{
  "id": "2501.02499",
  "version": "v1",
  "published": "2025-01-05T10:28:58.000Z",
  "updated": "2025-01-05T10:28:58.000Z",
  "title": "$q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials",
  "authors": [
    "Bakir Farhi"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper, we investigate a specific class of $q$-polynomial sequences that serve as a $q$-analogue of the classical Appell sequences. This framework offers an elegant approach to revisiting classical results by Carlitz and, more interestingly, to establishing an important extension of the Carlitz $q$-Bernoulli polynomials and numbers. In addition, we establish explicit series representations for our extended Carlitz $q$-Bernoulli numbers and express them in terms of $q$-Stirling numbers of the second kind. This leads to a novel formula that explicitly connects the Carlitz $q$-Bernoulli numbers with the $q$-Stirling numbers of the second kind.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-05T10:28:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A30",
      "11B68",
      "11B73",
      "05A40"
    ],
    "keywords": [
      "bernoulli numbers",
      "extended carlitz",
      "natural numbers",
      "consecutive powers",
      "second kind"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}