Received: 24 July 2018

DOI: 10.1111/sapm.12259

O R I G I N A L A R T I C L E

The Riemann–Hilbert analysis to thePollaczek–Jacobi type orthogonal polynomialsMin Chen1 Yang Chen2 En-Gui Fan1

1School of Mathematical Science, Fudan

University, Shanghai, P.R. China

2Faculty of Science and Technology,

Department of Mathematics, University of

Macau, Taipa, Macau, China

CorrespondenceEn-Gui Fan, School of Mathematical Science,

Fudan University, Shanghai 200433, P.R.

China.

Email: faneg@fudan.edu.cn

Funding InformationThe National Science Foundation of

China Project No. 11801083, No. 11671095,

No. 51879045; the Macau Science and Tech-

nology Development Fund Grant Number

FDCT 130/2014/A3, FDCT 023/2017/A1; the

National Science Foundation of China Grant

No. 11801083.

AbstractIn this paper, we study polynomials orthogonal with respect

to a Pollaczek–Jacobi type weight

𝑤𝑝𝐽(𝑥, 𝑡) = 𝑒

− 𝑡

𝑥 𝑥𝛼(1 − 𝑥)𝛽 , 𝑡 ≥ 0,𝛼 > 0, 𝛽 > 0, 𝑥 ∈ [0, 1].

The uniform asymptotic expansions for the monic orthog-

onal polynomials on the interval (0,1) and outside this

interval are obtained. Moreover, near 𝑥 = 0, the uniform

asymptotic expansion involves Airy function as 𝜍 = 2𝑛2𝑡 →∞, 𝑛 → ∞, and Bessel function of order 𝛼 as 𝜍 = 2𝑛2𝑡 →0, 𝑛 → ∞; in the neighborhood of 𝑥 = 1, the uniform

asymptotic expansion is associated with Bessel function of

order 𝛽 as 𝑛 → ∞. The recurrence coefficients and lead-

ing coefficient of the orthogonal polynomials are expressed

in terms of a particular Painlevé III transcendent. We also

obtain the limit of the kernel in the bulk of the spectrum.

The double scaled logarithmic derivative of the Hankel

determinant satisfies a 𝜎-form Painlevé III equation. The

asymptotic analysis is based on the Deift and Zhou's steep-

est descent method.

K E Y W O R D Sasymptotic analysis, mathematical physics

1 INTRODUCTION AND MAIN RESULTS

1.1 IntroductionThe study of the weighted orthogonal polynomials is not only motivated by questions from theapproximation theory,1 but also from the perspective of the random matrix theory because the joint

Stud Appl Math. 2019;1–39. wileyonlinelibrary.com/journal/sapm © 2019 Wiley Periodicals, Inc., A Wiley Company 1

2 CHEN ET AL.

probability density function can be expressed in terms of determinant of correlation kernel associatedwith the orthogonal polynomials, see.2,3

A weight function 𝑤(𝑥) defined on [−1, 1] or its subsets, belongs to the Szegő class as long as theweight function fulfills the following Szegő condition:

∫1

−1

ln𝑤(𝑥)√1 − 𝑥2

𝑑𝑥 > −∞.

The asymptotic theory of orthogonal polynomials for varying weights that satisfy the above Szegőcondition had been studied and developed by Szegő, see1 for more information.

The classical Hilb's and Plancherl–Rotach's formulas suggest that the local asymptotics in the hardand soft cases can be expressed in terms of Bessel and Airy functions, respectively. Kuijlaars and hiscollaborators studied orthogonal polynomials with respect to the modified Jacobi weight, which is theSzegő class and their applications in the random matrix theory4,5 by the Riemann–Hilbert approach.They gave the concrete construction of the parametrix at the spectrum edge by Bessel functions,4

while Deift and his collaborators constructed the parametrix at the soft spectrum edge in terms of Airyfunction.6,7

Actually, there are some situations and applications that require the associated weight functionswith singular behavior, especially in random matrix models and related fields. For instance, Berryand Shukla8 introduced the singularly perturbed Gaussion unitary ensemble to study the statisticsfor zeros of the Riemann Zeta function. The associated singularly perturbed Gaussion weight is

𝑤(𝑥, 𝑧, 𝑠) = exp(− 𝑧2

2𝑥2 +𝑠

𝑥− 𝑥2

2 ), 𝑥 ∈ ℝ, 𝑧 ∈ ℝ ⧵ {0}, 𝑠 ∈ [0,∞). Brightmore and coauthhors9 con-sidered the partition function of the singularly perturbed Gaussion unitary ensemble and found outthat its asymptotic behavior can be characterized by a particular Painlevé III equation. The singularly

perturbed Laguerre weight is 𝑤(𝑥, 𝑡) = 𝑥𝛼𝑒−𝑥− 𝑡

𝑥 , 𝛼 > 0, 𝑡 > 0, 𝑥 ∈ (0,∞), and there is an essential sin-gularity at the origin. This weight arises a calculation of the expectation values at finite temperaturein integrable quantum field theory by Lukyanov.10 Chen and Its11 showed that this weight associatedwith a certain linear statistics model, and investigated the Hankel determinant and relevant statisticquantities, and these can be expressed in terms of a Painlevé III equation for finite 𝑛 dimensional bythe ladder operator technique, the Riemann–Hilbert formation of the orthogonal polynomials and theJimbo–Miwa theory. Xu, Dai, and Zhao12 studied the limit behavior of the kernel, and it turns out to bea Painlevé III kernel, which can also translate to be the Bessel kernel and the Airy kernel with differentconditions. The double scaling limit of the Hankel determinant is also associated with the Painlevé IIIequation, see.13 For the case of higher order singularities in the weight, we refer.14,15

The Pollaczek polynomials are orthogonal to the following Pollaczek weight function, which alsoprovides an example of the non-Szegő class weight,

𝑤(𝑥, 𝑎, 𝑏) = 𝜋−1 exp ((2𝜃 − 𝜋)ℎ(𝜃))|Γ(1∕2 + 𝑖ℎ(𝜃))|2 ∼ exp(−𝑑(1 − 𝑥2)−

12), as 𝑥 → ±1,

where 𝜃 = arccos√𝑥, ℎ(𝜃) = 𝑎𝑥+𝑏

2√1−𝑥2

, 𝑎, 𝑏 ∈ 𝑅, and |𝑏| < 𝑎, 𝑑 is a positive constant, see.1 The uniform

asymptotics of the Pollaczek polynomials via the steep descendent method, see16 or.17

In this paper, we are interested in the orthogonal polynomials, unitary random matrices ensembleand the Hankel determinant associated with the following Pollaczek–Jacobi type weight,

𝑤𝑝𝐽(𝑥, 𝑡) = 𝑤(𝑥)𝑒−

𝑡

𝑥 , 𝑥 ∈ (0, 1), 𝑡 ≥ 0, (1)

CHEN ET AL. 3

where

𝑤(𝑥) = 𝑥𝛼(1 − 𝑥)𝛽 , 𝛼 > 0, 𝛽 > 0, (2)

is a shifted Jacobi weight function. Obviously, it is an non-Szegő class weight function due to the factor

𝑒− 𝑡

𝑥 , which introduces an infinitely strong zero at the origin.The monic polynomials 𝜋𝑛(𝑥) orthogonal with respect to the Pollaczek–Jacobi type weight satisfies

∫1

0𝜋𝑛(𝑥)𝜋𝑚(𝑥)𝑤𝑝𝐽

(𝑥, 𝑡)𝑑𝑥 = ℎ𝑛(𝑡)𝛿𝑛𝑚, (3)

with ℎ𝑛 is the square of the 𝐿2 norm and 𝜋𝑛(𝑥) satisfies the three-term recurrence relation as follows:

𝑥𝜋𝑛(𝑥) = 𝜋𝑛+1(𝑥) + 𝛼𝑛(𝑡)𝜋𝑛(𝑥) + 𝛽𝑛(𝑡)𝜋𝑛−1(𝑥). (4)

Moreover,

ℎ𝑛 = 𝛾−2𝑛 , (5)

with 𝛾𝑛 is the leading coefficient of the 𝑛th degree orthonormal polynomial associated with 𝑤𝑝𝐽(𝑥, 𝑡),

and 𝑃𝑛(𝑥) = 𝛾𝑛𝜋𝑛(𝑥).By the random matrix theory,2,3 the Pollaczek–Jacobi type unitary ensemble is a probability mea-

sure on the space of 𝑛 × 𝑛 Hermitian matrices, with eigenvalues 𝑥1, 𝑥2,… , 𝑥𝑛 in (0,1), then the jointprobability distribution function of the 𝑛 eigenvalues is given by

𝑝(𝑥1, 𝑥2,… , 𝑥𝑛)𝑑𝑥1 … 𝑑𝑥𝑛 =1𝑍𝑛

1𝑛!∏

1≤𝑗<𝑘≤𝑛(𝑥𝑗 − 𝑥𝑘)2

𝑛∏𝓁=1

𝑤𝑝𝐽(𝑥𝓁 , 𝑡)𝑑𝑥1 … 𝑑𝑥𝑛, (6)

where 𝑍𝑛 is called the partition function. The above joint probability density function can also writeas a determinant of correlation kernels as follows:

𝑝(𝑥1, 𝑥2,… , 𝑥𝑛) = det(𝐾𝑛(𝑥𝑘, 𝑥𝑗)

)1≤𝑘,𝑗≤𝑛,

where

𝐾𝑛(𝑥, 𝑦) ∶=√

𝑤𝑝𝐽(𝑥, 𝑡)√

𝑤𝑝𝐽(𝑦, 𝑡)

𝑛−1∑𝑗=0

𝜋𝑗(𝑥)𝜋𝑗(𝑦)ℎ𝑗

. (7)

With the Christoffel–Darboux formula,1 it rewrites as

𝐾𝑛(𝑥, 𝑦) =√

𝑤𝑝𝐽(𝑥, 𝑡)√

𝑤𝑝𝐽(𝑦, 𝑡)

𝜋𝑛(𝑥)𝜋𝑛−1(𝑦) − 𝜋𝑛(𝑦)𝜋𝑛−1(𝑥)(𝑥 − 𝑦)ℎ𝑛−1

. (8)

The above correlation kernel is also a reproducing kernel. It is interesting to investigate thegeneral behavior of local eigenvalue correlation kernel in the large 𝑛 limit. For example, Kui-jlaars and Vanlessen5 studied the modified Jacobi unitary ensemble (MJUE), which is given by the

4 CHEN ET AL.

modified Jacobi weight 𝑤(𝑥) = (1 − 𝑥)𝛼(1 + 𝑥)𝛽ℎ(𝑥), with 𝛼, 𝛽 > −1, ℎ(𝑥) is real analytic and posi-tive on [−1, 1]. They showed that the limit of the mean eigenvalue density is

lim𝑛→∞

1𝑛𝐾𝑛(𝑥, 𝑥) =

1

𝜋√1 − 𝑥2

, −1 < 𝑥 < 1,

the limiting kernel is the sine kernel in the bulk of the spectrum,

lim𝑛→∞

𝜋√1 − 𝑥2

𝑛𝐾𝑛

(𝑥 + 𝑢

𝜋√1 − 𝑥2

𝑛, 𝑥 + 𝑣

𝜋√1 − 𝑥2

𝑛

)= sin𝜋(𝑢 − 𝑣)

𝜋(𝑢 − 𝑣), 𝑥 ∈ (−1, 1), 𝑢, 𝑣 ∈ ℝ,

the Bessel kernel appears at the right spectrum edge, 𝑢, 𝑣 ∈ (0,∞),

lim𝑛→∞

12𝑛2

𝐾𝑛

(1 − 𝑢

2𝑛2, 1 − 𝑣

2𝑛2

)=

𝐽𝛼(√𝑢)√𝑣𝐽 ′

𝛼(√𝑣) −√𝑢𝐽 ′

𝛼(√𝑢)𝐽𝛼(

√𝑣)

2(𝑢 − 𝑣)= 𝕁𝛼(𝑢, 𝑣).

In the case of the Gaussian unitary ensemble (GUE), the limiting kernel becomes as the Airy kernel at

the edge of√2𝑛,

lim𝑛→∞

1

212 𝑛

16

𝐾𝑛

(√2𝑛 + 𝑥

212 𝑛

16

,√2𝑛 + 𝑦

212 𝑛

16

)= 𝐴𝑖(𝑥)𝐴𝑖′(𝑦) − 𝐴𝑖′(𝑥)𝐴𝑖(𝑦)

𝑥 − 𝑦,

see.2,18

The partition function 𝑍𝑛 in (6) can be expressed by the Hankel determinant 𝐷𝑛(𝑤𝑝𝐽, 𝑡), 𝑍𝑛 =

𝑛!𝐷𝑛(𝑤𝑝𝐽, 𝑡), and the Hankel determinant is given by

𝐷𝑛(𝑤𝑝𝐽, 𝑡) = 1

𝑛! ∫(0,1)𝑛∏

1≤𝑗<𝑘≤𝑛(𝑥𝑗 − 𝑥𝑘

)2 𝑛∏𝓁=1

𝑤𝑝𝐽(𝑥𝓁 , 𝑡)𝑑𝑥𝓁 =

𝑛−1∏𝑗=0

ℎ𝑗(𝑡), (9)

where ℎ𝑛(𝑡) satisfies (3).The Hankel determinant plays an important role in the study of random matrix model. For instance,

Chen and Its11 found that the Hankel determinant can be interpreted as the moment generating func-tion of linear statistics model; Texier and Majumdar19 described that the Hankel determinant may beinterpreted as the generating function for the distribution function of the Wigner delay time in chaoticcavities. More and interesting information about the connection between the Hankel determinant andthe Toeplitz determinant, we refer the paper20 and references therein.

To study the asymptotic behaviors of orthogonal polynomials and the Hankel determinant associ-ated with the Pollaczek–Jacobi type weight, we introduce a model Riemann–Hilbert problem (RHpfor short) for Ψ(𝜉, 𝜍), 𝜍 = 2𝑛2𝑡, and its existence and uniqueness had been proved by Xu, Dai, andZhao.12 Although its explicit solution of this model cannot write down for finite 𝜍 = 2𝑛2𝑡,Ψ(𝜉, 𝜍) canbe approximated by the Bessel model RHp as 𝜍 → 0 and the Airy model RHp as 𝜍 → ∞, separately.For the transition between these two regimes, see.12,13 We state main results as follows.

1.2 Asymptotics for monic orthogonal polynomialsThe strong asymptotic expansion in power of 1∕𝑛 for the monic orthogonal polynomials in terms ofSzegő function for polynomials outside the orthogonal interval.

CHEN ET AL. 5

Theorem 1. For 𝑧 ∈ ℂ ⧵ [0, 1], 𝜋𝑛(𝑧) has an asymptotic expansion as follows

𝜋𝑛(𝑧) = 2−(2𝑛+𝛼+𝛽+1)(𝜑(𝑧))2𝑛+𝛼+𝛽+1

2 (𝑧 − 1)−2𝛽+14 𝑧−

2𝛼+14 𝑒

𝑡

2𝑧(1 + 𝑂(𝑛−1)

), as 𝑛 → ∞, (10)

where 𝑡 > 0 and 𝜑(𝑧) = 2𝑧 − 1 + 2√𝑧(𝑧 − 1). The above expansion is uniformly for 𝑧 on compact

subsets of ℂ ⧵ [0, 1].

The strong asymptotic expansion for 𝜋𝑛(𝑧) in the bulk of the orthogonality interval (0,1) is given bythe following theorem:

Theorem 2. For 𝑥 ∈ (0, 1),

𝜋𝑛(𝑥) =2−(2𝑛+𝛼+𝛽)√

𝑤𝑝𝐽(𝑥, 𝑡)𝑥

14 (1 − 𝑥)

14

[(1 + 𝑂

(1𝑛

))cos((2𝑛 + 𝛼 + 𝛽 + 1) arccos

√𝑥 − 𝛽𝜋

2− 𝜋

4

)

+𝑂(1𝑛

)cos((2𝑛 + 𝛼 + 𝛽 − 1) arccos

√𝑥 − 𝛽𝜋

2+ 3𝜋

4

)], as 𝑛 → ∞, (11)

where 𝑤𝑝𝐽(𝑥, 𝑡) is the Pollaczek-Jacobi weight in (1). The above error terms hold uniformly for 𝑥 in

compact subsets of (0,1).

At the interval of (1 − 𝑟, 1), 𝑟 is small and positive, the strong asymptotic expansion in power of 1∕𝑛for the monic orthogonal polynomials is expressed by the Szegő function and the Bessel function oforder 𝛽.

Theorem 3. There exists a small and fixed 0 < 𝑟 ≪ 1∕2 so that 𝑥 ∈ (1 − 𝑟, 1),

𝜋𝑛(𝑥) =

√𝜋(𝑛 arccos

√𝑥) 1

2

22𝑛𝑥14 (1 − 𝑥)

14√

𝑤𝑝𝐽(𝑥, 𝑡)

×[(1 + 𝑂

(𝑛−1))(

cos 𝜃1𝐽𝛽(2𝑛 arccos√𝑥) + sin 𝜃1𝐽 ′

𝛽(2𝑛 arccos

√𝑥))

+𝑂(𝑛−1)(

cos 𝜃2𝐽𝛽(2𝑛 arccos√𝑥) + sin 𝜃2𝐽 ′

𝛽(2𝑛 arccos

√𝑥))]

, as 𝑛 → ∞, (12)

where the error terms are uniformly for 𝑥 in compact subsets of (1 − 𝑟, 1); 𝜃1, 𝜃2 are given by

𝜃1 = (𝛼 + 𝛽 + 1) arccos√𝑥, and 𝜃2 = (𝛼 + 𝛽 − 1) arccos

√𝑥, (13)

𝐽𝛽 is the Bessel function of order 𝛽,𝑤𝑝𝐽(𝑥, 𝑡) is given by (1).

For 𝑥 ∈ (0, 𝛿), 𝛿 is small and positive, the strong asymptotic expansion with error term 𝑂(𝜍) as𝜍 = 2𝑛2𝑡 → 0, involves the Bessel function of order 𝛼.

Theorem 4. There exists a small and fixed 𝛿, 0 < 𝛿 ≪ 1∕2, such that 𝑥 ∈ (0, 𝛿) and 𝜍 = 2𝑛2𝑡,

𝜋𝑛(𝑥) =

√𝜋(𝑛(𝜋 − 2 arccos

√𝑥)) 1

2

22𝑛+𝛼+𝛽𝑥14 (1 − 𝑥)

14√

2𝑤𝑝𝐽(𝑥, 𝑡)

exp⎛⎜⎜⎜⎝

−𝜍(𝑛(𝜋 − 2 arccos

√𝑥))2⎞⎟⎟⎟⎠

6 CHEN ET AL.

×{[

1 + 𝑂(1∕𝑛) + 𝑂(𝜍)][sin 𝜃3𝐽𝛼

(𝑛(𝜋 − 2 arccos

√𝑥))+ cos 𝜃3𝐽 ′

𝛼

(𝑛(𝜋 − 2 arccos

√𝑥))]

+ (𝑂 (1∕𝑛) + 𝑂(𝜍))[sin 𝜃4𝐽𝛼

(𝑛(𝜋 − 2 arccos

√𝑥))+ cos 𝜃4𝐽 ′

𝛼

(𝑛(𝜋 − 2 arccos

√𝑥))]}

, (14)

as 𝑛 → ∞, 𝜍 = 2𝑛2𝑡 → 0. The error terms hold uniformly for 𝑥 in compact subsets of (0, 𝛿)⋂

𝔻1,𝔻1 ={𝑥|𝑛(𝜋 − 2 arccos

√𝑥) ∈ (0,∞), 𝑛 → ∞}; 𝜃3 and 𝜃4 are given by

𝜃3 = (𝛼 + 𝛽 + 1) arccos√𝑥 − (𝛼 + 𝛽)𝜋

2, 𝜃4 = (𝛼 + 𝛽 − 1) arccos

√𝑥 − (𝛼 + 𝛽)𝜋

2, (15)

𝐽𝛼 is the Bessel function of order 𝛼,𝑤𝑝𝐽(𝑥, 𝑡) is given by (1).

As 𝜍 = 2𝑛2𝑡 → ∞, the strong asymptotic expansion can be calculated in terms of the Airy functionfor 𝑥 ∈ (0, 𝜀).

Theorem 5. There exists a small and fixed 𝜀, 0 < 𝜀 ≪ 1∕2, so that 𝑥 ∈ (0, 𝜀), as 𝑛 → ∞, 𝜍 =2𝑛2𝑡 → ∞,

𝜋𝑛(𝑥) =

√𝜋(𝑛(𝜋 − 2 arccos

√𝑥)) 1

2

22𝑛+𝛼+𝛽𝑥14 (1 − 𝑥)

14√

𝑤𝑝𝐽(𝑥, 𝑡)

×{[

1 + 𝑂(𝑛−2∕3

)][(sin 𝜃3𝑑11 + cos 𝜃3

(𝑛(𝜋 − 2 arccos

√𝑥))−1

(𝑏𝑑11 + 𝑑21))𝐴𝑖(𝜉)

+(sin 𝜃3𝑑12 + cos 𝜃3

(𝑛(𝜋 − 2 arccos

√𝑥))−1

(𝑏𝑑12 − 𝑑22))𝐴𝑖′(𝜉)

]+𝑂(𝑛−2∕3

)[(sin 𝜃4𝑑11 + cos 𝜃4

(𝑛(𝜋 − 2 arccos

√𝑥))−1

(𝑏𝑑11 + 𝑑21))𝐴𝑖(𝜉)

+(sin 𝜃4𝑑12 + cos 𝜃4

(𝑛(𝜋 − 2 arccos

√𝑥))−1

(𝑏𝑑12 − 𝑑22))𝐴𝑖′(𝜉)

]}, (16)

where 𝑏, 𝑑11, 𝑑12, 𝑑21, and 𝑑22 are given as follows:

𝑏 = 772(|𝜆| − 1)

, (17)

𝑑11 = (3∕2)1∕6𝜍−1∕9|𝜆|−1∕6 cos(𝛼

2arccos 1√|𝜆|

), (18)

𝑑22 = (3∕2)−1∕6𝜍1∕9|𝜆|1∕6 cos(𝛼

2arccos 1√|𝜆|

), (19)

𝑑12 = (3∕2)−1∕6𝜍−2∕9(|𝜆| − 1)−1∕2|𝜆|1∕6 sin(𝛼

2arccos 1√|𝜆|

), (20)

CHEN ET AL. 7

𝑑21 = (3∕2)1∕6𝜍2∕9(|𝜆| − 1)1∕2|𝜆|−1∕6 sin(𝛼

2arccos 1√|𝜆|

), (21)

and 𝜉 = (3∕2)2∕3(1 − |𝜆|)𝜍2∕9|𝜆|−2∕3, 𝜆 = 𝜍−2∕3𝑛2𝑓0(𝑧) (𝑓0(𝑧), see (75)). The error terms are uni-formly for 𝑥 in compact subsets of (0, 𝛿)

⋂𝔻2, 𝑡 ∈ (0, 𝑐], 𝑐 > 0;𝔻2 is given by

𝔻2 =

{𝑥||||(32)

23 (1 − |𝜆|)𝜍 2

9 ∈ (−∞, 0), |𝜆| = 𝜍−2∕3𝑛2(𝜋 − 2 arccos√𝑥)2, 𝜍 → ∞ 𝑛 → ∞

}. (22)

Moreover, 𝜃3, 𝜃4 are in (15), 𝐴𝑖 is the Airy function, 𝑤𝑝𝐽(𝑥, 𝑡) is in (1).

1.3 Asymptotic for the double-scaled Hankel determinantTo analyze the asymptotic behaviors of the Hankel determinant, the leading coefficient and the recur-rence coefficients of the orthogonal polynomials associated with the Pollaczek–Jacobi weight in (1),we first recall some results for fixed 𝑛 obtained by Chen and Dai21 with the ladder operator technique.

Theorem 6. (Chen and Dai21). Let 𝑤𝑝𝐽(𝑥, 𝑡) be in (1), and the logarithmic derivatives of the Hankel

determinant and auxiliary function 𝑟∗(𝑡) be defined as follows:

𝐻𝑛(𝑡) ∶= 𝑡𝑑

𝑑𝑡log𝐷𝑛(𝑤𝑝𝐽

, 𝑡), 𝑟∗(𝑡) = 𝑡

ℎ𝑛−1 ∫1

0𝜋𝑛−1(𝑥)𝜋𝑛(𝑥)𝑤𝑝𝐽

(𝑥, 𝑡) 1𝑥𝑑𝑥, (23)

the auxiliary quantities 𝑟∗𝑛(𝑡), 𝑟𝑛(𝑡), and 𝑅𝑛(𝑡) satisfy the following equations:

𝑟∗𝑛(𝑡) =𝑛𝑡 + 𝑡𝐻

′𝑛(𝑡)

2𝑛 + 𝛼 + 𝛽, (24)

𝑟𝑛(𝑡) =𝑛(𝑛 + 𝛼) + 𝑡𝐻

′𝑛(𝑡) −𝐻𝑛(𝑡)

2𝑛 + 𝛼 + 𝛽, (25)

and

𝑅𝑛(𝑡) =(2𝑛 + 1 + 𝛼 + 𝛽)[2𝑟2𝑛 + (𝑡 + 2𝛽 − 2𝑟∗)𝑟𝑛 + (2𝑛 + 𝛼)𝑟∗𝑛 − 𝑛𝑡 − 𝑡𝑟′𝑛(𝑡)]

2[(𝑟∗𝑛 − 𝑟𝑛)2 + (2𝑛 + 𝛼 − 𝑡)𝑟∗𝑛 + (𝛽 + 𝑡)𝑟𝑛 − 𝑛𝑡]. (26)

The recurrence coefficients 𝛼𝑛(𝑡) and 𝛽𝑛(𝑡) are expressed in terms of 𝑟∗𝑛(𝑡), 𝑟𝑛(𝑡), and 𝑅𝑛(𝑡) as follows:

(2𝑛 + 2 + 𝛼 + 𝛽)𝛼𝑛(𝑡) = 2(𝑟∗𝑛(𝑡) − 𝑟𝑛(𝑡)) + 𝑅𝑛(𝑡) − 𝛽 − 𝑡, (27)

and

[1 − (2𝑛 + 𝛼 + 𝛽)2]𝛽𝑛(𝑡) = −(𝑟∗𝑛(𝑡) − 𝑟𝑛(𝑡))2 − (𝛽 + 𝑡)𝑟𝑛(𝑡) + (𝑡 − 𝛼 − 2𝑛)𝑟∗𝑛 + 𝑛𝑡. (28)

Furthermore, let

𝑆𝑛(𝑡) =𝑅𝑛(𝑡)

2𝑛 + 1 + 𝛼 + 𝛽, (29)

8 CHEN ET AL.

then 𝑆𝑛(𝑡) satisfies the following Painlevé V equation:

𝑆′′𝑛 =

3𝑆𝑛 − 12𝑆𝑛(𝑆𝑛 − 1)

(𝑆′𝑛

)2 − 𝑆′𝑛

𝑡+(𝑆𝑛 − 1

)22𝑡2

(𝛿𝑛𝑆𝑛 −

𝛽2

𝑆𝑛

)+

𝛼𝑆𝑛

𝑡−

𝑆𝑛(𝑆𝑛 + 1)2(𝑆𝑛 − 1)

, (30)

where 𝛿𝑛 = (2𝑛 + 1 + 𝛼 + 𝛽)2 and the initial conditions

𝑆𝑛(0) = 1, 𝑆′(0) = 1∕𝛼. (31)

The quantity 𝐻𝑛(𝑡) satisfies a shifted Jimbo–Miwa–Okamoto 𝜎-form Painlevé equation(𝑡𝐻 ′′

𝑛

)2 = [𝑛(𝑛 + 𝛼 + 𝑏𝑡) −𝐻𝑛 + (𝛼 + 𝑡)𝐻 ′𝑛

]2 + 4𝐻 ′𝑛(𝑡𝐻

′𝑛 −𝐻𝑛)(𝛽 −𝐻 ′

𝑛), (32)

with the boundary conditions

𝐻𝑛(0) = 0, 𝐻 ′𝑛(0) = −𝑛(𝑛 + 𝛼 + 𝛽)∕𝛼. (33)

As the dimension 𝑛 → ∞, the double scaling limit of the Hankel determinant satisfies a 𝜎-formPainlevé III equation.

Theorem 7. Let 𝑤𝑝𝐽(𝑥, 𝑡) be the Pollaczek–Jacobi type weight in (1) and the double scaling scheme

be 𝜍 = 2𝑛2𝑡 as 𝑛 → ∞ and 𝑡 → 0, then the double scaling limit of the logarithmic derivative of theHankel determinant exists,

𝐻𝑛

(𝜍

2𝑛2

)= 1 − 4𝛼2 − 8𝑟(𝜍)

16

[1 + 𝑂

(𝑛−2

3)]

, as 𝑛 → ∞. (34)

If (𝜍) ∶= lim𝑛→∞

𝐻𝑛(𝜍

2𝑛2 ), then (𝜍) satisfies the following 𝜎-form Painlevé equation:

[𝜍′′(𝜍)

]2 + 4[′(𝜍)

]2[𝜍′(𝜍) −(𝜍)

]−[𝛼′(𝜍) + 2−1

]2 = 0, (35)

with the boundary conditions (𝜍) = 0,′(0) = −1∕2𝛼. Furthermore, one has

𝐷𝑛(𝑤𝑝𝐽, 𝑡) = 𝐷𝑛(𝑤, 0) exp

[(1 + 𝑂

(𝑛−2

3))

∫𝑡

0

1 − 4𝛼2 − 8𝑟(2𝑛2𝜂)16𝜂

𝑑𝜂

], (36)

where the above error terms hold uniformly for 𝑡 ∈ (0, 𝑐], 𝑐 > 0, and 𝜍 = 2𝑛2𝑡 → ∞ as 𝑡 → 𝑐−, andthe error terms can be replaced by 𝑂(𝑛−1) if 𝜍 = 2𝑛2𝑡 is finite; 𝑟(𝜍) satisfies the boundary conditionsin (81), and 𝐷𝑛(𝑤, 0) is given by

𝐷𝑛(𝑤, 0) = 𝐺(𝑛 + 1)𝐺(𝑛 + 1 + 𝛼)𝐺(𝑛 + 1 + 𝛽)𝐺(𝑛 + 1 + 𝛼 + 𝛽)𝐺(𝛼 + 1)𝐺(𝛽 + 1)𝐺(2𝑛 + 1 + 𝛼 + 𝛽)

, (37)

𝐺(𝑧) is the Barnes G-function, 𝐺(𝑧 + 1) = Γ(𝑧)𝐺(𝑧), with an initial condition of 𝐺(1) = 1, see thepaper22 for more information.

1.4 Asymptotics for recurrence coefficients and leading coefficientThe asymptotic expansions of the recurrence coefficients and leading coefficient of the associatedorthogonal polynomials use a solution of a third-order integrable ordinary differential equation (ODE),which is equivalent to a particular Painlevé III.

CHEN ET AL. 9

Theorem 8. Let the weight function 𝑤𝑝𝐽(𝑥, 𝑡) be defined in (1), then the recurrence coefficients

𝛼𝑛(𝑡), 𝛽𝑛(𝑡) in the three-term recurrence formula (4), and the leading coefficient 𝛾𝑛(𝑡) of the orthonormalpolynomials 𝑃𝑛(𝑥) = 𝛾𝑛(𝑡)𝜋𝑛(𝑥) have the following uniform asymptotic expansions:

𝛼𝑛(𝑡) =12+ 16𝜍𝑟′(𝜍) − 8𝑟(𝜍) − 4𝛽2 + 1

32𝑛2(1 + 𝑂

(𝑛−2

3))

, (38)

𝛽𝑛(𝑡) =116

− 16𝜍𝑟′(𝜍) − 8𝑟(𝜍) + 4𝛽2 − 1128𝑛2

(1 + 𝑂

(𝑛−2

3))

, (39)

𝛾𝑛(𝑡) = 𝛾𝑛(0)[1 + 8𝑟(𝜍) + 4𝛼2 − 1

64𝑛

(1 + 𝑂

(𝑛−2

3))]

, (40)

where 𝜍 = 2𝑛2𝑡, the error terms hold uniformly for 𝑡 ∈ (0, 𝑐], 𝑐 > 0, and 𝜍 → ∞ as 𝑡 → 𝑐−, the errorterms can be replaced by 𝑂(𝑛−1) as 𝜍 is finite; the initial condition is

𝛾𝑛(0) =

√Γ(2𝑛 + 1 + 𝛼 + 𝛽)Γ(2𝑛 + 2 + 𝛼 + 𝛽)

𝑛!Γ(𝑛 + 1 + 𝛼)Γ(𝑛 + 1 + 𝛽)Γ(𝑛 + 1 + 𝛼 + 𝛽). (41)

Moreover, 𝑟(𝜍) satisfies a third-order integrable differential equation

𝜍2𝑟′(𝜍)𝑟′′′(𝜍) − 𝜍2𝑟′′(𝜍)2 + 𝜍𝑟′(𝜍)𝑟′′(𝜍) − 𝜍𝑟′(𝜍)3 − 𝛼𝑟′(𝜍) + 1 = 0. (42)

With the aid of Theorem 8 and the asymptotic behaviors of 𝑟(𝜍) in (81), we have the followingcorollary:

Corollary 1. The asymptotic expansions of 𝛼𝑛(𝑡), 𝛽𝑛(𝑡),𝐻𝑛(𝑡), and 𝛾𝑛(𝑡)∕𝛾𝑛(0) are as follows.If 𝑛 → ∞ and 𝜍 = 2𝑛2𝑡 → 0+, then

𝛼𝑛(𝑡) =12+(

𝜍

4𝛼+ 𝛼2 − 𝛽2

8

)1𝑛2

+ 𝑂

(1

𝑛83

),

𝛽𝑛(𝑡) =116

−(

𝜍

16𝛼+ 2(𝛼2 + 𝛽2) − 1

64

)1𝑛2

+ 𝑂

(1

𝑛83

),

𝐻𝑛(𝑡) = − 12𝛼

𝜍 + (𝜍2) + 𝑂(𝑛−8

3),

𝛾𝑛(𝑡)𝛾𝑛(0)

= 1 + 𝜍

8𝛼𝑛(1 + (𝜍)) + 𝑂

(𝑛−5

3).

If 𝑛 → ∞, 𝑡 ∈ (0, 𝑐], 𝑐 > 0, and 𝑡 → 𝑐−, which also indicates 𝜍 = 2𝑛2𝑡 → ∞, then

𝛼𝑛(𝑡) =12+ 1

8(2𝑡)

23 𝑛

−23 + (𝑛−4

3),

𝛽𝑛(𝑡) =116

− 132

(2𝑡)23 𝑛

−23 + 𝑂

(𝑛−4

3),

10 CHEN ET AL.

𝐻𝑛(𝑡) = −34(2𝑡)

23 𝑛

43 + 𝑂

(𝑛

23),

𝛾𝑛(𝑡)𝛾𝑛(0)

= 1 + 316

(2𝑡)23 𝑛

13 + 𝑂

(𝑛−1

3).

1.5 UniversalityWe proof a universal behavior of the correlation kernel in the bulk of the spectrum as the size ofmatrices tends to infinity.

Theorem 9. Let 𝐾𝑛 be the kernel in (8), which is associated with the Pollaczek–Jacobi type weight in(1). Then the following results hold:

(1) For 𝑦 ∈ (0, 1), one has

1𝑛𝐾𝑛(𝑦, 𝑦) =

1𝜋√𝑦(1 − 𝑦)

+ 𝑂(1𝑛

), as 𝑛 → ∞, (43)

uniformly for 𝑦 in compact subsects of (0,1).(2) For 𝑎 ∈ (0, 1), and 𝑢, 𝑣 are finite real variables,

1𝑛𝜚(𝑎)

𝐾𝑛

(𝑎 + 𝑢

𝑛𝜚(𝑎), 𝑎 + 𝑣

𝑛𝜚(𝑎)

)= sin(𝜋(𝑢 − 𝑣))

𝜋(𝑢 − 𝑣)+ 𝑂(1𝑛

), as 𝑛 → ∞, (44)

where 𝜚(𝑦) = (𝜋√𝑦(1 − 𝑦))−1, and the above error term holds uniformly for 𝑎 in compact subsects of

(0,1) and 𝑢, 𝑣 in compact subsects of ℝ.

The rest of this paper is organized as follows. In Section 2, we present the RHp and the analysis ofthe monic orthogonal polynomials associated with the Pollaczek–Jacobi type weight. Section 3 liststhe proofs of the main results in this paper. Section 4 is our conclusion.

2 DEIFT–ZHOU STEEPEST DESCENT ANALYSIS

With the well-known characterization of orthogonal polynomials by Fokas, Its, and Kitaev in,23 theorthogonal polynomials with respect to the weight (1) can be characterized by an RHp for 𝑌 . We adoptthe steepest descent analysis of Deift and Zhou in,24 see also,6,7,25 to analyze the RHp for Y. Follow-ing the standard process, we obtain a series of invertible transformation 𝑌 → 𝑇 → 𝑆 → 𝑅, at last,the matrix function 𝑅 is close to the identity matrix. After that, taking a list of inverse transforma-tions, then one can obtain all the information of the orthogonal polynomials in the complex plane forlarge 𝑛.

2.1 RHp for 𝒀

The orthogonal polynomials with respect to the weight function 𝑤𝑝𝐽(𝑥, 𝑡) in (1) are described by the

following 2 × 2 matrix-valued function 𝑌 (𝑧).

(a) 𝑌 (𝑧) is analytic for 𝑧 ∈ ℂ ⧵ [0, 1].

CHEN ET AL. 11

(b) 𝑌 (𝑧) satisfies the jump condition

𝑌+(𝑥) = 𝑌−(𝑥)(1 𝑤𝑝𝐽

(𝑥, 𝑡)0 1

), for 𝑥 ∈ (0, 1).

(c) The asymptotic behavior of 𝑌 (𝑧) at infinity is

𝑌 (𝑧) =(𝐼 + (𝑧−1))( 𝑧𝑛 0

0 𝑧−𝑛

), 𝑧 → ∞.

(d) The asymptotic behavior of 𝑌 (𝑧) at the origin is

𝑌 (𝑧) =(𝑂(1) 𝑂(1)𝑂(1) 𝑂(1)

), 𝑧 → 0.

(e) The behavior of 𝑌 (𝑧) at 𝑧 = 1 and 𝛽 > 0 is

𝑌 (𝑧) =(𝑂(1) 𝑂(1)𝑂(1) 𝑂(1)

), 𝑧 → 1.

By the work of Fokas, Its, and Kitaev,23 the above RHp for 𝑌 (𝑧) has a unique solution,

𝑌 (𝑧) =⎛⎜⎜⎜⎝

𝜋𝑛(𝑧)12𝜋𝑖 ∫ 1

0𝜋𝑛(𝑥)𝑤𝑝𝐽

(𝑥,𝑡)𝑥−𝑧 𝑑𝑥

−2𝜋𝑖𝛾2𝑛−1𝜋𝑛−1(𝑧) −𝛾2

𝑛−1 ∫ 10

𝜋𝑛−1(𝑥)𝑤𝑝𝐽(𝑥,𝑡)

𝑥−𝑧 𝑑𝑥

⎞⎟⎟⎟⎠, (45)

where 𝜋𝑛(𝑧) is the monic polynomial, and 𝑃𝑛(𝑧) = 𝛾𝑛𝜋𝑛(𝑧) is the orthonormal polynomial with respectto the weight 𝑤𝑝𝐽

(𝑥, 𝑡) in (1).

2.2 RHp for 𝑻

To normalize the matrix function 𝑌 (𝑧) at infinity, we introduce the first transformation 𝑌 → 𝑇 , and 𝑇

is defined as follows:

𝑇 (𝑧) = 4𝑛𝜎3𝑌 (𝑧)𝑒−𝑡

2𝑧 𝜎3𝜑(𝑧)−𝑛𝜎3 , 𝑧 ∈ ℂ ⧵ [0, 1], (46)

where 𝜎3 = ( 1 00 −1 ) is the Pauli matrix, 𝑌 (𝑧) is the unique solution of the RHp for 𝑌 (𝑧), and

𝜑(𝑧) = 2𝑧 − 1 + 2√𝑧(𝑧 − 1). (47)

Then, 𝑇 (𝑧) solves the following RHp:

(a) 𝑇 (𝑧) is analytic for 𝑧 ∈ ℂ ⧵ [0, 1].(b) 𝑇 (𝑧) satisfies the following jump condition on (0, 1):

𝑇+(𝑥) = 𝑇−(𝑥)(𝜑+(𝑥)−2𝑛 𝑤(𝑥)

0 𝜑−(𝑥)−2𝑛

), 𝑥 ∈ (0, 1), (48)

where 𝑤(𝑥) is the shifted Jacobi weight in (2).

12 CHEN ET AL.

O

Σ1

Σ2

Σ3

1

F I G U R E 1 The contour Σ𝑆 =⋃3

𝑘=1 Σ𝑘 for 𝑆(𝑧)

(c) The behavior of 𝑇 (𝑧) at infinity is

𝑇 (𝑧) = 𝐼 + (𝑧−1), 𝑧 → ∞. (49)

(d) The asymptotic behavior of 𝑇 (𝑧) at the origin is

𝑇 (𝑧) =(𝑂(1) 𝑂(1)𝑂(1) 𝑂(1)

)𝑒− 𝑡

2𝑧 𝜎3 , 𝑧 → 0. (50)

(e) 𝑇 (𝑧) satisfies the following behavior as 𝑧 → 1,

𝑇 (𝑧) =(𝑂(1) 𝑂(1)𝑂(1) 𝑂(1)

). (51)

To remove the oscillation diagonal entries in the above jump matrix as 𝑥 in the interval (0,1), the con-tour can be deformed as an opening lens by the matrix factorization. We define the following piecewiseanalytic function 𝑆(𝑧).

2.3 RHp for 𝑺

We introduce the second transformation 𝑇 → 𝑆, and 𝑆(𝑧) is defined as

𝑆(𝑧) =

⎧⎪⎪⎨⎪⎪⎩

𝑇 (𝑧), for 𝑧 outside the lens shaped region,

𝑇 (𝑧)(

1 0−𝑤(𝑧)−1𝜑(𝑧)−2𝑛 1

), for 𝑧 in the upper lens region,

𝑇 (𝑧)(

1 0𝑤(𝑧)−1𝜑(𝑧)−2𝑛 1

), for 𝑧 in the lower lens region,

(52)

where arg 𝑧 ∈ (−𝜋, 𝜋).Combining the conditions (48), (50), and (51) of the RHp for 𝑇 and the definition of 𝑆 in (52), then

𝑆 satisfies the following RHp:

(a) 𝑆(𝑧) is analytic in ℂ ⧵ {∪3𝑘=1Σ𝑘}, illustrated in Figure 1.

CHEN ET AL. 13

(b) 𝑆+(𝑧) = 𝑆−(𝑧)𝐽𝑆 for 𝑧 ∈ {∪3𝑘=1Σ𝑘}, and the jump 𝐽𝑆 is given by

𝐽𝑆 (𝑧) =(

1 0𝑤(𝑧)−1𝜑(𝑧)−2𝑛 1

), for 𝑧 ∈ Σ1

⋃Σ3, (53)

𝐽𝑆 (𝑥) =(

0 𝑤(𝑥)−𝑤(𝑥)−1 0

), for 𝑥 ∈ Σ2 = (0, 1). (54)

(c) As 𝑧 → ∞, 𝑆(𝑧) has the following behavior:

𝑆(𝑧) = 𝐼 + (𝑧−1).(d) As 𝑧 → 0, the asymptotic behavior of 𝑆(𝑧) is sector wise as follows:

𝑆(𝑧) = (1)𝑒− 𝑡

2𝑧 𝜎3

⎧⎪⎪⎨⎪⎪⎩

𝐼, outside the lens region,(1 0

−𝑤(𝑧)−1𝜑(𝑧)−2𝑛 1

), in the upper lens region,(

1 0𝑤(𝑧)−1𝜑(𝑧)−2𝑛 1

), in the lower lens region.

(55)

(e) As 𝑧 → 1, 𝑆(𝑧) has the following behavior,

𝑆(𝑧) =

⎧⎪⎪⎨⎪⎪⎩

(𝑂(1) 𝑂(1)𝑂(1) 𝑂(1)

), outside the lens,

(|𝑧 − 1|−𝛽 1|𝑧 − 1|−𝛽 1

), inside the lens.

(56)

2.4 Global parametrixFor 𝑛 → ∞, due to the property of 𝜑(𝑧) in (47), the jump matrix 𝐽𝑆 in (53) on Σ1,Σ3 converges to theidentity matrix with an exponent small error terms. Then, 𝑆(𝑧) can be approximated by a solution ofthe following RHp for 𝑃 (∞)(𝑧):

(a) 𝑃 (∞)(𝑧) is analytic in ℂ ⧵ [0, 1].(b) 𝑃 (∞)(𝑧) satisfies the jump condition,

𝑃(∞)+ (𝑥) = 𝑃 (∞)

− (𝑥)(

0 𝑤(𝑥)−𝑤(𝑥)−1 0

), for 𝑥 ∈ (0, 1).

(c) The asymptotic behavior at infinity is

𝑃 (∞)(𝑧) = 𝐼 + (𝑧−1), 𝑧 → ∞.

The solution of the above RHp for 𝑃 (∞)(𝑧) is easy to verify as follows:

𝑃 (∞)(𝑧) = 𝐷(∞)−𝜎3𝑀−1𝑎(𝑧)−𝜎3𝑀𝐷(𝑧)𝜎3 , for 𝑧 ∈ ℂ⧵[0, 1], (57)

14 CHEN ET AL.

where 𝑀 = 𝐼+𝑖𝜎1√2, 𝑎(𝑧) = ( 𝑧−1

𝑧)14 , 𝐷(∞) = 2𝛼+𝜆, and

𝐷(𝑧) =

(√𝑧 − 1 +

√𝑧√

𝑧

)𝛼(√𝑧 − 1 +

√𝑧√

𝑧 − 1

)𝛽

= 𝜑(𝑧)𝛼+𝛽2 𝑧−

𝛼

2 (𝑧 − 1)−𝛽

2 , (58)

where 𝐷(𝑧) is the Szegő function and satisfies 𝐷+(𝑥)𝐷−(𝑥) = 𝑤(𝑥) for 𝑥 ∈ (0, 1).By some computations,

𝑆(𝑧)𝑃 (∞)(𝑧)−1 = 𝐼 + (𝑒−𝑐𝑛), 𝑛 → ∞, (59)

where the error term is uniformly for 𝑧 away from the end points 0 and 1. Due to the factor 𝑎(𝑧) in(57), 𝑃 (∞)(𝑧) has fourth-root singularities at 𝑧 = 0, 1, and the jump matrices 𝑆(𝑧)𝑃 (∞)(𝑧)−1 are notuniformly close to the unit matrix in the neighborhood of 0 and 1. So, it is needed to construct the localparametrix in the neighborhoods of these points.

2.5 Local parametrix 𝑷 (𝟏)(𝒛) at 𝒛 = 𝟏We construct the local parametrix 𝑃 (1)(𝑧) in the neighborhood of 𝑧 = 1, 𝑈 (1, 𝑟) = {𝑧 ∈ ℂ ∶|𝑧 − 1| < 𝑟} for a small and fixed 𝑟. The parametrix 𝑃 (1)(𝑧) satisfies the same jump relations as 𝑆

and is a solution of the following RHp:

(a) 𝑃 (1)(𝑧) is analytic in 𝑈 (1, 𝑟) ⧵ {⋃3

𝑘=1 Σ𝑘}, see Figure 1.

(b) 𝑃 (1)(𝑧) has the same jump conditions with 𝑆(𝑧) on 𝑈 (1, 𝑟)⋂

Σ𝑘, 𝑘 = 1, 2, 3, see (53) and (54).

(c) On the boundary 𝜕𝑈 (1, 𝑟), and 𝑛 → ∞,

𝑃 (1)(𝑧)𝑃 (∞)(𝑧)−1 = 𝐼 + (𝑛−1), (60)

where the error term is uniformly for 𝑧 ∈ 𝜕𝑈 (1, 𝑟) ⧵ Σ𝑘, 𝑘 = 1, 2, 3.

(d) The asymptotic behavior of 𝑃 (1)(𝑧) at 𝑧 = 1 is the same as 𝑆(𝑧) in (56).

To convert the jumps of the 𝑃 (1)(𝑧) to constant jumps, we introduce the following transformation:

𝑃 (1)(𝑧) = 𝑃 (1)(𝑧)𝑊 (𝑧)−𝜎3𝜑(𝑧)−𝑛𝜎3 , (61)

where the function 𝑊 (𝑧) is defined by

𝑊 (𝑧) =

{𝑒𝛽𝜋𝑖

2 𝑤(𝑧)12 , for Im𝑧 > 0,

𝑒−𝛽𝜋𝑖

2 𝑤(𝑧)12 , for Im𝑧 < 0,

(62)

and 𝑊+(𝑥)𝑊−(𝑥) = 𝑤(𝑥) for 𝑥 ∈ (0, 1).Therefore, the matrix-valued function 𝑃 (1)(𝑧) satisfies the following RHp:

(a) 𝑃 (1)(𝑧) is analytic in 𝑧 ∈ 𝑈 (1, 𝑟) ⧵ {⋃3

𝑘=1 Σ𝑘}, see Figure 1.

(b) 𝑃 (1)(𝑧) satisfies the following jump relations:

𝑃(1)+ (𝑧) = 𝑃 (1)

− (𝑧)(

1 0𝑒±𝛽𝜋𝑖 1

), (63)

CHEN ET AL. 15

where + for 𝑧 ∈ 𝑈 (1, 𝑟)⋂

Σ1 and − for 𝑧 ∈ 𝑈 (1, 𝑟)⋂

Σ3.

𝑃(1)+ (𝑥) = 𝑃 (1)

− (𝑥)(

0 1−1 0

), for 𝑥 ∈ 𝑈 (1, 𝑟)

⋂(0, 1). (64)

(c) For 𝛽 > 0 and 𝑧 → 1,

𝑃 (1)(𝑧) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(|𝑧 − 1| 𝛽2 |𝑧 − 1|− 𝛽

2|𝑧 − 1| 𝛽2 |𝑧 − 1|− 𝛽

2

), outside the lens,

(|𝑧 − 1|− 𝛽

2 |𝑧 − 1|− 𝛽

2|𝑧 − 1|− 𝛽

2 |𝑧 − 1|− 𝛽

2

), inside the lens.

(65)

Obviously, the above 𝑃 (1)(𝑧) is coherent with the Bessel model RHp Ψ𝐵(𝑧) in the Appendix A, So𝑃 (1)(𝑧) = 𝐸1(𝑧)Ψ𝐵(𝑛2𝑓1(𝑧)). Combining this fact with (61), then

𝑃 (1)(𝑧) = 𝐸1(𝑧)Ψ𝐵(𝑛2𝑓1(𝑧))𝑊 (𝑧)−𝜎3𝜑(𝑧)−𝑛𝜎3 , (66)

where

𝐸1(𝑧) = 𝑃 (∞)(𝑧)𝑊 (𝑧)𝜎3𝑀−1(𝑛2𝑓1(𝑧))14𝜎3 , (67)

and 𝑓1(𝑧) = (log𝜑(𝑧))2. The above 𝜑(𝑧), 𝑃 (∞)(𝑧), and 𝑊 (𝑧) can be found in (47), (57), and (62),respectively.

Remark 1. In fact 𝑓1(𝑧) is analytic across (0,1), and 𝑓1(1) = 0. The asymptotic behavior of 𝑓1(𝑧) as𝑧 → 1,

𝑓1(𝑧) = 4(𝑧 − 1)(1 − 1

3(𝑧 − 1) + ((𝑧 − 1)2)

),

𝑓 ′1(1) = 4, and 𝑓1(𝑧) is a conformal mapping in the neighborhood of 𝑧 = 1.By the expression of 𝐸1(𝑧) in (67), 𝐸1(𝑧) is analytic across (0,1). The following is readily verified:

the match condition of (60) with 𝑃 (∞)(𝑧) and 𝑃 (1)(𝑧) in (57) and (66), respectively.

2.6 Local parametrix 𝑷 (𝟎)(𝒛) at the originIn this subsection, the local parametrix 𝑃 (0)(𝑧) in the neighborhood of 𝑧 = 0, 𝑈 (0, 𝑟) = {𝑧 ∈ ℂ ∶|𝑧| < 𝑟} for a small and fixed 𝑟. 𝑃 (0)(𝑧) shares the jump conditions as 𝑆 and solves the followingRHp:

(a) 𝑃 (0)(𝑧) is analytic in 𝑈 (0, 𝑟) ⧵ {⋃3

𝑘=1 Σ𝑘}, see Figure 1.

(b) 𝑃 (0)(𝑧) has the same jump conditions with 𝑆(𝑧) on 𝑈 (0, 𝑟)⋂

Σ𝑘, 𝑘 = 1, 2, 3, see (53) and (54).

(c) On the boundary 𝜕𝑈 (0, 𝑟), and 𝑛 → ∞,

𝑃 (0)(𝑧)𝑃 (∞)(𝑧)−1 = 𝐼 + (𝑛−2∕3), (68)

where the error term is uniformly for 𝑧 ∈ 𝜕𝑈 (0, 𝑟) ⧵ Σ𝑘, 𝑘 = 1, 2, 3.

(d) The asymptotic behavior of 𝑃 (0)(𝑧) at 𝑧 = 0 is the same as 𝑆(𝑧) in (55).

16 CHEN ET AL.

It is convenient to introduce the following transformation for transforming jumps of 𝑃 (0)(𝑧) to con-stant jumps:

𝑃 (0)(𝑧) = 𝑃 (0)(𝑧)𝑊 (𝑧)−𝜎3𝜑(𝑧)−𝑛𝜎3 , (69)

where the function 𝑊 (𝑧) is defined by

𝑊 (𝑧) =

{𝑒−

𝛼𝜋𝑖

2 𝑤(𝑧)12 , for Im𝑧 > 0,

𝑒𝛼𝜋𝑖

2 𝑤(𝑧)12 , for Im𝑧 < 0,

(70)

and 𝑊+(𝑥)𝑊−(𝑥) = 𝑤(𝑥) for 𝑥 ∈ (0, 1).It turns out that 𝑃 (0)(𝑧) satisfies the RHp as follows:

(a) 𝑃 (0)(𝑧) is analytic for 𝑧 ∈ 𝑈 (0, 𝑟) ⧵ {⋃3

𝑘=1 Σ𝑘}, see Figure 1.

(b) 𝑃 (0)(𝑧) satisfies the following jump relations:

𝑃(0)+ (𝑧) = 𝑃 (0)

− (𝑧)𝐽 , (71)

where

𝐽 =

⎧⎪⎪⎨⎪⎪⎩

(1 0

𝑒∓𝛼𝜋𝑖 1

), −as 𝑧 ∈ 𝑈 (0, 𝑟)

⋂Σ1 and + as 𝑧 ∈ 𝑈 (0, 𝑟)

⋂Σ3,(

0 1−1 0

), as 𝑧 = 𝑥 ∈ 𝑈 (0, 𝑟)

⋂Σ2.

(72)

(c) The behavior of 𝑃 (0)(𝑧) as 𝑧 → 0,

𝑃 (0)(𝑧) = (1)𝑒− 𝑡

2𝑧 𝜎3𝑊 (𝑧)𝜎3𝜑(𝑧)𝑛𝜎3

⎧⎪⎪⎨⎪⎪⎩

𝐼, outside the lens region,(1 0

−𝑒−𝛼𝜋𝑖 1

), in the upper lens region,(

1 0𝑒𝛼𝜋𝑖 1

), in the lower lens region.

(73)

For later convenience, we define auxiliary function 𝑔(𝑧) as follows:

𝑔(𝑧) = log𝜑(𝑧) = log(2𝑧 − 1 + 2

√𝑧(𝑧 − 1)

), for 𝑧 ∈ ℂ ⧵ (−∞, 1]. (74)

Obviously, 𝑔(𝑧) is analytic in ℂ ⧵ (−∞, 1]. For 𝑥 ∈ (0, 1), 𝑔+(𝑥) = −𝑔−(𝑥), by 𝜑+(𝑥)𝜑−(𝑥) = 1. More-over, 𝑔(𝑧)2 is also analytic across (0, 1), 𝑔(0)2 = −𝜋2, and 𝑔(1)2 = 0.

Let

𝑓0(𝑧) = (𝑔(𝑧) − 𝑔(0))2 =(log 𝜑(𝑧)

𝜑(0)

)2, (75)

and 𝑓0(𝑧) has the asymptotic behavior of 𝑓0(𝑧) = −4𝑧(1 + 13𝑧 + 𝑂(𝑧2)) as 𝑧 → 0, because 𝑓0(𝑧) is

analytic in ℂ ⧵ (−∞, 0] and 𝑓 ′0(0) = −4, 𝑓0(𝑧) is a conformal mapping in a neighborhood of 𝑧 = 0.

CHEN ET AL. 17

Ω1

Ω4

Ω2

Ω3

O

γ3

γ2

γ1

F I G U R E 2 Contours of ∪3𝑗=1𝛾𝑗 . and regions Ω𝑗 , 𝑗 = 1,… , 4

Using the conformal mapping 𝜉 = 𝑛2𝑓0(𝑧), we transfer the above RHp for 𝑃 (0)(𝑧) in the 𝑧-plane toa model RHp in the 𝜉-plane.

We recall a model RHp for Ψ(𝜁 ), which is obtained by Xu, Dai, and Zhao in the study of correlationkernel associated with the singularly perturbed Laguerre weight, see12 for more information.

The RHp for Ψ(𝜁 ):

(a) Ψ(𝜁 ) is analytic in ℂ ⧵⋃3

𝑗=1 𝛾𝑗 , where 𝛾𝑗 as in Figure 2.

(b) Ψ(𝜁 ) satisfies the jump conditions as follows:

Ψ+(𝜁 ) = Ψ−(𝜁 )(

1 0𝑒±𝛼𝜋𝑖 1

), +as 𝜁 ∈ 𝛾1 and − as 𝜁 ∈ 𝛾3. (76)

Ψ+(𝜁 ) = Ψ−(𝜁 )(

0 1−1 0

), for 𝜁 ∈ 𝛾2. (77)

(c) For 𝜁 → ∞, the asymptotic behavior of Ψ(𝜁 ) is

Ψ(𝜁, 𝑠) =(𝐼 +

𝐶1(𝑠)𝜁

+ (

1𝜁2

))𝜁−

14𝜎3𝑀𝑒

√𝜁𝜎3 , arg 𝜁 ∈ (−𝜋, 𝜋), (78)

where 𝑀 = 𝐼+𝑖𝜎1√2, 𝜎1 = ( 0 1

1 0 ), 𝐶1(𝑠) = ( 𝑞(𝑠) −𝑖𝑟(𝑠)𝑖𝑡(𝑠) −𝑞(𝑠) ), 𝑞(𝑠) and 𝑡(𝑠) can be expressed by 𝑟(𝑠)

and its derivatives, see proposition 1 in.12

18 CHEN ET AL.

(d) For 𝜁 → 0, the asymptotic behavior of Ψ(𝜁 ) is as follows, the domains Ω𝑗 , 𝑗 = 1,… , 4 illustratedin Figure 2.

Ψ(𝜁, 𝑠) = 𝑄(𝑠)(𝐼 + (𝜁 ))𝑒 𝑠

𝜁𝜎3𝜁

𝛼

2 𝜎3

⎧⎪⎪⎨⎪⎪⎩

𝐼, 𝜁 ∈ Ω1⋃

Ω4,(1 0

−𝑒𝛼𝜋𝑖 1

), 𝜁 ∈ Ω2,(

1 0𝑒−𝛼𝜋𝑖 1

), 𝜁 ∈ Ω3,

(79)

with the argument arg 𝜁 ∈ (−𝜋, 𝜋) and 𝑄(𝑠) given by

𝑄(𝑠) =√

1 + 𝑞′(𝑠)2

(1 𝑖𝑟(𝑠)

1+𝑞′(𝑠)𝑖(1−𝑞′(𝑠))

𝑟′(𝑠) 1

)𝐶𝜎3 , (80)

with 𝐶 a nonzero constant, which can be found in,13 see (4.4) therein. Moreover, 𝑟(𝑠) satisfies theboundary conditions for 𝛼 > 0,

𝑟(𝑠) = 1 − 4𝛼28

+ 𝑠

𝛼+ 𝑂(𝑠2), 𝑠 → 0, and 𝑟(𝑠) = 3

2𝑠23 − 𝛼𝑠

13 + 𝑂(1), 𝑠 → +∞, (81)

see (1.14) in.13

Remark 2. The analysis is still valid for −1 < 𝛼 ≤ 0, and 𝑟(𝑠) satisfies the following boundary condi-

tions: 𝑟(𝑠) = 18 + 𝑂(𝑠 ln 𝑠), 𝑠 → 0, as 𝛼 = 0; 𝑟(𝑠) = 1−4𝛼2

8 + 𝑂(𝑠𝛼+1), 𝑠 → 0, as −1 < 𝛼 < 0.

By (2) and (47), two factors of 𝑊 (𝑧)𝜎3 and 𝜑(𝑧)𝑛𝜎3 in (73) can be approximated by (−𝑧)𝛼

2 𝜎3 and(−1)𝑛𝜎3 , as 𝑧 → 0. Comparing the RHp of (71)–(73) satisfied by 𝑃 (0)(𝑧) with the model RHp for

Ψ(𝜁, 𝑠), then 𝑃 (0)(𝑧) = 𝐸0(𝑧)Ψ(𝑛2𝑓0(𝑧), 2𝑛2𝑡)𝑒− 𝜋𝑖

2 𝜎3 . By (69), 𝑃 (0)(𝑧) can be constructed as follows:

𝑃 (0)(𝑧) = 𝐸0(𝑧)Ψ(𝑛2𝑓0(𝑧), 2𝑛2𝑡)𝑒− 𝜋𝑖

2 𝜎3𝑊 (𝑧)−𝜎3𝜑(𝑧)−𝑛𝜎3 , (82)

where

𝐸0(𝑧) = 𝑃 (∞)(𝑧)𝑒𝑛 log𝜑(0)𝜎3𝑊 (𝑧)𝜎3𝑒𝜋𝑖

2 𝜎3𝑀−1(𝑛2𝑓0(𝑧)) 14𝜎3, (83)

and 𝜑(𝑧), 𝑃 (∞)(𝑧),𝑊 (𝑧), 𝑓0(𝑧) are given by (47), (57), (70), and (75), respectively.It is readily to verify that 𝐸0(𝑧) is analytic in 𝑈 (0, 𝑟) with a weak singularity at 𝑧 = 0, and the

following match condition:

𝑃 (0)(𝑧)𝑃 (∞)(𝑧)−1 = 𝐼 + (1𝑛

), (84)

follows from (57) and (82), and the error term is uniformly for 𝑧 ∈ 𝜕𝑈 (0, 𝑟) ⧵ Σ𝑘, 𝑘 = 1, 2, 3.It is convenient to scale the variables 𝑧 and 𝑡 as follows

𝜉 = 𝑛2𝑓0(𝑧) ≈ −4𝑛2𝑧, as 𝑧 → 0; and 𝜍 = 2𝑛2𝑡. (85)

It's interesting to study the behavior of Ψ(𝜉, 𝜍), as 𝜍 → 0 and 𝜍 → ∞. These can also be found in,12 forthe completeness of this paper and convenience later, we restate these as follows, although the scalingprocess of variables and the background are totally different.

CHEN ET AL. 19

2.6.1 Approximation of 𝚿(𝝃, 𝝇) as 𝝇 → 𝟎In this subsection, we show that Ψ(𝜉, 𝜍) can be approximated by a modified Bessel model RHp as

𝜍 → 0. From (79), the singular factor 𝑒𝜍

𝜉𝜎3 in the model of Ψ(𝜉, 𝜍) becomes a regular part as 𝜍 → 0 for

bounded 𝜉. With the aid of solution (A.1) of the Bessel model RHp in Appendix A, in our case, wedefine

𝐺(𝜉, 𝜍) = 𝜋12𝜎3

⎛⎜⎜⎜⎝𝜉−

𝛼

2 𝐼𝛼

(√𝜉)

𝑖

𝜋𝜉

𝛼

2𝐴(√𝜉)

𝑖𝜋𝜉1−𝛼2 𝐼 ′𝛼

(√𝜉)

−𝜉𝛼+12 𝐵(√𝜉)

⎞⎟⎟⎟⎠(1 𝑓 (𝜉, 𝜍)0 1

)𝜉

𝛼

2 𝜎3𝑒𝜍

𝜉𝜎3 (−1)𝑛𝜎3𝐽 , (86)

where (−1)𝑛𝜎3 follows from 𝜑(𝑧) → −1 as 𝑧 → 0, 𝐽 is in (73), and for 𝛼 > 0, 𝛼 ∉ ℕ,

𝐴(𝑧) = 𝜋

2 sin(𝛼𝜋)𝐼−𝛼(𝑧), 𝐵(𝑧) = 𝜋

2 sin(𝛼𝜋)𝐼 ′−𝛼(𝑧);

for 𝛼 ∈ ℕ,

⎧⎪⎨⎪⎩𝐴(𝑧) = 1

2

(𝑧

2

)−𝛼 𝛼−1∑𝑗=0

𝐴1(𝑗)(𝑧2

4

)𝑗+ 1

2

(−𝑧2

)𝛼 ∞∑𝑗=0

𝐴2(𝑗)(𝑧2

4

)𝑗,

𝐵(𝑧) = 𝐴′(𝑧) + (−1)𝛼+1𝑧

𝐼𝛼(𝑧),

and 𝐴1(𝑘) =(−1)𝑘(𝛼−𝑘−1)!

Γ(𝑘+1) , 𝐴2(𝑘) =𝜓(𝑘+1)+𝜓(𝑘+𝛼+1)Γ(𝑘+1)Γ(𝑘+𝛼+1) , 𝜓(𝑥) = Γ′(𝑥)

Γ(𝑥) , these can be verified from the for-

mulas of (10.25.2),(10.27.4), and (10.31.1) in.26 Moreover,

𝑓 (𝜉, 𝜍) = − 14𝜋 sin(𝛼𝜋) ∫Γ

𝜏𝛼𝑒2𝜍𝜏

𝜏 − 𝜉𝑑𝜏 = 𝜉𝛼

2 sin(𝛼𝜋)𝑖(1 + 𝑂(𝜍𝜉−1)

), 𝛼 > 0, 𝛼 ∉ ℕ;

𝑓 (𝜉, 𝜍) = (−1)𝛼+1

2𝜋2 ∫Γ𝜏𝛼𝑒

2𝜍𝜏 ln(√

𝜏

2

)𝜏 − 𝜉

𝑑𝜏 =(−1)𝛼𝜉𝛼 ln

(√𝜉

2

)𝑖𝜋

(1 + 𝑂(𝜍𝜉−1) + 𝑂(𝜍𝛼+1 ln 𝜍𝜉−1)

),

𝛼 ∈ ℕ;

the above integration contour Γ and for the details of estimation for 𝑓 (𝜉, 𝜍), we refer section 5 in 12.With the aid of the above preparations, Ψ(𝜉, 𝜍) can be approximated by 𝐺(𝜉, 0) and 𝐺(𝜉, 𝜍) in the

following:

𝑅0𝑆 ={Ψ(𝜉, 𝜍)𝐺(𝜉, 0)−1, for |𝜉| > 𝑟,

Ψ(𝜉, 𝜍)𝐺(𝜉, 𝜍)−1, for |𝜉| < 𝑟,(87)

where 𝐺(𝜉, 0) is the solution of the usual Bessel model RHp (see Appendix A) except the factor (−1)𝑛𝜎3in (86) and the jump on the clockwise circle |𝜉| = 𝑟 satisfies

𝐽𝑅0𝑆={𝐼 + (𝜍) + (𝜍𝛼+1), for 𝛼 > 0, 𝛼 ∉ ℕ,𝐼 + (𝜍) + (𝜍𝛼+1 ln 𝜍), for 𝛼 ∈ ℕ, (88)

20 CHEN ET AL.

Π1

Π4

Π2

Π3

Π5

Π6

−I O

γ3 γ3

γ2

γ1

γ1

F I G U R E 3 Contours of 𝔾(𝜆). and regions Π𝑗 , 𝑗 = 1,… , 6

as 𝜍 → 0, and the above error terms hold uniformly for |𝜉| = 𝑟. In the spirt of the technique of estimationfor Cauchy operator norm in,6,27 one has

𝑅0𝑆 ={𝐼 + (𝜍) + (𝜍𝛼+1), for 𝛼 > 0, 𝛼 ∉ ℕ,𝐼 + (𝜍) + (𝜍𝛼+1 ln 𝜍), for 𝛼 ∈ ℕ, (89)

where the error terms are uniformly for |𝜉| = 𝑟.

2.6.2 Approximation of 𝚿(𝝃, 𝝇) as 𝝇 → ∞By the similar technique in,12 considering the transformation

Ψ(𝜉, 𝜍) exp

(−(𝜉 + 𝜍

23) 3

2𝜎3∕𝜉

),

and the above exponent factor is to counteract the singular factor 𝑒𝜍

𝜉𝜎3 in the model of Ψ(𝜉, 𝜍) as 𝜍 → ∞

and 𝜉 → 0. For this purpose, 𝜉 is scaled as 𝜍2∕3𝜆, then one defines 𝐺(𝜆) as follows:

𝔾(𝜆) ∶= 𝜍16𝜎3Ψ(𝜍2∕3𝜆, 𝜍)𝑒−𝜍

13 𝜃(𝜆)𝜎3 , 𝜃(𝜆) = (𝜆 + 1)

32

𝜆. (90)

𝔾(𝜆) satisfies the RHp as follows:

(a) 𝔾(𝜆) is analytic in ℂ⧵⋃3

𝑗=1 𝛾𝑗 , where 𝛾𝑗 is in Figure 3.

CHEN ET AL. 21

(b) 𝔾(𝜆) fulfills the following jump conditions:

𝔾+(𝜆) = 𝔾−(𝜆)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1 0

𝑒∓𝛼𝜋𝑖−2𝜍13 𝜃(𝜆) 1

), −as 𝜆 ∈ 𝛾1 and + as 𝜆 ∈ 𝛾3,

⎛⎜⎜⎝0 𝑒2𝜍

13 𝜃(𝜆)

−𝑒−2𝜍13 𝜃(𝜆) 0

⎞⎟⎟⎠, as 𝜆 ∈ (−1, 0),(0 1

−1 0

), as 𝜆 ∈ (−∞, 0).

(c) For 𝜆 → ∞,

𝔾(𝜆) = 𝜆−14𝜎3𝑀

(𝐼 + 𝜍

−13 𝑟(𝜍)2

(1 −𝑖−𝑖 −1

)𝜆−

12 + (𝜆−1)), arg 𝜁 ∈ (−𝜋, 𝜋),

where 𝑀 = 𝐼+𝑖𝜎1√2

, and 𝑟(𝜍) contains in 𝐶1, see (78).

(d) For 𝜆 → 0,

𝔾(𝜆) = (1)𝜆𝛼

2 𝜎3

⎧⎪⎨⎪⎩(

1 0

𝑒∓𝛼𝜋𝑖−2𝜍13 𝜃(𝜆) 1

), −as 𝜆 ∈ 𝜋6 and + as 𝜆 ∈ Π5,

𝐼, as 𝜆 ∈ Π1 ∪ Π4.

Note that 𝜍 is a large parameter, it is need to introduce the transformation as follows:

𝐻(𝜆) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

𝔾(𝜆)

(1 0

−𝑒−𝛼𝜋𝑖−2𝜍13 𝜃(𝜆) 1

), 𝜆 ∈ Π5,

𝔾(𝜆)

(1 0

𝑒𝛼𝜋𝑖−2𝜍13 𝜃(𝜆) 1

), 𝜆 ∈ Π6,

𝔾(𝜆), other domains.

Then, 𝐻(𝜆) satisfies the following RHp:

(a) 𝐻(𝜆) is analytic in ℂ ⧵ ��1 ∪ ��3 ∪ (−∞,−1) ∪ (−1, 0), see Figure 3.

(b) 𝐻(𝜆) fulfills the following jump conditions

𝐻+(𝜆) = 𝐻−(𝜆)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1 0

𝑒∓𝛼𝜋𝑖−2𝜍13 𝜃(𝜆) 1

), −as 𝜆 ∈ ��1 and + as 𝜆 ∈ ��3,(

𝑒𝛼𝜋𝑖 𝑒2𝜍13 𝜃(𝜆)

0 𝑒−𝛼𝜋𝑖

), as 𝜆 ∈ (−1, 0),(

0 1−1 0

), as 𝜆 ∈ (−∞, 0).

(c) For 𝜆 → ∞, the asymptotic behavior of 𝐻(𝜆) is the same as 𝔾(𝜆).

(d) For 𝜆 → 0,𝐻(𝜆) = (1)𝜆𝛼

2 𝜎3 .

22 CHEN ET AL.

The global parametric of 𝐻(𝜆) is given by

𝑁(𝜆) = (𝜆 + 1)−14𝜎3𝑀

(√𝜆 + 1 + 1√𝜆 + 1 − 1

)− 𝛼

2 𝜎3

, (91)

where 𝑁(𝜆) satisfies the following jump conditions:

𝑁+(𝜆) = 𝑁−(𝜆)⎧⎪⎨⎪⎩𝑒𝛼𝜋𝑖𝜎3 , as 𝜆 ∈ (−1, 0),(

0 1−1 0

), as 𝜆 ∈ (−∞, 0).

The local parametric can be constructed in terms of the Airy model PHp (see the Appendix B) asfollows:

𝐿(𝜆) = 𝐸01(𝜆)Ψ𝐴

(𝜍

29 𝑓01(𝜆)

)𝑒−𝜍

13 𝜃(𝜆)𝜎3𝑒±

𝛼𝜋𝑖

2 𝜎3 , ±Im𝜆 > 0 and 𝜆 ∈ 𝑈 (−1, 𝑟),

where

𝐸01(𝜆) = 𝑁(𝜆)𝑒∓𝛼𝜋𝑖

2 𝜎3𝑀−1(𝜍

29 𝑓01(𝜆)

) 14𝜎3

, for ± Im𝜆 > 0, and 𝜆 ∈ 𝑈 (−1, 𝑟), (92)

𝑓01(𝜆) =(−32𝜃(𝜆)) 2

3, 𝜃(𝜆) = (𝜆 + 1)

32

𝜆, (93)

To analysis the error terms, it is need to introduce the following transformation:

𝑅01(𝜆) ={𝐻(𝜆)𝑁(𝜆)−1, |𝜆 + 1| > 𝑟,

𝐻(𝜆)𝐿(𝜆)−1, |𝜆 + 1| < 𝑟.

For 𝜆 ∈ ��1, ��1 is illustrated in Figure 4, one has,

𝐽𝑅01= 𝑅01−(𝜆)

−1𝑅01+(𝜆) = 𝑁(𝜆)

(1 0

𝑒𝛼𝜋𝑖−2𝜍13 𝜃(𝜆) 1

)𝑁(𝜆)−1 = 𝐼 +

(𝑒−𝑐𝜍

13),

where 𝑁(𝜆) is in (91), Re𝜃(𝜆) > 0 for arg(𝜆 + 1) = 2𝜋3 and 𝑐 is a positive constant. The above estima-

tion is also valid for 𝜆 ∈ ��3.For 𝜆 ∈ (−1 + 𝑟, 0), see Figure 4, one finds,

𝐽𝑅01= 𝑁−(𝜆)

(1 𝑒𝛼𝜋𝑖+2𝜍

13 𝜃(𝜆)

0 1

)𝑁−(𝜆)−1 = 𝐼 +

(𝑒−𝑐𝜍

13),

where 𝑐 is a positive constant.For 𝜆 ∈ 𝜕𝑈 (−1, 𝑟), see Figure 4, one finds,

𝐽𝑅01= 𝐿(𝜆)𝑁(𝜆)−1 = 𝐼 +

𝑁(𝜆)𝑒∓𝛼𝜋𝑖

2 𝜎3𝐴1𝑒± 𝛼𝜋𝑖

2 𝜎3𝑁−1(𝜆)

(𝑓01(𝜆))32

𝜍−1

3 + (𝜍−23 ), ±Im𝜆 > 0,

= 𝐼 + (𝜍−13 ).

CHEN ET AL. 23

O−1 + r

γ3

γ1

F I G U R E 4 The contour Σ𝑅01of 𝑅01

Then, the jump 𝐽𝑅01satisfies the following behavior as 𝜍 → ∞,

𝐽𝑅01(𝜆) =

⎧⎪⎨⎪⎩𝐼 +

(𝑒−𝑐𝜍

13), Σ𝑅01

⧵ 𝜕𝑈 (−1, 𝑟),

𝐼 + (𝜍−13), 𝜕𝑈 (−1, 𝑟).

By the above behavior of 𝐽𝑅01and

𝑅01(𝑧) = 𝐼 + 12𝜋𝑖 ∫Σ𝑅01

𝑅01−(𝜏)(𝐽𝑅01

(𝜏) − 𝐼)

𝜏 − 𝑧𝑑𝜏, 𝑧 ∉ Σ𝑅01

,

then, we have

𝑅01(𝜆) = 𝐼 + (𝜍−13), (94)

where the error term is uniformly for 𝜆. Here, 𝜆 is away from the contour Σ𝑅01, see Figure 4.

Following the technique presented in,4 see section 8 therein, and the jump 𝐽𝑅01on 𝜕𝑈 (−1, 𝑟), one

obtains,

𝑅01(𝜆) = 𝐼 +(

𝑀1𝜆 + 1

+𝑀2

(𝜆 + 1)2

)𝜍−1

3 + (𝜍−23), (95)

where 𝑀1 =772 (

0 0𝑖 0 ) and 𝑀2 =

572 (

0 𝑖

0 0 ).

24 CHEN ET AL.

With the aid of the above analysis, one finds,

Ψ(𝜍23 𝜆, 𝜍) = 𝜍

−16𝜎3𝑅01(𝜆)𝐸01(𝜆)Ψ𝐴

(𝜍

29 𝑓01(𝜆)

)𝑒±

𝛼𝜋𝑖

2 𝜎3 , (96)

where ±Im𝜆 > 0, and 𝜆 ∈ 𝑈 (−1, 𝑟).From (82), (96), and (94), one is easy to verify the matching condition in (68).Comparing the above Equations (96) and (91) with (6.25) and (6.12) in,12 there are no lower trian-

gular constant matrices because that

(𝐼 + 𝜇𝜎−

)𝑥−

14𝜎3𝑀 = 𝑥−

14𝜎3𝑀

(𝐼 + (𝑥−1

2))

, 𝑥 → ∞,

with 𝜇 is a nonzero constant, see also,28 p. 11.

Remark 3. The singular factor 𝑒−𝑡∕𝑥, 𝑡 > 0, introduces an infinitely strong zero at 0 and has the effectof pushing the left end point of the equilibrium density away from 0 at a slow speed. If 𝐚 is left endpoint of the support of the density, then for 𝑡 > 0, and large 𝑛,

𝐚 = 𝑡23

(2(2𝑛 + 𝛼 + 𝛽))23

(1 + 𝑂(𝑛−2∕3)) = 𝜍23

4𝑛2(1 + 𝑂(𝑛−2)

), 𝜍 = 2𝑛2𝑡, (97)

which follows from a combination of (2.25) and (2.26) in.29 For the shifted Jacobi weight 𝑤(𝑥) =𝑥𝛼(1 − 𝑥)𝛽 , 𝐚 = 𝛼2∕3

(2𝑛+𝛼+𝛽)2 (1 + 𝑂(𝑛−2)).For 𝜍 → ∞, one recalls the conformal mapping 𝜉 = 𝑛2𝑓0(𝑧) = −4𝑛2𝑧(1 + 𝑂(𝑧)) in (85), rescaling

𝜉 = 𝜍23 𝜆 in (90), one finds 𝜉 = −𝜍

23 as 𝜆 = −1. Obviously, 𝜉 = 𝑛2𝑓0(𝐚) = −𝜍

23 and 𝐚 is in (97), and

this means the hard-edge of the spectrum turns out to be a soft edge behavior. Hence, the Airy modelappears in the neighborhood of 𝜆 = −1.

For 𝜍 → 0, the hard-edge of the spectrum keeps for 𝜉 = 𝑛2𝑓0(𝐚) = −𝜍23 → 0, so the Bessel model

appears in the neighborhood of origin and the minus sign turns its orientation.

2.7 RHp for 𝑹

With the aid of the global parametrix 𝑃 (∞) and local parametrixes in the neighborhoods of 𝑈 (0, 𝑟) and𝑈 (1, 𝑟), 𝑅(𝑧) is given by

𝑅(𝑧) =⎧⎪⎨⎪⎩𝑆(𝑧)𝑃 (∞)(𝑧)−1, 𝑧 ∈ ℂ ⧵ 𝑈 (1, 𝑟)

⋃𝑈 (0, 𝑟)

⋃∑𝑆,

𝑆(𝑧)𝑃 (1)(𝑧)−1, 𝑧 ∈ 𝑈 (1, 𝑟) ⧵∑

𝑆,

𝑆(𝑧)𝑃 (0)(𝑧)−1, 𝑧 ∈ 𝑈 (0, 𝑟) ⧵∑

𝑆 .

(98)

𝑅(𝑧) satisfies the RHp as follows:(𝑎)𝑅(𝑧) is analytic in ℂ ⧵ Σ𝑅, see Figure 5.(𝑏)𝑅(𝑧) satisfies the jump condition

𝑅+(𝑧) = 𝑅−(𝑧)𝐽𝑅(𝑧), (99)

CHEN ET AL. 25

0

Σ1

Σ3

1

F I G U R E 5 The contour for 𝑅(𝑧)

where

𝐽𝑅(𝑧) =⎧⎪⎨⎪⎩𝑃 (∞)(𝑧)𝐽𝑆 (𝑧)𝑃 (∞)−1(𝑧), 𝑧 ∈ Σ1

⋃Σ3,

𝑃 (0)(𝑧)𝑃 (∞)−1(𝑧), 𝑧 ∈ 𝜕𝑈 (0, 𝑟),𝑃 (1)(𝑧)𝑃 (∞)−1(𝑧), 𝑧 ∈ 𝜕𝑈 (1, 𝑟),

(100)

and 𝐽𝑆 (𝑧) denotes the jump matrices in (53).(𝑐) For 𝑧 → ∞, 𝑅(𝑧) = 𝐼 + (𝑧−1).By (59), (100) and the matching conditions (60), (68), one finds,

𝐽𝑅(𝑧) =⎧⎪⎨⎪⎩𝐼 + (𝑛−1), 𝑧 ∈ 𝜕𝑈 (1, 𝑟),𝐼 + (𝑛−2

3), 𝑧 ∈ 𝜕𝑈 (0, 𝑟),

𝐼 + (𝑒−𝑐𝑛), 𝑧 ∈ Σ1⋃

Σ3,

(101)

where 𝑐 is a positive constant, and the error term is uniformly for 𝑧 ∈ ℂ ⧵ Σ𝑅. Then, one finds

‖𝐽𝑅(𝑧) − 𝐼‖𝐿2∩𝐿∞(Σ𝑅) = (𝑛−23). (102)

Furthermore,

𝑅(𝑧) = 𝐼 + 12𝜋𝑖 ∫Σ𝑅

𝑅−(𝜏)(𝐽𝑅(𝜏) − 𝐼

)𝜏 − 𝑧

𝑑𝜏, 𝑧 ∉ Σ𝑅, (103)

by the method and procedure of estimation Cauchy operator norm as show in,6,27 one has

𝑅(𝑧) = 𝐼 + (𝑛−23), (104)

uniformly for 𝑧 ∈ ℂ.

3 THE PROOFS OF THEOREMS

It starts with the proof of Theorem 1 concerning the strong asymptotic expansion of the monic orthog-onal polynomials associated with the Pollaczek–Jacobi type weight.

26 CHEN ET AL.

Proof of Theorem 1. With the aid of the formulas of (46), (52), (57), and (98) for 𝑧 ∈ ℂ ⧵ [0, 1],

(𝑌11(𝑧)𝑌21(𝑧)

)= 4−𝑛𝜎3𝑅(𝑧)𝑃 (∞)(𝑧)𝑒

𝑡

2𝑧 𝜎3𝜑(𝑧)𝑛𝜎3(10

)

= 4−𝑛𝜎3𝑅(𝑧)

(2−(𝛼+𝛽) 𝑎(𝑧)

−1+𝑎(𝑧)2 𝐷(𝑧)𝑒

𝑡

2𝑧 𝜑(𝑧)𝑛

𝑖2(𝛼+𝛽) 𝑎(𝑧)−𝑎(𝑧)−1

2 𝐷(𝑧)𝑒𝑡

2𝑧 𝜑(𝑧)𝑛

),

where 𝜑(𝑧) and 𝐷(𝑧) are given by (47), (58), respectively. By the facts 𝑎(𝑧)−1 + 𝑎(𝑧) =𝜑(𝑧)

12 (𝑧 − 1)−

14 𝑧−

14 , 𝐷(𝑧) = 𝜑(𝑧)

𝛼+𝛽2 (𝑧 − 1)−

𝛽

2 𝑧−𝛼

2 , 𝑌11(𝑧) = 𝜋𝑛(𝑧), and after some calculations, oneobtains (10). ■

We continuous to consider the strong asymptotic expansion for the monic orthogonal polynomials𝜋𝑛(𝑧) for 𝑧 ∈ (0, 1).

Proof of Theorem 2. Let 𝕂 be a compact subset of (0,1). By the formulas (46), (52), (57), and (98),and taking inverse transformations from 𝑌 (𝑧) to 𝑅(𝑧), one obtains

(𝑌11(𝑥)𝑌21(𝑥)

)= 4−𝑛𝜎3𝑅(𝑧)2−(𝛼+𝛽)𝜎3

⎛⎜⎜⎝𝑎+(𝑥)−1+𝑎+(𝑥)

2 𝑖𝑎+(𝑥)−1−𝑎+(𝑥)

2𝑖𝑎+(𝑥)−𝑎+(𝑥)−1

2𝑎+(𝑥)−1+𝑎+(𝑥)

2

⎞⎟⎟⎠(

𝐷+(𝑥)𝜑+(𝑥)𝑛𝑒𝑡

2𝑥

𝐷+(𝑥)−1𝑤(𝑥)−1𝜑+(𝑥)−𝑛𝑒𝑡

2𝑥

).

From (57) and (58), one finds the following formulas:

𝑎+(𝑥)−1 + 𝑎+(𝑥) =𝑒𝑖(arccos

√𝑥−𝜋∕4)

𝑥1∕4(1 − 𝑥)1∕4, 𝑖(𝑎+(𝑥)−1 − 𝑎+(𝑥)) =

𝑒𝑖(− arccos√𝑥+𝜋∕4)

𝑥1∕4(1 − 𝑥)1∕4, (105)

𝐷+(𝑥) =𝑒𝑖((𝛼+𝛽) arccos

√𝑥−𝛽𝜋∕2)

𝑥𝛼∕2(1 − 𝑥)𝛽∕2, and 𝜑+(𝑥)𝑛 = 𝑒𝑖2𝑛 arccos

√𝑥, (106)

then the above matrix function becomes(𝑌11(𝑥)𝑌21(𝑥)

)= 1√

𝑤𝑝𝐽(𝑥, 𝑡)𝑥

14 (1 − 𝑥)

14

4−𝑛𝜎3𝑅(𝑧)2−(𝛼+𝛽)𝜎3

×⎛⎜⎜⎝

cos((2𝑛 + 𝛼 + 𝛽 + 1) arccos

√𝑥 − 𝛽𝜋

2 − 𝜋

4

)−𝑖 cos

((2𝑛 + 𝛼 + 𝛽 − 1) arccos

√𝑥 − 𝛽𝜋

2 − 𝜋

4

)⎞⎟⎟⎠.Moreover, 𝑅(𝑥) = 𝐼 + (𝑛−1) and the error term is uniform for 𝑥 ∈ 𝕂, then one is easy toverify (11). ■

We show that the strong asymptotic expansion for the monic polynomials is expressed by the Szegőfunction and the Bessel function of order 𝛽 on the interval (1 − 𝑟, 1), 𝑟 is small and positive.

CHEN ET AL. 27

Proof of Theorem 3. Taking a list of inverse transformations of 𝑌 (𝑧) → 𝑇 (𝑧) → 𝑆(𝑧) → 𝑅(𝑧), see(46), (52), (57), and (98), the parametrix 𝑃 (1)(𝑧) is in (66), 𝑊 (𝑧) is in (62), and Ψ𝐵(𝑧) is given by(A.1) in Appendix A, then one finds(𝑌11(𝑥)𝑌21(𝑥)

)= 4−𝑛𝜎3𝑅(𝑥)𝐸1(𝑥)Ψ𝐵−

(𝑛2𝑓1(𝑥))𝑊−(𝑥)−𝜎3𝜑−(𝑥)−𝑛𝜎3(

1 0−𝑤(𝑥)−1𝜑−(𝑥)−2𝑛 1

)𝜑−(𝑥)𝑛𝜎3𝑒

𝑡

2𝑥 𝜎3

(10

)

= 4−𝑛𝜎3𝑅(𝑥)𝐸1(𝑥)⎛⎜⎜⎝

𝑒𝑖𝛽𝜋

2 𝐼𝛽(√𝑛2𝑓1(𝑥))−

𝑖𝑒𝑖𝛽𝜋

2√𝑛2𝑓1(𝑥)−𝐼 ′𝛽(

√𝑛2𝑓1(𝑥))−

⎞⎟⎟⎠√𝜋(𝑤𝑝𝐽

(𝑥, 𝑡))−12

= 4−𝑛𝜎3𝑅(𝑥)𝐸1(𝑥)

(𝐽𝛽(2𝑛 arccos

√𝑥)

𝑖2𝑛 arccos√𝑥𝐽 ′

𝛽(2𝑛 arccos

√𝑥)

)√𝜋(𝑤𝑝𝐽

(𝑥, 𝑡))−12 , (107)

where 𝐽𝛽 is the Bessel function of order 𝛽, and we have used the facts

𝜑−(𝑥) = 𝑒−2𝑖 arccos√𝑥,

√𝑛2𝑓1(𝑥)− =

√𝑛2(ln𝜑(𝑥))2− = −2𝑛𝑖 arccos

√𝑥, (108)

and the formula 𝐼𝜃(𝑧)𝑒𝑖𝜋𝜃∕2 = 𝐽𝜃(𝑧𝑒𝑖𝜋

2 ) with arg 𝑧 ∈ (−𝜋, 𝜋∕2], see.26

To find out 𝐸1(𝑥), we compute 𝑃 (∞)+ (𝑥). By the definition of 𝑃 (∞)(𝑧) in (57) and some facts in (105)

and (106), it is easy to verify that

𝑃(∞)+ (𝑥) = 𝑒−

𝜋𝑖

4

2𝑥14 (1 − 𝑥)

14

2−(𝛼+𝛽)𝜎3(

𝑒𝑖𝜃1 𝑖𝑒−𝑖𝜃1

−𝑖𝑒𝑖𝜃2 𝑒−𝑖𝜃2

)𝑒−

𝛽𝜋𝑖

2 𝜎3𝑥−𝛼

2 𝜎3 (1 − 𝑥)−𝛽

2 𝜎3 , (109)

where 𝜃1 = (𝛼 + 𝛽 + 1) arccos√𝑥 and 𝜃2 = (𝛼 + 𝛽 − 1) arccos

√𝑥.

From (62) and (𝑛2𝑓1(𝑥))14𝜎3+ = 𝑒

𝜋𝑖

4 𝜎3 (2𝑛 arccos√𝑥)

12𝜎3 , it is readily found that

𝑊+(𝑥)𝜎3𝑀−1(𝑛2𝑓1(𝑥))14𝜎3+ = 𝑥

𝛼

2 𝜎3 (1 − 𝑥)𝛽

2 𝜎3𝑒𝛽𝑖

2 𝜎3𝑀−1𝑒𝜋𝑖

4 𝜎3 (2𝑛 arccos√𝑥)

12𝜎3 . (110)

Given the combination of the definition of 𝐸1(𝑧) in (67), (109), and (110), then one obtains

𝐸1(𝑥) =𝑒−

𝜋𝑖

4

2𝑥14 (1 − 𝑥)

14

2−(𝛼+𝛽)𝜎3(

cos 𝜃1 sin 𝜃1−𝑖 cos 𝜃2 −𝑖 sin 𝜃2

)𝑒𝜋𝑖

4 𝜎3 (2𝑛 arccos√𝑥)

12𝜎3 . (111)

Inserting (111) into (107), 𝑌11(𝑥) = 𝜋𝑛(𝑥), 𝑅(𝑥) = 1 + (𝑛−1) uniformly for 𝑥 in compact subsetsof (1 − 𝑟, 1), then one follows (12). ■

By the fact that the local parametric can be approximated by the Bessel model RHp as 𝜍 = 2𝑛2𝑡 → 0,we prove that for 𝑥 ∈ (0, 𝛿), 𝛿 is a small and positive constant, the strong asymptotic expansions ofmonic polynomials involve the Bessel function of order 𝛼 for 𝑥 ∈ (0, 𝛿).

28 CHEN ET AL.

Proof of Theorem 4. By a series of formulas (46), (52), (57), and (98), from 𝑌 (𝑧) to 𝑅(𝑧), one obtains

(𝑌11(𝑥)𝑌21(𝑥)

)= 𝑌+(𝑥)

(10

)= 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)Ψ−(𝜉, 𝜍)𝑒

(𝛼−1)𝜋𝑖2 𝜎3

(1 01 1

)(𝑤𝑝𝐽

(𝑥, 𝑡))−12𝜎3

(10

), (112)

where 𝐸0(𝑧) is in (83), 𝜉 = 𝑛2𝑓0(𝑧) (see (75)), and 𝜍 = 2𝑛2𝑡.As 𝜍 → 0,Ψ(𝜉, 𝜍) can be uniformly approximated by a modified Bessel model RHp, see (86), (87),

and (89), then

(𝑌11(𝑥)𝑌21(𝑥)

)= 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)𝑅0𝑆 (𝜍)𝐺−(𝜉, 𝜍)

(𝑒𝑖𝜋(𝛼−1)

2

𝑒−𝑖𝜋(𝛼−1)

2

)(𝑤𝑝𝐽

(𝑥, 𝑡))−12

= 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)𝑅0𝑆 (𝜍)𝜋12𝜎3

((−1)𝑛𝑒

𝜍

𝜉− 𝑒𝑖𝜋(𝛼−1)

2 𝐼𝛼(√𝜉)−

(−1)𝑛𝑖𝜋𝑒𝜍

𝜉− 𝑒𝑖𝜋(𝛼−1)

2√𝜉−𝐼

′𝛼(√𝜉)−

)(𝑤𝑝𝐽

(𝑥, 𝑡))−12

= 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)𝑅0𝑆 (𝜍)

(−𝑖𝐽𝛼(

√|𝜉|)√|𝜉|𝐽 ′𝛼(√|𝜉|)

)(−1)𝑛

√𝜋𝑒

𝜍

𝜉− (𝑤𝑝𝐽(𝑥, 𝑡))−

12 , (113)

where it has been used the similar technique in (107), and the facts that

𝜉− = 𝑛2𝑓0−(𝑥) = 𝑒−𝑖𝜋𝑛2(𝜋 − 2 arccos√𝑥)2, and

√|𝜉| = 𝑛(𝜋 − 2 arccos√𝑥), (114)

follow from (75).By the definition of 𝐸0(𝑧) in (83), 𝑃 (∞)

+ (𝑥) and 𝑊 (𝑧) are given by (109) and (70) separately. Onefinds,

𝐸0(𝑥) =(−1)𝑛𝑒−𝑖𝜋∕4√2𝑥1∕4(1 − 𝑥)1∕4

2−(𝛼+𝛽)𝜎3(− sin 𝜃3 cos 𝜃3𝑖 sin 𝜃4 −𝑖 cos 𝜃4

)𝑒−

𝑖𝜋

4 𝜎3(𝑛(𝜋 − 2 arccos

√𝑥)) 1

2𝜎3, (115)

where 𝜃3 and 𝜃4 can be found in (15).Substituting (114) and (115) into (113), and 𝑌11(𝑥) = 𝜋𝑛(𝑥), one obtains (14). Moreover,𝑅(𝑥) = 𝐼 +

(𝑛−1) uniformly for 𝑥 ∈ (0, 𝛿) with a small and fixed 0 < 𝛿 ≪ 1∕2 and 𝑅0𝑆 (𝜍) = 𝐼 + (𝜍) uniformlyfor 𝑥 ∈ 𝔻1 as 𝜍 = 2𝑛2𝑡 → 0, and 𝔻1 = {𝑥|𝑛(𝜋 − 2 arccos

√𝑥) ∈ (0,∞), 𝑛 → ∞}. ■

The local parametric used a model RHp for Ψ(𝜁 ) in Section 2.6, and this model can be approximatedby the Airy model RHp as 𝜍 = 2𝑛2𝑡 → ∞. We prove that the strong asymptotic expansion involves Airyfunction for 𝑥 ∈ (0, 𝜀).

Proof of Theorem 5. It is convenient to start with (112), and we rewrite it as follows:

(𝑌11(𝑥)𝑌21(𝑥)

)= 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)Ψ−(𝜉, 𝜍)𝑒

(𝛼−1)𝜋𝑖2 𝜎3

(1 01 1

)(𝑤𝑝𝐽

(𝑥, 𝑡))−12𝜎3

(10

), (116)

where 𝐸0(𝑧) is in (83), 𝜉 = 𝑛2𝑓0(𝑧) see (75), and 𝜍 = 2𝑛2𝑡.

CHEN ET AL. 29

As 𝜍 → ∞, 𝜉 is scaled by 𝜍2∕3𝜆, then Ψ(𝜍2∕3𝜆, 𝜍) can be uniformly approximated by the Airy modelRHp for 𝜍 → ∞, see (96), then one obtains,

(𝑌11(𝑥)𝑌21(𝑥)

)= 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)𝜍

−16𝜎3𝑅01(𝜍)𝐸01(𝜆)Ψ𝐴−(𝜍

29 𝑓01(𝜆))(−1)𝑛𝜎3

(−𝑖𝑖

)(𝑤𝑝𝐽

(𝑥, 𝑡))−12

= 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)𝜍−1

6𝜎3𝑅01𝐸01(𝜆)

(−𝑖𝐴𝑖−(𝜍

29 𝑓01(𝜆))

−𝐴𝑖′−(𝜍29 𝑓01(𝜆))

)(−1)𝑛

√2𝜋(𝑤𝑝𝐽

(𝑥, 𝑡))−12 ,

(117)

the above second equality has been used the solution of the Airy model RHp, see (B.3) in Appendix B.The factor (−1)𝑛𝜎3 is to counteract 𝜑(𝑧)𝑛𝜎3 in (73) as 𝑧 → 0.

By the expressions of 𝑅01 given by (95), one finds,

𝜍−1

6𝜎3𝑅01𝜍16𝜎3 =

(𝐼 + (𝜍−1∕3))( 1 0

−𝑖𝑏 1

), as 𝜍 → ∞, (118)

where 𝑏 is given by (17), the matrix 𝐴1 =148𝜎3 +

𝑖

8𝜎1 is given by (B.2) in Appendix B, 𝜃(𝜆) is in (93),and one used the facts as follows:

(𝜆 + 1)− 1

4𝜎3− = (|𝜆| − 1)−14𝜎3𝑒

𝑖𝜋

4 𝜎3 ,

(√𝜆 + 1 + 1√𝜆 + 1 − 1

)−

= exp

(−𝑖 arccos 1√|𝜆| + 𝑖𝜋

), (119)

and

(𝑓01−(𝜆)

)−32 = 2

3(|𝜆| − 1)−

32 |𝜆|𝑒 3𝜋𝑖

2 . (120)

Similarly, with the aid of the above facts, one also derives

𝜍−1

6𝜎3𝐸01(𝜆) =(

𝑑11 𝑖𝑑12𝑖𝑑21 𝑑22

), (121)

where 𝑑11, 𝑑12, 𝑑21, and 𝑑22 are given by (18)–(21), respectively. Substituting (115), (118), and (121)into (117), one follows (16).

By the steepest descent analysis of the RHp for the corresponding orthogonal polynomials, 𝑅(𝑥) =𝐼 + (𝑛−1) and the error term holds uniformly for 𝑥 ∈ (0, 𝜀), 𝑅01(𝜍) = 𝐼 + (𝜍−1∕3) uniformly for𝑥 ∈ 𝔻2 as 𝜍 = 2𝑛2𝑡 → ∞, and 𝔻2 is given by (22). ■

We collect some results derived by Chen and Dai21 for fixed 𝑛, and give a simple proof as follows.

Proof of Theorem 6. See propositions 2.3–5.3, lemmas 3.1, 3.4, 5.1, theorems 5.4–7.2 and their proofsin.21 Although the boundary conditions (31) for 𝑆𝑛(𝑡) and (33) for 𝐻𝑛(𝑡) are not given therein, onefollows 𝐻 ′

𝑛(0) and 𝑆′(0) from (2.18), (5.1), (7.1), and (7.7) in,21 respectively. ■

30 CHEN ET AL.

The Hankel determinant, the leading coefficients, and the recurrence coefficients of the orthogonalpolynomials associated with the Pollaczek–Jacobi weight can be expressed in terms of the matrix-valued function 𝑌 (𝑧) and these are given by the following lemma.

Lemma 1. The leading coefficient 𝛾𝑛(𝑡), the auxiliary function 𝑟∗(𝑡), and 𝐻𝑛(𝑡) are expressed in termsof matrix-valued function 𝑌 (𝑧) for finite 𝑛,

𝑟∗(𝑡) = −𝑡𝑌12(0)𝑌21(0), (122)

𝑑

𝑑𝑡

(𝛾𝑛(𝑡))−2 = −2𝜋𝑖𝑌11(0)𝑌12(0), (123)

𝐻 ′𝑛(𝑡) = −(2𝑛 + 𝛼 + 𝛽)𝑌12(0)𝑌21(0) − 𝑛, (124)

where 𝑌𝑖𝑗(𝑧) represents the (𝑖, 𝑗)-item of 𝑌 (𝑧) in (45).

Proof. By the eq. (2.18) in,21 one has

𝑟∗(𝑡) = 𝑡

ℎ𝑛−1 ∫1

0𝜋𝑛−1(𝑥)𝜋𝑛(𝑥)𝑤𝑝𝐽

(𝑥, 𝑡) 1𝑥𝑑𝑥 = 𝑡𝛾𝑛−1(𝑡)2𝜋𝑛−1(0)∫

1

0

𝜋𝑛(𝑥)𝑤𝑝𝐽(𝑥, 𝑡)

𝑥𝑑𝑥, (125)

using the facts that ℎ𝑛−1 = (𝛾𝑛−1(𝑡))−2 and

∫1

0

𝜋𝑛−1(𝑥) − 𝜋𝑛−1(0)𝑥

𝜋𝑛(𝑥)𝑤𝑝𝐽(𝑥, 𝑡)𝑑𝑥 = 0, (126)

which follows from the orthogonality in (3) and (𝜋𝑛−1(𝑥) − 𝜋𝑛−1(0))∕𝑥 is a polynomial of degree 𝑛 − 2.Comparing (125) with 𝑌 (𝑧) in (45), one follows (122). A combination of (122) and (24) gives (124).Similarly, taking derivatives with respect to 𝑡 on both sides of (3) and ℎ𝑛(𝑡) = (𝛾𝑛(𝑡))−2, one finds

𝑑

𝑑𝑡

(𝛾𝑛(𝑡))−2 = −𝜋𝑛(0)∫

1

0

𝜋𝑛(𝑥)𝑤𝑝𝐽(𝑥, 𝑡)

𝑥𝑑𝑥.

Comparing the LHS of the above equation with 𝑌 (𝑧) in (45) again, one obtains (123). ■

We consider that the limiting Hankel determinant satisfies a 𝜎-form Painlevé III equation and itsasymptotic expansion. Our method is based on the results for fixed 𝑛 and the steep-descent analysis.

Proof of Theorem 7. With the procedure of 𝑌 → 𝑇 → 𝑆 → 𝑅 in (46), (52), and (98), one finds,

𝑌+(𝑥) = 4−𝑛𝜎3𝑅(𝑥)𝐸0(𝑥)Ψ−(𝑛2𝑓0(𝑥), 𝜍)𝑒𝑖𝜋(𝛼−1)

2 𝜎3

(1 01 1

)(𝑤𝑝𝐽

(𝑥, 𝑡))−1

2𝜎3. (127)

To find out 𝑌 (0), we need to evaluate the asymptotic expansions of 𝐸0(𝑥) and Ψ−(𝑛2𝑓0(𝑥), 𝜍) as𝑥 → 0.

CHEN ET AL. 31

By (83), (57), and (70),

𝐸0(𝑧) = 2−(𝛼+𝛽)𝜎3𝑀−1(−1)𝑛𝜎3⎛⎜⎜⎜⎝

𝑖

2

(𝑧−1𝑧

)−14��(𝑧) 1

2

(𝑧−1𝑧

)−14��(𝑧)

−12

(𝑧−1𝑧

) 14��(𝑧) 𝑖

2

(𝑧−1𝑧

) 14��(𝑧)

⎞⎟⎟⎟⎠(𝑛2𝑓0(𝑧)

) 14𝜎3

= 2−(𝛼+𝛽)𝜎3𝑀−1(−1)𝑛𝜎3

×⎛⎜⎜⎝

(𝛼 + 𝛽)𝑧(𝑧 − 1)−14 (1 + 𝑂(𝑧)) (1 − 𝑧)−

14(1 − (𝛼+𝛽)2

2 𝑧 + 𝑂(𝑧2))

−(1 − 𝑧)14(1 − (𝛼+𝛽)2

2 𝑧 + 𝑂(𝑧2))

(𝛼 + 𝛽)(1 − 𝑧)14 (1 + 𝑂(𝑧))

⎞⎟⎟⎠×

(√2𝑛(1 + 𝑂(𝑧)) 0

0 1√2𝑛(1 + 𝑂(𝑧))

),

where ��(𝑧) ∶= (−𝜑(𝑧))𝛼 + 𝛽

2 − (−𝜑(𝑧))−𝛼 + 𝛽

2 = −2𝑖(𝛼 + 𝛽)√𝑧(1 + 𝑂(𝑧)), ��(𝑧) ∶= (−𝜑(𝑧))

𝛼+𝛽2 +

(−𝜑(𝑧))−𝛼+𝛽2 = 2 − (𝛼 + 𝛽)2𝑧 + 𝑂(𝑧2), and𝜑(𝑧) is given by (47), and 𝑓0(𝑧) = −4𝑧(1 + 𝑂(𝑧)) as 𝑧 → 0.

Then, one finds,

𝐸0(𝑥) = 2−(𝛼+𝛽)𝜎3𝐼 − 𝑖𝜎1√

2(−1)𝑛𝜎3

(0 1

−1 𝛼 + 𝛽

)(2𝑛)

12𝜎3 + (𝑥), 𝑥 → 0. (128)

With the aid of the asymptotic behavior of Ψ in (79) and (73), one follows that

Ψ−(𝜉, 𝜍)𝑒𝑖𝜋(𝛼−1)

2 𝜎3

(1 01 1

)(𝑤𝑝𝐽

(𝑥, 𝑡))−1

2𝜎3

= 𝑄(𝜍)(𝐼 + (𝜉))𝑒 𝜍

𝜉𝜎3 (−1)𝑛𝜎3𝜉

𝛼

2 𝜎3𝑒𝑖𝜋(𝛼−1)

2 𝜎3(𝑤𝑝𝐽

(𝑥, 𝑡))−1

2𝜎3, (129)

where 𝜍 = 2𝑛2𝑡 and 𝜉 = 𝑛2𝑓0(𝑧) play the role of a conformal mapping for 𝑧 ∈ 𝑈 (0, 𝑟), 𝑟 is sufficientlysmall and fixed.

As 𝑄(𝜍) is given by (80) and (128), as 𝑥 → 0, 𝑛 → ∞, one has

[𝐸0(0)𝑄(𝜍)

]11 = 𝑖(−1)𝑛

√𝑛2−(𝛼+𝛽)

√1 + 𝑞′(𝜍)

2𝐶(1 + 𝑂(𝑛−1)

), (130)

[𝐸0(0)𝑄(𝜍)

]12 = (−1)𝑛+1

√𝑛2−(𝛼+𝛽) 𝑟′(𝜍)√

2(1 + 𝑞′(𝜍))𝐶−1(1 + 𝑂(𝑛−1)

), (131)

[𝐸0(0)𝑄(𝜍)

]21 = (−1)𝑛+1

√𝑛2𝛼+𝛽

√1 + 𝑞′(𝜍)

2𝐶(1 + 𝑂(𝑛−1)

), (132)

where 𝐶 is an arbitrary nonzero constant.

32 CHEN ET AL.

Using the above formulas and 𝑅(𝑥) in (104), one obtains

𝑌11(0)𝑌12(0) = 4−2𝑛[𝐸0(0)𝑄(𝜍)

]11[𝐸0(0)𝑄(𝜍)

]12(1 + 𝑂(𝑛−

23 ))

= −𝑖2−4𝑛−2(𝛼+𝛽)−1𝑛𝑟′(𝜍)(1 + 𝑂(𝑛−23 )), (133)

𝑌12(0)𝑌21(0) =[𝐸0(0)𝑄(𝜍)

]12[𝐸0(0)𝑄(𝜍)

]21(1 + 𝑂(𝑛−

23 ))

= 2−1𝑛𝑟′(𝜍)(1 + 𝑂(𝑛−23 )), (134)

the above error terms hold uniformly for 𝑡 = (0, 𝑐], 𝑐 > 0, 𝜍 = 2𝑛2𝑡 → ∞, and 𝑛 → ∞.Substituting (134) into (124), one finds,

𝐻 ′𝑛(𝑡) = −𝑛2𝑟′(𝜍)(1 + 𝑂(𝑛−

23 )), 𝜍 = 2𝑛2𝑡. (135)

Integrating the above equation from 0 to 𝑡 on both sides and the boundary conditions 𝑟(0) = 1−4𝛼28

and𝐻𝑛(0) = 0, then one finds the limit of𝐻𝑛(𝜍

2𝑛2 ) as 𝑛 → ∞ exists and (34) is valid. With the definitionof 𝐻𝑛(𝑡) in (23) and integrating once again, then one derives the formula (36). Moreover, as 𝑡 = 0, thePollaczek–Jacobi weight reduces to the shifted Jacobi weight function (2) and its Hankel determinantcalculated by 𝐷𝑛(𝑤, 0) =

∏𝑛−1𝑗=0 ℎ𝑗(0), where ℎ𝑛(0) is given by

ℎ𝑛(0) =𝑛!Γ(𝑛 + 1 + 𝛼)Γ(𝑛 + 1 + 𝛽)Γ(𝑛 + 1 + 𝛼 + 𝛽)

Γ(2𝑛 + 1 + 𝛼 + 𝛽)Γ(2𝑛 + 2 + 𝛼 + 𝛽), (136)

it can be directly derived from (𝐶13) in.29

Inserting the facts of 𝐻𝑛(𝑡) = (𝜍)(1 + 𝑂(𝑛−23 )),𝐻 ′

𝑛(𝑡) = 2𝑛2′(𝜍)(1 + 𝑂(𝑛−23 )), and 𝐻 ′′

𝑛 (𝑡) =

4𝑛4′′(𝜍)(1 + 𝑂(𝑛−23 )) into (32), collecting the coefficients of 4𝑛4, then one obtains the 𝜎-form

Painlevé equation (35). The initial conditions follow from (33). ■

Remark 4. Let (𝜍) = (2𝜍) − 𝛼2∕4, then (𝜍) satisfies the Painlevé III 𝜎-form equation obtainedby Ohyama–Kawamuko–Sakai–Okamoto, see (18) in,30 see also the Remark 3 in.31

We note that (34) and (36) are the same as the double scaling limit of the Hankel determinant associ-ated with the singularly perturbed Laguerre weight, see theorem 1 in,13 the 𝜎-form Painlevé equation(35) is equivalent to the second-order eq. (30) in,13 but the double scaling scheme and initial data𝐷𝑛(𝑤, 0) are different from each other.

We show that the asymptotic behaviors of the recurrence coefficients 𝛼𝑛(𝑡), 𝛽𝑛(𝑡) in the three-termrecurrence formula (4), and the leading coefficient 𝛾𝑛(𝑡) of the orthonormal polynomials.

Proof of Theorem 8. To obtain the asymptotic behavior of the leading coefficient 𝛾𝑛(𝑡), we need toderive the asymptotic behavior of 𝑌 (𝑧) as 𝑧 → ∞.

Taking a series inverse transformations of 𝑌 → 𝑅, by (46), (52), and (98), then one obtains theexpression of 𝑌 (𝑧) as follows:

𝑌 (𝑧) = 4−𝑛𝜎3𝑅(𝑧)𝑃 (∞)(𝑧)𝑒𝑡

2𝑧 𝜎3𝜑(𝑧)𝑛𝜎3 . (137)

CHEN ET AL. 33

By (57), one follows the asymptotic expansion of 𝑃 (∞)(𝑧) as 𝑧 → ∞,

𝑃 (∞)(𝑧) = 𝐼 − 14(2−(𝛼+𝛽)𝜎3𝑀−1𝜎3𝑀2(𝛼+𝛽)𝜎3 + 𝜎3

)1𝑧+ (

1𝑧2

). (138)

From (102) and (103), one obtains,

𝑅(𝑧) = 𝐼 + (𝑛−23 𝑧−1), (139)

for large 𝑛, 𝑧 → ∞, and uniformly for 𝑡 ∈ (0, 𝑐], 𝑐 is a finite positive constant.With the aid of the general result of (3.30) in,27(𝛾𝑛(𝑡))2 = 𝑖(2𝜋[𝑌1]12)−1 and [𝑌1]12 is the (1,2)-entry

in the 𝑌1, which follows from

𝑌 (𝑧)𝑧−𝑛𝜎3 = 𝐼 +𝑌1𝑧

+ (

1𝑧2

).

Using the above four formulas, one obtains,(𝛾𝑛(𝑡))2 = 1

𝜋24𝑛+2(𝛼+𝛽)−1

(1 + 𝑂

(𝑛−2

3))

, (140)

uniformly for 𝑡 ∈ (0, 𝑐] and 𝑐 > 0.Collecting the quantity 𝑌11(0)𝑌12(0) in (133), the above approximation of (𝛾𝑛(𝑡))2, and (123), one

obtains,

𝑑

𝑑𝑡ln 𝛾𝑛(𝑡) =

14𝑛𝑟′(𝜍)

(1 + 𝑂

(𝑛−2

3))

. (141)

Integrating on both sides, one finds,

ln𝛾𝑛(𝑡)𝛾𝑛(0)

= 𝑟(𝜍) − 𝑟(0)4𝑛

(1 + 𝑂

(𝑛−2

3))

, (142)

where the boundary condition of 𝑟(0) is in (81). Initial data (41) follow from 𝛾𝑛(0) = ℎ𝑛(0)− 1

2 and(136). Then, Equation 40 is valid.

The asymptotic behavior of 𝛼𝑛(𝑡) in (38) follows from a combination of (24), (25), (27), (34), and(135). Moreover, substituting these four equations into (26), one obtains

𝑅𝑛(𝑡)2𝑛 + 1 + 𝛼 + 𝛽

= 1 + 𝜍𝑟′(𝜍)2𝑛2(1 + 𝑂

(𝑛−2

3))

, (143)

uniformly for 𝑡 ∈ (0, 𝑐] and 𝑐 > 0. With these asymptotic formulas, it is easy to verify the uniformasymptotic formula (39) by (28) for 𝛽𝑛(𝑡).

Similarly, the uniform asymptotic expansion for 𝑆𝑛(𝑡) follows from (29) and (143), and inserting itinto the Painlevé V equation (30), then the third-order integrable equation (42) is derived from collect-ing the coefficients of 𝑛−2. ■

Remark 5. As 𝑛 → ∞ and 𝑡 = 0, the asymptotic behavior of the recurrence coefficients in the above(38) and (39) are coherent with the classical theory in,32 see also29 and appendix C therein.

The above third-order differential equation is integrable, which had been shown in the paper,12 and𝑦(𝜍) = 𝜍𝑟′(𝜍) satisfies a Painlevé III equation.

We give a proof of Corollary 1 as follows.

34 CHEN ET AL.

The Proof of Corollary 1. The above asymptotic expansions follow form Equations 34, 39–42 and theboundary conditions (81) of 𝑟(𝜍). ■

We give the proof of universality concerning a universal behavior of the kernel in the bulk of thespectrum.

The Proof of Theorem 9. The kernel in (8) associated with the Pollaczek–Jacobi weight can also beexpressed in terms of 𝑌 (𝑧),

𝐾𝑛(𝑥, 𝑦; 𝑡) =

√𝑤𝑝𝐽

(𝑥, 𝑡)√

𝑤𝑝𝐽(𝑦, 𝑡)

2𝜋𝑖(𝑥 − 𝑦)(𝑌 −1+ (𝑦)𝑌+(𝑥)

)21

=

√𝑤𝑝𝐽

(𝑥, 𝑡)√

𝑤𝑝𝐽(𝑦, 𝑡)

2𝜋𝑖(𝑥 − 𝑦)(0 1)𝑌 −1+ (𝑦)𝑌+(𝑥)

(10

). (144)

To find 𝑌 (𝑧), we need to take a list of inverse transformations of 𝑌 → 𝑇 → 𝑆 → 𝑅 in (46), (52),and (98), then one finds,

𝑌+(𝑥)(𝑤𝑝𝐽(𝑥, 𝑡))

12

(10

)= 4−𝑛𝜎3𝑅(𝑥)𝐴(𝑥)𝐷+(𝑥)𝜎3

(1 0

𝑤(𝑥)−1𝜑+(𝑥)−2𝑛 1

)𝑒

𝑡

2𝑥 𝜎3𝜑+(𝑥)𝑛𝜎3 (𝑤𝑝𝐽(𝑥, 𝑡))

12

(10

)

= 4−𝑛𝜎3𝑅(𝑥)𝐴(𝑥)

(exp(− 𝑖𝛽𝜋

2 + 𝑖(𝛼 + 𝛽 + 2𝑛) arccos√𝑥)

exp( 𝑖𝛽𝜋2 − 𝑖(𝛼 + 𝛽 + 2𝑛) arccos√𝑥)

), (145)

where the following facts have been used:

𝜑+(𝑥) = exp(𝑖2 arccos√𝑥),

𝐷+(𝑥)𝑒𝑡

2𝑥 𝜑+(𝑥)𝑛 = (𝑤𝑝𝐽(𝑥, 𝑡))−

12 exp(−𝑖𝛽𝜋∕2 + 𝑖(𝛼 + 𝛽 + 2𝑛) arccos

√𝑥),

which follows from (47) and (58), 𝐴(𝑧) = 𝐷(∞)−𝜎3𝑀−1𝑎(𝑧)−𝜎3𝑀 , and 𝑀 = 𝐼+𝑖𝜎1√2, 𝑎(𝑧) =

( 𝑧−1𝑧)14 , 𝐷(∞) = 2𝛼+𝜆.

Obviously,

𝐴−1+ (𝑦)𝐴+(𝑥) = 𝐼 + 𝑂(𝑥 − 𝑦), (146)

where the error term is uniformly for 𝑥, 𝑦 in the compact subsets of (0,1). Substituting (104), (145),and (146) into (144), one finds,

𝐾𝑛(𝑥, 𝑦; 𝑡) =sin[(2𝑛 + 𝛼 + 𝛽)(arccos

√𝑦 − arccos

√𝑥)]

𝜋(𝑥 − 𝑦)+ 𝑂(1), (147)

let 𝑥 → 𝑦, then (43)is readily verified.

CHEN ET AL. 35

Assume 𝑎 ∈ (0, 1), and 𝑢, 𝑣 ∈ ℝ. If 𝑥 and 𝑦 are scaled as 𝑎 + 𝑢

𝑛𝜚(𝑎) and 𝑎 + 𝑣

𝑛𝜚(𝑎) in the above equation,

and 𝜚(𝑥) = (𝜋√𝑥(1 − 𝑥))−1. With the aid of the following fact:

arccos√𝑧 + 𝑠 = arccos

√𝑠 − 𝑧

2√𝑠(1 − 𝑠)

+ 𝑂(𝑧2),

one finds,

(2𝑛 + 𝛼 + 𝛽)(arccos

√𝑎 + 𝑣

𝑛𝜚(𝑎)− arccos

√𝑎 + 𝑢

𝑛𝜚(𝑎)

)= 𝜋(𝑢 − 𝑣) + 𝑂

(1𝑛

), (148)

uniformly for 𝑎 in compact subsects of (0,1) and 𝑢, 𝑣 in compact subsects of ℝ. It is easy to derive (44)by a combination of (147) and (148). ■

Remark 6. In fact, at the right hard-edge of the spectrum, the limit of the correlation kernel is the Besselkernel of order 𝛽, this is the same as the modified Jacobi weight;5 at the left hard-edge, the kernel inthe double-scaling limit can be described by a Painlevé III transcendent, namely, as the Painlevé IIIkernel and this is the same as the situation of the singularly perturbed Laguerre unitary ensemble in,12

although the scaling schemes are different from each other.

4 CONCLUSIONS

In this paper, we study the asymptotic expansions of the Pollaczek–Jacobi type orthogonal polynomi-als and the Pollaczek–Jacobi type unitary ensemble. The singular factor 𝑒−𝑡∕𝑥 in the weight (1) seemsas a bridge, which connects the Szegő class and non-Szegő class. The asymptotic expansions of thePollaczek–Jacobi type orthogonal polynomials involve the Airy function and its derivatives, near theessentially singular point as 𝜍 = 2𝑛2𝑡 → ∞ and it relates with the Bessel function as 𝜍 = 2𝑛2𝑡 → 0.Obviously, these are different from the Pollaczek orthogonal polynomials, see.16,17 Its asymptoticexpansions can be expressed in terms of Szegő function outside the orthogonal interval and in thebulk of the interval.

The asymptotic expansions of the recurrence coefficients and leading coefficient involve a particular

Painlevé III transcendent. It is worth mentioning that 𝛼𝑛 vary from 12 + 𝑂(𝑛−2) to 1

2 + 𝑂(𝑛−23 ), see

Corollary 1. Note that 12 + 𝑂(𝑛−2) is for the Jacobi polynomial and 1

2 + 𝑂(𝑛−1) is for the Pollaczekpolynomial. It is interesting that the double scaled Hankel determinant satisfies a 𝜎-form Painlevé IIIequation, and its asymptotic expansion is also found. Universality of the kernel in the bulk spectrumis also obtained.

ACKNOWLEDGMENTSThe authors would like to thank the referees for their careful work and helpful suggestions to improvethe quality of this paper. This work was supported by the National Science Foundation of Chinaunder Project No. 11801083, No. 11671095, No. 51879045. Yang Chen would like to thank theMacau Science and Technology Development Fund under grant number FDCT130/2014/A3, FDCT023/2017/A1, for generous support.

ORCIDYang Chen https://orcid.org/0000-0003-2762-7543

36 CHEN ET AL.

En-Gui Fan https://orcid.org/0000-0002-8816-8398

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How to cite this article: Chen M, Chen Y, Fan E-G. The Riemann–Hilbert anal-ysis to the Pollaczek–Jacobi type orthogonal polynomials. Stud Appl Math. 2019;1–39.https://doi.org/10.1111/sapm.12259

APPENDIX A: RHp FOR THE BESSEL MODELThe matrix-valued function Ψ𝐵(𝑧) satisfies the following RHp.

(a) Ψ𝐵(𝑧) is defined and analytic for ℂ ⧵ {⋃3

𝑘=1 𝛾𝑘}, where 𝛾𝑘 are illustrated in Figure 2,

(b) Ψ𝐵(𝑧) satisfies the following jump relations,

Ψ𝐵+(𝑧) = Ψ𝐵−(𝑧)(

1 0𝑒𝜂𝜋𝑖 1

), 𝑧 ∈ 𝛾1,

Ψ𝐵+(𝑥) = Ψ𝐵−(𝑥)(

0 1−1 0

), 𝑥 ∈ 𝛾2,

Ψ𝐵+(𝑧) = Ψ𝐵−(𝑧)(

1 0𝑒−𝜂𝜋𝑖 1

), 𝑧 ∈ 𝛾3.

(c) For 𝑧 → ∞,Ψ𝐵(𝑧) has the following behavior:

Ψ𝐵(𝑧) = 𝑧−14𝜎3𝑀

(𝐼 + 𝐵1𝑧

−12 + (𝑧−1))𝑒√𝑧𝜎3 , arg 𝑧 ∈ (−𝜋, 𝜋), 𝑧 → ∞.

38 CHEN ET AL.

(d) For −1 < 𝜂 < 0,Ψ𝐵(𝑧) has the following behavior:

Ψ𝐵(𝑧) = (|𝑧| 𝜂2 |𝑧| 𝜂2|𝑧| 𝜂2 |𝑧| 𝜂2

), as 𝑧 → 0.

For 𝜂 = 0,Ψ𝐵(𝑧) satisfies the following behavior:

Ψ𝐵(𝑧) = (log |𝑧| log |𝑧|log |𝑧| log |𝑧|

), as 𝑧 → 0.

For 𝜂 > 0,Ψ𝐵(𝑧) fulfills the following behavior:

Ψ𝐵(𝑧) =

⎧⎪⎪⎨⎪⎪⎩(|𝑧| 𝜂2 |𝑧|− 𝜂

2|𝑧| 𝜂2 |𝑧|− 𝜂

2

), for arg 𝑧 ∈

(−2𝜋

3 ,2𝜋3

)and 𝑧 → 1,

(|𝑧|− 𝜂

2 |𝑧|− 𝜂

2|𝑧|− 𝜂

2 |𝑧|− 𝜂

2

), for arg 𝑧 ∈

(−𝜋,−2𝜋

3

)⋃( 2𝜋3 , 𝜋)and 𝑧 → 1.

The solution of the above Bessel model PHp can be constructed by the Bessel functions and itsderivative as follows, see4 or,12,14

Ψ𝐵(𝑧) = 𝜋12𝜎3⎛⎜⎜⎝

𝐼𝛼

(√𝑧)

𝑖∕𝜋𝐾𝜂(√𝑧)

𝑖𝜋√𝑧𝐼 ′𝜂

(√𝑧)

−√𝑧𝐾 ′

𝜂(√𝑧)

⎞⎟⎟⎠⎧⎪⎪⎪⎨⎪⎪⎪⎩

𝐼, for arg 𝑧 ∈(−2𝜋

3 ,2𝜋3

),(

1 0−𝑒𝑖𝜂𝜋 1

), for arg 𝑧 ∈

(2𝜋3 , 𝜋),(

1 0𝑒−𝑖𝜂𝜋 1

), for arg 𝑧 ∈

(−𝜋,−2𝜋

3

).

(A.1)

APPENDIX B: RHp FOR THE AIRY MODELThe matrix valued function Ψ𝐴(𝑧) satisfies the following RHp:

(a) Ψ𝐴(𝑧) is defined and analytic for ℂ ⧵ {⋃4

𝑘=1 𝓁𝑘}, where 𝓁𝑘 are illustrated in Figure A.1.,

(b) Ψ𝐴(𝑧) satisfies the following jump relations:

Ψ𝐴+(𝑥) = Ψ𝐴−(𝑥)(1 10 1

), 𝑥 ∈ 𝓁1,

Ψ𝐴+(𝑧) = Ψ𝐴−(𝑧)(1 01 1

), 𝑧 ∈ 𝓁2,

Ψ𝐴+(𝑥) = Ψ𝐴−(𝑥)(

0 1−1 0

), 𝑧 ∈ 𝓁3.

Ψ𝐴+(𝑧) = Ψ𝐴−(𝑧)(1 01 1

), 𝑧 ∈ 𝓁4.

CHEN ET AL. 39

Ω1

Ω4

Ω2

Ω3

O

2πi3

−2πi3

�1

�4

�3

�2

F I G U R E A . 1 . Contours of ∪3𝑗=1𝓁𝑗 . and regions Ω𝑘, 𝑘 = 1,… , 4

(c) For 𝑧 → ∞,Ψ𝐴(𝑧) has the following behavior:

Ψ𝐴(𝑧) = 𝑧−14𝜎3𝑀

(𝐼 + 𝐴1𝑧

−32 + (𝑧−3))𝑒−2

3 𝑧32 𝜎3 , arg 𝑧 ∈ (−𝜋, 𝜋), 𝑧 → ∞, (B.1)

where 𝐴1 is given by

𝐴1 =148

𝜎3 + 𝑖18𝜎1. (B.2)

(d) For 𝑧 → 0,Ψ𝐴(𝑧) has the following behavior:

Ψ𝐴(𝑧) =((1) (1)(1) (1)

), as 𝑧 → 0.

The solution of the Airy model PHp can be expressed by the Airy functions and its derivative asfollows, see,27

Ψ𝐴(𝑧) = 𝑀𝐴

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(𝐴𝑖(𝑧) 𝐴𝑖(𝜔2𝑧)𝐴𝑖′(𝑧) 𝜔2𝐴𝑖′(𝜔2𝑧)

)𝑒− 𝜋𝑖

6 𝜎3 , for 𝑧 ∈ Ω1,(𝐴𝑖(𝑧) 𝐴𝑖(𝜔2𝑧)𝐴𝑖′(𝑧) 𝜔2𝐴𝑖′(𝜔2𝑧)

)𝑒− 𝜋𝑖

6 𝜎3

(1 0

−1 1

), for 𝑧 ∈ Ω2,(

𝐴𝑖(𝑧) −𝜔2𝐴𝑖(𝜔𝑧)𝐴𝑖′(𝑧) −𝐴𝑖′(𝜔𝑧)

)𝑒− 𝜋𝑖

6 𝜎3

(1 01 1

), for 𝑧 ∈ Ω3,(

𝐴𝑖(𝑧) −𝜔2𝐴𝑖(𝜔𝑧)𝐴𝑖′(𝑧) −𝐴𝑖′(𝜔𝑧)

)𝑒− 𝜋𝑖

6 𝜎3 , for 𝑧 ∈ Ω4,

(B.3)

where 𝑀𝐴 =√2𝜋𝑒

𝜋𝑖

6 ( 1 00 −𝑖 ), 𝐴𝑖(𝑧) is the Airy function, the domains 𝑧 ∈ Ω𝑘 are described in

Figure 4.