{
  "id": "math/0608398",
  "version": "v1",
  "published": "2006-08-15T17:25:25.000Z",
  "updated": "2006-08-15T17:25:25.000Z",
  "title": "Mixed powers of generating functions",
  "authors": [
    "Manuel Lladser"
  ],
  "comment": "14 pages",
  "journal": "Discrete Mathematics and Theoretical Computer Science Proceedings, AG, 171-182, 2006",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Given an integer m>=1, let || || be a norm in R^{m+1} and let S denote the set of points with nonnegative coordinates in the unit sphere with respect to this norm. Consider for each 1<= j<= m a function f_j(z) that is analytic in an open neighborhood of the point z=0 in the complex plane and with possibly negative Taylor coefficients. Given a vector n=(n_0,...,n_m) with nonnegative integer coefficients, we develop a method to systematically associate a parameter-varying integral to study the asymptotic behavior of the coefficient of z^{n_0} of the Taylor series of (f_1(z))^{n_1}...(f_m(z))^{n_m}, as ||n|| tends to infinity. The associated parameter-varying integral has a phase term with well specified properties that make the asymptotic analysis of the integral amenable to saddle-point methods: for many directions d in S, these methods ensure uniform asymptotic expansions for the Taylor coefficient of z^{n_0} of (f_1(z))^{n_1}...(f_m(z))^{n_m}, provided that n/||n|| stays sufficiently close to d as ||n|| blows up to infinity. Our method finds applications in studying the asymptotic behavior of the coefficients of a certain multivariable generating functions as well as in problems related to the Lagrange inversion formula for instance in the context random planar maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-15T17:25:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A16",
      "30B10",
      "41A60",
      "60C05"
    ],
    "keywords": [
      "generating functions",
      "mixed powers",
      "methods ensure uniform asymptotic expansions",
      "asymptotic behavior",
      "taylor coefficient"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8398L"
    }
  }
}