{
  "id": "2005.09590",
  "version": "v1",
  "published": "2020-05-19T17:13:35.000Z",
  "updated": "2020-05-19T17:13:35.000Z",
  "title": "$α$, $β$-expansions of the Riordan matrices of the associated subgroup",
  "authors": [
    "E. Burlachenko"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider the group of the matrices $\\left( 1,g\\left( x \\right) \\right)$ isomorphic to the group of formal power series $g\\left( x \\right)=x+{{g}_{2}}{{x}^{2}}+...$ under composition: $\\left( 1,{{g}_{2}}\\left( x \\right) \\right)\\left( 1,{{g}_{1}}\\left( x \\right) \\right)=\\left( 1,{{g}_{1}}\\left( {{g}_{2}}\\left( x \\right) \\right) \\right)$. Denote $P_{k}^{\\alpha }=\\left( 1,x{{\\left( 1-k\\alpha {{x}^{k}} \\right)}^{{-1}/{k}\\;}} \\right)$. Matrix $\\left( 1,g\\left( x \\right) \\right)$is decomposed into an infinite product of the matrices $P_{k}^{\\alpha }$ with suitable exponents in two ways: to left-handed and right-handed products with respect to the matrix $P_{1}^{{{\\alpha }_{1}}={{\\beta }_{1}}}$: $\\left( 1,g\\left( x \\right) \\right)=...P_{k}^{{{\\alpha }_{k}}}...P_{2}^{{{\\alpha }_{2}}}P_{1}^{{{\\alpha }_{1}}}=P_{1}^{{{\\beta }_{1}}}P_{2}^{{{\\beta }_{2}}}...P_{k}^{{{\\beta }_{k}}}...$. We obtain two formulas expressing the coefficients of the series ${{\\left( {g\\left( x \\right)}/{x}\\; \\right)}^{z}}$ in terms of the expansion coefficients ${{\\alpha }_{i}}$, ${{\\beta }_{i}}$ and introduce two one-parameter families of series $g_{\\alpha }^{\\left( t \\right)}\\left( x \\right)$ and $g_{\\beta }^{\\left( t \\right)}\\left( x \\right)$ associated with these expansions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-19T17:13:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riordan matrices",
      "associated subgroup",
      "formal power series",
      "infinite product",
      "expansion coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}