{
  "id": "1909.01508",
  "version": "v1",
  "published": "2019-09-04T00:49:45.000Z",
  "updated": "2019-09-04T00:49:45.000Z",
  "title": "Taylor coefficients of the Jacobi $θ_{3}\\left( q \\right)$ function",
  "authors": [
    "Tanay Wakhare",
    "Christophe Vignat"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $\\theta_{3}\\left(1\\right)$ to the case $\\theta_{3}\\left(q\\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are obtained by carefully studying the properties of the cumulants associated to a $\\theta_{3}$ (or discrete normal) distributed random variable. This article also states some congruence conjectures about integers sequences that generalize the one studied by D. Romik.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-04T00:49:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "taylor coefficients",
      "integers sequences",
      "congruence conjectures",
      "dan romik",
      "discrete normal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}