Spectral transformation associated with a perturbed $R_I$ type recurrence relation

TL;DR

This paper investigates spectral transformations of orthogonal polynomials satisfying a perturbed $R_I$ recurrence relation, analyzing their spectral properties, computational methods, and effects on associated functions and zeros.

Contribution

It introduces a spectral transformation framework for perturbed $R_I$ polynomials, compares transfer matrix and classical methods, and explores effects on Carathéodory functions and zeros.

Findings

Transfer matrix method is more efficient for perturbed polynomials.

Perturbations affect spectral properties and Carathéodory functions.

Zeros of perturbed polynomials interlace with classical ones.

Abstract

In this work, orthogonal polynomials satisfying $R_I$ type recurrence relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-\lambda_n (z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and $\mathcal{P}_0(z) = 1$ are analyzed when the recurrence coefficients are modified. The structural relationship between the perturbed and the unperturbed polynomials along with the spectral properties and spectral transformation of continued fraction are investigated. It is demonstrated that the transfer matrix method is computationally more efficient than the classical method for obtaining perturbed $R_I$ polynomials. Further, an interesting consequence of co-dilation on the Carath\'eodary function is presented. Finally, the study of co-recursion and co-dilation in connection to the unit circle is carried out with the help of an illustration. The interlacing and monotonicity of zeros between L-Jacobi polynomials and their perturbed forms are demonstrated.