Asymptotic zero behavior of laguerre polynomials with negative parameter

We consider Laguerre polynomials L_n^(αn) (nz) with varying negative parameters α_n, such that the limit A = -lim_n α_n/n exists and belongs to (0, 1). For A > 1, it is known that the zeros accumulate along an open contour in the complex plane. For every A ∈ (0, 1), we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit r = -lim _n 1/n log[dist(α_n, Z)] exists, we show that the zeros accumulate on γ_r ∪ [β₁, β₂] with β₁ and β₂ only depending on A. For r ∈ [0, ∞), γ_r is a closed loop encircling the origin, which for r = +γ, reduces to the origin. This shows a great sensitivity of the zeros to α_n's proximity to the integers. We use a Riemann-Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.