{
  "id": "1905.01821",
  "version": "v1",
  "published": "2019-05-06T04:27:32.000Z",
  "updated": "2019-05-06T04:27:32.000Z",
  "title": "Extensions of Bailey's $_6ψ_6$ series identity with seven free parameters",
  "authors": [
    "Chuanan Wei"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In the literature of basic hypergeometric series, Bailey's $_6\\psi_6$ series identity is very important. So finding the nontrivial extension of it is a quite significative work. In this paper, we establish, above all, a transformation formula involving two $_8\\psi_8$ series and a $_8\\phi_7$ series according to the analytic continuation argument. When the parameters are specified, it reduces to Bailey's $_6\\psi_6$ series identity and gives two different bilateral generalizations of the nonterminating form of Jackon's $_8\\phi_7$ summation formula. Subsequently, a new extension of Ramanujan's $_1\\psi_1$ series identity involving two $_2\\psi_2$ series and a $_2\\phi_1$ series is derived from this formula and a known transformation formula for bilateral basic hypergeometric series. Finally, a new proof for the known extension of Bailey's $_6\\psi_6$ series identity involving a $_8\\psi_8$ series and two $_8\\phi_7$ series is also offered via the analytic continuation argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-06T04:27:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "series identity",
      "seven free parameters",
      "analytic continuation argument",
      "bilateral basic hypergeometric series",
      "transformation formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}