{
  "id": "2010.01594",
  "version": "v1",
  "published": "2020-10-04T14:50:14.000Z",
  "updated": "2020-10-04T14:50:14.000Z",
  "title": "Congruences for a class of eta quotients and their applications",
  "authors": [
    "Shashika Petta Mestrige"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The partition function $ p_{[1^c\\ell^d]}(n)$ can be defined using the generating function, \\[\\sum_{n=0}^{\\infty}p_{[1^c{\\ell}^d]}(n)q^n=\\prod_{n=1}^{\\infty}\\dfrac{1}{(1-q^n)^c(1-q^{\\ell n})^d}.\\] In \\cite{P}, we proved infinite family of congruences for this partition function for $\\ell=11$. In this paper, we extend the ideas that we have used in \\cite{P} to prove infinite families of congruences for the partition function $p_{[1^c\\ell^d]}(n)$ modulo powers of $\\ell$ for any integers $c$ and $d$, for primes $5\\leq \\ell\\leq 17$. This generalizes Atkin, Gordon and Hughes' congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup $\\Gamma_0(\\ell)$. Finally we used these congruences to prove congruences and incongruences of the generalized Frobenius $\\ell$-color partitions, $\\ell-$regular partitions and $\\ell-$core partitions for $\\ell=5,7,11,13$ and $17$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-04T14:50:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eta quotients",
      "partition function",
      "applications",
      "infinite family",
      "color partitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}