{
  "id": "2210.07579",
  "version": "v1",
  "published": "2022-10-14T07:13:37.000Z",
  "updated": "2022-10-14T07:13:37.000Z",
  "title": "Summation from the viewpoint of distributions",
  "authors": [
    "Su Hu",
    "Min-Soo Kim"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "Let $\\{a_{1}, a_{2},\\ldots, a_{n},\\ldots\\}$ be a sequence of complex numbers. In this paper, inspired by a recent work of Sasane, we give an explanation of the sum $$a_{1}+2a_{2}+3a_{3}+\\cdots+na_{n}+\\cdots,$$ and more generally, for any $k\\in\\mathbb{N},$ the sum $$1^{k}a_{1} +2^{k}a_{2} +3^{k}a_{3} +\\cdots+n^{k}a_{n} +\\cdots,$$ from the viewpoint of distributions. As applications, we explain the following summation formulas \\begin{equation*} \\begin{aligned} 1^{k}-2^{k}+3^{k}-\\cdots&=-\\frac{E_{k}(0)}{2}, \\\\ 1^{k}+2^{k}+3^{k}+\\cdots&=-\\frac{B_{k+1}}{k+1}, \\\\ \\epsilon^{1}1^{k}+\\epsilon^{2}2^{k}+\\epsilon^{3}3^{k}+\\cdots&=-\\frac{B_{k+1}(\\epsilon)}{k+1}, \\end{aligned} \\end{equation*} where $E_{k}(0)$, $B_{k}$ and $B_{k}(\\epsilon)$ are the Euler polynomials at 0, the Bernoulli numbers and the Apostol-Bernoulli numbers, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-14T07:13:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46F12",
      "11B68",
      "11M06"
    ],
    "keywords": [
      "distributions",
      "euler polynomials",
      "complex numbers",
      "apostol-bernoulli numbers",
      "summation formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}