{
  "id": "2412.02257",
  "version": "v1",
  "published": "2024-12-03T08:32:37.000Z",
  "updated": "2024-12-03T08:32:37.000Z",
  "title": "Asymptotics for the reciprocal and shifted quotient of the partition function",
  "authors": [
    "Koustav Banerjee",
    "Peter Paule",
    "Cristian-Silviu Radu",
    "Carsten Schneider"
  ],
  "categories": [
    "math.NT",
    "cs.SC",
    "math.CO"
  ],
  "abstract": "Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition function, namely $p(n+k)/p(n)$ with $k\\in \\mathbb{N}$, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $p(n+k)$ and the multiplicative inverse $1/p(n)$, which is of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-03T08:32:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "05A20",
      "11P82"
    ],
    "keywords": [
      "partition function",
      "shifted quotient",
      "reciprocal",
      "error bounds",
      "main goal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}