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

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)-λ_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.

23 pages. arXiv admin note: text overlap with arXiv:2201.05422

Cited by in corpus (1)

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