{
  "id": "1305.6031",
  "version": "v1",
  "published": "2013-05-26T14:29:07.000Z",
  "updated": "2013-05-26T14:29:07.000Z",
  "title": "Congruences for Generalized Frobenius Partitions with an Arbitrarily Large Number of Colors",
  "authors": [
    "Frank G. Garvan",
    "James A. Sellers"
  ],
  "journal": "INTEGERS 14 (2014), Article A7",
  "categories": [
    "math.NT"
  ],
  "abstract": "In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\\phi_k(n)$ where $k\\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\\geq 0,$ $c\\phi_2(5n+3) \\equiv 0\\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. Work on Ramanujan--like congruence properties satisfied by the functions $c\\phi_k(n)$ continues, with recent works completed by Baruah and Sarmah as well as the author. Unfortunately, in all cases, the authors restrict their attention to small values of $k.$ This is often due to the difficulty in finding a \"nice\" representation of the generating function for $c\\phi_k(n)$ for large $k.$ Because of this, no Ramanujan--like congruences are known where $k$ is large. In this note, we rectify this situation by proving several infinite families of congruences for $c\\phi_k(n)$ where $k$ is allowed to grow arbitrarily large. The proof is truly elementary, relying on a generating function representation which appears in Andrews' Memoir but has gone relatively unnoticed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-26T14:29:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "arbitrarily large number",
      "colored generalized frobenius partition functions",
      "congruence properties",
      "generating function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6031G"
    }
  }
}