STRUCTURAL RELATIONS, ZEROS AND BIORTHOGONALITY OF MODIFIED RI AND RII TYPE POLYNOMIALS

Abstract

This dissertation is mainly concerned with perturbation in the recurrence coefficients of two specific types of recurrence relations called as RI and RII type recurrence relations in the literature. The generalized eigenvalue problem and continued fractions are associated with such recurrence relations, and hence they have found various applications in mathematical physics, approximation theory, etc. Two types of perturbation, viz., co-recursion and co-dilation, have been studied extensively in the theory of classical orthogonal polynomials and orthogonal polynomials on the unit circle. Analysis of the mathematical and computational outcome of applying these perturbations, either at the same level or at different levels, to the recurrence coefficients of RI and RII type recurrence relations is the aim of this thesis. Perturbations at the same level are experimented in the beginning, and at different levels are treated as subsequent developments. Using a forward recursion approach, the perturbed RI and RII polynomials are expressed as a non-linear combination of their original versions and corresponding associated polynomials. A transfer matrix approach is introduced to study the finite number of perturbations. This approach proved efficient in the computation of the perturbed RI and RII polynomials as it culminated the requirement of associated polynomials of the RI and RII type. A computational cost comparison between the forward recursion approach and the transfer matrix approach of determining the perturbed RI and RII polynomials is carried out, and the superiority of the transfer matrix approach over the forward recursion approach in both cases is established. Further, the computational complexity involved in computing perturbed RI polynomials by the forward recursion approach is compared to that of perturbed RII polynomials. A similar study is conducted for the transfer matrix approach, too. It is demonstrated that the perturbed RI and RII fractions (RI(z; μk, νk) and RII(z; μk, νk)) are pure rational spectral transformations of the one without perturbation (RI(z) and RII(z)). The interlacing, inclusion, and monotonicity properties of the zeros between each of the perturbed RI and RII polynomials and their corresponding unperturbed versions are examined. Relevant examples are used to study the implications of the above-mentioned perturbations in function theory and the theory of orthogonal polynomials on the unit circle. The concept of complementary chain sequence is explored in the context of RII polynomials, and a connection to the unit circle is presented. The co-recursive RII polynomials that arise by perturbing the first recurrence coefficient by μ are given by Pn+1(x, μ) = Pn+1(x) − μPn(x), n ≥ 0. Self perturbation of RII polynomials is defined by replacing constant μ with a non-zero sequence of real numbers αn, n ≥ 1. The generalized complementary Romanovski-Routh polynomials (GCRR polynomials), which are new in the literature, are constructed and the algebraic aspects of self perturbation of these polynomials are analyzed. A non-constant sequence that combines two GCRR polynomials for self perturbation is constructed such that the resultant is again a RII polynomial. The biorthogonal features of such a sequence of polynomials are inspected. Further, it is demonstrated that there is a triple interlacing of zeros between the perturbed and its constituent polynomials and that the zeros of two different sequences of polynomials produced using such perturbations are separated. Finally, the situation when the co-recursion and co-dilation in the RII type recurrence relation are not at the same level is dealt with. The structural relations between the so-obtained perturbed RII polynomials and the original ones are derived. The RII fraction with perturbation, say RII(x; μk, νk′ ), is obtained as the rational spectral transformation of the unperturbed one. The results derived for perturbed RII polynomials are used to deduce similar consequences for the RI polynomials and the real line. A quadrature rule is also constructed from perturbed RII type recurrence. Its accuracy is compared to the one existing in literature with the help of an illustration. Further scope of research is outlined wherever possible in this thesis.