{
  "id": "1605.01572",
  "version": "v1",
  "published": "2016-05-05T12:42:09.000Z",
  "updated": "2016-05-05T12:42:09.000Z",
  "title": "An algebra generated by $x - 2$",
  "authors": [
    "Hans-Christian Herbig",
    "Daniel Herden",
    "Christopher Seaton"
  ],
  "comment": "32 pages",
  "categories": [
    "math.CO",
    "math.AC",
    "math.AG"
  ],
  "abstract": "By a theorem of R. Stanley, a graded Cohen-Macaulay domain $A$ is Gorenstein if and only if its Hilbert series satisfies the functional equation \\[ \\operatorname{Hilb}_A(t^{-1})=(-1)^d t^{-a}\\operatorname{Hilb}_A(t), \\] where $d$ is the Krull dimension and $a$ is the a-invariant of $A$. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of $\\operatorname{Hilb}_A(t)$ at $t=1$. The main idea consists of examining the graded algebra $\\mathcal F=\\bigoplus_{r\\in \\mathbb{Z}}\\mathcal F_r$ of formal power series in the variable $x$ that fulfill the condition $\\varphi(x/(x-1))=(1-x)^r\\varphi(x)$. As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree $r=-(a+d)=0$, these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of H. W. Gould and L. Carlitz on power sums of symmetric number triangles is established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-05T12:42:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "11B68",
      "13H10",
      "13A50"
    ],
    "keywords": [
      "functional equation",
      "cubic relations",
      "euler polynomials",
      "hilbert series satisfies",
      "main idea consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}