Global asymptotics for Laguerre polynomials with large negative parameter -- a Riemann-Hilbert approach
Using the Riemann-Hilbert approach introduced by \textit{P. Deift} and \textit{X. Zhou} [Ann. Math. (2) 137, No. 2, 295--368 (1993; Zbl 0771.35042)], the asymptotic behaviour of the Laguerre polynomials \(L^{(\alpha_n)}_n(nz)\) as \(n\to\infty\), where \(\alpha_1\) is a sequence of negative numbers and \(-\alpha_n/n\) tends to a limit \(A> 1\) as \(n\to\infty\). An asymptotic expansion is obtained, which is uniformly valid in the upper half plane \(\mathbb C_+= \{z: \text{Im\,}z\geq 0\}\). A corresponding expansion is also obtained for the lower half plane \(\mathbb C_-= \{z:\text{Im\,}Z\geq 0\}\). The two expansions hold, in particular, in a region in the complex plane containing the curve \(\Gamma\), on which these polynomials are orthogonal.
scientific article; zbMATH DE number 5343451
scientific article