{
  "id": "1802.01374",
  "version": "v1",
  "published": "2018-02-05T13:02:33.000Z",
  "updated": "2018-02-05T13:02:33.000Z",
  "title": "Congruences for the Coefficients of the Powers of the Euler Product",
  "authors": [
    "Julia Q. D. Du",
    "Edward Y. S. Liu",
    "Jack C. D. Zhao"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\\prod _{n=1}^{\\infty}(1-q^n)^k=\\sum_{n=0}^{\\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\\alpha} n +\\frac{k(2^{2\\alpha}-1)}{3})$ $(1\\leq k\\leq 3)$ and $p_{3k} (3^{2\\beta}n+\\frac{k(3^{2\\beta}-1)}{8})$ $(1\\leq k\\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite families of congruences for $p_k(n)$ modulo any $m\\geq2$, where $1\\leq k\\leq 24$ and $3|k$ or $8|k$. Based on these congruences for $p_k(n)$, infinite families of congruences for many partition functions such as the overpartition function, $t$-core partition functions and $\\ell$-regular partition functions are easily obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-05T13:02:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euler product",
      "congruences",
      "coefficients",
      "infinite families",
      "regular partition functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}