### Abstract

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} ∪ [β_{1}, β_{2}] with β_{1} and β_{2} 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.

Original language | English (US) |
---|---|

Pages (from-to) | 497-523 |

Number of pages | 27 |

Journal | Constructive Approximation |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2004 |

### Fingerprint

### Keywords

- Nonlinear steepest descent
- Riemann-Hilbert problems
- Sensitivity to parameter

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics

### Cite this

*Constructive Approximation*,

*20*(4), 497-523. https://doi.org/10.1007/s00365-003-0536-3