{
  "id": "2301.10661",
  "version": "v1",
  "published": "2023-01-25T16:04:46.000Z",
  "updated": "2023-01-25T16:04:46.000Z",
  "title": "Values of $p$-adic hypergeometric functions, and $p$-adic analogue of Kummer's linear identity",
  "authors": [
    "Neelam Saikia"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime and $\\mathbb{F}_p$ be the finite field with $p$ elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the $p$-adic setting which we denote by ${_{3n-1}G_{3n-1}}(p, t),$ where $n\\geq1$ and $t\\in\\mathbb{F}_p$. These values are expressed in terms of numbers of zeros of certain polynomials over $\\mathbb{F}_p$. These results lead to certain $p$-adic analogues of classical hypergeometric identities. Namely, we obtain $p$-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zeros of these functions. For instance, if $n$ is odd then we obtain zeros of ${_{3n-1}G_{3n-1}}(p, t)=0$ under certain condition on $t$. In contrast we show that if $n$ is even then the function ${_{3n-1}G_{3n-1}}(p, t)$ has no non-trivial zeros for any prime $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-25T16:04:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic analogue",
      "adic hypergeometric functions",
      "kummers linear identity",
      "non-trivial zeros",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}