Chain sequences and Zeros of a perturbed $R_{II}$ type recurrence relation (2201.09409v1)

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.