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 semi-classical 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 semi-classical orthogonal polynomials, the recurrence
coefficients can no longer be expressed in an explicit form, but instead obeys systems
of non-linear 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 semi-classical orthogonal polynomials. We consider some semiclassical
Hermite and Laguerre weights using the Laguerre method, before applying both
methods to a semi-classical 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 Gel’fand-Levitan
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 semi-classical deformations.
In chapter 4, a new class of orthogonal polynomials are considered from a formal
approach: a family of two-variable 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.