Rational spectral transformation of continued fractions associated to a perturbed $R_I$ type recurrence relations

DOI10.1007/S40840-023-01561-8arXiv2201.05422MaRDI QIDQ6388256

Publication date: 14 January 2022

Abstract: n this work, orthogonal polynomials satisfying recurrence relation

m c P_{n + 1} (z) = (z - c_{n}) m c P_{n} (z) - l a m b d a_{n} (z - a_{n}) m c P_{n - 1} (z),

with

m c P_{- 1} (z) = 0

and

m c P_{0} (z) = 1

are analyzed when modifications of the recurrence coefficient is considered. Specifically, representation of new perturbed polynomials in terms of old unperturbed ones, behaviour of zeros and spectral transformation of Stieltjes function are given. Further, Toda lattice equations corresponding to perturbed system of recurrence coefficients are obtained. Finally, when

l a m b d a_{n}

is a positive chain sequence, co-dilation and its consequences are interpreted with the help of some illustrations.

Mathematics Subject Classification ID

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis (42C05) Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) (30C15) Continued fractions (11A55) Continued fractions; complex-analytic aspects (30B70)

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