北京雁栖湖应用数学研究院

This minicourse presents a background for our joint research project with Anton Selemenchuk and Anton Dzhamay. In the lectures we will recall classical continuous and discrete orthogonal polynomials that admit hypergeometric representation and Askey scheme of their classification. We will discuss the connection of the orthogonal polynomials to determinantal ensembles in probability theory. Some asymptotic results, such as convergence to Tracy-Widom distribution will be considered. After reviewing classical picture we will move to semiclassical orthogonal polynomials that are obtained from classical ones by some weight modification. In particular, we will discuss Christoffel, Geronimus and Uvarov transformations. We will review our recent results with Anton Selemenchuk on the asymptotics of polynomials, obtained from Krawtchouk polynomials by Christoffel transformation. If time permits, we will also discuss Fredholm determinants and their connection to Painleve equations.