{
  "abstract": "The massive Klein-Gordon equation on the Schwarzschild exterior reduces to a radial differential equation of confluent-Heun type. We present its exact reduction by rigorously retaining a subleading centrifugal term that preserves the horizon indicial exponents but critically shifts the accessory parameters. Within the Svartholm--Schmidt hypergeometric-expansion framework, we derive the three-term recurrence relation and establish the corrected continued-fraction compatibility condition matching the minimal tail to the lower-end recurrence. Crucially, we demonstrate that the physical compactification coordinate $z=(r-1)/r$ shifts the irregular singular point to $z=1$, inherently transforming the hypergeometric expansion into a five-term recurrence that cannot degenerate to three terms. To circumvent this obstruction, we construct the horizon-normalized physical branch via a direct Frobenius series at $z=0$. This representation exhibits high-order convergence and near-precision-floor residuals against the exact radial ODE, while clarifying why quasinormal-mode spectral selection remains a connection problem at the irregular singular endpoint.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38461",
  "author_total": 3,
  "authors": [
    {
      "name": "J. G. R. Valangelis"
    },
    {
      "name": "Ailton C. Nascimento"
    },
    {
      "name": "Helder A. S. Costa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vps40c6rbwrh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38461",
  "fulltext_url": null,
  "id": "3vps40c6rbwrh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38461",
  "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",
      "gr-qc"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7215091820",
  "pdf_url": "https://arxiv.org/pdf/2609.38461",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vps40c6rbwrh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:48Z",
      "native_id": "2609.38461",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38461"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Explicit Analytical Representations for the Schwarzschild Radial Equation via Hypergeometric and Frobenius Expansions",
  "title_text": "Explicit Analytical Representations for the Schwarzschild Radial Equation via Hypergeometric and Frobenius Expansions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:48Z",
  "url": "https://research.draconitic.com/paper/3vps40c6rbwrh",
  "venue": "arXiv",
  "versions": []
}