{
  "id": "2501.18744",
  "version": "v1",
  "published": "2025-01-30T20:49:13.000Z",
  "updated": "2025-01-30T20:49:13.000Z",
  "title": "On the $q$-factorization of power series",
  "authors": [
    "Robert Schneider",
    "Andrew V. Sills",
    "Hunter Waldron"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Any power series with unit constant term can be factored into an infinite product of the form $\\prod_{n\\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note $q$-analogues of our enumeration formulas.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-30T20:49:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B10",
      "11P81",
      "40A20"
    ],
    "keywords": [
      "power series",
      "factorization",
      "unit constant term",
      "partition enumeration functions",
      "enumeration formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}