Title:

On orthogonal polynomials and related discrete integrable systems

Orthogonal polynomials arise in many areas of mathematics and have been the subject of interest by many mathematicians. In recent years this interest has often arisen from outside the orthogonal polynomial community after their connection with integrable systems was found. This thesis is concerned with the different ways these connections can occur. We approach the problem from both perspectives, by looking for integrable structures in orthogonal polynomials and by using an integrable structure to relate different classes of orthogonal polynomials. In Chapter 2, we focus on certain classes of semiclassical orthogonal polynomials. For the classical orthogonal polynomials, the recurrence relations and differential equations are well known and easy to calculate explicitly using an orthogonality relation or generating function. However with semiclassical orthogonal polynomials, the recurrence coefficients can no longer be expressed in an explicit form, but instead obeys systems of nonlinear difference equations. These systems are derived by deriving compatibility relations between the recurrence relation and the differential equation. The compatibility problem can be approached in two ways; the first is the direct approach using the orthogonality relation, while the second introduces the Laguerre method, which derives a differential system for semiclassical orthogonal polynomials. We consider some semiclassical Hermite and Laguerre weights using the Laguerre method, before applying both methods to a semiclassical Jacobi weight. While some of the systems derived will have been seen before, most of them (at least not to our knowledge) have not been acquired from this approach. Chapter 3 considers a singular integral transform that is related to the Gelâ€™fandLevitan equation, which provides the inverse part of the inverse scattering method (a solution method of integrable systems). These singular integral transforms constitute a dressing method between elementary (bare) solutions of an integrable system to more complicated solutions of the same system. In the context of this thesis we are interested in adapting this method to the case of polynomial solutions and study dressing transforms between different families of polynomials, in particular between certain classical orthogonal polynomials and their semiclassical deformations. In chapter 4, a new class of orthogonal polynomials are considered from a formal approach: a family of twovariable orthogonal polynomials related through an elliptic curve. The formal approach means we are interested in those relations that can be derived, without specifying a weight function. Thus, we are mainly concerned with recursive structures, particularly on their explicit derivation so that a series of elliptic polynomials can be constructed. Using generalized Sylvester identities, recurrence relations are derived and we consider the consistency of their coefficients and the compatibility between the two relations. Although the chapter focuses on the structure of the recurrence relations, some applications are also presented.
