Chain sequences and zeros of polynomials related to a perturbed RII type recurrence relation

In this manuscript, new algebraic and analytic aspects of the orthogonal polynomials satisfying RII type recurrence relation given by Pn+1(x) = (x - cn)Pn(x) - lambda n(x2 + omega 2)Pn-1(x), n >= 0, omega is an element of R\{0}, with P-1(x) = 0 and P0(x) = 1 are investigated when the recurrence coefficients are perturbed. Specifically, representation of new perturbed polynomials (co-polynomials of RII 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. Further, it is shown that the transfer matrix approach is computationally more efficient than the classical approach. The consequences of such perturbations on the unit circle are analysed. A particular perturbation in the chain sequence called complementary chain sequences and its effect on the corre-sponding reflection coefficients is also studied. An illustration involving complementary Romanovski-Routh polynomials is presented.(c) 2022 Elsevier B.V. All rights reserved.

Automated summary

In this manuscript, new algebraic and analytic aspects of orthogonal polynomials satisfying the RII type recurrence relation are investigated. The representation of new perturbed polynomials in terms of original ones, as well as the interlacing and monotonicity properties of their zeros, are studied. The transfer matrix approach is used to obtain new structural relations, and its computational efficiency is compared to the classical approach. The consequences of the perturbations on the unit circle and the effect of a particular perturbation called complementary chain sequences on reflection coefficients are analyzed.