Chain sequences and Zeros of a perturbed $R_{II}$ type recurrence relation

TL;DR

This paper investigates algebraic and analytic properties of orthogonal polynomials satisfying a perturbed $R_{II}$ recurrence relation, focusing on zero interlacing, chain sequences, and effects on Verblunsky coefficients.

Contribution

It introduces new representations of perturbed $R_{II}$ polynomials, analyzes zero interlacing and monotonicity, and explores the impact of chain sequence perturbations on the unit circle and Verblunsky coefficients.

Findings

Representation of perturbed polynomials in terms of original ones

Interlacing and monotonicity of zeros under perturbation

Effects of chain sequence perturbations on Verblunsky coefficients

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.