Yu-Qiu Zhao

An Infinite Asymptotic Expansion for the Extreme Zeros of the Pollaczek Polynomials

In this paper, we first establish an integral expression for the Pollaczek polynomials from a generating function. By applying a canonical transformation to the integral and carrying out a detailed analysis of the integrand, we derive a uniform asymptotic expansion for in terms of the Airy function and its derivative, in descending powers of N. The uniformity is in an interval next to the turning point, with M being a constant. The coefficients of the expansion are analytic functions of a parameter that depends only on t where, and not on the large parameter N. From the expansion of the polynomials we obtain an asymptotic expansion in powers of for the largest zeros. As a special case, a four-term approximation is provided for comparison and illustration. The method used in this paper seems to be applicable to more general situations.

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

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 Szego class. In some earlier literature involving Szego 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-Szego 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.

Full asymptotic expansions of the Landau constants via a difference equation approach

Abstract We derive full asymptotic expansions for the Landau constants G n as n → ∞ . Some of the expansions are not new, yet all the coefficients of the expansions are given iteratively in an explicit manner, and are more efficiently evaluated as compared with the known results. We obtain the asymptotic formulas, old and new, by applying the theory of Wong and Li for second-order linear difference equations. In deriving the expansions, we have also confirmed a conjecture made by Nemes and Nemes.

Asymptotics of Landau Constants with Optimal Error Bounds

We study the asymptotic expansion for the Landau constants

Uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an accumulation point

G_n

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

Gn,

Asymptotic distributions of the zeros of certain classes of gauss hypergeometric polynomials

\begin{aligned} \pi G_n\sim \ln N + \gamma +4\ln 2 + \sum _{s=1}^\infty \frac{\beta _{2s}}{ N^{2s}},\quad n\rightarrow \infty , \end{aligned}