{"paper":{"title":"Asymptotic zero behavior of Laguerre polynomials with negative parameter","license":"","headline":"","cross_cats":["math.CV"],"primary_cat":"math.CA","authors_text":"A.B.J. Kuijlaars, K.T-R McLaughlin","submitted_at":"2002-05-15T18:08:02Z","abstract_excerpt":"We consider Laguerre polynomials $L_n^{(\\alpha_n)}(nz)$ with varying negative parameters $\\alpha_n$, such that the limit $A = -\\lim_n \\alpha_n/n$ 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 $A \\in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit $r= - \\lim_n \\frac{1}{n} \\log \\dist(\\alpha_n, \\mathbb Z)$ exists, we show that the zeros accumulate on $\\Gamma_r \\cup [\\beta_1,\\beta_2]$ with $\\beta_1$ and $\\beta_2$ only depending on $A$. For $r "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0205175","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0205175/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}