{
  "id": "math/0212035",
  "version": "v1",
  "published": "2002-12-03T02:36:42.000Z",
  "updated": "2002-12-03T02:36:42.000Z",
  "title": "Numerical Computation of \\prod_{n=1}^\\infty (1 - tx^n)",
  "authors": [
    "Alan D. Sokal"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "I present and analyze a quadratically convergent algorithm for computing the infinite product \\prod_{n=1}^\\infty (1 - tx^n) for arbitrary complex t and x satisfying |x| < 1, based on the identity \\prod_{n=1}^\\infty (1 - tx^n) = \\sum_{m=0}^\\infty {(-t)^m x^{m(m+1)/2} \\over (1-x)(1-x^2) ... (1-x^m)} due to Euler. The efficiency of the algorithm deteriorates as |x| \\uparrow 1, but much more slowly than in previous algorithms. The key lemma is a two-sided bound on the Dedekind eta function at pure imaginary argument, \\eta(iy), that is sharp at the two endpoints y=0,\\infty and is accurate to within 9.1% over the entire interval 0 < y < \\infty.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-12-03T02:36:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33F05",
      "05A30",
      "11F20",
      "11P82",
      "33D99",
      "65D20",
      "82B23"
    ],
    "keywords": [
      "numerical computation",
      "dedekind eta function",
      "pure imaginary argument",
      "infinite product",
      "arbitrary complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....12035S"
    }
  }
}