{
  "id": "2303.11468",
  "version": "v1",
  "published": "2023-03-20T21:52:22.000Z",
  "updated": "2023-03-20T21:52:22.000Z",
  "title": "Special functions and dual special functions in Drinfeld modules of arbitrary rank",
  "authors": [
    "Giacomo Hermes Ferraro"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a previous paper, the author showed in the context of Drinfeld-Hayes modules that the product between a special function and a Pellarin zeta function is rational, and that the latter can be interpreted as a dual special function. Both results are generalized in this paper. We use a very simple functorial point of view to interpret special functions and dual special functions in Drinfeld modules of arbitrary rank, allowing us to define a universal special function $\\omega_\\phi$ and a universal dual special function $\\zeta_\\phi$. In analogy to the Drinfeld-Hayes case, we prove that the latter can be expressed as an Eisenstein-like series over the period lattice, and that the scalar product $\\omega_\\phi\\cdot\\zeta_\\phi$, an element of $\\mathbb{C}_\\infty\\hat\\otimes\\Omega$, is rational. We also describe the module of special functions in the generality of Anderson modules, as already done by Gazda and Maurischat, and answer a question posed in that same paper about the invertibility of special functions in the context of Drinfeld-Hayes modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-20T21:52:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G09"
    ],
    "keywords": [
      "drinfeld modules",
      "arbitrary rank",
      "drinfeld-hayes modules",
      "universal dual special function",
      "pellarin zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}