{
  "id": "1709.08550",
  "version": "v1",
  "published": "2017-09-25T15:20:00.000Z",
  "updated": "2017-09-25T15:20:00.000Z",
  "title": "Asymptotic formulae for Eulerian series",
  "authors": [
    "Nian Hong Zhou"
  ],
  "comment": "3^3 pages",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "Let $(a;q)_{\\infty}$ be the $q$-Pochhammer symbol and $\\mathrm{li}_2(x)$ be the dilogarithm function. Let $\\prod_{\\alpha,\\beta,\\gamma}$ be a finite product with every triple $(\\alpha,\\beta,\\gamma)\\in(\\mathbb{R}_{>0})^3$ and $S_{\\alpha\\beta\\gamma}\\in\\mathbb{R}$. Also let the triple $(A,B,v)\\in\\left(\\mathbb{R}_{>0}\\times\\mathbb{R}^2\\right)\\cup\\left(\\{0\\}^2\\times\\mathbb{R}_{>0}\\right)\\cup\\left(\\{0\\}\\times\\mathbb{R}_{<0}\\times\\mathbb{R}\\right)$. In this work, we let $z=e^v$, denote by $H_{-1}(u)=vu-Au^2+\\sum_{\\alpha}\\mathrm{li}_2(e^{-\\alpha u})\\sum_{\\beta,\\gamma} \\beta^{-1}S_{\\alpha\\beta\\gamma}$ and consider the Eulerien series \\[\\mathcal{H}(z;q)=\\sum_{m=0}^{\\infty}\\frac{q^{Am^2+Bm}z^{m}}{\\prod\\limits_{\\alpha,\\beta,\\gamma}(q^{\\alpha m+\\gamma};q^{\\beta})_{\\infty}^{S_{\\alpha\\beta\\gamma}}}.\\] We prove that if there exist an $\\varepsilon>0$ such that $H_{-1}(u)$ is an increasing function on $[0,\\varepsilon)$, then as $q\\rightarrow 1^-$, \\[\\mathcal{H}(z;q)=\\left(1+o\\left(|\\log q|^p\\right)\\right)\\int\\limits_{0}^{\\infty}\\frac{q^{Ax^2+Bx}z^{x}}{\\prod\\limits_{\\alpha,\\beta,\\gamma}(q^{\\alpha x+\\gamma};q^{\\beta})_{\\infty}^{S_{\\alpha\\beta\\gamma}}}\\,dx\\] holds for each $p\\ge 0$. We also obtain full asymptotic expansions for $\\mathcal{H}(z;q)$ which satisfy above condition as $q\\rightarrow 1^{-}$. The complete asymptotic expansions for related basic hypergeometric series could be derived as special cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-25T15:20:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P82",
      "11F27",
      "33D15",
      "41A58"
    ],
    "keywords": [
      "eulerian series",
      "asymptotic formulae",
      "full asymptotic expansions",
      "complete asymptotic expansions",
      "related basic hypergeometric series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}