{
  "id": "2301.09587",
  "version": "v1",
  "published": "2023-01-23T17:40:22.000Z",
  "updated": "2023-01-23T17:40:22.000Z",
  "title": "On some Binomial Coefficient Identities with Applications",
  "authors": [
    "Necdet Batir",
    "Sezer Sorgunand Sevda Atpinar"
  ],
  "comment": "Submitted",
  "categories": [
    "math.CO"
  ],
  "abstract": "We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \\begin{align*} \\sum_{k=0}^{n}\\binom{\\alpha}{n-k}\\binom{\\beta+k}{k}x^k &=\\sum_{k=0}^{n}(-1)^{n+k}\\binom{\\beta-\\alpha+n}{n-k}\\binom{\\beta+k}{k}(x+1)^k, \\end{align*} where $n$ is a non-negative integer and $\\alpha$ and $\\beta$ are complex numbers, which are not negative integers. Our approach is based on a particularly interesting combination of the Taylor theorem and the Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to Alzer and Kouba, and offer a new binomial sum identity. Furthermore, as applications, we give many harmonic number sum identities. As examples, we prove that \\begin{equation*} H_n=\\frac{1}{2}\\sum_{k=1}^{n}(-1)^{n+k}\\binom{n}{k}\\binom{n+k}{k}H_k \\end{equation*} and \\begin{align*} \\sum_{k=0}^{n}\\binom{n}{k}^2H_kH_{n-k}=\\binom{2n}{n} \\left((H_{2n}-2H_n)^2+H_{n}^{(2)}-H_{2n}^{(2)}\\right). \\end{align*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-23T17:40:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "05A19"
    ],
    "keywords": [
      "binomial coefficient identities",
      "applications",
      "harmonic number sum identities",
      "binomial sum identity",
      "taylor theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}