{
  "id": "1309.5693",
  "version": "v2",
  "published": "2013-09-23T04:37:19.000Z",
  "updated": "2015-04-27T04:50:42.000Z",
  "title": "Monotonicity and log-behavior of some functions related to the Euler Gamma function",
  "authors": [
    "Bao-Xuan Zhu"
  ],
  "comment": "to appear in Proceedings of the Edinburgh Mathematical Society (2015)",
  "categories": [
    "math.CO"
  ],
  "abstract": "The aim of this paper is to develop analytic techniques to deal with certain monotonicity of combinatorial sequences. (1) A criterion for the monotonicity of the function $\\sqrt[x]{f(x)}$ is given, which is a continuous analog for one result of Wang and Zhu. (2) The log-behavior of the functions $\\theta(x)=\\sqrt[x]{2 \\zeta(x)\\Gamma(x+1)}$ and $F(x)=\\sqrt[x]{\\frac{\\Gamma(ax+b+1)}{\\Gamma(c x+d+1)\\Gamma(e x+f+1)}}$ is considered, where $\\zeta(x)$ and $\\Gamma(x)$ are the Riemann zeta function and the Euler Gamma function, respectively. As consequences, the strict log-concavities of the function $\\theta(x)$ (a conjecture of Chen {\\it et al.}) and $\\{\\sqrt[n]{z_n}\\}$ for some combinatorial sequences (including the Bernoulli numbers, the Tangent numbers, the Catalan numbers, the Fuss-Catalan numbers and some Binomial coefficients) are demonstrated. In particular, this contains some results of Chen {\\it et al.}, Luca and St\\u{a}nic\\u{a}. (3). By researching logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequence $\\{\\frac{(n_{0}+ia)!}{(k_0+ib)!(\\overline{k_0}+i\\overline{b})!}\\}_{i\\geq0}$ is proved. This generalizes two results of Chen {\\it et al.} that both the Catalan numbers $\\frac{1}{n+1}\\binom{2n}{n}$ and central binomial coefficients $\\binom{2n}{n}$ are infinitely log-monotonic and strengths one result of Su and Wang that $\\binom{dn}{\\delta n}$ is log-convex in $n$. (4) The asymptotically infinite log-monotonicity of derangement numbers is showed. (5)The logarithmically complete monotonicity of functions $1/\\sqrt[x]{a \\zeta(x+b)\\Gamma(x+c)}$ and $\\sqrt[x]{\\rho\\prod_{i=1}^n\\frac{\\Gamma(x+a_i)}{\\Gamma(x+b_i)}}$ is also obtained, which generalizes the results of Lee and Tepedelenlio\\v{g}lu, Qi and Li.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-23T04:37:19.000Z",
      "title": "Analytic approaches to monotonicity and log-behavior of combinatorial sequences",
      "abstract": "We develop analytic techniques to deal with monotonicity of combinatorial sequences of the forms $\\{\\sqrt[n]{z_n}\\}$ and $\\{\\sqrt[n+1]{z_{n+1}}/\\sqrt[n]{z_n}\\}$. We not only give a criterion for the monotonicity of the function $\\sqrt[x]{f(x)}$, but also nearly prove a conjecture of Chen {\\it et al.} on the strict log-concavity of the function $\\theta(x)=\\sqrt[x]{2 \\zeta(x)\\Gamma(x+1)}$, where $\\zeta(x)$ is the Riemann zeta function and $\\Gamma(x)$ is the Euler Gamma function. Further, we consider the log-behavior of the function $F(x)=\\sqrt[x]{\\frac{\\Gamma(ax+b+1)}{\\Gamma(c x+d+1)\\Gamma(e x+f+1)}}.$ As applications, we can obtain strict log-concavities of $\\{\\sqrt[n]{z_n}\\}$ for some combinatorial sequences, including the Bernoulli numbers, the Tangent numbers, the Catalan numbers, the Fuss-Catalan numbers, the Binomial coefficients $\\binom{2n}{n}$, $\\binom{3n}{n}$, $\\binom{4n}{n}$, $\\binom{5n}{n}$, $\\binom{5n}{2n}$ and so on. In particular, this implies some known results of Chen {\\it et al.}, Luca and St\\u{a}nic\\u{a}.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-27T04:50:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20",
      "26A48"
    ],
    "keywords": [
      "combinatorial sequences",
      "analytic approaches",
      "monotonicity",
      "log-behavior",
      "strict log-concavity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.5693Z"
    }
  }
}