Uniform asymptotics and zeros of the associated Pollaczek polynomials

2020 ◽  
Vol 145 (4) ◽  
pp. 625-646 ◽  
Author(s):  
Xiao‐Min Huang ◽  
R. Wong
Keyword(s):  
Uniform Asymptotics ◽  
Pollaczek Polynomials

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

2011 ◽  
Vol 09 (04) ◽  
pp. 447-480 ◽  
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.

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

2008 ◽  
Vol 464 (2096) ◽  
pp. 2091-2112 ◽  
Author(s):  
Jian-Rong Zhou ◽  
Yu-Qiu Zhao
Keyword(s):  
Complex Plane ◽  
Airy Function ◽  
Asymptotic Approximations ◽  
Uniform Asymptotics ◽  
Recurrence Coefficients ◽  
Special Values ◽  
Pollaczek Polynomials

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.

Uniform asymptotics for Meixner–Pollaczek polynomials with varying parameters

2011 ◽  
Vol 349 (19-20) ◽  
pp. 1031-1035 ◽  
Author(s):  
Jun Wang ◽  
Weiyuan Qiu ◽  
Roderick Wong
Keyword(s):  
Uniform Asymptotics ◽  
Varying Parameters ◽  
Pollaczek Polynomials

Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials

2018 ◽  
Vol 11 (3) ◽  
pp. 29-39
Author(s):  
E. I. Jafarov ◽  
A. M. Jafarova ◽  
S. M. Nagiyev
Keyword(s):  
Recurrence Relations ◽  
Pollaczek Polynomials
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