are uniformly close to $\boldsymbol{I}$ along any
sequence $\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$. Indeed, it follows from
RHP-${\boldsymbol{P}_{0}}$(d) and RHP-${\boldsymbol{P}_{i}}$(d),
$i\in\\{1,2,3\\}$, that the jump of $\boldsymbol{Z}(z)$ on $\partial U$ can be
estimated as

(5.19)
$\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\varepsilon}\big{(}n^{-1/3}\big{)}\quad\text{on}\quad\partial
U,$

where the constants in $\boldsymbol{\mathcal{O}}_{\varepsilon}(\cdot)$ are
independent of $n$ but do depend on $\varepsilon$. Furthermore, we get from
(5.7) as well as (2.7) that the jump of $\boldsymbol{Z}(z)$ on
$\Gamma_{i}^{\pm}\setminus\overline{U}$ satisfies

(5.20)
$\boldsymbol{I}+\rho_{i}^{-1}(s)\Phi^{-2n}(s)\boldsymbol{M}^{\mathsf{reg}}(s)\left(\begin{matrix}0&0\\\
1&0\end{matrix}\right)(\boldsymbol{M}^{\mathsf{reg}})^{-1}(s)=\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\varepsilon}\big{(}e^{-cn}\big{)}$

for some constant $c>0$, where we also used the fact that the arcs
$\Gamma_{i}^{\pm}$ lie fixed distance away from $\Delta$ (again, the constant
in $\boldsymbol{\mathcal{O}}_{\varepsilon}(\cdot)$ is independent of $n$ but
depends on $\varepsilon$).

Now, if it holds that $\kappa_{i,j}(z)\equiv 1$, then $\boldsymbol{P}_{0}(z)$
is an exact parametrix and $\boldsymbol{Z}(z)$ does not have jumps on
$\Delta_{i}^{\star}$. Below, we assume that all of these functions are not
identically $1$, the case where only some of them are can be considered
analogously. Let $p$ be the largest integer such that
$|1-\kappa_{i,j}(s)|=\mathcal{O}(|s-a_{0}|^{p})$ as $s\to a_{0}$ for all the
pairs $i,j$. It necessarily holds that $p\geq 1$. It follows from
RHP-${\boldsymbol{P}_{0}}$(b) that the jump of $\boldsymbol{Z}(z)$ on
$\Delta_{i}^{\star}$ can be written as

(5.21)
$\boldsymbol{I}+\left(\frac{1-\kappa_{i,i-1}(s)}{\rho_{i+1+}(s)}+\frac{1-\kappa_{i,i+1}(s)}{\rho_{i-1-}(s)}\right)\boldsymbol{P}_{0+}(s)\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\boldsymbol{P}_{0+}^{-1}(s)\\\
=\boldsymbol{I}+\mathcal{O}\big{(}|s-a_{0}|^{p}\big{)}\boldsymbol{E}_{0}^{\mathsf{reg}}(s)\boldsymbol{\Phi}_{\alpha_{0}}\left(n^{2/3}\zeta(s);0\right)\boldsymbol{J}\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\boldsymbol{J}^{-1}\boldsymbol{\Phi}_{\alpha_{0}}^{-1}\left(n^{2/3}\zeta(s);0\right)\boldsymbol{E}_{0}^{\mathsf{reg}}(s)^{-1},$

where we used estimates
$|\rho_{i}(s)|\sim|s-a_{0}|^{\alpha_{0}}\sim|r(s)|^{2}$, see (5.12) and
RHP-${\boldsymbol{P}_{0}}$(b), as well as (5.14) (notice also that we do not
need to take boundary values as the relevant entries of
$\boldsymbol{\Phi}_{\alpha_{0}}$ are analytic across the considered arcs). Let
$Z=n^{2/3}\zeta(s)$. We shall estimate the expression in (5.21) separately in
two regimes: when $|Z|>1$ and when $|Z|\leq 1$. Uniformity of the asymptotics
in RHP-${\boldsymbol{\Phi}}_{\alpha}$(d) and (5.15) yield that

(5.22)
$\boldsymbol{\Phi}_{\alpha_{0}}\left(Z;0\right)\boldsymbol{J}\left(\begin{matrix}0&0\\\
1&0\end{matrix}\right)\boldsymbol{J}^{-1}\boldsymbol{\Phi}_{\alpha_{0}}^{-1}\left(Z;0\right)=\mathcal{O}\left(|Z|^{1/2}e^{-(4/3)|Z|^{3/2}}\right){=\mathcal{O}\big{(}|Z|^{-p}\big{)}}$

uniformly for $|Z|>1$, where one needs to observe that $Z^{3/2}>0$ for
$s\in\Delta_{1}^{\star}$ and $Z^{3/2}<0$ for
$s\in\Delta_{2}^{\star}\cup\Delta_{3}^{\star}$, see (5.9) (recall also that
$\mathrm{det}(\boldsymbol{\Phi}_{\alpha_{0}}(Z;0))\equiv 1$), and notice that
$|Z|^{p+1/2}e^{-(4/3)|Z|^{3/2}}$ is a bounded above positive function of $|Z|$
on $[1,\infty)$. On the other hand, we get from
RHP-${\boldsymbol{\Phi}}_{\alpha}$(c) that

(5.23)
$\boldsymbol{\Phi}_{\alpha_{0}}\left(Z;0\right)\boldsymbol{J}\left(\begin{matrix}0&0\\\
1&0\end{matrix}\right)\boldsymbol{J}^{-1}\boldsymbol{\Phi}_{\alpha_{0}}^{-1}\left(Z;0\right)=\\\
\begin{cases}{\mathcal{O}}(1)|Z|^{\alpha_{0}\sigma_{3}/2}{\mathcal{O}}(1)|Z|^{-\alpha_{0}\sigma_{3}/2}\mathcal{O}(1),&\alpha_{0}\neq
0,\\\
{\mathcal{O}}(1)\mathcal{O}(\log|Z|){\mathcal{O}}(1)\mathcal{O}(\log|Z|)\mathcal{O}(1),&\alpha_{0}=0,\end{cases}=\begin{cases}\mathcal{O}\left(|Z|^{-|\alpha_{0}|}\right),&\alpha_{0}\neq
0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\mathcal{O}\left(\log^{2}|Z|\right),&\alpha_{0}=0,\end{cases}{=\mathcal{O}\big{(}|Z|^{-p}\big{)}}$

uniformly for $|Z|\leq 1$ because $|Z|^{p-|\alpha_{0}|}$ and
$|Z|^{p}\log^{2}|Z|$ are bounded positive functions of $|Z|$ on $[0,1]$ since
$p\geq 1$ and $|\alpha_{0}|\leq 1$. Recall that $|z-a_{0}|/|\zeta(z)|$ is
bounded in $U_{0}$ and $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$ is an analytic
matrix function with the same determinant as
$\boldsymbol{M}^{\mathsf{reg}}(z)$. Definition (5.16), estimates (5.7), and
the maximum modulus principle yield that
$\boldsymbol{E}_{0}^{\mathsf{reg}}(z)=\boldsymbol{\mathcal{O}}_{\varepsilon}(1)n^{\sigma_{3}/6}$
and
$\boldsymbol{E}_{0}^{\mathsf{reg}}(z)^{-1}=\boldsymbol{\mathcal{O}}_{\varepsilon}(1)n^{-\sigma_{3}/6}$.
It now follows from (5.21)–(5.23) that the jump of $\boldsymbol{Z}(z)$ on
$\cup\Delta_{i}^{\star}$ can be estimated as

(5.24)
${\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\varepsilon}\bigg{(}n^{1/3}\max_{s\in\cup\Delta_{i}^{\star}}|s-a_{0}|^{p}|Z|^{-p}\bigg{)}=\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\varepsilon}\big{(}n^{-(2p-1)/3}\big{)},}$

where $Z=n^{2/3}\zeta(s)$. As $p\geq 1$, the error rate in (5.24) is not worse
than $\boldsymbol{\mathcal{O}}_{\varepsilon}(n^{-1/3})$.

Finally, by arguing as in [5, Theorem 7.103 and Corollary 7.108], see also [9,
Theorem 8.1], we obtain from (5.19), (5.20), and (5.24) that the matrix
$\boldsymbol{Z}(z)$ exists for all
$n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ large enough and that

$\|\boldsymbol{Z}_{\pm}-\boldsymbol{I}\|_{2,\Sigma}=\mathcal{O}_{\varepsilon}\big{(}n^{-1/3}\big{)}.$

Since the jumps of $\boldsymbol{Z}(z)$ on $\Sigma$ are restrictions of
holomorphic matrix functions, the standard deformation of the contour
technique and the above estimate yield that

(5.25)
$\boldsymbol{Z}=\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\delta,\varepsilon}\big{(}n^{-1/3}\big{)}\quad\text{uniformly
in}\quad\overline{\mathbb{C}}\setminus\\{a_{0}\\}.$

###  5.6. Proof of Theorem 2

Let $\boldsymbol{Z}(z)$ be a solution of RHP-${\boldsymbol{Z}}$, in which case
(5.18) holds. Given a closed set
$K\subset\overline{\mathbb{C}}\setminus\Delta$, the contour $\Sigma$ can
always be adjusted so that $K$ lies in the exterior domain of $\Sigma$. Then
it follows from (5.3) that $\boldsymbol{Y}(z)=\boldsymbol{X}(z)$ on $K$.
Formulae (3.4) and (3.5) now follow immediately from (5.2), (5.4) and (5.25)
since

$w^{i-1}(z)\left[\left(\boldsymbol{ZN}_{*}\right)(z)\right]_{1i}=(1+\upsilon_{n1}(z))\Psi_{n}\big{(}z^{(i-1)}\big{)}+\upsilon_{n2}(z)\Psi_{n-1}\big{(}z^{(i-1)}\big{)}$

for $z\in K$, where $1+\upsilon_{n1}(z),\upsilon_{n2}(z)$ are the first row
entries of $\boldsymbol{Z}(z)$. Similarly, if $K$ is a compact subset of
$\Delta^{\circ}$, the lens $\Sigma$ can be arranged so that $K$ does not
intersect $\overline{U}$. As before, we get from (5.2), (5.3), and (5.18) that

(5.26)
$\gamma_{n}^{-1}Q_{n}(z)=\Big{(}\big{(}1+\upsilon_{n1}(z)\big{)}\Psi_{n}(z)+\upsilon_{n2}(z)\Psi_{n-1}(z)\Big{)}\pm\\\
(\rho_{i}w)^{-1}(z)\Big{(}\big{(}1+\upsilon_{n1}(z)\big{)}\Psi_{n}^{*}(z)+\upsilon_{n2}(z)\Psi_{n-1}^{*}(z)\Big{)}$

for $z\in\Omega_{i\pm}\setminus\overline{U}$. The top formula in (3.6) now
follows by taking the trace of the right-hand side of the above equality on
$K$ and using the top relation in (2.3) (which yields that
$\Psi_{n\pm}^{*}(s)=\Psi_{n\mp}(s)(\rho w_{+})(s)$ for $s\in\Delta^{\circ}$).
Since $R_{n}(z)=[\boldsymbol{X}]_{12}(z)$ for
$z\in\Omega_{i\pm}\setminus\overline{U}$, the bottom relation in (3.6) is even
simpler to derive.

##  6\. Proof of Theorem 3

The main difference between proofs of Theorem 2 and 3 is that we no longer can
use the matrix $\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;0)$ and shall substitute
it by $\boldsymbol{\Phi}_{\alpha_{0}}^{\mathsf{sing}}(\zeta;0)$, which has
slightly different behavior at infinity. This change necessitates
modifications in matrices $\boldsymbol{M}^{\mathsf{reg}}(z)$ and
$\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$.

###  6.1. On Matrices $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$

To be able to introduce necessary modifications of
$\boldsymbol{M}^{\mathsf{reg}}(z)$ and $\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$,
we need to discuss in more detail the behavior of
$\boldsymbol{E}_{0}^{\mathsf{reg}}(z)$ at $a_{0}$. To this end, let us write

$\boldsymbol{E}_{0}^{\mathsf{reg}}(z)=\left(\begin{matrix}E_{n,1}(z)&E_{n,2}(z)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
E_{n,3}(z)&E_{n,4}(z)\end{matrix}\right)n^{\sigma_{3}/6}.$

It follows readily from (5.16) and (5.4) that

(6.1) $\left\\{\begin{array}[]{rcl}\displaystyle
E_{n,i}(z)&:=&\big{(}\mathrm{i}\zeta^{1/2}(z)\big{)}^{1-i}G(z)\big{(}T_{n}(z)-(-1)^{i}(HT_{n}^{*})(z)\big{)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
E_{n,i+2}(z)&:=&\big{(}\mathrm{i}\zeta^{1/2}(z)\big{)}^{1-i}G(z)\left(\widetilde{T}_{n-1}(z)-(-1)^{i}\big{(}H\widetilde{T}_{n-1}^{*}\big{)}(z)\right),\end{array}\right.\quad
z\in S_{2},$

for $i\in\\{1,2\\}$, where we set
$\widetilde{T}_{m}({\boldsymbol{z}}):=(T_{m}/\Phi)({\boldsymbol{z}})$, and

(6.2) $G(z):=\frac{(\zeta^{1/4}S_{\rho}r)(z)}{\sqrt{2}},\quad
H(z):=\frac{1}{\mathrm{i}\left((S_{\rho}r)^{2}w\right)(z)},\quad z\in S_{2}.$

Notice that the functions $G(z)$ and $H(z)$ remain bounded as $S_{2}\ni z\to
a_{0}$ by (2.14) and the very definitions of $r(z)$ in (5.12) and of
$\zeta(z)$ in (5.8). We are interested in the quantity

(6.3)
$D_{n}:=E_{n,3}(a_{0})E_{n,1}^{\prime}(a_{0})-E_{n,1}(a_{0})E_{n,3}^{\prime}(a_{0}).$

Since $E_{n,i}(z)$ are analytic at $a_{0}$ and the functions
$T_{n}({\boldsymbol{z}})$ are analytic at $\boldsymbol{a}_{0}$, the lifts of
$G(z)$ and $H(z)$ to $\pi^{-1}(S_{2})\cap\mathfrak{S}^{(0)}$ admit analytic
continuations to some neighborhood of $\boldsymbol{a}_{0}$. Thus, the
quantities $G(a_{0})$ and $H(a_{0})$ are well-defined. In fact, since
$E_{n,2}(a_{0})$ and $E_{n,4}(a_{0})$ are finite while
$T_{n}(a_{0})=T_{n}^{*}(a_{0})$ and
$\widetilde{T}_{n}(a_{0})=\widetilde{T}_{n}^{*}(a_{0})$, it must follow from
(6.1) that

(6.4) $H(z)=1+h_{1}(z-a_{0})^{1/2}+h_{2}(z-a_{0})+\cdots$

(from now on all the stated expansions are assumed to hold for $z\in S_{2}\cap
U_{0}$), where we use the same determination of $(z-a_{0})^{1/2}$ as in
(2.28). It also clearly follows from (2.28) and (4.18) that

$\widetilde{T}_{n}(z)=t_{0}^{(n)}+t_{1}^{(n)}(z-a_{0})^{1/2}+t_{2}^{(n)}(z-a_{0})+\cdots,$

(that is, the first three coefficients in the Puiseux series of
$\widetilde{T}_{n}(z)$ and of $T_{n}(z)$ are the same). Observe also that

$T_{n}^{*}(z)=t_{0}^{(n)}-t_{1}^{(n)}(z-a_{0})^{1/2}+t_{2}^{(n)}(z-a_{0})+\cdots.$

Let us write $G(z)=g_{0}+g_{1}(z-a_{0})^{1/2}+g_{2}(z-a_{0})+\cdots$. Notice
that $g_{0}\neq 0$ by (2.14). Then a straightforward computation shows that

(6.5)
$E_{n,2k+1}(z)=2g_{0}t_{0}^{(n-k)}+\left(\big{(}2g_{2}+g_{1}h_{1}+g_{0}h_{2}\big{)}t_{0}^{(n-k)}+\right.\\\
+\left.g_{0}\left(2t_{2}^{(n-k)}-h_{1}t_{1}^{(n-k)}\right)\right)(z-a_{0})+\cdots$

for $k\in\\{0,1\\}$ (in particular, it must hold that $2g_{1}+g_{0}h_{1}=0$).
Hence, we get from the definitions of the numbers $D_{n}$ in (6.3) and the
sequence $\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$ in (2.29) that

(6.6) $|D_{n}|=2|g_{0}|^{2}|\mathrm{det}(\boldsymbol{T}_{n})|\geq
2|g_{0}|^{2}\varepsilon>0,\quad n\in\mathbb{N}_{\varepsilon}^{\mathsf{sing}}.$

Let us also justify the remark made after Proposition 5. Analysis similar to
the one above yields that

$E_{n,2k+2}(a_{0})=-\mathrm{i}g_{0}\zeta_{1}\left(2t_{1}^{(n-k)}-h_{1}t_{0}^{(n-k)}\right),\quad\zeta_{1}:=\left(\lim_{z\to
a_{0}}\frac{z-a_{0}}{\zeta(z)}\right)^{1/2},$

for $k\in\\{0,1\\}$. It further follows from (5.16) and the display after
(5.4) that

$\left(\gamma_{n}\gamma_{n-1}^{*}\right)^{-1}=\mathrm{det}\left(\boldsymbol{E}_{0}^{\mathsf{reg}}(z)\right)=\big{(}E_{n,1}E_{n,4}-E_{n,2}E_{n,3}\big{)}(z),$

which is true for any $z\in U_{0}$. Therefore, taking $z=a_{0}$ in the above
equality gives that

(6.7)
$\left(\gamma_{n}\gamma_{n-1}^{*}\right)^{-1}=-4\mathrm{i}\zeta_{1}g_{0}^{2}\left(t_{0}^{(n)}t_{1}^{(n-1)}-t_{0}^{(n-1)}t_{1}^{(n)}\right).$

The desired claim now follows from (5.5)–(5.6).

###  6.2. Global and Local Parametrices

Recall the definition of $\boldsymbol{M}^{\mathsf{reg}}(z)$ in (5.4). The new
global parametrix is now defined as
$\boldsymbol{N}(z)=\boldsymbol{CM}^{\mathsf{sing}}(z)\Phi^{n\sigma_{3}}(z)$,
where

(6.8)
$\boldsymbol{M}^{\mathsf{sing}}(z):=\left(\boldsymbol{I}+(z-a_{0})^{-1}\boldsymbol{L}\right)\boldsymbol{M}^{\mathsf{reg}}(z)$

and the matrix $\boldsymbol{L}$ is given by

(6.9)
$\boldsymbol{L}:=\frac{1}{D_{n}}\left(\begin{matrix}-(E_{n,1}E_{n,3})(a_{0})&E_{n,1}^{2}(a_{0})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-E_{n,3}^{2}(a_{0})&(E_{n,1}E_{n,3})(a_{0})\end{matrix}\right).$

Notice that
$\boldsymbol{M}^{\mathsf{reg}}(\infty)=\boldsymbol{M}^{\mathsf{sing}}(\infty)=\boldsymbol{C}^{-1}$
and, since $\boldsymbol{L}$ has zero trace and determinant,
$\mathrm{det}(\boldsymbol{M}^{\mathsf{reg}}(z))=\mathrm{det}(\boldsymbol{M}^{\mathsf{sing}}(z))$.
Observe also that the just defined matrix $\boldsymbol{N}(z)$ satisfies
RHP-${\boldsymbol{N}}$(a,b) and RHP-${\boldsymbol{N}}$(c) around
$a_{1},a_{2},a_{3}$. However, its behavior at $a_{0}$ is obviously different.
Most importantly for us, it is still true that

(6.10) $\big{|}(\boldsymbol{M}^{\mathsf{sing}})^{\pm
1}(z)\big{|}=\boldsymbol{\mathcal{O}}_{K,\varepsilon}(1)$

for $z\in K$ as $\mathbb{N}_{\varepsilon}^{\mathsf{sing}}\ni n\to\infty$,
where $K$ is any compact set that avoids $a_{0},a_{1},a_{2},a_{3}$. Indeed,
since
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}\subseteq\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$,
estimates (5.7) still hold. It was computed in the previous subsection that

(6.11) $\big{|}E_{n,2k+1}(a_{0})\big{|}=2\big{|}g_{0}t_{0}^{(n-k)}\big{|}\leq
2|g_{0}|M_{R}$

for all $n\in\mathbb{N}$, where the last conclusion is a consequence of
(2.26). Thus, (6.10) follows from (6.6).

Local parametrix $\boldsymbol{P}_{0}(z)$ is now constructed similarly to
(5.14) as

(6.12)
$\boldsymbol{P}_{0}(z)=\boldsymbol{E}_{0}^{\mathsf{sing}}(z)\boldsymbol{\Phi}_{\alpha_{0}}^{\mathsf{sing}}\big{(}n^{2/3}\zeta(z);0\big{)}\boldsymbol{J}(z)r^{-\sigma_{3}}(z),$

where $\boldsymbol{\Phi}_{\alpha_{0}}^{\mathsf{sing}}(\zeta;x)$ is the
solution of RHP-${\boldsymbol{\Phi}}_{\alpha}^{\mathsf{sing}}$ from Appendix B
and $\boldsymbol{E}_{0}^{\mathsf{sing}}(z)$ is given by

(6.13)
$\boldsymbol{E}_{0}^{\mathsf{sing}}(z):=\Big{(}\boldsymbol{I}+(z-a_{0})^{-1}\boldsymbol{L}\Big{)}\boldsymbol{E}_{0}^{\mathsf{reg}}(z)\big{(}n^{2/3}\zeta(z)\big{)}^{-\sigma_{3}}.$

It follows from (5.16), (B.12), and (6.8) that (5.17) holds with
$\boldsymbol{M}^{\mathsf{reg}}(z)$ replaced by
$\boldsymbol{M}^{\mathsf{sing}}(z)$. Hence, provided
$\boldsymbol{E}_{0}^{\mathsf{sing}}(z)$ is analytic in $U_{0}$, the just
constructed matrix $\boldsymbol{P}_{0}(z)$ solves RHP-${\boldsymbol{P}_{0}}$
with $\boldsymbol{M}^{\mathsf{reg}}(z)$ replaced by
$\boldsymbol{M}^{\mathsf{sing}}(z)$ in RHP-${\boldsymbol{P}_{0}}$(d) (again,
entries of $\boldsymbol{L}$ are bounded by (6.6) and (6.11) along
$\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$). To show analyticity of
$\boldsymbol{E}_{0}^{\mathsf{sing}}(z)$ in $U_{0}$, observe that

$\boldsymbol{E}_{0}^{\mathsf{sing}}(z)=\left(\begin{matrix}1-\frac{(E_{n,1}E_{n,3})(a_{0})}{(z-a_{0})D_{n}}&\frac{E_{n,1}^{2}(a_{0})}{(z-a_{0})D_{n}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\frac{E_{n,3}^{2}(a_{0})}{(z-a_{0})D_{n}}&1+\frac{(E_{n,1}E_{n,3})(a_{0})}{(z-a_{0})D_{n}}\end{matrix}\right)\left(\begin{matrix}\frac{E_{n,1}(z)}{z-a_{0}}&(z-a_{0})*\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\frac{E_{n,3}(z)}{z-a_{0}}&(z-a_{0})*\end{matrix}\right)\left(\frac{n^{1/2}\zeta(z)}{z-a_{0}}\right)^{-\sigma_{3}}$

by (6.9). Recall that $\zeta(z)$ has a simple zero at $a_{0}$ and therefore
the last matrix above is analytic at $a_{0}$. Thus, we only need to
investigate analyticity of the first column of the product of the first two
matrices above. The $(1,1)$-entry of this product is equal to

$\frac{1}{z-a_{0}}\left(E_{n,1}(z)-E_{n,1}(a_{0})\frac{E_{n,3}(a_{0})E_{n,1}(z)-E_{n,1}(a_{0})E_{n,3}(z)}{(z-a_{0})D_{n}}\right),$

which is indeed analytic at $a_{0}$ by the very choice of $D_{n}$ in (6.3).
The fact that $(2,1)$-entry of the product is analytic at $a_{0}$ can be
checked analogously.

###  6.3. A Discussion

In this subsection we briefly digress from the proof of Theorem 3 and provide
a broader view of the construction of global and local paramatrices presented
in this and the previous sections.

In this section we are forced to use matrix function
$\boldsymbol{\Phi}_{\alpha_{0}}^{\mathsf{sing}}(\zeta;0)$ instead of
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;0)$ as the latter does not exist in the
considered case due to polar singularities at $x=0$ of the functions
$U_{k}(x)$ from (B.1). However, one can clearly see from (B.10) that
$\boldsymbol{\Phi}_{\alpha_{0}}^{\mathsf{sing}}(\zeta;x)$ is actually well-
defined around zero as long as $U_{1}(0)\neq 0$. A natural question arises
whether one could use
$\boldsymbol{\Phi}_{\alpha_{0}}^{\mathsf{sing}}(\zeta;0)$ instead of
$\boldsymbol{\Phi}_{\alpha_{0}}(\zeta;0)$ in all cases leading to
$U_{1}(0)\neq 0$. Let us explain that this is indeed possible.

If we want to use $\boldsymbol{\Phi}_{\alpha_{0}}^{\mathsf{sing}}(\zeta;0)$ as
model local parametrix, we do need to introduce modified global parametrix
$\boldsymbol{M}^{\mathsf{sing}}(z)$ via (6.8) with $\boldsymbol{L}$ given by
(6.9), but differently defined constants $D_{n}$. Indeed, if $U_{1}(0)\neq 0$,
then (B.12) no longer holds and needs to be replaced by

$\left(\boldsymbol{I}-\frac{1}{\zeta}\left(\begin{matrix}0&0\\\
2^{-1/3}\mathrm{i}U_{1}^{-1}(0)&0\end{matrix}\right)\right)\boldsymbol{\Phi}_{\alpha}^{\mathsf{sing}}(\zeta;0)=\frac{\zeta^{3\sigma_{3}/4}}{\sqrt{2}}\left(\begin{matrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{matrix}\right)\left(\boldsymbol{I}+\boldsymbol{\mathcal{O}}\Big{(}\zeta^{-1/2}\Big{)}\right)\exp\left\\{-\frac{2}{3}\zeta^{3/2}\sigma_{3}\right\\}$

(in fact, the above formula extends (B.12) since $U_{1}^{-1}(0)=0$ for weights
in $W^{\mathsf{sing}}$). This change necessitates the following modification
in the definition of $\boldsymbol{E}_{0}^{\mathsf{sing}}(z)$:

$\displaystyle\boldsymbol{E}_{0}^{\mathsf{sing}}(z)$
$\displaystyle:=\left(\boldsymbol{I}+\frac{\boldsymbol{L}}{z-a_{0}}\right)\boldsymbol{E}_{0}^{\mathsf{reg}}(z)\big{(}n^{2/3}\zeta(z)\big{)}^{-\sigma_{3}}\left(\boldsymbol{I}-\frac{2^{-1/3}\mathrm{i}U_{1}^{-1}(0)}{n^{2/3}\zeta(s)}\left(\begin{matrix}0&0\\\
1&0\end{matrix}\right)\right)$
$\displaystyle=\left(\boldsymbol{I}+\frac{\boldsymbol{L}}{z-a_{0}}\right)\left[\left(\begin{matrix}\frac{E_{n,1}(z)}{z-a_{0}}&(z-a_{0})E_{n,2}(z)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\frac{E_{n,3}(z)}{z-a_{0}}&(z-a_{0})E_{n,4}(z)\end{matrix}\right)+n^{1/3}\Lambda(z)\left(\begin{matrix}E_{n,2}(z)&0\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
E_{n,4}(z)&0\end{matrix}\right)\right]\left(\frac{n^{1/2}\zeta(z)}{z-a_{0}}\right)^{-\sigma_{3}},$

where
$\Lambda(z):=-2^{-1/3}\mathrm{i}U_{1}^{-1}(0)\big{(}\zeta(z)/(z-a_{0})\big{)}^{2}$.
Therefore, $\boldsymbol{E}_{0}^{\mathsf{sing}}(z)$ is analytic at $a_{0}$ if

$\frac{\boldsymbol{L}}{(z-a_{0})^{2}}\left(\begin{matrix}E_{n,1}(z)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
E_{n,3}(z)\end{matrix}\right)+\frac{1}{z-a_{0}}\left(\begin{matrix}E_{n,1}(z)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
E_{n,3}(z)\end{matrix}\right)+\frac{n^{1/3}\Lambda(z)\boldsymbol{L}}{z-a_{0}}\left(\begin{matrix}E_{n,2}(z)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ E_{n,4}(z)\end{matrix}\right)$

is analytic at $a_{0}$. To achieve the latter, one needs to take
$\boldsymbol{L}$ as in (6.9) with

$D_{n}=n^{1/3}\Lambda(0)\big{(}E_{n,3}E_{n,2}-E_{n,1}E_{n,4}\big{)}(a_{0})+\big{(}E_{n,3}E_{n,1}^{\prime}-E_{n,1}E_{n,3}^{\prime}\big{)}(a_{0}).$

Observe that when $U_{1}^{-1}(0)=0$, $\Lambda(0)=0$ and we recover the
original definition of $D_{n}$ in (6.3). Hence, when $U_{1}^{-1}(0)\neq 0$, it
follows from (6.7) and the display before that $D_{n}$ is non-zero for all
$n\in\mathbb{N}_{\varepsilon}^{\mathsf{reg}}$ large enough and the above
construction goes through. It also holds in this case that
$\boldsymbol{L}=\boldsymbol{\mathcal{O}}\big{(}n^{-1/3}\big{)}$ and therefore
matrices $\boldsymbol{M}^{\mathsf{reg}}(z)$ and
$\boldsymbol{M}^{\mathsf{sing}}(z)$ are asymptotically the same.

###  6.4. Proof of Theorem 3

Consider now RHP-${\boldsymbol{Z}}$ with
$\boldsymbol{N}_{*}(z)=\boldsymbol{M}^{\mathsf{sing}}(z)\Phi^{n\sigma_{3}}(z)$
and $\boldsymbol{P}_{0}(z)$ given by (6.12). Again, let us estimate the size
of the jump matrices in RHP-${\boldsymbol{Z}}$(b). It follows from (6.10) that
estimates (5.19)–(5.20) remain valid in the present case. Using (B.12) instead
of RHP-${\boldsymbol{\Phi}}_{\alpha}$(d) gives the same estimate as in (5.22),
one only needs to use boundedness of $|Z|^{p+3/2}e^{-(4/3)|Z|^{3/2}}$. Since
RHP-${\boldsymbol{\Phi}}_{\alpha}^{\mathsf{sing}}$(c) is the same as
RHP-${\boldsymbol{\Phi}}_{\alpha}$(c), the estimate (5.23) remain unchanged.
It follows from the definition of $\boldsymbol{E}_{0}^{\mathsf{sing}}(z)$ in
(6.13), its holomorphy in $U_{0}$, the maximum modulus principle, and (6.10)
that
$\boldsymbol{E}_{0}^{\mathsf{sing}}(z)=\boldsymbol{\mathcal{O}}_{\varepsilon}(1)n^{-\sigma_{3}/2}$
and
$\boldsymbol{E}_{0}^{\mathsf{sing}}(z)^{-1}=\boldsymbol{\mathcal{O}}_{\varepsilon}(1)n^{\sigma_{3}/2}$.
Therefore, (5.24) now is replaced by

$\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\varepsilon}\big{(}n\max|s-a_{0}|^{p}|Z|^{-p}\big{)}=\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\varepsilon}\big{(}n^{-(2p-3)/3}\big{)}.$

Recall that $p$ is the largest integer such that
$|1-\kappa_{i,j}(s)|=\mathcal{O}(|s-a_{0}|^{p})$ as $s\to a_{0}$. It can be
readily checked that condition (3.7) implies that $p\geq 2$ and therefore
$(2p-3)/3\geq 1/3$. Thus, the jump matrices in RHP-${\boldsymbol{Z}}$(b) still
satisfy the estimate
$\boldsymbol{I}+\boldsymbol{\mathcal{O}}_{\varepsilon}(n^{-1/3})$. Hence, we
again can conclude that $\boldsymbol{Z}(z)$ exists for all
$n\in\mathbb{N}_{\varepsilon}^{\mathsf{sing}}$ large enough and satisfies
(5.25).

As in the proof of Theorem 2, given a closed set
$K\subset\overline{\mathbb{C}}\setminus\Delta$, the contour $\Sigma$ can
always be adjusted so that $K$ lies in the exterior domain of $\Sigma$. Then
we get from (5.3) and (5.18) that

$\boldsymbol{Y}(z)=\boldsymbol{X}(z)=\boldsymbol{C}\boldsymbol{M}^{\mathsf{sing}}(z)\Phi^{n\sigma_{3}}(z),\quad
z\in K.$

Therefore, we get from (5.2) that

$\gamma_{n}^{-1}Q_{n}(z)=\big{(}1+\upsilon_{n1}^{\mathsf{sing}}(z)\big{)}\left(\left(1+\frac{[\boldsymbol{L}]_{11}}{z-a_{0}}\right)\Psi_{n}(z)+\frac{[\boldsymbol{L}]_{12}}{z-a_{0}}\Psi_{n-1}(z)\right)\\\
+\upsilon_{n2}^{\mathsf{sing}}(z)\left(\left(1+\frac{[\boldsymbol{L}]_{22}}{z-a_{0}}\right)\Psi_{n-1}(z)+\frac{[\boldsymbol{L}]_{21}}{z-a_{0}}\Psi_{n}(z)\right),\quad
z\in K,$

where $1+\upsilon_{n1}^{\mathsf{sing}}(z),\upsilon_{n2}^{\mathsf{sing}}(z)$
are the first row entries of $\boldsymbol{Z}(z)$. For $i\in\\{1,2\\}$, let

(6.14)
$\upsilon_{ni}(z):=\upsilon_{ni}^{\mathsf{sing}}(z)\left(1+\frac{[\boldsymbol{L}]_{ii}}{z-a_{0}}\right)+\upsilon_{n3-i}^{\mathsf{sing}}(z)\frac{[\boldsymbol{L}]_{3-ii}}{z-a_{0}}.$

It follows from (6.9), (6.11), and (6.6) that $\upsilon_{ni}(z)$ satisfy
bounds as in (3.5). Moreover, we have that

$\gamma_{n}^{-1}Q_{n}(z)=\left(1+\frac{[\boldsymbol{L}]_{11}}{z-a_{0}}+\upsilon_{n1}(z)\right)\Psi_{n}(z)+\left(\frac{[\boldsymbol{L}]_{12}}{z-a_{0}}+\upsilon_{n2}(z)\right)\Psi_{n-1}(z),\quad
z\in K.$

We further get from (6.9), (6.5), and (6.6) that

$[\boldsymbol{L}]_{11}=-\frac{(E_{n,1}E_{n,3})(a_{0})}{D_{n}}=-\frac{2t_{0}^{(n-1)}t_{0}^{(n)}}{\mathrm{det}(\boldsymbol{T}_{n})}\quad\text{and}\quad[\boldsymbol{L}]_{12}=\frac{E_{n,1}^{2}(a_{0})}{D_{n}}=\frac{2(t_{0}^{(n)})^{2}}{\mathrm{det}(\boldsymbol{T}_{n})}.$

This finishes the proof of the asymptotic formula for $Q_{n}(z)$ while the
formula for $(wR_{n})(z)$ can be shown absolutely analogously. Similarly, we
get from (5.2), (5.3), and (5.18) with $\boldsymbol{M}^{\mathsf{reg}}(z)$
replaced by $\boldsymbol{M}^{\mathsf{sing}}(z)$ that

$\gamma_{n}^{-1}Q_{n}(z)=\left(\left(1+\frac{[\boldsymbol{L}]_{11}}{z-a_{0}}+\upsilon_{n1}(z)\right)\Psi_{n}(z)+\upsilon_{n2}(z)\Psi_{n-1}(z)\right)\\\
\pm(\rho_{i}w)^{-1}(z)\left(\left(1+\frac{[\boldsymbol{L}]_{11}}{z-a_{0}}+\upsilon_{n1}(z)\right)\Psi_{n}^{*}(z)+\upsilon_{n2}(z)\Psi_{n-1}^{*}(z)\right)$

for $z\in\Omega_{i}^{\pm}\setminus\overline{U}$ with $\upsilon_{ni}(z)$ as in
(6.14). After that the proof of the analog of the first relation in (3.6)
proceeds exactly as after (5.26). The proof of the second relation in (3.6)
can be obtained analogously.

##  Appendix A Riemann-Hilbert Problem for Painlevé II

###  A.1. Lax Pair and the Corresponding Riemann-Hilbert Problem

The material of this section originates in [8]. However, we essentially follow
the presentation in [9, Section 5.0], see also [9, Section 1.0] for the
relevant facts of the general theory of differential equations. Let $q(s)$ be
a solution of Painlevé II equation

(A.1) $q^{\prime\prime}(s)=sq(s)+2q^{3}(s)-\nu,\quad\nu>-1/2.$

It is known that $q(s)$ is a meromorphic function in the entire complex plane.
For each $s$, which is not a pole of $q$, consider the following system of
differential equations:

(A.2)
$\partial_{\tau}\boldsymbol{\Psi}(\tau;s)=\boldsymbol{A}^{FN}(\tau;s)\boldsymbol{\Psi}(\tau;s),$

where

(A.3)
$\boldsymbol{A}^{FN}(\tau;s):=\left(\begin{matrix}-\mathrm{i}(4\tau^{2}+s+2q^{2}(s))&4\tau
q(s)+2\mathrm{i}q^{\prime}(s)+\nu/\tau\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
4\tau
q(s)-2\mathrm{i}q^{\prime}(s)+\nu/\tau&\mathrm{i}(4\tau^{2}+s+2q^{2}(s))\end{matrix}\right)$

and

(A.4)