{
  "abstract": "An $(α,β)$ Somos-$4$ sequence $S_n$ is defined by the recurrence $S_nS_{n-4}=αS_{n-1}S_{n-3}+βS_{n-2}^2$ ($n\\geq 4$), with suitable initial values, where $α$ and $β$ are constant parameters. A widely studied question is the following: When does the Hankel transform of a generating function become an $(α,β)$ Somos-4 sequence? In particular, how can $α$ and $β$ be derived for such a function? A sufficient condition for this problem has been established by Wang and Zhang. In this paper, we obtain the following three main results. (i): We extend the Wang--Zhang sufficient condition by working over the rational function field. Then we combine this result with the Sulanke--Xin quadratic transformation to resolve all of Barry's currently unsolved $(α,β)$ Somos-4 conjectures, which arise in diverse contexts, including generalized Catalan recurrences, Riordan arrays, generalized Bernstein arrays, and elliptic curves. (ii): We show that the odd and even subsequences of an $(α,β)$ Somos-4 sequence are again $(α,β)$ Somos-4 sequences with transformed parameters. This is employed to establish Barry's Hurwitz transform conjecture. (iii): Using the theory of orthogonal polynomials, we prove a Hankel determinant formula and thereby prove a conjecture related to the $(α,β)$ Somos-4 sequence. In addition, we prove some conjectures on formulas for periodic Hankel determinants.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.08703",
  "author_total": 3,
  "authors": [
    {
      "name": "Feihu Liu"
    },
    {
      "name": "Ying Wang"
    },
    {
      "name": "Zihao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ysnhdqhc30k5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.08703",
  "fulltext_url": null,
  "id": "0ysnhdqhc30k5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.08703",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7202174952",
  "pdf_url": "https://arxiv.org/pdf/2608.08703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ysnhdqhc30k5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2608.08703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.08703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hankel Transform and $(α,β)$ Somos-4 Sequences",
  "title_text": "Hankel Transform and (α,β) Somos-4 Sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/0ysnhdqhc30k5",
  "venue": "arXiv",
  "versions": []
}