{
  "abstract": "We consider singular (aka genus 0) walks in the quarter plane and their associated generating functions $Q(x, y, t)$, which enumerate the walks starting from the origin, of fixed endpoint (encoded by the spatial variables $x$ and $y$) and of fixed length (encoded by the time variable $t$). We first prove that the previous series can be extended up to a universal value of $t$ (in the sense that this holds for all singular models), namely $t = 1/2$ , and we provide a probabilistic interpretation of $Q(x, y, 1/2)$. As a second step, we refine earlier results in the literature and show that $Q(x, y, t)$ is indeed differentially transcendental for any $t \\in (0, 1/2]$. Moreover, we prove that $Q(x, y, 1/2)$ is strongly differentially transcendental. As a last step, we show that for certain models the series expansion of $Q(x, y, 1/2)$ is directly related to Bernoulli numbers. This provides a second proof of its strong differential transcendence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "Alin Bostan"
    },
    {
      "name": "Lucia Di Vizio"
    },
    {
      "name": "Kilian Raschel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/98x6y34ysm2c0.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "98x6y34ysm2c0",
  "issns": [
    "0030-8730",
    "1945-5844"
  ],
  "landing_url": "https://hal.science/hal-05172463v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05172463/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": "Mathematical Sciences Publishers",
  "ris": "https://research.draconitic.com/paper/98x6y34ysm2c0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "hal-05172463",
      "source": "hal",
      "url": "https://hal.science/hal-05172463v2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Singular walks in the quarter plane and Bernoulli numbers",
  "title_text": "Singular walks in the quarter plane and Bernoulli numbers",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:00Z",
  "url": "https://research.draconitic.com/paper/98x6y34ysm2c0",
  "venue": "Pacific Journal of Mathematics",
  "versions": []
}