{
  "id": "2409.08213",
  "version": "v1",
  "published": "2024-09-12T16:53:53.000Z",
  "updated": "2024-09-12T16:53:53.000Z",
  "title": "On determinants involving $(\\frac{j+k}p)\\pm(\\frac{j-k}p)$",
  "authors": [
    "Deyi Chen",
    "Zhi-Wei Sun"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\\frac {j+k}p)\\pm(\\frac{j-k}p)$, where $(\\frac{\\cdot}p)$ denotes the Legendre symbol. When $p\\equiv1\\pmod4$, we determine the characteristic polynomials of the matrices $$\\left[\\left(\\frac{j+k}p\\right)+\\left(\\frac{j-k}p\\right)\\right]_{1\\le j,k\\le n}\\ \\ \\text{and}\\ \\ \\left[\\left(\\frac{j+k}p\\right)-\\left(\\frac{j-k}p\\right)\\right]_{1\\le j,k\\le n},$$ and also prove that \\begin{align*} &\\ \\left|x+\\left(\\frac{j+k}p\\right)+\\left(\\frac{j-k}p\\right)+\\left(\\frac jp\\right)y+\\left(\\frac kp\\right)z+\\left(\\frac{jk}p\\right)w\\right|_{1\\le j,k\\le n} \\\\=&\\ (-p)^{(p-5)/4}\\left(\\left(\\frac{p-1}2\\right)^2wx-\\left(\\frac{p-1}2y-1\\right)\\left(\\frac{p-1}2z-1\\right)\\right), \\end{align*} which was previously conjectured by the second author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-12T16:53:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A15",
      "11C20",
      "15A15",
      "15A18"
    ],
    "keywords": [
      "legendre symbol",
      "odd prime",
      "second author",
      "characteristic polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}