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In this paper we study the asymptotics (as $n\to \infty$) of the sequences ofLaguerre polynomials with varying complex parameters $\alpha$ depending on thedegree $n$. More precisely, we assume that $\alpha_n = n A_n, $ and $ \lim_nA_n=A \in \mathbb{C}$. This study has been carried out previously only for$\alpha_n\in \mathbb{R}$, but complex values of $A$ introduce an asymmetry thatmakes the problem more difficult.  The main ingredient of the asymptotic analysis is the right choice of thecontour of orthogonality, which requires the analysis of the global structureof trajectories of an associated quadratic differential on the complex plane,which may have an independent interest.  While the weak asymptotics is obtained by reduction to the theorem ofGonchar--Rakhmanov--Stahl, the strong asymptotic results are derived via thenon-commutative steepest descent analysis based on the Riemann-Hilbertcharacterization of the Laguerre polynomials.

Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters