Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight
Function

Recurrence Coefficients for Orthogonal Polynomials
with a Logarithmic Weight Function††This paper is a contribution to the
Special Issue on Evolution Equations, Exactly Solvable Models and Random
Matrices in honor of Alexander Its’ 70th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Its.html

Percy DEIFT a and Mateusz PIORKOWSKI b

P. Deift and M. Piorkowski

a) Department of Mathematics, Courant Institute of Mathematical Sciences,
a) New York University, 251 Mercer Str., New York, NY 10012, USA
<EMAIL_ADDRESS>
b) Department of Mathematics, Katholieke Universiteit Leuven,
b) Celestijnenlaan 200B, 3001 Leuven, Belgium<EMAIL_ADDRESS>
Received July 19, 2023, in final form January 01, 2024; Published online
January 10, 2024

We prove an asymptotic formula for the recurrence coefficients of orthogonal
polynomials with orthogonality measure $\log\bigl{(}\frac{2}{1-x}\bigr{)}{\rm
d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a
conjecture by Magnus and extends earlier results by Conway and one of the
authors. The proof relies on the Riemann–Hilbert method. The main difficulty
in applying the method to the problem at hand is the lack of an appropriate
local parametrix near the logarithmic singularity at $x=+1$.

orthogonal polynomials; Riemann–Hilbert problems; recurrence coefficients;
steepest descent method

42C05; 34M50; 45E05; 45M05

##  1 Introduction

###  1.1 Background

Figure 1: Plot of the weight function $w(x)$.

In this paper, we study orthogonal polynomials with orthogonality measure
$w(x){\rm d}x$ given by

$\displaystyle w(x){\rm d}x=\log\left(\frac{2}{1-x}\right){\rm d}x,\qquad
x\in[-1,1).$ (1.1)

Note that $w(x)$ has a logarithmic singularity for $x\to+1$ and a simple zero
at $x=-1$, see Figure 1. Denote by $\\{p_{n}\\}_{n=0}^{\infty}$ the
corresponding orthonormal polynomials,

$\displaystyle\int_{-1}^{1}p_{m}(x)p_{n}(x)w(x){\rm d}x=\delta_{mn},\qquad
m,n\in\mathbb{N}.$

The polynomials $\\{p_{n}\\}_{n=0}^{\infty}$ satisfy the three terms
recurrence relation given by

$\displaystyle
xp_{n}(x)=b_{n}p_{n+1}(x)+a_{n}p_{n}(x)+b_{n-1}p_{n-1}(x),\qquad n\geq 1,$

where $a_{n}\in\mathbb{R}$ and $b_{n}>0$. Note that our notation for $a_{n}$,
$b_{n}$ is the same as in [4, 5] but opposite to the one in [12, 15]. Listed
below are the first few recurrence coefficients for the weight function $w$.
The large $n$ asymptotics of the recurrence coefficients $a_{n}$, $b_{n}$ are
the main focus of the present work. To be precise we will prove the following
result.

######  Theorem 1.1.

The recurrence coefficients $\\{a_{n}\\}_{n=0}^{\infty}$,
$\\{b_{n}\\}_{n=0}^{\infty}$ of the orthogonal polynomials with orthogonality
measure $w(x){\rm d}x$, with $w$ given in (1.1), satisfy

$\displaystyle
a_{n}=\frac{1}{4n^{2}}-\frac{3}{16n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right)\qquad\text{as}\quad
n\to\infty$ (1.2)

and

$\displaystyle
b_{n}=\frac{1}{2}-\frac{1}{16n^{2}}-\frac{3}{32n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right)\qquad\text{as}\quad
n\to\infty.$ (1.3)

Comparing (1.2) and (1.3) with Table 1, we see that, already for $n=4$,
$a_{n}$ and $b_{n}^{2}$ are close to their limiting values, $0$ and
$\frac{1}{4}$ respectively, up to the second digit.

$n$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$a_{n}$ | $\frac{1}{2}$ | $\frac{1}{14}$ |  $\frac{263}{9058}$ $\approx 0.029$ |  $\frac{1995511}{126347454}$ $\approx 0.016$ |  $\frac{436364251361}{43886567673522}$ $\approx 0.010$
$b_{n}^{2}$ | $\frac{7}{36}$ | $\frac{2588}{11025}$ |  $\frac{71180289}{293026300}$ $\approx 0.243$ |  $\frac{1329399823424}{5405644687527}$ $\approx 0.246$ |  $\frac{39672481023099631594375}{160381475127054568640484}$ $\approx 0.247$

Table 1: First recurrence coefficients for the weight function $w$.

Theorem 1.1 is a special case of a conjecture by Magnus, who analyzed in [15]
a few examples of continuous weight functions with logarithmic singularities
at the edge and in the bulk of the support, among others $-\log(t)$ with
$t\in(0,1]$ which is equivalent to (1.1) after the affine change of variables
$x=1-2t$ (see the remark below). More generally, for the case of a logarithmic
singularity at the edge of the support, Magnus considered $w_{M}(x)$, now
supported, without loss of generality, on $x\in(-1,1)$, satisfying the
following two conditions:

  * •

$w_{M}(x)/(1+x)^{\beta}$ has a positive finite limit for $x\to-1$,

  * •

$w_{M}(x)/[-(1-x)^{\alpha}\log(1-x)]$ has a positive finite limit for
$x\to+1$,

where $\alpha,\beta>-1$. He conjectured based on numerical evidence that the
recurrence coefficients $a_{M,n}$ and $b_{M,n}$ of the corresponding
orthogonal polynomials satisfy for $n\to\infty$,

$\displaystyle a_{M,n}=\frac{\beta^{2}-\alpha^{2}}{4n^{2}}+\frac{2B}{n^{2}\log
n}+\frac{2C}{n^{2}\log^{2}n}+o\big{(}(n\log n)^{-2}\big{)}$ (1.4)

and

$\displaystyle
b_{M,n}=\frac{1}{2}-\frac{\alpha^{2}+\beta^{2}-\frac{1}{2}}{8n^{2}}+\frac{B}{n^{2}\log
n}+\frac{C}{n^{2}\log^{2}n}+o\big{(}(n\log n)^{-2}\big{)}.$ (1.5)

Additionally, it was conjectured that for the special case $w_{M}=w$ from
(1.1), $B=0$ and $C=-\frac{3}{32}$ holds, which is confirmed by Theorem 1.1.
Note that for $w$, we have $\alpha=0$ and $\beta=1$.

######  Remark 1.2.

For a general orthogonality measure ${\rm d}\mu(x)$ supported on $x\in(-1,1)$
with recurrence coefficients $A_{n}$, $B_{n}$, the orthogonality measure
defined by ${\rm d}\widetilde{\mu}(t):={\rm d}\mu(1-2t)$ with ${t\in(0,1)}$
leads to recurrence coefficients given by

$\displaystyle\widetilde{A}_{n}=\frac{1}{2}-\frac{A_{n}}{2},\qquad\widetilde{B}_{n}=\frac{B_{n}}{2}.$

###  1.2 State of the art

Earlier work on Magnus’ conjecture was done by Conway and one of the authors
in [4] using Riemann–Hilbert (RH) techniques. The weight function considered
therein had the form

$\displaystyle w_{k}(x)=\log\left(\frac{2k}{1-x}\right),\qquad
x\in[-1,1),\quad k>1.$ (1.6)

The authors prove the conjecture for this special case corresponding to
$\alpha=\beta=0$, and also obtain $B=0$ and $C=-\frac{3}{32}$, further
suggesting that these constants do not depend on the behaviour of the weight
function away from the logarithmic singularity.

######  Theorem 1.3 ([4]).

The recurrence coefficients $\big{\\{}a_{n}^{(k)}\big{\\}}_{n=0}^{\infty}$,
$\big{\\{}b_{n}^{(k)}\big{\\}}_{n=0}^{\infty}$ of the orthogonal polynomials
with orthogonality measure $w_{k}(x){\rm d}x$, with $w_{k}$ given in (1.6),
satisfy

$\displaystyle
a_{n}^{(k)}=-\frac{3}{16n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right)\qquad\text{as}\quad
n\to\infty$

and

$\displaystyle
b_{n}^{(k)}=\frac{1}{2}+\frac{1}{16n^{2}}-\frac{3}{32n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right)\qquad\text{as}\quad
n\to\infty.$

Note that for $k>1$, the weight function $w_{k}(x)$ has a positive finite
value at $x=-1$, while $\lim_{k\to 1+}w_{k}(x)=w(x)$ for $x\in[-1,1)$. Hence,
we see different $1/n^{2}$-terms in Theorem 1.1 compared to Theorem 1.3,
consistent with the conjectural (1.4) and (1.5). Also observe that terms of
order $1/\big{(}n^{2}\log^{2}n\big{)}$ are identical in both theorems, in
particular they do not depend on $k$. These terms can be interpreted as
contributions from the logarithmic singularity at $x=+1$, which does not
depend on $k\geq 1$.

For $k<1$, the weight function $w_{k}$ would have a simple zero in the
interior of $(-1,1)$ and the uniqueness of the corresponding polynomials is no
longer guaranteed. We will not deal with this case in the present paper.

The main difficulty encountered in [4] was the lack of a known parametrix in
the vicinity of the logarithmic singularity, a key ingredient in the usual
nonlinear steepest descent analysis. Hence, the authors relied instead on a
technically involved comparison to the Legendre problem with weight function
$w_{Leg}(x)\equiv 1$, $x\in(-1,1)$. Surprisingly, their argument could not be
generalized in an obvious way to the weight function $w=\lim_{k\to 1}w_{k}$,
due to the appearance of a simple zero of $w$ at $x=-1$. While, an analogous
comparison to the Jacobi problem with weight function $w_{Jac}(x)=1+x$ (or
something similar) seems suggestive, significant challenges remain due to the
presence of the simple zero. In particular, the crucial Theorem 4.7 in [4]
requires a different proof in this case: now one needs to control the
behaviour of the Cauchy operator acting on spaces with Muckenhoupt weights.

Orthogonal polynomials with logarithmic weight functions, have applications to
both pure mathematics and physics. In particular, apart from logarithmic
singularities, these weight functions also tend to have zeros (see [15,
Section 8] and [20]). Such applications motivate us in the present paper to
extend the results in [4] to the weight function $w$ in (1.1), having both a
logarithmic singularity and a simple zero.

######  Remark 1.4.

One might think, a priori, that the vanishing of a weight $w(x)$ at a point
should not give rise to serious technical difficulties. Naively, it would
appear that only singularities in the weight, and not zeros, should present
obstacles. In this regard, we recall the hope and the prophecy of Lenard’s
1972 paper [13]:

> “It is the author’s hope that a rigorous analysis will someday carry the
> results to the point where the true role of the zeros of the generating
> function will be understood. When that day comes a capstone will have been
> put on a beautiful edifice to whose construction many contributed and whose
> foundations lie in the studies of Gabor Szegő half a century ago”.

###  1.3 Relation of the present work to [4]

Significant parts of the analysis performed in the present paper are based on
the analysis introduced in [4]. Hence, we will repeatedly refer to that paper
for proofs of certain statements. This is justified by the fact that the
majority of estimates found in [4] do not depend on the distinction $k>1$ and
$k=1$ in (1.6). Thus, the proofs of many of the results will also hold for the
weight function (1.1) that we are interested in. There are however certain
propositions which have their analogs in [4], but still deserve a separate
proof due to some minor differences. These are Proposition 2.7 which is the
analog of Proposition 2.5 in [4], and Proposition 7.5 which is the analog of
Propositions 5.3 and 5.4 in [4]. In the case of Proposition 2.7, it is
necessary to prove a slightly more general result than Proposition 2.5 in [4].
Meanwhile, Proposition 6.5 contains an application of Proposition 2.7 in its
more general form, and uses different RH solutions than Propositions 5.3 and
5.4 in [4]. Both results are proven in Appendix A.

Finally, let us reiterate that the analog of Theorem 4.7 in [4] concerning the
uniform boundedness of the inverse of a certain singular integral operator
requires a completely different approach to the case $k=1$, due to the
appearance of a simple zero in the weight function $w$ at $x=-1$. This
necessitates the construction of an appropriate local parametrix in the
vicinity of the zero, which is then used to invert the RH problem locally.
While the local parametrix is well-known and can be expressed in terms of
Bessel functions, see [12, Section 6], its appearance significantly
complicates the analysis that follows. Crucially, the method of proving
Theorem 4.7 in [4] is no longer sufficient in this new setting. In fact, the
material found in Sections 4–6 is entirely devoted to formulating and proving
Theorems 6.1 and 6.2, which are the analogs of Theorem 4.7 in [4]. Here, a key
role is played by Proposition 4.6, which in a sense localizes the effect that
the logarithmic singularity has on the uniform invertibility of the associated
singular integral operator. Interestingly, Theorem 4.7 in [4] itself plays a
crucial role in the proofs of these results. Sections 4–6 contain the main
novelties of the present work.

###  1.4 Outline of the paper

In the following, we will briefly summarize the content of each section.

  * •

In Section 2, we introduce two auxiliary weight functions, the model and the
Legendre weight function, together with related quantities, which will be
relevant for the RH analysis. We also list certain estimates and asymptotic
results which will be used in later sections.

  * •

In Section 3, we introduce the Fokas–Its–Kitaev RH problem for orthogonal
polynomials. We proceed to perform the necessary conjugation and deformation
steps to arrive at three distinct RH problems amenable to asymptotics
analysis: the logarithmic, model and Legendre RH problems. We then state the
relation between solutions of these RH problems and the corresponding
recurrence coefficients.

  * •

In Section 4, we derive an explicit formula for the Legendre resolvent, that
is, the inverse of a singular integral operator associated to the Legendre RH
problem. This formula is not used in [4].

  * •

In Section 5, we perform a detailed analysis of the known local parametrices
for the RH problem near the point $-1$, which can be constructed explicitly
using Bessel functions, as in [12, Section 6]. This leads, in particular, to
uniform asymptotics on their growth as $n\to\infty$.

  * •

In Section 6, we introduce modified versions of the three previously mentioned
RH problems using the appropriate local parametrices from Section 3 around
$z=-1$. These modified RH problems are better suited when comparing their
associated resolvents. Thus, by showing the uniform invertibility of the
modified Legendre resolvent, we obtain the uniform invertibility of the
modified logarithmic and model resolvents, thereby proving an analog of
Theorem 4.7 in [4].

  * •

In Section 7, we derive an asymptotic formula expressing the difference
between the recurrence coefficients for the logarithmic weight and the
recurrence coefficients for the model weight. The aforementioned uniform
invertibility of the associated resolvents plays a crucial part in this
argument. As the asymptotics of the recurrence coefficients for the model
weight is known, Theorem 1.1 follows.

  * •

In Appendix A, we provide some proofs of more technical nature that are
omitted from the main text.

It is also worth mentioning that a common step in the RH analysis – the
construction of a local parametrix – is not performed in the vicinity of the
logarithmic singularity at $+1$. The reason is, quite simply, as in [4], that
we were unable to find the local parametrix in the presence of such a
singularity. Similar instances in which the local parametrix was not
constructed explicitly can be found in [6, Section 5] and [11], where non-
constructive Fredholm methods were used instead. For a discussion of RH
problems without explicitly solvable local parametrices, see [19].

###  1.5 Notation

Throughout this paper, all contours that arise are finite unions of smooth and
oriented arcs, with a finite number of points of (self)intersection. More
details can be found in the book [3] which treats a more general class of so-
called Carleson contours.

Let $\Gamma\subset\mathbb{C}$ be such a contour and $m$ an analytic function
on $\mathbb{C}\setminus\Gamma$. For $s\in\Gamma$, we will denote by
$m_{\pm}(s)$ the limit of $m(z)$ as $z\to s\pm$, provided this limit exists.
The notation $z\to s\pm$ denotes a nontangential limit in
$\mathbb{C}\setminus\Gamma$ to $s\in\Gamma$, from the $+$, resp. $-$, side of
the contour, see Figure 2. Recall that as $\Gamma$ is taken to be oriented,
this notion is well-defined away from the points of intersection. Everything
generalizes to matrix-valued functions $m$ in a straightforward manner.

$\Gamma$$m(z)$$+$$-$$+$$-$$+$$-$$+$$-$$m_{-}(s)$$m_{+}(s)$$\bullet$$s$ Figure
2: A contour with a point of intersection.

We will denote by $\mathbb{C}_{\pm}$ the upper, resp. lower open half plane of
$\mathbb{C}$ and use the notation $\overline{\mathbb{C}}$ for the Riemann
sphere $\mathbb{C}\cup\\{\infty\\}$. Unless specified otherwise, $z^{1/2}$,
$z\in\mathbb{C}\setminus(-\infty,0)$ will denote the principal branch of the
square root.

For two sequences $A_{n}$ and $B_{n}$ in a normed space, we will use the
notation $A_{n}\lesssim B_{n}$ if there exists a $c>0$ and $N\in\mathbb{N}$
such that $\|A_{n}\|\leq c\|B_{n}\|$ for all $n\geq N$. Equivalently, we will
sometimes use the notation $A_{n}=O(B_{n})$. A similar definition holds if $n$
is substituted for a continuous variable, e.g., $f(x)\lesssim g(x)$
(equivalently $f(x)=O(g(x))$) for $x\to x_{0}$ means $\|f(x)\|\leq c\|g(x)\|$
for some $c>0$ and all $x$ satisfying $|x-x_{0}|\leq\varepsilon$.

Finally, for a $d\times d$ dimensional measurable matrix-valued function
$f(s)$, $s\in\Gamma$, we write $f\in L^{p}(\Sigma)$, with $p\in[1,\infty)$, if
and only if

$\displaystyle\|f\|_{L^{p}(\Gamma)}:=\Biggl{(}\int_{\Gamma}\sum_{i,j=1}^{d}|f_{ij}(s)|^{p}|{\rm
d}s|\Biggr{)}^{\frac{1}{p}}<\infty,$

where $|{\rm d}s|$ denotes the arc length measure of $\Gamma$, and $f\in
L^{\infty}(\Gamma)$ if and only if

$\displaystyle\|f\|_{L^{\infty}(\Gamma)}:=\max_{i,j=1\dots
d}\\{\|f_{ij}\|_{L^{\infty}(\Gamma)}\\}<\infty.$

In particular, $\|f\|_{L^{2}(\Gamma)}$ denotes the $L^{2}$-norm on $\Gamma$ of
the Hilbert–Schmidt norm of $f(s)$. Generalizations to weighted $L^{p}$-spaces
are introduced in Section 4.

##  2 Auxiliary functions used in the RH analysis

###  2.1 The model and Legendre weight functions

To obtain the recurrence coefficients related to the weight function $w$, we
will have to compare with a different _model_ weight function given by

$\displaystyle\widehat{w}(x)=(1+x)\mathrm{e}^{d_{0}x},\qquad x\in[-1,1],$

where $d_{0}\in\mathbb{R}$ is determined via (2.7). As we will see, the choice
of $d_{0}$ gives an error estimate in (2.13) of order
$O\big{(}|1+z|^{3/2}\big{)}$, rather that $O\big{(}|1+z|^{1/2}\big{)}$, as in
part 3 of Proposition 2.3 in [4]. This extra decay as $z\to-1$ considerably
simplifies the proof of the key Lemma 7.6.

Note that $\widehat{w}$ can be analytically extended to an entire function.
Moreover, $\widehat{w}(x)$ has a simple zero at $x=-1$ and a finite positive
value at $x=+1$. As it lies in the class of weight functions considered in
[12], the corresponding RH analysis is well understood. Observe that our
requirements on $\widehat{w}$ do not specify it uniquely, hence one could have
performed the comparison argument with other choices of model weight functions
as well.

As we will heavily rely on the arguments found in [4], we will also introduce
the Legendre weight function

$\displaystyle\widetilde{w}(x)=1,\qquad x\in[-1,1].$

As shown in [4, Theorem 4.7] (see also Theorem 4.2), the weight
$\widetilde{w}$ gives rise to a singular integral operator defined in Section
4, with an inverse that is uniformly bounded as $n\to\infty$. However,
$\widetilde{w}$ does not approximate the logarithmic weight function $w$ for
$x\to-1$, due to the presence of a simple zero at that point. In contrast, the
weight $\widehat{w}$ approximates the logarithmic weight function $w$ for
$x\to-1$, however for $\widehat{w}$ the analogous singular integral operator
is not invertible in $L^{2}$ (see property (iv) in RH problem (see Section
3.4) showing that the solution will not be square integrable). This is the
essential technical difficulty that we face in this paper.

###  2.2 The Szegő functions

To perform the nonlinear steepest descent analysis, we need to define the
Szegő function $F$ associated to the logarithmic weight function $w$:

$\displaystyle
F(z)=\exp\Bigg{(}\dfrac{\big{(}z^{2}-1\big{)}^{1/2}}{2\pi}\int_{-1}^{1}\dfrac{\log
w(s)}{\sqrt{1-s^{2}}}\dfrac{{\rm d}s}{z-s}\Bigg{)},\qquad
z\in\mathbb{C}\setminus[-1,1].$ (2.1)

Here $\big{(}z^{2}-1\big{)}^{1/2}$ is uniquely specified as an analytic
function having a branch cut along $(-1,1)$, and
$\big{(}z^{2}-1\big{)}^{1/2}\approx z$ for $z\to\infty$. Analogously, we
define $\widehat{F}$ to be the Szegő function associated to the model weight
function $\widehat{w}$:

$\displaystyle\widehat{F}(z)=\exp\Bigg{(}\dfrac{\big{(}z^{2}-1\big{)}^{1/2}}{2\pi}\int_{-1}^{1}\dfrac{\log\widehat{w}(s)}{\sqrt{1-s^{2}}}\dfrac{{\rm
d}s}{z-s}\Bigg{)},\qquad z\in\mathbb{C}\setminus[-1,1].$

Note that as $\log\widetilde{w}\equiv 0$, the Szegő function for the Legendre
weight $\widetilde{w}$ is trivial: $\widetilde{F}\equiv 1$.

The Szegő functions $F$, $\widehat{F}$ satisfy the following properties which
are crucial for the RH analysis.

######  Proposition 2.1.

The functions $F,\widehat{F}\colon\mathbb{C}\setminus[-1,1]\to\mathbb{C}$
satisfy the following properties:

  1. $(i)$

$F(z)$, $\widehat{F}(z)$ are analytic for $z\in\mathbb{C}\setminus[-1,1]$,
with $F(\overline{z})=\overline{F(z)}$ and
$\widehat{F}(\overline{z})=\overline{\widehat{F}(z)}$,

  2. $(ii)$

$\lim_{z\to\infty}F(z)=F_{\infty}\in\mathbb{R}_{+}$,
$\lim_{z\to\infty}\widehat{F}(z)=\widehat{F}_{\infty}\in\mathbb{R}_{+}$,

  3. $(iii)$

$F_{+}(x)F_{-}(x)=w(x)$, $\widehat{F}_{+}(x)\widehat{F}_{-}(x)=\widehat{w}(x)$
for $x\in[-1,1)$,

  4. $(iv)$

$|F_{\pm}(x)|^{2}=w(x)$,
$\big{|}\widehat{F}_{\pm}(x)\big{|}^{2}=\widehat{w}(x)$ for $x\in[-1,1)$.

Note that at $x=-1$, the limits of $F(z)$, $\widehat{F}(z)$ are equal to $0$.
In particular, they are independent of the path $z\to-1$, and we write
$F_{\pm}(-1)=\lim_{z\to-1}F(z)=\lim_{x\downarrow-1}F_{\pm}(x)=0$ and
$\widehat{F}_{\pm}(-1)=\lim_{z\to-1}\widehat{F}(z)=\lim_{x\downarrow-1}\widehat{F}_{\pm}(x)=0$
in this case. We also introduce the function $\phi$ given by (see [4,
Proposition 2.1])

$\displaystyle\phi(z)=z+\big{(}z^{2}-1\big{)}^{1/2},\qquad
z\in\mathbb{C}\setminus[-1,1].$

With this choice, we see that $\phi$ defines a biholomorphism between
$\mathbb{C}\cup\\{\infty\\}\setminus[-1,1]$ and
${\mathbb{C}\cup\\{\infty\\}\setminus\\{z\colon|z|\leq 1\\}}$, mapping
$\infty$ to itself. In particular, the function $\phi$ satisfies

$\displaystyle|\phi(z)|>1,\qquad z\in\mathbb{C}\setminus[-1,1],$ (2.2)

and

$\displaystyle\phi(z)=2z+O\left(\frac{1}{z}\right)\qquad\mbox{as}\quad
z\to\infty.$

Moreover, the inequality (2.2) holds uniformly away from the interval
$[-1,1]$, while on the interval we have

$\displaystyle\lim_{z\to x}|\phi(z)|=1,\qquad x\in[-1,1].$ (2.3)

Additionally, as $\phi(\overline{z})=\overline{\phi(z)}$ for
$z\in\mathbb{C}\setminus[-1,1]$, it follows from (2.3) that

$\displaystyle\phi_{+}(x)\phi_{-}(x)=1,\qquad x\in(-1,1).$

Near the points $z=\pm 1$, we have

$\displaystyle\begin{split}\phi(z)&=1+\sqrt{2}(z-1)^{1/2}+O(|z-1|),\qquad
z\to+1,\\\ \phi(z)&=-1\pm\sqrt{2}\mathrm{i}(z+1)^{1/2}+O(|z+1|),\qquad
z\to-1,\quad z\in\mathbb{C}_{\pm}.\end{split}$ (2.4)

For the subsequent analysis, it is necessary to understand the behaviour of
the functions $\frac{F^{2}}{w}(z)$, $\frac{\widehat{F}^{2}}{\widehat{w}}(z)$
near the points $z=\pm 1$, as these functions show up in the jump matrices of
the corresponding RH problems, see Section 3.

######  Proposition 2.2.

The function
$\frac{F^{2}}{w}\colon\mathbb{C}\setminus[-1,\infty)\to\mathbb{C}$ satisfies

$\displaystyle\frac{F^{2}}{w}(z)=1\mp\frac{\mathrm{i}\pi}{w(z)}-\frac{\pi^{2}}{2w^{2}(z)}+O\left(\frac{1}{\log^{3}(z-1)}\right)$
(2.5)

uniformly for any path $z\to+1$ in $\mathbb{C}\setminus[-1,\infty)$, where the
$-$ is taken for $\mathop{\mathrm{Im}}(z)>0$ and $+$ for
$\mathop{\mathrm{Im}}(z)<0$, and also

$\displaystyle\frac{F^{2}}{w}(z)=\phi(z)^{-1}\exp\bigl{\\{}-\big{(}z^{2}-1\big{)}^{1/2}\big{(}d_{0}+O(|z+1|)\big{)}\bigr{\\}},$
(2.6)

uniformly for any path $z\to-1$ in $\mathbb{C}\setminus[-1,\infty)$, where

$\displaystyle
d_{0}=\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{\log\big{(}w(\zeta)/(1+\zeta)\big{)}}{(\zeta^{2}-1)^{1/2}}\frac{{\rm
d}\zeta}{\zeta+1}$ (2.7)

and $\gamma$ is an oriented contour originating from the point $\zeta=1$,
going anticlockwise around the interval $[-1,1]$ and ending again at the point
$\zeta=1$ as depicted in Figure 3.

(3.5,0, -1,1.2, -4.8,0, -1,-1.2, 3.5,0) $+1$$-1$$\times$$z$$\gamma$ Figure 3:
The contour $\gamma$ encircling the point $z$.

###### Proof.

For the proof of statement (2.5), see [4, Proposition A.1]. Statement (2.6) is
a special case of [12, Lemma 6.6], but with an additional restriction on the
choice of the contour $\gamma$ stemming from the logarithmic singularity of
$w$. Due to this technicality, we repeat the proof found therein.

First, let us note that if $F_{1}$ is the Szegő function of a weight $w_{1}$
and $F_{2}$ the Szegő function of a weight $w_{2}$, then the Szegő function
$F_{12}$ of the product $w_{1}w_{2}$ is given by the product of the individual
Szegő functions: $F_{12}=F_{1}F_{2}$. As the Szegő function of a Jacobi weight
$w_{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta}$ with $\alpha,\beta>-1$, is
given by (see [12, Remark 5.1])

$\displaystyle
F_{\alpha,\beta}(z)=\frac{(z-1)^{\alpha/2}(z+1)^{\beta/2}}{\phi(z)^{\frac{\alpha+\beta}{2}}},$

we can conclude that

$\displaystyle F(z)={}$ $\displaystyle\frac{(z+1)^{1/2}}{\phi(z)^{1/2}}$
$\displaystyle\times\exp\Bigg{(}\dfrac{\big{(}z^{2}-1\big{)}^{1/2}}{2\pi}\int_{-1}^{1}\dfrac{\log(w(s)/(1+s))}{\sqrt{1-s^{2}}}\dfrac{{\rm
d}s}{z-s}\Bigg{)},\qquad z\in\mathbb{C}\setminus[-1,1].$ (2.8)

Again, $(z+1)^{1/2}$ and $\phi(z)^{1/2}$ denote the principal branches (a
simple calculation shows that $\phi(z)\not\in(-\infty,-1]$ for
$z\in\mathbb{C}\setminus(-\infty,1]$). Moreover, for $x<-1$,
$\big{(}\phi(x)^{1/2}\big{)}_{+}=-\big{(}\phi(x)^{1/2}\big{)}_{-}$ as
$\phi(z)\sim 2z$ for $z\to\infty$. Thus $(z+1)^{1/2}/\phi(z)^{1/2}$ is indeed
analytic in $\mathbb{C}\setminus[-1,1]$.

To analyze the argument of the exponential in (2.8), we choose for a fixed
$z\in\mathbb{C}\setminus[-1,1]$ a contour $\gamma$ as shown in Figure 3. Then
a residue calculation shows that

$\displaystyle\frac{1}{\pi}\int_{-1}^{1}\frac{\log(w(s)/(1+s))}{\sqrt{1-s^{2}}}\frac{{\rm
d}s}{z-s}$
$\displaystyle\qquad=\frac{\log(w(z)/(1+z))}{\big{(}z^{2}-1\big{)}^{1/2}}-\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{\log(w(\zeta)/(1+\zeta))}{\big{(}\zeta^{2}-1\big{)}^{1/2}}\frac{{\rm
d}\zeta}{\zeta-z}.$ (2.9)

Here we note the $w(z)/(1+z)$ is analytic and non-zero in the simply connected
region $\mathbb{C}\setminus[1,\infty)$, and hence $\log(w(z)/(1+z))$ exists
and is analytic in this region. Moreover, the integrand in (2.9) is integrable
along $\gamma$ as long as $z\not\in\gamma$.

Note also that as $\log(w(\zeta)/(1+\zeta))$ has an iterated logarithmic
singularity at $\zeta=1$, we cannot deform $\gamma$ away from this point.
Plugging (2.8) and (2.9) into the definition of $\frac{F^{2}}{w}$, we obtain

$\displaystyle\frac{F^{2}}{w}(z)=\phi(z)^{-1}\exp\Biggl{(}-\frac{\big{(}z^{2}-1\big{)}^{1/2}}{2\pi\mathrm{i}}\int_{\gamma}\frac{\log(w(\zeta)/(1+\zeta))}{\big{(}\zeta^{2}-1\big{)}^{1/2}}\frac{{\rm
d}\zeta}{\zeta-z}\Biggr{)}.$

We compute the Taylor expansion around $\zeta=-1$ to obtain

$\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{\log(w(\zeta)/(1+\zeta))}{\big{(}\zeta^{2}-1\big{)}^{1/2}}\frac{{\rm
d}\zeta}{\zeta-z}=\sum_{k=0}^{\infty}d_{k}(z+1)^{k},$

where $d_{k}$ is given by

$\displaystyle
d_{k}=\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{\log\big{(}w(\zeta)/(1+\zeta)\big{)}}{(\zeta^{2}-1)^{1/2}}\frac{{\rm
d}\zeta}{(\zeta+1)^{k+1}}.$

This finishes the proof. ∎

The analog of Proposition 2.2 for $\frac{\widehat{F}^{2}}{\widehat{w}}$ is
more elementary.

######  Proposition 2.3.

The function
$\frac{\widehat{F}^{2}}{\widehat{w}}\colon\mathbb{C}\setminus[-1,1]\to\mathbb{C}$
satisfies

$\displaystyle\frac{\widehat{F}^{2}}{\widehat{w}}(z)=1+O\big{(}|z-1|^{1/2}\big{)}$
(2.10)

uniformly for any path $z\to+1$ in $\mathbb{C}\setminus[-1,1]$ and can be
written as

$\displaystyle\frac{\widehat{F}^{2}}{\widehat{w}}(z)=\phi(z)^{-1}\exp\bigl{\\{}-\big{(}z^{2}-1\big{)}^{1/2}d_{0}\bigr{\\}}.$
(2.11)

###### Proof.

Statement (2.10) follows directly from [12, Lemma 6.4], while statement (2.11)
can be obtained with the same line of reasoning as in Proposition 2.2, but now
the integral simplifies:

$\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{\log(\widehat{w}(\zeta)/(1+\zeta))}{\big{(}\zeta^{2}-1\big{)}^{1/2}}\frac{{\rm
d}\zeta}{\zeta-z}=\frac{d_{0}}{2\pi\mathrm{i}}\int_{\gamma}\frac{\zeta}{\big{(}\zeta^{2}-1\big{)}^{1/2}}\frac{{\rm
d}\zeta}{\zeta-z}.$

In particular, we can deform the contour $\gamma$ to infinity leading to
$\int_{\gamma}\frac{\zeta}{(\zeta^{2}-1)^{1/2}}\frac{{\rm
d}\zeta}{\zeta-z}=2\pi\mathrm{i}$ and finishing the proof. ∎

From now, on estimates of the type (2.10) are always understood to be uniform
for any path taken in the appropriate domain. The following two corollaries
contain important estimates related to the behaviour of the functions
$\frac{F^{2}}{w}$ and $\frac{\widehat{F}^{2}}{\widehat{w}}$ near the critical
points $\pm 1$.

######  Corollary 2.4 ([4, Proposition 2.4]).

For $x\to 1$ such that $x>1$, we have

$\displaystyle\frac{F^{2}}{w_{+}}(x)+\frac{F^{2}}{w_{-}}(x)-2=-\frac{3\pi^{2}}{\log^{2}\frac{2}{x-1}}+O\left(\frac{1}{\log^{3}(x-1)}\right).$
(2.12)

######  Corollary 2.5.

For $z\to-1$, we have

$\displaystyle\frac{F^{2}}{w}(z)-\frac{\widehat{F}^{2}}{\widehat{w}}(z)=O\big{(}|z+1|^{3/2}\big{)}.$
(2.13)

###### Proof.

The statement follows from (2.6) and (2.11). ∎

######  Remark 2.6.

As noted earlier, the estimate (2.13) is the motivation for considering the
particular weight function $\widehat{w}(x)$ instead of the simpler Jacobi
weight function $1+x$.

For later analysis, we will also need the following technical result.

######  Proposition 2.7.

Fix $R>0$. Let $r_{n},\tilde{r}_{n}\in(0,1)$, $n\in\mathbb{N}$ be two
sequences satisfying $r_{n},\tilde{r}_{n}\to 0$, such that
$n\big{|}\frac{r_{n}}{\tilde{r}_{n}}-1|<R$. Then

$\displaystyle\frac{F^{2}}{w_{+}}(1+r_{n})-\frac{F^{2}}{w_{+}}(1+\tilde{r}_{n})+\frac{F^{2}}{w_{-}}(1+r_{n})-\frac{F^{2}}{w_{-}}(1+\tilde{r}_{n})$
$\displaystyle\qquad=O(r_{n}\log|\log
r_{n}|)+O\left(\frac{1}{n\log^{3}r_{n}}\right)+O\big{(}n^{-2}\big{)},$

where the implied constants in the $O$-terms depend only on $R$.

###### Proof.

For the proof see Proposition A.1. ∎

######  Remark 2.8.

The original formulation found in [4, Proposition A.4] assumes that
$r_{n},\tilde{r}_{n}=\smash{O\big{(}\frac{1}{n^{2}}\big{)}}$, but does not
take into account the fact that the error term will additionally depend on the
convergence rate of the sequences $r_{n}$, $\tilde{r}_{n}$, i.e., on the bound
$C_{r,\tilde{r}}>0$ such that
$r_{n},\tilde{r}_{n}<\frac{C_{r,\tilde{r}}}{n^{2}}$. This fact however becomes
crucial in the proof of Proposition 7.3 [4, Proposition C.4]. To capture the
dependence of the error term on the precise decay rate of $r_{n}$,
$\tilde{r}_{n}$, we have chosen to express the error in terms of $r_{n}$ and
$n$, though one could have used $C_{r,\tilde{r}}$ and $n$ instead. Luckily,
the gap in the original formulation can be easily filled in as shown in
Proposition A.1.

##  3 The Riemann–Hilbert problem for orthogonal polynomials

In the following section, we recall the celebrated Fokas–Its–Kitaev
characterization of orthogonal polynomials via RH problems [10]. We will state
the problem explicitly in the case where the weight function is $w(x)$,
$x\in[-1,1]$, but similar characterizations hold for the other weight
functions $\widehat{w}$ and $\widetilde{w}$.

###  3.1 Fokas–Its–Kitaev RH problem for the logarithmic weight

Find a $2\times 2$ matrix-valued function
$Y=Y^{(n)}\colon\mathbb{C}\setminus[-1,1]\to\mathbb{C}^{2\times 2}$ satisfying
the following properties:

  1. (i)

$Y(z)$ is analytic for $z\in\mathbb{C}\setminus[-1,1]$,

  2. (ii)

$Y$ satisfies the jump condition

$\displaystyle Y_{+}(s)=Y_{-}(s)\begin{pmatrix}1&w(s)\\\
0&1\end{pmatrix},\qquad s\in[-1,1],$

  3. (iii)

$Y(z)\begin{pmatrix}z^{-n}&0\\\ 0&z^{n}\end{pmatrix}=I+O\big{(}z^{-1}\big{)}$,
as $z\rightarrow\infty$,

  4. (iv)

$Y$ is bounded away from the points $\pm 1$, and has the following behaviours
near the points $\pm 1$:

$\displaystyle Y(z)=\begin{pmatrix}O(1)&O\big{(}\log^{2}|z-1|\big{)}\\\
O(1)&O\big{(}\log^{2}|z-1|\big{)}\end{pmatrix},\qquad z\to+1,$

and

$\displaystyle Y(z)=\begin{pmatrix}O(1)&O(1)\\\ O(1)&O(1)\end{pmatrix},\qquad
z\to-1.$

The condition (iv) for $z\to+1$ has been shown in [4, Section 3.1] for the
case of the weight function having a logarithmic singularity, while the
behaviour for $z\to-1$ is a special case of the algebraic-type singularity of
the weight function treated in [12, Section 2].

To understand the connection between the RH problem for $Y$ and orthogonal
polynomials with respect to the orthogonality measure $w(x){\rm d}x$ on
$[-1,1]$, we need to introduce the Cauchy operator $\mathcal{C}_{[-1,1]}$ on
the interval $[-1,1]$:

$\displaystyle\mathcal{C}_{[-1,1]}\colon\ L^{2}([-1,1])$
$\displaystyle\to\mathcal{O}(\mathbb{C}\setminus[-1,1]),$ $\displaystyle f(s)$
$\displaystyle\mapsto\mathcal{C}_{[-1,1]}(f)(z)=\frac{1}{2\pi\mathrm{i}}\int_{-1}^{1}\frac{f(s)}{s-z}{\rm
d}s.$

Here $\mathcal{O}(\mathbb{C}\setminus[-1,1])$ denotes the space of functions
holomorphic in $\mathbb{C}\setminus[-1,1]$. When $\mathcal{C}_{[-1,1]}$ is
applied to matrix-valued functions, it is understood to act componentwise.
Cauchy operators can also be defined on different contours, as in Section 4.

We can now state the seminal result by Fokas, Its and Kitaev which
characterizes orthogonal polynomials via a RH problem:

######  Theorem 3.1 ([10]).

The RH problem for $Y$ is solved uniquely by

$\displaystyle
Y(z)=\begin{pmatrix}\pi_{n}(z)&\mathcal{C}_{[-1,1]}(\pi_{n}w)(z)\\\
-2\pi\mathrm{i}\gamma^{2}_{n-1}\pi_{n-1}(z)&-2\pi\mathrm{i}\gamma^{2}_{n-1}\mathcal{C}_{[-1,1]}(\pi_{n-1}w)(z)\end{pmatrix},$

where $\pi_{n}(z)$ is the $n$-th monic orthogonal polynomial with respect to
the weight function $w(x)$ and $\gamma_{n}>0$ is the leading coefficients of
the orthonormal polynomial $p_{n}$, meaning $p_{n}=\gamma_{n}\pi_{n}$.

The fact that the Fokas–Its–Kitaev RH problem characterizes the corresponding
orthogonal polynomials has lead to numerous new results, in particular in the
case where the associated weight function satisfies local analyticity
properties (see, e.g., [2, 5, 6, 7, 12]), but also in the case of nonanalytic
weights (see [1, 11, 16, 17, 21]). Instrumental in the derivation of those
results has been the nonlinear steepest descent method, first presented in [8]
to study the long-time asymptotics of the mKdV equation and later generalized
to the Fokas–Its–Kitaev RH problem in [2, 6, 7]. It turns out that the
recurrence coefficients can also be simply expressed in term of $Y$ (see [5]):

######  Proposition 3.2.

Let $Y_{1}^{(n)}\in\mathbb{C}^{2\times 2}$ be given through the expansion

$\displaystyle
Y^{(n)}(z)z^{-n\sigma_{3}}=I+\frac{Y_{1}^{(n)}}{z}+O\left(\frac{1}{z^{2}}\right)\qquad\text{as}\quad
z\to\infty.$

Then the recurrence coefficients $a_{n}$ and $b_{n-1}$ can be extracted from
$Y_{1}^{(n)}$ via the formulas

$\displaystyle
a_{n}=\big{(}Y_{1}^{(n)}\big{)}_{11}-\big{(}Y_{1}^{(n+1)}\big{)}_{11}\qquad\text{and}\qquad
b_{n-1}^{2}=\big{(}Y_{1}^{(n)}\big{)}_{12}\big{(}Y_{1}^{(n)}\big{)}_{21}.$

The nonlinear steepest descent analysis for weight functions supported on a
single finite interval has been performed in [12], and in the following we
shall repeat the conjugation and deformation steps found therein. In the first
step one normalizes the RH problem at infinity through an appropriate
conjugation, viz.,

$\displaystyle
T(z):=\left(\frac{2}{F_{\infty}}\right)^{n\sigma_{3}}Y(z)\left(\frac{F(z)}{\phi^{n}(z)}\right)^{\sigma_{3}},\qquad
z\in\mathbb{C}\setminus[-1,1].$ (3.1)

Then $T$ turns out to be the unique solution of the following RH problem.

###  3.2 Normalized Fokas–Its–Kitaev RH problem for the logarithmic weight

Find a $2\times 2$ matrix-valued function
$T=T^{(n)}\colon\mathbb{C}\setminus[-1,1]\to\mathbb{C}^{2\times 2}$ satisfying
the following properties:

  1. (i)

$T(z)$ is analytic for $z\in\mathbb{C}\setminus[-1,1]$,

  2. (ii)

$T$ satisfies the jump condition

$\displaystyle
T_{+}(s)=T_{-}(s)\begin{pmatrix}\frac{F_{+}^{2}}{w}(s)\phi_{+}^{-2n}(s)&1\\\
0&\frac{F_{-}^{2}}{w}(s)\phi_{-}^{-2n}(s)\end{pmatrix},\qquad s\in[-1,1],$
(3.2)

  3. (iii)

$T(z)=I+O\big{(}z^{-1}\big{)}$, as $z\rightarrow\infty$.

  4. (iv)

$T$ is bounded away from the points $\pm 1$, and has the following behaviours
near the points $\pm 1$:

$\displaystyle
T(z)=\begin{pmatrix}O\big{(}\log^{1/2}|z-1|\big{)}&O\big{(}\log^{3/2}|z-1|\big{)}\\\
O\big{(}\log^{1/2}|z-1|\big{)}&O\big{(}\log^{3/2}|z-1|\big{)}\end{pmatrix},\qquad
z\to+1$

and

$\displaystyle
T(z)=\begin{pmatrix}O\big{(}|z+1|^{1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\\\
O\big{(}|z+1|^{1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\end{pmatrix},\qquad
z\to-1.$ (3.3)

Note that we follow the convention found in [4] where the matrix $T$ is
conjugated by the Szegő function $F$, as in (3.1). Hence, the matrix $T$ found
in [12, equation (3.1)] differs from ours in that respect. In our case the
inclusion of the Szegő function $F$ in the jump matrices (3.2) will play a
crucial role in regularizing the jump matrices of the RH problems and thus
enabling us to make the comparison argument in Section 6.1 effective.

However, as the weight function $w_{k}$ from (1.6) is nonvanishing at $z=-1$,
the matrix $T$ in (3.1) also differs crucially in its behaviour as $z\to-1$
from the one found in [4, equation (3.7)]. The reason is that our logarithmic
weight function $w$ has a simple zero at $z=-1$, implying by item (iv) in
Proposition 2.1 that $|F(z)|=O\big{(}|z+1|^{1/2}\big{)}$ as $z\to-1$. This
induces the $O\big{(}|z+1|^{-1/2}\big{)}$-behaviour in $T$ as $z\to-1$.
Crucially, the entries of $T_{\pm}$ (and later $Q_{\pm}$) will not be square
integrable, meaning that the $L^{2}$-theory used in [4] will not be applicable
directly. We will circumvent this difficulty by defining certain ⋆RH problems
in Section 5 through inverting locally by appropriate local parametrix
solutions near the endpoint $z=-1$.

In the second step, we will use the following factorization of the jump matrix
(3.2):

$\displaystyle\begin{pmatrix}\frac{F_{+}^{2}}{w}(s)\phi_{+}^{-2n}(s)&1\\\
0&\frac{F_{-}^{2}}{w}(s)\phi_{-}^{-2n}(s)\end{pmatrix}=\begin{pmatrix}1&0\\\
\frac{F_{-}^{2}}{w}(s)\phi_{-}^{-2n}(s)&1\end{pmatrix}$
$\displaystyle\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\begin{pmatrix}1&0\\\
\frac{F_{+}^{2}}{w}(s)\phi_{+}^{-2n}(s)&1\end{pmatrix}.$

Next, we introduce the oriented _lens_ jump contour
$\Sigma=\Sigma_{1}\cup\Sigma_{2}\cup(-1,1+\delta)$, see Figure 4, where
$\delta>0$ is fixed and $n$-independent.

(0,2.5, 3.5,1.5, 7,2.5) (0,2.5, 3.5,3.5, 7,2.5)
$\Sigma_{1}$$\Sigma_{2}$$\Omega_{0}$$\Omega_{1}$$\Omega_{2}$$1+\delta$$-1$$1$$\bullet$
Figure 4: Lens-shaped jump contour $\Sigma$.

Note that the matrices

$\displaystyle\begin{pmatrix}1&0\\\
\frac{F_{+}^{2}}{w}(s)\phi_{+}^{-2n}(s)&1\end{pmatrix},\qquad\begin{pmatrix}1&0\\\
\frac{F_{-}^{2}}{w}(s)\phi_{-}^{-2n}(s)&1\end{pmatrix},\qquad s\in(-1,1)$

can be analytically continued to $z\in\Omega_{1}$ and $z\in\Omega_{2}$,
respectively. Hence, we can define the following matrix-valued function
$Q\colon\mathbb{C}\setminus\Sigma\to\mathbb{C}^{2\times 2}$:

$\displaystyle Q(z)=\begin{cases}T(z),&z\in\Omega_{0},\\\
T(z)\begin{pmatrix}1&0\\\
-\frac{F^{2}}{w}(z)\phi^{-2n}(z)&1\end{pmatrix},&z\in\Omega_{1},\vspace{1mm}\\\
T(z)\begin{pmatrix}1&0\\\
\frac{F^{2}}{w}(z)\phi^{-2n}(z)&1\end{pmatrix},&z\in\Omega_{2}.\end{cases}$
(3.4)

Then $Q$ will be the solution to the following RH problem.

###  3.3 Logarithmic RH problem

Find a $2\times 2$ matrix-valued function
$Q\colon\mathbb{C}\setminus\Sigma\to\mathbb{C}^{2\times 2}$ satisfying the
following properties:

  1. (i)

$Q(z)$ is analytic for $z\in\mathbb{C}\setminus\Sigma$,

  2. (ii)

$Q$ satisfies the jump condition

$\displaystyle Q_{+}(s)=Q_{-}(s)v(s),\qquad s\in\Sigma,$

where

$\displaystyle v(s)=\begin{cases}\begin{pmatrix}1&0\\\
\frac{F^{2}}{w}(s)\phi^{-2n}(s)&1\end{pmatrix},&\mbox{for}\
s\in\Sigma_{1}\cup\Sigma_{2},\vspace{1mm}\\\ \begin{pmatrix}1&0\\\
\big{(}\frac{F^{2}}{w_{+}}(s)+\frac{F^{2}}{w_{-}}(s)\big{)}\phi^{-2n}(s)&1\end{pmatrix},&\mbox{for}\
s\in(1,1+\delta),\vspace{1mm}\\\ \begin{pmatrix}0&1\\\
-1&0\end{pmatrix},&\mbox{for}\ s\in(-1,1),\end{cases}$ (3.5)

  3. (iii)

$Q(z)=I+O\big{(}z^{-1}\big{)}$, as $z\rightarrow\infty$,

  4. (iv)

$Q$ is bounded away from the points $\pm 1$, and has the following behaviours
near the points $\pm 1$:

$\displaystyle
Q(z)=\begin{pmatrix}O\big{(}\log^{3/2}|z-1|\big{)}&O\big{(}\log^{3/2}|z-1|\big{)}\\\
O\big{(}\log^{3/2}|z-1|\big{)}&O\big{(}\log^{3/2}|z-1|\big{)}\end{pmatrix},\qquad
z\to+1$

and

$\displaystyle
Q(z)=\begin{pmatrix}O\big{(}|z+1|^{-1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\\\
O\big{(}|z+1|^{-1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\end{pmatrix},\qquad
z\to-1.$

Note that $Q_{\pm}\not\in L^{2}(\Sigma)$ due to its behaviour as $z\to-1$,
which is caused by the simple zero of $w$ at that point.

For the asymptotic analysis in Section 7, we will need the analog of the
logarithmic RH problem stated for the model weight function $\widehat{w}$. The
derivation from the Fokas–Its–Kitaev formulation remains unchanged except for
the use of the functions $\widehat{w}$ and $\widehat{F}$ instead of $w$ and
$F$. The behaviour near $z\to\pm 1$ can be read off from [12, Section 4],
after taking into account the behaviour of the Szegő function $\widehat{F}$.

###  3.4 Model RH problem

Find a $2\times 2$ matrix-valued function
${\widehat{Q}}\colon\mathbb{C}\setminus\Sigma\to\mathbb{C}^{2\times 2}$
satisfying the following properties:

  1. (i)

${\widehat{Q}}(z)$ is analytic for $z\in\mathbb{C}\setminus\Sigma$,

  2. (ii)

${\widehat{Q}}$ satisfies the jump condition

$\displaystyle{\widehat{Q}}_{+}(s)={\widehat{Q}}_{-}(s){\widehat{v}}(s),\qquad
s\in\Sigma,$

where

$\displaystyle{\widehat{v}}(s)=\begin{cases}\begin{pmatrix}1&0\\\
\frac{\widehat{F}^{2}}{\widehat{w}}(s)\phi^{-2n}(s)&1\end{pmatrix},&\mbox{for}\
s\in\Sigma_{1}\cup\Sigma_{2},\vspace{1mm}\\\ \begin{pmatrix}1&0\\\
2\frac{\widehat{F}^{2}}{\widehat{w}}(s)\phi^{-2n}(s)&1\end{pmatrix},&\mbox{for}\
s\in(1,1+\delta),\vspace{1mm}\\\ \begin{pmatrix}0&1\\\
-1&0\end{pmatrix},&\mbox{for}\ s\in(-1,1),\end{cases}$ (3.6)

  3. (iii)

${\widehat{Q}}(z)=I+O\big{(}z^{-1}\big{)}$, as $z\rightarrow\infty$,

  4. (iv)

${\widehat{Q}}$ is bounded away from the points $\pm 1$, and has the following
behaviours near the points $\pm 1$:

$\displaystyle{\widehat{Q}}(z)=\begin{pmatrix}O(\log|z-1|)&O(\log|z-1|)\\\
O(\log|z-1|)&O(\log|z-1|)\end{pmatrix},\qquad z\to+1$

and

$\displaystyle{\widehat{Q}}(z)=\begin{pmatrix}O\big{(}|z+1|^{-1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\\\
O\big{(}|z+1|^{-1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\end{pmatrix},\qquad
z\to-1.$

Note that the jump matrix ${\widehat{v}}$ simplifies compared to $v$, as the
weight function $\widehat{w}$ is continuous (in fact analytic) across
$(1,1+\delta)$. As for $Q$, due to the simple zero of $\widehat{w}$ at $z=-1$,
${\widehat{Q}}$ will not be square integrable near that point.

Note that the weight function $\widehat{w}$ falls into the class of _modified
Jacobi weight functions_ considered in [12]. As such, an asymptotic series
expansion in powers of $n^{-1}$ for the recurrence coefficients
$\widehat{a}_{n}$, $\widehat{b}_{n}$ can be explicitly computed. Note however
that we use a different convention here than in [12], i.e., the roles of
$\widehat{a}_{n}$, $\widehat{b}_{n}$ are interchanged. We write down the
expansion up to the $n^{-2}$-term:

######  Corollary 3.3 ([12, Theorem. 1.10]).

The recurrence coefficients $\widehat{a}_{n}$, $\widehat{b}_{n}$ associated to
the weight function $\widehat{w}$ satisfy

$\displaystyle\widehat{a}_{n}=\frac{1}{4n^{2}}+O\left(\frac{1}{n^{3}}\right),\qquad\widehat{b}_{n}=\frac{1}{2}-\frac{1}{16n^{2}}+O\left(\frac{1}{n^{3}}\right).$

To compute the asymptotics of the recurrence coefficients $a_{n}$, $b_{n}$, we
will first compute the asymptotics of $a_{n}-\widehat{a}_{n}$,
$b_{n}^{2}-\widehat{b}_{n}^{2}$ and then use Corollary 3.3. Hence, we will
need an analog of Proposition 3.2 above for the differences of recurrence
coefficients, which we express in terms of $Q$ and ${\widehat{Q}}$.

######  Proposition 3.4 ([4, Proposition 3.6]).

Let $Q_{1}^{(n)}$ and ${\widehat{Q}}_{1}^{(n)}$ be given through the expansion

$\displaystyle
Q^{(n)}=I+\frac{Q_{1}^{(n)}}{z}+O\left(\frac{1}{z^{2}}\right)\qquad\text{as}\quad
z\to\infty,$

and

$\displaystyle{\widehat{Q}}^{(n)}=I+\frac{{\widehat{Q}}_{1}^{(n)}}{z}+O\left(\frac{1}{z^{2}}\right)\qquad\text{as}\quad
z\to\infty.$

Then the differences $a_{n}-\widehat{a}_{n}$ and
$b_{n}^{2}-\widehat{b}_{n}^{2}$ can be expressed via

$\displaystyle
a_{n}-\widehat{a}_{n}=\big{(}Q_{1}^{(n)}\big{)}_{11}-\big{(}{\widehat{Q}}_{1}^{(n)}\big{)}_{11}-\big{(}\big{(}Q_{1}^{(n+1)}\big{)}_{11}-\big{(}{\widehat{Q}}_{1}^{(n+1)}\big{)}_{11}\big{)},$
(3.7)

and

$\displaystyle b_{n-1}^{2}-\widehat{b}_{n-1}^{2}={}$
$\displaystyle\big{(}\big{(}Q_{1}^{(n)}\big{)}_{12}-\big{(}{\widehat{Q}}_{1}^{(n)}\big{)}_{12}\big{)}\big{(}\big{(}Q_{1}^{(n)}\big{)}_{21}-\big{(}Q_{1}^{(n+1)}\big{)}_{21}\big{)}$
$\displaystyle+\big{(}{\widehat{Q}}_{1}^{(n)}\big{)}_{12}\big{[}\big{(}\big{(}Q_{1}^{(n)}\big{)}_{21}-\big{(}Q_{1}^{(n+1)}\big{)}_{21}\big{)}-\big{(}\big{(}{\widehat{Q}}_{1}^{(n)}\big{)}_{21}-\big{(}{\widehat{Q}}_{1}^{(n+1)}\big{)}_{21}\big{)}\big{]}.$
(3.8)

The usefulness of Proposition 3.4 comes from the fact that
$Q^{(n)}_{1}-{\widehat{Q}}^{(n)}_{1}$ has a simple integral representation.

######  Proposition 3.5 ([4, Proposition 4.9]).

The following formula holds:

$\displaystyle
Q^{(n)}_{1}-{\widehat{Q}}^{(n)}_{1}=-\frac{1}{2\pi\mathrm{i}}\int_{\Sigma}Q^{(n)}_{-}(s)\big{(}v^{(n)}(s)-{\widehat{v}}^{(n)}(s)\big{)}\big{[}{\widehat{Q}}^{(n)}_{-}(s)\big{]}^{-1}{\rm
d}s.$ (3.9)

###### Proof.

In the following, we drop the subscript $(n)$ for better readability, i.e.,
write $Q_{1}$ for $Q^{(n)}_{1}$ and so on. Let us define the matrix-valued
function $X(z)=Q(z)\big{[}{\widehat{Q}}(z)\big{]}^{-1}$ for
$z\in\mathbb{C}\setminus\Sigma$. Note that

$\displaystyle
X(z)=I+\frac{Q_{1}-{\widehat{Q}}_{1}}{z}+O\left(\frac{1}{z^{2}}\right)\qquad\text{as
}z\to\infty.$ (3.10)

Moreover, $X$ satisfies the jump condition

$\displaystyle X_{+}(s)=X_{-}(s)v_{X}(s),\qquad s\in\Sigma,$

where $v_{X}={\widehat{Q}}_{-}v{\widehat{v}}^{-1}{\widehat{Q}}_{-}^{-1}$. We
claim that $X_{\pm}\in L^{1}(\Sigma)$. To see this let us first introduce the
analog of $T$ for the weight function $\widehat{w}$:

$\displaystyle\widehat{T}(z)=\left(\frac{2}{\widehat{F}_{\infty}}\right)^{n\sigma_{3}}\widehat{Y}(z)\left(\frac{\widehat{F}(z)}{\phi^{n}(z)}\right)^{\sigma_{3}}.$

Here $\hat{Y}$ is the solution to the Fokas–Its–Kitaev problem for the weight
function $\widehat{w}$. Then for ${z\in\Omega_{0}}$ (cf. Figure 4), we have

$\displaystyle X(z)=T(z)\big{[}\widehat{T}(z)\big{]}^{-1}=O(1),\qquad
z\in\Omega_{0},\quad z\to-1,$

where we have used (3.3) and its analog (see [12, Section 2])

$\displaystyle\widehat{T}(z)=\begin{pmatrix}O\big{(}|z+1|^{1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\\\
O\big{(}|z+1|^{1/2}\big{)}&O\big{(}|z+1|^{-1/2}\big{)}\end{pmatrix},\qquad
z\to-1.$ (3.11)

Note that the analog of (3.4) remains valid

$\displaystyle{\widehat{Q}}(z)=\begin{cases}\widehat{T}(z),&z\in\Omega_{0},\\\
\widehat{T}(z)\begin{pmatrix}1&0\\\
-\frac{\widehat{F}^{2}}{\widehat{w}}(z)\phi^{-2n}(z)&1\end{pmatrix},&z\in\Omega_{1},\vspace{1mm}\\\
\widehat{T}(z)\begin{pmatrix}1&0\\\
\frac{\widehat{F}^{2}}{\widehat{w}}(z)\phi^{-2n}(z)&1\end{pmatrix},&z\in\Omega_{2}.\end{cases}$

Thus for $z\in\Omega_{1}\cup\Omega_{2}$, we have

$\displaystyle X(z)=T(z)\begin{pmatrix}1&0\\\
\mp\big{(}\frac{F^{2}}{w}(z)-\frac{\widehat{F}^{2}}{\widehat{w}}(z)\big{)}\phi^{-2n}(z)&1\end{pmatrix}\big{[}\widehat{T}(z)\big{]}^{-1}=O(1),$
$\displaystyle z\in\Omega_{1}\cup\Omega_{2},\qquad z\to-1,$

where the $+$, resp. $-$ sign refer to $\Omega_{1}$ and $\Omega_{2}$. Apart
from (3.3) and (3.11), we have used in addition (2.13). As $X$ can have only
logarithmic singularities for $z\to 1$ and otherwise the limits to the contour
are analytic, it follows that $X_{\pm}\in L^{1}(\Sigma)$.

Next, we can use the Sokhotski–Plemelj theorem to conclude that $X$ can be
represented as

$\displaystyle X(z)$
$\displaystyle=I+\frac{1}{2\pi\mathrm{i}}\int_{\Sigma}\frac{X_{+}(s)-X_{-}(s)}{s-z}{\rm
d}s$
$\displaystyle=I+\frac{1}{2\pi\mathrm{i}}\int_{\Sigma}\frac{Q_{-}(s)\big{(}v(s)[{\widehat{v}}(s)]^{-1}-I\big{)}\big{[}{\widehat{Q}}_{-}(s)\big{]}^{-1}}{s-z}{\rm
d}s$
$\displaystyle=I+\frac{1}{2\pi\mathrm{i}}\int_{\Sigma}\frac{Q_{-}(s)(v(s)-{\widehat{v}}(s))[{\widehat{Q}}_{-}(s)]^{-1}}{s-z}{\rm
d}s,$

where we made use of the particular form of the jump matrices $v$ and
${\widehat{v}}$. This representation together with (3.10) implies (3.9). ∎

Finally, we recall the Legendre RH problem taken from [4, Section 3.4]. While
this RH problem does not approximate the logarithmic RH problem globally, it
does so near the logarithmic singularity and additionally gives rise to a
singular integral operator whose inverse is uniformly bounded as $n\to\infty$
(see Theorem 4.2). The existence of a RH problem with these properties will be
crucial for the proof of Theorem 6.2. Note that the Szegő function for the
Legendre weight is just given by $\widetilde{F}\equiv 1$.

###  3.5 Legendre RH problem

Find a $2\times 2$ matrix-valued function
${\widetilde{Q}}\colon\mathbb{C}\setminus\Sigma\to\mathbb{C}^{2\times 2}$
satisfying the following properties (see [4, Section 3.4]):

  1. (i)

${\widetilde{Q}}(z)$ is analytic for $z\in\mathbb{C}\setminus\Sigma$,

  2. (ii)

${\widetilde{Q}}$ satisfies the jump condition

$\displaystyle{\widetilde{Q}}_{+}(s)={\widetilde{Q}}_{-}(s){\widetilde{v}}(s),\qquad
s\in\Sigma,$

where

$\displaystyle{\widetilde{v}}(s)=\begin{cases}\begin{pmatrix}1&0\\\
\phi^{-2n}(s)&1\end{pmatrix},&\mbox{for}\
s\in\Sigma_{1}\cup\Sigma_{2},\vspace{1mm}\\\ \begin{pmatrix}1&0\\\
2\phi^{-2n}(s)&1\end{pmatrix},&\mbox{for}\ s\in(1,1+\delta),\vspace{1mm}\\\
\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},&\mbox{for}\ s\in(-1,1),\end{cases}$

  3. (iii)

${\widetilde{Q}}(z)=I+O\big{(}z^{-1}\big{)}$ as $z\rightarrow\infty$,

  4. (iv)

${\widetilde{Q}}(z)$ is bounded away from the points $\pm 1$, and has the
following behaviours near the points $\pm 1$:

$\displaystyle{\widetilde{Q}}(z)=\begin{pmatrix}O(\log|z-1|)&O(\log|z-1|)\\\
O(\log|z-1|)&O(\log|z-1|)\end{pmatrix},\qquad z\to+1$

and

$\displaystyle{\widetilde{Q}}(z)=\begin{pmatrix}O(\log|z+1|)&O(\log|z+1|)\\\
O(\log|z+1|)&O(\log|z+1|)\end{pmatrix},\qquad z\to-1.$

The weight function $\widetilde{w}(x)=1$, $x\in(-1,1)$ lies in the class of
weight functions considered in [12], and it follow from the calculations
therein that the solution ${\widetilde{Q}}$ can be globally approximated with
arbitrary small errors. Moreover, unlike the logarithmic and model RH problems
found in Sections 3.3 and 3.4, the Legendre RH problem can be stated with a
weaker $L^{2}$-condition instead of condition (iv) (cf. [4, Proposition 3.2]),
as follows.

######  Proposition 3.6.

The matrix-valued function ${\widetilde{Q}}$ is the unique solution of the
Legendre RH problem with the condition _(iv)_ being replaced by the condition
${\widetilde{Q}}_{\pm}\in L^{2}([-1,1])$.

###### Proof.

Let $L$ be another solution of the Legendre RH problem, but with $L_{\pm}\in
L^{2}([-1,1])$ instead of condition (iv). Then $\det L$ will be a holomorphic
function in $\mathbb{C}\setminus[-1,1]$ with $\det L_{+}=\det L_{-}$ on
$\Sigma$ and $\det L_{\pm}\in L^{1}(\Sigma)$. By Morera’s theorem, it follows
that $\det L$ is in fact an entire function with $\lim_{z\to\infty}\det
L(z)=1$. Hence by Liouville’s theorem $\det L\equiv 1$ in $\mathbb{C}$.

We conclude that $L$ is invertible in $\mathbb{C}\setminus\Sigma$ and we can
define ${\widetilde{Q}}[L]^{-1}$. As with the determinant,
${\widetilde{Q}}[L]^{-1}$ will have no jump across the contour $\Sigma$:
$\big{(}{\widetilde{Q}}[L]^{-1}\big{)}_{+}=\big{(}{\widetilde{Q}}[L]^{-1}\big{)}_{-}$.
Moreover, by the $L^{2}$-condition for ${\widetilde{Q}}$ and $L$, we have that
$\big{(}{\widetilde{Q}}[L]^{-1}\big{)}_{\pm}\in L^{1}(\Sigma)$. It follows
that ${\widetilde{Q}}[L]^{-1}$ can be extended to an entire function. By
Liouville’s theorem, we have that
${\widetilde{Q}}[L]^{-1}\equiv\lim_{z\to\infty}{\widetilde{Q}}(z)[L(z)]^{-1}=I$,
hence ${\widetilde{Q}}=L$. ∎

##  4 An explicit formula for the Legendre resolvent

In this section, we will associate to the Legendre RH problem a singular
integral operator. First, let us define the Cauchy operator on $\Sigma$ by

$\displaystyle\mathcal{C}_{\Sigma}\colon\
L^{2}(\Sigma)\to\mathcal{O}(\mathbb{C}\setminus\Sigma),\qquad
f(s)\mapsto\mathcal{C}_{\Sigma}(f)(z)=\frac{1}{2\pi\mathrm{i}}\int_{\Sigma}\frac{f(s)}{s-z}{\rm
d}s,$

where $\mathcal{O}(\mathbb{C}\setminus\Sigma)$ denotes the set of analytic
functions on the open set $\mathbb{C}\setminus\Sigma$. For $f\in
L^{2}(\Sigma)$, we further define the two Cauchy boundary operators by

$\displaystyle\mathcal{C}_{\Sigma}^{\pm}(f)(s)=\lim_{z\to
s\pm}\mathcal{C}_{\Sigma}(f)(z).$ (4.1)

In our setting, the curve $\Sigma$ is clearly a Carleson curve and hence the
limit in (4.1) exists for almost all $s\in\Sigma$ and satisfies
$\mathcal{C}_{\Sigma}^{\pm}(f)\in L^{2}(\Sigma)$, see [3] for more details. We
define the two Cauchy boundary operators via

$\displaystyle\mathcal{C}_{\Sigma}^{\pm}\colon\ L^{2}(\Sigma)\to
L^{2}(\Sigma).$

These are bounded operators on $L^{2}(\Sigma)$, cf. Theorem 4.1 below. Note
that $C^{\pm}_{\Sigma}$ satisfy the important identity
$C^{+}_{\Sigma}-C^{-}_{\Sigma}=1$.

More generally, the mapping in (4.1) induces a bounded operator on certain
weighted $L^{p}$-spaces. To be precise let $\Gamma$ be an oriented composed
locally rectifiable curve (see [3, Section 1]) and $p\in(1,\infty)$. Given a
weight function $r\colon\Gamma\to\mathbb{R}$, $r\geq 0$, define the Banach
space $L^{p}(\Gamma,r)$ of all measurable functions $f$ on $\Gamma$, such that
the norm

$\displaystyle\|f\|_{L^{p}(\Gamma,r)}=\bigg{(}\int_{\Gamma}|f(s)|^{p}r(s)^{p}|{\rm
d}s|\bigg{)}^{\frac{1}{p}}$

remains finite. Note that there is a $p$-th power of $r$ in the integral. We
say that $r$ is a _Muckenhoupt weight_ if $r\in L^{p}_{\rm loc}(\Gamma)$,
$1/r\in L^{q}_{\rm loc}(\Gamma)$ and

$\displaystyle\sup_{s\in\Gamma}\sup_{\rho>0}\bigg{(}\frac{1}{\rho}\int_{\Gamma\cap
D(s,\rho)}r(s^{\prime})^{p}|{\rm
d}s^{\prime}|\bigg{)}^{\frac{1}{p}}\bigg{(}\frac{1}{\rho}\int_{\Gamma\cap
D(s,\rho)}r(s^{\prime})^{-q}|{\rm d}s^{\prime}|\bigg{)}^{\frac{1}{q}}<\infty,$

where $D(s,\rho)$ is the open disc around $s$ of radius $\rho$ and
$1/p+1/q=1$. For any $p\in(1,\infty)$ we denote the set of all Muckenhoupt
weights by $A_{p}(\Gamma)$. The following results holds.

######  Theorem 4.1.

Let $p\in(1,\infty)$ and let $\Gamma$ be an oriented composed locally
rectifiable curve. Assume $r\colon\Gamma\to\mathbb{R}$, $r\geq 0$ is a given
weight. Then the mappings

$\displaystyle f\mapsto\lim_{z\to s\pm}\int_{\Gamma}\frac{f(s)}{s-z}{\rm d}s$
(4.2)

define bounded operators from $L^{p}(\Gamma,r)\to L^{p}(\Gamma,r)$ if and only
if $r$ is a Muckenhoupt weight, i.e., $r\in A_{p}(\Gamma)$.

The proof can be found in [3, Theorem 4.15], for more material on this topic
with emphasis on RH theory see [14]. We will abuse notation and denote the
mapping (4.2) by $\mathcal{C}^{\pm}_{\Gamma}$ irrespectively of the choice of
domain $L^{p}(\Gamma,r)$.

Next, let us define the operator

$\displaystyle\mathcal{C}_{{\widetilde{v}}}\colon\ L^{2}(\Sigma)\to
L^{2}(\Sigma),\qquad f\mapsto\mathcal{C}_{\Sigma}^{-}(f({\widetilde{v}}-I)),$

and consider the following singular integral equation in $L^{2}(\Sigma)$

$\displaystyle(1-\mathcal{C}_{{\widetilde{v}}}){\widetilde{\mu}}=I.$ (4.3)

Note that as the contour is bounded, $I$ is indeed an element of
$L^{2}(\Sigma)$. Equation (4.3) is, in fact, equivalent to the Legendre RH
problem. More explicitly, any solution ${\widetilde{\mu}}$ will give rise to a
solution $L=I+\mathcal{C}_{\Sigma}({\widetilde{\mu}}({\widetilde{v}}-I))$ of
the Legendre RH problem with the condition $L_{\pm}\in L^{2}(\Sigma)$ instead
of condition (iv), as can be verified by direct computation. By Proposition
3.6, the solution to the Legendre RH problem (see Section 3.5) exists and is
unique, hence we must have ${\widetilde{Q}}=L$ implying

$\displaystyle{\widetilde{Q}}=I+\mathcal{C}_{\Sigma}({\widetilde{\mu}}({\widetilde{v}}-I)).$
(4.4)

Moreover, from the Sokhotski–Plemelj formula

$\displaystyle{\widetilde{Q}}=I+\mathcal{C}_{\Sigma}\big{(}{\widetilde{Q}}_{+}-{\widetilde{Q}}_{-}\big{)}=I+\mathcal{C}_{\Sigma}\big{(}{\widetilde{Q}}_{-}({\widetilde{v}}-I)\big{)},$

it follows, after taking the minus limit to the contour $\Sigma$, that
${\widetilde{\mu}}:={\widetilde{Q}}_{-}$ is indeed a solution to (4.3).
Moreover, any solution ${\widetilde{\mu}}$ of (4.3) must be equal to
${\widetilde{Q}}_{-}$ as can be seen from (4.4) and

$\displaystyle{\widetilde{Q}}_{-}=I+\mathcal{C}_{\Sigma}^{-}({\widetilde{\mu}}({\widetilde{v}}-I))=I+\mathcal{C}_{{\widetilde{v}}}{\widetilde{\mu}}={\widetilde{\mu}}.$

Together these arguments imply that (4.3) has a unique solution and hence
$1-\mathcal{C}_{{\widetilde{v}}}$ must be injective. In [4], it has been shown
that $1-\mathcal{C}_{{\widetilde{v}}}$ is in fact uniformly invertible for
$n\to\infty$, as described in the following result.

######  Theorem 4.2 ([4, Theorem 4.5]).

The operator $1-\mathcal{C}_{{\widetilde{v}}}$ is invertible for all
sufficiently large $n$. Moreover, the operator bound of
$(1-\mathcal{C}_{{\widetilde{v}}})^{-1}$ as an operator $L^{2}(\Sigma)\to
L^{2}(\Sigma)$ remains uniformly bounded for $n\to\infty$.

Theorem 4.2 played a central role in [4]. The uniform invertibility of the
operator $1-\mathcal{C}_{{\widetilde{v}}}$ will also be crucial in the
approach presented here and is the motivation for introducing the Legendre RH
problem in addition to the logarithmic and model RH problems. However, in
order to use Theorem 4.2, we will need to derive an explicit representation of
the operator $(1-\mathcal{C}_{{\widetilde{v}}})^{-1}$. To accomplish this, we
recall the definition of an _inhomogeneous RH problem of the first kind_ , as
introduced in [9, Section 2.6]. This notion has been instrumental in the proof
of Theorem 4.2 in [4]. In the following, $h$ will denote a matrix-valued
function on a contour $\Gamma$, with $h^{\pm 1}\in L^{\infty}(\Gamma)$.

### Inhomogeneous RH problem of the first kind

For a given $g\in L^{2}(\Gamma)$, one seeks an $f\in L^{2}(\Gamma)$, such that
$m_{\pm}=\mathcal{C}_{\Sigma}^{\pm}(f)+g$ satisfies the jump relation:

$\displaystyle m_{+}(s)$ $\displaystyle=m_{-}(s)h(s),\qquad s\in\Gamma.$ (4.5)

A similar notion of an inhomogeneous RH problem of the second kind can be
found in [9, Section 2] but will not be needed here.

The importance of the above inhomogeneous RH problem comes from the following
result proven in [9, Proposition 2.6].

######  Theorem 4.3.

The mapping $1-\mathcal{C}_{h}\colon L^{2}(\Gamma)\to L^{2}(\Gamma)$ is
invertible if and only if the corresponding inhomogeneous RH problem of the
first kind is uniquely solvable for each $g\in L^{2}(\Gamma)$. Moreover, the
inverse satisfies $\big{\|}(1-\mathcal{C}_{h})^{-1}\big{\|}_{L^{2}(\Gamma)\to
L^{2}(\Gamma)}\leq c$ if and only if $\|m_{-}\|_{L^{2}(\Gamma)}\leq
c\|g\|_{L^{2}(\Gamma)}$ for all ${g\in L^{2}(\Gamma)}$ and the same constant
$c>0$.

######  Remark 4.4.

Note that by (4.5), $m_{+}=m_{-}h$ on $\Gamma$, hence
$\|m_{-}\|_{L^{2}(\Gamma)}\leq c\|g\|_{L^{2}(\Gamma)}$ implies
$\|m_{\pm}\|_{L^{2}(\Gamma)}\leq c^{\prime}\|g\|_{L^{2}(\Gamma)}$ with
$c^{\prime}=c\|h\|_{L^{\infty}(\Gamma)}$.

As a corollary we obtain.

######  Corollary 4.5.

Given a sequence $h_{n}$ of matrix-valued functions with $h_{n}^{\pm 1}\in
L^{\infty}(\Gamma)$, the operator $(1-\mathcal{C}_{h_{n}})^{-1}$ is uniformly
bounded if and only if the corresponding inhomogeneous RH problems are
uniquely solvable with $\|m_{-}\|_{L^{2}(\Gamma)}\leq c\|g\|_{L^{2}(\Gamma)}$,
and $c>0$ independent of $n\in\mathbb{N}$ and $g\in L^{2}(\Gamma)$.

Note that Theorem 4.2 is proven in [4, Section 4.2] via Corollary 4.5. We will
use Theorems 4.2 and 4.3 to derive an explicit expression for the inverse
$(1-\mathcal{C}_{{\widetilde{v}}})^{-1}$ in Proposition 4.6 below. This allows
us to identify locally the contribution of the logarithmic singularity to the
uniform boundedness of $(1-\mathcal{C}_{{\widetilde{v}}})^{-1}$, which is
central to our approach as it enables us to prove Theorem 6.2.

######  Proposition 4.6.

The inverse of $1-\mathcal{C}_{{\widetilde{v}}}$ has the explicit form

$\displaystyle(1-\mathcal{C}_{{\widetilde{v}}})^{-1}\colon\ L^{2}(\Sigma)\to
L^{2}(\Sigma),\qquad g\mapsto
g+\mathcal{C}_{\Sigma}^{-}\big{(}g({\widetilde{v}}-I){\widetilde{Q}}_{+}^{-1}\big{)}{\widetilde{Q}}_{-}.$
(4.6)

###### Proof.

From Theorem 4.3, we know that for $g\in L^{2}(\Sigma)$, the (unique)
solvability of the equation

$\displaystyle(1-\mathcal{C}_{{\widetilde{v}}})\psi=g,\quad g\in
L^{2}(\Sigma),$ (4.7)

in $L^{2}(\Sigma)$ is equivalent to the (unique) solvability of the following
inhomogeneous RH problem.

### Inhomogeneous Legendre RH problem

For a given $g\in L^{2}(\Sigma)$, one seeks an $f\in L^{2}(\Sigma)$, such that
$m_{\pm}=\mathcal{C}_{\Sigma}^{\pm}(f)+g$ satisfies the jump relation:

$\displaystyle m_{+}(s)$ $\displaystyle=m_{-}(s){\widetilde{v}}(s),\qquad
s\in\Sigma.$

Note that Theorem 4.2 together with Corollary 4.5 imply that there exists a
constant $c$ independent of $n\in\mathbb{N}$ and $g\in L^{2}(\Sigma)$ such
that the inhomogeneous Legendre RH problem has a unique solution $m_{\pm}$
with $\|m_{-}\|_{L^{2}(\Sigma)}\leq c\|g\|_{L^{2}(\Sigma)}$. We shall briefly
recall the exact relation between (4.7) and the inhomogeneous Legendre RH
problem (cf. [9, Section 2]), as it will be needed later in the proof. First,
note that if we have a solution $m_{\pm}$ to the inhomogeneous Legendre RH
problem, we must have $f=m_{+}-m_{-}=m_{-}({\widetilde{v}}-I)$. If we now set
$\psi:=m_{-}=\mathcal{C}_{\Sigma}^{-}(f)+g\in L^{2}(\Sigma)$, it follows that

$\displaystyle(1-\mathcal{C}_{{\widetilde{v}}})\psi$
$\displaystyle=\psi-\mathcal{C}_{\Sigma}^{-}(\psi({\widetilde{v}}-I))=\mathcal{C}_{\Sigma}^{-}(f)+g-\mathcal{C}_{\Sigma}^{-}(m_{-}({\widetilde{v}}-I)))=g.$

On the other hand, having a solution $\psi$ to the integral equation (4.7), we
can define $m_{\pm}:=\mathcal{C}_{\Sigma}^{\pm}(\psi({\widetilde{v}}-I))+g$
and compute, using the Sokhotski–Plemelj (SP) formula,

$\displaystyle m_{+}$
$\displaystyle=\mathcal{C}_{\Sigma}^{+}(\psi({\widetilde{v}}-I))+g\overset{\text{SP}}{=}\psi({\widetilde{v}}-I)+\mathcal{C}_{{\widetilde{v}}}(\psi)+g=\psi{\widetilde{v}}=m_{-}{\widetilde{v}},$
(4.8)

as $m_{-}=\mathcal{C}_{{\widetilde{v}}}\psi+g=\psi$. We can now derive an
expression for the operator $(1-\mathcal{C}_{{\widetilde{v}}})^{-1}$. Assume
$g\in L^{2}(\Sigma)$ is given and let $\psi\in L^{2}(\Sigma)$ be the unique
solution of $(1-\mathcal{C}_{{\widetilde{v}}})\psi=g$. Then
$m_{\pm}=\mathcal{C}_{\Sigma}^{\pm}(f)+g$ with $f=\psi({\widetilde{v}}-I)$
solves the corresponding inhomogeneous RH problem, as we have seen in equation
(4.8). We want to find an expression for $\psi=m_{-}$ in terms of $g$. To
derive (4.6), we start with

$\displaystyle
m_{+}=m_{-}{\widetilde{v}},\qquad\mathcal{C}_{\Sigma}^{+}(f)+g=(\mathcal{C}_{\Sigma}^{-}(f)+g)\tilde{v},\qquad\big{(}\mathcal{C}_{\Sigma}^{+}(f)+g\big{)}{\widetilde{Q}}_{+}^{-1}=(\mathcal{C}_{\Sigma}^{-}(f)+g){\widetilde{Q}}_{-}^{-1},$
$\displaystyle\mathcal{C}_{\Sigma}^{+}(f){\widetilde{Q}}_{+}^{-1}-\mathcal{C}_{\Sigma}^{-}(f){\widetilde{Q}}_{-}^{-1}=g\big{(}{\widetilde{Q}}_{-}^{-1}-{\widetilde{Q}}_{+}^{-1}\big{)}.$
(4.9)

Observe that the left- and right-hand sides in the last line might not lie in
$L^{2}(\Sigma)$. However, using property (iv) of ${\widetilde{Q}}$ from the RH
problem (see Section 3.5), we see that they lie in $L^{2-\epsilon}(\Sigma)$
for any $\epsilon\in(0,1)$. Hence, if we define
$H=\mathcal{C}_{\Sigma}(f){\widetilde{Q}}^{-1}$, we see that $H$ is analytic
in $\mathbb{C}\setminus\Sigma$, satisfies $H_{\pm}\in L^{2-\epsilon}(\Sigma)$
and vanishes at $\infty$. By the Sokhotski–Plemelj formula, it follows that
$H=\mathcal{C}_{\Sigma}(H_{+}-H_{-})$. Thus, applying $\mathcal{C}_{\Sigma}$,
which is understood to act on the space $L^{2-\epsilon}(\Sigma)$, in the last
line of (4.9), we obtain

$\displaystyle\mathcal{C}_{\Sigma}(f){\widetilde{Q}}^{-1}=\mathcal{C}_{\Sigma}\big{(}\underbrace{g\big{(}{\widetilde{Q}}_{-}^{-1}-{\widetilde{Q}}_{+}^{-1}\big{)}}_{g(\tilde{v}-I){\widetilde{Q}}_{+}^{-1}}\big{)},\qquad\mathcal{C}_{\Sigma}(f)=\mathcal{C}_{\Sigma}\big{(}g({\widetilde{v}}-I){\widetilde{Q}}_{+}^{-1}\big{)}{\widetilde{Q}},$
$\displaystyle\mathcal{C}_{\Sigma}^{-}(f)+g=g+\mathcal{C}_{\Sigma}^{-}\big{(}g({\widetilde{v}}-I){\widetilde{Q}}_{+}^{-1}\big{)}{\widetilde{Q}}_{-},\qquad\psi=g+\mathcal{C}_{\Sigma}^{-}\big{(}g({\widetilde{v}}-I){\widetilde{Q}}_{+}^{-1}\big{)}{\widetilde{Q}}_{-}.$

Note here that
$\mathcal{C}_{\Sigma}^{-}\big{(}g({\widetilde{v}}-I){\widetilde{Q}}_{+}^{-1}\big{)}{\widetilde{Q}}_{-}$
is a priori a function in $L^{2-\epsilon}(\Sigma)$, however as
$\psi=(1-\mathcal{C}_{{\widetilde{v}}})^{-1}g\in L^{2}(\Sigma)$ by Theorem
4.2, we conclude that indeed
$\mathcal{C}_{\Sigma}^{-}\big{(}g({\widetilde{v}}-I){\widetilde{Q}}_{+}^{-1}\big{)}{\widetilde{Q}}_{-}=\psi-g\in
L^{2}(\Sigma)$.

We have thus proved that $(1-\mathcal{C}_{{\widetilde{v}}})^{-1}$ must indeed
have the formed stated in (4.6). ∎

##  5 Local parametrices around the point $\boldsymbol{z=-1}$

In the following, we will use appropriate local parametrices $P$,
${\widehat{P}}$ and ${\widetilde{P}}$ to invert the three RH problems
introduced in Section 3, locally near $z=-1$. We denote the modified RH
problems with ⋆RH.

The explicit construction of the local parametrices is taken from [12,
equation (6.50)] and can be given in terms of Bessel functions and the Szegő
functions corresponding to the three weights. To define these parametrices, we
first need a local $n$-dependent change of variables $z\to\zeta$. Following
[12, Section 6], we choose a sufficiently small neighbourhood $U$ of the point
$z=-1$ and define the mapping

$\displaystyle\xi\colon\ U\to\mathbb{C},\qquad
z\mapsto\xi(z)=\frac{\log^{2}(-\phi(z))}{4}.$ (5.1)

Note that for $s\in U\cap[-1,1]$,
$\log^{2}(-\phi_{+}(s))=\log^{2}\bigl{(}-\phi^{-1}_{-}(s)\bigr{)}=\log^{2}(-\phi_{-}(s))$
implying that $\xi$ is indeed well-defined and holomorphic.

$-1$$z\to\xi$$0$ Figure 5: The change of variables $z\to\xi$.

Now using (2.4), we see that $\xi^{\prime}(-1)=-1/2$, meaning that for $U$
sufficiently small, $\xi$ will define a biholomorphic mapping between $U$ and
its image $\xi(U)$. Introduce now $\zeta=\zeta^{(n)}(z)=n^{2}\xi(z)$ for $z\in
U$ together with $\Sigma_{\Psi}^{(n)}=n^{2}\xi(U\cap\Sigma)$. We can assume
that $\Sigma$ has been chosen such that $\Sigma_{\Psi}^{(n)}$ can be extended
to $\Sigma_{\Psi}\supset\Sigma_{\Psi}^{(n)}$ consisting of three straight line
segments $\gamma_{i}$, $i=1,2,3$, originating from $\zeta=0$ at the angles
$\pm\frac{2\pi}{3}$ and $\pi$, see Figure 6. Accordingly, we will regard
$\zeta$ as a variable in the whole complex plane.

$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\mathbb{R}$$0$0.05,0.05 (3.4,2.5,
6.7,2.5)  Figure 6: Contour for the local parametrix problems in the
$\zeta$-plane.

Next, we shall define two piecewise holomorphic functions
$\Psi_{\nu}\colon\mathbb{C}\setminus\Sigma_{\Psi}\to\mathbb{C}^{2\times 2}$
for $\nu=0,1$ (see [12, equation (6.51)]):

$\displaystyle\Psi_{\nu}(\zeta)=\begin{cases}\begin{pmatrix}I_{\nu}\big{(}2\zeta^{1/2}\big{)}&-\frac{\mathrm{i}}{\pi}K_{\nu}\big{(}2\zeta^{1/2}\big{)}\\\
-2\pi\mathrm{i}\zeta^{1/2}I_{\nu}^{\prime}\big{(}2\zeta^{1/2}\big{)}&-2\zeta^{1/2}K_{\nu}^{\prime}\big{(}2\zeta^{1/2}\big{)}\end{pmatrix},\qquad|\arg\zeta|<\frac{2\pi}{3},\vspace{1mm}\\\
\begin{pmatrix}\frac{1}{2}H^{(1)}_{\nu}\big{(}2(-\zeta)^{1/2}\big{)}&-\frac{1}{2}H^{(2)}_{\nu}\big{(}2(-\zeta)^{1/2}\big{)}\\\
-\pi\zeta^{1/2}(H^{(1)}_{\nu})^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}&\pi\zeta^{1/2}(H^{(2)}_{\nu})^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}\end{pmatrix}\mathrm{e}^{\frac{1}{2}\pi\mathrm{i}\nu\sigma_{3}},\\\
\qquad\frac{2\pi}{3}<\arg\zeta<\pi,\vspace{1mm}\\\
\begin{pmatrix}\frac{1}{2}H^{(2)}_{\nu}\big{(}2(-\zeta)^{1/2}\big{)}&\frac{1}{2}H^{(1)}_{\nu}\big{(}2(-\zeta)^{1/2}\big{)}\\\
\pi\zeta^{1/2}(H^{(2)}_{\nu})^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}&\pi\zeta^{1/2}(H^{(1)}_{\nu})^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}\end{pmatrix}\mathrm{e}^{-\frac{1}{2}\pi\mathrm{i}\nu\sigma_{3}},\\\
\qquad-\pi<\arg\zeta<-\tfrac{2\pi}{3}.\end{cases}$

Here the functions $I_{\nu}$, $K_{\nu}$ with $\nu\in\mathbb{C}$ are the
familiar _modified Bessel functions_. Generally, these are holomorphic
functions in the domain $z\in\mathbb{C}\setminus(-\infty,0]$ and have a branch
cut along the negative real axis. In the special case $\nu\in\mathbb{Z}$,
$I_{\nu}$ is entire.

Analogously, the functions $H^{(1)}_{\nu}$, $H^{(2)}_{\nu}$ with
$\nu\in\mathbb{C}$ are holomorphic for $z\in\mathbb{C}\setminus(-\infty,0]$
and have a branch cut on the negative real axis. They are the _Bessel
functions of the third kind_ , also known as the _Hankel functions_.
Properties of these function can be found in [18, Section 10]. In the
following, we will be interested in the behaviour of $\Psi_{\nu}$ as $\zeta\to
0$, which can be deduced from the following lemma:

######  Lemma 5.1 ([18, Section 10]).

The following asymptotic formulas hold uniformly for any path $\zeta\to 0$:

$\displaystyle
I_{0}\big{(}2\zeta^{1/2}\big{)},K_{0}\big{(}2\zeta^{1/2}\big{)},H^{(1)}_{0}\big{(}2(-\zeta)^{1/2}\big{)},H^{(2)}_{0}\big{(}2(-\zeta)^{1/2}\big{)}=O(\log|\zeta|),$
$\displaystyle
I_{0}^{\prime}\big{(}2\zeta^{1/2}\big{)},K_{0}^{\prime}\big{(}2\zeta^{1/2}\big{)},\big{(}H^{(1)}_{0}\big{)}^{\prime}\big{(}2(-\zeta)^{1/2}\big{)},\big{(}H^{(2)}_{0}\big{)}^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}=O\left(\frac{1}{|\zeta|^{1/2}}\right),$
(5.2)

and

$\displaystyle
I_{1}\big{(}2\zeta^{1/2}\big{)},K_{1}\big{(}2\zeta^{1/2}\big{)},H^{(1)}_{1}\big{(}2(-\zeta)^{1/2}\big{)},H^{(2)}_{1}\big{(}2(-\zeta)^{1/2}\big{)}=O\left(\frac{1}{|\zeta|^{1/2}}\right),$
$\displaystyle
I_{1}^{\prime}\big{(}2\zeta^{1/2}\big{)},K_{1}^{\prime}\big{(}2\zeta^{1/2}\big{)},\big{(}H^{(1)}_{1}\big{)}^{\prime}\big{(}2(-\zeta)^{1/2}\big{)},\big{(}H^{(2)}_{1}\big{)}^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}=O\left(\frac{1}{|\zeta|}\right).$
(5.3)

The following asymptotic formulas hold uniformly for $\zeta\to\infty$ in the
prescribed sectors for any $\delta>0$ and $\nu=0,1$:

$\displaystyle
I_{\nu}\big{(}2\zeta^{1/2}\big{)}\mathrm{e}^{-2\zeta^{1/2}},I_{\nu}^{\prime}\big{(}\zeta^{1/2}\big{)}\mathrm{e}^{-2\zeta^{1/2}}=O\left(\frac{1}{|\zeta|^{1/4}}\right),$
$\displaystyle
K_{\nu}\big{(}2\zeta^{1/2}\big{)}\mathrm{e}^{2\zeta^{1/2}},K_{\nu}^{\prime}\big{(}2\zeta^{1/2}\big{)}\mathrm{e}^{2\zeta^{1/2}}=O\left(\frac{1}{|\zeta|^{1/4}}\right),\qquad|\arg\zeta|<\pi-\delta,$
$\displaystyle
H^{(1)}_{\nu}\big{(}2(-\zeta)^{1/2}\big{)}\mathrm{e}^{-2\mathrm{i}(-\zeta)^{1/2}},(H^{(1)}_{\nu})^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}\mathrm{e}^{-2\mathrm{i}(-\zeta)^{1/2}}=O\left(\frac{1}{|\zeta|^{1/4}}\right),\qquad\arg\zeta\not=0,$
$\displaystyle
H^{(2)}_{\nu}\big{(}2(-\zeta)^{1/2}\big{)}\mathrm{e}^{2\mathrm{i}(-\zeta)^{1/2}},(H^{(2)}_{\nu})^{\prime}\big{(}2(-\zeta)^{1/2}\big{)}\mathrm{e}^{2\mathrm{i}(-\zeta)^{1/2}}=O\left(\frac{1}{|\zeta|^{1/4}}\right),\qquad\arg\zeta\not=0.$
(5.4)

In the formulas (5.2)–(5.4), $(\cdot)^{1/2}$ denotes the principal branch.

In [12, Section 6], it is shown that $\Psi_{\nu}$ satisfies the following jump
conditions across the contours $\gamma_{i}$:

$\displaystyle\Psi_{\nu,+}(\zeta)=\Psi_{\nu,-}(\zeta)\begin{cases}\begin{pmatrix}1&0\\\
\mathrm{e}^{\nu\pi\mathrm{i}}&1\end{pmatrix},&\zeta\in\gamma_{1},\vspace{1mm}\\\
\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},&\zeta\in\gamma_{2},\vspace{1mm}\\\
\begin{pmatrix}1&0\\\
\mathrm{e}^{-\nu\pi\mathrm{i}}&1\end{pmatrix},&\zeta\in\gamma_{3}.\end{cases}$

In the following, we will use $\Psi_{\nu}$, $\nu=0,1$ to write down three
local parametrices $P$, ${\widehat{P}}$, ${\widetilde{P}}$ around $z=-1$ for
the three RH problems defined in Section 3 (see [12, equation (6.52)]):

$\displaystyle P(z)=E(z)(2\pi
n)^{\sigma_{3}/2}\Psi_{1}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}\left(\frac{F(z)}{W(z)}\right)^{\sigma_{3}},$
$\displaystyle{\widehat{P}}(z)=\widehat{E}(z)(2\pi
n)^{\sigma_{3}/2}\Psi_{1}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}\left(\frac{\widehat{F}(z)}{\widehat{W}(z)}\right)^{\sigma_{3}},\qquad
z\in U\setminus\Sigma,$ $\displaystyle{\widetilde{P}}(z)=\widetilde{E}(z)(2\pi
n)^{\sigma_{3}/2}\Psi_{0}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}.$

Here $W=\sqrt{-w}$, $\widehat{W}=\sqrt{-\widehat{w}}$ are chosen to have a
branch cut on $(-1,\infty)\cap U$ and to be positive on $(-\infty,-1)\cap U$.
The matrix-valued functions $E$, $\widehat{E}$ and $\widetilde{E}$ are in fact
holomorphic for $z$ in $U$. More explicitly, we have (see [12, equation
(6.53)])

$\displaystyle
E(z)=N(z)\left(\frac{W(z)}{F(z)}\right)^{\sigma_{3}}\frac{1}{\sqrt{2}}\begin{pmatrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{pmatrix}\xi(z)^{\sigma_{3}/4}$ (5.5)

which is in fact holomorphic for $z\in U$. Here $N$ is the outer parametrix
solution

$\displaystyle
N(z)=\begin{pmatrix}\dfrac{a(z)+a(z)^{-1}}{2}&&\dfrac{a(z)-a(z)^{-1}}{2\mathrm{i}}\\\
\dfrac{a(z)-a(z)^{-1}}{-2\mathrm{i}}&&\dfrac{a(z)+a(z)^{-1}}{2}\end{pmatrix},$
(5.6)

where

$\displaystyle a(z)=\left(\dfrac{z-1}{z+1}\right)^{1/4}$

with a branch cut on $(-1,1)$ and $a(\infty)=1$. Similar formulae can be
obtained for $\widehat{E}$ and $\widetilde{E}$ by substituting in (5.5)
$\widehat{F}$, $\widehat{W}$ and $\widetilde{F}\equiv 1$, $\widetilde{W}\equiv
1$, respectively. Crucially however, the outer parametrix $N$ is the same for
all three problems. One can check that the determinants of all three
parametrices are constant equal to $1$ inside $U$, cf. [12, Section 7].
Furthermore, $E$, $\widehat{E}$ and $\widetilde{E}$ are analytic and bounded
in $U$, the singularity of $a(z)$ at $z=-1$ being compensated by the factor
$\xi(z)^{\sigma_{3}/4}$.

######  Lemma 5.2.

The matrix-valued functions $P$, ${\widehat{P}}$ and ${\widetilde{P}}$ defined
in $U\setminus\Sigma$, satisfy the following conditions:

  1. $(i)$

For $s\in U\cap\Sigma$,

$\displaystyle
P_{+}(s)=P_{-}(s)v(s),\qquad{\widehat{P}}_{+}(s)={\widehat{P}}_{-}(s){\widehat{v}}(s),\qquad{\widetilde{P}}_{+}(s)={\widetilde{P}}_{-}(s){\widetilde{v}}(s).$
(5.7)

  2. $(ii)$

For $s\in\partial U$ $($see _[12, equation (6.41)]_$)$,

$\displaystyle
P(s),{\widehat{P}}(s),{\widetilde{P}}(s)=N(s)+O\big{(}n^{-1}\big{)}.$ (5.8)

  3. $(iii)$

For $z\in U$, we have uniformly $($see _[12, equation (7.10)]_$)$

$\displaystyle{\widetilde{Q}}(z)\big{[}{\widetilde{P}}(z)\big{]}^{-1}=I+O\big{(}n^{-1}\big{)},\qquad{\widehat{Q}}(z)\big{[}{\widehat{P}}(z)\big{]}^{-1}=I+O\big{(}n^{-1}\big{)}.$
(5.9)

  4. $(iv)$

For $z\in U$,

$\displaystyle
P(z),{\widehat{P}}(z)=O\big{(}\max\big{\\{}|z+1|^{-1/4},n^{-1/2}|z+1|^{-1/2}\big{\\}}\big{)},$
(5.10)

and

$\displaystyle{\widetilde{P}}(z)=O\big{(}|z+1|^{-1/4}\big{)}$ (5.11)

uniformly as $n\to\infty$.

###### Proof.

A detailed derivation of the local parametrices can be found in [12, Section
6] together with a proof of properties (i), (ii), for property (iii) see [12,
Section 7]. Note that while the weight function $w$ does not fall into the
class of weight functions considered in [12] due to the logarithmic
singularity at $z=+1$, the local construction and estimation of the left
parametrix near $z=-1$ found therein remains unchanged.

Regarding point (iv), we start with the properties of $P$ and ${\widehat{P}}$.
Noting that $E(z)$ and $\widehat{E}(z)$ are holomorphic, $n$-independent and
have unit determinants, hence it is enough to consider $E^{-1}(z)P(z)$ and
$\widehat{E}^{-1}(z){\widehat{P}}(z)$ instead of $P(z)$ and ${\widehat{P}}(z)$
in (5.10). Analogously, it follows from (2.6) and (2.11) that
$\big{(}\frac{F}{W}\big{)}^{\sigma_{3}}$ and
$\big{(}\frac{\widehat{F}}{\widehat{W}}\big{)}^{\sigma_{3}}$ are
$n$-independent and bounded in a neighbourhood of $z=-1$, hence they also do
not contribute in (5.10).

It remains to study

$\displaystyle(2\pi
n)^{\sigma_{3}/2}\Psi_{1}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}$

which is equal to both $E^{-1}(z)P(z)\big{(}\frac{F}{W}\big{)}^{-\sigma_{3}}$
and
$\widehat{E}^{-1}(z){\widehat{P}}(z)\big{(}\frac{\widehat{F}}{\widehat{W}}\big{)}^{-\sigma_{3}}$.
It follows from the definition of $\xi$ in (5.1) that
$[-\phi(z)]^{-n\sigma_{3}}=\mathrm{e}^{-2\sqrt{n^{2}\xi(z)}\sigma_{3}}$ where
the square root has a branch cut along $z>-1$. Writing $\zeta=n^{2}\xi(z)$ and
assuming $|z+1|\gtrsim O\big{(}n^{-2}\big{)}$, we see that $|\zeta|\gtrsim
O(1)$ and using the estimates in (5.4) we conclude that

$\displaystyle(2\pi
n)^{\sigma_{3}/2}\Psi_{1}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}$
$\displaystyle\quad=(2\pi
n)^{\sigma_{3}/2}\begin{pmatrix}O\big{(}|\zeta|^{-1/4}\big{)}&O\big{(}|\zeta|^{-1/4}\big{)}\\\
O\big{(}|\zeta|^{1/4}\big{)}&O\big{(}|\zeta|^{1/4}\big{)}\end{pmatrix}$
$\displaystyle\quad=\begin{pmatrix}O\big{(}|z+1|^{-1/4}\big{)}&O\big{(}|z+1|^{-1/4}\big{)}\\\
O\big{(}|z+1|^{1/4}\big{)}&O\big{(}|z+1|^{1/4}\big{)}\end{pmatrix},\qquad|z+1|\gtrsim
O\big{(}n^{-2}\big{)}.$ (5.12)

For $|z+1|\lesssim O\big{(}n^{-2}\big{)}$, we use the estimates (5.3) instead
to conclude

$\displaystyle(2\pi
n)^{\sigma_{3}/2}\Psi_{1}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}$
$\displaystyle\quad=(2\pi
n)^{\sigma_{3}/2}\begin{pmatrix}O\big{(}|\zeta|^{-1/2}\big{)}&O\big{(}|\zeta|^{-1/2}\big{)}\\\
O\big{(}|\zeta|^{-1/2}\big{)}&O\big{(}|\zeta|^{-1/2}\big{)}\end{pmatrix}$
$\displaystyle\quad=\begin{pmatrix}O\big{(}n^{-1/2}|z+1|^{-1/2}\big{)}&O\big{(}n^{-1/2}|z+1|^{-1/2}\big{)}\\\
O\big{(}n^{-3/2}|z+1|^{-1/2}\big{)}&O\big{(}n^{-3/2}|z+1|^{-1/2}\big{)}\end{pmatrix},\qquad
z\in|z+1|\lesssim O\big{(}n^{-2}\big{)}.$ (5.13)

Note that in this case $[-\phi(z)]^{-n\sigma_{3}}=O(1)$, hence this term does
not contribute. One checks that indeed for $|z+1|\sim n^{-2}$ the bounds in
(5.12) and (5.13) are of the same order.

The proof of (5.11) works in a similar fashion. Again the holomorphic
prefactor $\widetilde{E}$ can be ignored. For $|z+1|\gtrsim
O\big{(}n^{-2}\big{)}$, we get as before

$\displaystyle(2\pi
n)^{\sigma_{3}/2}\Psi_{0}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}$
$\displaystyle\quad=\begin{pmatrix}O\big{(}|z+1|^{-1/4}\big{)}&O\big{(}|z+1|^{-1/4}\big{)}\\\
O\big{(}(|z+1|^{1/4}\big{)}&O\big{(}(|z+1|^{1/4}\big{)}\end{pmatrix},\qquad|z+1|\gtrsim
O\big{(}n^{-2}\big{)}.$

However, for $|z+1|\lesssim O\big{(}n^{-2}\big{)}$, we get different
asymptotics after applying (5.2). We obtain

$\displaystyle(2\pi
n)^{\sigma_{3}/2}\Psi_{0}\big{(}n^{2}\xi(z)\big{)}[-\phi(z)]^{-n\sigma_{3}}$
$\displaystyle\quad=\begin{pmatrix}O\big{(}n^{1/2}\big{|}\log\big{(}n^{2}(z+1)\big{)}\big{|}\big{)}&O\big{(}n^{1/2}\big{|}\log\big{(}n^{2}(z+1)\big{)}\big{|}\big{)}\\\
O\big{(}n^{-1/2}\big{)}&O\big{(}n^{-1/2}\big{)}\end{pmatrix},\quad|z+1|\lesssim
O\big{(}n^{-2}\big{)}.$ (5.14)

Now observe that for $|z+1|\lesssim O\big{(}n^{-2}\big{)}$, we have trivially
$n^{2}|z+1|\lesssim O(1)$ and thus

$\displaystyle\big{|}\log\big{(}n^{2}(z+1)\big{)}\big{|}\lesssim\big{|}n^{2}(z+1)\big{|}^{-1/4}.$

This estimate, together with $n^{-1/2}\lesssim|z+1|^{-1/4}$ (in fact
$n^{1/2}\lesssim|z+1|^{-1/4}$ holds), implies that the matrix entries in
(5.14) can be bounded by $O\big{(}|z+1|^{-1/4}\big{)}$ uniformly as
$n\to\infty$, showing (5.11) and finishing the proof. ∎

The fact that all three parametrices display the same asymptotic behaviour for
$|z+1|\gtrsim O\big{(}n^{-2}\big{)}$ is consistent with the matching condition
(5.8) which is the same in all three cases. Note that for a fixed $n$,
${\widetilde{P}}(z)$ has only a logarithmic singularity near $z=-1$, but the
order in (5.11) is necessary to obtain a uniform bound for $n\to\infty$.

######  Corollary 5.3.

For $z$ in a neighbourhood $U_{+1}$ of $+1$, the matrix-valued function
${\widetilde{Q}}$ satisfies the asymptotics

$\displaystyle{\widetilde{Q}}(z)=O\big{(}|z-1|^{-1/4}\big{)}$ (5.15)

uniformly for $n\to\infty$. Moreover, ${\widetilde{Q}}$, and its boundary
values ${\widetilde{Q}}_{\pm}$ on $\Sigma$, are bounded in
$\overline{\mathbb{C}}:=\mathbb{C}\cup\\{\infty\\}$ away from small
neighbourhoods of $\pm 1$, uniformly for $n\to\infty$.

###### Proof.

Analogously to the local parametrix for the Legendre problem at $z\\!=\\!-1$,
${{\widetilde{P}}_{-1}(z)\\!:=\\!{\widetilde{P}}(z)}$, one can construct a
local parametrix ${\widetilde{P}}_{+1}(z)$ for the Legendre problem near the
point $z=+1$ with local jumps inside an open neighbourhood $U_{+1}$ of $+1$ as
depicted in Figure 7.

$\big{(}\begin{smallmatrix}1&0\\\
\phi^{-2n}&1\end{smallmatrix}\big{)}$$\big{(}\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\big{)}$$\big{(}\begin{smallmatrix}1&0\\\
\phi^{-2n}&1\end{smallmatrix}\big{)}$$+1$$U_{+1}$0.05,0.05 (3.4,2.5, 6.4,2.5)
Figure 7: The jump conditions for ${\widetilde{P}}_{+1}$.

In fact, we have
${\widetilde{P}}_{+1}(z)=\sigma_{3}{\widetilde{P}}_{-1}(-z)\sigma_{3}$, see
[4, equation B.10]. Hence, it follows from Lemma 5.2 (iv) that

$\displaystyle{\widetilde{P}}_{+1}(z)=O\big{(}|z-1|^{-1/4}\big{)}$ (5.16)

uniformly for $n\to\infty$. After deforming the local contour such that it
matches locally with $\Sigma$ as depicted in Figure 8, we can analytically
continue ${\widetilde{P}}_{+1}$ as necessary to obtain a _deformed_ local
parametrix ${\widetilde{P}}_{+1}^{\text{def}}$, which would satisfy locally
the same jump conditions as ${\widetilde{Q}}$.

$\big{(}\begin{smallmatrix}1&0\\\
\phi^{-2n}&1\end{smallmatrix}\big{)}$$\big{(}\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\big{)}$$\big{(}\begin{smallmatrix}1&0\\\
\phi^{-2n}&1\end{smallmatrix}\big{)}$$\big{(}\begin{smallmatrix}1&0\\\
2\phi^{-2n}&1\end{smallmatrix}\big{)}$$+1$$\boldsymbol{\cdot}$$1+\delta$$U_{+1}$0.05,0.05
(5,2.5, 6.4,2.5) (4.6,1.7, 4.1, 2, 4, 2.5)  Figure 8: The jump conditions for
${\widetilde{P}}_{+1}^{\text{def}}$.

Because the jump matrices ${\widetilde{v}}$ and their analytic continuations
are uniformly bounded near $z=+1$, ${\widetilde{P}}_{+1}^{\text{def}}$ would
continue to satisfy the estimate (5.16)

$\displaystyle{\widetilde{P}}_{+1}^{\text{def}}(z)=O\big{(}|z-1|^{-1/4}\big{)}$
(5.17)

uniformly for $n\to\infty$. As the contour deformations are local, we also
know that the matching condition (5.9) remains unchanged, at least on
$\partial U_{+1}$:

$\displaystyle{\widetilde{Q}}(s)\big{[}{\widetilde{P}}_{+1}^{\text{def}}(s)\big{]}^{-1}=I+O\big{(}n^{-1}\big{)},\qquad
s\in\partial U_{+1}.$ (5.18)

However, as
${\widetilde{Q}}\big{[}{\widetilde{P}}_{+1}^{\text{def}}\big{]}^{-1}$ has no
jumps inside $U_{+1}$, we can extend (5.18) to all of $U_{+1}$ by the maximum
modulus principle for holomorphic functions. Together with (5.17), the
estimate (5.15) follows.

Regarding the second statement, it has been shown in [12, Section 7] that for
a wide class of weight functions including $\widetilde{w}$, the outer
parametrix solution $N$ is an approximation to the exact RH solution, in our
case ${\widetilde{Q}}$, uniformly in $z$ as long we stay away from the points
$\pm 1$:

$\displaystyle\big{|}{\widetilde{Q}}(z)-N(z)\big{|}=O\big{(}n^{-1}\big{)},\qquad
z\mbox{ staying away from }\pm 1.$

As can be seen from (5.6), $N$ is $n$-independent and bounded away from the
points $\pm 1$. This finishes the proof. ∎

##  6 Modified RH problems

We are now in a position to define modified versions of the three RH problems
found in Section 3, which will be referred to as the respective ⋆-analogs. The
three ⋆RH problems are introduced implicitly by defining their respective
solutions. Here $U$ is a small neighbourhood of $-1$, as before.

$\displaystyle Q^{\star}(z)$
$\displaystyle=\begin{cases}Q(z),&z\in\mathbb{C}\setminus(\Sigma\cup U),\\\
Q(z)[P(z)]^{-1},&z\in U,\end{cases}$ $\displaystyle{\widehat{Q}^{\star}}(z)$
$\displaystyle=\begin{cases}{\widehat{Q}}(z),&z\in\mathbb{C}\setminus(\Sigma\cup
U),\\\ {\widehat{Q}}(z)[{\widehat{P}}(z)]^{-1},&z\in U,\end{cases}$
$\displaystyle{\widetilde{Q}^{\star}}(z)$
$\displaystyle=\begin{cases}{\widetilde{Q}}(z),&z\in\mathbb{C}\setminus(\Sigma\cup
U),\\\ {\widetilde{Q}}(z)[{\widetilde{P}}(z)]^{-1},&z\in U.\end{cases}$

Note that because of (5.7), the jumps on $U\cap\Sigma$ cancel out, which means
that all three solutions can be uniquely defined on all of $U$. We will denote
the new contour, which is the same for all three problems, by
$\Sigma^{\star}=(\Sigma\setminus U)\cup\partial U$ and assume that $\partial
U$ is oriented clockwise. The corresponding jump matrices are denoted by
$v^{\star}$, ${\widehat{v}^{\star}}$ and ${\widetilde{v}^{\star}}$, hence

$\displaystyle Q^{\star}_{+}(s)=Q^{\star}_{-}(s)v^{\star}(s),\qquad
s\in\Sigma^{\star},$

and so on. Note also that the normalization at infinity remains unchanged and
the jump matrices on $\partial U$ are just the corresponding local
parametrices:

$\displaystyle
v^{\star}(s)=P(s),\qquad{\widehat{v}^{\star}}(s)={\widehat{P}}(s),\qquad{\widetilde{v}^{\star}}(s)={\widetilde{P}}(s),\qquad
s\in\partial U.$

Now consider the singular integral operator

$\displaystyle 1-\mathcal{C}_{{\widetilde{v}^{\star}}}\colon\
L^{2}(\Sigma^{\star})\to L^{2}(\Sigma^{\star}),\qquad f\mapsto
f-\mathcal{C}_{\Sigma^{\star}}^{-}(f({\widetilde{v}^{\star}}-I)).$

We claim that this operator is invertible and that the inverse is given by

$\displaystyle(1-\mathcal{C}_{{\widetilde{v}^{\star}}})^{-1}\colon\
L^{2}(\Sigma^{\star})\to L^{2}(\Sigma^{\star}),\qquad f\mapsto
f+\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}f({\widetilde{v}^{\star}}-I)[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}.$

At the moment, it is not even clear whether the above operator is well-defined
as a map from $L^{2}(\Sigma^{\star})$ to itself, as
${\widetilde{Q}^{\star}}_{\pm}$ has a logarithmic singularity near $+1$.

To prove the claim, we proceed as follows: First, we partition the jump
contour $\Sigma^{\star}=\underbrace{\partial
U}_{\Sigma^{\ell}}\,\cup\,\underbrace{\Sigma\setminus U}_{\Sigma^{r}}$ as
shown in Figure 9.

(0.85,2, 4,1.5, 7,2.5)(0.85,3, 4,3.5,
7,2.5)$1+\delta$$-1$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U}$$1$$\bullet$$\bullet$$\Sigma^{\ell}$$\Sigma^{r}$
Figure 9: The jump contour $\Sigma^{\star}=\Sigma^{\ell}\cup\Sigma^{r}$.

Next, we decompose the operator $\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}\
\cdot\
({\widetilde{v}^{\star}}-I)[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}$.
Setting ${\widetilde{u}^{\star}}={\widetilde{v}^{\star}}-I$, we obtain

$\displaystyle\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}\ \cdot\
{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}={}$
$\displaystyle\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}\ \cdot\
\chi_{\Sigma^{\ell}}{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}+\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}\
\cdot\
\chi_{\Sigma^{r}}{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}$
$\displaystyle={}$ $\displaystyle\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}\
\cdot\
\chi_{\Sigma^{\ell}}{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}+\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}\
\cdot\
\chi_{\Sigma^{r}}{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}\chi_{\Sigma^{r}}$
$\displaystyle+\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}\ \cdot\
\chi_{\Sigma^{r}}{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}\chi_{\Sigma^{\ell}},$

where $\chi_{\Sigma^{j}}$ is the characteristic function of $\Sigma^{j}$ for
$j=\ell,r$. By definition, we have
${\widetilde{Q}^{\star}}_{\pm}\chi_{\Sigma^{r}}={\widetilde{Q}}_{\pm}\chi_{\Sigma^{r}}$.
Note that the mapping

$\displaystyle f\mapsto
f\chi_{\Sigma^{\ell}}{\widetilde{u}^{\star}}\big{[}{\widetilde{Q}^{\star}}_{+}\big{]}^{-1}$

is an operator uniformly bounded in $n$ from $L^{2}(\Sigma^{\star})\to
L^{2}\big{(}\Sigma^{\star},|z-1|^{-1/4}\big{)}$, as
$\big{[}{\widetilde{Q}^{\star}}_{+}\big{]}^{-1}$ converges uniformly on
$\Sigma^{\ell}=\partial U$ to the outer parametrix $N$, see (5.8) and (5.9).
Here and in the following, we refer to _uniform boundedness in $n$_ of some
operator $T=T_{n}$, to the operator norm $\|T_{n}\|$ being bounded by an
$n$-independent constant, cf. Theorem 4.2. As $|z-1|^{-1/4}\in
A_{2}(\Sigma^{\star})$ (see Theorem 4.1), we have that the mapping

$\displaystyle
f\mapsto\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}f\chi_{\Sigma^{\ell}}{\widetilde{u}^{\star}}\big{[}{\widetilde{Q}^{\star}}_{+}\big{]}^{-1}\big{)}$

defines a uniformly bounded operator from $L^{2}(\Sigma^{\star})\to
L^{2}\big{(}\Sigma^{\star},|z-1|^{-1/4}\big{)}$. Finally, using
${\widetilde{Q}^{\star}}={\widetilde{Q}}$ in $\mathbb{C}\setminus U$ and the
estimate (5.15), we conclude that

$\displaystyle
f\mapsto\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}f\chi_{\Sigma^{\ell}}{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}$

defines a uniformly bounded operator from $L^{2}(\Sigma^{\star})\to
L^{2}(\Sigma^{\star})$.

The uniform boundedness of the mapping

$\displaystyle
f\mapsto\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}f\chi_{\Sigma^{r}}{\widetilde{u}^{\star}}[{\widetilde{Q}^{\star}}_{+}]^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}\chi_{\Sigma^{r}}$

as an operator from $L^{2}(\Sigma^{\star})\to L^{2}(\Sigma^{\star})$ follows
directly from Proposition 4.6 together with Theorem 4.2.

Finally, the mapping

$\displaystyle f\mapsto
f\chi_{\Sigma^{r}}{\widetilde{u}^{\star}}\big{[}{\widetilde{Q}^{\star}}_{+}\big{]}^{-1}$

defines a uniformly bounded operator from $L^{2}(\Sigma^{\star})$ to
$L^{2}\big{(}\Sigma^{\star},|z-1|^{1/4}\big{)}$. As $|z-1|^{1/4}\in
A_{2}(\Sigma^{\star})$, the mapping

$\displaystyle
f\mapsto\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}f\chi_{\Sigma^{r}}{\widetilde{u}^{\star}}\big{[}{\widetilde{Q}^{\star}}_{+}\big{]}^{-1}\big{)}$

is a uniformly bounded operator from $L^{2}(\Sigma^{\star})\to
L^{2}\big{(}\Sigma^{\star},|z-1|^{1/4}\big{)}$ as well. Moreover, the
multiplication with ${\widetilde{Q}^{\star}}_{-}\chi_{\Sigma^{\ell}}$ defines
a uniformly bounded operator from
$L^{2}\big{(}\Sigma^{\star},|z-1|^{1/4}\big{)}\to L^{2}(\Sigma^{\star})$,
implying that

$\displaystyle
f\mapsto\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}f\chi_{\Sigma^{r}}{\widetilde{u}^{\star}}\big{[}{\widetilde{Q}^{\star}}_{+}\big{]}^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}\chi_{\Sigma^{\ell}}$

is a uniformly bounded operator from $L^{2}(\Sigma^{\star})\to
L^{2}(\Sigma^{\star})$. We obtain:

######  Theorem 6.1.

The inverse of the operator $1-\mathcal{C}_{{\widetilde{v}^{\star}}}$ on
$L^{2}(\Sigma^{\star})$ exists and is given by

$\displaystyle(1-\mathcal{C}_{{\widetilde{v}^{\star}}})^{-1}\colon\
L^{2}(\Sigma^{\star})\to L^{2}(\Sigma^{\star}),\ \ f\mapsto
f+\mathcal{C}_{\Sigma^{\star}}^{-}\big{(}f({\widetilde{v}^{\star}}-I)\big{[}{\widetilde{Q}^{\star}}_{+}\big{]}^{-1}\big{)}{\widetilde{Q}^{\star}}_{-}.$

Moreover, the operator norm is uniformly bounded as $n\to\infty$.

###### Proof.

That the mapping is indeed uniformly bounded follows from the preceding
argument. The fact that it coincides with the inverse of
$1-\mathcal{C}_{{\widetilde{v}^{\star}}}$ follows from a computation analogous
to the one given in the proof of Proposition 4.6. ∎

###  6.1 Uniform invertibility of the ⋆resolvents

The uniform boundedness stated in Theorem 6.1 can now be extended to the
uniform invertibility of the operators
$1-\mathcal{C}_{{\widetilde{v}^{\star}}}$,
$1-\mathcal{C}_{{\widehat{v}}^{\star}}$. Note that the jump matrices satisfy

$\displaystyle\|v^{\star}-{\widetilde{v}^{\star}}\|_{L^{\infty}(\Sigma^{\star})},\
\|{\widehat{v}^{\star}}-{\widetilde{v}^{\star}}\|_{L^{\infty}(\Sigma^{\star})}\to
0$ (6.1)

for $n\to\infty$. In fact, we have

$\displaystyle
v^{\star}-{\widetilde{v}^{\star}}=\begin{cases}\begin{pmatrix}0&0\\\
\big{(}\frac{F^{2}}{w}(s)-1\big{)}\phi^{-2n}(s)&0\end{pmatrix},&s\in(\Sigma_{1}\cup\Sigma_{2})\cap\Sigma^{r},\vspace{1mm}\\\
\begin{pmatrix}0&0\\\
\big{(}\frac{F^{2}}{w}_{+}(s)+\frac{F^{2}}{w}_{-}(s)-2\big{)}\phi^{-2n}(s)&0\end{pmatrix},&s\in(1,1+\delta),\\\
0,&s\in(-1,1)\cap\Sigma^{r},\\\
P(s)-{\widetilde{P}}(s),&s\in\Sigma^{\ell},\end{cases}$ (6.2)

and the corresponding claim in (6.1) for the difference
$v^{\star}-{\widetilde{v}^{\star}}$ follows from (2.2), Corollary 2.4 and
(5.8). An analog formula of the form (6.2) can be written down for
${\widehat{v}^{\star}}-{\widetilde{v}^{\star}}$:

$\displaystyle{\widehat{v}^{\star}}-{\widetilde{v}^{\star}}=\begin{cases}\begin{pmatrix}0&0\\\
\big{(}\frac{\widehat{F}^{2}}{\widehat{w}}(s)-1\big{)}\phi^{-2n}(s)&0\end{pmatrix},&s\in(\Sigma_{1}\cup\Sigma_{2})\cap\Sigma^{r},\vspace{1mm}\\\
\begin{pmatrix}0&0\\\
2\big{(}\frac{\widehat{F}^{2}}{\widehat{w}}(s)-1\big{)}\phi^{-2n}(s)&0\end{pmatrix},&s\in(1,1+\delta),\\\
0,&s\in(-1,1)\cap\Sigma^{r},\\\
{\widehat{P}}(s)-{\widetilde{P}}(s),&s\in\Sigma^{\ell}.\end{cases}$

Here (6.1) follows as before after using estimate (2.10) instead of (2.5).

Thus it follows that

$\displaystyle\|(1-\mathcal{C}_{v^{\star}})-(1-\mathcal{C}_{{\widetilde{v}^{\star}}})\|_{L^{2}(\Sigma^{\star})\to
L^{2}(\Sigma^{\star})}\to
0,\qquad\|(1-\mathcal{C}_{{\widehat{v}}^{\star}})-(1-\mathcal{C}_{{\widetilde{v}^{\star}}})\|_{L^{2}(\Sigma^{\star})\to
L^{2}(\Sigma^{\star})}\to 0$

as $n\to\infty$. As the operator $1-\mathcal{C}_{{\widetilde{v}^{\star}}}$ is
uniformly invertible, a standard argument (see, e.g., [4, Theorem 4.7]) shows
that the operators $1-\mathcal{C}_{v^{\star}}$,
$1-\mathcal{C}_{{\widehat{v}}^{\star}}$ are also uniformly invertible for $n$
large enough. We summarize:

######  Theorem 6.2.

The operators $1-\mathcal{C}_{v^{\star}}$ and
$1-\mathcal{C}_{{\widehat{v}}^{\star}}$ are invertible for $n$ large enough
and the operator norms of their inverses are uniformly bounded as
$n\to\infty$.

Note that Theorem 6.2 is the analog of Theorem 4.2 initially proved in [4,
Theorem 4.5], but for the logarithmic weight function and on a contour
avoiding the problematic point $z=-1$. Interestingly Theorem 4.2 played a
crucial role in the proof of Theorem 6.2.

##  7 Asymptotic Analysis

The following section will be largely based on [4, Section 5] and culminates
in the proof of Theorem 1.1.

###  7.1 Some norm estimates

Let us define $\mu^{\star}_{n}=Q_{-}^{\star(n)}$ and
$\widehat{\mu}^{\star}_{n}=\widehat{Q}_{-}^{\star(n)}$, where in the following
we will make the $n$-dependence explicit. In particular, we will denote the
jump matrices with a subscript $n$ and the Cauchy operator with a superscript
$(n)$ for notational convenience, i.e., $v^{\star}_{n}$,
$\mathcal{C}_{\Sigma}^{(n)}$ and so on. We will now prove an analog of [4,
Proposition 5.1] in the ⋆-case.

######  Proposition 7.1.

The following estimates hold for $n\to\infty$:

  1. $(i)$

$\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\star})}=O\big{(}\frac{1}{n^{1/2}\log^{2}n}\big{)}$,

  2. $(ii)$

$\|\mu^{\star}_{n}-\widehat{\mu}^{\star}_{n}\|_{L^{2}(\Sigma^{\star})}=O\big{(}\frac{1}{n^{1/2}\log^{2}n}\big{)}$,

  3. $(iii)$

$\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})-\widehat{\mu}^{\star}_{n+1}(v^{\star}_{n+1}-{\widehat{v}^{\star}}_{n+1})\|_{L^{2}(\Sigma^{\star})}=O\big{(}\frac{1}{n^{3/2}\log^{2}n}\big{)}$,

  4. $(iv)$

$\|(\mu^{\star}_{n}-\widehat{\mu}^{\star}_{n})-(\mu^{\star}_{n+1}-\widehat{\mu}^{\star}_{n+1})\|_{L^{2}(\Sigma^{\star})}=O\big{(}\frac{1}{n^{3/2}\log^{2}n}\big{)}$.

###### Proof.

The proof is for the most part taken from [4, Proposition B.2], with the only
difference being the contribution from $\Sigma^{\ell}=\partial U$ instead of
$\Sigma\cap U$.

For (i), let us write

$\displaystyle\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\star})}^{2}=\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\ell})}^{2}+\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{r})}^{2}.$
(7.1)

As
${\widehat{\mu}}_{n}|_{s\in\Sigma^{r}}=\widehat{\mu}^{\star}_{n}|_{s\in\Sigma^{r}}$,
$v_{n}|_{s\in\Sigma^{r}}=v^{\star}_{n}|_{s\in\Sigma^{r}}$ and
${\widehat{v}}_{n}|_{s\in\Sigma^{r}}={\widehat{v}^{\star}}_{n}|_{s\in\Sigma^{r}}$,
one can conclude from [4, equation 5.1] that

$\displaystyle\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{r})}=O\left(\frac{1}{n^{1/2}\log^{2}n}\right).$
(7.2)

Hence it remains to consider the term
$\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\ell})}$.
Recall that for $s\in\Sigma^{\ell}$ we have $v^{\star}_{n}(s)=P^{(n)}(s)$ and
${\widehat{v}^{\star}}_{n}(s)={\widehat{P}}^{(n)}(s)$ and so it follows from
Lemma 5.2 (ii) that

$\displaystyle\|v^{\star}_{n}-{\widehat{v}^{\star}}_{n}\|_{L^{\infty}(\Sigma^{\ell})}=O\big{(}n^{-1}\big{)}.$
(7.3)

Moreover, from Lemma 5.2 (iii) it also follows that
$\widehat{\mu}^{\star}_{n}(s)={\widehat{Q}}^{(n)}(s)\big{[}{\widehat{P}}^{(n)}(s)\big{]}^{-1}$
is uniformly bounded for $s\in\Sigma^{\ell}$. Hence we conclude

$\displaystyle\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\ell})}=O\bigl{(}n^{-1}\bigr{)}$

which together with (7.2) proves (i).

Point (ii) follows from (i) by considering

$\displaystyle\mu^{\star}_{n}-\widehat{\mu}^{\star}_{n}$
$\displaystyle=\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}I-\big{(}1-\mathcal{C}_{{\widehat{v}}^{\star}}^{(n)}\big{)}^{-1}I=\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}\big{(}\mathcal{C}_{v^{\star}}^{(n)}-\mathcal{C}_{{\widehat{v}}^{\star}}^{(n)}\big{)}\big{(}1-\mathcal{C}_{{\widehat{v}}^{\star}}^{(n)}\big{)}^{-1}I$
$\displaystyle=\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}\big{(}\mathcal{C}_{v^{\star}}^{(n)}-\mathcal{C}_{{\widehat{v}}^{\star}}^{(n)}\big{)}\widehat{\mu}^{\star}_{n}=\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})).$

As $\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}$,
$\mathcal{C}_{\Sigma^{\star}}^{-}$ are bounded operators (uniformly in $n$),
it follows that

$\displaystyle\|\mu^{\star}_{n}-\widehat{\mu}^{\star}_{n}\|_{L^{2}(\Sigma^{\star})}\lesssim\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\star})}=O\left(\frac{1}{n^{1/2}\log^{2}n}\right)$

showing (ii).

For (iii), we will again decompose the norm as in (7.1). As before, following
the arguments found in [4, Proposition B.2], we conclude that

$\displaystyle\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})-\widehat{\mu}^{\star}_{n+1}(v^{\star}_{n+1}-{\widehat{v}^{\star}}_{n+1})\|_{L^{2}(\Sigma^{r})}=O\left(\frac{1}{n^{3/2}\log^{2}n}\right).$

For the remaining term, we will use the fact that the local parametrices
$P^{(n)}(s)$, ${\widehat{P}}^{(n)}(s)$ have an infinite series expansion in
powers of $n^{-1}$ which is uniform on $\Sigma^{\ell}$, see [12, equation
(8.2)]:

$\displaystyle
v_{n}^{\star}(s)=P^{(n)}(s)\sim\left(I+\sum_{k=1}^{\infty}\frac{\Delta_{k}(s)}{n^{k}}\right)N(s),$
$\displaystyle{\widehat{v}^{\star}}_{n}(s)={\widehat{P}}^{(n)}(s)\sim\left(I+\sum_{k=1}^{\infty}\frac{\widehat{\Delta}_{k}(s)}{n^{k}}\right)N(s),$

where $\Delta_{k}$ and $\widehat{\Delta}_{k}$ are $n$ independent and can be
explicitly computed. In particular,

$\displaystyle
v^{\star}_{n}(s)-v^{\star}_{n+1}(s)=O\big{(}n^{-2}\big{)},\qquad{\widehat{v}^{\star}}_{n}(s)-{\widehat{v}^{\star}}_{n+1}(s)=O\big{(}n^{-2}\big{)}$
(7.4)

uniformly for $s\in\Sigma^{\ell}$. Hence, using the fact
$\widehat{\mu}^{\star}_{n}(s)=I+O\big{(}n^{-1}\big{)}$ uniformly for
$s\in\Sigma^{\ell}$ (see (5.9)), we can write

$\displaystyle\|\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})-\widehat{\mu}^{\star}_{n+1}(v^{\star}_{n+1}-{\widehat{v}^{\star}}_{n+1})\|_{L^{2}(\Sigma^{\ell})}\lesssim\|(\widehat{\mu}^{\star}_{n}-\widehat{\mu}^{\star}_{n+1})(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\ell})}+O\big{(}n^{-2}\big{)}.$

Additionally, it follows that
$\widehat{\mu}^{\star}_{n}(s)-\widehat{\mu}^{\star}_{n+1}(s)=O\big{(}n^{-1}\big{)}$,
(even $O\big{(}n^{-2}\big{)}$ see [12, equation (8.7)]) uniformly for
$s\in\Sigma^{\ell}$. Together with (7.3), we can conclude that

$\displaystyle\|(\widehat{\mu}^{\star}_{n}-\widehat{\mu}^{\star}_{n+1})(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})\|_{L^{2}(\Sigma^{\ell})}=O\big{(}n^{-2}\big{)},$

which proves (iii).

In order to prove (iv), we first write

$\displaystyle(\mu^{\star}_{n}-\widehat{\mu}^{\star}_{n})-(\mu^{\star}_{n+1}-\widehat{\mu}^{\star}_{n+1})={}$
$\displaystyle\big{(}\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}I-\big{(}1-\mathcal{C}_{{\widehat{v}}^{\star}}^{(n)}\big{)}^{-1}I\big{)}$
$\displaystyle-\big{(}\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}I-\big{(}1-\mathcal{C}_{{\widehat{v}}^{\star}}^{(n+1)}\big{)}^{-1}I\big{)}$
$\displaystyle={}$
$\displaystyle\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n}))$
$\displaystyle-\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n+1}(v^{\star}_{n+1}-{\widehat{v}^{\star}}_{n+1}))$
$\displaystyle={}$
$\displaystyle\big{(}\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}-\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}\big{)}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n}))$
$\displaystyle+\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})-\widehat{\mu}^{\star}_{n+1}(v^{\star}_{n+1}-{\widehat{v}^{\star}}_{n+1})).$

From the uniform boundedness of
$\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}$,
$\mathcal{C}_{\Sigma^{\star}}^{-}$ and point (iii), it follows that

$\displaystyle\big{\|}\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})-\widehat{\mu}^{\star}_{n+1}(v^{\star}_{n+1}-{\widehat{v}^{\star}}_{n+1}))\big{\|}_{L^{2}(\Sigma^{\star})}=O\left(\frac{1}{n^{3/2}\log^{2}n}\right).$

For the remaining term, we have

$\displaystyle\big{(}\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}-\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}\big{)}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n}))$
$\displaystyle\qquad=\big{(}\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}\big{(}\mathcal{C}_{v^{\star}}^{(n)}-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}\big{)}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n})).$
(7.5)

Now observe that

$\displaystyle\big{\|}\mathcal{C}_{v^{\star}}^{(n)}-\mathcal{C}_{v^{\star}}^{(n+1)}\big{\|}_{L^{2}(\Sigma^{\star})\to
L^{2}(\Sigma^{\star})}\lesssim\|v^{\star}_{n}-v^{\star}_{n+1}\|_{L^{\infty}(\Sigma^{\star})}.$
(7.6)

It has been shown in [4, p. 54] that
$\|v^{\star}_{n}-v^{\star}_{n+1}\|_{L^{\infty}(\Sigma^{r})}=O\big{(}n^{-1}\big{)}$,
hence with (7.4) it follows that

$\displaystyle\|v^{\star}_{n}-v^{\star}_{n+1}\|_{L^{\infty}(\Sigma^{\star})}=O\big{(}n^{-1}\big{)},$

which implies using the bound (7.6)

$\displaystyle\big{\|}\mathcal{C}_{v^{\star}}^{(n)}-\mathcal{C}_{v^{\star}}^{(n+1)}\big{\|}_{L^{2}(\Sigma^{\star})\to
L^{2}(\Sigma^{\star})}=O\big{(}n^{-1}\big{)}.$

Plugging this estimate together with (i) into (7.5), we conclude that

$\displaystyle\big{\|}\big{(}\big{(}1-\mathcal{C}_{v^{\star}}^{(n)}\big{)}^{-1}-\big{(}1-\mathcal{C}_{v^{\star}}^{(n+1)}\big{)}^{-1}\big{)}\mathcal{C}_{\Sigma^{\star}}^{-}(\widehat{\mu}^{\star}_{n}(v^{\star}_{n}-{\widehat{v}^{\star}}_{n}))\big{\|}_{L^{2}(\Sigma^{\star})}=O\left(\frac{1}{n^{3/2}\log^{2}n}\right),$

which implies (iv) and finishes the proof. ∎

We immediately get from Proposition 7.1:

######  Corollary 7.2.

The following estimates hold for $n\to\infty$:

  1. $(i)$

$\|{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n})\|_{L^{2}(\Sigma^{r})}=O\big{(}\frac{1}{n^{1/2}\log^{2}n}\big{)}$,

  2. $(ii)$

$\|\mu_{n}-{\widehat{\mu}}_{n}\|_{L^{2}(\Sigma^{r})}=O\big{(}\frac{1}{n^{1/2}\log^{2}n}\big{)}$,

  3. $(iii)$

$\|{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n})-{\widehat{\mu}}_{n+1}(v_{n+1}-{\widehat{v}}_{n+1})\|_{L^{2}(\Sigma^{r})}=O\big{(}\frac{1}{n^{3/2}\log^{2}}\big{)}$,

  4. $(iv)$

$\|(\mu_{n}-{\widehat{\mu}}_{n})-(\mu_{n+1}-{\widehat{\mu}}_{n+1})\|_{L^{2}(\Sigma^{r})}=O\big{(}\frac{1}{n^{3/2}\log^{2}n}\big{)}$.

Note that we restricted the path of integration to $\Sigma^{r}$. The
contributions coming from $\Sigma\cap U$ turn out to be of smaller order as
will be shown in Lemma 7.6.

To obtain an asymptotic formula for the recurrence coefficients, we need to
obtain an asymptotic formula for (3.9) which in our current notation reads

$\displaystyle
Q_{1}^{(n)}-{\widehat{Q}}_{1}^{(n)}=-\frac{1}{2\pi\mathrm{i}}\int_{\Sigma}{\mu}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}^{-1}_{n}{\rm
d}s.$ (7.7)

The key in proving Theorem 1.1 lies in the following proposition.

######  Proposition 7.3.

For $n\to\infty$ the following estimates hold:

$\displaystyle
Q_{1}^{(n)}-{\widehat{Q}}_{1}^{(n)}=\frac{3}{16n\log^{2}n}\begin{pmatrix}-1&\mathrm{i}\\\
\phantom{-}\mathrm{i}&1\end{pmatrix}+O\left(\frac{1}{n\log^{3}n}\right)$

and

$\displaystyle
Q_{1}^{(n)}-{\widehat{Q}}_{1}^{(n)}-\big{(}Q_{1}^{(n+1)}-{\widehat{Q}}_{1}^{(n+1)}\big{)}=\frac{3}{16n^{2}\log^{2}n}\begin{pmatrix}-1&\mathrm{i}\\\
\phantom{-}\mathrm{i}&1\end{pmatrix}+O\left(\frac{1}{n^{2}\log^{3}n}\right).$
(7.8)

We will make use of two important results stated in [4, Section 5.2], but
restricted to the contour $\Sigma^{r}$:

######  Proposition 7.4.

The following estimates hold:

$\displaystyle\int_{\Sigma^{r}}{\mu}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s=\int_{\Sigma^{r}}{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s+O\left(\frac{1}{n\log^{4}n}\right),$ (7.9)

and

$\displaystyle\int_{\Sigma^{r}}{\mu}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}^{-1}_{n}{\rm
d}s-\int_{\Sigma^{r}}{\mu}_{n+1}\big{(}v_{n+1}-{\widehat{v}}_{n+1}\big{)}{\widehat{\mu}}^{-1}_{n+1}{\rm
d}s$
$\displaystyle\qquad=\int_{\Sigma^{r}}{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}^{-1}_{n}{\rm
d}s-\int_{\Sigma^{r}}{\widehat{\mu}}_{n+1}\big{(}v_{n+1}-{\widehat{v}}_{n+1}\big{)}{\widehat{\mu}}^{-1}_{n+1}{\rm
d}s+O\left(\frac{1}{n^{2}\log^{4}n}\right).$ (7.10)

###### Proof.

For (7.9), observe that

$\displaystyle\int_{\Sigma^{r}}{\mu}_{n}$
$\displaystyle(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s=\int_{\Sigma^{r}}{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s+\int_{\Sigma^{r}}({\mu}_{n}-{\widehat{\mu}}_{n})(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s.$ (7.11)

As $\det{\mu}=\det{\widehat{\mu}}\equiv 1$ and $v-{\widehat{v}}$ has a nonzero
entry only in the $21$-entry, it follows through explicit calculation that
$\|(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}\|_{L^{2}(\Sigma^{r})}=\|{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n})\|_{L^{2}(\Sigma^{r})}$.
Now using estimates (i) and (ii) from Corollary 7.2, it follows that the error
term in (7.11) can be bounded by

$\displaystyle\bigg{|}\int_{\Sigma^{r}}({\mu}_{n}-{\widehat{\mu}}_{n})(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s\bigg{|}\lesssim\|\mu_{n}-{\widehat{\mu}}_{n}\|_{L^{2}(\Sigma^{r})}\big{\|}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}\big{\|}_{L^{2}(\Sigma^{r})}=O\left(\frac{1}{n\log^{4}n}\right).$

In a similar fashion for (7.10), we can write

$\displaystyle\int_{\Sigma^{r}}{\mu}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}^{-1}_{n}{\rm
d}s-\int_{\Sigma^{r}}{\mu}_{n+1}(v_{n+1}-{\widehat{v}}_{n+1}){\widehat{\mu}}^{-1}_{n+1}{\rm
d}s$
$\displaystyle\qquad=\int_{\Sigma^{r}}{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}^{-1}_{n}{\rm
d}s-\int_{\Sigma^{r}}{\widehat{\mu}}_{n+1}(v_{n+1}-{\widehat{v}}_{n+1}){\widehat{\mu}}^{-1}_{n+1}{\rm
d}s$
$\displaystyle\qquad\phantom{=}{}+\int_{\Sigma^{r}}({\mu}_{n}-{\widehat{\mu}}_{n})\big{[}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}^{-1}_{n}-(v_{n+1}-{\widehat{v}}_{n+1}){\widehat{\mu}}^{-1}_{n+1}\big{]}{\rm
d}s$
$\displaystyle\qquad\phantom{=}{}+\int_{\Sigma^{r}}\big{[}({\mu}_{n}-{\widehat{\mu}}_{n})-({\mu}_{n+1}-{\widehat{\mu}}_{n+1})\big{]}(v_{n+1}-{\widehat{v}}_{n+1}){\widehat{\mu}}^{-1}_{n+1}{\rm
d}s.$ (7.12)

We can now estimate the two error terms in (7.12), using Corollary 7.2,

$\displaystyle\bigg{|}\int_{\Sigma^{r}}({\mu}_{n}-{\widehat{\mu}}_{n})\big{[}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}^{-1}_{n}-(v_{n+1}-{\widehat{v}}_{n+1}){\widehat{\mu}}^{-1}_{n+1}\big{]}{\rm
d}s\bigg{|}$
$\displaystyle\qquad\lesssim\|\mu_{n}-{\widehat{\mu}}_{n}\|_{L^{2}(\Sigma^{r})}\|{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n})-{\widehat{\mu}}_{n+1}(v_{n+1}-{\widehat{v}}_{n+1})\|_{L^{2}(\Sigma^{r})}=O\left(\frac{1}{n^{2}\log^{4}n}\right),$

and

$\displaystyle\bigg{|}\int_{\Sigma^{r}}\big{[}({\mu}_{n}-{\widehat{\mu}}_{n})-({\mu}_{n+1}-{\widehat{\mu}}_{n+1})\big{]}(v_{n+1}-{\widehat{v}}_{n+1}){\widehat{\mu}}^{-1}_{n+1}{\rm
d}s\bigg{|}$
$\displaystyle\qquad\lesssim\|(\mu_{n}-{\widehat{\mu}}_{n})-(\mu_{n+1}-{\widehat{\mu}}_{n+1})\|_{L^{2}(\Sigma^{r})}\|{\widehat{\mu}}_{n+1}(v_{n+1}-{\widehat{v}}_{n+1})\|_{L^{2}(\Sigma^{r})}=O\left(\frac{1}{n^{2}\log^{4}n}\right),$

proving (7.10). ∎

The next result, stated in [4, Section 5], contains the leading order term in
the integrals found in Proposition 7.4:

######  Proposition 7.5.

The following estimates holds:

$\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\Sigma^{r}}{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s=\frac{3}{16n\log^{2}n}\begin{pmatrix}\phantom{-}1&-\mathrm{i}\\\
-\mathrm{i}&-1\end{pmatrix}+O\left(\frac{1}{n\log^{3}n}\right)$

and

$\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\Sigma^{r}}{\widehat{\mu}}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s-\frac{1}{2\pi\mathrm{i}}\int_{\Sigma^{r}}{\widehat{\mu}}_{n+1}(v_{n+1}-{\widehat{v}}_{n+1}){\widehat{\mu}}^{-1}_{n+1}{\rm
d}s$
$\displaystyle\qquad=\frac{3}{16n^{2}\log^{2}n}\begin{pmatrix}\phantom{-}1&-\mathrm{i}\\\
-\mathrm{i}&-1\end{pmatrix}+O\left(\frac{1}{n^{2}\log^{3}n}\right).$

###### Proof.

The proof is essentially given in [4, Propositions C.3 and C.4]. The main
difference is the occurrence of ${\widehat{\mu}}_{n}$ instead of
${\widetilde{\mu}}_{n}={\widetilde{Q}}^{(n)}_{-}$. We provide the full proof
of both formulas in Proposition A.2. ∎

Note that in the formula (7.7) the integral is over $\Sigma$, while in the
Propositions 7.4 and 7.5 the integrals are over $\Sigma^{r}$. It remains to
show that the remaining integral over $\Sigma^{U}=\Sigma\cap U$, which is
localized around the point $-1$, is negligible. This is the content of the
next lemma.

######  Lemma 7.6.

For $n\to\infty$, we have

$\displaystyle\int_{\Sigma^{U}}{\mu}_{n}(v_{n}-{\widehat{v}}_{n}){\widehat{\mu}}_{n}^{-1}{\rm
d}s=O\left(\frac{1}{n^{3}}\right).$ (7.13)

###### Proof.

First, we need to show that $Q^{\star}(z)=Q(z)[P(z)]^{-1}$ for $z\in U$ is
uniformly bounded as $n\to\infty$. Note that this statement is more intricate
than the corresponding statements for
${\widehat{Q}}(z)\big{[}{\widehat{P}}(z)\big{]}^{-1}$ and
${\widetilde{Q}}(z)\big{[}{\widetilde{P}}(z)\big{]}^{-1}$, as the logarithmic
weight function is not covered in [12] and hence we do not have equation (5.9)
for $Q(z)[P(z)]^{-1}$ at our disposal. Recall that for
${s\\!\in\\!\Sigma^{\ell}=\partial U}$, we have

$\displaystyle\mu^{\star}(s)=Q^{\star}_{-}(s)=Q(s)\big{[}P(s)\big{]}^{-1}.$
(7.14)

From (7.14), we see that on $\partial U$, $\mu^{\star}$ is in fact the
restriction of the analytic function $Q^{\star}(z)=Q(z)[P(z)]^{-1}$, $z\in U$.
Moreover, $\|\mu^{\star}\|_{L^{2}(\Sigma^{U})}=O(1)$ for $n\to\infty$, as
$\mu^{\star}=(1-\mathcal{C}_{v^{\star}})^{-1}I$ and
${(1-\mathcal{C}_{v^{\star}})^{-1}}$ is uniformly bounded on
$L^{2}(\Sigma^{\star})$. Thus, using Cauchy’s integral formula, we get for
$z\in U$

$\displaystyle|Q^{\star}(z)|=\bigg{|}\frac{1}{2\pi\mathrm{i}}\int_{\partial
U}\frac{\mu^{\star}(s){\rm d}s}{s-z}\bigg{|}\lesssim
O\left(\frac{1}{\mathop{\mathrm{dist}}(z,\partial U)}\right).$

We have some freedom to choose $U$, so if necessary we can shrink it and
conclude

$\displaystyle|Q^{\star}(z)|=O(1),\qquad z\in U,$ (7.15)

uniformly as $n\to\infty$. Let us now rewrite (7.13), where we drop the
$n$-dependence for better readability:

$\displaystyle\int_{\Sigma^{U}}{\mu}(v-{\widehat{v}}){\widehat{\mu}}^{-1}{\rm
d}s=\int_{\Sigma^{U}}Q^{\star}(s)P_{-}(s)(v(s)-{\widehat{v}}(s))\big{[}{\widehat{P}}_{-}(s)\big{]}^{-1}\big{[}{\widehat{Q}^{\star}}(s)\big{]}^{-1}{\rm
d}s.$ (7.16)

We now list bounds for each of the factors in the integrand:

  * •

(2.4) and (2.13) imply that for some $c>0$

$\displaystyle v(s)-{\widehat{v}}(s)=\begin{cases}\begin{pmatrix}0&0\\\
\big{(}\frac{F^{2}}{w}(s)-\frac{\widehat{F}^{2}}{\widehat{w}}(s)\big{)}\phi^{-2n}(s)&0\end{pmatrix}\\\
\quad=O\big{(}|s+1|^{3/2}\mathrm{e}^{-cn|s+1|^{1/2}}\big{)},\qquad
s\in\Sigma^{U}\setminus(-1,1),\\\ 0,\qquad
s\in\Sigma^{U}\cap(-1,1),\end{cases}$

  * •

(5.9) and (7.15) implies that $[Q^{\star}(z)]^{\pm
1},[{\widehat{Q}^{\star}}(z)]^{\pm 1}=O(1)$,

  * •

and (5.10) implies that $\big{[}P_{-}(z)\big{]}^{\pm
1},\big{[}{\widehat{P}}_{-}(z)\big{]}^{\pm 1}=O\big{(}|z+1|^{-1/2}\big{)}$.

Taking the contributions of all factors in (7.16) into account we see that the
integrand can be estimated by
$O\big{(}|z+1|^{1/2}\mathrm{e}^{-cn|z+1|^{1/2}}\big{)}$, which precisely
integrates to the error in (7.13). ∎

Propositions 7.4 and 7.5 together with Lemma 7.6 now readily imply Proposition
7.3.

###  7.2 Asymptotics of the recurrence coefficients

Next, we will derive the asymptotics for the recurrence coefficients stated in
Theorem 1.1. This subsection follows the same line of reasoning as [4, Section
5.2].

Recall the result of Corollary 3.3, which states that the recurrence
coefficients of the orthogonal polynomials with weight function $\widehat{w}$
satisfy

$\displaystyle\widehat{a}_{n}=\frac{1}{4n^{2}}+O\left(\frac{1}{n^{3}}\right),\qquad\widehat{b}_{n}=\frac{1}{2}-\frac{1}{16n^{2}}+O\left(\frac{1}{n^{3}}\right).$
(7.17)

With the help of Proposition 7.3, we can now expand (3.7):

$\displaystyle a_{n}-\widehat{a}_{n}$
$\displaystyle=\big{(}Q_{1}^{(n)}\big{)}_{11}-\big{(}{\widehat{Q}}_{1}^{(n)}\big{)}_{11}-\big{(}\big{(}Q_{1}^{(n+1)}\big{)}_{11}-\big{(}{\widehat{Q}}_{1}^{(n+1)}\big{)}_{11}\big{)}$
$\displaystyle=\Bigg{(}\frac{3}{16n\log^{2}n}\begin{pmatrix}-1&\mathrm{i}\\\
\phantom{-}\mathrm{i}&1\end{pmatrix}+O\left(\frac{1}{n\log^{3}n}\right)\Bigg{)}_{11}$
$\displaystyle=-\frac{3}{16n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right).$

Now substituting the asymptotic formula (7.17) for $\widehat{a}_{n}$ implies
(1.2).

The proof of the asymptotic formula (1.3) is more involved. First recall [12,
Section 8], in which it is shown that for $z$ away from $\pm 1$, we have

$\displaystyle{\widehat{Q}}^{(n)}(z)=\left(I+\frac{R_{1}(z)}{n}+Er(z,n)\right)N(z),$
(7.18)

where $R_{1}(z)$, $Er(z,n)$ are matrix-valued functions, holomorphic for
$z\in\Omega_{0}$ (cf. Figure 4), satisfying

$\displaystyle|R_{1}(z)|\leq\frac{c_{1}}{|z|},\qquad|Er(z,n)|\leq\frac{c_{2}}{|z|n^{2}},\qquad
z\to\infty$ (7.19)

for some $c_{1},c_{2}>0$. Importantly, $R_{1}$ is $n$-independent. As a
consequence of (7.18) and (7.19), we obtain

$\displaystyle\big{|}{\widehat{Q}}^{(n+1)}(z)-{\widehat{Q}}^{(n)}(z)\big{|}\lesssim\frac{c_{2}}{|z|n^{2}},\qquad
z\to\infty,$

from which

$\displaystyle{\widehat{Q}}^{(n)}_{1}-{\widehat{Q}}^{(n+1)}_{1}=O\left(\frac{1}{n^{2}}\right)$
(7.20)

follows. Additionally, direct computation leads to
$N_{12}(z)=-\frac{1}{2\mathrm{i}z}+O\big{(}z^{-2}\big{)}$ implying

$\displaystyle\big{(}{\widehat{Q}}^{(n)}_{1}\big{)}_{12}=-\frac{1}{2\mathrm{i}}+O\left(\frac{1}{n}\right).$
(7.21)

Now using (7.8), we conclude from (7.20) that

$\displaystyle Q^{(n)}_{1}-Q^{(n+1)}_{1}$
$\displaystyle={\widehat{Q}}^{(n)}_{1}-{\widehat{Q}}^{(n+1)}_{1}+O\left(\frac{1}{n^{2}\log^{2}n}\right)=O\left(\frac{1}{n^{2}}\right).$
(7.22)

Regarding formula (3.8), we now obtain using (7.21) and (7.22), together with
Proposition 7.3:

$\displaystyle b_{n}^{2}-\widehat{b}_{n}^{2}={}$
$\displaystyle\big{(}\big{(}Q_{1}^{(n+1)}\big{)}_{12}-\big{(}{\widehat{Q}}_{1}^{(n+1)}\big{)}_{12}\big{)}\big{(}\big{(}Q_{1}^{(n+1)}\big{)}_{21}-\big{(}Q_{1}^{(n+2)}\big{)}_{21}\big{)}$
$\displaystyle+\big{(}{\widehat{Q}}_{1}^{(n)}\big{)}_{12}\big{[}\big{(}\big{(}Q_{1}^{(n+1)}\big{)}_{21}-\big{(}Q_{1}^{(n+2)}\big{)}_{21}\big{)}-\big{(}\big{(}{\widehat{Q}}_{1}^{(n+1)}\big{)}_{21}-\big{(}{\widehat{Q}}_{1}^{(n+2)}\big{)}_{21}\big{)}\big{]}$
$\displaystyle={}$
$\displaystyle\left(\frac{3\mathrm{i}}{16n\log^{2}n}+O\left(\frac{1}{n\log^{3}n}\right)\right)O\left(\frac{1}{n^{2}}\right)$
$\displaystyle+\left(-\frac{1}{2\mathrm{i}}+O\left(\frac{1}{n}\right)\right)\left(\frac{3\mathrm{i}}{16n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right)\right)$
$\displaystyle={}$
$\displaystyle-\frac{3}{32n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right).$
(7.23)

Moreover, we have

$\displaystyle
b_{n}^{2}-\widehat{b}_{n}^{2}=\big{(}b_{n}-\widehat{b}_{n}\big{)}\big{(}b_{n}+\widehat{b}_{n}\big{)}$
$\displaystyle=\big{(}b_{n}-\widehat{b}_{n}\big{)}\big{(}2\widehat{b}_{n}+b_{n}-\widehat{b}_{n}\big{)}$
$\displaystyle=\big{(}b_{n}-\widehat{b}_{n}\big{)}\big{(}1+b_{n}-\widehat{b}_{n}+O\big{(}n^{-2}\big{)}\big{)}$
(7.24)

As $\widehat{b}_{n}=\frac{1}{2}+O\big{(}n^{-2}\big{)}$, see Corollary 3.3, and
$b_{n}>0$, we have that
$\big{(}1+b_{n}-\widehat{b}_{n}+O\big{(}n^{-2}\big{)}\big{)}>1/2-\epsilon$,
which with (7.23) implies that

$\displaystyle b_{n}-\widehat{b}_{n}=O\left(\frac{1}{n^{2}\log^{2}n}\right).$
(7.25)

Substituting (7.25) once again into the term
$\big{(}1+b_{n}-\widehat{b}_{n}+O\big{(}n^{-2}\big{)}\big{)}$, we conclude
from (7.24) that in fact

$\displaystyle
b_{n}^{2}-\widehat{b}_{n}^{2}=\big{(}b_{n}-\widehat{b}_{n}\big{)}\big{(}1+O\big{(}n^{-2}\big{)}\big{)},$

which with (7.23) implies

$\displaystyle
b_{n}-\widehat{b}_{n}=-\frac{3}{32n^{2}\log^{2}n}+O\left(\frac{1}{n^{2}\log^{3}n}\right).$

That together with the asymptotic formula (7.17) implies (1.3) finishing the
proof of Theorem 1.1.

##  Appendix A Proofs of certain propositions

The following appendix contains proofs of Propositions 2.7 and 7.5. These
differ in certain details from the analogous proofs in [4], hence are included
here.

######  Proposition A.1.

Fix $R>0$. Let $r_{n},\tilde{r}_{n}\in(0,1)$, $n\in\mathbb{N}$ be two
sequences satisfying $r_{n},\tilde{r}_{n}\to 0$, such that
$n\big{|}\frac{r_{n}}{\tilde{r}_{n}}-1|<R$. Then

$\displaystyle\frac{F^{2}}{w_{+}}(1+r_{n})-\frac{F^{2}}{w_{+}}(1+\tilde{r}_{n})+\frac{F^{2}}{w_{-}}(1+r_{n})-\frac{F^{2}}{w_{-}}(1+\tilde{r}_{n})$
$\displaystyle\qquad=O(r_{n}\log|\log
r_{n}|)+O\left(\frac{1}{n\log^{3}r_{n}}\right)+O\left(\frac{1}{n^{2}}\right),$

where the implied constants in the $O$-terms depend only on $R$.

###### Proof.

Let us assume without loss of generality $r_{n}\geq\tilde{r}_{n}$. It follows
from definition (2.1) that

$\displaystyle\log F(1+r_{n})$
$\displaystyle=\big{(}(1+r_{n})^{2}-1\big{)}^{1/2}\frac{1}{2\pi\mathrm{i}}\int_{-1}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm d}s}{s-(1+r_{n})}$
$\displaystyle=\big{(}\sqrt{2}r_{n}^{1/2}+O\big{(}r_{n}^{3/2}\big{)}\big{)}\frac{1}{2\pi\mathrm{i}}\int_{-1}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm d}s}{s-(1+r_{n})}.$

Moreover, by [4, equation A.11], we have

$\displaystyle\int_{-1}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm
d}s}{s-(1+r_{n})}=O\left(\frac{\log|\log r_{n}|}{r_{n}^{1/2}}\right),$

implying that, in fact,

$\displaystyle\log
F(1+r_{n})=\sqrt{2}r_{n}^{1/2}\frac{1}{2\pi\mathrm{i}}\int_{-1}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm
d}s}{s-(1+r_{n})}+O(r_{n}\log|\log r_{n}|).$

In particular, we obtain

$\displaystyle\log\frac{F(1+r_{n})}{F(1+\tilde{r}_{n})}={}$
$\displaystyle\sqrt{2}r_{n}^{1/2}\frac{1}{2\pi\mathrm{i}}\int_{0}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm d}s}{s-(1+r_{n})}$
$\displaystyle-\sqrt{2}\tilde{r}_{n}^{1/2}\frac{1}{2\pi\mathrm{i}}\int_{0}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm d}s}{s-(1+\tilde{r}_{n})}$
$\displaystyle+\frac{1}{2\pi\mathrm{i}}\int_{-1}^{0}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\Bigg{[}\frac{\sqrt{2}r_{n}^{1/2}}{s-(1+r_{n})}-\frac{\sqrt{2}\tilde{r}_{n}^{1/2}}{s-(1+\tilde{r}_{n})}\Bigg{]}{\rm
d}s$ $\displaystyle+O(r_{n}\log|\log r_{n}|).$ (A.1)

The term in the bracket can be estimated as follow:

$\displaystyle\Bigg{|}\frac{\sqrt{2}r_{n}^{1/2}}{s-(1+r_{n})}-\frac{\sqrt{2}\tilde{r}_{n}^{1/2}}{s-(1+\tilde{r}_{n})}\Bigg{|}$
$\displaystyle=\Bigg{|}\sqrt{2}\frac{r_{n}^{1/2}(s-(1+\tilde{r}_{n}))-\tilde{r}_{n}^{1/2}(s-(1+r_{n}))}{(s-(1+r_{n}))(s-(1+\tilde{r}_{n}))}\Bigg{|}$
$\displaystyle=\Bigg{|}\sqrt{2}\frac{\big{(}s-1+r_{n}^{1/2}\tilde{r}_{n}^{1/2}\big{)}\big{(}r_{n}^{1/2}-\tilde{r}_{n}^{1/2}\big{)}}{(s-(1+r_{n}))(s-(1+\tilde{r}_{n}))}\Bigg{|}$
$\displaystyle=\underbrace{\Bigg{|}\sqrt{2}\frac{\big{(}s-1+r_{n}^{1/2}\tilde{r}_{n}^{1/2}\big{)}\tilde{r}_{n}^{1/2}}{(s-(1+r_{n}))(s-(1+\tilde{r}_{n}))}\Bigg{|}}_{=O\big{(}r_{n}^{1/2}\big{)}}\cdot\underbrace{\left|\left(\frac{r_{n}}{\tilde{r}_{n}}\right)^{1/2}-1\right|}_{<\frac{R}{n}}$
$\displaystyle=O\left(\frac{r_{n}^{1/2}}{n}\right),$

where we used that $s\in(-1,0)$. Thus (A.1) can be rewritten as

$\displaystyle\log\frac{F(1+r_{n})}{F(1+\tilde{r}_{n})}={}$
$\displaystyle\sqrt{2}r_{n}^{1/2}\frac{1}{2\pi\mathrm{i}}\int_{0}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm d}s}{s-(1+r_{n})}$
$\displaystyle-\sqrt{2}\tilde{r}_{n}^{1/2}\frac{1}{2\pi\mathrm{i}}\int_{0}^{1}\frac{\log
w(s)}{\big{(}s^{2}-1\big{)}_{+}^{1/2}}\frac{{\rm d}s}{s-(1+\tilde{r}_{n})}$
$\displaystyle+O(r_{n}\log|\log r_{n}|)+O\left(\frac{r_{n}^{1/2}}{n}\right),$

Now, after the change of variables $t=1-s$ and some algebraic manipulation, we
get (cf. proof of [4, Proposition A.4])

$\displaystyle\log\frac{F(1+r_{n})}{F(1+\tilde{r}_{n})}={}$
$\displaystyle\frac{r_{n}^{1/2}}{2\pi}\int_{0}^{1}\frac{\log
w(1-t)}{\sqrt{t}}\frac{{\rm
d}t}{t+r_{n}}-\frac{\tilde{r}_{n}^{1/2}}{2\pi}\int_{0}^{1}\frac{\log
w(1-t)}{\sqrt{t}}\frac{{\rm d}t}{t+\tilde{r}_{n}}$
$\displaystyle+\frac{1}{2\pi}H(r_{n},\tilde{r}_{n})+O(r_{n}\log|\log
r_{n}|)+O\left(\frac{r_{n}^{1/2}}{n}\right),$ (A.2)

where

$\displaystyle H(r_{n},\tilde{r}_{n})={}$
$\displaystyle\big{(}r_{n}^{1/2}-\tilde{r}_{n}^{1/2}\big{)}\int_{0}^{1}\log
w(1-t)\left[\frac{\sqrt{t}}{\sqrt{2}\sqrt{2-t}+(2-t)}\right]\frac{{\rm
d}t}{t+r_{n}}$ $\displaystyle+\tilde{r}_{n}^{1/2}\int_{0}^{1}\log
w(1-t)\left[\frac{\sqrt{t}}{\sqrt{2}\sqrt{2-t}+(2-t)}\right]\left(\frac{1}{t+r_{n}}-\frac{1}{t+\tilde{r}_{n}}\right){\rm
d}t.$

Thus, we can estimate

$\displaystyle|H(r_{n},\tilde{r}_{n})|\leq{}$
$\displaystyle\big{|}r_{n}^{1/2}-\tilde{r}_{n}^{1/2}\big{|}\int_{0}^{1}|\log
w(1-t)|\left[\frac{\sqrt{t}}{\sqrt{2}\sqrt{2-t}+(2-t)}\right]\frac{{\rm
d}t}{t}$ $\displaystyle+\tilde{r}_{n}^{1/2}\int_{0}^{1}|\log
w(1-t)|\left[\frac{\sqrt{t}}{\sqrt{2}\sqrt{2-t}+(2-t)}\right]\frac{|\tilde{r}_{n}-r_{n}|{\rm
d}t}{|t+r_{n}|\cdot|t+\tilde{r}_{n}|}$ $\displaystyle\leq{}$ $\displaystyle
c\tilde{r}_{n}^{1/2}\left|\frac{r_{n}^{1/2}}{\tilde{r}_{n}^{1/2}}-1\right|+\tilde{r}_{n}^{1/2}\left|\frac{\tilde{r}_{n}-r_{n}}{\tilde{r}_{n}}\right|\int_{0}^{1}|\log
w(1-t)|\left[\frac{\sqrt{t}}{\sqrt{2}\sqrt{2-t}+(2-t)}\right]\frac{{\rm
d}t}{t}$ $\displaystyle\leq{}$ $\displaystyle
c\tilde{r}_{n}^{1/2}\underbrace{\left|\frac{r_{n}^{1/2}}{\tilde{r}_{n}^{1/2}}-1\right|}_{<\frac{R}{n}}+c\tilde{r}_{n}^{1/2}\underbrace{\left|\frac{r_{n}}{\tilde{r}_{n}}-1\right|}_{<\frac{R}{n}}=O\left(\frac{r_{n}^{1/2}}{n}\right).$

So, we see that $H(r_{n},\tilde{r}_{n})$ can be included in the error term
$O\big{(}\frac{r_{n}^{1/2}}{n}\big{)}$.

For the remaining integrals in (A.2), we obtain after performing the change of
variables $t\to r_{n}t$ and $t\to\tilde{r}_{n}t$ in the first and second
integral, respectively,

$\displaystyle r_{n}^{1/2}\int_{0}^{1}\frac{\log w(1-t)}{\sqrt{t}}\frac{{\rm
d}t}{t+r_{n}}-\tilde{r}_{n}^{1/2}\int_{0}^{1}\frac{\log
w(1-t)}{\sqrt{t}}\frac{{\rm d}t}{t+\tilde{r}_{n}}$
$\displaystyle\qquad=\int_{0}^{1/r_{n}}\frac{\log
w(1-r_{n}t)}{\sqrt{t}}\frac{{\rm
d}t}{t+1}-\int_{0}^{1/\tilde{r}_{n}}\frac{\log
w(1-\tilde{r}_{n}t)}{\sqrt{t}}\frac{{\rm d}t}{t+1}$
$\displaystyle\qquad=\int_{0}^{1/r_{n}}\frac{\big{[}\log w(1-r_{n}t)-\log
w(1-\tilde{r}_{n}t)\big{]}}{\sqrt{t}}\frac{{\rm
d}t}{t+1}-\int_{1/r_{n}}^{1/\tilde{r}_{n}}\frac{\log
w(1-\tilde{r}_{n}t)}{\sqrt{t}}\frac{{\rm d}t}{t+1}.$ (A.3)

Using the fact that $|\log w(1-\tilde{r}_{n}t)|$ is uniformly bounded for
$t\in\big{[}\frac{1}{r_{n}},\frac{1}{\tilde{r}_{n}}\big{]}$, the last integral
in (A.3) can be estimated via

$\displaystyle\left|\int_{1/r_{n}}^{1/\tilde{r}_{n}}\frac{\log
w(1-\tilde{r}_{n}t)}{\sqrt{t}}\frac{{\rm d}t}{t+1}\right|$
$\displaystyle\leq\left\|\frac{1}{\sqrt{t}}\frac{1}{t+1}\right\|_{L^{\infty}(1/r_{n},1/\tilde{r}_{n})}\int_{1/r_{n}}^{1/\tilde{r}_{n}}|\log
w(1-\tilde{r}_{n}t)|{\rm d}t$ $\displaystyle\leq
cr_{n}^{3/2}\left|\frac{1}{\tilde{r}_{n}}-\frac{1}{r_{n}}\right|=cr_{n}^{1/2}\left|\frac{r_{n}}{\tilde{r}_{n}}-1\right|=O\left(\frac{r_{n}^{1/2}}{n}\right),$

and hence can be again included in the
$O\big{(}\frac{r_{n}^{1/2}}{n}\big{)}$-term in (A.2).

Let us now consider the remaining integral in the last line of (A.3):

$\displaystyle\int_{0}^{1/r_{n}}\big{[}\log w(1-r_{n}t)-\log
w(1-\tilde{r}_{n}t)\big{]}\frac{{\rm d}t}{t^{3/2}+t^{1/2}}$
$\displaystyle\qquad=\int_{0}^{1/r_{n}}\log\left(\frac{w(1-r_{n}t)}{w(1-\tilde{r}_{n}t)}\right)\frac{{\rm
d}t}{t^{3/2}+t^{1/2}}$ (A.4)

Now define
$a=a(r_{n},\tilde{r}_{n};n):=n\big{(}\frac{r_{n}}{\tilde{r}_{n}}-1\big{)}\in[-R,R]$.
Note that for $t\in\big{(}0,\frac{1}{r_{n}}\big{)}$, we have $\log
2\leq\log\frac{2}{\tilde{r}_{n}t}$, hence

$\displaystyle\frac{w(1-r_{n}t)}{w(1-\tilde{r}_{n}t)}$
$\displaystyle=\frac{\log\frac{2}{r_{n}t}}{\log\frac{2}{\tilde{r}_{n}t}}=1+\frac{\log\frac{2}{r_{n}t}-\log\frac{2}{\tilde{r}_{n}t}}{\log\frac{2}{\tilde{r}_{n}t}}=1+\frac{\log\frac{\tilde{r}_{n}}{r_{n}}}{\log\frac{2}{\tilde{r}_{n}t}}$
$\displaystyle=1+\frac{\log(1+\frac{a}{n})}{\log\frac{2}{\tilde{r}_{n}t}}=1+\frac{a}{n\log\frac{2}{\tilde{r}_{n}t}}+O\left(\frac{1}{n^{2}}\right).$

Thus, we can estimate the integrand of (A.4) by

$\displaystyle\log\frac{w(1-r_{n}t)}{w(1-\tilde{r}_{n}t)}=\frac{a}{n\log\frac{2}{\tilde{r}_{n}t}}+O\left(\frac{1}{n^{2}}\right),$

where the $O\big{(}\frac{1}{n^{2}}\big{)}$-term is uniform for
$t\in\big{[}0,\frac{1}{r_{n}}\big{]}$. Substituting this into (A.4), we obtain

$\displaystyle\int_{0}^{1/r_{n}}\left(\frac{a}{n\log\frac{2}{\tilde{r}_{n}t}}+O\left(\frac{1}{n^{2}}\right)\right)\frac{{\rm