1 Introduction

In a recent paper[8],Clarkson et al.made a detailed study of the monic orthogonal polynomials associated with the generalized Freud weight

where and t∈R.In particular,they showed that these polynomials satisfy a differential-difference equation and a second-order linear differential equation.The coefficients in these equations involve the coefficientβn(t;α)in the three-term recurrence relation

where S−1(x;t)=0 and S0(x;t)=1.The recurrence coefficient βn(t;α)satisfies the first discrete Painlevé equation d P I,and is related to solutions of the fourth Painlevé equation P IV.As a follow-up,in[7]Clarkson and Jordaan investigated the asymptotic behavior of the polynomials S n(x;t),as well as the recurrence coefficient βn(t;α),when the degree n,or the parameter t,tends to infinity.However,with regard to the polynomial S n(x;t),what they gave is a solutionto an approximate equation(see[7,(4.12)]),and not an asymptotic approximation to the polynomial S n(x;t).

In the present paper,we are mainly concerned with the problem of finding asymptotic formulas for the monic polynomials S n(x;t),as n→∞,for all values of t,whether t is fixed or varying.This problem has actually already been studied by several people in special cases.For instance,whenα=0 in(1.1)and the weight function is given in the form

asymptotic formulas(as n→∞)of the associated monic orthogonal polynomials have been given by Bleher and Its in the case andTheir method makes use of a WKB formula(see[4,(7.5)])from the differential equation theory.Without loss of generality,one can assume the constant g in(1.3)to be equal to 1.To simplify the calculation,one may also take the parameter N to be equal to the degree n.Thus,in[19]Wong and Zhang investigated the problem in the case when the weight function is given by w(x)=e−n V(x)with

and h= −2.At about the same time,results for the same case(i.e.,h= −2)but with a much more general V(x)was obtained by Claeys et al.[5].For fixed t,asymptotic formulas for the monic polynomials S n(x;t)in(1.2)can be written out directly from the results in[20].Quite recently,in[2]and[3],Bertola and Tovbis obtained the asymptotics of the recurrence coefficients for the orthogonal polynomials associated with the weight(1.3),where g∈C,h=1 and the variable x is on the rays arg x= θ(for fixedθ)in the complex plane.As a follow-up of the investigations in[2]and[3],one may also consider the case when the weight function has a singularity at the origin like the one in(1.1).But,we will leave this problem to a future investigation.

In this paper,we shall study the behavior of S n(x;t)as n→∞,for all values of t.Our approach is completely based on the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou[12],and will not make use of any asymptotic result from differential equation theory.We divide our discussion into three cases:(i)c>−2,(ii)c= −2,and(iii)c< −2,where is a constant and N=n+α.As we shall show,in the first two cases,the support of the associated equilibrium measureµt is a single interval,whereas in the third case the support ofµt consists of two intervals.In all three cases,we will present infinite asymptotic expansions of S n(z;t)for z in various different regions.These regions together cover the whole complex z-plane.

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2 Riemann-Hilbert Problem

For notational convenience,from here on we will useπn(z)to denote the monic orthogonal polynomials in(1.2),and write p n(z)= γnπn(z)for the orthogonal polynomials with respect to the weight in(1.1).Consider the following Riemann-Hilbert problem(RHP for short)for a 2×2 matrix-valued function Y:

Recall that andαt in(3.18).From(3.9)it follows that

(Y a)Y(z)is analytic in

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(Y b)Y(z)takes continuous boundary values Y+(x)and Y−(x)such that

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for x∈R{0},where w(x)is the weight function given by(1.1).(Y c)For z∈CR,

as z→∞.

Theorem 2.1 The unique solution to the above RHP is given by

as z→0.

The following well-known theorem is due to Fokas,Its and Kitaev[15].

(Y d)For z∈CR,

where and

is the Cauchy transform of f.

To asymptotically evaluate the solution of the RHP for Y,we will first follow the nonlinear steepest descent method introduced by Deift and Zhou[12],and further developed in[9–11].This method consists of a sequence of transformations

which ultimately leads to a RHP that can be solved explicitly.The transformation Y→U is just a rescaling,and the transformation U→T is a normalization process.The transformation T→V involves a factorization of the jump matrix and deformation of contours so that V can be approximated by an exact solution to a RHP for n large.The final transformation V→S consists of some local analysis and the construction of parametrices.Since every step of the transformation is explicit and reversible,one can obtain the asymptotics of Y,and hence of πn(z),by a sequence of inverse transformations.

We define the first transformation by

where N=n+α.Then,U satisfies the following RHP:

(U a)U(z)is analytic in CR;

(U b)U(z)takes continuous boundary values U+(x)and U−(x)such that

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for x∈R{0},

(U c)For z∈CR,

as z→∞.

(U d)U(z)has the same behavior as Y(z)for z near the origin.

We next derive an explicit formula for the probability density functionµt(s)in(3.1).From(3.12),one observes

For convenience,we have scaled the variable z by multiplying it by

3 Normalization and Auxiliary Functions

In order to normalize the behavior of U(z)at z=∞,we first need to introduce some notations.Letαt be a positive real number,and denote by µt(s)a probability density function supported on and

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The explicit formulas forαt and µt(s)will be determined later.

Next,we introduce the so-called g-function which is the logarithmic potential ofµt(s);that is

where for each s we view log(z−s)as an analytic function of the variable z with a branch cut along(−∞,s].

Similarly,we also define

where for each s we view log(z−s)as an analytic function of the variable z with a branch cut along[s,∞).

From(3.1)–(3.2),it is easy to check that the g-function satisfies the jump conditions

and

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we introduce the transformation

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On account of(3.2)and(3.4),one readily sees that e ng(z)can be analytically extended to C[−αt,αt]and

飞行时间测距是通过测定UWB脉冲信号从UWB标签到UWB基站的信号往返时间从而确定其距离的，这种方法解决常规时间到达（Time of Arrival，TOA）以及时间到达差（Time Difference of Arrival，TDOA）方法中需要UWB标签与基站保持时间同步的问题。该测距原理是标签先向基站发送测距序列，其中可包含要求应答等信息，基站经过一个固定延时之后转发信号[12]，标签接收后计算一次来回程时间，标签到基站i距离di可用式（1）表示：

By adopting the convention thatσ3 denotes the Pauli matrix

where l N is a constant to be determined.A straightforward calculation shows that T(z)is the unique solution of the following RHP:

(T a)T(z)is analytic in CR;

(T b)For x∈R{0},

where

(T c)T(z)behaves like the identity matrix at infinity:

for z∈CR;

(T d)T(z)has the following behavior for z∈CR as z→0

for i≥ 1,[x]denotes the largest integer≤x,and takes branch cut along[−a,a]and behaves like z as z→∞.It then follows from(3.15)and(3.20)that

for x ∈ (−αt,αt).It follows from(3.9)by substituting x= αt in(3.5)that

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Diff erentiating(3.9)yields

where

For convenience,we set

where is analytic inand behaves like z as z → ∞.Since

we can solve this scalar RHP to give

or,equivalently,

From(3.1)and(3.12),it is easily seen that G(z)→0 andas z→∞.Hence,we have from(3.15)

and

Since the integrand is an odd function,we find that(3.16)is trivially true.By a change of variable s= αt sinθ,it follows that

From Theorem 2.1 and(2.1),it is easily seen that

for x ∈ (−αt,αt),where P.V.denotes the Cauchy principal value(see[1,p.518]).Therefore,

To evaluate G(z)in(3.15),we first note that for any integer m ≥ 1 and z∈ C[−a,a]with a>0,Lemma 1 in[20]gives

where

We proceed to seek for a probability density functionµt(s)in(3.1)and a constant l N mentioned above so that J12 in jump condition(T b)becomes 1 for x ∈ (−αt,αt).Thus,we set

A combination of(3.19)and(3.22)gives that

for x ∈ (−αt,αt),whereαt is shown in(3.18).

By the Plemelj formula(see[1,p.518]),we obtain

3.1 Case for c>−2

In this subsection,we consider the first case.Thus,we have

We first proceed to calculate l N in(3.10).By(3.1)–(3.2),the function g(z)− log(z+ αt)is analytic inand

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Let us define

where is analytic inand behaves like z asIt can be readily verified that for x ∈ (−αt,αt),

It is easy to see from(3.23)thatµt(x)is not always larger than or equal to 0 for all t∈ R.Indeed,ifthen there existsδ>0 such thatµt(x)<0 for x ∈ (−δ,δ)⊂(−αt,αt),which contradicts our assumption that µt(x)is positive in the support.This fact shows that the support ofµt(x)consists of more than one interval ifIn the following,we divide our discussion into three cases:c> −2,c= −2(i.e.,the critical case)and c<−2,where c=tN−1 2 is a constant.In the last case,the support ofµt consists of two intervals.

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Now,let z→∞;on account of(3.25),we have

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Let the integrals on the left-hand side of(3.27)be denoted by I1,I2 and I3,respectively.It is easily seen that

and

To evaluate I1,we note that for any 0≤a

(see[18,Lemma 1]).Hence,by taking s=x+αt,we obtain

Inserting(3.28)–(3.31)into(3.27)gives

Letwhere is analytic inand behaves likes z as z → ∞.Clearly,ν±(x)= ±πiµt(x)for x ∈ [−αt,αt].Define

where the path of integration from αt to z lies entirely in the regionexcept for the initial pointαt.

Similarly,we define

where the path of integration from −αt to z lies entirely in the region z∈ C[−αt,∞),except for the initial point−αt.

The functionsφ(z)anddefined above will play an important role in our argument,and some of their properties are given below.For proofs of these results,see[20].

Prop osition 3.1 LetandWe have

for z∈ C±.The mapping properties ofφ(z)are given by

and

for z large enough.Similarly,the mapping properties ofare given by

and

for z large enough.

Prop osition 3.2 With constant l N given in(3.32)and defined by

the following connection formulas between the g-function(e g-function)andφ-functionfunction)hold

Furthermore,we have

and

3.2 Case for c=−2

Now,we consider the second caseWith

we have from(3.23)

With andthe formula in(3.32)gives

Letwhere is analytic inand behaves likes z as z → ∞.Clearly,forDefine

where the path of integration from to z lies entirely in the regionexcept for the initial point

Similarly,we define

where the path of integration from to z liesentirely in the regionexcept for the initial point

The functionsφ(z)anddefined above play the same role asφ(z)anddefined in(3.33)–(3.34)for case(i).The following are some of their properties.

Prop osition 3.3 Denotingandwe have

for z∈ C±.The mapping properties ofφ(z)are given by

If z∈C+,we have

as z→0 and

for z large enough.Similarly,the mapping properties ofare given by

If z∈C+,we have

as z→0 and

for z large enough.

Prop osition 3.4 With constant l N given in(3.51)and defined by

the following connection formulas between the g-functionandφ-functionfunction)hold

and

The material in this and the previous section parallels that in[19,§3].

3.3 Case for c<−2

In this third case,the equilibrium measure will be defined on two intervals,and we denote them by E:=[−b,−a]∪ [a,b]with 0t(s)supported on E,i.e.,µt(s)≥ 0 and

The explicit formula for a,b,andµt(s)will be determined later.

Now,we repeat the analysis at the beginning of this section from(3.1)to(3.12)with the interval[−αt,αt]replaced by E.With this minor change,we define

From(3.71)–(3.72),it is easily checked that

Also,on account of(3.72)–(3.73),one readily sees that e ng(z)can be analytically extended to C[−b,b]and(3.6)holds.As in the previous cases,we introduce the transformation T(z)given in(3.8),and obtain its associated RHP stated in(T a)-(T d).To determine a probability density functionµt(s)in(3.71)and a constant l N mentioned above so that the entry J12 in jump condition(T b)becomes 1 for x∈E,we set

for x∈E.By substituting x=b in(3.73),it follows from(3.74)that

Differentiating(3.74)yields

where

Corresponding to(3.13),we set

whereis analytic in CE and behaves like z2 as z→∞(see[9,p.171]).Since

solving this scalar RHP gives

or equivalently

Recall(3.71)and(3.77).Since G(z)→0 andaswe have from(3.80)

for j=0,1,and

(see[9,p.172]).Equations(3.81)–(3.82)are equivalent to

and

Now we have two equations with two unknowns a and b.By a change of variable

we obtain after a long and complicated calculation

(see[4,p.218]).We next derive an explicit formula for the probability density functionµt(s)in(3.71).It is well known that

To evaluate G(z)in(3.80),we need Lemma 1 in[20],which states that for b>a>0,we have

Recall that

wheretakes branch cut along[a,b]and behaves like z as z→ ∞.It then follows from(3.80)and(3.86)–(3.87)that

which gives

for x∈E,where a,b are given in(3.85).

Next,we proceed to calculate l N in(3.75).By(3.71)–(3.72),the functionis analytic in CE and

Let us define

whereis analytic in CE and behaves like z2 as z→∞.It can be readily verified that for x∈E,

By the Plemelj formula(see[1,p.518]),we obtain

From(3.74),it follows

which is equivalent to

Now,let z→∞;on account of(3.90),we have

Let the integrals on the left-hand side of(3.92)be denoted by I1,I2 and I3,respectively.As in Subsections 3.1–3.2,we have

and

Taking s=b2−x2,we obtain

Inserting(3.93)–(3.95)into(3.92)gives

As before,letwhereisanalytic in CE and behaves likes z2 as z→ ∞.Clearly,ν±(x)= ±πiµt(x)for x ∈ E.Define

where the path of integration from b to z lies entirely in the region z∈ C(−∞,b],except for the initial point b.Similarly,define

where the path of integration from −b to z lies entirely in the region z ∈ C[−b,∞),except for the initial point−b.Moreover,we define

where the path of integration from a to z lies entirely in the region z∈ C(−∞,−a)∪(a,∞),except for the initial point a.The following two results correspond to Propositions 3.1–3.2 for case(i),and to Propositions 3.3–3.4 for case(ii).

Prop osition 3.5 The mapping properties ofφ(z)are given by

and

for z large enough.Similarly,the mapping properties ofare given by

and

for z large enough.Moreover,the mapping properties ofφ(z)are given by

and

for z large enough.

Prop osition 3.6 The following connection formula holds between the g-function andφfunction

Furthermore,we have for z∈C±,

4 Contour Deformation

In this section,we consider the problem of contour deformation in three separate cases.We give more details in the first case;for the other two cases,we only present the conclusions since the results can be obtained in the same manner as in the first case.

4.1 Case for c>−2

With the properties ofφ(z)and eφ(z)established in Proposition 3.1,the jump matrix for T(z)in condition(T b)can be written as follows:

for x ∈ (0,αt),

for x∈ (−αt,0),

for x ∈ (αt,∞),

for x∈ (−∞,−αt).

Note that for the jump matrix on(0,αt),we have the following factorization:

The first and the third matrices on the right-hand side of(4.5)have the analytic continuation

on both sides of(0,αt).When x ∈ (−αt,0),we have a similar result with φ(x)replaced byBased on the factorization in(4.5),we transform the RHP for T(z)into a RHP for V(z)given below,by opening a lens around(−αt,αt)going through the origin(see Figure 1).The precise shape of the lens will be specified later;for now we choose it to be contained in the region of analyticity ofφ(z)and eφ(z).

Let denote the lens-shaped contour shown in Figure1.Thesecond transformation T→V is then defined by

where regions I,II,III,IV arealso shown in Figure1.Furthermore,wedefine thejump matrices

It is readily verified that V satisfies the conditions of the following RHP:

(V a)V(z)is analytic in

(V b)for

(V c)V(z)behaves like the identity matrix at infinity:

for z∈C(Σ∪R);

(V d)ifα<0,

ifα>0,

From Proposition 3.1 and the behavior ofin(3.33)near z= αt,the lens-shaped regions can be chosen sufficiently small so that

for z∈I∪II.Similarly,it can be shown that

for z∈III∪IV.These together with(3.36)and(3.40)imply that the jump matrix J V(z)tends exponentially to the identity matrix asforWhen z ∈ (−αt,0)∪ (0,αt),J V(z)is the constant matrix given by(4.11).It is therefore natural to suggest that for large n,the solution of the RHP for V(z)may behave asymptotically like the solution of the following RHP for V∞(z):

(V∞,a)V∞(z)is analytic in C[−αt,αt];

(V∞,b)for x ∈ (−αt,0)∪(0,αt),

(V∞,c)V∞(z)behaves like the identity matrix at infinity:

for z ∈ C[−αt,αt].

The construction of V∞(z)was done in[9–11],and we have

where

with a branch cut along[−αt,αt]and β(z)→ 1 as z→ ∞.It is worthwhile to point out that V∞(z)has the factorization

where σ3 is the Pauli matrix given in(3.7).By(4.6)and(4.12)–(4.13),it is clear that T ∼ V as n→∞.Hence,it follows that

On account of(3.8),we can work backwards to get

where l N is given in(3.32).

Let L(z)represent the right-hand side of(4.18).By(3.47),it is easily verified that

for z∈ CR,where log(−z)is analytic in z∈ C[0,∞).Hence,we obtain from(3.44),(3.47)and(4.19)that

for z∈CR.

4.2 Case for c=−2

Here,we follow the arguments given in Subsection 4.1,and transform the RHP for T(z)into the RHP for V(z)by opening a lens passing through−√2,0 and√2 as shown in Figure 1.By the same reasoning given in Subsection 4.1,it is natural to suggest that for large n,the solution of the RHP for V(z)should behave asymptotically like the solution of the following RHP for V∞(z):

(V∞,a)V∞(z)is analytic in

(V∞,b)for

(V∞,c)V∞(z)behaves like the identity matrix at infinity:

This problem can be solved explicitly,and its solution is given by

where

with a branch cut alongand β(z)→ 1 as z→ ∞ (see(4.14)–(4.15)).Again,we note that V∞(z)has the factorization given in(4.16).

4.3 Case for c<−2

With the properties ofandestablished in Proposition 3.6,the jump matrix for T(z)in condition(T b)can be written as follows:

for x∈(a,b),

for x ∈ (−b,−a),

for x∈(b,∞),

for x∈ (−∞,−b),

for x∈(−a,0),

for x∈(0,a).

From(4.5),the transformation T→V is then defined by

Furthermore,we define the jump matrix

Since for for x<−b andφ(x)>0 for x>b,as in the previous cases we expect the solution of the RHP for V(z)to behave asymptotically like the solution of the following RHP for V∞(z):

(V∞,a)V∞(z)is analytic in C[−b,b];

(V∞,b)for x∈E,

for x∈ (−a,0),

and for x∈(0,a),

(V∞,c)for z∈CE,

We construct V∞(z)in terms of the Szegö function D associated with eαπi on E,which is an analytic function on C[−b,b]satisfying D+(x)D−(x)=1 for x ∈ E,andfor x ∈ [−a,0]andfor x ∈ [0,a].We seek the function D(z)in the formThen the problem is reduced to constructing a scalar functionwhich is an analytic function on CE satisfying for x ∈ E,andfor x ∈ (−a,0)and for x ∈ (0,a).We can solve this scalar RHP by the Plemelj formula to yield

This gives

and

Therefore,the RHP for V∞(z)can be transformed into the RHP forwith the jump matrixon E,via the formula

Thus,we have the RHP for

is analytic in C[−b,b];

for x∈E,

and for x∈(−a,a){0},

for z∈CE,

The RHP forcan be solved explicitly by

where

with a branch cut along E and as z → ∞.(This can actually be derived by using an elliptic theta function(see[16,pp.15–16]).It is worthwhile to note thatcan be written as

whereσ3 is the Pauli matrix given in(3.7).Since T∼V as n→∞,it follows that

(see(4.17)).On account of(3.8),we can work backwards to get

where l N is given in(3.96).

5 Construction of the Parametrices

In this section,we will construct an approximation U∗(z)to the solution of the RHP for U(z)for large n.We again divide our discussion into three cases.For the first case,we bring in Bessel functions in the formation of a parametrix in a region containing the origin.For the second case,we introduce solutions associated with Painlevé II equation and construct the model RHP in the formation of a parametrix in a region containing the origin.For the third case,no special attention is required in the neighborhood of the origin,since it now lies outside the support of the equilibrium measure.

For regions outside the origin,we make use of Airy functions in all three cases.We give more details for the first case,and present only the conclusions for the two other cases.

5.1 Case for c>−2

Due to the singularity of|x|2αin the weight function,special attention must be paid to the neighborhood of the origin,which we will discuss first.The arguments in this subsection parallel those in[20].

5.1.1 Parametrix in the neighborhood of the origin

Let Uδbe a small disk with center at 0 and radiusδ>0.We seek a 2×2 matrix-valued function P(z)in Uδ,which has the same jumps as U(z)and matches the behavior of U(z)on the boundary∂Uδof the disk(see(4.18)).That is,we wish to find a 2×2 matrix-valued function that satisfies the following RHP:

(P a)P(z)is analytic in UδR;

(P b)for x ∈ (−δ,δ){0},

(P c)for z∈ ∂UδR,P(z)satisfies the matching condition

as n→∞.

Denote by

the analytic continuation of the functionto the whole complex plane with a branch cut along the negative real-axis.It is easily seen that the jump matrix in(P b)has the following factorization:

In order to solve the RHP for P(z),we first transform it into the RHP for

Clearly,the function on the right-hand side of P(z)in(5.3)is the analytic continuation of the functionto the whole complex plane with a branch cut along the negative real-axis.Now,let us consider the RHP for e P:

is analytic in UδR;

)for x ∈ (−δ,δ){0},

satisfies the matching condition

as n→∞(see(3.45)and(4.18)).

To solve the above RHP for,we first consider the matrix-valued function

where takes the branch cut along the negative real-axis,and Iν,Kν are modified Bessel functions defined in the complex plane with cut along the negative real-axis.Recall that

where m is an integer.It is easily shown that

Furthermore,from the well-known formulas

as z→∞inwe have

asζ→∞.Next,we introduce the function

which is analytic in a neighborhood of the origin by Proposition 3.1.In fact,it follows from(3.23)and(3.33)that

as z→0.A comparison of conditionsandforwith(5.5)and(5.7)shows that

where

for z∈C+and

for z∈ C−.The jump condition(V∞,b)for(V∞)implies that E0(z)is actually analytic in Uδ.Indeed,with the aid of(4.14)and(5.11)–(5.12),we have the explicit formula

where

is analytic in C(−∞,−αt]∪ [αt,∞),where αt is shown in(3.18).Finally,a combination of(5.3)and(5.10)gives

5.1.2 Parametrices outside Uδ

For z outside theδ-neighborhood Uδof the origin,we will construct the parametrices by using Airy functions and elementary functions.To facilitatethe following discussions,we divide the complex plane into four parts:UδandΩi,i=1,2,3,by the contours(see Figure 3).Note thatand Γ3∪Γ4 is the complex conjugation ofΓ1∪Γ2.The curveΓ1 is chosen so that the functionφ(z)is one-to-one inΩ1∩C+and satisfies 0for z∈Ω1∩C+.Similarly,we chooseΓ2 such that for

In view of(4.16),we can rewrite(4.18)as

and from(3.45)we obtain

To find an approximation to U(z),we first look for a matrix which is asymptotic to

From Proposition 3.1,it is clear that the function defined by

is analytic in C(−∞,−αt],whereφ(z)is defined in(3.33)and depends on N.In particular,by the construction ofΩ1,we have for z∈Ω1∩C+,

Also,forFrom the asymptotic behavior of the Airy function(see[17,p.392]),we have

where.It is then immediate that for z∈Ω1∩C+,

Similarly,forwe also have−πN(z)<0 and

For z∈Ω3,there is a result corresponding to(5.15)withφ(z)and l N replaced byandrespectively.Indeed,it follows from(3.46),(4.16)and(4.20)that

Let

which is analytic in C[αt,∞).Also,note that

for z∈Ω3.Hence,as before,it can be shown that the matrix

is the leading term in the asymptotic expansion of the matrices

for z∈Ω3∈C+,and

for z∈Ω3∈C−.Define the matrix function

An appeal to the formulas

and

shows that Q(z)satisfies

The above heuristic argument now suggests that U(z)is asymptotically approximated by

where

for z∈Ω1,and

for z∈ Ω3.Note that the functionsandαlog(−z)}are analytic continuations of

to the cut planes C(−∞,0]and C[0,∞),respectively.Hence,from(5.27)it follows that

On the other hand,sinceandare analytic in C(−∞,−αt]and C[αt,∞),respectively,one easily sees thatfor x ∈ (δ,∞)andfor x ∈ (−∞,−δ).From(5.28)and(5.31),it can be shown that

for all x ∈ R[−δ,δ].Furthermore,P∗(z)has the same large z behavior as U(z)shown in(U c)(see[20,(5.38)and(5.39)]).

In summary,it is now natural to suggest that U(z)is asymptotically approximated by

5.2 Case for c=−2

In this subsection,we first introduce the Ψα function associated with Painlevé II equation and then construct the model RHP to be used in the formation of a parametrix in a region containing the origin.

5.2.1 Ψα functions for Painlevé II equation and model RHP

LetΨα(ζ,s)be a 2× 2 complex-valued matrix function.We consider the following linear differential equations:

and

where u and v are functions of s.The compatibility condition forΨαshows that u(s)should satisfy the Painlevé equation

and v(s)=u′(s).Now,we focus on equation(5.35)and view u,v,s as parameters.In each sector

for k=1,···,6,there exists a unique solution Ψα,k of(5.35)such that

asζ→∞ in each section S k.

Since Ψα,k(ζ,s)are solutions of the same linear differential equation,they are related by the so-called Stokes matrices A k,i.e.,

and

Here,each A k is a triangular matrix of the form

where a k,k=1,2,···,6,are complex numbers satisfying

The above properties can be found in[14,p.164].The entries a k,which are also called Stokes multipliers,are in general dependent on the parameters s,u and v.However,Flaschka and Newell[13]showed that if u(s)is a solution of Painlevé II equation(5.37)and v(s)=u′(s),then a k are constants.In particular,for the Hastings-Mcleod solution of Painlevé II equation,which satisfies the boundary conditions

and

and

the choice is

(see[6,p.604]).If we set

then it follows from above equations thatsatisfies the following RHP:

is analytic in CR;

forζ∈R,

asζ→∞,

Since all solutions to the second Painlevé equation are meromorphic functions with infinite number of poles,the RHP forshown above is solvable if and only if s does not belong to the set of poles of u(s).

With the above preparation,we are now ready to formulate a model RHP for our later use.Letθ∈R,and define

It is then easily seen that is a solution of the following RHP:

is analytic in CR;

forζ∈ R,

where the real line is oriented from−∞to∞,φand defined in(3.52)and(3.53),respectively;

asζ→∞,

In the following subsection,we will show how Ψ(ζ;s,θ)is involved in the construction of a parametrix in the neighborhood of z=0.

5.2.2 Parametrix in the neighborhood of the origin

Let Uδbe a domain in the complex plane containing the origin,the size of which will be determined later.Wefirst wish to find a matrix-valued function Q(z)that satisfies the following RHP:

(Q a)Q(z)is analytic in UδR;

(Q b)for x∈Uδ∩R,

whereφandare defined in(3.52)and(3.53).

(Q c)Q(z)satisfies the matching condition

as n→ ∞ for z∈UδR,where V∞ is the matrix given in(4.21)–(4.22).

A comparison of conditionsand(Q b)invokes us to introduce the mapping

By Proposition 3.3,we see that the right-hand side of(5.42)is analytic in

on account of(ν)+= −(ν)− forand can be written as

where

with branch cuts alongIt is clear that

and

as z→0.To make the mapping(5.42)one-to-one at least in a neighborhood of zero,it is natural to set

Since s=0∈R,the RHP foris solvable(see the comment followingThus,is known.On account ofand(Q c),we set

where

The jump condition(V∞,b)for V∞ shows that E0(z)is actually analytic inIn fact,it follows from(4.21)and(5.13)that we have the explicit formula

where

is analytic in

In the literature,one may find the use of a“double-scaling”approach;that is,c= −2+ρN−ǫ for some constantsρandǫ>0.In this case,the parametrix in the neighborhood of the origin can still be constructed in terms of the function Ψα(ζ,s)for the model RHP associated with Painlevé II equation.However,the variable s depends on the parameter c;in our case(i.e.,c=−2),we have s=0(see[4,6]).

5.2.3 Parametrices outside the origin

For z outside theδ-neighborhood Uδof the origin,we will(as before)construct the parametrices by using Airy functions and elementary functions.To facilitate discussion,we divide the complex plane into four parts:UδandΩi,i=1,2,3(see Figure 3).

Letβ(z)be given as in(5.48),and let Q(z)be given as in(5.45)withφ(z)andin(3.52)and(3.53),respectively.As in Subsection 5.1,we expect U(z)to be asymptotically approximated by

where

for z∈Ω1,and

for z ∈ Ω3,and l N and are shown in(3.51)and(3.67).V∞ is the matrix given in(4.21)–(4.22).Note that U∗(z)also satisfies the jump condition in(U b)for U(z)in Section 2.

5.3 Case for c<−2

Note that unlike cases(i)and(ii),here no special attention is needed for the singularity of the weight function at the origin.The main reason is that the interval(−a,a),and hence the origin,lies outside the support of the equilibrium measure.Therefore,the orthogonal polynomials

in(1.2)have no zero in a small neighborhood of the origin.

In this subsection,we will construct an approximation U∗(z)to the solution of the RHP for U(z)as n→∞,by using only Airy functions and elementary functions.To facilitate the following discussions,we divide the complex plane into four parts:Ωi,i=1,···,4(see Figure 4).

In the view of(4.35),we can rewrite(4.37)as

and from(3.111)we obtain

To find an approximation to U(z),we first look for a matrix which is asymptotic to

From Proposition 3.5,it is clear that the function defined by

is analytic in C(−∞,a],whereφ(z)is defined in(3.97)and depends on N.In particular,for z∈Ω1∩C+,we have

Also,foras.Now,as in Subsection 5.1.2,we have from the asymptotic behavior of the Airy function(see[17,p.392]),

for z∈Ω1∩C+,where

Also,we have

and

for.There is a result corresponding to(5.52)withφ(z)replaced byfor z∈Ω3.Indeed,from(3.111)and(4.35),it follows that

Let

which is analytic in C[−a,∞).Also,note that

for z∈Ω3.Hence,as before,it can be shown that the matrix

is the leading term in the asymptotic expansion of the matrices

for z∈Ω3∩C+,and

for z∈Ω3∩C−.For z∈Ω4 and z∈Ω2,we replaceφ(z)andand define

which is analytic in C(−∞,−b]∪[b,∞)and

for z∈Ω4 and z∈Ω2.Thus,let us define the matrix function

The identities

and

show that Q(z)satisfies

The above heuristic argument suggests that U(z)is asymptotically approximated by

where

for z∈Ω1,and

for z∈ Ω3,whereand

for z∈Ω4∩C±,

for.From(5.66)it can be shown that

for all x∈ R{0}.Furthermore,U∗(z)has the same large z behavior as U(z)shown in(U c).

6 Uniform Asymptotic Expansions

In this section,we will present uniform asymptotic expansions of the monic polynomials πn(x)in Theorem 2.1.As in the previous sections,there are three cases to be considered,but in all three cases,we will use the same notations with different meaning.

6.1 Case for c>−2

To derive such expansions,we define the matrix-valued function

Since U(z)and U∗(z)havethe samejump condition on(−δ,δ){0},S(z)is analytic in Uδ{0}.In fact,0 is a removable singularity of S(z),(see[20,(6.2)–(6.9)]).Also,it is easily seen that S(z)satisfies the following RHP:

(S a)S(z)is analytic for z∈CΓ,whereΓis the contour shown in Figure 3.

(S b)for z∈Γ,

where;

(S c)for z∈ CΓ,

as z→∞.

To solve this problem,we first derive the asymptotic expansion of the jump matrix J(z)as n→∞.This is done in exactly the same manner as in[20],and we have

where the coefficients J k(z)are explicitly given 2×2 matrices;for details,see[20,p.750].

An appeal to Theorem 3 in[20]shows that the solution of the RHP for S has the asymptotic expansion

as n→∞uniformly for z∈CΓ,where N=n+αand the coefficient functions S k(z)can be determined recursively by

for z∈CΓ.

The proofs of the following theorems can be carried out along the same lines as given in[19–20].

Theorem 6.1 LetΩi,i=1,2,3,and Uδbe the regions shown in Figure 3.With N= and f N(z)defined in(3.32)and(5.17),the asymptotic expansion of the polynomialπn(N 14 z)is given by

where A(z,N)and B(z,N)are analytic inΩ1 and have asymptotic expansions

and

uniformly for z ∈ Ω1.In(6.6)–(6.7),the coefficient functionrefers to the element in the i th row and j th column of the matrix S k(z),which is given in(6.4).The functionβ(z)is given in(4.15).

Similarly,withgiven in(5.22),we have

whereandare analytic inΩ3 and have asymptotic expansions

and

uniformly for z∈Ω3.

Letφ(z)be defined as in(3.33).We have

where A 1(z,N)and B 1(z,N)are analytic functions of z inΩ2.In addition,they have asymptotic expansions

and

uniformly inΩ2.

Finally,when z∈ Uδ,we have

where ηN(z)is defined in(5.8),A 0(z,N)and B 0(z,N)are analytic functions in Uδ with asymptotic expansions

as n→∞.The leading coefficients are given by

where E 0(z)is given in(5.13).

6.2 Cases for c≤−2

Similarly,we have the following two theorems on global asymptotic expansions of the polynomialsπn(x)in Theorem 2.1.

Theorem 6.2 LetΩi,i=1,2,3,and Uδbe the regions shown in Figure 3,with N=n+α, and f N(z)defined in(3.51)and(5.50).The asymptotic expansion of the polynomiais given

where A(z,N)and B(z,N)are analytic inΩ1 and have asymptotic expansions

and

uniformly for z ∈ Ω1.In(6.16)–(6.17),the function β(z)is given in(4.22).The coefficient functionrefers to the element in the i th row and j th column of the matrix S k(z),which is given in(6.4).In(6.4),the coefficients J k(z)are 2×2 matrices explicitly given in[19,p.148].

Letφ(z)be defined as in(3.52).We have

where A 1(z,N)and B 1(z,N)are analytic functions of z inΩ2.In addition,they have asymptotic expansions

and

uniformly inΩ2.

Let(ψ1(ζ,s),ψ2(ζ,s))be the unique solution of equation(5.35)characterized by the asymptotic behavior

asζ→ ∞ for 0

as n→∞ uniformly for z∈Uδ,where

with

and

When z∈Ω3,a corresponding asymptotic expansion can be obtained by using the reflection formula

Theorem 6.3 Let Ωi,i=1,···,4 be the regions shown in Figure 4,with N=n+ α, and f N(z)defined in(3.96)and(5.54),respectively.The asymptotic expansion of the polynomialis given by

where A(z,N)and B(z,N)are analytic inΩ1 and have asymptotic expansions

and

uniformly for z∈Ω1.Furthermore,

the coefficient functions and W ij in(6.23)–(6.24),i,j=1,2,refer to the elements in the i th row and j th column of the matrices S k(z)and W(z),respectively.The function eγ(z)is given in(4.34).Moreover,D(z)and D∞ are given in(4.31)–(4.32).

Similarly,withgiven in(5.63),we have

where A(z,N)and B(z,N)are analytic inΩ2 and have asymptotic expansions

and

uniformly for z ∈ Ω2.In(6.26)–(6.27),

When z∈Ω3 and z∈Ω4,a corresponding asymptotic expansion can be obtained by using the reflection formula

By Theorem 2.1,there exists a 2×2 matrix Y1 such that

(see condition(Y c)).The recurrence coefficient βn(t;α)in(1.2)is related to Y1 by βn(t;α)=(Y1)12(Y1)21(see,e.g.,[6,(5.2)]).Thus,the asymptotics of the recurrence coefficient βn(t;α)can also be derived by using the asymptotics of Y as n→∞.

Acknowledgement The first author is grateful to Dan Dai,Yu Lin,Xiang-Sheng Wang and Lun Zhang for useful discussions.

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