{
  "id": "2305.06072",
  "version": "v1",
  "published": "2023-05-10T11:53:02.000Z",
  "updated": "2023-05-10T11:53:02.000Z",
  "title": "Explicit, recurrent, determinantal expressions of the $k$th power of formal power series and applications to the generalized Bernoulli numbers",
  "authors": [
    "Said Zriaa",
    "Mohammed Mouçouf"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the unit of any formal power series. We apply these results to the generalized Bernoulli numbers and Bernoulli numbers, we derive new closed-form expressions and some recursive relations of these famous numbers. In addition, we present several identities in determinant form. Using these results, an elegant generalization of a well known identity of Euler is presented. We also note some connections between the Stirling numbers of the second kind and the generalized Bernoulli numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-10T11:53:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "formal power series",
      "generalized bernoulli numbers",
      "th power",
      "determinantal expressions",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}