## Abstract

The Pollaczek weight is an example of the non-Szegö class. In this paper, we investigate the asymptotics of the Pollaczek polynomials via the Riemann–Hilbert approach. In the analysis, the original endpoints ±1 of the orthogonal interval are shifted to the Mhaskar–Rakhmanov–Saff numbers *α*_{n} and *β*_{n}. It is also shown, by analysing the singularities of the *ϕ*-function, that the endpoint parametrices constructed in terms of the Airy function are bound to be local. Asymptotic approximations are obtained in overlapping regions that cover the whole complex plane. The approximations, some special values and the leading and recurrence coefficients are compared with the known results.

## 1. Introduction

There have been various methods to study the asymptotic behaviours of orthogonal polynomials, such as the methods using differential equations, definite integrals and difference equations. The last decade saw the development of a novel approach, serving the same purpose. The approach is based on an observation made by Fokas *et al*. (1992), which links orthogonal polynomials with matrix-valued Riemann–Hilbert problems, involving only the curves of orthogonality and the weight functions. The nonlinear steepest descent method of Deift & Zhou (1993) then plays a central role in extracting the asymptotic approximations. The readers are referred to the pioneering work of Bleher & Its (1999) and Deift *et al*. (1999*a*,*b*) for the development of the steepest descent method and for its applications to orthogonal polynomials, with mathematical physical background such as nonlinear partial differential equations and random matrix theory.

Recently, Kuijlaars *et al*. (2004) have considered the modified Jacobi polynomials via the Riemann–Hilbert approach. The results in Kuijlaars *et al*. (2004) are applied to the random matrix theory by Kuijlaars & Vanlessen in a follow-up paper (Kuijlaars & Vanlessen 2002). The modified Jacobi weight supported on [−1,1] is a typical example of the Szegö class. In the investigation (Kuijlaars *et al*. 2004), the asymptotic approximation of the polynomials outside of [−1,1] is obtained in terms of the Szegö function, and the parametrices at the endpoints ±1 are constructed in Bessel functions. The procedure works in Kuijlaars *et al*. (2004) and seems to be applicable to some other Szegö-class weights.

The Pollaczek polynomials *P*_{n}(*x*; *a*, *b*) also furnish an orthogonal system on [−1,1], with weight function(1.1)where *θ*=arccos *x* for *x*∈(−1,1) and , and *a* and *b* are real constants such that |*b*|<*a*. The relation of orthogonality is(1.2)for *n*, *m*=0, 1, 2, …. The highest coefficient *k*_{n} of *P*_{n}(*x*; *a*, *b*) is (see Szegö 1975, pp. 393–396)(1.3)It has long been observed that the Pollaczek polynomials show in many aspects a singular behaviour compared with the classical polynomials (Szegö 1975). One example is the formula(1.4)for the extreme zeros, where denotes the zeros of *P*_{n}(*x*; *a*, *b*), with , and *ν* is fixed (see Novikoff 1954, compare Zhou & Zhao 2007 for refined behaviours of the extreme zeros). Consequently, we have the gap(1.5)which is noticeable comparing with, say, the Jacobi polynomials . Adapting the same notations *x*_{νn} and *θ*_{νn}, for the Jacobi polynomials we have where *j*_{α,ν} denotes the *ν*-th positive zero of the Bessel function *J*_{α}(*x*) (cf. Abramowitz & Stegun 1972, p. 797). Thus one has in this case which means that, for the Pollaczek polynomials, the gap between the largest zero and the right endpoint is much wider than those in classical cases such as Jacobi polynomials. The situation is similar as the left endpoint is concerned. See Szegö (1975, pp. 399–400) for other properties of the Pollaczek polynomials indicating a singular behaviour.

Ismail (1994) and Bo & Wong (1996) have investigated the large-*n* asymptotic behaviour of these polynomials, aiming partially at finding the next term in the asymptotic expansion of *θ*_{νn}. Both the authors use integrals. But Ismail's approach uses the confluent Horn function while Bo and Wong use the Airy function in their derivation of the uniform asymptotic expansion for the Pollaczek polynomials. Both eqn (63) in Zhou & Zhao (2007) and formula (6.11) below agree with the result given by Bo & Wong (1996). The term uniform is used here for the *θ*-interval of the validity of the expansion, despite the fact that it is narrow, covers a turning point . To extend the interval of validity, Zhao & Zhou (2006) have made an effort to obtain a universal expansion in the whole interval *θ*∈(0,*π*). This has partially been achieved by appealing to the parabolic cylinder functions, but the terms other than the leading term seem impossible to obtain. Therefore, it is natural to turn to newly developed methods such as the Riemann–Hilbert approach.

It is readily seen from (1.1) that as *x*→1^{−} and as *x*→−1^{+}, hence we have(1.6)that is, the Szegö condition is not fulfilled. Thus the Pollaczek weight provides an example of the non-Szegö class.

We note that (i) as we mentioned previously, the Pollaczek polynomials exhibit a wider gap between the largest zero and the endpoint of the interval when compared with the Jacobi polynomials, and (ii) the weight function is actually of *C*^{∞}() if we zero-extend it to \[−1,1]. The previous observation inspires us to consider this case as if we are dealing with the polynomials orthogonal on the whole . We then set to calculate the Mhaskar–Rakhmanov–Saff (MRS) numbers *α*_{n} and *β*_{n}, the *g*-function and the *ϕ*-function. Detailed analysing of the *ϕ*-function *ϕ*_{n} shows that there are infinite many singularities of it locating on both (−∞,−1] and [1,∞), and possibly finite singularities on the upper and symmetrically on the lower half-plane. The parametrices at *α*_{n} and *β*_{n} then are restricted to be local, more precisely, in neighbourhoods of size *O*(1/*n*).

The main objective of the present investigation is to demonstrate an application of the Riemann–Hilbert approach to non-Szegö class, using the system of the Pollaczek polynomials as an example. The results obtained are the asymptotic approximations in overlapping regions that cover the whole complex *z*-plane. The local parametrices at *α*_{n} and *β*_{n} are constructed in terms of the Airy functions. The main idea, as mentioned earlier, is to consider the weight function on the whole . The endpoints are then shifted to *α*_{n} and *β*_{n}, while, in this case, the original endpoints ±1 are of no significance in the approximations.

Such narrowing of interval, of an amount small yet observable, seems to be a common feature of the non-Szegö cases, as the Riemann–Hilbert approach is applied.

The rest of this paper is arranged as follows. In §2, we derive with some details infinite asymptotic expansions of the MRS numbers. In §3, several auxiliary functions such as the *g*-function and the *ϕ*-function are studied. The analysis of the singularities of *ϕ*_{n} is also carried out in this section. Section 4 will be devoted to transformations now routine in the Riemann–Hilbert approach. The parametrices are constructed in §5 and the approximations are stated in §6. A brief comparison of the asymptotic approximations, some special values and the leading and recurrence coefficients with known results are also attached to this last section.

## 2. The MRS numbers

The *g*-function can be determined by the MRS numbers (Rakhmanov 1982; Mhaskar & Saff 1984) *α*_{n} and *β*_{n}, and the corresponding equilibrium measure *ψ*_{n}(*s*) d*s* as follows:(2.1)where the branch of ln (*z*−*s*) is chosen such that for . The function *g*_{n}(*z*) is analytic in the cut plane , and behaves like ln *z* for large |*z*|.

We proceed to determine *α*_{n}, *β*_{n} and *ψ*_{n}(*x*) by applying one of the phase conditions(2.2)where *l*_{n} is a constant to be determined.

Denote(2.3)It is readily seen that *G*_{n}(*z*) is analytic in and is equal to away from the cut plane (−∞,*β*_{n}]. Assuming that the boundary value of *G*_{n}(*z*) can be continued to (*α*_{n},β_{n}), the function *G*_{n} satisfies the scalar Riemann–Hilbert problem(2.4)with(2.5)Solving (2.4) while requiring it to have at most weak singularities at the endpoints *α*_{n} and *β*_{n} and to be bounded at infinity gives(2.6)Expanding *G*_{n}(*z*) at ∞ and comparing the leading coefficients with (2.5) yields(2.7)and(2.8)From (2.7) and (2.8), we will determine the unique existence of *α*_{n} and *β*_{n} for large *n*. The density function *ψ*_{n}(*s*) can be determined accordingly.

First, noting that and that one can write(2.9)and(2.10)with and , it is easily shown that *α*_{n}→−1 and *β*_{n}→1 as *n*→∞. In fact, from (2.7) to (2.10) we haveandwhich implies the unique solvability of *α*_{n} and *β*_{n} for large *n*. Indeed, we have(2.11)

We go further to obtain the latter terms of the asymptotic expansions for *α*_{n} and *β*_{n} in descending powers of *n*. To do so, we split the integration interval [*α*_{n},β_{n}] in (2.7) and (2.8) to . We obtain from (2.7), (2.9) and (2.10) the following relation:(2.12)where *A*_{1}, *B*_{1} and *C*_{1} are asymptotically of the form . It is verified by careful calculation that all the corresponding coefficients in *A*_{1} and *B*_{1} cancel, which makes the logarithmic terms in (2.12) cancel each other. Hence, instead of (2.12), we have the modified version(2.13)up to all orders of 1/*n*. Similarly, from (2.8) we have(2.14)up to all orders of 1/*n*, where and . The latter terms can then be determined iteratively by matching the equal orders of 1/*n* in these coupled equations. Hence we have derived expansions for 1+*α*_{n} and 1−*β*_{n} as stated in lemma 2.1.

*The MRS numbers α*_{n} *and* *β*_{n} *possess the following asymptotic expansions in descending powers of n*.(2.15)*where* *and* , *and the pairs of constants* (*a*_{j}, *b*_{j}) *are determined iteratively for* *j*=2, 3, ….

## 3. The auxiliary functions

### (a) The g-function and related functions

The function *G*_{n}(*z*) defined in (2.3) is analytic in and behaves like at ∞, thus making it a good global quantity. While, on the contrary, the function *ϕ*_{n}(*z*) makes itself a good local quantity at *z*=*β*_{n} and is defined in a way as follows.

First, we assume that there exists a function *ν*_{n}(*z*), analytic in , such that(3.1)where *Ω* is a neighbourhood of [*α*_{n},*β*_{n}]. The existence of *ν*_{n}(*z*) and the determination of *Ω* will be provided in the next subsection. Then one has(3.2)which is an analytic function in . Similarly, for later use, we define a function analytic in as(3.3)

We introduce yet another function *Φ*_{n}(*z*) as(3.4)which, like *g*_{n}(*z*), is also an analytic function in . The derivative of vanishes in the cut plane, and thus we have(3.5)by analytic continuation, where .

Now *g*_{n}(*z*) and *ϕ*_{n}(*z*) have jumps 2*πi* and −2*πi*, respectively, along . While for *x*∈(*α*_{n},β_{n}), in view of (3.1) we haveHence has only possibly isolated singularities at {*α*_{n},*β*_{n}} in *Ω*. If *g*_{n}(*z*) and *ϕ*_{n}(*z*) possess only weak singularities at these points, which is true in quite general cases, then the singularities at *α*_{n} and *β*_{n} are removable. We havewhere *E*_{n}(*z*) is an analytic function in the neighbourhood *Ω* of [*α*_{n},*β*_{n}].

### (b) The singularities of ϕ_{n}

It is crucial to determine the singularities of *ϕ*_{n}. This can indeed be turned into a problem of determining an analytic continuation of ln*w*(*x*) to a neighbourhood *Ω* of [*α*_{n},*β*_{n}], for we will show later that(3.6)The logarithm of the weight function can be written as(3.7)where and are analytic functions in , such that and for *θ*∈(0,*π*) or, equivalently, . It is worth mentioning that for . Therefore, *φ*(*z*) maps the domain onto the upper half complex plane, hence the function is a single-valued analytic function in and maps this domain onto the strip . It is also easily verified by boundary correspondence that the function *h*_{0}(*z*) maps the upper half *z*-plane conformally onto the cut right half-plane . Thus, noting the obvious fact that for , we see thatFrom (3.6) and the global analyticity of *Φ*_{n}(*z*), we see that *ϕ*_{n}(*z*) can be extended analytically to where ln*w*(*z*) is analytic. This indicates that the neighbourhood *Ω* can be chosen as , excluding the singularities of ln*w*(*z*). We note that by this way the neighbourhood *Ω* and the singularities of *ϕ*_{n}(*z*) are independent of *n*, for ln*w*(*z*) and its singularities do not depend on *n*.

We proceed to determine the singularities of ln*w*(*z*), and hence of *ϕ*_{n}(*z*). From (3.7), the endpoints ±1 are branch points of ln*w*(*z*) and *ϕ*_{n}(*z*). To find out other singularities, we recall that the gamma function possesses no zeros on the finite plane, with simple poles at all non-positive integers. The function *ϕ*_{n}(*z*) has branch points at where the gamma functions in (3.7) have simple poles, and there are no other singularities in the principal branch.

Thus the singularities are wherefor *k*=0,1,2,…. To locate the singularities, we take the first equation as an example. Settingand substituting them back into the equation, we have(3.8)the first of which implies that (i) *ρ*_{A}=ρ_{B} or (ii) with integers *l*. In case (i) the singularities, if any, locate onwhich is a circle centred at , with radius . So if *b*>0, the circle is on the left-hand side of the imaginary axis; if *b*<0, it crosses the interval [−1,1] at , thus includes within the interior the interval [*β*_{n},1]; the circle degenerates to the imaginary axis as *b*→0. Also, all the singularities are in a bounded domain. In the present case, from the second equation in (3.8) one has , which has no solution if and describes a circle passing through three points ±1 and otherwise. The two circles mentioned intersect exactly at one point *z*_{k} on the upper half-plane for each integer k, such that . The cause for proving no solution is that for Im *z*>0.

In case (ii), we must have since *θ*_{A}, *θ*_{B}∈[0,*π*]. Noting that for the upper half-plane Im *z*>0 and for the interval −1<*z*<1, the only possibilities are . The elementary calculation shows that there is one singularity for , which is if *b*>0 or if *b*<0. If there is an integer *k,* such that , then there is a singularity corresponding to this *k*, namely . For those infinitely many *k* such that , there are a pair of singularities , located on the upper edges of (−∞,−1) and (1,+∞,). It is worth noting that all the singularities are not located in (−1,1).

Similar analysis carried out for the lower half-plane with similar outcome. We summarize the results as follows; see also figure 1 below for an illustration.

*When* *with a non-negative integer k*_{0}, *there are k*_{0} *pairs of singularities of ϕ*_{n}(*z*) *locating away from* (−1,1), *on the circle* *centred at* , *with radius* . *There are also infinite many singularities of ϕ*_{n}(*z*) *located on* (1,∞) *and* (−∞,−1), *with*, *respectively*, *accumulating points* 1 *and* −1.

We are now in a position to show that (3.6) is actually valid for , where, as mentioned before, the neighbourhood *Ω* of [*α*_{n},*β*_{n}] is the domain independent of *n*, in which ln*w*(*z*) is analytic. In view of (3.2), the existence and analyticity of *ν*_{n}(*z*) can be obtained by taking derivatives on both sides of (3.6) for , and by verifying (3.1) via recalling that for *x*∈(*α*_{n},*β*_{n}) and regarding (2.4) as .

### (c) An expansion of ϕ_{n}*(*z*)* at z=β_{n}

We begin by expanding *ϕ*_{n}(*z*) for *z*∈(*β*_{n},1). From (1.1) and (2.6), we have the following expansion for *G*_{n}(*z*) as *z*∈(*β*_{n},1):(3.9)where *G*_{n,1} and *G*_{n,2} are analytic functions for *n* fixed. Thus from (3.4), we have(3.10)where *Φ*_{n,1} and *Φ*_{n,2} are also analytic functions at the origin with the radius of convergence depending on *n*. It is worth noting that the functions and take real values for *z*∈(*β*_{n},1). For , with *ϵ*(*n*) small positive, we have the boundary values asIn view of (3.1), (3.2) and (3.6), we have for *x*∈(*α*_{n},β_{n}), thus(3.11)is valid for , where *Ω* is the neighbourhood of [*α*_{n},*β*_{n}]. What remains is to evaluate the coefficients *ϕ*_{2,j}(*n*) of(3.12)To do so, resuming *z*∈(*β*_{n},1), we evaluate the integral in (2.6). The functions *f*_{P} and *f*_{N} in (2.9) and (2.10) can be expressed as sums of a regular function at *β*_{n} with a function exponentially small at *x*=1^{−} and *x*=−1^{+}, respectively. By splitting the integration interval into , and picking up all the integer powers of *z*−*β*_{n} in the expansion of the integral in (2.6), via very careful calculation we get , where , and the coefficients satisfy for arbitrary constant 0<*r*<1, where *M* may depend on *r* but not on *n*. Thus we have lemma 3.2 as follows.

*The function ϕ*_{n}(*z*) *possesses the following expansion*:(3.13)*where* ; ; *c*_{ϕ,0}=1; *and* *with constants M and r independent of n*, 0<*r*<1 *arbitrary*. *It also holds the following asymptotic expansion for large n*:(3.14)*where each ξ*_{k}(*τ*), *independent of n*, *is an analytic function of τ and is equal to* *for* |*τ*|<*r*, *for arbitrary constant r*∈(0,1), *with* .

As can be seen from the derivation of (3.13), each coefficient *c*_{ϕ,j}(*n*) can be expressed in convergent power series in 1+*α*_{n} and 1−*β*_{n}, with a factor . In view of (2.15), we have (3.14).

Near *z*=*α*_{n}, a similar expansion in descending powers of *n* for can also be established, with coefficients in terms of this time.

## 4. The transformations *Y*→*T*→*S*

Initially, the Riemann–Hilbert problem for *Y* is analytic in

*Y*(*z*) is analytic in ;*Y*(*z*) satisfies the jump condition(4.1)*Y*(*z*) has the asymptotic behaviour at infinity as(4.2)and*Y*(*z*) has at most weak singularities at*z*=±1.

It is known that the Riemann–Hilbert problem has a unique solution(4.3)where *π*_{n} is the monic polynomial, that is, , and(4.4)is the Cauchy-type integral (see Fokas *et al*. 1992). Here and hereafter we suppress the obvious dependence of *Y* on *n*, as we will do with *T*, *S*, *N*, *P* and *R*.

The first of a series of transformations is *Y*→*T*(4.5)whereThis furnishes a normalization at infinity, such that

*T*(*z*) is analytic in ;(4.6)

, as

*z*→∞; and*T*(*z*) has at most weak singularities at*z*=±1.

The function *g*_{n}(*z*) is determined as in the previous sections. It can be verified that a set of phase conditions is satisfied by *g*_{n}.

Next we show that the functionis real and negative for *x*∈(*β*_{n},1), and goes monotonically from 0 to −∞ as *x* goes rightward along the interval. To do so, we note that the specific weight function introduced in (1.1) satisfies(4.7)This fact can be verified via analysing in details along subintervals if *b*=0, and along if *b*>0, where , and totally parallel if *b*<0.

Upon (4.7), taking derivative of −2*nϕ*_{n}(*x*) and applying the mean value theorem, we have(4.8)for *x*∈(*β*_{n},1), where *s*_{0}∈(*α*_{n},β_{n}) and *x*_{1}∈(*s*_{0},*x*).

Next, another transformation *T*→*S* is based on a factorization of the jump matrix(4.9)for *x*∈(*α*_{n},β_{n}). It is appropriate to define(4.10)with the regions illustrated in figure 2. Then we have yet another Riemann–Hilbert problem satisfied by *S*, which is

*S*(*z*) is analytic in , where are indicated in figure 2;(4.11)

(4.12)and

*S*(*z*) has at most weak singularities at*z*=±1,*α*_{n},*β*_{n}, whereandObviously, we can evaluate*Y*by solving out*S*.

We are now in a position to solve the following limiting Riemann–Hilbert problem for *N*(*z*)

*N*(*z*) is analytic in ;(4.13)

(4.14)and

*N*(*z*) has at most singularities of order less than 3 at*z*=*α*_{n}and*z*=*β*_{n}.

One solution to this problem is(4.15)where ; ; ; ; and , the branch is chosen such that ϱ_{n}(*x*) is real positive for *x*>*β*_{n}. The constants *B*_{j} and are to be determined in the next section.

We chose the unusual definition of *N* in (4.15) in order to simplify the matching condition (5.6) below. Although the first two matrices on the right-hand side of (4.15) do not commute, their commutator is only *O*(1/*n*) on the circles and . Hence, the non-commutativity does not matter, in view of the asymptotic expansion (5.20) of the solution *R*(*z*) to the final Riemann–Hilbert problem for *R*.

## 5. Local parametrix and the transformation *S*→*R*

The construction of the parametrix at *z*=*β*_{n} is different from those in, for example, Deift *et al*. (1999*b*), Kuijlaars *et al*. (2004) and Wang & Wong (2005) in that the neighbourhood is of the size , which means that it shrinks as *n* increases. Constant attention has to be paid to the dependence on *n* of the quantities involved.

We introduce the auxiliary contour illustrated in figure 3, which divides the complex plane into four regions I, II, III and IV. Denote and define (cf. Deift *et al*. 1999, (7.9))(5.1)Further, from lemma 3.2 we see that(5.2)is an analytic function in a neighbourhood *U*_{β}(*r*) of *β*_{n}, where the branch is chosen such that *λ*_{n}(*z*) takes positive real values for *z*>*β*_{n}. Clearly we have and . The function *λ*_{n} maps the re-scaled neighbourhood , 0<*r*<1, conformally to a size *O*(1) neighbourhood of the origin in the *λ*-plane, where .

Introduce(5.3)which is analytic in the interior of *U*_{β}(*r*), where *f*_{R}(τ) is the regular part of , in which *ξ*_{0}(*τ*) is defined in (3.14), and *c*_{1} will appear in (5.8). We also denote the singular part by for later use, where the coefficients are and . This is the specific *f*_{S}(*τ*) in (4.15). With all these preparation done, we readily find that(5.4)which solves the following Riemann–Hilbert problem for the parametrix:

*P*(*z*) is analytic in (figure 3);*P*(*z*) satisfies the same jump conditions as*S*in*U*_{β}(*r*), i.e.(5.5)and*P*(*z*) fulfils the matching condition along ∂*U*_{β}(*r*) as(5.6)

The verification of the matching condition (5.6) is done in a manner similar to that given in Deift *et al*. (1999). Note that although *N*^{−1}(*z*) has a high-order singularity at *z*=*β*_{n}, we are only concerned with *z* on the circle . We also note that for *P* to satisfy (5.6), we have to specify the coefficients *B*_{j} in (4.15) as mentioned previously. We also note that in the verification of (P1)–(P3), use has been made of the fact that the Airy function satisfies(5.7)and that for ,(5.8)and(5.9)with (cf. Abramowitz & Stegun 1972, p. 446–448)Indeed, instead of (5.6), from (3.14) we have the refined formula(5.10)for or, equivalently, |*τ*|=*r*, where and *r*∈(0,1), and *Δ*_{k}(τ) are functions depending only on *τ*, but not on *n*.

Totally parallel, in terms of the function defined in (3.3), we can also construct the parametrix at *z*=α_{n}. An expansion similar to (5.10) now can be written as(5.11)for or, equivalently, , where , are independent of *n*. The coefficients and in (4.15) are determined by requiring and by noticing that

Now we bring in the final transformation by defining(5.12)and comparing figure 4 for the regions involved, where *U*_{β}(*r*) *U*_{α}(*r*) are, respectively, the small discs centred at *β*_{n} and *α*_{n}. So defined, the matrix-valued function *R*(*z*) satisfies a Riemann–Hilbert problem on the remaining contours illustrated in figure 4, as follows:

*R*(*z*) is analytic in (figure 4);*R*(*z*) satisfies the jump conditions(5.13)whereand*R*(*z*) demonstrates the behaviour at infinity as(5.14)

Taking (4.8) and (3.14) into account and applying a similar argument to , we know that both *J*_{S,4} and *J*_{S,5} are asymptotically *I*, plus exponentially small errors for large *n*, where the positive number *m*_{0} depends only on *r*. We proceed to discuss *J*_{S,1} and *J*_{S,3} along the corresponding portions of . Precise analysis is needed to show that *J*_{S,1} and *J*_{S,3} are also of the form of *I* plus exponentially small terms. First, we describe the closed domains *Dl*, *Dc* and *Dr*, illustrated in figure 4*a*. is a rectangle, is part of a triangle, so is , where *r*, *δ* and *δ*′ are appropriate positive constants not depending on *n*.

*There exists a constant M*_{0}>0, *independent of n*, *such that*(5.15)*for* *and n large*.

From the Cauchy–Riemann condition, . Recalling that , we have(5.16)

It follows from (4.8) and the fact as *x*→1^{−} and −1^{+} that there exists a positive constant *C*_{0}, such that for *x*∈(−1,1), while the imaginary part vanishes. Hence there exists a convex subdomain *Ω*_{1} of *Ω*, such that , and and for *z*∈*Ω*_{1}, where the positive number *C*_{1} can be made small by shrinking *Ω*_{1}. Determined by *w*, *Ω*_{1} does not depend on *n*. Accordingly we have for *s*, *z*∈*Ω*_{1}. We also have by making *δ*′ appropriately small. Having had all these, one gets(5.17)

When *z*∈*Dc*, we split the integration interval into , with 0<*δ*_{1}<*δ*. Obviously, the real part of the integration along is positive. Sincefor some positive constants *C*_{2} and *M*_{0}, we have(5.18)for *n* large enough, where the *O*(1) term, related to , is uniformly bounded in *Dc*.

As for *z*∈*Dl*, this time we split the interval to subintervals and , with *r*_{1} small as compared with *r*. Again the real part of the integral along the latter is positive. Noting the facts that along the left subintervals, and for , and for *t* in a small-opening sector with Re*t*≥*r* and , and that for *z*+1 small in a small-opening sector, we then havefor some positive constant *C*_{3} determined by *w* and . Hence for *z*∈*Dl*, we have(5.19)

The case *z*∈*Dr* can be discussed in the same way, with an estimate like (5.19) obtained, *α*_{n} being replaced by *β*_{n}. Summing up all these yields (5.15). Thus completes the proof of lemma 5.1. ▪

It now clearly follows from lemma 5.1 that on illustrated in figure 4, uniformly we have for some positive constant *C*_{4}. The same result is valid for . Thus, we have for . The asymptotic expansions for *R* are readily constructed (cf. Deift *et al*. 1999*b*; Wang & Wong 2005) as(5.20)where the coefficients can be obtained iteratively for *k*=1, 2,…,(5.21)We note that *R*_{k}(*n*,*z*) (and their boundary values) are uniformly bounded on the whole complex *z*-plane. The uniformity is in both *n* and *z*.

## 6. The asymptotic approximations

Tracing all the way back from *R* to *Y*, we can obtain the asymptotic approximation for *Y*_{11}, and thus for *P*_{n}(*z*; *a*, *b*), on the complex plane. Owing to the obvious symmetry and the reflection formula (Szegö 1975)(6.1)we need only to expand on part of the upper half-plane.

*For r belonging to a compact subinterval of* (0,1), *it holds* (*figure 5*) *the following*:

*for z*∈*A*_{r},(6.2)*where*,*and k*_{n}*is given in*(*1.3*),*for z*∈*B*_{r},(6.3)*where*,*and**for*,(6.4)*where*,*and the function λ*_{n}(*z*)*is defined in*(*5.2*).

We proceed to compare the theorem with known results. The first of them is Szegö (1975, p. 395, (5.1)), obtained by applying Darboux's method. Using the variables in this paper, it reads(6.5)for fixed *z* outside of the closed interval [−1,1], where, as in (3.7), . According to Szegö (1975), the appearance of indicates a singular behaviour compared with the classical polynomials.

Now we derive (6.5) from (6.2). For *z* keeping a positive distance *δ* from [−1,1], both *f*_{S} and are of *O*(1/*n*), hence we have . To express *g*_{n}(*z*), we introduce the Szegö function(6.6)which is analytic in , satisfying for *x*∈(−1,1) and that everywhere. We also have . One can write , with . Taking (2.6) into account, elementary calculation yields(6.7)for general weight *w*, where (cf. (2.2)). Now we bring in the Szegö function and switch the integral on [*α*_{n},β_{n}] to ones on , and obtain that(6.8)Matching the constant terms on both sides of (6.8) then gives(6.9)Detailed analysis shows that the last term on the right-hand side of (6.8) can be expressed as . Thus substituting (6.6), (6.8) and (6.9) into (6.2) gives (6.5) exactly. To some extent this justifies the necessity of bringing in the MRS numbers in the present case.

Next we turn to the *O*(1/*n*) neighbourhoods of *β*_{n}. From (3.5), (3.6) and (6.9), in general, we have(6.10)where ln *w*(*z*) is the analytic continuation of ln *w*(*x*) in a neighbourhood of (−1,1), as introduced in (3.6) and (3.7). Accordingly, we have .

Now for , we have , and , hence(6.11)with , which agrees with the rigorous results of Bo & Wong (1996, p. 169, (6.7)), and Zhou & Zhao (2007, p. 274, (63)), obtained via the uniform expansion of integrals. It is worth mentioning that the turning points and in Bo & Wong (1996) and Zhao & Zhou (2006) are asymptotically *β*_{n} and *α*_{n}, respectively, with errors of order *O*(1/*n*^{2}).

The approximation on real axis can be obtained along the same line as Deift *et al*. (1999*b*). We take the case *x*=1 as an example. Splitting the integration interval to and taking into account the fact that for *s*∼1^{−}, we see that the integral in (6.7) is of the form for *z*=1. Thus, from (6.7), (6.9) and (6.2) we have(6.12)in which the leading term is exactly what Szegö (1975, p. 396, (5.2)) states.

The approximations (6.2)–(6.4) can readily be refined to asymptotic expansions in descending integer powers of *n*. The location of the zeros can then be determined similar to (Deift *et al*. 1999*b*). There have been results on zeros of the Pollaczek polynomials (cf. Ismail 1994; Bo & Wong 1996; Zhou & Zhao 2007). We would not persuade that here in this paper.

Lastly, we calculate the highest coefficients of the polynomials and the recurrence coefficients in , where , which furnish the corresponding orthonormal system. These coefficients are already known explicitly. We do so for a consistency check. Following the steps leading to (6.2), we have(6.13)for *z* keeps a constant distance from [−1,1], where . From the relations and (cf. (4.3)), we see that(6.14)in agreement with (1.3) and the recurrence relation (cf. Szegö 1975, p. 393, (1.7))(6.15)

We complete our paper by appending a short discussion. For non-Szegö-class polynomials, it is quite general that the weights can be extended to *C*^{∞} functions on the whole real line. We demonstrated in this paper a treatment initiated by employing the MRS numbers, taking the places of the original endpoints. The situation differs from the finite-interval case as considered in Kuijlaars *et al*. (2004) in that such shift of the ends have to occur, to explain the singular behaviours of the polynomials. It is also different from the infinite-interval case (Deift *et al*. 1999*b*) in that, as mentioned earlier, the parametrization has to be local: of size *O*(1−*β*_{n}), since the function *ϕ*_{n} naturally has singularities at the original endpoints ±1.

The approach used here could be applied to other non-Szegö-class weights (Zhao 2006). We also would like to point out that several ideas seem to be applicable even to some Szegö-class weights.

## Acknowledgments

The authors are grateful to the referees for their very helpful comments and suggestions on an earlier version of the paper. This research was partially supported by the National Natural Science Foundation of China, GuangDong Natural Science Foundation and a fund from the Advanced Research Center of ZhongShan University.

## Footnotes

- Received December 29, 2007.
- Accepted March 14, 2008.

- © 2008 The Royal Society