In this work, orthogonal polynomials satisfying recurrence relation, with P −1 (z) = 0 and P 0 (z) = 1 are analyzed when modifications of the recurrence coefficient is considered. Specifically, representation of new perturbed polynomials in terms of old unperturbed ones, behaviour of zeros and spectral transformation of Stieltjes function are given. Further, Toda lattice equations corresponding to perturbed system of recurrence coefficients are obtained. Finally, when λ n is a positive chain sequence, co-dilation and its consequences are interpreted with the help of some illustrations.
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.
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