{
  "id": "2307.03086",
  "version": "v1",
  "published": "2023-07-06T15:57:51.000Z",
  "updated": "2023-07-06T15:57:51.000Z",
  "title": "New series involving binomial coefficients",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper, we deduce several new identities on infinite series with denominators of summands containing both binomial coefficients and linear parts. We mainly evaluate the sums $$\\sum_{k=1}^\\infty\\frac{x_0^k}{(2k-1)\\binom{3k}k},\\ \\sum_{k=0}^\\infty\\frac{x_0^k}{(3k+1)\\binom{3k}k} \\ \\text{and}\\ \\sum_{k=0}^\\infty\\frac{x_0^k}{(3k+2)\\binom{3k}k}$$ for any $x_0\\in(-27/4,27/4)$. In particular, we have $$\\sum_{k=1}^\\infty\\frac{(\\frac{9-3\\sqrt3}2)^k}{(2k-1)\\binom{3k}k}=(\\sqrt3-1)\\log(\\sqrt3+1).$$ We also prove that $$\\sum_{k=1}^\\infty \\frac{\\sum_{k<j\\le 3k}1/j}{(2k-1)2^k\\binom{3k}k}=\\frac{\\pi^2}{36}-\\frac{\\pi}{18} -\\frac2{15}(G+\\log^22)+\\frac{\\log2}3,$$ where $G$ is the Catalan constant. In addition, we pose some new conjectural series identities involving binomial coefficients; for example, we conjecture that $$\\sum_{k=1}^\\infty\\frac{\\binom{2k}k^3}{4096^k}\\bigg(9(42k+5)\\sum_{j=0}^{k-1}\\frac1{(2j+1)^4}+\\frac{25}{(2k+1)^3}\\bigg)=\\frac 56\\pi^3.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-06T15:57:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "11B65",
      "11A07",
      "11B68",
      "33B15"
    ],
    "keywords": [
      "binomial coefficients",
      "conjectural series identities",
      "catalan constant",
      "infinite series",
      "linear parts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}