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 bywhere λ n is a positive chain sequence and a n , b n , c n are sequences of real or complex numbers with P −1 (x) = 0 and 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.