{
  "id": "2410.18058",
  "version": "v1",
  "published": "2024-10-23T17:32:19.000Z",
  "updated": "2024-10-23T17:32:19.000Z",
  "title": "On the Heine Binomial Operators",
  "authors": [
    "Ronald Orozco López"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we introduce the Heine binomial operators H$_{n}(bD_{q})$ based on $q$-differential operator $D_{q}$. The motivation for introducing the operators H$_{n}(bD_{q})$ is that their limit turns out to be the $q$-exponential operator T$(bD_{q})$ given by Chen. The Hahn polynomials $\\Phi_{m}^{(q^n)}(b,x|q)$ can easily be represented by using the operators H$_{n}(bD_{q})$. Here, we derive $q$-exponential and ordinary generating function, Mehler's formula, Rogers formula, and other identities for the polynomials $\\Phi_{m}^{(q^n)}(b,x|q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-23T17:32:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A30",
      "33D45"
    ],
    "keywords": [
      "heine binomial operators",
      "differential operator",
      "exponential operator",
      "mehlers formula",
      "ordinary generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}