Global asymptotics of the Szegő–Askey polynomials

The Szegő–Askey polynomials are orthogonal polynomials on the unit circle. In this paper, we study their asymptotic behavior by knowing only their weight function. Using the Riemann–Hilbert method, we obtain global asymptotic formulas in terms of Bessel functions and elementary functions for z in two overlapping regions, which together cover the whole complex plane. Our results agree with those obtained earlier by Temme [Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986) 369–376]. Temme's approach started from an explicit expression of the Szegő–Askey polynomials in terms of an2F1-function, and followed by integral methods.