Asymptotics for Laguerre polynomials with large order and parameters

We study the asymptotic behavior of Laguerre polynomials Ln(αn)(z) as n→∞n→∞, where αn/nαn/n has a finite positive limit or the limit is +∞+∞. Applying the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems, we derive the uniform asymptotics of such polynomials, which improves on the results of Bosbach and Gawronski (1998). In particular, our theorem is useful to obtain the asymptotics of complex Hermite polynomials and related double integrals.