{
  "id": "1806.04424",
  "version": "v1",
  "published": "2018-06-12T10:01:16.000Z",
  "updated": "2018-06-12T10:01:16.000Z",
  "title": "Partition implications of a new three parameter $q$-series identity",
  "authors": [
    "Atul Dixit",
    "Bibekananda Maji"
  ],
  "comment": "30 pages, submitted for publication",
  "categories": [
    "math.CO"
  ],
  "abstract": "A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews, Garvan and Liang, which itself generalizes the famous result of Fokkink, Fokkink and Wang. This three-parameter identity also leads to several new weighted partition identities as well as a natural proof of a recent result of Garvan. This natural proof gives interesting number-theoretic information along the way. We also obtain a new result consisting of an infinite series involving a special case of Fine's function $F(a,b;t)$, namely, $F(0,q^n;cq^n)$. For $c=1$, this gives Andrews' famous identity for $\\mathrm{spt}(n)$ whereas for $c=-1, 0$ and $q$, it unravels new relations that the divisor function $d(n)$ has with other partition-theoretic functions such as the largest parts function $\\mathrm{lpt}(n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-12T10:01:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "11P84",
      "05A17"
    ],
    "keywords": [
      "series identity",
      "partition implications",
      "three-parameter identity",
      "natural proof",
      "largest parts function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}