{
  "id": "2409.09809",
  "version": "v1",
  "published": "2024-09-15T18:03:52.000Z",
  "updated": "2024-09-15T18:03:52.000Z",
  "title": "Explicit Expressions for Iterates of Power Series",
  "authors": [
    "Beauduin Kei"
  ],
  "comment": "14 pages, to be published",
  "categories": [
    "math.CO",
    "math.CA"
  ],
  "abstract": "In this paper, we present five different formulas for both discrete and fractional iterations of an invertible power series $f$ utilizing a novel and unifying approach from umbral calculus. Established formulas are extended, and their proofs simplified, while new formulas are introduced. In particular, through the use of $q$-calculus identities, we eliminate the requirement for $f'(0)$ to equal $1$ and, consequently, the corresponding new expressions for the iterative logarithm are derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-15T18:03:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B12",
      "05A40",
      "05A30",
      "13F25"
    ],
    "keywords": [
      "explicit expressions",
      "invertible power series",
      "fractional iterations",
      "umbral calculus",
      "calculus identities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}