Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter

Abstract

Abstract. We study the asymptotic behavior of Laguerre polynomials L (αn) n (nz) as n → ∞, where α n is a sequence of negative parameters such that −α n /n tends to a limit A > 1 as n → ∞. These polynomials satisfy a non-hermitian orthogonality on certain contours in the complex plane. This fact allows the formulation of a Riemann-Hilbert problem whose solution is given in terms of these Laguerre polynomials. The asymptotic analysis of the Riemann-Hilbert problem is carried out by the steepest descent method of Deift and Zhou, in the same spirit as done by Deift et al. for the case of orthogonal polynomials on the real line. A main feature of the present paper is the choice of the correct contour.

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“…In [27], the leading term of the asymptotic expansion was obtained for α = −An, where A > 1. We consider a somewhat more general situation, α = −An − A 1 , where A 1 ∈ R, and using Theorem 1 we obtain more exact asymptotic formulae.…”

“…In [27] it is shown that Σ = Γ ∪ Σ 1 ∪ Σ 2 , where Γ is an arc that connects In [27] it is shown that the support of the equilibrium measure in the field…”

Uniform asymptotics for Jacobi polynomials with varying large negative parameters— a Riemann-Hilbert approach

“…The proof is rather routine, and we give only a brief version of it. For details, see [9] or [10]. Y (z) obviously satisfies the condition in (Y a ).…”

“…Thus, the final result usually consists of a set of asymptotic expansions, each valid in a different region; cf. [3] and [10]. In this paper, we shall present a modification of this method, and construct an asymptotic expansion for P (α n ,β n ) n (z), which holds uniformly in the upper half-plane C + = {z ∈ C : Im z ≥ 0}, where α n and β n are given in (1.2).…”

Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

“…This technique originated with Deift and Zhou [15] and was applied to the asymptotics of orthogonal polynomials by Deift et al [12,13,14]. See also [7,8,4,20,21,22,23,24] for recent developments. The Riemann-Hilbert problem for multiple orthogonal polynomials was given by Van Assche et al [39].…”