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

摘要

The Pollaczek weight is an example of the non-Szego 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 alpha(n) and beta(n). It is also shown, by analysing the singularities of the phi-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.