q-orthogonal polynomials and the Riemann-Hilbert problem

Abstract

Orthogonal polynomials have been studied since the nineteenth century, however many of their characteristics still remain unexplored. In this thesis we focus on q-orthogonal polynomials and endeavour to explain their properties using their formulation as a Riemann-Hilbert problem (RHP). The work we present is a compilation of original research completed by the author and their supervisor over the course of their PhD. The majority of these original results have been published, the thesis ties together these publications and, mostly in the final two chapters, presents new results. Compared to previous results in the literature we were able to more accurately describe the zeros, L_2 norm and recurrence coefficients of q-orthogonal polynomials as their degree tends to infinity. We were also able to better answer the question of uniqueness of positive solutions to corresponding discrete Painlev'e equations. In order to obtain our asymptotic results from the RHP a number of new techniques were required. These techniques mostly revolved around q-calculus. In particular, we constructed new functions and exploited their properties to solve RHPs and q-difference equations arising in Chapters 3 to 6