{
  "id": "2201.09409",
  "version": "v1",
  "published": "2022-01-24T01:17:24.000Z",
  "updated": "2022-01-24T01:17:24.000Z",
  "title": "Chain sequences and Zeros of a perturbed $R_{II}$ type recurrence relation",
  "authors": [
    "Vinay Shukla",
    "A. Swaminathan"
  ],
  "comment": "23 pages. arXiv admin note: text overlap with arXiv:2201.05422",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this manuscript, new algebraic and analytic aspects of the orthogonal polynomials satisfying $R_{II}$ type recurrence relation given by \\begin{align*} \\mathcal{P}_{n+1}(x) = (x-c_n)\\mathcal{P}_n(x)-\\lambda_n (x-a_n)(x-b_n)\\mathcal{P}_{n-1}(x), \\quad n \\geq 0, \\end{align*} where $\\lambda_n$ is a positive chain sequence and $a_n$, $b_n$, $c_n$ are sequences of real or complex numbers with $\\mathcal{P}_{-1}(x) = 0$ and $\\mathcal{P}_0(x) = 1$ are investigated when the recurrence coefficients are perturbed. Specifically, representation of new perturbed polynomials (co-polynomials of $R_{II}$ type) in terms of original ones with the interlacing and monotonicity properties of zeros are given. For finite perturbations, a transfer matrix approach is used to obtain new structural relations. Effect of co-dilation in the corresponding chain sequences and their consequences onto the unit circle are analysed. A particular perturbation in the corresponding chain sequence called complementary chain sequences and its effect on the corresponding Verblunsky coefficients is also studied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-24T01:17:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "30C15",
      "15A24"
    ],
    "keywords": [
      "type recurrence relation",
      "corresponding chain sequence",
      "transfer matrix approach",
      "complementary chain sequences",
      "orthogonal polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}