{
  "id": "1406.3801",
  "version": "v1",
  "published": "2014-06-15T08:29:45.000Z",
  "updated": "2014-06-15T08:29:45.000Z",
  "title": "Ramanujan-type Congruences for Overpartitions Modulo 5",
  "authors": [
    "William Y. C. Chen",
    "Lisa H. Sun",
    "Rong-Hua Wang",
    "Li Zhang"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $\\overline{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\\overline{p}(4n+3)\\equiv 0 \\pmod{8}$ for $n\\geq 0$. They also conjectured that $\\overline{p}(40n+35)\\equiv 0 \\pmod{40}$ for $n\\geq 0$. Chen and Xia proved this conjecture by using the $(p,k)$-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that $\\overline{p}(5n)\\equiv (-1)^{n}\\overline{p}(4\\cdot 5n) \\pmod{5}$ for $n \\geq 0$ and $\\overline{p}(n)\\equiv (-1)^{n}\\overline{p}(4n)\\pmod{8}$ for $n \\geq 0$ by using the relation of the generating function of $\\overline{p}(5n)$ modulo $5$ found by Treneer and the $2$-adic expansion of the generating function of $\\overline{p}(n)$ due to Mahlburg. As a consequence, we deduce that $\\overline{p}(4^k(40n+35))\\equiv 0 \\pmod{40}$ for $n,k\\geq 0$. Furthermore, applying the Hecke operator on $\\phi(q)^3$ and the fact that $\\phi(q)^3$ is a Hecke eigenform, we obtain an infinite family of congrences $\\overline{p}(4^k \\cdot5\\ell^2n)\\equiv 0 \\pmod{5}$, where $k\\ge 0$ and $\\ell$ is a prime such that $\\ell\\equiv3 \\pmod{5}$ and $\\left(\\frac{-n}{\\ell}\\right)=-1$. Moreover, we show that $\\overline{p}(5^{2}n)\\equiv \\overline{p}(5^{4}n) \\pmod{5}$ for $n \\ge 0$. So we are led to the congruences $\\overline{p}\\big(4^k5^{2i+3}(5n\\pm1)\\big)\\equiv 0 \\pmod{5}$ for $n, k, i\\ge 0$. In this way, we obtain various Ramanujan-type congruences for $\\overline{p}(n)$ modulo $5$ such as $\\overline{p}(45(3n+1))\\equiv 0 \\pmod{5}$ and $\\overline{p}(125(5n\\pm 1))\\equiv 0 \\pmod{5}$ for $n\\geq 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-15T08:29:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "ramanujan-type congruences",
      "overpartitions modulo",
      "generating function",
      "conjecture",
      "theta functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.3801C"
    }
  }
}