{
  "id": "2401.14197",
  "version": "v1",
  "published": "2024-01-25T14:18:15.000Z",
  "updated": "2024-01-25T14:18:15.000Z",
  "title": "Proof of conjectures on series with summands involving $ \\binom{2k}{k}8^k/(\\binom{3k}{k}\\binom{6k}{3k})$",
  "authors": [
    "Zhi-Wei Sun",
    "Yajun Zhou"
  ],
  "comment": "23 pages, 5 tables. A sequel to arXiv:2401.12083v1",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\\binom{2k}k8^k/(\\binom{3k}k\\binom{6k}{3k})$. For example, we prove that \\[\\sum_{k=0}^\\infty\\frac{(350k-17)\\binom{2k}k8^k} {\\binom{3k}k\\binom{6k}{3k}}=15\\sqrt2\\,\\pi+27\\] and \\[\\sum_{k=1}^\\infty\\frac{\\left\\{(5k-1)\\left[16\\mathsf H_{2k-1}^{(2)}-3\\mathsf H_{k-1}^{(2)}\\right]-\\frac{12(6k-1)}{(2k-1)^2}\\right\\}\\binom{2k}k8^k} {k(2k-1)\\binom{3k}k\\binom{6k}{3k}}=\\frac{\\pi^3}{12\\sqrt2},\\] where $\\mathsf H^{(2)}_m$ denotes the second-order harmonic number $\\sum_{0<j\\leq m}\\frac1{j^2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-25T14:18:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33B15",
      "33B30",
      "33F10",
      "11M32",
      "11B65"
    ],
    "keywords": [
      "cyclotomic multiple zeta values",
      "conjectures",
      "second-order harmonic number",
      "conjectural identities",
      "infinite series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}