{
  "id": "2403.05708",
  "version": "v1",
  "published": "2024-03-08T22:53:00.000Z",
  "updated": "2024-03-08T22:53:00.000Z",
  "title": "Extending the Meijer $G$-function",
  "authors": [
    "Dmitrii Karp",
    "Alexey Kuznetsov"
  ],
  "comment": "27 pages, 2 figures",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "By replacing the Euler gamma function by the Barnes double gamma function in the definition of the Meijer $G$-function, we introduce a new family of special functions, which we call $K$-functions. This is a very general class of functions, which includes as special cases Meijer $G$-functions (thus also all hypergeometric functions ${}_p F_q$) as well as several new functions that appeared recently in the literature. Our goal is to define the $K$-function, study its analytic and transformation properties and relate it to several functions that appeared recently in the study of random processes and the fractional Laplacian. We further introduce a generalization of the Kilbas-Saigo function and show that it is a special case of $K$-function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-08T22:53:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C60",
      "33B15"
    ],
    "keywords": [
      "euler gamma function",
      "barnes double gamma function",
      "special cases meijer",
      "fractional laplacian",
      "random processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}