Discrete Painlevé Equations and Orthogonal Polynomials
Over the last decade it became clear that discrete Painlevé equations appear in a wide range of interesting applications. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question. In this work we illustrate this procedure by considering two examples. The first example comes from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this example we study orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form. This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK). The second example is also related to the theory of orthogonal polynomials but the motivation comes from the Probability Theory. We consider the problem of tiling a hexagon by lozenges with some generalized weight and are interested in computing important statistical properties of this model called the gap probabilities. This model can be related to a q-Racah discrete orthogonal polynomial ensemble, and this computation is again done with the help of discrete Painlevé equation. This is a joint work with Alisa Knizel (Columbia University).