{
  "abstract": "Let $ν_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\\bigl(1+ν_2(m)/2\\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\\prod_{m\\ge 1}(1-q^{2N_m})$ and $\\prod_{r\\ge 1}\\bigl(q^{2^r r};q^{2^{r+1}r}\\bigr)_\\infty$. Our method is to regard the exponent map $m\\mapsto m(ν_2(m)+2)$ as a dynamical system on the positive integers: the factors of the associated quotient link into chains along its forward orbits, and the three orbits seeded at $1$, $4$ and $5$ are pairwise disjoint and telescope to exactly $1/\\bigl((1-q)(1-q^4)(1-q^5)\\bigr)$. The exponent triple $(1,4,5)$ is admissible in the sense of earlier work by Liu, which yields the desired factorization into two series with nonnegative coefficients. This orbit telescoping technique appears to be a mechanism complementary to the Pólya--Szeg\\H o criterion that underlies most existing positivity results of this kind in the literature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25739",
  "author_total": 2,
  "authors": [
    {
      "name": "Manjil P. Saikia"
    },
    {
      "name": "Abhishek Sarma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/807ewt9wcbdjv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25739",
  "fulltext_url": null,
  "id": "807ewt9wcbdjv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25739",
  "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",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214117394",
  "pdf_url": "https://arxiv.org/pdf/2609.25739",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/807ewt9wcbdjv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25739",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25739"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the positivity of truncated pentagonal number series and some conjectures of Merca",
  "title_text": "On the positivity of truncated pentagonal number series and some conjectures of Merca",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/807ewt9wcbdjv",
  "venue": "arXiv",
  "versions": []
}