{
  "id": "2307.02638",
  "version": "v1",
  "published": "2023-07-05T20:13:57.000Z",
  "updated": "2023-07-05T20:13:57.000Z",
  "title": "Note on expanding implicit functions into formal power series by means of multivariable Stirling polynomials",
  "authors": [
    "Alfred Schreiber"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Starting from the representation of a function $f(x,y)$ as a formal power series with Taylor coefficients $f_{m,n}$, we establish a formal series for the implicit function $y=y(x)$ such that $f(x,y)=0$ and the coefficients of the series for $y$ depend exclusively on the $f_{m,n}$. The solution to this problem provided here relies on using partial Bell polynomials and their orthogonal companions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-05T20:13:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F25",
      "11B83",
      "05A19",
      "11C08"
    ],
    "keywords": [
      "formal power series",
      "expanding implicit functions",
      "multivariable stirling polynomials",
      "partial bell polynomials",
      "taylor coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}