Uniform asymptotics of the Pollaczek polynomials via the Riemann–Hilbert approach

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.

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UNIFORM ASYMPTOTICS OF A SYSTEM OF SZEGÖ CLASS POLYNOMIALS VIA THE RIEMANN–HILBERT APPROACH

Analysis and Applications ◽

10.1142/s0219530511001947 ◽

2011 ◽

Vol 09 (04) ◽

pp. 447-480 ◽

Cited By ~ 5

Author(s):

JIAN-RONG ZHOU ◽

SHUAI-XIA XU ◽

YU-QIU ZHAO

Keyword(s):

Monic Polynomial ◽

Large Degree ◽

Asymptotic Formulas ◽

Uniform Asymptotics ◽

Singular Behavior ◽

Classical Orthogonal Polynomials ◽

Recurrence Coefficients ◽

Polynomial Degree ◽

Pollaczek Polynomials ◽

Class Polynomials

We study the uniform asymptotics of a system of polynomials orthogonal on [-1, 1] with weight function w(x) = exp {-1/(1 - x2)μ}, 0 < μ < 1/2, via the Riemann–Hilbert approach. These polynomials belong to the Szegö class. In some earlier literature involving Szegö class weights, Bessel-type parametrices at the endpoints ±1 are used to study the uniform large degree asymptotics. Yet in the present investigation, we show that the original endpoints ±1 of the orthogonal interval are to be shifted to the MRS numbers ±βn, depending on the polynomial degree n and serving as turning points. The parametrices at ±βn are constructed in shrinking neighborhoods of size 1 - βn, in terms of the Airy function. The polynomials exhibit a singular behavior as compared with the classical orthogonal polynomials, in aspects such as the location of the extreme zeros, and the approximation away from the orthogonal interval. The singular behavior resembles that of the typical non-Szegö class polynomials, cf. the Pollaczek polynomials. Asymptotic approximations are obtained in overlapping regions which cover the whole complex plane. Several large-n asymptotic formulas for πn(1), i.e. the value of the nth monic polynomial at 1, and for the leading and recurrence coefficients, are also derived.

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Hyperasymptotics for integrals with finite endpoints

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1992.0156 ◽

1992 ◽

Vol 439 (1906) ◽

pp. 373-396 ◽

Cited By ~ 14

Keyword(s):

Asymptotic Expansion ◽

Complex Plane ◽

Airy Function ◽

Error Function ◽

Steepest Descent ◽

Similar Form ◽

Complete Asymptotic Expansion ◽

Complementary Error Function ◽

Plane Passing ◽

Method Of Steepest Descent

Berry & Howls (1991) (hereinafter called BH) refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp {– kf ( z )} along doubly infinite contours in the complex plane passing over saddlepoints of f ( z ). Here we derive analogous results for integrals with integrands of a similar form, but whose local expansions in powers of 1/ k are made about the finite endpoints of semi-infinite contours of integration. We treat endpoints where f ( z ) behaves locally linearly or quadratically. Generically, local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f ( z ). We derive ‘resurgence relations’ which express the original integral exactly as a truncated endpoint expansion plus a remainder, involving the global saddle structure of f ( z ) via integrals through certain ‘adjacent’ saddles. The saddles adjacent to the endpoint are determined by a topological rule. If the least term of the endpoint expansion is the N 0 ( k ) th, summing to here calculates the endpoint integral up to an error of approximately exp ( – N 0 ( k )). We develop a scheme, involving iteration of the new resurgence relations with a similar one derived in BH, which can reduce this error down to exp( – 2.386 N 0 ( k )). This ‘hyperasymptotic’ formalism parallels that of BH and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane, provided that it does not coincide with other saddles. We illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc( x ) and a constructed ‘incomplete’ Airy function.

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GLOBAL ASYMPTOTICS OF STIELTJES–WIGERT POLYNOMIALS

Analysis and Applications ◽

10.1142/s0219530513500280 ◽

2013 ◽

Vol 11 (05) ◽

pp. 1350028 ◽

Cited By ~ 3

Author(s):

Y. T. LI ◽

R. WONG

Keyword(s):

Complex Plane ◽

Airy Function ◽

The Other ◽

Asymptotic Formulas ◽

Global Asymptotics

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials Sn(z; q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function Aq(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function Aq(x) and the ordinary Airy function Ai (x) as q → 1.

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