Zhi-Tao WEN Roderick WONG Shuai-Xia XU

(Dedicated to Professor Philippe G.Ciarlet on the occasion of his 80th birthday)

1 Introduction

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

whereand 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 PI,and is related to solutions of the fourth Painlevé equation PIV.As a follow-up,in[7]Clarkson and Jordaan investigated the asymptotic behavior of the polynomials Sn(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 Sn(x;t),what they gave is a solutionto an approximate equation(see[7,(4.12)]),and not an asymptotic approximation to the polynomial Sn(x;t).

In the present paper,we are mainly concerned with the problem of finding asymptotic formulas for the monic polynomials Sn(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

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 Sn(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 Sn(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,whereis a constant and N=n+α.As we shall show,in the first two cases,the support of the associated equilibrium measureµtis a single interval,whereas in the third case the support ofµtconsists of two intervals.In all three cases,we will present infinite asymptotic expansions of Sn(z;t)for z in various different regions.These regions together cover the whole complex z-plane.

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 pn(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:

(Ya)Y(z)is analytic in

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

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

as z→∞.

(Yd)For z∈CR,

as z→0.

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

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

whereand

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 transformationby

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

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

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

for x∈R{0},

(Uc)For z∈CR,

as z→∞.

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

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

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.Letαtbe a positive real number,and denote by µt(s)a probability density function supported onand

The explicit formulas forαtand 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

we introduce the transformation

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

By adopting the convention thatσ3denotes the Pauli matrix

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

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

(Tb)For x∈R{0},

where

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

for z∈CR;

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

We proceed to seek for a probability density functionµt(s)in(3.1)and a constant lNmentioned above so that J12in jump condition(Tb)becomes 1 for x ∈ (−αt,αt).Thus,we set

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

Diff erentiating(3.9)yields

where

For convenience,we set

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= αtsinθ,it follows that

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

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

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

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

3.1 Case for c>−2

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

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

Let us define

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

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

Let the integrals on the left-hand side of(3.27)be denoted by I1,I2and 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

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

Similarly,we define

where the path of integration from −αtto 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 3.1LetandWe 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 3.2With constant lNgiven in(3.32)anddefined 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

we have from(3.23)

Withandthe formula in(3.32)gives

Similarly,we define

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 3.3Denotingandwe 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 3.4With constant lNgiven in(3.51)anddefined 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 0

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 eng(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(Ta)-(Td).To determine a probability density functionµt(s)in(3.71)and a constant lNmentioned above so that the entry J12in jump condition(Tb)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

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

which gives

for x∈E,where a,b are given in(3.85).

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

Let us define

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,I2and 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

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 3.5The 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 3.6The 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(Tb)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).

Figure 1 The lens-shaped contourΣgoing through the origin.

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:

(Va)V(z)is analytic in

(Vb)for

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

for z∈C(Σ∪R);

(Vd)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 JV(z)tends exponentially to the identity matrix asforWhen z ∈ (−αt,0)∪ (0,αt),JV(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 σ3is 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 lNis 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∞,c)V∞(z)behaves like the identity matrix at infinity:

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

where

4.3 Case for c<−2

With the properties ofandestablished in Proposition 3.6,the jump matrix for T(z)in condition(Tb)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).

Figure 2 The lens-shaped contourΣ = Σ1∪···∪ Σ4.

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

Furthermore,we define the jump matrix

Sinceforfor 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 eαπion 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 satisfyingfor x ∈ E,andfor x ∈ (−a,0)andfor 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 andas 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

whereσ3is 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 lNis 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:

(Pa)P(z)is analytic in UδR;

(Pb)for x ∈ (−δ,δ){0},

(Pc)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(Pb)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

wheretakes the branch cut along the negative real-axis,and Iν,Kνare 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 αtis shown in(3.18).Finally,a combination of(5.3)and(5.10)gives

5.1.2 Parametrices outside Uδ

Figure 3 Contour Γi,i=1,···,8 and the domains Ω1,Ω2,Ω3.

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−π

For z∈Ω3,there is a result corresponding to(5.15)withφ(z)and lNreplaced 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

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

for all x ∈ R[−δ,δ].Furthermore,P∗(z)has the same large z behavior as U(z)shown in(Uc)(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 Ψαfunction 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,kof(5.35)such that

asζ→∞ in each section Sk.

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

and

Here,each Akis a triangular matrix of the form

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

The above properties can be found in[14,p.164].The entries ak,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 akare 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 thatis a solution of the following RHP:

is analytic in CR;

forζ∈ R,

where the real line is oriented from−∞to∞,φanddefined 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:

(Qa)Q(z)is analytic in UδR;

(Qb)for x∈Uδ∩R,

whereφandare defined in(3.52)and(3.53).

(Qc)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(Qb)invokes us to introduce the mapping

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

where

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(Qc),we set

where

where

In the literature,one may find the use of a“double-scaling”approach;that is,c= −2+ρN−ǫfor 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 lNandare 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(Ub)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).

Figure 4 The domains Ω1,···,Ω4.

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∈Ω4and z∈Ω2,we replaceφ(z)andand define

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

for z∈Ω4and 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(Uc).

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:

(Sb)for z∈Γ,

where;

(Sc)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 Jk(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 Sk(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].

where A(z,N)and B(z,N)are analytic inΩ1and 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 Sk(z),which is given in(6.4).The functionβ(z)is given in(4.15).

Similarly,withgiven in(5.22),we have

whereandare analytic inΩ3and have asymptotic expansions

and

uniformly for z∈Ω3.

Letφ(z)be defined as in(3.33).We have

where A1(z,N)and B1(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),A0(z,N)and B0(z,N)are analytic functions in Uδwith asymptotic expansions

as n→∞.The leading coefficients are given by

where E0(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.

where A(z,N)and B(z,N)are analytic inΩ1and 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 Sk(z),which is given in(6.4).In(6.4),the coefficients Jk(z)are 2×2 matrices explicitly given in[19,p.148].

Letφ(z)be defined as in(3.52).We have

where A1(z,N)and B1(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

where A(z,N)and B(z,N)are analytic inΩ1and have asymptotic expansions

and

uniformly for z∈Ω1.Furthermore,

the coefficient functionsand Wijin(6.23)–(6.24),i,j=1,2,refer to the elements in the i th row and j th column of the matrices Sk(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 inΩ2and have asymptotic expansions

and

uniformly for z ∈ Ω2.In(6.26)–(6.27),

When z∈Ω3and z∈Ω4,a corresponding asymptotic expansion can be obtained by using the reflection formula

By Theorem 2.1,there exists a 2×2 matrix Y1such that

(see condition(Yc)).The recurrence coefficient βn(t;α)in(1.2)is related to Y1by 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→∞.

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

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