{
  "abstract": "Let $\\varrho\\neq -1$ be a complex number with modulus $|\\varrho|=1$ and $K(x,y)=P_1(x)+P_2(y)$ for two polynomials $P_1$ and $P_2$. We consider the non-oscillatory solutions such that $λ(f)<\\infty$ of the generalized Hill equation $f''-K(e^z,e^{-\\varrho z})f=0$ ($\\sharp$). When $\\varrho=1$, the Hill equation $f''-K(e^z,e^{-z})f=0$ ($†$) is also written as $x^2u''-[K(x,x^{-1})-1/4]u=0$ ($‡$). We point out that there is a full correspondence between the non-oscillatory solutions of equation ($†$) and the Liouvillian solutions of equation ($‡$). Then this paper has two purposes. First, we show that if equation ($\\sharp$) has a nonzero non-oscillatory solution, then $\\varrho=1$. To this end, we solve entire solutions of a general Tumura--Clunie type differential equation. Second, for the particular Hill equation $f''-(e^{\\mathbf{k}z}+b_{\\mathbf{s}}e^{\\mathbf{s}z}+b_0)f=0$, where $\\mathbf{k}>\\mathbf{s}\\geq 1$ are integers and $b_{\\mathbf{s}}\\not=0$, we use Kovacic's algorithms to determine the non-oscillatory solutions with relatively few zeros.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24008",
  "author_total": 1,
  "authors": [
    {
      "name": "Yueyang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/by3wemd2r895j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24008",
  "fulltext_url": null,
  "id": "by3wemd2r895j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24008",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2604|Meromorphic and Entire Functions"
    ]
  ],
  "openalex": "W7214046999",
  "pdf_url": "https://arxiv.org/pdf/2609.24008",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/by3wemd2r895j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.24008",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24008"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On non-oscillatory solutions of the generalized Hill equation",
  "title_text": "On non-oscillatory solutions of the generalized Hill equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/by3wemd2r895j",
  "venue": "arXiv",
  "versions": []
}