{
  "id": "2408.15212",
  "version": "v1",
  "published": "2024-08-27T17:19:31.000Z",
  "updated": "2024-08-27T17:19:31.000Z",
  "title": "Chebyshev approximation of $x^m (-\\log x)^l$ in the interval $0\\le x \\le 1$",
  "authors": [
    "Richard J. Mathar"
  ],
  "comment": "9 pages, no figures. Integrals table in the anc directory",
  "categories": [
    "math.CA"
  ],
  "abstract": "The series expansion of $x^m (-\\log x)^l$ in terms of the shifted Chebyshev Polynomials $T_n^*(x)$ requires evaluation of the integral family $\\int_0^1 x^m (-\\log x)^l dx / \\sqrt{x-x^2}$. We demonstrate that these can be reduced by partial integration to sums over integrals with exponent $m=0$ which have known representations as finite sums over polygamma functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-27T17:19:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A09",
      "41A10",
      "G.1.2"
    ],
    "keywords": [
      "chebyshev approximation",
      "polygamma functions",
      "series expansion",
      "finite sums",
      "partial integration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}