{
  "id": "2410.07922",
  "version": "v1",
  "published": "2024-10-10T13:47:46.000Z",
  "updated": "2024-10-10T13:47:46.000Z",
  "title": "Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers",
  "authors": [
    "S. R. Mane"
  ],
  "comment": "18 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are presented, using Fuss-Catalan numbers. Properties of the roots are enumerated. We quantify the accuracy of asymptotic approximations for $F_k(n)$ for $n\\gg1$. Our results subsume and extend some results published by previous authors. We also comment on the use of multinomial sums for the $k$-generalized Fibonacci numbers and related sequences. We also employ the generating function to express $F_k(n)$ as a concise (non-nested) sum of binomial coefficients for arbitrary $k$. Finally, we present a basis (or `fundamental solutions') to solve the above recurrence for arbitrary initial conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-10T13:47:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11B37",
      "39A06"
    ],
    "keywords": [
      "generalized fibonacci numbers",
      "fuss-catalan numbers",
      "arbitrary initial conditions",
      "explicit expressions",
      "results subsume"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}