Norbert Wiener Center for Harmonic Analysis and Applications

Department of Mathematics, University of Maryland, College Park, Maryland, USA

National Institute of Standards and Technology, Gaithersburg, Maryland, USA

# Orthogonal Polynomials and Special Functions Summer School

OPSF-S6

## University of Maryland, College Park, MD

July 11-July 15, 2016

Department of Mathematics, University of Maryland, College Park, Maryland, USA

National Institute of Standards and Technology, Gaithersburg, Maryland, USA

OPSF-S6

July 11-July 15, 2016

This program is for graduate students and post-docs. We expect to be able to fund up a number of students, early career researchers and students from third-world countries.

The OPSF summer schools are organized by the Orthogonal Polynomials,
Special Functions and Applications (OPSFA)
Steering Committee.
The OPSF-S6 program consists of a one-week summer
school for graduate students and early career researchers to be held in Summer 2016
on the campus of the University of Maryland. It will focus on orthogonal
polynomials and special functions, and feature lectures delivered by top researchers in
their fields.

**John Benedetto**, Department of Mathematics and the Norbert Wiener Center, University of Maryland**Stephen Casey**, Department of Mathematics and Statistics, American University**Howard Cohl**, Applied and Computational Mathematics Division, National Institute of Standards and Technology**Mourad Ismail**, Department of Mathematics, University of Central Florida**Daniel Lozier**, Applied and Computational Mathematics Division, National Institute of Standards and Technology**Kasso Okoudjou**, Department of Mathematics and the Norbert Wiener Center, University of Maryland

- Antonio Durán
**Exceptional Orthogonal Polynomials**,

Departamento de Análisis Mathemático, Universidad de Sevilla, Sevilla, Spain

ABSTRACT: We will consider the two more important extensions of the classical and classical discrete orthogonal polynomials. Namely: Krall or bispectral polynomials which, besides the orthogonality, are also common eigenfunctions of higher order differential or difference operators; and exceptional polynomials which have recently appeared in connection with quantum mechanic models associated to certain rational perturbations of the classical potentials. We also explore the relationship between both extensions and how they can be used to expand Askey tableau. - Mourad Ismail
**Theory and Applications of q-Series**,

Department of Mathematics, University of Central Florida, Orlando, Florida, USA

ABSTRACT: We develop the theory of q-series based on q-Taylor Analysis. This will take us through the Sears and Watson transformations. We will also cover q-orthogonal polynomials and biorthogonal rational functions. As applications we will derive the Rogers-Ramanujan identities and some of their generalizations. - Erik Koelink
**Spectral Theory and Special Functions**,

Department of Mathematics, Radboud Universiteit Nijmegen, Nijmegen, The Netherlands

ABSTRACT: Many special functions are eigenfunctions to explicit operators, such as difference and differential operators. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. One of the best known cases is the proof of Favard’s theorem for orthogonal polynomials, and we start with this case. This approach will then be extended to other situations. - Hjalmar Rosengren
**Elliptic Hypergeometric Functions**,

Chalmers University of Technology and University of Gothenburg, Goteborg, Sweden

ABSTRACT: Elliptic hypergeometric functions are a relatively recent class of special functions. Although some examples can be found in the physics literature from the 1980’s, their mathematical theory has only been developed during the last twenty years. We will give an introduction to elliptic hypergeometric series and integrals and discuss some relations to other topics such as solvable lattice models. - Jiang Zeng
**Orthogonal Polynomials and Combinatorics**,

Institut Camille Jordan Universite Claude Bernard Lyon-I, Villeurbanne, Lyon, France

ABSTRACT: We present the interplay between orthogonal polynomials and combinatorics by studying combinatorial aspects of the polynomials themselves and their moments. The six lectures will be roughly divided as follows: 1. Basic enumerative combinatorics; 2. Classical orthogonal polynomials as enumerative polynomials.; 3. Flajolet-Viennot’s theory of general orthogonal polynomials (I); 4. Flajolet-Viennot’s theory of general orthogonal polynomials (II); 5. Moments of q-orthogonal polynomials; and 6. Linearization coefficients of q-orthogonal polynomials.

TIME |
SUMMARY |
LECTURER |

8:30 - 8:45 am | WELCOME | |

8:45 - 9:35 am | q-Taylor series and Summation theorems |
Ismail |

9:35 – 9:55 am | COFFEE BREAK (20 minutes) | |

9:55 – 10:45 am | Transformations and q-Hermite Polynomials |
Ismail |

10:45 – 11:30 am | EXERCISES (45 minutes) | Ismail |

11:30 – 1:00 pm | LUNCH (1 hour 30 minutes) | |

1:00 – 1:50 pm | Elliptic Hypergeometric Functions I | Rosengren |

1:50 – 2:10 pm | COFFEE BREAK (20 minutes) | |

2:10 – 3:00 pm | Elliptic Hypergeometric Functions II | Rosengren |

3:00 – 3:45 pm | EXERCISES (45 minutes) | Rosengren |

3:45 – 3:55 pm | INTERMISSION (10 minutes) | |

3:55 – 4:45 pm | Spectral Theory and Special Functions I | Koelink |

4:45 – 5:05 pm | BREAK (20 minutes) | |

5:05 – 5:55 pm | Spectral Theory and Special Functions II | Koelink |

5:55 – 6:40 pm | EXERCISES (45 minutes) | Koelink |

TIME |
SUMMARY |
LECTURER |

8:45 - 9:35 am | The classical and classical discrete families | Durán |

9:35 – 9:55 am | COFFEE BREAK (20 minutes) | |

9:55 – 10:45 am | The Askey tableau. Krall and exceptional polynomials. Darboux Transforms | Durán |

10:45 – 11:30 am | EXERCISES (45 minutes) | Durán |

11:30 – 1:00 pm | LUNCH (1 hour 30 minutes) | |

1:00 – 1:50 pm | Orthogonal Polynomials and Combinatorics I | Zeng |

1:50 – 2:10 pm | COFFEE BREAK (20 minutes) | |

2:10 – 3:00 pm | Orthogonal Polynomials and Combinatorics II | Zeng |

3:00 – 3:45 pm | EXERCISES (45 minutes) | Zeng |

3:45 – 3:55 pm | INTERMISSION (10 minutes) | |

3:55 – 4:45 pm | The Askey-Wilson polynomials and Operators | Ismail |

4:45 – 5:05 pm | BREAK (20 minutes) | |

5:05 – 5:55 pm | Rodrigues formulas and summation theorems | Ismail |

5:55 – 6:40 pm | EXERCISES (45 minutes) | Ismail |

TIME |
SUMMARY |
LECTURER |

8:45 - 9:35 am | Elliptic Hypergeometric Functions III | Rosengren |

9:35 – 9:55 am | COFFEE BREAK (20 minutes) | |

9:55 – 10:45 am | Elliptic Hypergeometric Functions IV | Rosengren |

10:45 – 11:30 am | EXERCISES (45 minutes) | Rosengren |

11:30 – 1:00 pm | LUNCH (1 hour 30 minutes) | |

1:00 – 1:50 pm | Spectral Theory and Special Functions III | Koelink |

1:50 – 2:10 pm | COFFEE BREAK (20 minutes) | |

2:10 – 3:00 pm | Spectral Theory and Special Functions IV | Koelink |

3:00 – 3:45 pm | EXERCISES (45 minutes) | Koelink |

3:45 – 3:55 pm | INTERMISSION (10 minutes) | |

3:55 – 4:45 pm | D-operators |
Durán |

4:45 – 5:05 pm | BREAK (20 minutes) | |

5:05 – 5:55 pm | Constructing Krall polynomials by using D-operators |
Durán |

5:55 – 6:40 pm | EXERCISES (45 minutes) | Durán |

TIME |
SUMMARY |
LECTURER |

8:45 - 9:35 am | Orthogonal Polynomials and Combinatorics III | Zeng |

9:35 – 9:55 am | COFFEE BREAK (20 minutes) | |

9:55 – 10:45 am | Orthogonal Polynomials and Combinatorics IV | Zeng |

10:45 – 11:30 am | EXERCISES (45 minutes) | Zeng |

11:30 – 1:00 pm | LUNCH (1 hour 30 minutes) | |

1:00 – 1:50 pm | q-series identities and integrals, old and new |
Ismail |

1:50 – 2:10 pm | COFFEE BREAK (20 minutes) | |

2:10 – 3:00 pm | Applications | Ismail |

3:00 – 3:45 pm | EXERCISES (45 minutes) | Ismail |

3:45 – 3:55 pm | INTERMISSION (10 minutes) | |

3:55 – 4:45 pm | Elliptic Hypergeometric Functions V | Rosengren |

4:45 – 5:05 pm | BREAK (20 minutes) | |

5:05 – 5:55 pm | Elliptic Hypergeometric Functions VI | Rosengren |

5:55 – 6:40 pm | EXERCISES (45 minutes) | Rosengren |

TIME |
SUMMARY |
LECTURER |

8:45 - 9:35 am | Spectral Theory and Special Functions V | Koelink |

9:35 – 9:55 am | COFFEE BREAK (20 minutes) | |

9:55 – 10:45 am | Spectral Theory and Special Functions VI | Koelink |

10:45 – 11:30 am | EXERCISES (45 minutes) | Koelink |

11:30 – 1:00 pm | LUNCH (1 hour 30 minutes) | |

1:00 – 1:50 pm | First expansion of the Askey tableau. Exceptional polynomials: discrete case | Durán |

1:50 – 2:10 pm | COFFEE BREAK (20 minutes) | |

2:10 – 3:00 pm | Exceptional polynomials: continuous case. Second expansion of the Askey tableau | Durán |

3:00 – 3:45 pm | EXERCISES (45 minutes) | Durán |

3:45 – 3:55 pm | INTERMISSION (10 minutes) | |

3:55 – 4:45 pm | Orthogonal Polynomials and Combinatorics V | Zeng |

4:45 – 5:05 pm | BREAK (20 minutes) | |

5:05 – 5:55 pm | Orthogonal Polynomials and Combinatorics VI | Zeng |

5:55 – 6:40 pm | EXERCISES (45 minutes) | Zeng |

Any opinions, findings
and conclusions or recomendations expressed in this material are those
of the author(s) and do not necessarily reflect the views of the
National Science Foundation (NSF).