This Summer School consists of:

**Three lecture courses**(by Kerstin Jordaan, Nalini Joshi and Walter Van Assche) which will be supplemented by daily tutorial sessions.**Three guest lectures**, by Andy Hone, Andrei Martinez-Finkelshtein and Adri Olde Daalhuis.

“**Properties of Orthogonal Polynomials**”

by **Kerstin Jordaan** (University of South Africa, South Africa)

Abstract. In these lectures, an introduction will be given to the theory of orthogonal polynomials. We discuss basic concepts and known properties of orthogonal polynomials within the context of applications. The lectures aim to show, by means of accessible examples, that interesting research problems arise from asking questions about the characteristic properties satisfied by classical orthogonal polynomials and their extensions or generalisations. Topics to be covered include the three term recurrence relation and recurrence coefficients, the spectral theorem, Hankel determinants, Jacobi matrices, Rodrigues type formulae, Padé approximants to hypergeometric functions, the Askey scheme of hypergeometric polynomials, properties of zeros of orthogonal polynomials, Markov’s monotonicity theorem, quasi-orthogonal polynomials, Pearson’s equation and semi-classical orthogonal polynomials.

Lecture notes: Jordaan assignment 1 (tutorial 1); Jordaan slides Lecture 1; Jordaan slides Lecture 2; Jordaan slides Lecture 3; Jordaan slides Lecture 4; Jordaan slides Lecture 5

“**Discrete Painleve Equations**

by **Nalini Joshi** (University of Sydney, Australia)

Abstract. In these lectures, an introduction will be given to discrete Painlevé equations, which are discrete analogues of the well known Painlevé equations. The solutions of the Painlevé equations appear as universal nonlinear mathematical models in many applications. The study of their discrete versions stands at the leading edge of this field and has created an extraordinarily rich vein of new ideas and methods over the past 20 years. It is now clear that these discrete versions will be useful and applicable, yet their properties are not well known to most mathematicians and mathematical scientists. This course of lectures will provide an introduction to nonlinear discrete equations, an overview of properties of discrete Painlevé equations, a toolbox to describe their phase space in terms of singularities and geometry, and how this may be mined to provide asymptotic behaviours of solutions. The lectures will provide training and information for students, who may encounter the nonlinear discrete equations, and in particular, the discrete Painlev´e equations, as mathematical or physical models in the course of their research.

Lecture notes: Joshi_opsfa_tutorials; Joshi_Lecture0; Joshi Lectures 1 to 4 OPSFA

“**Multiple Orthogonal Polynomials**”

by **Walter Van Assche** (KU Leuven, Belgium)

Abstract. Multiple orthogonal polynomials are polynomials of one variable that satisfy orthogonality conditions with respect to r > 1 measures. They appeared as denominators of Hermite-Pad´e approximants to several functions in the 19th century and for that reason they are also known as Hermite-Padé polynomials. Other names used in the literature are polyorthogonal polynomials and d-orthogonal polynomials, which are basically the type II multiple orthogonal polynomials near the diagonal (the so-called stepline) and d corresponds to the number of measures (which we denote by r ). Multiple orthogonal polynomials were studied in the 20th century by Mahler who obtained many properties, in particular the existence and normality (perfect systems). Analytic properties, in particular the asymptotic behavior of the zeros, were obtained at the end of the 20th century. In the 21st century multiple orthogonal polynomials started to appear in various random matrix models, non-intersecting random paths, and other determinantal processes. In particular the average characteristic polynomial of a random matrix often turns out to be a multiple orthogonal polynomial so that the eigenvalues of a random matrix are expected to behave like the zeros of a multiple orthogonal polynomial. Multiple orthogonal polynomials also are useful to describe certain discrete integrable processes. In these lectures an introduction and a discussion of the most important families of multiple orthogonal polynomials will be given together with their known properties and applications.

Lecture notes:** ** Van Assche Lecture 1; Van Assche – Lecture 2 ; Van Assche – Lecture 3 ; Van Assche – Lecture 4 ; Van Assche – Lecture 5

# —— GUEST LECTURES ——

# Guest Lecture by Andy Hone

Title:** Continued fractions and nonlinear recurrences**

**Abstract**: Three-term linear recurrences are a central part of the theory of continued fractions, in description of convergents, as well as being a feature of orthogonal polynomials. This talk will consider some examples of nonlinear recurrences associated with continued fractions. In the setting of the real numbers, some new examples of transcendental numbers with an explicit continued fraction expansion will be provided. The case of function fields associated with algebraic curves will also be mentioned, pointing out the connection with certain discrete integrable systems, Hankel determinants and orthogonal polynomials.

# Guest Lecture by Andrei Martínez Finkelshtein

Title:* Vector equilibrium and asymptotics of zeros of multiple orthogonal polynomials*

**Abstract**: It is known since the end of the 20th century that the analytic properties, in particular the asymptotic behavior of the zeros, of families of polynomials that satisfy orthogonality conditions with respect to several measures (multiple orthogonal polynomials or MOPS) have an electrostatic description in terms of vector-valued measures. On the real line these measures typically provide a global minimum of an associated energy functional, but when we deal with non-hermitian orthogonality, a more general notion of equilibrium must be considered. The first part of the lecture contains a general introduction to these notions, while in the second one we discuss the case of multiple orthogonal polynomials with respect to a cubic weight, where the integration goes along non-homotopic paths on the plane. For their asymptotic description we need to analyze saddle points of an energy functional in which the mutual interaction comprises both attracting and repelling forces. The resulting measures can be characterized by a cubic algebraic equation (spectral curve) whose solutions are appropriate combinations of the Cauchy transform of its components. In particular, these measures are supported on a finite number of analytic arcs that are trajectories of a quadratic differential globally defined on a three-sheeted Riemann surface. The complete description of the so-called critical graph for such a differential (and its dynamics as a function of the parameters of the problem) is the key ingredient of the asymptotic analysis of the MOPS.

This is talk is partially based on the joint work with Guilherme L. F. Silva (U. Michigan, Ann Arbor).

# Guest Lecture by Adri Olde Daalhuis

Title: *Exponential Asymptotics and Resurgence*

Abstract: Exponential asymptototics has been a very active area of research in the last 3 decades. It started with fundamental work by Berry, Ecalle and Kruskal. Small exponentials are usually responsible for the divergence of asymptotic series. Resurgence enables the divergent tails to be decoded to yield these exponentials. Including these small exponentials leads to exponentially-improved asymptotics at several levels: hyperasymptotics. Via resurgence we are now able to compute the so-called connection coefficients, or Stokes multipliers. I will demonstrate the latest progress of exponential asymptotics for ordinary differential equations and partial differential equations.

These new techniques give us also better representations for the remainders in the asymptotic expansions and this has led recently to much sharper error bounds for the asymptotic expansions of many special functions.

The smoothing of the Stokes phenomenon via an error function was introduced by Berry in 1989. I will discuss the higher order Stokes phenomenon and its smoothing via combinations of error functions.