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 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.