London Mathematical Society Research School

Orthogonal Polynomials and Special Functions

26th–30th June, 2017

School of Mathematics, Statistics & Actuarial ScienceSibson Building, Parkwood Road

University of KentCanterbury, CT2 7FS, UK

OrganisersPeter Clarkson (University of Kent, Canterbury, UK)Ana Loureiro (University of Kent, Canterbury, UK)

Information Booklet

Welcome to the London Mathematical Society Research School Orthogonal Polynomials andSpecial Functions. The Research School is also an activity of the SIAM Activity Group on OrthogonalPolynomials and Special Functions; this is the seventh School on Orthogonal Polynomials and SpecialFunctions. The activity group promotes basic research in orthogonal polynomials and special functions;further the application of this subject in other parts of mathematics, and in science and industry. Italso encourages and supports the exchange of information, ideas, and techniques between workers inthis field and other mathematicians and scientists.

The Research School consists of three courses by eminent Mathematicians in the field of orthogonalpolynomials and special functions, each with five lectures and two tutorial sessions, supplemented bythree guest lectures.

We thank the London Mathematical Society (LMS) for their generous funding of the ResearchSchool. We are grateful to Elizabeth Fisher (LMS) and Louisa Harvey (University of Kent) for theirassistance in the organisation of the Research School.

We hope that you’ll enjoy the Research School, learn much and enhance your appreciation of thewonderful world of orthogonal polynomials and special functions.

Peter Clarkson and Ana Loureiro

Course LecturersKerstin Jordaan University of South Africa, Pretoria, South AfricaNalini Joshi University of Sydney, AustraliaWalter Van Assche University of Leuven, Belgium

Guest LecturersAndrew Hone University of Kent, Canterbury, UKAndrei Martınez-Finkelshtein University of Almerıa, SpainAdri Olde Daalhuis University of Edinburgh, UK

Tutorial AssistantsAlfredo Deano University of Kent, Canterbury, UKAndrew Hone University of Kent, Canterbury, UKAna Loureiro University of Kent, Canterbury, UK

Local Organising CommitteeClaire CarterPeter ClarksonAlfredo DeanoAna LoureiroElizabeth MansfieldJohn PearsonIan Wood

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OPSF Research School TimetableMonday Tuesday Wednesday Thursday Friday26 June 27 June 28 June 29 June 30 June

08:00–09:00 Registration& Opening

09:00–10:00 Lecture Lecture Lecture Lecture LectureJordaan 1 Van Assche 2 Van Assche 4 Joshi 5 Jordaan 5

10:00–11:00 Lecture Lecture Lecture Lecture LectureJoshi 1 Jordaan 2 Joshi 4 Jordaan 4 Van Assche 5

11:00–11:30 Coffee break Coffee break Coffee break Coffee break Coffee break11:30–12:30 Tutorial Tutorial Lecture Tutorial Q&A session

Jordaan 1 Joshi 1 Jordaan 3 Van Assche 212:30–14:00 Lunch Lunch Lunch Lunch Lunch14:00–15:00 Lecture Lecture Tutorial Aurora event Guest Lecture

Van Assche 1 Joshi 3 Joshi 2 (Joshi) Martınez-Finkelshtein15:00–16:00 Tutorial Lecture Tutorial

Van Assche 1 Van Assche 3 Jordaan 216:00–16:30 Coffee break Coffee break Coffee break16:30–17:30 Lecture Guest Lecture Guest Lecture

Joshi 2 Hone Olde DaalhuisDrinks Reception School Dinner

Kerstin Jordaan Nalini Joshi Walter Van Assche

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Lecture Course 1:“Properties of orthogonal polynomials”Kerstin Jordaan (University of South Africa, Pretoria, South Africa)Abstract. In these lectures, an introduction will be given to the theory of orthogonal polynomials. Wewill discuss basic concepts and classical properties of orthogonal polynomials within the context of ap-plications. The lectures aim to show, by means of accessible examples, that interesting research prob-lems arise from asking questions about the characteristic properties satisfied by classical orthogonalpolynomials and their extensions or generalisations. Topics to be covered include the three term recur-rence relation and recurrence coefficients, the spectral theorem, Hankel determinants, Jacobi matrices,Rodrigues type formulae, Pade approximants to hypergeometric functions, the Askey scheme of hyper-geometric polynomials, contiguous relations, properties of zeros of orthogonal polynomials, Markov’smonotonicity theorem, Sturm’s convexity theorem, quasi-orthogonal polynomials, linear combinationsof orthogonal polynomials, Pearson’s equation and semi-classical orthogonal polynomials.

Lecture Course 2:“Discrete Painleve Equations”Nalini Joshi (University of Sydney, Australia)Abstract. In these lectures, an introduction will be given to discrete Painleve equations, which arediscrete analogues of the well known Painleve equations. The solutions of the Painleve equationsappear as universal nonlinear mathematical models in many applications. The study of their discreteversions stands at the leading edge of this field and has created an extraordinarily rich vein of newideas and methods over the past 20 years. It is now clear that these discrete versions will be usefuland applicable, yet their properties are not well known to most mathematicians and mathematicalscientists. This course of lectures will provide an introduction to nonlinear discrete equations, anoverview of properties of discrete Painleve equations, a toolbox to describe their phase space in terms ofsingularities 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 nonlineardiscrete equations, and in particular, the discrete Painleve equations, as mathematical or physicalmodels in the course of their research.

Lecture Course 3:“Multiple Orthogonal Polynomials”Walter Van Assche (University of Leuven, Belgium)Abstract. Multiple orthogonal polynomials are polynomials of one variable that satisfy orthogonal-ity conditions with respect to r > 1 measures. They appeared as denominators of Hermite-Padeapproximants to several functions in the 19th century and for that reason they are also known asHermite-Pade polynomials. Other names used in the literature are polyorthogonal polynomials andd-orthogonal polynomials, which are basically the type II multiple orthogonal polynomials near thediagonal (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 prop-erties, in particular the existence and normality (perfect systems). Analytic properties, in particularthe asymptotic behaviour of the zeros, were obtained at the end of the 20th century. In the 21st centurymultiple orthogonal polynomials started to appear in various random matrix models, non-intersectingrandom paths, and other determinantal processes. In particular the average characteristic polynomialof a random matrix often turns out to be a multiple orthogonal polynomial so that the eigenvalues of arandom matrix are expected to behave like the zeros of a multiple orthogonal polynomial. Multiple or-thogonal polynomials also are useful to describe certain discrete integrable processes. In these lecturesan introduction and a discussion of the most important families of multiple orthogonal polynomials willbe given together with their known properties and applications.

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Guest Lecture 1:“Continued fractions and nonlinear recurrences ”Andrew Hone (University of Kent, Canterbury, UK)Abstract. Three-term linear recurrences are a central part of the theory of continued fractions, indescription of convergents, as well as being a feature of orthogonal polynomials. This talk will con-sider some examples of nonlinear recurrences associated with continued fractions. In the setting ofthe real numbers, some new examples of transcendental numbers with an explicit continued fractionexpansion will be provided. The case of function fields associated with algebraic curves will also bementioned, pointing out the connection with certain discrete integrable systems, Hankel determinantsand orthogonal polynomials.

Guest Lecture 2:“Vector equilibrium and asymptotics of zeros of multiple orthogonal polynomials”Andrei Martınez-Finkelshtein (University of Almerıa, Spain)Abstract. It is known since the end of the 20th century that the analytic properties, in particularthe asymptotic behaviour of the zeros, of families of polynomials that satisfy orthogonality conditionswith respect to several measures (multiple orthogonal polynomials) have an electrostatic description interms of vector-valued measures. On the real line these measures typically provide a global minimumof an associated energy functional, but when we deal with non-hermitian orthogonality, a more generalnotion of equilibrium must be considered. The first part of the lecture contains a general introductionto these notions, while in the second one we discuss the case of multiple orthogonal polynomials withrespect to a cubic weight, where the integration goes along non-homotopic paths on the plane. For theirasymptotic description we need to analyze saddle points of an energy functional in which the mutualinteraction comprises both attracting and repelling forces. The resulting measures can be character-ized by a cubic algebraic equation (spectral curve) whose solutions are appropriate combinations ofthe Cauchy transform of its components. In particular, these measures are supported on a finite num-ber of analytic arcs that are trajectories of a quadratic differential globally defined on a three-sheetedRiemann surface. The complete description of the so-called critical graph for such a differential (andits dynamics as a function of the parameters of the problem) is the key ingredient of the asymptoticanalysis of the multiple orthogonal polynomials.

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

Guest Lecture 3:“Exponential Asymptotics and Resurgence”Adri Olde Daalhuis (University of Edinburgh, UK)Abstract. Exponential asymptototics has been a very active area of research in the last 3 decades. Itstarted with fundamental work by Berry, Ecalle and Kruskal. Small exponentials are usually respon-sible for the divergence of asymptotic series. Resurgence enables the divergent tails to be decoded toyield these exponentials. Including these small exponentials leads to exponentially-improved asymp-totics at several levels: hyperasymptotics. Via resurgence we are now able to compute the so-calledconnection coefficients, or Stokes multipliers. I will demonstrate the latest progress of exponentialasymptotics for ordinary differential equations and partial differential equations.

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

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

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InformationRegistrationRegistration will take place 08:00–09:00 on Monday 3rd July in the Atrium in the Sibson building,where the Research School is taking place.

VenueThe Research School take place at Sibson Building (C2 on the campus map):

Sibson Building, Parkwood RoadUniversity of KentCanterbury, CT2 7FS

Floor plans for the ground and first floors of the Sibson building

AccommodationAccommodation is located in Turing College (N7 on the campus map).

• Check-in is from 14:00 on your day of arrival and check-out is by 10:00 on your day of departure.Please visit Turing Reception (N7 on the campus map) upon your arrival to collect your roomkeys.

• Turing Reception will be open from 08:00 to 22:00 daily. If you are due to check-in to your accom-modation after reception has closed you will need to contact Campus Security upon your arrivalwho will open reception. An internal phone is available outside reception to dial through to our24hr security team.

• Bedrooms are serviced daily and provided with towels, linen and complimentary toiletries. Teaand coffee making facilities are also provided.

• Delegates are asked to familiarise themselves with the University’s evacuation procedures, whichare displayed on the back of the door in each bedroom.

• Breakfast is provided in Hut 8, Turing College between 07:45 and 09:30 and you will be requiredto show your room key to prove residency.

• An evening meal will be served in Rutherford Dining Hall from 18:30–19:30, on Sunday 25thJune. Evening meals will be served in Hut 8, Turing College from 18:30–19:30, on Monday 26th-Wednesday 28th June.

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Lectures & TutorialsThe lectures will take place in Sibson Seminar Room 6, on the first floor of the Sibson building. Tuto-rials will take place in Sibson Lecture Theatre 2, on the first floor of the Sibson building.

Coffee Breaks & LunchCoffee breaks will take place on the first floor of Sibson building throughout the Research School.Lunch will be served in the Sibson Atrium throughout the Research School.

Evening mealsAn evening meal will be served in Rutherford Dining Hall from 18:30–19:30, on Sunday 25th June.Evening meals will be served in Hut 8, Turing College from 18:30–19:30, on Monday 26th-Wednesday28th June.

Welcome Reception – Monday 26th JuneOn Monday 26th June there will be a welcome reception in the Sibson Atrium from 17:30–18:30.

Aurora @Kent Event – Thursday 29th JuneOn Thursday 29th June there will be the Aurora @Kent Event Making waves: an interview withProfessor Nalini Joshi, 14:00-15:00, in Sibson Lecture Theatre 3, on the first floor of the Sibsonbuilding.

Research School Dinner – Thursday 29th JuneThe Research School dinner will take place on Thursday 6th July in the Darwin Conference Suite,which is on the ground floor of Darwin College, University of Kent. Pre-dinner drinks will be servedfrom 19:30 with dinner at 20:00.

Sibson CafeThe Sibson Cafe is located on the ground floor of the Sibson building, which is open 09:00–15:00 duringthe Research School. A range of hot and cold drinks, sandwiches, salads and snacks are available.

The Sibson Building

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Internet AccessWireless zones are available in teaching and social areas on campus. It is a condition of connectingto the internet using the Kent network that all laptops should be running current anti-virus softwareand the latest security updates.

Use the University of Kent WiFi service eduroam if you are from an eduroam institution. Dele-gates are not from an eduroam institution will be supplied with a computer log-in in order to accessthe internet during their stay. Log-ins and instructions (for the University network) will be suppliedduring registration. Alternatively, anyone can pick up WiFi Guest, free public WiFi provided by TheCloud, on campus.How to get on visitor WiFi:

• find WiFi Guest in your list of available WiFi networks and select it;• log in, or register if you’re a first time user, to gain internet access.

SmokingThe University operates a non-smoking policy in all areas.

MessagesBetween the hours of 09:00 and 17:00 (BST) messages for delegates can be left with the Conferenceteam on +44 (0)1227 828000 or the SMSAS General Office +44 (0)1227 827181. Outside of these hoursurgent messages only can be left with Campus Security on +44 (0)1227 823300

Campus facilitiesSet in over 450 acres of beautiful parkland overlooking the historical city of Canterbury, the Universityof Kent campus provides a perfect location for conferences and events guests. We have a wide range ofaward-winning restaurants and bars, a cinema, theatre, shops, banking, as well as a gym and outdoorsports facilities.

Historic and Cosmopolitan CanterburyThe University of Kent campus is just a 20 minute walk or short bus journey into the city. Canterburyhas something for everyone to enjoy – whether you want to take a boat ride on the River Stour, shop ’tilyou drop at big name stores and independent boutiques or sample some of the local culinary delightsin award-winning restaurants and bars.

Tourist Information about Canterbury can be downloaded here.A map of Canterbury can be downloaded here.A map of Canterbury City Centre can be downloaded here.A guide to top attractions in Canterbury, including a map, can be downloaded here.

Health/Medical InsuranceDelegates are advised to ensure that they have adequate medical insurance to cover the period of theirstay in the United Kingdom.

Electrical Voltage

United Kingdom standard voltage is 240v, with squarethree-pin plugs — you may need to have an adaptorwith you.

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ParticipantsAtef Alaya University of Gabes, Tunisia

& Umm Al-Qura University, Makkah, Saudi ArabiaGerardo Ariznabarreta Universidad Complutense de Madrid, SpainDimitrios Askitis University of Copenhagen, DenmarkAhmad Barhoumi Indiana University-Purdue University Indianapolis, USAArpad Baricz Babes-Bolyai University, Cluj-Napoca, Romania

& Obuda University, Budapest, HungaryKiran Kumar Behera Indian Institute of Technology Roorkee, IndiaGeoffroy Bergeron University of Montreal, CanadaOksana Bihun University of Colorado, Colorado Springs, USAEric-Olivier Bosse Universite de Montreal, CanadaCaroline Brett Lancaster University, UKJian Cao Hangzhou Normal University, ChinaAndrew Celsus University of Cambridge, UKYoon Choun University of Kent, Canterbury, UKSourav Das Indian Institute of Technology Roorkee, IndiaAbel Dıaz Gonzalez Universidad Carlos III de Madrid, SpainEllen Dowie University of Kent, Canterbury, UKAlexander Dyachenko Technische Universitat Berlin, GermanyJulien Gaboriaud Universite de Montreal, CanadaJuan Carlos Garcıa Ardila Universidad Carlos III de Madrid, SpainMarıa Angeles Garcıa-Ferrero ICMAT, Madrid, SpainPeter Gill Australian National University, AustraliaJean Carlo Guella Universidade de Sao Paulo, BrazilJesse Hayhow Queen Mary University of London, UKWei Ying Hu City University of Hong Kong, ChinaAbey Sherif Kelil University of Pretoria, South AfricaRostyslav Kozhan Uppsala University, SwedenJean-Michel Lemay Universite de Montreal, CanadaHelder Lima University of Kent, Canterbury, UKJordan Makwana University of Kent, Canterbury, UKJuan Francisco Manas Manas Universidad de Almerıa, SpainMisael Marriaga Universidad Carlos III de Madrid, SpainHoury Melkonian Heriot-Watt University, Edinburgh, UKSheila Menchavez Mindanao State University, PhilippinesProscovia Nabasinga Miria UTAMU, UgandaSengul Nalci Tumer Izmir Institute of Technology, TurkeyHasen Ozturk University of Reading, UKAbdelsadek Saib University of Tebessa, AlgeriaLuis SarmentoStella Schindler Washington University in St. Louis, USAAlisa Sheinkman National Research University High School of Economics, RussiaMikhail Tyaglov Shanghai Jiao Tong University, ChinaTanay Wakhare University of Maryland, College Park, USAYian Yao Georgia Institute of Technology, Atlanta, USAYanely Zaldivar Gerpe Universidad Carlos III de Madrid, SpainMichele Zadra University of Kent, Canterbury, UKWeiwei Zhang Georgia Institute of Technology, Atlanta, USA

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